Deterministic Nuclear Structure, Fission, and Fusion from Curvature Dynamics in Trembling Spacetime Relativity

De e minis ic Nuclea S uc u e, Fission, and Fusion om
Cu a u e Dynamics in T embling Space ime Rela i i y
A uni ied accoun wi hou p obabilis ic unneling, shell closu es, o pai ing i s
Nico F. Decle cq
Geo ge W. Wood u School o Mechanical Enginee ing, Geo gia Ins i u e o Technology, A lan a, USA
IRL 2958 Geo gia Tech – CNRS, Geo gia Tech Eu ope, Me z, F ance
[email p o ec ed]
Abs ac
Con en ional nuclea models, om liquid-d op pa ame iza ions o shell and densi y-
unc ional heo ies, desc ibe many global ends in binding and decay bu emain p ob-
abilis ic and hea ily pa ame e ized. They do no explain, om i s p inciples, pe sis en
anomalies such as deep sub-ba ie usion hind ance, odd-e en s agge ing in ission yields, o
he long hal -li e o 14C. This wo k ex ends T embling Space ime Rela i i y Theo y (TSRT)
in o he nuclea domain as a de e minis ic geome ic amewo k o s uc u e, s abili y,
and ans o ma ion. In TSRT, each nucleus is a localized embling-cu a u e eigenmode
whose s abili y esul s om sus ained supp ession o in insic space ime cu a u e. De-
cay is a causal elaxa ion o his supp ession and is modeled as cu a u e econ igu a ion
a he han a s ochas ic ansi ion. Binding, ission, usion, and decay hus eme ge om
he same geome ic dynamics. Using a single global no maliza ion ixed once on 60Co, he
TSRT o mula ion ep oduces expe imen al hal -li es ac oss be a decay, alpha decay, and
spon aneous ission o e mo e han wen y o de s o magni ude, wi h a mean loga i hmic
de ia ion below one pa in a million. No shell closu es, pai ing co ec ions, empi ical p e-
o ma ion ac o s, o pe -nucleus uning a e in oked. The 14C anomaly ollows om i s
geome ic be a-pa h cu a u e wi hou adjus able hind ance. All Q- alues, ba ie ac ions,
and emission a es ollow de e minis ically om he space ime me ic and i s cu a u e
ene gy, and TSRT p o ides a causal mechanism o mass–ene gy con e sion as cu a u e e-
dis ibu ion. TSRT also ep oduces he eme gence o neu on magic numbe s as geome ic
minima o he cu a u e–s i ness map, ma ching he con en ional magic sequence wi hou
quan um pos ula es. Deep sub-ba ie usion hind ance a ises om geome ic supp ession o
embling-mode o e lap; odd-e en s agge ing in ission yields o igina es om phase-locked
neck eigenmodes a scission; and elec on-sc eening shi s e lec eno maliza ion o nea -
ield elec omagne ic cu a u e. These e ec s a e consis en wi h he embling-space ime
geome y ha also unde lies a omic s uc u e, pho on emission, and g a i a ional edshi .
Quan i a i ely, TSRT ep oduces absolu e nuclea hal -li es om milliseconds up o abou
en quin illion yea s, wi h mean loga i hmic de ia ion below one million h, using only h ee
global cons an s ixed once pe mode. This accu acy su passes con en ional mic oscopic and
empi ical models—which ypically yield mean loga i hmic de ia ions be ween one en h and
one hund ed h—by oughly en housand imes, showing ha nuclea decay is a de e min-
is ic geome ic phenomenon a he han a s ochas ic p ocess. All physical quan i ies a e
exp essed in absolu e SI and nuclea uni s, wi h TSRT calib a ion cons an s ied o un-
damen al cu a u e and li e ime ancho s wi hou empi ical scaling. TSRT also ep oduces
absolu e ission ene ge ics, yielding a o al ene gy elease o 170.0 MeV o he mal-neu on-
induced U-235 (n, ) wi hou any pe -obse able uning. By de i ing nuclea binding, decay,
eac ion imescales, and mass–ene gy con e sion om a single geome ic p inciple, TSRT
es ablishes a uni ied, p edic i e, and ep oducible desc ip ion o nuclea s abili y and ans-
o ma ion. I b idges mic oscopic nuclea s uc u e wi h mac oscopic ela i is ic consis ency,
showing ha mass–ene gy balance and decay kine ics a e na u al mani es a ions o cu a u e
edis ibu ion in embling space ime.
1
2
Con en s
1 In oduc ion 8
2 Co e No a ion, Con en ions and Me ic Signa u e 15
3 The Genesis o Space ime and Elemen a y Pa icles in TSRT 16
4 TSRT Nuclea Founda ions: E ec i e Cu a u e Po en ial and Mo ion 18
4.1 Single-nucleon mo ion in a embling nuclea ield . . . . . . . . . . . . . . . . . . 19
5 Binding Ene gy Sys ema ics and Radii om T embling Geome y 21
5.1 Sa u a ion and he B/A cu e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 Cha ge adii and scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6 De o ma ion Landscapes and Fission Ba ie s 25
6.1 Geome ic de o ma ion coo dina e and ba ie o ma ion . . . . . . . . . . . . . . 25
6.2 Fission ba ie s om cu a u e ac ion . . . . . . . . . . . . . . . . . . . . . . . . 28
6.3 Ac inide benchma ks: 235U and 239Pu ........................ 29
7 Fusion Ba ie s, P ope -Time Tunneling, and S-Fac o T ends 30
7.1 Fusion a es and as ophysical S- ac o . . . . . . . . . . . . . . . . . . . . . . . . 31
7.2 Cu a u e-enhanced Coulomb ba ie . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.3 WKB-like unneling om p ope - ime ac ion . . . . . . . . . . . . . . . . . . . . 34
8 TSRT Desc ip ion o S ong and Weak Nuclea Fo ces 37
8.1 The S ong In e ac ion as Cu a u e Sa u a ion . . . . . . . . . . . . . . . . . . . 37
8.2 The Weak In e ac ion as Cu a u e Recon igu a ion . . . . . . . . . . . . . . . . 39
8.3 Complemen a i y o S ong and Weak Fo ces . . . . . . . . . . . . . . . . . . . . 40
9 S abili y and Ins abili y o A omic Nuclei 40
9.1 Geome ic O igin o Nuclea S abili y . . . . . . . . . . . . . . . . . . . . . . . . 43
9.2 De e minis ic Li e ime Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
9.3 De e minis ic Li e ime De i a ion . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
9.4 Mode-Speci icFac o s ................................. 45
9.5 Calib a ions and Global Cons an s . . . . . . . . . . . . . . . . . . . . . . . . . . 46
9.6 Rep esen a i e Benchma ks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
9.7 P edic i e C i e ia o Nuclea Li e imes . . . . . . . . . . . . . . . . . . . . . . . 49
9.8 A De e minis ic βMa ix Elemen in TSRT and he Case o 14C......... 50
9.9 Geome ic Pic u e o S abili y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
9.10 Fusion-d i en ins abili y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
9.11 TSRT Li e ime Model and Calib a ion . . . . . . . . . . . . . . . . . . . . . . . . 53
10 Nuclea Fission in TSRT 55
10.1 Cu a u e S ess in Hea y Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
10.2 De e minis ic Clea age o he Nuclea Mode . . . . . . . . . . . . . . . . . . . . . 59
10.3 Ene gy Release in Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
10.4InducedFission..................................... 61
10.5 Compa ison wi h Expe imen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
10.6 Odd–E en S agge ing (OES) in Fission F agmen Yields . . . . . . . . . . . . . . 63
10.7 OES Model: Two Bessel Packe s wi h Localized Neck Bias . . . . . . . . . . . . . 64
10.8 Geome ic Syn hesis o Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
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11 Mass–Ene gy Con e sion in Fission and he TSRT Explana ion 67
11.1 Con en ional View o Mass De ec . . . . . . . . . . . . . . . . . . . . . . . . . . 67
11.2 TSRT In e p e a ion: Cu a u e Redis ibu ion . . . . . . . . . . . . . . . . . . . 68
11.3 Geome ic Equi alence o E=mc2.......................... 69
11.4 Dis ibu ion o Released Ene gy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
11.5 Compa ison wi h Expe imen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
11.6 Geome ic Signi icance o Mass–Ene gy Con e sion . . . . . . . . . . . . . . . . . 72
12 Nuclea Fusion in TSRT 72
12.1 Con en ional Desc ip ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
12.2 TSRT View: Cu a u e Concen a ion . . . . . . . . . . . . . . . . . . . . . . . . 74
12.3 Ba ie and De e minism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
12.4 Ene gy Release Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
12.5 Nume ical Example: Deu e ium–T i ium Fusion . . . . . . . . . . . . . . . . . . . 78
12.6S ella FusionChains.................................. 80
12.7 P ope -Time Ac ion and he Low-Ene gy Exponen ial . . . . . . . . . . . . . . . 81
12.8 Elec on–Sc eening Shi s in Ul a–Low–Ene gy Fusion . . . . . . . . . . . . . . . 83
12.9 Case S udy: Deep Sub-Ba ie Fusion Hind ance . . . . . . . . . . . . . . . . . . 85
12.10Geome ic Syn hesis o Fusion in TSRT . . . . . . . . . . . . . . . . . . . . . . . 88
13 Compa ison wi h Quan um Nuclea Models 89
13.1 S anda d Quan um App oaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
13.2 TSRT s. Liquid-D op Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
13.3TSRT s.ShellModel ................................. 91
13.4FusionandTunneling ................................. 92
13.5 Decay Channels and De e minis ic TSRT Li e imes . . . . . . . . . . . . . . . . . 93
13.6Nume icalBenchma ks................................. 93
13.7Geome icAd an ages................................. 94
14 B oade Implica ions and Fu u e Wo k 95
14.1 In eg a ion in o Fundamen al Physics . . . . . . . . . . . . . . . . . . . . . . . . . 95
14.2 As ophysical and Cosmological Connec ions . . . . . . . . . . . . . . . . . . . . . 95
14.3 Expe imen al and Nume ical Valida ion . . . . . . . . . . . . . . . . . . . . . . . 95
14.4 T anspo mapping: ecoil, Dopple , g a i a ional . . . . . . . . . . . . . . . . . . 96
14.5 Cha ge-Sensed Elec omagne ic Cu a u e T anspo . . . . . . . . . . . . . . . . 96
14.6 Fu u e Theo e ical De elopmen s . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
14.7Pe spec i e ....................................... 97
15 Calib a ed Ancho s o Absolu e Uni s in TSRT (Deu e on o Fission Chain) 97
15.1 Ene gy mapping: absolu e by cons uc ion . . . . . . . . . . . . . . . . . . . . . . 97
15.2 Cu a u e s eng h: ixed once and eused . . . . . . . . . . . . . . . . . . . . . . 98
15.3 Li e ime scale: a single physical ancho . . . . . . . . . . . . . . . . . . . . . . . . 98
15.4 On he embling ampli ude hAi: no an independen i . . . . . . . . . . . . . . 98
15.5 Gamma– ay no maliza ion (E2) and in e nal con e sion . . . . . . . . . . . . . . 98
15.6P ac icalsumma y ................................... 99
15.7 Ou look: emo ing ancho s en i ely . . . . . . . . . . . . . . . . . . . . . . . . . . 99
16 Conclusions 99
17 Acknowledgmen s 101
18 Appendices 103
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
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A Me hods: Nume ical P ocedu es, Pa ame e s, and Calib a ions 105
B Ex ended Tables and Da a Compa isons 108
B.1 Binding-ene gy compa ison along ep esen a i e chains . . . . . . . . . . . . . . . 108
B.2 Cha ge adii i s and esiduals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
B.3 Ac inide ission-ba ie benchma ks . . . . . . . . . . . . . . . . . . . . . . . . . . 109
B.4 Elec on sc eening: gamma- ay lines and be a spec a . . . . . . . . . . . . . . . 109
B.5 Deep sub-ba ie hind ance: uni e sal slope pai . . . . . . . . . . . . . . . . . . . 109
B.6 Odd–e en s agge ing (OES) da a and TSRT en elope . . . . . . . . . . . . . . . . 109
B.7 Li e ime ables and s abili y-map exce p s . . . . . . . . . . . . . . . . . . . . . . 110
C TSRT Nuclea Geome y: De ini ions and No a ion 111
C.1 Me ic Con en ion and P ope Time . . . . . . . . . . . . . . . . . . . . . . . . . 111
C.2 T embling De ia ion, Cu a u e Measu es, and Va iance . . . . . . . . . . . . . . 111
C.3 Binding and S abili y Func ionals . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
C.4 Ac ion Decomposi ion and S abili y Maps . . . . . . . . . . . . . . . . . . . . . . 112
C.5 Uni s and de ini ions used h oughou . . . . . . . . . . . . . . . . . . . . . . . . 113
Uni s and de ini ions used h oughou . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
D Geome ic O igin o Nuclea Magic Numbe s in TSRT 114
D.1 Wha isbeingcompu ed................................114
D.2 Cu a u e eigenmodes in a ini e embling domain . . . . . . . . . . . . . . . . . 114
D.3 Degene acy and cumula i e occupa ion . . . . . . . . . . . . . . . . . . . . . . . . 115
D.4 How we popula e he able ( oo s, degene acies, cumula i e) . . . . . . . . . . . . 115
D.5 Why a small co ec ion is needed (spin–geodesic spli ing) . . . . . . . . . . . . . 116
D.6 How o ep oduce he numbe s (sc ip s and s eps) . . . . . . . . . . . . . . . . . 118
D.7 Link o main ex and o he appendices . . . . . . . . . . . . . . . . . . . . . . . 118
D.8 Summa y o he eade ................................119
E Calib a ion, Physical Cons an s, and No maliza ions 120
E.1 Cons an sandUni s ..................................120
E.2 Ene gy no maliza ion used in compu a ions . . . . . . . . . . . . . . . . . . . . . 121
E.3 MATLAB: Cons an s and No maliza ion . . . . . . . . . . . . . . . . . . . . . . . 121
F Nume ical Disc e iza ion and In eg a ion Schemes 123
F.1 Spa ial G ids and Quad a u e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
F.1.1 MATLAB: G id and Quad a u e Se up . . . . . . . . . . . . . . . . . . . . 124
F.2 Con e gence and Tole ances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
F.2.1 MATLAB: Con e gence Tes Rou ine . . . . . . . . . . . . . . . . . . . . 126
F.3 MATLAB: G id + In eg a ion U ili ies . . . . . . . . . . . . . . . . . . . . . . . . 127
G T embling Fields o Nucleons and Composi e Nuclei 129
G.1 Pa ame ic scala baseline o nucleons . . . . . . . . . . . . . . . . . . . . . . . . 129
G.1.1 MATLAB: Nucleon T embling Field Gene a o . . . . . . . . . . . . . . . 129
G.2 Composi e con igu a ions and phase locking . . . . . . . . . . . . . . . . . . . . . 130
G.2.1 MATLAB: Composi e nucleus ield builde . . . . . . . . . . . . . . . . . . 131
G.3 MATLAB: Field Gene a o s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
H Cu a u e Measu es and Ene gy Func ionals 132
H.1 F om ξµν o a scala measu e K............................132
H.2 Binding, ission, and usion ene ge ics . . . . . . . . . . . . . . . . . . . . . . . . . 133
H.3 MATLAB: Cu a u e and Ene gies . . . . . . . . . . . . . . . . . . . . . . . . . . 133
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
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I S abili y Maps and Li e ime Es ima o s 135
I.1 Cu a u e-Slope S abili y Map (De e minis ic ∆K2Analysis) . . . . . . . . . . . 135
I.2 Mode-awa e Q- alues om TSRT bindings . . . . . . . . . . . . . . . . . . . . . . 136
I.3 Ene ge ics and cu a u e supp ession . . . . . . . . . . . . . . . . . . . . . . . . . 137
I.4 Mode-awa e Q- alues: implemen a ion pa h . . . . . . . . . . . . . . . . . . . . . 137
I.5 Pe -mode li e ime p edic ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
I.6 Pipelineand iles....................................137
I.7 Nume icalsa egua ds..................................137
J Kinema ic T anspo (Recoil, Dopple , G a i y) 138
J.1 Two–body ecoil ....................................138
J.2 Dopple shi (sou ce o obse e mo ion) . . . . . . . . . . . . . . . . . . . . . . 139
J.3 G a i a ional edshi (op ional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
J.4 O de ing and p ac ical applica ion . . . . . . . . . . . . . . . . . . . . . . . . . . 140
K Cha ge-Sensed Elec omagne ic Cu a u e T anspo (CSCT) 141
K.1 Se upandde ini ions..................................141
K.2 Closed- o m nea - ield model and scaling . . . . . . . . . . . . . . . . . . . . . . . 142
K.3 Implemen a ion ecipe (used in igu es/ ables) . . . . . . . . . . . . . . . . . . . . 144
K.4 Wo kedexamples....................................144
K.5 No esoncalib a ion ..................................144
L Binding-Ene gy Benchma ks Ac oss he Valley o S abili y 146
L.1 Me hodology ......................................146
L.2 Rep esen a i e Resul s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
L.3 Discussion........................................148
L.4 Rep oducibili y.....................................148
L.5 MATLAB: Binding Plo P oduce and Co e Rou ine . . . . . . . . . . . . . . . . 148
M Fission: Su ace Cu a u e G adien and Bi u ca ion 151
M.1 Compu ing ∆Ksu ace ..................................151
M.2 Bi u ca ion Pa h and Ene gy Release . . . . . . . . . . . . . . . . . . . . . . . . . 152
M.3 Neu on-cap u e inc emen and h eshold c ossing . . . . . . . . . . . . . . . . . . 153
M.4 MATLAB: Fission Rou ines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
N De i a ion o he Neck-Mode Model o Odd–E en S agge ing 157
N.1 Geome y, no a ion, and he OES indica o . . . . . . . . . . . . . . . . . . . . . 157
N.2 Cylind ical neck modes and bounda y locking . . . . . . . . . . . . . . . . . . . . 157
N.3 F om neck modes o an OES ampli ude . . . . . . . . . . . . . . . . . . . . . . . 158
N.4 Fini e neck ex en and Gaussian damping . . . . . . . . . . . . . . . . . . . . . . 158
N.5 Pa i y locking and ac ion gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
N.6 A localized cu a u e “bias” and why i is geome ic . . . . . . . . . . . . . . . . . 159
N.7 Collec ing e ms: de i a ion o he wo king model . . . . . . . . . . . . . . . . . . 159
N.8 Connec ion o expe imen and ep oducibili y . . . . . . . . . . . . . . . . . . . . 160
O Fusion: Geodesic O e lap C i e ion and Ene gy 161
O.1 De e minis ic geodesic o e lap: de ini ion and ma ch o he main ex . . . . . . 161
O.2 F om o e lap o Ve and he ansmission exponen . . . . . . . . . . . . . . . . . 162
O.3 Fusion ene gy om cu a u e concen a ion: de ini ion, ma ch, and nume ics . . 162
O.4 Rep oducibili y, g ids, and con e gence . . . . . . . . . . . . . . . . . . . . . . . . 163
O.5 MATLAB:FusionRou ines ..............................164
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025

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P TSRT E alua ion o As ophysical S-Fac o s 166
P.1 F om c oss sec ion o S- ac o : de ini ion and de i a ion . . . . . . . . . . . . . . 166
P.2 TSRT o igin o he exponen ial and mapping o S(E)...............167
P.3 Asymp o ics and he TSRT in a ian s Ξ1,Ξ2....................168
P.4 Calib a ion o he uni e sal slopes (α, β).......................168
P.5 Nume ical se up and link o he TSRT ac ion . . . . . . . . . . . . . . . . . . . . 169
P.6 Rep esen a i e Resul s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
P.7 Discussion........................................170
P.8 Rep oducibili y.....................................171
P.9 MATLAB: TSRT S- ac o Sc ip . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Q Deep Sub-Ba ie Fusion Hind ance: TSRT Implemen a ion and Valida ion 175
Q.1 Theo y- o-Algo i hm Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Q.2 Nume ical De ails and De e minism . . . . . . . . . . . . . . . . . . . . . . . . . 176
Q.3 MATLAB: Fusion Hind ance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
R Seconda y Ta ge s: TSRT Ad an ages Beyond Hind ance 180
R.1 Odd–E en S agge ing (OES) Ranking . . . . . . . . . . . . . . . . . . . . . . . . 180
R.2 MATLAB: OES om Expe imen al Da a . . . . . . . . . . . . . . . . . . . . . . 180
R.3 Elec on–Sc eening in TSRT: De i a ion, Da a, and Code . . . . . . . . . . . . . 183
R.3.1 No a ion and p elimina ies . . . . . . . . . . . . . . . . . . . . . . . . . . 183
R.3.2 De i a ion o he ene gy-dependen sc eening . . . . . . . . . . . . . . . . 183
R.3.3 No a ion and Uni s o Sc eening Analysis . . . . . . . . . . . . . . . . . . 186
R.3.4 Somme eld pa ame e and TSRT enhancemen a io . . . . . . . . . . . . 187
R.3.5 Eme gen embling modula ion and Bessel o m . . . . . . . . . . . . . . 188
R.3.6 Da a able (Pd hos d(d, p) ) .........................190
R.3.7 MATLAB lis ing and wo k low . . . . . . . . . . . . . . . . . . . . . . . . 190
S En i onmen , De e minism, and Valida ion P o ocols 191
S.1 MATLAB Ve sion and Toolboxes . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
S.2 De e minismandSeeds ................................191
S.3 Valida ionChecklis ..................................192
T Rep oducibili y: Figu e and Table Gene a ion 194
T.1 En i onmen , cons an s, and uni s . . . . . . . . . . . . . . . . . . . . . . . . . . 194
T.2 Mas e build ......................................194
T.3 A i ac map ......................................194
T.4 Odd–E en S agge ing igu e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
T.5 Sc eening enhancemen igu e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
T.6 Hind ance igu e ....................................195
T.7 Binding cu es and ables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
T.8 FusionQ able .....................................195
T.9 Fission compa ison able . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
T.10 S- ac o slopes and sys em cu es . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
T.11 Con e gence plo s (op ional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
T.12 De e minism, ilenames, and audi s . . . . . . . . . . . . . . . . . . . . . . . . . . 196
U Sensi i i y and Unce ain y Analyses 197
U.1 Pa ame e Scans ....................................197
U.2 MATLAB:ScanSkele on ...............................198
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
7
V Uni s, Con e sions, and Sign-Con en ion No es 200
V.1 Uni sandCon e sions.................................200
V.2 Physical Cons an s Used in Compu a ions . . . . . . . . . . . . . . . . . . . . . . 201
V.3 Me ic Sign and Reade Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
W Expe imen al Da a Tables 203
W.1 Comp ehensi e Da a P o enance Map (All Resul s) . . . . . . . . . . . . . . . . . 203
W.2 Binding Ene gies: Small Ve ba im Exce p ( o Rep oducibili y) . . . . . . . . . . 204
W.3 Cha ge Radii: Small Ve ba im Exce p ( o Rep oducibili y) . . . . . . . . . . . . 204
W.4 Sensi i i y Summa y (Rep esen a i e) . . . . . . . . . . . . . . . . . . . . . . . . 204
W.5 Calib a ion Ancho : γ(E2) in 156Gd .........................205
W.6 Odd–E en S agge ing in 235UFissionYields.....................205
W.7 Sc eened d(d, p) S- ac o (Pdhos )..........................205
W.8 Hinde ed Hea y–Ion Fusion: De i a ion o he TSRT Asymp o ic Fo m . . . . . . 205
X TSRT Pa ame e s and Cons an s Used in All Figu es and Calcula ions 208
Y MATLAB P ocedu es and Rep oducibili y Wo k low 210
Y.1 MATLAB Code Index and Thema ic Map . . . . . . . . . . . . . . . . . . . . . . 211
Y.2 Global Cons an s and Ini ializa ion . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Y.3 Con e gence and Robus ness Scans . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Y.4 Binding-Ene gy and Radius Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . 218
Y.5 Elec on Sc eening in d(d, p) (Pdhos ) .......................223
Y.6 Hea y-Ion Fusion Hind ance (S- ac o )........................225
Y.7 Odd–E en S agge ing (OES) o 235U(n h, )Yields..................231
Y.8 E2 Gamma Calib a ion and Ra es . . . . . . . . . . . . . . . . . . . . . . . . . . 234
Y.9 Li e ime Pipeline and Benchma ks . . . . . . . . . . . . . . . . . . . . . . . . . . 238
Y.10 Diagnos ics and Valida ion Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . 240
Y.11 S abili y Maps and Diagnos ic Lines . . . . . . . . . . . . . . . . . . . . . . . . . 241
Y.12 Nume ical Helpe s (Quad a u e) . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
Y.13 Co e Field Gene a o s (suppo ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Y.14 CSCT / Nea -Field Elec omagne ism Tables (op ional b anch) . . . . . . . . . . 254
Y.15 Fission Ene ge ics and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Y.16 Fission Pa h Co ela ions (op ional) . . . . . . . . . . . . . . . . . . . . . . . . . 274
Y.17 Pa ame e Scan U ili ies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
Y.18 Mas e O ches a o (Run-All) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
Index o MATLAB Files 348
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
8
1 In oduc ion
This wo k is dedica ed o he luminous legacy o Si Isaac New on, Albe Eins ein, and Lise
Mei ne , isiona ies who, om s a es o soli ude, con inemen , and exile, disce ned he unda-
men al p inciples ha bind he cosmos, om he celes ial o he nuclea .1
The s abili y o a omic nuclei has long been ea ed as an eme gen s a is ical p ope y o he
s ong and weak in e ac ions.2In s anda d nuclea models, decay p obabili ies a e in oduced
h ough ba ie -pene a ion o malisms and s ochas ic ansi ion a es. Wi hin he T embling
Space ime Rela i i y Theo y (TSRT), howe e , decay is a de e minis ic geome ic phenomenon:
i ep esen s he elaxa ion o cu a u e s ess in a localized embling-space ime con igu a ion.
Unlike empi ical ela ions such as Viola–Seabo g [10] o mode n mac oscopic–mic oscopic de-
cay models [11], which ypically de ia e om expe imen al hal -li es by 10−1–10−2in log10 scale,
TSRT ep oduces measu ed li e imes ac oss all decay channels wi h a mean loga i hmic de ia ion
below 10−6. This p ecision is achie ed wi hou any nucleus-speci ic i ing o channel-dependen
pa ame e s: each mode (β−,α, and spon aneous ission) ollows om he same de e minis ic
cu a u e dynamics. The con as in p edic i e powe ma ks a decisi e imp o emen o e exis -
ing li e ime sys ema ics and alida es TSRT as a ully quan i a i e, pa ame e - ee desc ip ion
o nuclea decay.
A di ec e i ica ion o his absolu e scaling is p o ided by he mal-neu on-induced ission
o 235U, o which he TSRT pipeline yields ETSRT
iss = 170.028 MeV, in quan i a i e ag eemen
wi h he expe imen al o al ene gy elease (170 ±3 MeV; Table 42 (p. 257)). This esul
1The his o y o physics is o en o cla i y eme ging om obscu i y, and ew embody his u h mo e p o oundly
han New on, Eins ein, and Mei ne . Thei g ea es insigh s we e no bo n in com o bu we e o ged in he
c ucible o pe sonal and p o essional ad e si y, eminding us ha he pa h o undamen al u h is o en walked
alone. Si Isaac New on †(1643–1727), ollowing he dea h o his a he and he es angemen o his mo he ’s
ema iage, u ned his soli ude in o a p o ound inwa d jou ney. F om his seclusion, he shaped ou unde s anding
o he uni e se, laying he e y ounda ions o physics and calculus [1,2]. Albe Eins ein †(1879–1955), g appling
wi h he silen g ie o his daugh e ’s loss and he p o essional con inemen o a pa en cle k’s o ice, held s ead as
o his pu sui o u h [3, 4]. The e, he e ined New on’s legacy and unco e ed he deep uni y o mass and
ene gy [5], a p inciple ha would become a co ne s one o he nuclea age. Lise Mei ne †(1878–1968), o ced
in o exile om Nazi Ge many and s ipped o he posi ion, ca ied he puzzle o nuclea ission in he mind. In a
o eign coun y, wi h only heo e ical ools, she p o ided he i s co ec in e p e a ion o he p ocess, calcula ing
he immense ene gy eleased and gi ing physical eali y o Eins ein’s amous equa ion [6].
Toge he , hei li es o m a powe ul na a i e o insigh w es ed om ad e si y, p o ing ha he deepes
u hs o na u e a e o en e ealed o hose who pe sis agains he cu en . As wi h he nuclea o ces hey
helped un eil, i s ha nessed o des uc ion [7], he deepe lesson is no one o annihila ion, bu o balance
and po en ial. F om he i e o ission, we a e guided owa d li e-sus aining applica ions [8] and he p omise o
usion [9]. The powe inhe en in na u e challenges humani y o choose wisdom o e de as a ion.
Du ing he Be lin yea s, Mei ne wo ked alongside James F anck †(1882–1964), in he closely kni physics
communi y; h ough F anck, and he subsequen supe iso y line descending ia Egon A. Hiedemann †(1900–
1969), and Mack A. B eazeale †(1930–2009) o he p esen au ho , he academic genealogy o his wo k aces
back o ha same gene a ion o a omic pionee s. Thei expe imen al app oach was passed on when B eazeale
in oduced he au ho o expe imen al physics in he ea ly 2000s. I was in B eazeale’s labo a o y ha he
au ho i s encoun e ed a ema kable op ical lens, sal aged om a decommissioned U.S. spy plane—a elic o
Cold Wa ensions o e he e y nuclea a senal Mei ne ’s science had unlocked, now epu posed o sha pen he
ocus o peace ul inqui y. The e, oo, he i s wo ked wi hin a Fa aday cage, i s silence se e ing all cell phone
and adio links; he ee ie quie was s angely amilia , a di ec echo o his childhood, uning his sho wa e adio
o he c ackling oices om behind he I on Cu ain and wonde ing a he sec e wo ld hey con eyed. Al hough
he au ho ’s own esea ch has elied chie ly on expe imen al me hods in enginee ing physics and ul asonics, his
endu ing ascina ion has emained wi h heo e ical physics, o which he p esen wo k belongs; in his espec , i
also con inues a second supe iso y line, h ough Oswald J. Le oy †(1936–2022), in an academic genealogy ha
includes Hen i Poinca é †(1854-1912).
By u he unco e ing he sec e s o he nucleus, his wo k seeks o honou hei adi ion: o pu sue cla i y
whe e he way is na ow, and o us ha na u e, when app oached wi h esol e and e hical cou age, e eals i s
cohe ence e en om he deepes obscu i y.
2Fo TSRT cu a u e-based s abili y c i e ia see Appendix C.
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
9
ollows au oma ically om he cu a u e-based ene gy mapping and he one- ime scission-wid h
calib a ion (Lis ings 44 (p. 297) and 29 (p. 263)), wi hou any pe -obse able adjus men .
Global liquid-d op i s, mac oscopic–mic oscopic maps, and shell/DFT app oaches ep oduce
many ends bu emain p obabilis ic and pa ame e -hea y; on ep esen a i e li e ime bench-
ma ks hei mean de ia ions a e ypically a he en h- o-hund ed h-o -a-decade le el in log
space. The TSRT law, calib a ed once on a single weak ancho and a single scale pe decay
mode, achie es a mean absolu e de ia ion o o de one-million h o a decade ac oss β,α, and
SF cases p esen ed he e, wi hou pe -nuclide adjus men s.
TSRT is a de e minis ic geome ic amewo k in which space ime ca ies a small, causal,
locally co ela ed embling o he me ic. Conc e ely, dynamics p oceed along u u e-di ec ed
imelike geodesics o gµν =ηµν +ξµν, whe e he de ia ion ξµν encodes he embling ield
and espec s p ope - ime causali y (dτ2=gµνdxµdxν>0). Ma e co esponds o localized
embling eigenmodes, and he amilia in e ac ions a ise as egimes o cu a u e: g a i y om
global cu a u e, elec omagne ism om long- ange oscilla o y embling, he s ong in e ac ion
om cu a u e sa u a ion unde o e lap, and he weak in e ac ion om cu a u e econ igu-
a ion when local symme y ails o be main ained. Quan um-signa u e phenomena (spec a,
blackbody law, in e e ence, and en anglemen ) ollow no om p obabilis ic pos ula es bu
om de e minis ic co ela ions among geodesic cong uences. Obse ables a e compu ed om
cu a u e-based ac ion and ene gy unc ionals wi h a single calib a ion ha p opaga es ac oss
domains [12–17].
"While his wo k uses he amewo k o TSRT o p opose a new uni ica ion o nuclea physics,
i does so wi h he deep humili y bo n om con empla ing he monumen al insigh s o he pas
cen u y. We a e bu la ecoming a ele s on a pa h ca ed by gian s, and ou iew is as only
because o he heigh s o which hey ca ied us."
This a icle applies TSRT’s causal embling geome y o nuclei,3using he same s uc u es
de ined in he ounda ional pape s (me ic decomposi ion, p ope - ime causali y, eme gen ac ion
scale, and co puscula ene gy ela ions) [12–18]. Classical nuclea models (liquid d op, shell,
mac–mic) a e e e ed o only as benchma ks o compa ison.
The mo i a ion o his wo k has bo h scien i ic and pe sonal o igins and de elops he nuclea
consequences o TSRT.4
Wi hin he uni ied geome ic TSRT pic u e, a single once-calib a ed cu a u e s eng h, one
li e ime ancho , and exac SI con e sion ac o s connec TSRT di ec ly o measu ed nuclea
3Disclaime : This open undamen al esea ch a icle de elops T embling Space ime Rela i i y Theo y (TSRT)
a a undamen al and heo e ical le el. All esul s conce n he ma hema ical and physical ounda ions o space ime
geome y and i s explana o y scope o nuclea phenomena. The con en is limi ed o gene al scien i ic heo y
and does no p esen , p opose, o imply any echnology o echnological applica ion.
4The au ho mos humbly obse es a pa allel be ween he genesis o his wo k and ha o Pablo Picasso’s
Gue nica. Fo Picasso in his Pa is s udio, he ca alys was he bombing o Gue nica, an a oci y ha ans o med
a o mal commission o he 1937 Wo ld’s Fai in o a aw, u gen ac o wi ness. Simila ly, o he au ho , a
o ma i e pe iod o in ellec ual dissa is ac ion du ing his s udies a he Ca holic Uni e si y o Leu en, while
engaging wi h he cou ses o KUL P o esso J. Coussemen and he ounda ional ex book o Ghen Uni e si y
P o esso K. Heyde, e ealed a ield seemingly cons ained by agmen ed models and a lack o a cohe en , uni ied
heo y. This ini ial disappoin men , bo n o a s uden ’s sea ch o deepe answe s ha emained elusi e, was la e
c ys allized by a isi o Hi oshima in 2004. Tha ca aly ic shock ans o med a gene al in ellec ual us a ion
in o he speci ic, bu ning desi e o unde s and he a omic nucleus ha unde pins his esea ch. Jus as Picasso
could no ha e o eseen he legacy o Gue nica, he au ho did no an icipa e ha his in ellec ual jou ney, bo n
o bo h solemn wi ness and a esol e o add ess pas unce ain ies, would culmina e in wha he now ega ds as
his mos de ini i e con ibu ion o he ield. This wo k, he e o e, is p esen ed no me ely as a inding, bu as a
pe sonal scien i ic ho izon, a ep esen a ion o he limi o he au ho ’s own capaci y o a p oblem whose po en ial
o deepe unde s anding emains, as always, in ini e. The au ho he eby in i es he physics communi y and, in
pa icula , nuclea physicis s, o con inue his wo k and o apply i o speci ic opics and esea ch p oblems in
hei ields.
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
16
We adop he (+,−,−,−)me ic signa u e h oughou .7P ope ime and he embling
me ic decomposi ion a e gi en by
dτ2=gµν dxµdxν, gµν =ηµν +ξµν, dτ2>0( imelike).(4)
All global no maliza ion and calib a ion cons an s appea once in Appendix A. These include
he cu a u e–ene gy con e sion cons an CE(Equa ion (6)), he TSRT adius scale TSRT
0
(Equa ion (14)), and he global li e ime scale Cλ(Equa ion (2)). No cons an is in oduced
locally in any sec ion wi hou c oss- e e ence o his mas e lis .
Unless s a ed o he wise, he speed o ligh cis e ained explici ly o main ain dimensional
anspa ency. Nume ical wo k ollows a consis en uni sys em based on MeV and MeV/c2
(1 u c2= 931.49410242 MeV). Fo compac ness, accompanying exp essions in na u al uni s
(c= 1) may be shown alongside ull SI/ ela i is ic o ms; hese a e p o ided pu ely o algeb aic
cla i y and a e ne e used o nume ical subs i u ion.
Table 7: Mas e no a ion employed h oughou he manusc ip . Topic-speci ic e inemen s ap-
pea in Table 19 (p. 113). All exp essions use he me ic signa u e (+,−,−,−). Comple e uni
de ini ions: Appendix V.
Symbol Meaning
Z, N, A P o on numbe , neu on numbe , mass numbe (A=Z+N)
τP ope ime (mono onic in TSRT once causal o ien a ion is ixed)
uµFou - eloci y along a ( u u e-di ec ed) geodesic
Kµν T embling-cu a u e (cu a u e-s ess) enso used in TSRT ene gy/ac ion unc ionals
hK2iLocal cu a u e a iance (TSRT)
∆K2(Z, N)Cu a u e-supp ession measu e (s abili y diagnos ic)
SiT embling ac ion o an isola ed nucleon eigenmode
Sin Leading o e lap (cu a u e-sa u a ion) con ibu ion o he ac ion
δSco Highe -o de in e e ence co ec ions (mul ipole/phase)
RcCu a u e/o e lap (locking) scale; e ec i e s ong- ange indica o
Asa Sa u a ion ampli ude o embling modes a s abili y h eshold
ℓEM, ℓCElec omagne ic / colo embling cohe ence leng hs
λBRela i e de B oglie scale eco e ed by TSRT ac ion h esholds
κneck Local scission-neck cu a u e (con ols OES ampli ude)
∆(3)(Z)Th ee-poin odd–e en s agge ing indica o on cha ge yields
∆Se en–odd P ope - ime ac ion gap be ween neighbo ing e en/odd agmen geodesics
O[ΞA,ΞB]O e lap unc ional be ween embling con igu a ions ΞA,ΞB
Oc i C i ical o e lap h eshold o usion onse
R⋆Sepa a ion a which O=Oc i
∆UeTSRT geome ic sc eening shi (de e minis ic EM embling pola iza ion)
ηSomme eld pa ame e Z1Z2e2/(~ )
~TSRT TSRT ac ion uni (equals ~in he low-cu a u e coa se-g ained limi )
3 The Genesis o Space ime and Elemen a y Pa icles in TSRT
TSRT ad ances a single, de e minis ic geome ic amewo k om which space ime, pa icles,
and in e ac ions eme ge cohe en ly [12–18]. The s a ing poin is a causally o ien ed me ic
wi h a small, local, de e minis ic “ embling”: once a ime o ien a ion is ixed, p ope ime τ
inc eases mono onically along u u e-di ec ed imelike geodesics. On cosmological scales, he
coa se-g ained imp in o hese local co ela ions yields an e ec i e global low ha looks like
Hubble expansion; he same mechanism ha s abilizes mic oscopic modes hus se s he la ge-
scale s uc u e o he obse ed 4D space ime [16]. In his iew, “da k” phenomenology e lec s
geome ic bookkeeping o embling cu a u e a he han addi ional ma e sec o s.
7When compa ing wi h sou ces using (−,+,+,+), adjus signs in line elemen s, cu a u e scala s, and ac ion
densi ies acco dingly. TSRT’s use o p ope - ime causali y ollows [12].
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025

17
Elemen a y pa icles a e modeled no as poin objec s bu as localized eigenmodes o em-
bling cu a u e. Thei mass, spin, and cha ges a e geome ic in a ian s o he mode’s s uc u e;
amilies a ise om a ini e ca alog o admissible opologies and symme ies [12,17]. Composi e
sys ems (a oms and nuclei) a e bound con igu a ions o such modes whose s abili y is diagnosed
by educ ions in app op ia e cu a u e measu es (see Sec ion 4). Pho ons co espond o p opa-
ga ing elec omagne ic embling modes [13,18]; neu inos appea as neu al geodesic agmen s
pe mi ed by cu a u e econ igu a ion cons ain s [12,15].
Wi hin his geome ic pic u e he ou in e ac ions a e no pos ula es bu egimes o he
same embling cu a u e. G a i a ion is global cu a u e accumula ion (long- ange cohe ence);
elec omagne ism is long- ange oscilla o y co ela ion ca ied by cu a u e wa es (pho ons) [13];
he s ong in e ac ion is cu a u e sa u a ion, whe e local o e lap o nucleonic modes supp esses
cu a u e a iance and locks a igh ly bound con igu a ion; and he weak in e ac ion is econ-
igu a ion, whe e locking symme ies ail and he sys em elaxes de e minis ically along causal
geodesics, expo ing cha ge and lep on numbe ia escaping agmen s. Phenomena o en spli
be ween “quan um” and “ ela i is ic” domains a e he eby uni ied: supe posi ion and unnel-
ing pos ula es a e eplaced by causal geodesic co ela ions and ac ion-minimizing econ igu a-
ions [12,15,18].
Mass–ene gy and binding ollow om he same cu a u e accoun ing. The e ec i e ine ial
mass o a mode is a unc ional o i s embling cu a u e, and he binding ene gy o a composi e
is he educ ion o ha measu e when isola ed modes lock cohe en ly. Fo nuclei, he empi ical
mass–ene gy de ec is eco e ed as he di e ence be ween he sum o isola ed nucleon eigenmasses
and he mass o he bound con igu a ion (Equa ion (9) in Sec ion 4). The global sys ema ics o
nuclea binding used la e a e summa ized by
B(A, Z) = Bgeom(A, Z) + ∆Bshell(A, Z) + ∆Bpai (A, Z),(5)
in oduced o mally in Sec ion 5 (Equa ion (10)). He e Bgeom cap u es sa u a ion om cu a u e
locking, while shell8and pai ing9 e ms a ise om symme y/phase co ec ions o he unde lying
embling modes.
Fo o ien a ion, we w i e A o mass numbe and B o o al nuclea binding ene gy; B/A
is he binding pe nucleon. In bo h da a and TSRT, B/A peaks in he i on egion, A≃56–62
(Sec ion 5).10 Geome ically, his maximum is he balance poin whe e sho - ange cu a u e
sa u a ion is s onges while long- ange Coulomb cu a u e emains modes ; below i he assem-
bly is unde -sa u a ed, and abo e i su ace and Coulomb cos s e ode he gain.
8In con en ional nuclea s uc u e, a “shell” deno es a se o single-pa icle o bi als in a mean- ield po en ial,
g ouped by quan um numbe s (e.g. nℓj) and sepa a ed by ene gy gaps. Magic numbe s a ise whe e la ge gaps yield
enhanced s abili y. In TSRT, a “shell” co esponds ins ead o a amily o localized embling eigenmodes whose
cu a u e pa e ns main ain cohe ence unde sa u a ion. The TSRT analogue o a shell gap is a geome ic gap:
a disc e e jump in he cu a u e-locking pa e n ha locally supp esses mode ea angemen . Bo h iewpoin s
iden i y special nucleon numbe s wi h enhanced s abili y, bu TSRT a ibu es he e ec o cu a u e opology
a he han quan ized o bi als. The nume ical shell co ec ions in Equa ion (5) he e o e map di ec ly on o
geome ic cohe ence h esholds a he han quan um le el spacings.
9In con en ional heo ies, “Pai ing” e e s o he empi ical endency o like nucleons (pp o nn) o o m co -
ela ed J= 0 spin-single pai s, lowe ing he ene gy o e en–e en nuclei and p oducing odd–e en s agge ing
in masses and yields. In TSRT, pai ing e lec s a cu a u e-cohe ence e ec : when wo like embling modes
o e lap wi h opposi e spa ial/phase o ien a ion, hei local cu a u e s esses pa ially cancel, educing he e -
ec i e ac ion. E en–e en nuclei minimize cu a u e s ess geome ically; odd–Aand odd–odd sys ems e ain
uncompensa ed cu a u e. The phenomenology (enhanced s abili y o e en–e en nuclei, s agge ing wi h pa i y
o Zand N) is sha ed in bo h iews, bu TSRT explains i as a de e minis ic geome ic cancella ion a he
han a quan um-co ela ion ene gy. Thus ∆Bpai in Equa ion (5) plays he same ole nume ically while ha ing
a cu a u e-based o igin.
10When he ex men ions “B/A @A≈60” o colloquially “B/A = 60,” i is sho hand o he peak o B/A
occu s nea mass numbe A≈60 (i on g oup). Nume ically, B/A a he peak is ∼8–9MeV pe nucleon in
con en ional uni s. I TSRT no malized uni s a e used elsewhe e, he con e sion is gi en in Appendix A oge he
wi h he no maliza ion cons an .
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
18
The channels mos ele an o his pape ollow immedia ely. P o ons and neu ons a e he
nucleonic embling eigenmodes whose o e lap builds nuclei; pho ons encode eleased elec o-
magne ic cu a u e in adia i e elaxa ions; and elec ons wi h (an i)neu inos a e he geodesic
agmen s p oduced in weak econ igu a ions du ing βp ocesses. In wha ollows we use his
dic iona y consis en ly: s ong sa u a ion and he e ec i e cu a u e po en ial (Sec ion 4); de-
o ma ion, ba ie s, and ission (Sec ion 6 and Sec ion 10); and he usion p og am, ba ie s,
p ope - ime ansmission, and nea - ield sc eening (Sec ion 7 and Sec ion 12). This b ie e-
cap p o ides he concep ual b idge om i s p inciples o he nuclea esul s de eloped in he
emainde o he pape .
4 TSRT Nuclea Founda ions: E ec i e Cu a u e Po en ial and
Mo ion
Consis ency wi h implemen a ion: The e ec i e cu a u e ene gy and locking cons uc ions de-
sc ibed he e a e implemen ed by s nucleon ield.m, s composi e ield.m,
s cu a u e enso .m, and s ene gy omcu a u e.m. No i ed, pe -nucleus po en ial is
in oduced in code; all e ms o igina e om he TSRT unc ionals summa ized in Appendix H
and cons an s in Appendix E.
Suppo ing appendices: Appendix C (de ini ions and locking), Appendix H (ene gy unc ionals),
Appendix F (schemes), and Appendix E (calib a ions).
Co e MATLAB: s nucleon ield.m, s composi e ield.m, s cu a u e enso .m,
s ene gy omcu a u e.m.
In TSRT,11 e e y undamen al pa icle is desc ibed as a localized embling eigenmode o
space ime cu a u e12 [12,17].
Wi hin his amewo k, embling space ime en o ces causali y by p ohibi ing backwa d e o-
lu ion o τ.13 The ou undamen al in e ac ions hen eme ge no as ex e nal pos ula es bu as
dis inc geome ic cons ain s o embling cu a u e: elec omagne ism as long- ange oscilla o y
co ela ion, g a i a ion as global cu a u e accumula ion, and he s ong and weak in e ac ions
as sho - ange cu a u e-binding and cu a u e- econ igu a ion mechanisms [12].
F om his pe spec i e, an a omic nucleus is no a collec ion o independen nucleons bound
by an ex e nally de ined po en ial.14 Ins ead, i is a sel -consis en geome ic con igu a ion o
embling modes in which he local cu a u e ields o p o ons and neu ons in e lock o o m
a s able causal s uc u e. The s ong in e ac ion eme ges as a di ec consequence o cu a u e
11In TSRT: (1) Pa icles a e localized embling eigenmodes o space ime cu a u e, espec ing he (+,−,−,−)
me ic and causal o ien a ion ixed h oughou ; (2) nuclea binding eme ges when cu a u e–o e lap sa u a es a
TSRT locking c i e ion (Appendix C, X); (3) weak p ocesses a e de e minis ic cu a u e econ igu a ions along
p ope ime, no s ochas ic ansi ions; (4) a single calib a ion cons an (Appendix A) se s absolu e scales and
is ne e e- i pe nucleus o channel. All de i a ions ollow hese assump ions and a e c oss- e e enced o he
appendices o ull ep oducibili y.
12Th oughou his wo k we adop he (+,−,−,−)me ic signa u e. This choice is pu ely con en ional: one
could equally well use (−,+,+,+), p o ided all de ini ions a e adjus ed consis en ly. The wo signa u es yield
iden ical physical p edic ions, di e ing only in he sign pa e ns ha appea in he ma hema ical exp essions. In
ei he con en ion, p ope ime τad ances mono onically along u u e-di ec ed imelike geodesics once a causal
o ien a ion is ixed. Wha he (+,−,−,−)con en ion makes explici is ha τ emains posi i e along causal
e olu ion, while in he opposi e con en ion he same physical beha io appea s wi h in e ed algeb aic signs.
Thus he me ic choice does no al e he unde lying physics, bu only he symbolic o m by which we ep esen
i .
13Rega dless o signa u e in Equa ion (4).
14The locking c i e ion and cu a u e-o e lap mechanics a e e iewed in Appendix C and Appendix X
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
19
sa u a ion:15 when wo o mo e nucleonic embling modes a e b ough wi hin a c i ical p ox-
imi y, hei local oscilla o y geome ies o e lap, leading o a collec i e minimum in he e ec i e
embling ac ion. This geome ic esonance en o ces a igh ly bound con igu a ion analogous o
a po en ial well, bu undamen ally de e minis ic and me ic-based.
The weak in e ac ion, in u n, a ises om pe missible econ igu a ions o embling geodesics
when cu a u e symme y condi ions a e no pe ec ly me .16 Fo example, be a decay co e-
sponds in TSRT o a local embling ins abili y in which a nucleonic mode elaxes by emi ing a
causal geodesic agmen , iden i ied a mac oscopic scales as an elec on o neu ino [15]. Thus,
nuclea s abili y is egula ed no by p obabilis ic unneling, bu by he de e minis ic geome y
o cu a u e-bounded embling pa hs.
We will epea edly use he cu a u e-ene gy unc ional
E[Ξ] = CEZV
Kµν(Ξ) uµuνdV, (6)
wi h CE ixed by he calib a ion p o ocol in Appendix A. The mass–ene gy link we employ la e
(e.g., in Equa ion (108)) is gi en in Sec ion 11.2, Equa ion (119). The enso Kµν and i s scala
no m a e de ined in Appendix C (Uni s and de ini ions).
4.1 Single-nucleon mo ion in a embling nuclea ield
We model single-nucleon dynamics by he embling-de o med geodesic equa ion
d2xµ
dτ2+ Γµ
αβη+ξnucldxα
dτ
dxβ
dτ = 0,(7)
whe e xµ(τ)deno es he space ime wo ldline o he nucleon pa ame e ized by i s p ope ime τ,
and Γµ
αβ[η+ξnucl]a e he connec ion coe icien s associa ed wi h he local me ic gµν =ηµν +
ξnucl,µν.17 He e, ηµν ep esen s he backg ound Minkowski me ic and ξnucl,µν he embling
de ia ion ield gene a ed by he collec i e nuclea cu a u e. The ield ξnucl is cons ained
by p ope - ime causali y and ene gy–momen um conse a ion, ollowing he same geome ic
pos ula es as in Re e ences [12,14].
Equa ion (7) exp esses he causal mo ion o a nucleon as a ee geodesic in a locally de o med
space ime whose cu a u e oscilla es a he embling equency de e mined by he nuclea con-
igu a ion. Physically, he e m Γµ
αβ[η+ξnucl]ac s as an e ec i e in e nal o ce desc ibing how
he nucleon’s wo ldline esponds o local cu a u e oscilla ions a he han o ex e nal po en-
ials. The esul ing e ec i e po en ial Ve he e o e eme ges geome ically om he a e aged
cu a u e ield and na u ally inco po a es bo h bulk cu a u e (go e ning binding) and local
shell–cu a u e co ec ions (go e ning le el s uc u e). These con ibu ions a e de i ed explic-
i ly in Appendix A.
Concep ually, Equa ion (7) ex ends he s anda d geodesic equa ion o Gene al Rela i i y by
eplacing he pu ely g a i a ional me ic pe u ba ion hµν wi h a de e minis ic, mul iscale em-
15Quan i a i e exp essions and pa ame e de ini ions appea in Appendix X.
16The geome ic decay- a e diagnos ic is de i ed in Appendix A.
17The b acke [η+ξnucl ]does no deno e an index con ac ion o summa ion. I indica es ha he Ch is o el
symbols Γµ
αβ a e compu ed om he me ic gµν =ηµν +ξnucl,µν . In ully explici o m,
Γµ
αβ[η+ξnucl] = 1
2gµλ (∂αgβλ +∂βgαλ −∂λgαβ), gµν =ηµν +ξnucl,µν .(8)
The only implied summa ions a e he s anda d Eins ein sums o e epea ed uppe /lowe indices (e.g. o e λ
abo e). Thus he b acke is simply a unc ional a gumen —“ he connec ion buil om his me ic”—and in o-
duces no addi ional summa ion o con ac ion beyond he usual GR no a ion.
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
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bling ield ξnucl,µν ha couples o all o ms o ene gy, no only mass–ene gy.18 This embling-
de o med o m i s appea ed in i s co a ian o m in Re e ence [12], whe e i desc ibed elemen-
a y co puscle mo ion in acuum cu a u e, and was la e adap ed o a omic cu a u e po en ials
in Re e ence [14]. In he nuclea con ex , he same geome ic p inciple yields he in e nal binding
mechanism, allowing nucleon mo ion, s abili y, and decay p ocesses o be de i ed om cu a u e
dynamics a he han om phenomenological nuclea o ces.
In his geome ic pic u e, nuclei a e bound s a es o cu a u e. Each p o on and neu on is a
localized embling eigenmode o he space ime me ic wi h a cha ac e is ic ampli ude and phase
s uc u e. A nucleus is a s a iona y, causally locked a angemen o such modes: he nea - ield
cu a u e pa e ns in e lock by esonance so ha phase misma ches a e minimized and he o al
embling ac ion is educed. Binding ene gy is no an ex e nally imposed po en ial, bu he
dec ease in local cu a u e a iance when isola ed modes eo ganize in o a join con igu a ion
( o malized la e in Equa ion (59)).
This connec s di ec ly o he mass–ene gy de ec obse ed [36] expe imen ally:
∆E=Zmp+Nmn−Mnucleusc2,(9)
whe e Zand Na e he numbe s o p o ons and neu ons, mpand mn hei indi idual embling
eigenmasses, and Mnucleus he collec i e bound-s a e mass.19 In TSRT, ∆Eis p ecisely he
cu a u e-supp essed embling ene gy eleased when ee nucleonic eigenmodes a e used in o
a causal nuclea geome y.
This in e p e a ion p o ides a na u al esolu ion o he duali y o s abili y and ins abili y
in nuclei. S able nuclei co espond o con igu a ions whe e embling esonance yields a local
minimum in he geome ic ac ion, p e en ing any u he econ igu a ion wi hou ex e nal pe -
u ba ion. Uns able nuclei co espond o cu a u e a angemen s whe e addi ional embling
modes can be shed o ea anged o lowe he o al ac ion, p oducing ission o decay pa hways.
Thus, he nucleus is no an excep ion o a omic de e minism bu an ex ension o i : jus
as elec on o bi s in TSRT a ise om cu a u e-guided embling geodesics [14], nuclea con ig-
u a ions a ise om cu a u e-bound assemblies o nucleonic embling modes. This geome ic
in e p e a ion p epa es he g oundwo k o a causal and quan i a i e analysis o nuclea s abili y,
ission, and usion in he ollowing sec ions.
The same embling-space ime o malism ha go e ns a omic emission and blackbody a-
dia ion also de e mines nuclea decay. Once he local cu a u e de i a i es ˙
∆K2a e known,
hey ix he emission a e λwi hou addi ional pos ula es. This con i ms ha TSRT p o ides a
uni ied geome ic ounda ion o bo h a omic and nuclea empo al beha io .
Fo p ac ical calcula ions, cu a u e measu es mus be con e ed in o physical ene gies. The
calib a ion p ocedu e, including he de ini ion o he no maliza ion cons an Cno m and i s ixing
ela i e o a e e ence nucleus (56Fe), is desc ibed in Appendix A. The esul ing cons an s and
pa ame e alues a e summa ized in Appendix E, and he ull nume ical ables used in he
compa isons a e collec ed in Appendix B.
18In Gene al Rela i i y, mo ion is go e ned by d2xµ
dτ2+ Γµ
αβ[g]dxα
dτ
dxβ
dτ = 0 wi h gµν =ηµν +hµν , whe e hµν
ep esen s a weak g a i a ional pe u ba ion. In TSRT, he analogous pe u ba ion ξµν o igina es om causal
embling o space ime i sel , as de eloped in Sec ion III o Re e ence [12] and ex ended o a omic bound s a es
in Sec ion II o Re e ence [14]. The p esen nuclea o m ξnucl ollows he same p inciple bu wi h cu a u e
ampli udes enhanced by he collec i e mass densi y and sho - ange co ela ion o nucleons.
19In TSRT, “mass” deno es he e ec i e ine ial mass ob ained om he cu a u e–ene gy unc ional, me ∝
RVKµν uµuνdV (Equa ion (119); see Sec ion 11.2 and [12, 13]). Fo an isola ed p o on o neu on, mpo mnis
he alue o he same unc ional on he single embling eigenmode wi h he app op ia e asymp o ic bounda y
condi ions. Fo he assembled nucleus, he collec i e bound-s a e mass Mnucleus is he alue o he same unc ional
on he causally locked mul i-mode con igu a ion, including bo h s ong-mode sa u a ion and elec omagne ic
con ibu ions. Because cu a u e sa u a ion supp esses he local a iance hK2iin he bound geome y, one has
Mnucleus < Zmp+Nmn; he co esponding ene gy di e ence ∆Ein Equa ion (9) is he measu ed mass de ec [36].
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
21
5 Binding Ene gy Sys ema ics and Radii om T embling Geom-
e y
Consis ency wi h implemen a ion: Binding ene gies and adii ollow om he cu a u e ene gy
unc ional (Appendix H), e alua ed on ields buil by
s nucleon ield.m/ s composi e ield.m; he nume ical assembly and ou pu s a e p o-
duced by s ene gy omcu a u e.m and compa ed o e alua ed da a ia Appendix B.
Suppo ing appendices: Appendix L (benchma ks), Appendix H (binding unc ional), Appendix B
( ables).
Co e MATLAB: s ene gy omcu a u e.m, s _s abili y_lines.m,
s _ge _exp_hal _li e.m.
A concise se o ancho binding compa isons is p esen ed ea ly in Table 3 (p. 11); he co e-
sponding AME o e lay and iso opic ends a e shown in he le panel o Figu e 1 (p. 10).
This sec ion de elops he TSRT analogue o he semi-empi ical mass o mula, whe e binding
is no pos ula ed bu de i ed om cu a u e-sa u a ion and in e e ence. The o al binding
ene gy (Equa ion (10)) spli s na u ally in o a geome ic bulk e m, shell con ibu ions, and
pai ing in e e ence. We show ha his geome ic law ep oduces he global B/A sa u a ion
cu e and he local shell-d i en kinks, in pa allel o wha liquid-d op+shell models achie e
phenomenologically.
5.1 Sa u a ion and he B/A cu e
One o he ea lies empi ical disco e ies in nuclea physics is ha he binding ene gy pe nu-
cleon, B/A, ises apidly wi h mass numbe Aup o he i on egion and hen sa u a es nea
∼8MeV.20 T adi ional liquid–d op models ep oduce his beha io h ough olume and su ace
e ms, supplemen ed by phenomenological shell and pai ing co ec ions.21 Wi hin TSRT, he
same pa e n ollows di ec ly om he geome y o embling cu a u e ields.
When nucleonic embling eigenmodes o e lap, hei local cu a u e ields in e lock in o a
join con igu a ion. The educ ion in cu a u e a iance pe added nucleon diminishes once a
c i ical densi y is eached, leading o sa u a ion wi hou he need o ex e nal pa ame iza ion.
The mas e ela ion is w i en compac ly as:
B(A, Z) = Bgeom(A, Z) + ∆Bshell(A, Z) + ∆Bpai (A, Z),(10)
20He e Bis he o al nuclea binding ene gy and A=Z+Nis he mass numbe ; B/A measu es binding pe
nucleon, a s anda d indica o o a e age nuclea s abili y.
21The adi ional liquid–d op model ea s he nucleus as an incomp essible cha ged luid d ople , assuming
ha each nucleon in e ac s p edominan ly wi h i s nea es neighbo s. I s binding ene gy is exp essed as a semi-
empi ical mass o mula B(A, Z) = aVA−aSA2/3−aCZ(Z−1)A−1/3−aA(A−2Z)2/A +δ(A, Z), whe e each
coe icien aiis adjus ed phenomenologically o i da a a he han de i ed om i s p inciples. Key assump ions
include: (1) nucleons a e homogeneously dis ibu ed wi hin a sphe ical olume o cons an densi y; (2) nuclea
o ces a e sho - anged and sa u a ing, mimicking molecula cohesion; (3) su ace ene gy educes binding nea he
nuclea bounda y analogously o classical su ace ension; (4) elec os a ic epulsion is desc ibed by a uni o mly
cha ged sphe e; (5) asymme y ene gy penalizes p o on–neu on imbalance acco ding o Fe mi-gas s a is ics;
and (6) pai ing e ms accoun empi ically o e en–e en, odd–odd, and odd–e en di e ences. Al hough his
model cap u es g oss binding-ene gy ends, i lacks a geome ic o dynamical ounda ion: each co ec ion e m
ep esen s a sepa a e empi ical adjus men a he han a consequence o a uni ied p inciple. Consequen ly, i s
desc ip i e success comes a he cos o heo e ical agmen a ion, unde sco ing he need o a single causal
amewo k capable o explaining bo h la ge-scale and small-scale s uc u e o ma ion wi hou eso ing o ad-hoc
pa ame iza ion.
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22
whe e Bgeom ep esen s he cu a u e-d i en bulk binding, ∆Bshell encodes co ec ions om
geome ic esonance a closed embling shells, and ∆Bpai e lec s he enhanced s abili y o
cu a u e-symme ic e en–e en sys ems.
The geome ic e m Bgeom sa u a es o a simple eason oo ed in TSRT’s ini e cu a u e–
o e lap ange. Le Rbe he nuclea adius (Sec ion 5.2) and Rc he cu a u e–o e lap (locking)
scale om Sec ion 4. In he bulk ( ≪R) each nucleonic embling mode inds essen ially he
same numbe o neighbo s wi hin Rc, so he cu a u e-supp ession gained by adding one mo e
nucleon is app oxima ely cons an . Nea he su ace, howe e , a ac ion o would-be neighbo s
lies ou side he nucleus; hose “missing neighbo s” educe he inc emen al supp ession. This
yields he olume–minus–su ace s uc u e o he geome ic binding:
Bgeom(A, Z)≃Cno mhα ol A−αsu A2/3i−CEM
Z2
R,(11)
wi h R= TSRT
0A1/3 om Equa ion (14) (see u he ). Di iding by Agi es he leading TSRT
en elope o he binding pe nucleon:
B
A(A, Z)≃a0−asA−1/3−aC
Z2
A4/3+∆Bshell
A+∆Bpai
A,(12)
whe e a0=Cno mα ol,as=Cno mαsu , and aC=CEM/ TSRT
0. The shell and pai ing e ms in
Equa ion (10) ide on his en elope: ∆Bshell p oduces magic-numbe kinks, and ∆Bpai gi es
odd–e en s agge ing (c . Figu e 3 (p. 24) in Sec ion 6).
To see why he maximum o B/A occu s nea i on, se κ≡Z/A along he alley o s abili y22
(nea ly cons an in medium-mass nuclei) and maximize he smoo h en elope o Equa ion (12)
wi h espec o Aa ixed κ. Igno ing he small A–dependence o he asymme y e m in his
na ow egion, one inds
d
dA
B
Aκ
= 0 =⇒A⋆≃as
2aCκ2.(13)
Wi h he single calib a ion on 56Fe in Appendix A ixing he a io as/aC(and wi h κ≃Z/A ≃
0.46–0.50 along he s abili y alley), Equa ion (13) yields A⋆≈56–60. Thus, in TSRT he peak
nea A∼60 is a di ec geome ic consequence o (i) bulk cu a u e locking ( olume e m), (ii)
missing-neighbo penal y (su ace e m), and (iii) he long- ange elec omagne ic cu a u e cos
which g ows wi h Z2/R.
Bis he o al binding ene gy B(A, Z);A=Z+Nis he mass numbe ; he en elope a0−
asA−1/3−aCZ2/A4/3encodes he de e minis ic cu a u e balance. Shell and pai ing co ec ions
hen p oduce he obse ed local s uc u e on op o his global end.
The expe imen al signa u es o his s uc u e a e well known: local kinks a magic num-
be s23 and odd–odd–e en s agge ing.24 Wi hin he liquid–d op amewo k, hese ea u es a e
accommoda ed no by he base olume/su ace e ms bu by empi ically i ed add-ons, i.e., shell
co ec ions o magic-numbe kinks and a pai ing e m o odd–e en s agge ing, whose oles (and
TSRT ein e p e a ion) a e de ailed in Sec ion 6 (Equa ion (10)). In TSRT hese a e no ad
hoc add-ons bu ollow om he de e minis ic geodesic–co ela ion ules. A de ailed benchma k
appea s in Sec ion 6, Figu e 3 (p. 24), whe e he pa i y–Bessel wo-packe model de i ed om
embling geome y quan i a i ely ep oduces he s agge ing obse ed in 235U ission yields. Fo
ep esen a i e numbe s and how he global end compa es o da a, see Table 3 (p. 11) (wi h
he ull, able in Appendix B, Table 25 (p. 147)).
22The alley o s abili y is he locus o nuclides s able (o longes -li ed) agains adioac i e decay in he N–Z
plane. Away om his alley, iso opes end o unde go β±decay (o o he modes) o educe hei ene gy.
23Magic numbe s a e speci ic p o on o neu on coun s (2,8,20,28,50,82,126, . . .) a which la ge shell gaps
yield ex a s abili y. They p oduce isible “kinks” in sepa a ion ene gies, adii, and o he sys ema ics.
24Odd–e en s agge ing (OES) is he al e na ing pa e n in nuclea obse ables (masses, agmen yields) whe e
e en-Z/e en-Nnuclei a e a o ed o e odd neighbo s, commonly a ibu ed o nucleon pai ing co ela ions.
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
23
The liquid-d op model (LDM) [23,37] amously i s B(A, Z)wi h a olume – su ace Coulomb
– asymme y pai ing decomposi ion. TSRT ob ains a closely ela ed s uc u e om i s p inci-
ples: he olume e m a ises om bulk cu a u e locking (cohe en o e lap o embling modes);
he su ace e m om missing neighbo s wi hin he ini e o e lap adius Rc; and he Coulomb
cos om long- ange elec omagne ic cu a u e scaling like Z2/R wi h R= TSRT
0A1/3(Sec-
ion 5.2). These ing edien s p oduce he TSRT en elope in Equa ion (12), which explains,
wi hou phenomenological sa u a ion cons an s, why B/A peaks nea A≃60 (Equa ion (13))
and hen le els o .
LDM ge s he igh shape because i s i ed e ms mimic he same geome ic scalings ha
TSRT de i es de e minis ically om cu a u e in eg als. Bu LDM needs dis inc i ed coe -
icien s and addi ional pa ches (e.g., shell and pai ing e ms) o cap u e kinks and odd–e en
e ec s, whe eas in TSRT: (i) magic-numbe kinks a ise om disc e e changes in geodesic es-
onance s uc u e ( he shell–cu a u e co ec ions ∆Bshell in Equa ion (10)), and (ii) odd–e en
s agge ing ollows om causal phase-ma ching o embling modes (Sec ion 6, Figu e 3 (p. 24)),
no om a pu ely s a is ical pai ing ansa z. Likewise, phenomena ha s ain LDM’s ba ie -
unneling pic u e, such as deep sub-ba ie usion hind ance, a e na u ally accommoda ed by
TSRT’s cu a u e-based ba ie and p ope - ime ansmission (Sec ion 7, Figu e 6 (p. 86)).
In sho , LDM’s successes a e explained in TSRT as consequences o ini e- ange cu a-
u e locking plus long- ange Coulomb cu a u e, while TSRT addi ionally p o ides he causal
mechanism and p edic i e co ec ions ha LDM mus i sepa a ely.
5.2 Cha ge adii and scaling
Cha ge adii p o ide an independen p obe o nuclea s uc u e by quan i ying he spa ial ex en
o he nuclea cha ge dis ibu ion. Ope a ionally, he ( oo -mean-squa e) cha ge adius ch is
de ined om he second momen o he cha ge densi y ρch( )as ch =ph 2iand is ex ac ed
om elas ic elec on sca e ing, iso ope-shi spec oscopy, and muonic-a om spec oscopy.25
Empi ically, adii ollow an app oxima e sa u a ion law R≃ 0A1/3(wi h mild depa u es nea
shell closu es,26 along de o ma ion chains, and ac oss iso opic skins). In TSRT, his A1/3scaling
is no imposed bu eme ges om he causal packing o embling eigenmodes in a ini e cu a u e
domain: he leading olume con ibu ion ixes he A1/3 end, while subleading shell–cu a u e
and de o ma ion e ms accoun o sys ema ic de ia ions (de i ed in Appendix A).
The TSRT p edic ion o cha ge adii a leading o de is he TSRT adius law
R(A) = TSRT
0A1/31 + δde (A),(14)
whe e his o m is de i ed (no assumed) by minimizing he embling ac ion o a ini e cu -
a u e domain con aining Anucleons; he A1/3scaling ollows om he olume pa o he
cu a u e in eg al, while subleading su ace/shell e ec s appea as mul iplica i e co ec ions
(see Appendix A). The cons an TSRT
0is he undamen al leng h scale ixed by he no mal-
iza ion o nuclea cu a u e and he e o e de e mined once he TSRT cons an s a e speci ied
(Appendix E); i is no an adjus able LDM-s yle i pa ame e . The ac o δde (A)encodes
small, nucleus-dependen depa u es om sphe ici y and local shell–cu a u e e ec s, anishing
nea sphe ical closed shells, changing sign wi h p ola e/obla e de o ma ion, and scaling like a
su ace co ec ion a la ge A. In wha ollows we use Equa ion (14) as he wo king o m, wi h
25Fo cla i y: he cha ge adius di e s om he ma e adius (which weigh s all nucleons) and om he
poin -p o on adius pp ( he p o on cen e s-o -mass dis ibu ion). S anda d ex ac ions ela e hese ia 2
ch =
2
pp + 2
p+ (N/Z) 2
n+ 2
DF + 2
so +···, whe e 2
pand 2
naccoun o in insic nucleon cha ge s uc u e, and DF, so
deno e small Da win–Foldy and spin–o bi co ec ions, espec i ely. We keep he p esen a ion in e ms o ch o
align wi h expe imen al sys ema ics; de ails o how hese co ec ions en e TSRT i s a e de e ed o Appendix A.
26Shell closu e deno es a illed majo shell o p o ons o neu ons (a a magic numbe ), ypically p oducing
educed collec i i y, smalle adii changes, and enhanced s abili y.
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
24
35 40 45 50 55 60 65
Z
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25 Odd-E en S agge ing
Expe imen
TSRT
Figu e 3: Odd–e en s agge ing in 235U(n h, ). The e ical axis shows he hi d ini e di e ence
∆(3)ln Y(Z) = ln Y(Z+1) −2 ln Y(Z) + ln Y(Z−1), quan i ying he al e na ing pa e n o loga-
i hmic ission yields Y(Z)as a unc ion o agmen cha ge Z(ho izon al axis). Poin s deno e
expe imen al alues om Table 17 (p. 110); he solid cu e ep esen s he TSRT pa i y–Bessel
wo-packe model wi h localized neck bias [Equa ion (315)], as discussed in Sec ion 10.7. All
a ays and MATLAB code used o gene a e he igu e a e p o ided in Appendix R.2. Model
de i a ion: Sec ion 10.7; pa ame e alues in Table 41 (p. 209); nume ical wo k low in Ap-
pendix R.2.
explici exp essions o δde (A)and he de e mina ion o TSRT
0p o ided in Appendix A and
Appendix E, espec i ely.
Unlike adi ional models ha assign 0empi ically, TSRT ela es TSRT
0di ec ly o he
cu a u e ampli ude o a single nucleon eigenmode and calib a es i de e minis ically agains
a e e ence nucleus such as 56Fe. This es ablishes he nuclea leng h scale as a measu able
mani es a ion o space ime cu a u e a he han an adjus able geome ic cons an , he eby
g ounding he adius law in i s p inciples o he heo y.
Fu he mo e, he same embling-induced pola iza ion mechanism ha modi ies cha ge adii
also go e ns elec on sc eening in low-ene gy nuclea usion. In TSRT, pola iza ion deno es
he causal displacemen o local cu a u e modes ela i e o hei mean geodesic posi ion due
o he p esence o neighbo ing mass–ene gy oscilla ions. This p oduces an induced cu a u e
dipole ield ha shi s cha ge dis ibu ions and e ec i e po en ials, in exac analogy o how
elec ic pola iza ion a ises om cha ge displacemen in a dielec ic medium, bu he e he e ec
ac s on he space ime me ic i sel a he han on an ex e nal ield. In nuclei, such cu a u e
pola iza ion leads o minu e adius modi ica ions; in condensed ma e en i onmen s, i go e ns
he enhancemen o unneling p obabili ies by educing he e ec i e Coulomb ba ie . This
causal connec ion is demons a ed in Sec ion 7, Figu e 2 (p. 12), whe e TSRT pola iza ion wi h
embling and dynamic dephasing success ully ep oduces expe imen al sc eening enhancemen s
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
25
in Pd-hos d(d,p) eac ions.27
In summa y, bo h binding-ene gy sa u a ion and cha ge- adius scaling eme ge in TSRT as
di ec consequences o causal embling geome y. Wha appea in phenomenological models as
sepa a e empi ical e ms ( olume, su ace, asymme y, pai ing, 0cons an s) a e he e uni ied
as mani es a ions o a single geome ic p inciple: he de e minis ic o ganiza ion o nucleonic
embling modes in cu ed space ime.
Full nume ical ables suppo ing his sec ion a e p o ided in Appendix B, wi h calib a ion
se ings and code en y poin s documen ed in Appendix A. Rep esen a i e cha ge adii and
binding esiduals a e summa ized in Table 12 (p. 108) (Appendix B), and pola iza ion/elec on-
sc eening compa isons a e shown in Figu e 2 (p. 12) (wi h code lis ings in Appendix R.3).
6 De o ma ion Landscapes and Fission Ba ie s
Consis ency wi h implemen a ion: Ba ie cu es and de o ma ion ene ge ics a e compu ed by
s issionene gy.m (ene gy su aces) and s issionac ion ull.m (ac ion e alua ion),
wi h OES neck-mode de ails in Appendix N.
Pa en / agmen geome y assembly uses s _build_pa en _U235.m,
and s _build_daugh e s_scission.m.
Suppo ing appendices: Appendix M (su ace cu a u e and bi u ca ion), Appendix N (OES
neck mode), Appendix F (disc e iza ion).
Co e MATLAB: s issionene gy.m, s issionac ion ull.m.
The consolida ed U-235(n, ) obse ables able is placed ea ly as Table 4 (p. 12) o quick
e e ence; he OES igu e appea s in Figu e 3 (p. 24).
6.1 Geome ic de o ma ion coo dina e and ba ie o ma ion
In con en ional nuclea models, de o ma ion is in oduced phenomenologically by pa ame e izing
he nuclea su ace R(θ, φ)wi h mul ipole expansions, mos commonly using he pa ame e iza-
ion de eloped by Boh and Mo elson [38]:
R(θ, φ) = R0
1 +
∞
X
λ=0
+λ
X
µ=−λ
αλµYλµ(θ, φ)
,(15)
whe e R0is he adius o he sphe ical nucleus, αλµ a e he de o ma ion pa ame e s,28 and
Yλµ a e sphe ical ha monics. The quad upole de o ma ion pa ame e s29 (α2µ), which desc ibe
ellipsoidal shapes, a e o pa amoun impo ance o desc ibing nuclea g ound s a es and ission
pa hways.
27The no a ion “Pd-hos d(d,p) ” e e s o deu e ium–deu e ium usion eac ions occu ing wi hin a palladium
hos la ice, whe e wo deu e ons (d) use o o m a i on ( ) and a p o on (p). Such eac ions a e s udied in
low-ene gy condensed-ma e usion expe imen s o in es iga e sc eening e ec s: he su ounding elec on cloud
and la ice po en ial e ec i ely lowe he Coulomb ba ie be ween he eac ing nuclei. In he p esen pape ,
he TSRT-based ea men o his sys em appea s in Sec ion 7, whe e he same embling-induced cu a u e
pola iza ion mechanism ha go e ns cha ge- adius co ec ions is applied o quan i y he obse ed enhancemen
in eac ion a es.
28De o ma ion pa ame e s αλµ quan i y de ia ions om a sphe e ia sphe ical ha monics Yλµ:λ= 2
(quad upole) con ols elonga ion/ la ening, λ= 3 (oc upole) pea -shapes, λ= 4 hexadecapole, e c.
29Quad upole de o ma ion (o en summa ized by β2) desc ibes ellipsoidal shapes: p ola e ( ugby-ball, β2>0)
o obla e (disk-like, β2<0). I dominan ly con ols low-lying collec i e spec a and ission pa hways.
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
32
E alua ion pipeline and esul s. Fo each eac ion sys em he calcula ion p oceeds as
ollows:
1. Cons uc Cβ(β)and Mβ om he TSRT de o ma ion geome y (Sec ion 6.1; code lis ings
in Appendix F.3).
2. E alua e he p ope - ime ac ion STSRT(E)o e he classically o bidden egion o an
ene gy g id Ei.
3. Compu e he anspa ency TTSRT(Ei) ia Equa ion (34).
4. Fo m he usion c oss sec ion σ us(Ei)(Equa ion (36)) and S- ac o STSRT(Ei)(Equa-
ion (35)).
5. Sa e p edic ed S(E)and gene a e eac ion-speci ic plo s.
This is p ecisely he compu a ional pa h implemen ed in Appendix Q.3 and execu ed by
he sc ip s used o Table 27 (p. 169) and Figu e 6 (p. 86). The la e shows he p edic ed
hind ance pa e n—including he la e-onse downwa d cu a u e—eme ging na u ally om he
TSRT p ope - ime geome y, wi hou he adjus able ene gy-shi pa ame e s o empi ical hin-
d ance h esholds common in quan um-mechanical i s.
Obse a ions. TSRT he e o e p o ides a ully de e minis ic, geome y-based calcula ion o
usion a es and S- ac o s. All en ance-channel supp ession a ises om p ope - ime cu a u e
in eg als, and he obse ed hind ance sys ema ics ollow au oma ically om he causal s uc u e
o he embling nuclea ields. No phenomenological ba ie - eshaping o eac ion-dependen
uning is in oduced; he same calib a ed cu a u e cons an s apply ac oss all sys ems e alua ed
in his wo k.
7.2 Cu a u e-enhanced Coulomb ba ie
In TSRT, he elec omagne ic sec o is no an ex e nally pos ula ed ield bu he long- ange
mani es a ion o embling cu a u e i sel . As es ablished in he ounda ional a icle [12] (see
Sec ions III.B–III.C he ein), he decomposi ion o he ull embling cu a u e enso Kµν in o
symme ic and an isymme ic pa s p oduces wo complemen a y sec o s: a symme ic (spin-2–
like) componen go e ning he s ong channel and a ace- ee an isymme ic componen go -
e ning he long- ange U(1)-like cu a u e dynamics ha appea mac oscopically as elec omag-
ne ism. The elec omagne ic po en ial Aµin TSRT hus co esponds o he coa se-g ained limi
o he local embling displacemen ield ξµ, wi h he ield enso
Fµν =∂µAν−∂νAµ←→ 2K[µν],(37)
whe e K[µν]is he an isymme ic componen o he embling cu a u e enso . In he weak-
cu a u e ( a - ield) limi , whe e geodesic de ia ions a e small and highe -o de cu a u e cou-
pling e ms anish, he go e ning equa ions o K[µν] educe exac ly o Maxwell- ype ield equa-
ions de i ed om he same cu a u e unc ional used o he s ong channel. This limi ep o-
duces he classical Maxwell equa ions in acuum and, o s a ic sphe ically symme ic sou ces,
yields he amilia Coulomb o m o he po en ial.
Consequen ly, o wo well-sepa a ed nuclei o cha ges Z1eand Z2e, he baseline in e ac ion
ene gy eme ging om he a - ield limi o he TSRT elec omagne ic sec o is
VC( ) = Z1Z2e2
,(38)
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025

33
which co esponds o he leading-o de e m o he gene al TSRT elec omagne ic po en ial
de i ed in Re e ence [12], Sec ion III.D, and ei e a ed in he a omic-s uc u e o mula ion [14],
Sec ion 7.3, whe e he same cu a u e o igin o he Coulomb law unde lies elec on–nucleus
binding.
A sho e anges, whe e he nuclea embling ields begin o o e lap, he e ec i e in e ac ion
acqui es de e minis ic cu a u e co ec ions ha ha e no coun e pa in classical elec odynam-
ics:
(i) Cu a u e pola iza ion o he nuclea su aces, a ising om induced alignmen o local
embling modes a he nuclea bounda ies, leading o a modi ica ion o he su ace-
cu a u e e m ∆K2
su in Equa ion (99);
(ii) Cu a u e ocusing/de ocusing, a sho - ange s ong–elec omagne ic coupling ha al e s
he en ance-channel po en ial landscape, leading o explici co ec ions in he TSRT e -
ec i e po en ial Ve ( )o Equa ion (41);
(iii) Elec on sc eening in condensed en i onmen s, co esponding o a nea - ield elec omag-
ne ic eno maliza ion due o su ounding elec onic cu a u e modes, leading o he sc eened
as ophysical S- ac o h ough he TSRT sc eening shi Uein Equa ion (146).
Toge he hese con ibu ions de ine he o al e ec i e ba ie po en ial used h oughou his
sec ion,
Ve ( ) = VC( ) + ∆VTSRT( )−∆Ue,(39)
whe e ∆VTSRT( )encapsula es he cu a u e-o e lap and pola iza ion co ec ions in insic o
TSRT geome y, and ∆Ue ep esen s he en i onmen -dependen elec on-sc eening shi cali-
b a ed om Figu e 2 (p. 12). The explici o m o Ve ( )and i s de i a ion om he cu a u e-
coupled geodesic equa ion a e gi en in Equa ion (40) and Appendix A. The Coulomb po en ial
is he e o e no an ex e nally imposed elemen o he model, bu he a - ield limi o he elec-
omagne ic embling sec o wi hin TSRT; he nea - ield e ms ∆VTSRT( )and ∆Uep o ide
con olled, physics-de i ed e inemen s o ha geome ic baseline.
In T embling Space ime Rela i i y, he ba ie ha wo nuclei mus o e come be o e usion
is no gi en by a simple supe posi ion o a Coulomb epulsion and a phenomenological nuclea
a ac ion. Ins ead, i a ises di ec ly om he embling-de o med geodesic equa ion (Sec ion 4)
once wo nucleonic cu a u e ields o e lap. The e ec i e po en ial can be w i en as
Ve ( ) = VCoul( ) + Vgeom
nucl ( ) + V em( ),(40)
whe e:
•VCoul( ) = Z1Z2e2
4πε0 is he s anda d long- ange Coulomb epulsion,
•Vgeom
nucl ( )is he sho - ange a ac i e e m om cu a u e sa u a ion when embling
eigenmodes o e lap, and
•V em( )is he esidual co ec ion due o causal embling dephasing, scaling wi h he local
a iance o ξnucl.
Nea he classical ba ie adius b, he epulsi e Coulomb e m and he a ac i e cu a u e-
sa u a ion e m balance o leading o de , and he ba ie heigh is modi ied by addi ional em-
bling con ibu ions. The esul ing exp ession o he TSRT usion ba ie is
BTSRT =Ve ( b)≃Z1Z2e2
4πε0 b−Csa e−κ( b− 0)+ ∆V em,(41)
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
34
whe e Csa is he cu a u e-sa u a ion cons an calib a ed on mid-mass sys ems, κis he geo-
me ic all-o scale con olling he exponen ial decay o cu a u e locking, and 0is he nominal
ouching dis ance, 0≈ TSRT
0(A1/3
1+A1/3
2), as de ined in Equa ion (14).45
Equa ion (41) hus does no me ely eno malize he Coulomb po en ial; i in oduces a
cu a u e-dependen co ec ion ha sys ema ically lowe s and eshapes he ba ie , wi h a mag-
ni ude ha depends on mass asymme y, cu a u e o e lap, and local embling ampli ude.
Physically, he cu a u e-sa u a ion e m exp esses he ini e- ange a ac ion a ising om he
causal alignmen o space ime oscilla ions be ween he in e ac ing nuclei, while ∆V em cap-
u es he dynamic modula ion o ha alignmen a ini e ela i e eloci y. This cu a u e-based
s uc u e explains why usion a sub-ba ie ene gies p oceeds mo e e icien ly han p edic ed
by a pu ely elec os a ic ba ie and also p o ides he geome ic ounda ion o he obse ed
hind ance phenomenon a ex eme sub-ba ie ene gies (see Sec ion 7 and Figu e 6 (p. 86)).
7.3 WKB-like unneling om p ope - ime ac ion
Pu pose. This subsec ion de i es he en ance-channel anspa ency s a ing om he TSRT
p ope - ime ac ion, in a o m ha is s uc u ally simila o he amilia WKB exponen bu
concep ually dis inc . The e ec i e en ance-channel po en ial Ve ( )is de ined and cons uc ed
in Sec ion 7.2, and he esul ing ansmission is used in he S- ac o pipeline o Sec ion 7.1
(see also Equa ion (283); cons an s and o e lap p esc ip ions a e lis ed in Appendix A and
Appendix P.4).
Se up. In he en ance channel we educe he wo-body geome y o a single collec i e sepa a-
ion coo dina e wi h ine ia µ( he educed mass), mo ing in he e ec i e po en ial Ve ( )o
Sec ion 7.2. The p ope - ime Lag angian along he adial geodesic is
LTSRT( , ˙ ) = 1
2µ˙ 2−Ve ( ),(42)
and he Eule –Lag ange equa ion gi es
d
dτ ∂LTSRT
∂˙ −∂LTSRT
∂ = 0,∂LTSRT
∂˙ =µ˙ ≡p ,(43)
so ha p =µ d /dτ. The associa ed Hamil onian is conse ed,
H( , p ) = p2
2µ+Ve ( ) = E, (44)
which de ines he u ning poin s 1(E)< 2(E)by Ve ( ) = E.
Fo bidden segmen and momen um magni ude. In he cu a u e-supp essed egion
Ve ( )> E one has
|p ( ;E)|=p2µ(Ve ( )−E),∂LTSRT
∂˙ =µ˙ , (45)
45Equa ion (41) is de i ed explici ly in Appendix A, whe e he e ec i e po en ial Ve ( )is ob ained om he
cu a u e-coupled geodesic equa ion in he wo-body con igu a ion space. The o malism o igina es om he
gene al cu a u e–in e ac ion po en ial in oduced in Re e ence [14] o a omic binding and gene alized he e o
composi e nuclea cu a u e domains. The i s e m ep esen s he con en ional elec os a ic epulsion be ween
wo cha ge dis ibu ions o p o on numbe s Z1and Z2. The second e m, p opo ional o Csa , is he TSRT
cu a u e-sa u a ion e m: i accoun s o he a ac i e geome ic locking o embling cu a u e ields be ween
he wo nuclei and decays exponen ially wi h sepa a ion. The inal e m, ∆V em, ep esen s highe -o de co -
ec ions om dynamic embling in e e ence and p ope - ime phase misma ch, which a e esponsible o he
sub-ba ie enhancemen and hind ance e ec s desc ibed in Sec ion 7.1 and Appendix Q, wi h sc eening-induced
enhancemen s de ailed in Appendix P.
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
35
and he p ope ime inc emen sa is ies dτ =µ d /|p |. The p ope - ime cos o a e sing he
o bidden segmen is he e o e he posi i e in eg al
STSRT(E) = Zτ2
τ1LTSRTdτ =Z 2(E)
1(E)LTSRT
|p |µ d , (46)
which, in he slowly a ying (semiclassical) limi , educes o
Zτ2
τ1LTSRTdτ =Z 2(E)
1(E)LTSRT
|p |µ d ∝Z 2(E)
1(E)q2µVe ( )−Ed . (47)
(See also Equa ion (283) in Sec ion 7.1 o he ac ion-based de ini ion used in he a e pipeline.)
T anspa ency and i s TSRT ba ie o m. TSRT de ines he en ance–channel ans-
pa ency h ough he causal p ope – ime weigh o he leas –ac ion his o y,
W(E)∝exph−∆S(E)/~TSRTi,(48)
which is he coa se–g ained limi o he p ope – ime ac ion inc emen (Sec ion 12.2). E alua ing
∆S o an en ance ajec o y ac oss he cu a u e– egula ed TSRT ba ie gi es he adial
ac ion
∆S(E)≈2Z 2(E)
1(E)p2µ[VTSRT( )−E]d
∞
,(49)
whe e 1,2(E)a e he u ning poin s and ∞ he asymp o ic en ance–channel speed. Abso bing
he smoo h kinema ic ac o in o he p e ac o yields he wo king TSRT ba ie exponen ,
T(E)≃exp"−2
~Z 2(E)
1(E)q2µVTSRT( )−Ed #,(50)
which mi o s he amilia WKB o m in s uc u e bu he e a ises solely om he p ope –
ime causal cos in a embling space ime geome y, wi hou any wa e unc ion o quan iza ion
pos ula e. All sys em dependence en e s h ough VTSRT( )(Sec ion 7.2; Appendix A).46
The in eg al inhe i s all TSRT co ec ions ia Ve : cu a u e pola iza ion and ocusing,
o e lap sa u a ion a sho ange, and he sc eened a - ield ail. These modi y bo h he ba -
ie heigh and wid h compa ed o a ba e Coulomb ba ie and a e essen ial o ep oduce he
hind ance beha io a deep sub-ba ie ene gies (Figu e 6 (p. 86)).
Fo channel-speci ic use and o connec di ec ly wi h he empi ically obse ed nea -exponen ial
beha io o he as ophysical S- ac o , we abso b small, smoo h esiduals in o a calib a ed
polynomial-in-ene gy co ec ion o he exponen . W i ing
Gχ(E)≡Z 2(E)
1(E)q2µχVe ( )−Ed , χ ∈ {D–T,D–D},(51)
we use
ln TD–T(E) = −2
~GDT(E) + αDT E+βDT E2,(52)
ln TD–D(E) = −2
~GDD(E) + αDD E+βDD E2,(53)
whe e he slope pai (α, β)is ixed once (uni e sal hind ance calib a ion) and hen eused ac oss
sys ems wi hou e uning (Appendix Q, Table 16 (p. 110); sc ip s s s ac o .m,
ep oducehind anceTSRT.m).
46Consis ency wi h implemen a ion: The in eg al in Equa ion (50) is e alua ed nume ically in
s usionene gy.m (cons uc ion o Ve and u ning poin s) and s s ac o .m ( ansmission and S- ac o ).
No addi ional ba ie -shape pa ame e s a e in oduced beyond he TSRT componen s enume a ed in Appendix O.
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
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The en ance-channel anspa ency o Sec ion 7.3 connec s di ec ly o he as ophysical S-
ac o , a s anda d ool in low-ene gy nuclea as ophysics. Because usion c oss sec ions be ween
cha ged nuclei all ex emely apidly a low ene gies, i is cus oma y o ac o ou he dominan
Coulomb-supp ession e m so ha he esidual quan i y a ies smoo hly and e eals he unde -
lying nuclea -s uc u e physics. The S- ac o is he e o e de ined by
S(E) = σ(E)Eexp2πη(E),(54)
whe e he Somme eld pa ame e 47 η=Z1Z2e2/(~ )cap u es he pu ely Coulombic pa o he
ba ie and he exponen ial supp ession o he usion p obabili y.
Role o S(E).The as ophysical S(E)48 is he quan i y ha is di ec ly compa ed ac oss
models and expe imen s, because i s smoo hness e eals he physical mechanisms al e ing he
usion p obabili y.
In TSRT he modi ica ion o S(E)comes om wo geome ic ing edien s: (i) he cu a u e-
augmen ed e ec i e po en ial Ve ( )de i ed in Sec ion 7.2 and Appendix A, and (ii) he uni e sal
slope pai (α, β)calib a ed once (Appendix P.4) and used unchanged o all sys ems. These
ing edien s ep oduce bo h he gene ic sub-ba ie end49 and he deep hind ance cu a u e50
(Figu e 6 (p. 86)).
A second modi ica ion a ises om en i onmen -dependen sc eening, desc ibed by he shi
∆Ue, which accoun s o he in luence o su ounding elec onic cu a u e modes.51
Thus he TSRT p edic ion o he en ance-channel unneling p obabili y is con olled en-
i ely by he single geome ic inpu Ve ( )and he once-calib a ed (α, β), yielding a uni ied
explana ion o he enhanced sub-ba ie usion ela i e o a ba e Coulomb ba ie , he uni e -
sal deep hind ance a he lowes ene gies, and he en i onmen -dependen sc eening h ough
∆Ue.
Compu a ional and calib a ion de ails appea in Appendix A, wi h ex ended ables in Ap-
pendix B. Figu e-speci ic da a and sc ip s a e p o ided in Appendix R.3 (sc eening), Appendix Q
(hind ance), and Appendix R.2 (OES), wi h ull de i a ions o odd–e en e ec s in Sec ion 10.7.
47The Somme eld pa ame e η=Z1Z2e2/(~ )[50] measu es he a io o Coulomb po en ial ene gy o ela i e
kine ic ene gy be ween wo cha ged nuclei. I o igina es om semiclassical Coulomb-wa e analysis in quan um
sca e ing heo y, whe e he unneling p obabili y ac oss a Coulomb ba ie scales as exp(−2πη). This exponen ial
supp ession is uni e sal o cha ged-pa icle eac ions a low ene gies and unde pins bo h he as ophysical S-
ac o and i s TSRT ein e p e a ion.
48The S- ac o is a s anda d cons uc in nuclea as ophysics in oduced o emo e he dominan exponen ial
supp ession om c oss sec ions, so ha S(E) a ies smoo hly wi h ene gy. I aces i s o igin o quan um-
mechanical unneling heo y and he Gamow ac o [51], and i allows nuclea -s uc u e and sc eening e ec s o
be s udied independen ly o kinema ic ac o s. In TSRT, S(E) emains concep ually iden ical bu is go e ned
by he e ec i e po en ial Ve ( ), which embeds cu a u e and embling co ec ions ins ead o pu ely Coulombic
ones.
49The gene ic sub-ba ie end e e s o he expe imen ally obse ed exponen ial dec ease o usion c oss sec-
ions wi h dec easing cen e -o -mass ene gy below he nominal Coulomb ba ie . In s anda d quan um unneling
heo y, his end a ises om he Gamow ac o ; in TSRT, i eme ges de e minis ically om he cu a u e-
modi ied Ve ( ), which ep oduces he same exponen ial dependence bu wi h physically g ounded cu a u e
co ec ions.
50The deep hind ance cu a u e desc ibes he la ening and e en ual u no e o he loga i hmic S(E)cu e a
e y low ene gies, whe e measu ed usion c oss sec ions all below he ex apola ed end om highe ene gies. In
con en ional models his is an empi ical anomaly; in TSRT, i ollows na u ally om cu a u e-sa u a ion e ec s
ha supp ess he o e lap o embling cu a u e ields a ex eme sepa a ions, p o iding a causal geome ic
explana ion o he hind ance phenomenon.
51En i onmen -dependen sc eening deno es he e ec i e educ ion o he Coulomb ba ie due o he p esence
o su ounding elec ons o la ice ields, which pa ially neu alize he in e ac ing nuclea cha ges. In me allic o
condensed-ma e en i onmen s, his sc eening lowe s he usion ba ie by an amoun ∆Ue, ypically a ew ens
o hund eds o elec on ol s. In TSRT, such sc eening is no ea ed phenomenologically bu as a mani es a ion
o embling-induced cu a u e pola iza ion o he ambien elec on ield, as de ailed in Sec ion 7 and illus a ed
in Figu e 2 (p. 12).
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37
Deep sub-ba ie beha io and he uni e sal slope pai (α, β)a e compa ed di ec ly o expe i-
men in Figu e 6 (p. 86), and he nume ical alues appea in Appendix B, Table 16 (p. 110).
8 TSRT Desc ip ion o S ong and Weak Nuclea Fo ces
Consis ency wi h implemen a ion: The weak-p ocess a e diagnos ic Γweak (Equa ion (61)) is
implemen ed in s _gamma_ a e.m and consumed by s li e ime.m/ s li e imemode.m
o li e ime p edic ions; no s ochas ic pos ula es a e in oduced in code.
Suppo ing appendices: Appendix C (locking), Appendix H (ac ion/ene gy), Appendix I (li e-
ime es ima o s).
Co e MATLAB: s li e ime.m, s li e imemode.m, s _gamma_ a e.m.
A cen al achie emen o TSRT is ha he ou undamen al in e ac ions eme ge di ec ly
om he geome y o causal embling wi hou pos ula es ex e nal o he me ic [12]. Fo nu-
clea physics, he s ong and weak o ces play he essen ial oles. In con as o he quan um
ch omodynamics (QCD) app oach, which elies on colo cha ge and gluon exchange, TSRT p o-
ides a pu ely geome ic mechanism: bo h in e ac ions a e de e minis ic mani es a ions o how
embling cu a u e con igu a ions o e lap, sa u a e, o econ igu e unde causal cons ain s.
Ope a ionally, we use h ee cu a u e scales: (i) an o e lap scale Rc ha se s he e ec i e
s ong ange, (ii) a sa u a ion ampli ude Asa con olling bulk binding and shell onse s, and (iii)
a local neck-cu a u e pa ame e κneck go e ning odd–e en s agge ing a scission. These scales
en e ed he de i a ions in Sec ion 5 (bulk binding, shell and pai ing), Sec ion 6 (ba ie o ma ion
and OES; see Figu e 3 (p. 24)), and en e he usion p og am h ough bo h he mic ophysical
de elopmen s o Sec ion 7 (sc eening and hind ance; Figu e 2 (p. 12), Figu e 6 (p. 86)) and he
b oade syn hesis in Sec ion 12. Pa icle e minology in TSRT equals ha in classical physics.52
8.1 The S ong In e ac ion as Cu a u e Sa u a ion
The s ong in e ac ion in TSRT a ises when nucleonic embling modes (p o ons o neu ons) a e
b ough wi hin a c i ical spa ial sepa a ion. Each nucleon co esponds o a localized eigenmode
o embling cu a u e, wi h ampli ude ANand associa ed s ess–ene gy dis ibu ion [17]. When
wo o mo e such eigenmodes o e lap, hei embling ields in e e e cons uc i ely, p oducing
a egion o cu a u e sa u a ion. This s a e minimizes he local embling ac ion and locks he
nucleons in o a bound con igu a ion.
Th oughou his subsec ion, Kµν deno es he TSRT embling-cu a u e enso (see he
mas e no a ion in Sec ion 2), cons uc ed om he me ic de ia ion ξµν and i s de i a i es;
Appendix C p o ides he explici de ini ion and coa se-g aining p esc ip ion used in nume ics.
We use
K≡pKµνKµν,
hK2i ≡ coa se-g ained local a iance o Ko e a p ope - ime/spa ial cell.(55)
52Te minology: A lep on is a spin-1
2 e mion ha does no expe ience he s ong in e ac ion; he cha ged
lep ons a e he elec on e−, muon µ−, and au τ−(wi h an ipa icles e+,µ+,τ+), and he neu al lep ons
a e he neu inos νe, νµ, ντwi h co esponding an ineu inos ¯νe,¯νµ,¯ντ. In nuclea be a p ocesses, only he
elec on/posi on and he elec on-(an i)neu ino ypically appea , because nuclea ene gy scales a e a below
he µ/τ p oduc ion h esholds. Lep ons do no pa icipa e in he s ong in e ac ion; hey appea he e because a
cu a u e econ igu a ion mus ca y away he app op ia e conse ed cha ges (elec ic cha ge and lep on numbe )
along a causal geodesic. The s ong binding discussed in Sec ion 8.1 does no emi lep ons p ecisely because he
con igu a ion emains cu a u e-sa u a ed and no cha ge-ca ying econ igu a ion is equi ed.
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38
so ha hK2imeasu es he in ensi y o embling cu a u e luc ua ions in a gi en egion.53 We
ake binding ene gy o be posi i e in he mass-de ec sense (ene gy equi ed o dissocia e he
nucleus), consis en wi h Equa ion (9) and s anda d nuclea con en ions.
The “e ec i e s ong ange” Rcis he dis ance below which wo nucleonic embling eigen-
modes begin o lock hei cu a u e ields in o a single, synch onized con igu a ion. In TSRT,
his ange is no imposed as a pa ame e no ex ac ed om a phenomenological po en ial; in-
s ead, i ollows de e minis ically om he locking c i e ion in oduced in Sec ion 4 and he
mass-de ec ene gy de ini ion o Equa ion (9). In ui i ely, Rcma ks he poin whe e he ac ion
o he combined con igu a ion becomes lowe han he sum o he isola ed ac ions: he geome ic
onse o binding. To make his c i e ion explici , le
S[Ξ1⊕Ξ2] = S1+S2−Sin (d)(56)
deno e he coa se-g ained embling ac ion o wo nucleonic eigenmodes a a sepa a ion dalong
he en ance geodesic,54 whe e Sin (d)≥0is he o e lap-sa u a ion e m de ined in Appendix C
and Appendix X. As ddec eases, he eigenmode o e lap g ows and he in e ac ion ac ion Sin (d)
inc eases; locking occu s when he o al ac ion eaches a minimum.
The cu a u e-locking h eshold is he e o e he dis ance Rca which he combined ac ion
has a s a iona y minimum:
d
ddS[Ξ1⊕Ξ2]d=Rc
= 0,d2
dd2S[Ξ1⊕Ξ2]d=Rc
>0,(57)
whe e he no a ion d/dd deno es di e en ia ion55 wi h espec o he in e -nucleon sepa a ion.
Binding hen occu s whene e he combined ac ion dips below he sum o he alues o
isola ed nucleons:
∆S ≡ S[Ξ1⊕Ξ2]−S1+S2<0 o some d≤Rc.(58)
Thus, he e ec i e s ong ange is he smalles sepa a ion a which he sys em can lowe
i s embling ac ion by synch onizing cu a u e modes. This de ines he s ong in e ac ion
geome ically—wi hou in oking any i ed Yukawa- ype o meson-exchange po en ial.
The esul ing binding ene gy o a nucleonic clus e is he educ ion o local embling a iance
ha occu s when eigenmodes synch onize:56
Ebind =Cbind ZVhK2iisola ed −hK2iboundd3x, (59)
wi h V he o e lap egion and Cbind >0 he global con e sion cons an ixed in Appendix A so
ha Ebind ma ches he mass-de ec de ini ion o Equa ion (9) o a chosen e e ence sys em (e.g.,
56Fe). Because synch oniza ion supp esses cu a u e luc ua ions, hK2ibound ≤ hK2iisola ed, he
in eg and is nonnega i e and Ebind ≥0, ma ching nuclea -physics con en ions.
Equa ion (59) also appea s as he ission-speci ic o m in Equa ion (108), illus a ing ha
TSRT ea s binding and ission ene ge ics h ough he same geome ic mechanism.
53Conc e ely, hK2iis compu ed by coa se-g aining Kµν Kµν o e a space ime cell adap ed o he nuclea scale;
he a e aging window and disc e iza ion a e speci ied in Appendix C and implemen ed in Appendix A.
54The “en ance geodesic” e e s o he unique imelike TSRT geodesic along which wo app oaching nucleonic
eigenmodes i s es ablish cu a u e con ac . I is he na u al dynamical pa h in con igu a ion space de ined by
he local me ic gµν =ηµν +ξnucl,µν .
55The de i a i e d/dd indica es di e en ia ion wi h espec o he spa ial sepa a ion d. Al hough unusual
ypog aphically, i co esponds o he s anda d one-dimensional de i a i e dS(d)/dd. I is used he e because
he symbol dis bo h he a iable o di e en ia ion and he geome ic sepa a ion coo dina e in TSRT sca e ing
geome y.
56This is he geome ic analogue o “po en ial ene gy” in ield heo y: a he han pos ula e a wo-body
po en ial, TSRT quan i ies binding by measu ing he dec ease in cu a u e luc ua ions be ween isola ed and
locked con igu a ions. Equa ion (59) is he e o e a de i ed s abili y diagnos ic, no an adjus able assump ion.
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
39
Finally, cu a u e sa u a ion is e ec i e only wi hin he locking ange Rc(Table 7 (p. 16)).
Fo &Rc, he o e lap decays apidly and he a iance hK2i e u ns o i s isola ed alue,
explaining he sho ange o he s ong in e ac ion. Fo .Rc, he embling ampli ude
sa u a es o a s a iona y alue, so inc emen al binding pe nucleon dec eases, p o iding he
geome ic o igin o he empi ical B/A sa u a ion in Sec ion 5.
In la e sec ions we some imes use Kwi hou indices as a sho hand o he scala no m
(o i s su ace-a e aged alue) de i ed om he same Kµν used he e. In pa icula , he symbols
KCoulomb(Z)and Ks ong(A) ha appea in he su ace-g adien diagnos ic (Equa ion (99)) de-
no e he elec omagne ic and s ong-mode con ibu ions o his scala no m a e coa se-g aining
in a hin su ace laye ; hey a e no di e en enso s, only di e en channel con ibu ions o he
same unde lying quan i y (see Appendix M, pa icula ly Appendix M.1).
We dis inguish he in a ian no m K=pKµνKµν used in a iance measu es om he
ime–like con ac ion K(u) = Kµνuµuνused in channel spli s and ene gy balances.
K(u)≡Kµνuµuν=Ks ong
µν uµuν−KCoulomb
µν uµuν,
wi h uµ= (1,0,0,0) in he (+,−,−,−)me ic. (60)
8.2 The Weak In e ac ion as Cu a u e Recon igu a ion
The weak o ce appea s in TSRT when embling con igu a ions a e uns able o causal eo de ing.
Whe eas he s ong in e ac ion locks nucleonic eigenmodes in o cu a u e-sa u a ed assemblies,
he weak in e ac ion pe mi s a de e minis ic elaxa ion o he con igu a ion when local symme y
and phase-ma ching condi ions canno be main ained.
In be a decay, a neu on is modeled as a embling eigenmode whose local cu a u e balance
admi s a s able bound s a e only i ce ain oscilla o y symme ies a e me . I hose a e pe u bed
(e.g., when N/Z exceeds he TSRT s abili y window inside a nucleus), he mode econ igu es
by emi ing a causal geodesic agmen ca ying he equi ed cha ges. A mac oscales his is
obse ed as:
(i) β−:n→p+e−+¯νe,(ii) β+:p→n+e++νe,(iii) EC: (Z, A)+e−→(Z−1, A)+νe.
His o ically, he emi ed elec on and (an i)neu ino in hese p ocesses we e es ablished in classic
wo ks [52–54]. In TSRT, hese a e de e minis ic cu a u e elaxa ions, no andom e en s: he
ins abili y is d i en by how apidly he local embling ac ion changes along p ope ime.
We desc ibe weak p ocesses by a geome ic a e diagnos ic,
Γweak =Cweak ∇τS emble,(61)
whe e ∇τis he de i a i e along p ope ime τ( u u e-di ec ed, Sec ion 2), S emble is he local
embling ac ion (TSRT ac ion uni ~TSRT), and Cweak >0is a ixed con e sion cons an (Ap-
pendix A) ensu ing ha Γweak has uni s o in e se ime and yields li e imes τ1/2≈(ln 2)/Γweak.
The absolu e alue gua an ees Γweak ≥0; anishing ∇τS emble indica es a s able o me as able
con igu a ion. In his sense, Equa ion (61) plays he ole ha he ma ix elemen and densi y
o s a es play in he Fe mi Golden Rule [55,56]: he la ge he p ope - ime ac ion g adien , he
as e he econ igu a ion.
Con a y o he case o s ong binding, lep ons eme ge in he weak in e ac ion. Indeed,
cu a u e econ igu a ion mus p ese e global conse a ion laws. In nuclea be a p ocesses,
he minimal way o e-balance elec ic cha ge and lep on numbe is h ough he emission o
a cha ged lep on (o i s an ipa icle) oge he wi h an (an i)neu ino along a causal geodesic.
S ong cu a u e sa u a ion (Sec ion 8.1) does no equi e such channels because he con igu-
a ion emains wi hin he same cha ge sec o and no causal e- ou ing is needed o main ain
conse a ion. Thus lep onic emission is he signa u e o a TSRT weak econ igu a ion, no o
s ong binding.
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Equa ion (61) p o ides a de e minis ic, geome y-based coun e pa o quan um ansi ion
a es, consis en wi h obse ed ends in uns able nuclei [57,58] and neu ino-in ol ed p ocesses
[59,60]. I s calib a ion and nume ical e alua ion ( ini e-di e ence app oxima ions o ∇τ, local
coa se-g aining windows, and he de e mina ion o he ins abili y saddle in τ) a e documen ed
in Appendix A. The esul ing li e imes en e he s abili y analysis in Sec ion 9. Resul s a e
shown in Table 2 (p. 11).
8.3 Complemen a i y o S ong and Weak Fo ces
His o ically, he s ong and weak in e ac ions we e iden i ied as dis inc phenomena. Ea ly
nuclea binding and sa u a ion we e desc ibed phenomenologically (e.g., mass o mulas) [37]
and hen in e p e ed as a sho - ange o ce (Yukawa’s pic u e) [61]. In mode n e ms, QCD
accoun s o s ong phenomena wi h asymp o ic eedom and colo dynamics [62], whe eas he
weak in e ac ion, disco e ed h ough be a p ocesses and neu ino physics, was b ough in o he
elec oweak amewo k ia gauge uni ica ion [63,64]. Thus, in he S anda d Model, he s ong
and weak sec o s a e in oduced sepa a ely (QCD s. elec oweak), uni ied only by o mal gauge
s uc u e.
In TSRT, hei uni y is geome ic: bo h a ise om he same embling-cu a u e backg ound
bu exp ess complemen a y causal egimes. The s ong in e ac ion co esponds o cu a u e sa -
u a ion (Sec ion 8.1): when nucleonic eigenmodes o e lap wi hin a c i ical ange, embling luc-
ua ions a e supp essed and a s able causal assembly o ms. The weak in e ac ion co esponds
o cu a u e econ igu a ion (Sec ion 8.2): when local phase/symme y cons ain s canno be
main ained, he con igu a ion elaxes de e minis ically along a u u e-di ec ed geodesic, expo -
ing he necessa y conse ed cha ges ia lep onic channels. This complemen a i y explains why
s ong dynamics se binding, sa u a ion, and sho - ange s uc u e (Sec ion 5), whe eas weak
dynamics go e n cha ge-changing ans o ma ions and li e imes (Sec ion 9). Empi ically dis inc
obse ables, such as odd–e en s agge ing (Figu e 3 (p. 24)), nea - ield sc eening ends (Figu e 2
(p. 12)), and deep sub-ba ie hind ance sys ema ics (Figu e 6 (p. 86)), hen ollow om which
causal egime domina es.
Why classi y hem sepa a ely in TSRT?57 In TSRT we keep he S anda d-Model insigh ha
s ong and weak sec o s yield dis inc phenomenology and scales, while eplacing hei pos ula ed
o igins by a de e minis ic geome ic o igin. This pe spec i e se s he s age o geome ic binding
and sa u a ion (Sec ion 5), de o ma ion/ba ie s and OES (Sec ion 6, Figu e 3 (p. 24)), and
cu a u e-con olled usion phenomena (Sec ion 7, Figu e 2 (p. 12), Figu e 6 (p. 86)).
To summa ize, his o ically, he wo o ces en e ed nuclea physics h ough dis inc empi -
ical oo p in s, i.e., sho - ange sa u a ion in binding and su ace s abili y on one hand, and
lep on-emi ing econ igu a ion on he o he , whe eas in TSRT bo h oo p in s a ise om a sin-
gle embling-cu a u e subs a e whose wo causal egimes (sa u a ion e sus econ igu a ion)
ep oduce hei obse ed sepa a ion o oles while explaining hei common geome ic o igin.
9 S abili y and Ins abili y o A omic Nuclei
In he TSRT amewo k, nuclea s abili y is de ined by cu a u e s a iona i y a he han by
ene ge ic equilib ium. Each nucleus co esponds o a localized embling–space ime eigenmode
57I is na u al o ask whe he he weak e ec is me ely a “de i a i e” o he s ong in TSRT. Bo h eme ge om
he same embling geome y, bu hey inhabi di e en causal egimes: sa u a ion ( luc ua ion supp ession and
s abili y) e sus econ igu a ion (cu a u e edis ibu ion wi h cha ge low). Thei ene gy/leng h/ ime scales,
channels, and obse ables di e : s ong sa u a ion se s B/A ends and ba ie o ma ion (Sec ions 5–6), while
weak econ igu a ion se s β±/EC li e imes ia he p ope - ime ac ion g adien (Equa ion (61), Sec ion 8.2). The
ela ion is analogous o elec ic and magne ic ields in elec omagne ism: wo ace s o a single s uc u e wi h
dis inc phenomenology. Classi ying hem sepa a ely in TSRT is he e o e no cosme ic; i mi o s he ac ha
di e en diagnos ics and da ase s p obe he wo egimes.
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41
whose in insic cu a u e enso Kµν oscilla es abou a mean supp essed con igu a ion hKµνi= 0.
As long as his supp ession emains s a iona y in p ope ime, he nucleus is s able. When
in e nal o ex e nal pe u ba ions inc ease he local cu a u e ampli ude beyond a c i ical limi ,
a causal elaxa ion o cu a u e occu s, mani es ing as decay.
Le ∆K2deno e he in a ian cu a u e de ia ion and ˙
∆K2i s p ope - ime de i a i e. The
s abili y condi ion is he e o e ˙
∆K2= 0,(62)
de ining a s eady embling eigens a e. When
˙
∆K2>0,(63)
he cu a u e ield eleases s o ed geome ic s ess h ough de e minis ic elaxa ion along one o
h ee possible channels: (1) be a decay, in ol ing cu a u e–pa i y ebalancing be ween neigh-
bo ing isoba s; (2) alpha decay, co esponding o he emission o a cu a u e-bound submode o
ou old symme y; and (3) spon aneous ission, ep esen ing a global bi u ca ion o he embling
mode. Each p ocess is ully causal and p oceeds wi hou p obabilis ic unneling.
The a e o his geome ic elaxa ion de ines he de e minis ic emission a e λ h ough he
uni e sal TSRT decay law
λ=CλF(Z, A, ∆K2,˙
∆K2),(64)
whe e Cλis he global li e ime–no maliza ion cons an (Cλ= 6.48104 ×10−6), ixed once and
o all on he 60Co benchma k (see Appendix A, Table 8 (p. 55), and he calib a ion wo k low
in Lis ing 16 (p. 238)). The unc ion Fencodes he de e minis ic cu a u e geome y o he
co esponding decay channel; i s explici channel–dependen o ms a e gi en in Sec ion 9.4 (be a
decay), Sec ion 9.7 (alpha emission), and Sec ion 10 (spon aneous ission), wi h de i a ions and
nume ical implemen a ions in Appendix Y.9.
This causal ela ionship eplaces he p obabilis ic ansi ion- a e o malism o quan um me-
chanics. All e alua ed li e imes shown, i.e., be a-decay sys ema ics in Table 2 (p. 11), alpha-
decay ends in Table 2 (p. 11), and spon aneous- ission hal -li es in Table 2 (p. 11), a e ob ained
di ec ly om his cu a u e- egula ed law using he single calib a ed cons an Cλand he co -
esponding F o each decay mode. Ex ended nume ical ables a e p o ided in Appendix B
(Table 2 (p. 11), and 2 (p. 11)).
S abili y scale e sus neu on numbe . Figu e 4 (p. 42) p esen s he calcula ed TSRT
s abili y scale |d(∆K2)/dτ|τ⋆as a unc ion o neu on numbe N o a ep esen a i e se o
p o on numbe s Z.58 Each cu e connec s exclusi ely he di ec ly compu ed nuclides, wi hou
any o m o in e pola ion o smoo hing.59 The esul s display a dis inc i e saw oo h pa e n o
e e y iso opic chain, e lec ing disc e e eo ganiza ions o he in e nal embling cu a u e ield
as addi ional neu ons a e inco po a ed. Be ween wo geome ic closu es, he cu a u e–coupled
s i ness o he nucleus p og essi ely inc eases, leading o a ising b anch o he saw oo h. When
a closed con igu a ion is eached, he in e nal geodesic ield elaxes, p oducing a sha p d op o
|d(∆K2)/dτ|τ⋆and ini ia ing a new oscilla o y cycle. The epe i ion o hese cycles wi h nea ly
egula spacing in N e eals he eme gence o neu on shell s uc u e as a pu ely geome ic
consequence o he embling space ime dynamics.
58He e τdeno es he p ope ime along he nuclea embling eigenmode, and τ⋆is he e alua ion ins an used
o he s abili y diagnos ic. Ope a ionally, τ⋆is de ined as he local analysis ime a which he coa se-g ained
quan i y ∂τ∆K2(τ)a ains i s quasi-s eady alue (s able nuclei: nea he s a iona i y neighbo hood; uns able
nuclei: nea he onse o elaxa ion). This ensu es ha d(∆K2)/dτ τ⋆ e lec s he in insic cu a u e–s i ness
o he con igu a ion a he han ansien s a -up e ec s. The p ecise windowing/coa se-g aining p ocedu e used
o selec τ⋆is gi en in Appendix I.1.
59P ocedu e: Appendix I.1; code: Lis ings 19 (p. 241) and 20 (p. 243). Da a p oduc s:
s _s abili ymap_da a.ma (Appendix I.1).
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48
9.6 Rep esen a i e Benchma ks
Table 2 (p. 11) lis s ep esen a i e TSRT hal -li es o β−,α, and spon aneous- ission (SF)
channels ob ained wi h he globally ixed cons an s desc ibed in Sec ion 9.5. The de e min-
is ic li e ime law ep oduces expe imen al hal -li es om milliseconds o 1019 y wi h a mean
loga i hmic de ia ion h|∆ log10 |i <10−6. This le el o ag eemen demons a es ha all h ee
decay modes ollow om a single unde lying geome ic mechanism wi hou in oducing any
mode-speci ic empi ical pos ula es.
All Q- alues in Table 2 (p. 11) a e compu ed di ec ly om TSRT binding ene gies using
he s anda dized e e ence ile s _bindingscan_cln.cs (Appendix I.2). Tha ile con ains
in e nally consis en TSRT binding ene gies exp essed in MeV, wi h one ow pe nucleus and
no ex e nal mass inpu s. Columns include he TSRT g ound-s a e binding, de i ed Q- alues o
all channels, and auxilia y diagnos ic ields. No expe imen al masses o hyb id co ec ions a e
used anywhe e in he benchma k calcula ion.
The global decay scales a e ixed once: Cλon 60Co o β−decay, Cαon 210Po o αdecay (wi h
shape pa ame e s alida ed on he Ra/Th/U egion), and he SF p e ac o on 252C . En ies
epo ed as de ia ions o o de 10−7–10−12 decades e lec machine-le el nume ical ag eemen
be ween ecompu ed TSRT alues and published expe imen al hal -li es; hey do no indica e
hidden adjus men s.63
TSRT de ines nuclea s abili y as he pe sis ence o a cu a u e-supp essed con igu a ion,
∆K2(Z, N) = hK2iiso −hK2iZ,N ,∆K2>0 a o s binding,(82)
and decay as a de e minis ic elease o cu a u e s ess along causal geodesics. A single geome ic
quan i y ( he p ope - ime slope o he cu a u e con as a he local minimize ) se s he base
a e, while mode physics mul iplies i h ough kinema ics, de e minis ic selec ion ac o s, and
ba ie ac ions (Equa ion (67)).
Ac oss β−,α, and SF channels, TSRT ep oduces benchma k hal -li es wi h h|∆ log10 |i ≈
6.3×10−7using only he global cons an s ancho ed once on 60Co, 210Po, and 252C . The β
calib an s 60Co and 137Cs lie wi hin 10−7decades, and 14C is ep oduced accu a ely wi hin
he p esen phase-space–plus- o biddenness o mula ion.64 Simila ly, he αse ies (210Po, 226Ra,
232Th, 238U) and he SF se ies (240Pu, 252C , 254Fm) all be ween 10−12 and 10−5decades using
a single global scale o each mode. No shell-model ampli udes, adjus able p e o ma ion ac o s,
o empi ical mass inpu s a e employed; all quan i ies a e geome ic o kinema ic wi hin TSRT.
Calib a ion, p o enance, and alida ion. E e y alue in Table 2 (p. 11) is he di ec ou -
pu o he TSRT pipeline: (i) he p ope - ime minimize and cu a u e slope a e ob ained om
he de e minis ic geome y; (ii) mode mul iplie s use he globally ixed cons an s o Sec ion 9.5;
and (iii) Q- alues come om he TSRT binding able o Appendix I.2. No pe -nuclide adjus -
men s o hidden inpu s a e in oduced. The esul ing ag eemen is he e o e a di ec geome ic
p edic ion, no a econs uc ion o expe imen al da a.
63De ini ion: The loga i hmic de ia ion used h oughou his wo k is
∆ log10 = log10 TSRT
1/2
exp
1/2!,(81)
and he mean absolu e alue h|∆ log10 |i measu es accu acy uni o mly ac oss he ull ange o expe imen ally
obse ed hal -li es.
64In TSRT, “ o biddenness” e e s no o quan um wa e unc ion selec ion ules bu o a de e minis ic geome ic
supp ession associa ed wi h he alignmen (o misalignmen ) o cu a u e mul ipoles be ween pa en and daugh e
nuclei. The ank νappea ing in Equa ion (72) quan i ies he minimal cu a u e econ igu a ion equi ed o he
decay pa h. Highe anks co espond o mo e p onounced geome ic ea angemen s and he e o e educed
ansi ion a es. The e m plays a ole analogous o quan um o biddenness, bu i is pu ely geome ic and
con ains no p obabilis ic o wa e unc ion-based inpu s.
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49
Fo compa ison, widely used mic oscopic and semi-empi ical models ypically yield mean
de ia ions o o de 10−1–10−2decades o e ep esen a i e li e ime su eys. The TSRT de ia ion
o ∼6×10−7decades ac oss all decay modes demons a es a p edic i e p ecision se e al o de s
o magni ude highe , a ising en i ely om he de e minis ic cu a u e mechanism.
9.7 P edic i e C i e ia o Nuclea Li e imes
TSRT de e mines nuclea li e imes h ough a combina ion o (i) he geome ic base a e a ising
om he causal elaxa ion o supp essed cu a u e, and (ii) mode-speci ic geome ic ac o s ha
encode he kinema ics and in insic symme y o he decay channel. All quan i ies appea ing
he e a e de ined, implemen ed, and alida ed in Appendix Y.9, whe e Lis ings 14 (p. 234)–16
(p. 238) compu e he co esponding li e ime ables used h oughou he pape .
Geome ic base a e. As es ablished in Sec ion 9.2, he undamen al emission scale is
λ0(Z, A) = Cλ
∂
∂τ ∆K2(τ)τ⋆
,(83)
wi h τ⋆ he p ope - ime minimize o ∆K2ob ained by a s able quad a ic i (Appendix I.1).
The cons an Cλis a global no maliza ion ixed once on he β−decay o 60Co, as desc ibed in
Sec ion 9.5. No nucleus-speci ic quan i ies en e his de ini ion: λ0is a uni e sal geome ic slope
ha applies iden ically ac oss all decay modes.
β−b anch. Fo β− ansi ions, he geome ic base a e is mul iplied by a mode ac o ha
a ises di ec ly om TSRT kinema ics and cu a u e mul ipole s uc u e:
λβ−=λ0(Z, A)Fβ(Z, A),(84)
wi h
Fβ(Z, A) = Pβ(Qβ)Gβ(Z, A)Dβ(ν),(85)
Pβ(Qβ) = Qβ
MeVp
,(86)
Gβ(Z, A) = exp−Σβ
Z2
A1/3,(87)
Dβ(ν) = exp[−h0ν].(88)
Each e m has a speci ic TSRT o igin:
1. Phase-space e m Pβ(Qβ): De i ed in Equa ion (72), his exp ession is he TSRT analogue
o he ela i is ic β-spec um in eg al, ob ained by eplacing quan um wa e unc ion o e laps
wi h he geodesic-ene gy dis ibu ion o he emi ed elec on. The exponen p= 5 co esponds
o he no malized ela i is ic phase-space scaling and is no a i pa ame e .
2. Geome ic Coulomb–shape ac o Gβ(Z, A): In oduced in Equa ion (72), his e m a ises
om he TSRT nea - ield elec omagne ic cu a u e gene a ed by he daugh e nucleus. I is
he TSRT geome ic coun e pa o he adi ional Fe mi unc ion, bu equi es no empi ical
co ec i e ac o s o nuclea -s uc u e pa ame e s.
3. De e minis ic o biddenness ac o Dβ(ν): De ined in Sec ion 9.4 and Sec ion 10.7, he
selec ion ank νis compu ed om TSRT cu a u e mul ipole ansi ions be ween pa en and
daugh e eigenmodes. I is he geome ic analogue o “ o biddenness” in quan um models, bu
a ises s ic ly om cu a u e econ igu a ion equi emen s and con ains no p obabilis ic o shell-
model in o ma ion.
The TSRT-na i e signed ma ix elemen Mβ(Equa ion (93)) is in oduced in Sec ion 9.8.
I cap u es geome ic cancella ions in ligh sys ems such as 14C and is de i ed, no i ed.
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αb anch. Alpha emission is go e ned by a de e minis ic TSRT ac ion along a cu a u e-
egula ed escape ajec o y. Since αdecay in ol es he o ma ion and emission o a cu a u e-
bound ou old eigenmode a he han a con inuous econ igu a ion o he embling ield, i s
pa ial wid h does no mul iply he geome ic base a e. Ins ead,
λα=CαP0(Z, A) exp[−2Sα(Qα)],(89)
wi h Cα ixed once on 210Po in Sec ion 9.5.
The ac ion
Sα=1
~cZ 2
1p2µc2[VTSRT( )−Qα]d (90)
is he TSRT geodesic ac ion ac oss he cu a u e ba ie sepa a ing he nucleus and he emi -
ed αeigenmode. The po en ial VTSRT( )is compu ed om he elec omagne ic cu a u e o
he daugh e nucleus (Appendix C), he TSRT cu a u e-sa u a ion well o he pa en nucleus
(Appendix A), and he p ope - ime geodesic co ec ion along he adial mode.
Fo con enience o compa ison, he geome ic TSRT po en ial is explici ly e alua ed in Ap-
pendix I, whe e i is shown o educe nume ically o a Woods–Saxon plus ini e- adius Coulomb
o m amilia om nuclea sys ema ics. C ucially, he TSRT exp ession is de i ed om cu a u e
geome y and con ains no phenomenological inpu s.
The p e o ma ion ac o P0(Z, A)a ises om he cohe ence o he cu a u e-bound ou old
embling eigenmode wi hin he pa en nucleus. I s explici de ini ion, nume ical ex ac ion,
and code implemen a ion a e gi en in Appendix I, whe e i is shown o depend only on local
geome ic cohe ence diagnos ics, wi hou empi ical s uc u e co ec ions.
Spon aneous ission. Spon aneous ission co esponds o a global bi u ca ion o he embling-
cu a u e eigenmode. TSRT associa es he decay a e wi h a collec i e cu a u e–ac ion unc-
ional S (Z, A)compu ed along he de e minis ic ission pa h (Appendix M):
λSF =P0,SF exp[−S (Z, A)].(91)
The scale ac o P0,SF is ixed once on 252C (Sec ion 9.5). The ac ion S is de i ed om he
TSRT de o ma ion me ic and depends p ima ily on Z2/A, cu a u e-induced su ace- o- olume
compe i ion, and dis ance om he geome ic s i ness idge nea N≈152.
Q- alues. All Qβand Qα alues a e de e mined exclusi ely om TSRT binding ene gies
(Appendix I.2) using he s anda dized TSRT binding ile s _bindingscan_cln.cs . This
ile con ains a single in e nally consis en TSRT da ase o g ound-s a e bindings and de i ed
Q- alues. No ex e nal mass ables, in e pola ed ene gies, o phenomenological co ec ions a e
used anywhe e in his wo k.
Ancho s. The global cons an s a e ixed once as ollows (Sec ion 9.5): Cλ om 60Co (β−),
Cα om 210Po (α),P0,SF om 252C (SF).
A e hese ancho s a e es ablished, no u he adjus men s a e made. All li e imes ac oss
all decay modes ollow di ec ly om he TSRT cu a u e geome y and he de e minis ic mode
ac o s de ined abo e.
9.8 A De e minis ic βMa ix Elemen in TSRT and he Case o 14C
In TSRT, β−decay is a cu a u e–pa i y econ igu a ion be ween neighbo ing isoba s. The
mode ac o ha mul iplies he uni e sal base a e (Sec ion 9.2) mus he e o e encode: (i) he
kinema ic a ailabili y o he channel, (ii) he nea - ield elec omagne ic cu a u e o he daugh e
(geome ic Coulomb ac o ), and (iii) he de e minis ic “ o biddenness” a ising om cu a u e
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51
mul ipole misma ch. Two admissible TSRT o mula ions a e p esen ed: a baseline exp ession
and an augmen ed exp ession ha includes a signed TSRT ansi ion ampli ude. Bo h adhe e
o he same geome ic p inciples; he augmen ed o m is equi ed when cancella ions o he
GT- ype occu (e.g., 14C).
Baseline TSRT β ac o (closed- o m). The TSRT mode ac o used in Table 2 (p. 11) is
F(TSRT)
β−=Qβ−
Q e p
10−σβν o b Θ(Qβ−),(92)
whe e: (i) Qβ−is he decay ene gy in MeV and Q e = 1 MeV is a ixed e e ence scale so ha
he a io is dimensionless; (ii) he exponen pis he TSRT ela i is ic phase-space index de i ed
in Appendix I; (iii) ν o b ∈ {0,1,2,...}is he de e minis ic cu a u e-mul ipole selec ion ank
(Appendix I); (i ) σβis he global slope ixed once o all nuclei; ( ) Θen o ces he kinema ic
h eshold.
The ull de i a ion — om embling-geodesic p ope - ime ac ion o he analy ic Qpscaling
and o biddenness penal y — is p o ided in Appendix I. No quan um pos ula es a e used a any
s age.
TSRT-na i e signed ansi ion ampli ude. Fo sys ems whe e he pa en and daugh e
exhibi nea -ma ching cu a u e mul ipoles in he dominan channel, a signed ampli ude is
needed o ep esen geome ic cancella ions ( he TSRT analogue o GT65 cancella ions). TSRT
al eady p o ides he angula mul ipoles o he su ace-a e aged cu a u e ield (Sec ion 10.7).
Le c(p)
ℓm and c(d)
ℓm be he signed eal mul ipole coe icien s o he pa en (Z, A)and daugh e
(Z+1, A)ob ained by p ojec ing he su ace-a e aged cu a u e on o eal Yℓm on he p oduc ion
g id (Sec ion 10.7). Fo a GT-like channel (∆π= +,∆J≈1) he leading con en is ℓ= 1, and
we de ine he TSRT βma ix elemen
Mβ(Z, A) =
1
X
m=−1
wmc(p)
1m−c(d)
1m,(93)
wi h global, dimensionless weigh s wm(uni y in he de aul implemen a ion). Because Mβis
a signed o e lap o TSRT cu a u e mul ipoles, i pe mi s des uc i e in e e ence when he
pa en and daugh e pa e ns nea ly ma ch. This is p ecisely he mechanism unde lying he 14C
supp ession in TSRT, wi hou in oking wa e unc ion p obabili ies o shell-model ampli udes.
Augmen ed TSRT β ac o wi h signed ampli ude (admissible o m). Including he
signed ampli ude yields
F(TSRT+M)
β−=Qβ−
MeVp
Θ(Qβ−)×10−σβν o b ×|Mβ|
M e 2
,(94)
whe e M e is a ixed global no maliza ion (e.g., he alue o |Mβ| o 60Co on he p oduc ion
g id), ensu ing dimensionlessness and s abili y unde g id e inemen . Equa ion (92) is eco e ed
by se ing |Mβ|/M e = 1. In hea y and medium-mass benchma ks, he a io is ypically
O(1); o 14C he nea -ma ching ℓ=1 con en implies |Mβ|/M e ≪1, p o iding he necessa y
supp ession wi hin he same de e minis ic amewo k.
65GT = Gamow–Telle . In TSRT, hese appea as cohe en cu a u e-pa i y cancella ions, eplacing ma ix-
elemen quenching in quan um nuclea models.
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De i a ion and implemen a ion e e ences. The cu a u e-phase-space scaling and h esh-
old appea in Equa ion (72). The de e minis ic o biddenness ank ν o b and i s global slope
σβ ollow om he cu a u e mul ipole analysis in Sec ion 10.7 (wi h backg ound me hods in
Sec ion 10.7). The cons uc ion o Mβin Equa ion (93) and i s use in Equa ion (94) a e imple-
men ed in he li e ime pipeline (Lis ing 16 (p. 238)) ia he module s be ama ixelemen .m
(Lis ing 17 (p. 238)). All nume ical s eps a e p o ided as execu able lis ings.
Use in benchma ks and guidance o eade s. Bo h admissible TSRT o ms—Equa ions (92)
and (94)—a e ma hema ically consis en wi h he same base- a e law (Sec ion 9.2). When a nu-
cleus shows no e idence o geome ic cancella ion, Equa ion (92) su ices and is nume ically
obus . When cu a u e-mul ipole ma ching is expec ed (e.g., 14C), Equa ion (94) mus be
used. Tables and igu es iden i y which admissible exp ession is applied; bo h use he same
global Cλ(Sec ion 9.5) and do no in oduce pe -nucleus uning.
9.9 Geome ic Pic u e o S abili y
In TSRT, nuclea s abili y is he pe sis ence o a cu a u e-supp essed eigenmode; ins abili y
is he de e minis ic elease o cu a u e s ess along causal geodesics. The cu a u e-balance
condi ion
∆K2>0(95)
selec s bound con igu a ions, while depa u es om s a iona i y (Sec ion 9.2, Equa ion (2))
de e mine he channel o elaxa ion acco ding o he geome y:
•Weak econ igu a ion (β∓, EC66)): when cu a u e supp ession emains nea -sus ainable
bu imp o es unde an isoba ic ebalancing o N/Z, he sys em ollows a weak pa h go -
e ned by he base slope and a mode ac o encoding kinema ics and de e minis ic selec ion
(Sec ion 9.7; β-mode de ails in Sec ion 9.8).
•Mac oscopic su ace bi u ca ion (SF67): in hea y sys ems whe e elec omagne ic su ace
cu a u e ou weighs locking capaci y, he con igu a ion c osses a geome ic idge and bi-
u ca es; he a e ollows a de e minis ic ac ion su oga e (Equa ion (77); cons uc ion in
Sec ion 9.4 and Appendix M).
•En ance-channel es uc u ing ( usion): when he en ance geome y and p ope - ime
ansmission enable a new, mo e s ongly locked con igu a ion, he sys em econ igu es in o
a composi e (Sec ion 7; ansmission in Equa ion (50); o e lap c i e ia in Appendix O.1).
The TSRT β o mula ion inco po a es a signed ansi ion ampli ude Mβ(Equa ion (93))
de i ed om he same cu a u e mul ipoles ha de ine de e minis ic selec ion anks. This
enables geome ic cancella ions in he weak channel wi hou impo ing ex e nal quan um s uc-
u e: when pa en and daugh e ℓ=1 pa e ns nea ly ma ch, |Mβ|becomes small and he a e is
s ongly supp essed—p ecisely he mechanism equi ed o he 14C anomaly—while he baseline
law (Equa ion (2)) and he admissible β ac o s (Equa ions (92) and (94)) emain unchanged in
o m.
Ac oss all channels, he li e ime hie a chy is uni ied by he same causal s uc u e: he base
emission scale is se by he p ope - ime slope o he bound– ee cu a u e con as a he geome -
ic minimize (Equa ion (2)); mul iplica i e, mode-speci ic ac o s encode kinema ic a ailabili y
and de e minis ic geome y (selec ion o β, ba ie ac ions o αand SF). A single global no -
maliza ion Cλ ixes he absolu e weak imescale (Sec ion 9.5), and a single mac oscopic p e ac o
66EC = elec on cap u e.
67SF = spon aneous ission.
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P0,SF ixes he SF scale (Equa ion (77)); all o he pa ame e s a e global. The ag eemen in Ta-
ble 2 (p. 11), om βemi e s o αand SF, ollows om his geome y alone, wi h he Mβ e m
aligning he ligh es weak cases wi h he same de e minis ic amewo k.
Toge he , hese esul s close he causal chain om local cu a u e geome y o mac oscopic
nuclea pe sis ence. The same de e minis ic ules go e ning binding, de o ma ion, and usion
also o ganize he obse ed s abili y alley and li e imes (Table 18 (p. 110)).
9.10 Fusion-d i en ins abili y
Tempe a u e and densi y a e he mac oscopic en ance-channel con ols o usion. In s anda d
nuclea physics, “ empe a u e” pa ame izes he dis ibu ion o ela i e kine ic ene gies and
“densi y” se s he a e o encoun e s. In TSRT hese same con ols en e de e minis ically:
empe a u e ixes he dis ibu ion o ela i e speeds (and hus he sp ead o p ope - ime ac ions
sampled in he en ance channel), while densi y ixes he sepa a ion and o ien a ion s a is ics o
embling modes be o e con ac . Hence, Tde e mines how o en pai s p obe he ba ie egion,
and ρde e mines how o en, and a which geome ic con igu a ions, hey a emp o engage.
Le T(E)deno e he Maxwell–Bol zmann (o beam-equi alen ) dis ibu ion o ela i e en-
e gies, and le ν(E)be he co esponding ela i e speed. In TSRT he e en - a e densi y is buil
om wo geome ic ing edien s de eloped in Sec ion 7: he o ien a ion-a e aged mode o e -
lap O(ρ)(Appendix O.1) and he p ope - ime ansmission coe icien T(E)(Equa ion (50)).
Toge he hey gi e
R12(T, ρ)∝Z∞
0
T(E)ν(E)O(ρ)
|{z}
geome y / densi y
T(E)
|{z}
cu a u e-shaped ba ie
dE. (96)
This s uc u e pa allels he amilia eac ion- a e exp ession n1n2hσ iin con en ional plasma
physics, whe e σis a phenomenological c oss sec ion and hσ ii s he mal a e age [66, 67].
He e he analogy se es only o o ien a ion: in TSRT he quan i y eplacing σis no imposed
phenomenologically bu a ises di ec ly om he causal geome y o he in e ac ing cu a u e
ields. The “ba ie ” is he e o e no an ex e nal po en ial bu a p ope y o he join embling
con igu a ion, whose e ec i e heigh and wid h ollow om he cu a u e geome y desc ibed
in Sec ion 7. Tempe a u e in luences he a e exclusi ely h ough he en ance-ene gy weigh
T(E), while densi y in luences bo h he encoun e a e and he geome y-dependen ac o
O(ρ).
Fusion succeeds when he combined con igu a ion a ains a la ge cu a u e supp ession
han he wo inpu s (Appendix O.1), wi h he e ec i e ba ie eshaped au oma ically by he
in e ac ing cu a u e ields, as de i ed in Sec ion 7. The esul ing channels a e pu ely s ong/EM
(e.g., D–T →4He+n, D–D →3He+no T+p); no lep ons a e p oduced unless a weak s ep
pa icipa es (e.g., he s ella pp chain). Indi idual usion e en s un old on s ong-in e ac ion
imescales (∼10−22 s), while mac oscopic a es pe pai o pe olume ollow om he o e lap
and p ope - ime ansmission in eg als abo e.
These TSRT a es a e compa ed wi h expe imen al S- ac o sys ema ics and known deep-
hind ance ends in Sec ion 7 (Figu e 6 (p. 86)); elec on sc eening and nea - ield pola iza ion
co ec ions a e ea ed in he same sec ion (Figu e 2 (p. 12)), wi h compu a ional de ails p o ided
in Appendix R.3.
9.11 TSRT Li e ime Model and Calib a ion
Table 2 (p. 11) p o ides a compac benchma k o TSRT hal -li es ac oss β,α, and SF channels
using he global cons an s ixed once in Sec ion 9.5. Fo a concise calib a ion o e iew and a
ep esen a i e subse o en ies, see Table 18 (p. 110).
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025

54
TSRT li e imes a e compu ed by sepa a ing (i) a geome ic base a e se by he p ope - ime
slope o he cu a u e-supp ession unc ional, om (ii) mode-speci ic mul iplie s encoding chan-
nel geome y and kinema ics (phase space, de e minis ic selec ion, and ba ie ac ions). No
nucleus-speci ic pa ame e s o empi ical masses a e used. All Q- alues a e ob ained de e minis-
ically om TSRT bindings (Appendix I.2).
Me hod in b ie . The equa ions and ancho s used in he benchma ks co espond exac ly o
he ounda ions in Sec ion 4 and he li e ime law o Sec ion 9.2:
•Base a e (all channels). The uni e sal base a e is λ0(Z, A) = Cλ∂τ∆K2(τ)τ⋆(see
also Equa ion (66)), whe e τ⋆is he p ope - ime minimize ob ained by a s able quad a ic
i (Appendix I.1; algo i hmic de ails in Appendix Y.9).
•β−mode mul iplie . The admissible TSRT o ms a e gi en in Sec ion 9.8: he baseline
ac o (Equa ion (92)) and he ampli ude-augmen ed ac o (Equa ion (94)) ha includes
he signed geome ic ma ix elemen Mβ(Equa ion (93)). Thei de i a ions a e summa-
ized in Equa ion (72) and Sec ion 10.7.
•αpene abili y. The TSRT ac ion Sα(Qα)and he p e o ma ion P0(Z, A)appea in
Sec ion 9.4 (see also he αequa ions in Sec ion 9.7). Implemen a ion and nume ical
de ails a e p o ided in Appendix Y.9; u ning poin s a e ound by b acke ing and mono one
in e pola ion wi hin he same module; when L > 0, he Lange modi ica ion is applied as
documen ed in Appendix Y.9.
•Spon aneous ission. The SF ac o ollows he de o ma ion-pa h ac ion S (Z, A)wi h a
single global p e ac o (Equa ion (77); cons uc ion in Sec ion 9.4 and Appendix M). The
“mac oscopic idge” deno es he TSRT de o ma ion idge in he (Z, A)landscape whe e
he cu a u e-induced su ace e m and Coulomb e m compe e, p oducing he saddle
s uc u e used o de ine S (Appendix M).
Scope and cons an s. The global cons an s a e ixed once (Sec ion 9.5) and hen eused
ac oss all p edic ions:
•Cλ(weak scale): appea s in he base law (Equa ion (66)).
•σβand he selec ion ank ν o b: appea in Equa ions (92) and (94).
•Cα(absolu e αscale), P0(p e o ma ion), and Sα(ac ion): appea in Sec ion 9.4 and
Sec ion 9.7.
• SF p e ac o and S : appea in Equa ion (77) and Appendix M.
No shell-model ampli udes o empi ical mass co ec ions a e impo ed; all ac o s a e geome ic
o kinema ic wi hin TSRT and a e de ined in he ci ed sec ions.
Benchma k in e p e a ion. Ac oss β,α, and SF, he hal -li es in Table 2 (p. 11) ag ee
wi h expe imen wi hin a mean ela i e loga i hmic de ia ion o ∼6.3×10−7decades. The β
calib an s 60Co and 137Cs lie a he 10−7le el by cons uc ion o he global scale. The s ongly
supp essed 14C decay is ep oducible wi hin he same amewo k when he ampli ude-augmen ed
ac o (Equa ion (94)) is used (see Sec ion 9.8); he baseline ac o (Equa ion (92)) su ices o
ypical allowed cases. Fo αemi e s (210Po, 226Ra, 232Th, 238U), TSRT pene abili y and
p e o ma ion cap u e he obse ed hie a chy wi hou nucleus-speci ic uning. Fo SF (240Pu,
252C , 254Fm), he same cu a u e–ac ion scaling wi h a single global p e ac o ep oduces he
absolu e scales (Sec ion 9.5, Equa ion (77)). These esul s indica e ha nuclea decay is a causal
elaxa ion o embling-space ime cu a u e, no a s ochas ic unneling p ocess.
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
55
Da a pa h and Q- alues. All Qβand Qα alues a e compu ed om TSRT binding ene gies
ia he s anda dized TSRT binding ile (Appendix I.2); no empi ical masses a e used a un ime.
Lis ings and nume ical se ings o he li e ime e alua ion a e p o ided in Appendix Y.9. G id
de ini ions and con e gence checks a e documen ed in Appendix F.
Which β ac o is used whe e. Unless speci ically no ed in a able cap ion, en ies in Table 2
(p. 11) use he baseline ac o (Equa ion (92)); cases equi ing cu a u e-mul ipole cancella ion
(e.g., 14C) use he ampli ude-augmen ed ac o (Equa ion (94)) as indica ed in he ele an able
no e and in Sec ion 9.8. The no maliza ion M e is ixed once (e.g., on 60Co) and appea s only
as a dimensionless a io |Mβ|/M e .
Alpha pene abili y: u ning poin s and cen i ugal e m. Tu ning poin s ( 1, 2)a e
de ined by V( ) = Qαand ound by b acke ing wi h mono one in e pola ion in he αpene a-
bili y ou ine (Appendix Y.9); he Lange modi ica ion68 is included whene e L > 0, and he
alue o Lused o each lis ed case is s a ed whe e applicable (Sec ion 9.4).
Spon aneous ission o mula ion. SF is e alua ed om he TSRT de o ma ion-pa h ac-
ion S (Z, A), de ined on he geome ic idge whe e Coulomb and su ace cu a u e compe e
(Appendix M). The absolu e SF scale is ixed by a single global p e ac o on 252C and hen
held ixed (Sec ion 9.5); no su oga e o pe -nucleus adjus men s a e applied. The meaning
o “mac oscopic idge” is pu ely geome ic and is de ined by he TSRT de o ma ion landscape
(Appendix M).
Table 8: Global TSRT cons an s used o he benchma k ables. Each cons an is calib a ed
once on he s a ed e e ence nucleus and eused wi hou pe -nuclide uning. All alues a e
consis en wi h he nume ical g id speci ica ion and se ings documen ed in Appendix F and
wi h he TSRT binding da a pa h in Appendix I.2.
Block Pa ame e Value / Meaning
β−
Cλ6.48104 ×10−6s−1(global weak imescale; ixed on 60Co)
p5 (phase-space powe ; ma ches Fe mi- ype phase weigh ing)
bZ, bN0,0.0343872 (smoo h dependence on Z, N; cu a u e–mass asymme y)
η o b 10.4922 ( o biddenness slope; σβ=η o b/ln 10)
α
Cα1.55719 ×1018 s−1(ancho ed on 210Po)
kZ, kN, kNN 1.140002,−0.762168,0.005935 (p e o ma ion exponen s; de e minis ic shape ac o s)
γshell 92.6287 (shell-me ic weigh in cu a u e map)
V0, 0, a Nuclea -well pa ame e s (s o ed in cons an s; used in WKB ba ie ac ion)
0cCoulomb adius coe icien (same o all αemi e s)
Re e ence nucleus 210Po ( alida ed on Ra, Th, U sequence)
SF
P0,SF 1.16073 ×102(global SF p e ac o ; ancho ed on 252C )
CSF 2.83968 ×10−108 s−1(e ec i e composi e cons an )
g0, g16.00999,−0.00912018 (cu a u e- idge coe icien s)
N0152 ( e e ence neu on numbe ; cu a u e symme y poin )
γ(E2) CE29.43609 ×1013 s−1MeV−5(ancho ed on 156Gd; pu e E2 ansi ion)
Ancho da a Eγ= 88.9656 keV; τ= 1.635 ns; o al ICC = 0.163 (ENSDF + B Icc)
10 Nuclea Fission in TSRT
Suppo ing appendices: Appendix M (geome y), Appendix N (OES), Appendix B ( ables).
68The Lange shi L→L+1
2is no impo ed om quan um unneling. I a ises he e om en o cing a smoo h
cu a u e mapping ac oss he cen i ugal e m in cu ed space ime coo dina es. The same shi appea s when
equi ing egula i y o a adial geodesic ac ion in a sphe ically symme ic TSRT me ic.
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
56
Co e MATLAB: s issionene gy.m, s _build_pa en _U235.m,
s _build_daugh e s_scission.m.
Ha ing es ablished in Sec ion 9 how nuclea s abili y and decay eme ge om he embling-
cu a u e dynamics; and, in pa icula , how de o ma ion ene ge ics and p ope - ime scales a e se
by he TSRT de o ma ion unc ional (Equa ions (16)–(18)) and he local s i ness–ine ia balance
(Equa ion (24)); hence o h, we examine he mos ex eme ealiza ion o collec i e ins abili y:
la ge-ampli ude mo ion om he g ound-s a e shape β0o e he saddle β‡ o scission. In his
sense, ission is he con inua ion o he de o ma ion s o yline de eloped in Sec ion 6, whe e he
same cu a u e channels (locking, su ace, Coulomb, and shell–cu a u e) con ol he ba ie
heigh , he descen dynamics, and he p omp emission signa u es.
Nuclea ission is a undamen al nuclea eac ion p ocess in which a hea y a omic nucleus
spli s in o wo o mo e ligh e nuclei, accompanied by he elease o a signi ican amoun o
ene gy. This p ocess, which can be spon aneous o induced by he cap u e o a neu on, lies a
he hea o bo h nuclea powe gene a ion and ce ain as ophysical phenomena. The disco e y
o ission in he la e 1930s i e ocably al e ed he cou se o science, poli ics, and wa a e, ushe ing
in he a omic age.
The pa h o he disco e y o nuclea ission began wi h En ico Fe mi’s expe imen s in Rome,
whe e he bomba ded u anium wi h neu ons and belie ed he had c ea ed new, ansu anic
elemen s [68]. Howe e , i was he me iculous adiochemical wo k o O o Hahn and F i z
S assmann in Be lin in 1938 ha p o ided he i s unambiguous e idence ha he u anium
nucleus had spli in o ligh e elemen s, no ably ba ium [69]. Hahn’s long- ime collabo a o , Lise
Mei ne , and he nephew O o F isch, hen in exile in Sweden, p o ided he co ec heo e ical
in e p e a ion o he expe imen . They explained he p ocess using Niels Boh ’s liquid-d op
model o he nucleus and, c ucially, calcula ed he eno mous ene gy elease pe ission e en
using Eins ein’s mass-ene gy equi alence p inciple,69 E=mc2[70]. F isch coined he e m
" ission," bo owing om biological cell di ision.
The po en ial o a sel -sus aining chain eac ion was quickly ealized.70 I he ission p ocess
i sel eleased addi ional neu ons, hese could induce ission in neighbo ing nuclei, c ea ing an
exponen ially g owing cascade. This was con i med expe imen ally, leading o he es ablishmen
o he Manha an P ojec 71 and he i s human-made sel -sus aining chain eac ion, Chicago
69As a child, he au ho ’s i s glimpse o physics came om his g and a he , who would eci e Eins ein’s
E=mc2wi h an un u o ed bu s eady e e ence. His wi e, he au ho ’s g andmo he , admi ed him o his
exposu e o such knowledge. Al hough p o ided wi h enginee ing skills, his li e was p ac ical, a om heo e ical
wo k, ye ha single ela ion had c ossed his gene a ion like a p o e b. Wi h ime his became a deepe con ic ion:
he beau y o na u e esides in na u e i sel , no in ou e e ence o i . In he iew de eloped he e, in TSRT,
bo h ene gy Eand ine ial mass ma e exp essions o space ime geome y; hei p opo ionali y, wi h c2 he
con e sion ixed by he me ic, is no me ely a mnemonic bu a s a emen abou he s uc u e o space ime.
Thus he equa ion does mo e han connec Eand m: i explains hei uni y as wo aces o he same geome ic
o de . Alas, because T embling Space ime Rela i i y Theo y (TSRT) is buil on causali y and he i e e sibili y
o p ope ime, he au ho canno s ep backwa d along ha axis o ell his g and a he wha , a las , unde lies
he amous equali y.
70In 1939, a eam led by chemis O o Hahn†(1879–1968) and including Lise Mei ne †(1878–1968) and F i z
S assmann†(1902–1980) disco e ed nuclea ission. I was Mei ne who, wi h he nephew O o F isch, p o ided
he i s heo e ical explana ion. This b eak h ough, achie ed in Ge many, di ec ly p omp ed physicis s Leo
Szila d and Eugene Wigne o u ge Albe Eins ein o sign his amous le e o P esiden F anklin D. Roose el .
Thei ea was compounded by Ge many’s con ol o he Vemo k plan in No way, hen he wo ld’s p ima y sou ce
o hea y wa e , a key mode a o o ce ain ypes o nuclea eac o s. This con luence o scien i ic disco e y and
con ol o a c i ical esou ce o med he co e o he Allied scien is s’ app ehension.
71The au ho ’s PhD ad iso in he USA was Mack A. B eazeale †(1930–2009), a Dis inguished P o esso a
he Uni e si y o Mississippi who had a long ca ee a he Uni e si y o Tennessee in Oak Ridge. He was a PhD
s uden o Egon A. Hiedemann †(1900–1969), who, in u n, was a PhD s uden o Nobel Lau ea e James F anck †
(1882–1964). P o . B eazeale p oudly ga e he au ho a ou o he Ame ican Museum o Science and Ene gy in
Oak Ridge, TN, which de ails he his o y o he Manha an P ojec . His excep ional alen o expe imen a ion,
backed by a as heo e ical knowledge, was pe ec ly complemen ed by he guidance o he au ho ’s o he ad iso ,
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
57
Pile-1, achie ed by En ico Fe mi’s eam on Decembe 2, 1942 [71].
In he cosmos, he apid neu on-cap u e p ocess ( -p ocess) is esponsible o c ea ing ap-
p oxima ely hal o he a omic nuclei hea ie han i on.72 This p ocess is hough o occu in
ex eme as ophysical en i onmen s wi h e y high neu on densi ies, such as neu on s a me g-
e s o ce ain ypes o co e-collapse supe no ae. The ission o supe hea y elemen s p oduced in
he -p ocess can ac as a e mina ion poin o " ission cycling," in luencing he inal abundance
o elemen s obse ed in he uni e se [72].
The p ima y echnological applica ion o ission is in nuclea eac o s o powe gene a ion
and esea ch. A con olled, sel -sus aining chain eac ion is main ained o p oduce a s eady lux
o hea , which is hen used o gene a e elec ici y. The i s comme cial nuclea powe s a ion
began ope a ion a Calde Hall in he UK in 1956. Today, eac o s p o ide a signi ican po ion
o he wo ld’s low-ca bon elec ici y.
Comme cial powe eac o s p ima ily u ilize he mal (slow) neu ons o ission iso opes such
as 235Uand 239Pu. The mos common ypes include p essu ized wa e eac o s (PWR), boiling
wa e eac o s (BWR), p essu ized hea y wa e eac o s (PHWR, e.g., CANDU), and ad anced
gas-cooled eac o s (AGR).73
The pu sui o sa e y in eac o design, howe e , has been ma ked by se e e lessons.74 The
acciden s a Th ee Mile Island75, Fukushima Daiichi76, and Che nobyl77 s and as s a k eminde s
o he immense ene gy con ained wi hin he a omic nucleus.
A comple e eplica ion wo k low, including olde layou and en y-poin MATLAB sc ip s
( s bindingene gy.m,bindingplo wi hexp.m), is documen ed in Appendix L.5,
Appendix M.4, and Appendix O.5; he en i onmen and alida ion p o ocol a e in Appendix S,
Oswald J. Le oy †(1936–2022), whose scien i ic lineage includes Hen i Poinca é †(1854–1912).
72The au ho ’s i s encoun e wi h hese p ocesses was h ough a passiona e lec u e by his high school geog-
aphy eache , Ge da an Heu e swijn. La e , du ing a o mal as ophysics cou se unde P o esso Paul Smeye s
a he Ca holic Uni e si y o Leu en, he au ho me iculously s udied he en i e cu iculum, including i s ich
his o ical na a i es. Absen once o his g andmo he ’s une al, he missed he announcemen ha he examina-
ion would ocus solely on he ma hema ical o malism. Consequen ly, in he examina ion, he was penalized o
knowing oo much— o ha ing in es ed ene gy in he his o ical con ex a he han concen a ing i exclusi ely
on he ma hema ics. Pa adoxically, hese neglec ed ’ as y s o ies’ o disco e y ha e p o ided endu ing inspi-
a ion, while he ma hema ical de ails, i al as hey a e, ha e aded. This expe ience unde sco ed a p o ound
u h: ma hema ics is he language we use o desc ibe ou in ui ion, bu he s o ies o scien i ic inqui y a e wha
igni e and sus ain he passion o esea ch. Wha seemed a was ed e o became he mos aluable pa o his
as ophysical educa ion.
73P essu ized wa e eac o s (PWR) a e he mos widesp ead ype, using high-p essu e wa e as bo h coolan
and mode a o ; he p ima y coolan loop ans e s hea o a seconda y loop o gene a e s eam. Boiling wa e
eac o s (BWR) a e simila o PWRs bu ope a e a a lowe p essu e, allowing he coolan o boil di ec ly in he
eac o co e; he s eam p oduced hen d i es he u bine di ec ly. P essu ized hea y wa e eac o s (PHWR),
such as he CANDU design, use hea y wa e (D2O) as a mode a o ; his design pe mi s he use o na u al
(unen iched) u anium as uel. Ad anced gas-cooled eac o s (AGR), de eloped in he UK, use ca bon dioxide as
a coolan and g aphi e as a mode a o .
74As many youngs e s aised in he 1980s, he au ho was p o oundly shocked by he disas e in Che nobyl.
I was no me ely a news i em; i was an e en ha seeped in o he daily li e, blu ing he lines be ween he
abs ac ea o he Cold Wa and a angible, in isible h ea . The e en un olded behind he I on Cu ain,
a ealm o sec ecy and ideological opposi ion, which made he ensuing cloud o nuclea ma e ial d i ing o e
Wes e n Eu ope all he mo e e i ying. I was a s a k, physical demons a ion ha he bo de s we ough o e ,
he ideologies we we e augh o ea , we e meaningless o he indi e en laws o physics and me eo ology.
75Th ee Mile Island, Pennsyl ania, USA, 1979. A pa ial co e mel down due o a combina ion o equipmen
ailu e and human e o . I esul ed in no di ec a ali ies bu had a p o ound impac on public opinion and
nuclea egula ion.
76Fukushima P e ec u e, Japan, 2011. A s a ion blackou igge ed by a sunami led o co e mel downs in
h ee eac o s and he elease o adioac i e ma e ial.
77The Che nobyl disas e (Che nobyl, Uk ainian SSR, 1986) was caused by a ca as ophic powe su ge du ing
a sa e y es , leading o an explosion, i e, and he elease o massi e amoun s o adioac i e ma e ial. In
he a e ma h, in e na ional humani a ian e o s b ough child en om a ec ed a eas o o he coun ies o
ecupe a ion. Re lec ing his, wo such child en egula ly s ayed wi h he au ho ’s amily in Belgium, expe iencing
pe iods o peace and espi e away om he con amina ed en i onmen .
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
64
da ase s, he yields a e no malized such ha
X
Z
Y(Z) = 1,(112)
so ha Y(Z) ep esen s he p obabili y o ob aining a agmen wi h cha ge Zin a gi en ission
eac ion.
To quan i y odd–e en s agge ing, one in oduces a ini e-di e ence ope a o ha highligh s
local al e na ions while supp essing smoo h backg ound ends. The s anda d choice is he
h ee-poin indica o
∆(3)
Zln Y(Z) = 1
2ln Y(Z+1) −2 ln Y(Z) + ln Y(Z−1),(113)
which ampli ies he pa i y-al e na ing s uc u e (e en s. odd Z) while canceling b oad sys em-
a ic a ia ions in Y(Z).
10.7 OES Model: Two Bessel Packe s wi h Localized Neck Bias
He e, we p esen he de e minis ic TSRT model ha ep oduces he odd–e en s agge ing (OES)
pa e n shown in Figu e 3 (p. 24), oge he wi h i s geome ic de i a ion and pa ame e in-
e p e a ion. All in e media e a ays, sc ip s, and i ed alues a e p o ided in Appendix R.2
(Lis ing 13 (p. 231)); expe imen al inpu s appea in Table 17 (p. 110).
Du ing he inal s ages o ission, he pa en nucleus elonga es un il a na ow scission neck
o ms be ween he wo eme ging agmen s. Mic oscopic simula ions (e.g., ime-dependen mean-
ield and ene gy-densi y- unc ional s udies [81,82]) consis en ly indica e ha his neck becomes
hin, s ongly cu ed, and dynamically uns able jus p io o sepa a ion. Wi hin TSRT, he neck
has a p ecise causal ole: i is a ini e, app oxima ely cylind ical co ido h ough which embling
cu a u e p opaga es subjec o bounda y-induced con inemen . As he neck adius app oaches
he cu a u e-sa u a ion scale, axial s anding pa e ns o he embling ield a e suppo ed and
cons ained by he local bounda y geome y. These neck-con ined pa e ns possess de ini e
e lec ion symme y ac oss he neck plane (e en o odd wi h espec o neck e lec ion), and ha
symme y con ols whe he he daugh e cu a u es close cohe en ly (e en symme y) o wi h
a phase in e sion (odd symme y). This symme y-con olled closu e modula es he agmen -
cha ge dis ibu ion in an al e na ing manne and is he geome ic o igin o OES in TSRT.
F om neck con inemen o he wo king exp ession. Unde cylind ical con inemen , he
ans e se s uc u e o a neck-con ined embling pa e n is well app oxima ed by Bessel unc-
ions Jn( adial sepa a ion in cylind ical coo dina es). A pu ely pe iodic Bessel con en , how-
e e , would gene a e long- ange oscilla ions in Z, inconsis en wi h he obse ed locali y o he
OES modula ion. In TSRT, ini e- ange cu a u e supp ession p o ides he equi ed localiza-
ion: neck-suppo ed modes a e damped away om he geome ic cen e o scission, and a small,
localized, pa i y-neu al cu a u e dep ession may a ise om neck aniso opy a he poin o up-
u e. These wo geome ic e ec s a e encoded as Gaussian en elopes mul iplying sho Bessel
packe s, plus a na ow, pa i y-neu al bias cen e ed nea he empi ically obse ed dip.
Pu ing hese elemen s oge he (explici ly shown in Appendix R.2) yields he OES obse -
able ∆(3)
Zln Y(Z)(Equa ion (113)) in he compac o m:
∆(3)
Zln Y(Z) = (−1)Z+φ
"exp−Z−Zc
L122
X
n=0
cnJn
k1(Z−Zc)
+ exp−Z−Zc
L222
X
n=0
dnJn
k2(Z−Zc)#
+Bexph−Z−Z0
L02i.
(114)
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025

65
which is algeb aically equi alen o Equa ion (315). The ac o s ha e di ec geome ic mean-
ings:
•Zcis he cha ge a he neck cen e (geome ic scission cen oid in Z).
•k1,2a e neck eigen-wa enumbe s se by cylind ical con inemen o he embling ield.
•L1,2a e ini e- ange damping leng hs gene a ed by cu a u e sa u a ion away om he
neck egion.
•cn, dna e symme y-de e mined modal weigh s o he sho Bessel packe s (n= 0,1,2
su ices o he obse ed bandwid h).
•φ∈ {0,1}is he neck- e lec ion symme y phase (e en/odd) ha se s he al e na ing sign
h ough (−1)Z+φ.
•Bis a small, pa i y-neu al bias (aniso opy) localized a (Z0, L0), ep esen ing a de e -
minis ic cu a u e dep ession a scission.
None o hese quan i ies is “empi ical” in o igin: cylind ical con inemen , ini e- ange damp-
ing, symme y weigh s, and he pa i y phase a ise om he same cu a u e geome y used
h oughou he pape ; hei nume ical alues a e cons ained by he obse ed scission con igu a-
ion and a e ixed once o he ull Z ange used in he compa ison (no pe -Z e uning). A mo e
de ailed de i a ion o he neck con inemen , damping, and bias e ms is p o ided in Appendix N
and connec s back o TSRT’s gene al cu a u e o malism [12,17].
Con as wi h s a is ical o pai ing-based pic u es. Con en ional explana ions o OES
o en in oke s ochas ic pai ing o le el-densi y e ec s. By con as , TSRT a ibu es OES o
de e minis ic neck-mode symme y unde cu a u e con inemen . The Bessel packe s encode
he local, geome y-bounded embling con en ; he pa i y ac o cap u es he neck- e lec ion
symme y; and he Gaussian en elopes (and he na ow bias) implemen ini e- ange cu a-
u e supp ession and local aniso opy. No quan um-p obabilis ic supe posi ion is assumed o
equi ed; “supe posi ion” he e deno es classical linea combina ion o neck-con ined cu a u e
pa e ns.
Calib a ion, usage, and esul s behind Figu e 3 (p. 24). Fo 235U(n h, ) (expe imen al
Y(Z)inpu s in Table 17 (p. 110)), we de e mine a single, sel -consis en pa ame e se o e
he ange Z= 35 ...61, hen compu e ∆(3)ln Y(Z) ia Equa ion (113) and o e lay he TSRT
p edic ion om Equa ion (114). The i ed alues used o gene a e Figu e 3 (p. 24) a e:
Zc= 50.016, k1= 0.1456, L1= 20.68, k2= 0.1884, L2= 30.00, φ = 0,
B=−0.1033, Z0= 41.466, L0= 2.911,
(c0, c1, c2) = (8.087,−0.600,0.200),(d0, d1, d2) = (−8.083,−0.0579,21.461).
These cons an s a e applied uni o mly ac oss he s a ed Zwindow wi h no local adjus men .
The esul ing cu e in Figu e 3 (p. 24) ep oduces he obse ed al e na ing pa e n, including he
dip nea Z≈40–45, while a oiding spu ious long- ange oscilla ions. The physical in e p e a ion
is anspa en : he na ow dep ession is explained by he localized, pa i y-neu al bias B < 0
(neck aniso opy), and he absence o long- ange inging ollows om he ini e damping leng hs
L1,2, which e lec he locali y o cu a u e co ela ions in he scission neck.
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66
Rep oducibili y. The en i e wo k low is documen ed end- o-end. The sc ip oesmake ig.m
(Lis ing 13 (p. 231)) loads he expe imen al a ays (Table 17 (p. 110)), compu es ∆(3)ln Y(Z)
ia Equa ion (113), e alua es Equa ion (114) on he same geome ic Zg id, and pe o ms a
wo-s age cons ained i (coa se seeding ollowed by bounded lsqnonlin), expo ing .pd /.eps
ou pu s. No in e media e o hidden iles a e equi ed, and no pe -Z e uning is pe o med. The
neck-model de ails and pa ame e oles a e summa ized in Appendix N; b oade ission-geome y
con ex appea s in Sec ion 10.
10.8 Geome ic Syn hesis o Fission
Fission in TSRT is a cohe en sequence o de e minis ic geome ic ansi ions go e ned by he
causal e olu ion o cu a u e imbalance and sa u a ion. The p ocess begins wi h he build–up
o su ace cu a u e s ess in hea y nuclei (Sec ion 10.1), quan i ied by he diagnos ic o Equa-
ion (99). As ∆Ksu ace app oaches he local sa u a ion capaci y Ksa (Table 7 (p. 16); me hods
in Appendix M), he con igu a ion eaches a de e minis ic h eshold. Once he h eshold is
c ossed, he mode bi u ca es (Sec ion 10.2) and wo causally sepa a ed cu a u e wells o m,
co esponding o he daugh e agmen s; he pos –saddle e olu ion om β‡ o scission p oceeds
on s ong–in e ac ion imescales se by Equa ion (24), i.e. ∼10−21–10−22 s.
The ene gy elease ollows om he geome ic cu a u e di e ence o malized in Equa-
ion (108) (Sec ion 10.3). Using he calib a ed cu a u e–supp ession cons an s (Appendix C.4)
and he e alua ion pipeline in Appendix M (Appendices M.1–M.2), he absolu e elease o
he mal–neu on–induced 235U(n, ) e alua es o
ETSRT
iss = 170.028 MeV,
as epo ed in Table 42 (p. 257) (gene a ed by Lis ing 28 (p. 257)). This ep oduces he empi ical
O(200 MeV) scale wi hou in oking s ochas ic “mass de ec s”: he ene gy is he de e minis ic
educ ion o in eg a ed cu a u e s ess be ween he compac pa en and he inal agmen s.
Induced ission en e s he same amewo k: neu on cap u e o ex e nal exci a ion inc eases
∆Ksu ace beyond Ksa , ini ia ing h eshold c ossing de e minis ically (Sec ion 10.4). Inne and
ou e ba ie loca ions and heigh s ollow om he s a iona i y analysis o Ede (β)(Sec ion 6,
Equa ions (27)–(28)); benchma k alues o 235U and 239Pu a e summa ized in Appendix B,
Table 13 (p. 109), wi h unce ain y p opaga ion speci ied in Appendix Y.3 and Appendix B.7.
Fine–s uc u e in agmen yields, no ably odd–e en s agge ing (OES), a ises om de e -
minis ic neck–geome y e ec s. The pa i y–locked neck–mode model (Sec ions 10.6–10.7) uses
wo sho Bessel packe s wi h ini e– ange damping and a localized, pa i y–neu al neck bias
[Equa ion (114)]. The single, ixed pa ame e se epo ed benea h Figu e 3 (p. 24) ( i ed once
o e Z= 35 ...61) ep oduces he obse ed al e na ing pa e n in 235U(n h, ), including he
localized dep ession nea Z≈40–45, while a oiding long– ange inging. All a ays and sc ip s
a e p o ided in Appendix R.2 (Lis ing 13 (p. 231)); expe imen al inpu s appea in Table 17
(p. 110).
Taken oge he , hese esul s show ha TSRT uni ies global obse ables ( h esholds, ba ie
sys ema ics, and ene gy elease) and ine–s uc u e e ec s (OES) wi hin a single de e minis ic
cu a u e geome y. Momen um and angula –momen um balances ollow om he co a ian
cons uc ion, and o al–ene gy conse a ion is en o ced by he cu a u e–mass equi alence used
in Equa ion (108) (Sec ion 11.2).
This geome ic syn hesis also p o ides a na u al b idge o usion in Sec ion 12: whe eas
ission co esponds o cu a u e bi u ca ion once sa u a ion is exceeded, usion co esponds o
cu a u e consolida ion by cons uc i e o e lap o embling domains. The de e minis ic accoun
o en ance–channel ansmission and S– ac o ends is de eloped in Sec ion 7.
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67
11 Mass–Ene gy Con e sion in Fission and he TSRT Explana-
ion
Be o e add essing he con en ional no ion o mass de ec , i is use ul o cla i y how he ela ion
E=mc2, o , mo e p ecisely in TSRT, he geome ic iden i y be ween cu a u e ene gy and
ine ial mass, en e s he p esen manusc ip . Al hough he o mal de i a ion o cu a u e–mass
equi alence appea s la e in Sec ion 11.3, he equi alence has al eady been used implici ly in
se e al ea lie sec ions, whe e i unc ions as an ope a ional ool o compu ing ission ene ge ics.
The i s such usage occu s in he ission–ene gy balance o Sec ion 10.3, whe e he eleased
ene gy is exp essed h ough he cu a u e di e ence be ween he pa en mode and i s geodesic
agmen s [Equa ion (108)]. The e he ac o c2en e s no as an ex e nally imposed con e sion
ule, bu as he na u al scale ela ing p ope - ime cu a u e ac ion o ine ial esponse in TSRT.
As no ed in he commen a y ollowing Equa ion (108), his usage is consis en wi h, and la e
jus i ied by, he o mal cu a u e–mass ela ion de eloped in Sec ions III B–C o Re e ence [12].
The same geome ic iden i y unde lies he h eshold condi ions and ene gy accoun ing in
induced ission (Sec ion 10.4), whe e cu a u e sa u a ion a he su ace igge s de e minis ic
clea age, and in he scission analysis o Appendices M–M.2, whe e he eo ganiza ion o em-
bling cu a u e de e mines agmen kine ic ene gies and p omp emissions. I also o ms he
concep ual b idge o he usion analysis in Sec ion 12, in which cu a u e consolida ion a he
han bi u ca ion p oduces he associa ed ene ge ic signa u es.
The pu pose o he p esen sec ion is he e o e wo old. Fi s , i consolida es hese ea lie uses
o cu a u e–mass equi alence in o a uni ied TSRT explana ion o nuclea ene gy elease. Second,
i con as s his geome ic in e p e a ion wi h he s anda d “mass de ec ” pic u e, p epa ing
he g ound o he o mal de i a ion ha ollows in Sec ion 11.3. Only wi h his dis inc ion
in place can he TSRT iewpoin be clea ly unde s ood: nuclea ene gy does no a ise om
he disappea ance o mass bu om he causal econ igu a ion o embling cu a u e, wi h c2
eme ging as he in insic geome ic scale linking p ope - ime cu a u e ac ion o ine ial ene gy.
11.1 Con en ional View o Mass De ec
In s anda d nuclea physics [19,83], he ene gy eleased du ing ission is a ibu ed o a “mass
de ec ,” e alua ed h ough he mass–ene gy equi alence p inciple [5]:
∆m c2=mnucleus −m agmen s −mneu onsc2,(115)
whe e he educ ion in es mass is in e p e ed as he nuclea binding ene gy, o en modeled
phenomenologically h ough he liquid–d op o mula [37]. This iewpoin is empi ically success-
ul [19,84]: he obse ed ∼200 MeV eleased in hea y-nucleus ission is nume ically consis en
wi h he di e ence be ween abula ed masses be o e and a e he e en .
Wha he con en ional pic u e p o ides, howe e , is p ima ily an accoun ing iden i y. I
s a es how much ene gy appea s bu no why he nucleus eleases ha ene gy o by wha in e nal
mechanism he es mass is educed. E en in quan um-mechanical o quan um– ield– heo e ic
ea men s, he ela ion E=mc2is inse ed as a ounda ional p inciple a he han de i ed
om a causal dynamical mechanism: mass change is acknowledged h ough mass ables, and
he excess is assigned o kine ic and adia i e channels, bu he unde lying p ocess by which he
nuclea con igu a ion eo ganizes o con e in e nal s ess in o ou going ene gy is no speci ied
a a geome ic o mechanis ic le el.
TSRT app oaches his ques ion om a di e en di ec ion. Ra he han in e p e ing nu-
clea ene ge ics h ough mass de ici s, TSRT a ibu es he eleased ene gy o a de e minis ic
econ igu a ion o embling-space ime cu a u e. In his geome ic amewo k, he dec ease
o e ec i e ine ial mass is no a p imi i e assump ion bu he ine i able consequence o how
cu a u e densi y edis ibu es when a single, high-s ess cu a u e con igu a ion sepa a es in o
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
68
wo lowe -s ess agmen s. The cu a u e–mass equi alence used in ission (and o malized in
Sec ion 11.3) hus p o ides a causal explana ion o he ene gy libe a ed in Equa ion (115),
eplacing he bookkeeping in e p e a ion o mass de ec wi h a geome ic mechanism oo ed in
TSRT dynamics.
11.2 TSRT In e p e a ion: Cu a u e Redis ibu ion
In TSRT, mass is no a p imi i e p ope y bu he mani es a ion o localized embling cu a u e
modes bound by causal geodesics (see Sec ion 4 and Re e ences [12,14,16]). The appa en “loss
o mass” in ission co esponds, in his amewo k, o a edis ibu ion o space ime cu a u e: a
highly sa u a ed bound s a e elaxes in o agmen s whose embling cu a u e densi y is lowe .
The geome ic ene gy con en o a nucleus is de ined by he con ac ion o he embling
cu a u e enso wi h he p ope – ime geodesic low:
E∝ZV
Kµν uµuνdV, (116)
whe e Kµν is he embling cu a u e enso , uµ he uni ou – eloci y along causal geodesics,
and V he nuclea olume.85 Equa ion (116) is he same as Equa ion (123) in he con ex o
nuclea usion.
Al hough Equa ion (116) is w i en wi h a p opo ionali y sign, TSRT ixes he p opo ion-
ali y cons an uniquely h ough he cu a u e–mass equi alence de i ed in Sec ions III B–C
o Re e ence [12]. The e, he p ope – ime a ia ion o he embling ac ion iden i ies he es
ene gy o a localized cu a u e con igu a ion as
E=c2ZV
KµνuµuνdV, (117)
so ha he amilia ac o c2eme ges no om dimensional analysis o empi ical impo , bu
om he geome ic iden i y linking cu a u e p ojec ion, p ope – ime ac ion, and ine ial e-
sponse. Thus, whene e one compu es a di e ence o cu a u e–ene gy in eg als be ween wo
con igu a ions, he con e sion o obse able ene gy necessa ily includes he mul iplying ac o
c2. The appea ance o c2in he ission con ex he e o e ollows di ec ly and ine i ably om he
ounda ional TSRT cu a u e–ene gy ela ion, and is no an addi ional pos ula e in oduced a
his s age.
Du ing ission, he ele an obse able is he change in his cu a u e–ene gy in eg al be ween
he ini ial nucleus and i s daugh e agmen s, equi alen o Equa ion (108):
∆E= ∆ZV
KµνuµuνdV c2,(118)
whe e ∆indica es sub ac ion o he inal ( agmen ed) con igu a ion om he ini ial bound
con igu a ion. Thus he eleased ission ene gy is no c ea ed ex nihilo bu is he geome ic
mani es a ion o educed cu a u e s ess once he nucleus econ igu es in o less sa u a ed modes.
This o mula ion di ec ly mo i a es he de ini ion o he e ec i e mass o a bound sys em as
me ∝ZV
KµνuµuνdV, (119)
85Fo eade s amilia wi h classical gene al ela i i y, he s uc u e o Equa ion (116) is delibe a ely eminiscen
o he s anda d ene gy de ini ion in cu ed space ime. In GR, he local ene gy densi y measu ed by an obse e
wi h ou – eloci y uµis ob ained by con ac ing he s ess–ene gy enso wi h he obse e ’s mo ion: E∝
RVTµν uµuνdV . In TSRT, he embling cu a u e enso Kµν eplaces Tµν as he undamen al quan i y: ene gy
is no in oduced as a sepa a e sou ce e m, bu is iden i ied di ec ly wi h cu a u e luc ua ions o space ime
i sel . This geome ic ein e p e a ion, de eloped in de ail in [12], allows nuclea binding and elease o ene gy
o be unde s ood wi hou in oking quan um pos ula es, as di ec mani es a ions o cu a u e edis ibu ion.
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
69
a ela ion de eloped u he in Sec ion 13. He e, he p opo ionali y cons an is ixed by calib a-
ion o a omic binding scales (Appendix H), ensu ing consis ency ac oss a omic, nuclea , and
cosmological domains.86
Fo explici nume ical cons uc ion o Kµν om he unde lying embling enso ξµν, in-
cluding s ep–by–s ep de i a ions and MATLAB implemen a ions, see Appendix H, especially
Appendix H.1 and he ou ines in Appendix H.3.
11.3 Geome ic Equi alence o E=mc2
He e, we highligh he impo ance o wha is explained in Sec ion 11.2.
One o he mos celeb a ed esul s in physics is he mass–ene gy ela ion E=mc2[5]. In
s anda d ea men s his o mula is in oduced empi ically, ei he h ough kinema ic a gumen s
o by appealing o he ene gy balance o nuclea eac ions. Wi hin TSRT, howe e , he ela ion
a ises di ec ly and necessa ily om he embling cu a u e amewo k, and is he e o e no longe
a pos ula e bu an eme gen iden i y.
As shown in Equa ion (118), he ene gy o a bound con igu a ion is p opo ional o he
in eg al o embling cu a u e p ojec ed along he p ope - ime geodesic low. This immedia ely
de ines an e ec i e ine ial mass as in Equa ion (119). Thus he empi ical success o E= ∆mc2
ollows because ission and usion co espond o cu a u e edis ibu ion, which changes he
e ec i e ine ial mass o he sys em.
The c ucial concep ual ad ance is ha mass is no undamen al in TSRT: i is a de i ed p op-
e y o space ime cu a u e unde embling dynamics. This pe spec i e ex ends beyond nuclea
physics. The same cu a u e–mass co espondence unde lies TSRT explana ions o a omic bind-
ing scales [14], black–hole en opy and Hawking emission [16], and he Planck pos ula e o ligh
quan a [13]. In e e y case, he ine ial pa ame e mis e ealed as sho hand o a embling
cu a u e in eg al, and E=mc2as a geome ic iden i y a he han an unexplained axiom.
By embedding Eins ein’s o mula wi hin a uni ied geome ic amewo k, TSRT bo h p e-
se es i s uni e sali y and emo es i s a bi a iness: he iconic E=mc2is no longe a s a ing
assump ion bu he ine i able ou come o causal embling geome y.
11.4 Dis ibu ion o Released Ene gy
The cu a u e educ ion quan i ied in Equa ion (118) does no appea in a single channel bu is
edis ibu ed among se e al physically dis inc ca ie s. The ela i e con ibu ions can be e al-
ua ed di ec ly om he TSRT cu a u e in eg als and a e summa ised in Table 42 (p. 257), wi h
he unde lying nume ical ou ines p o ided in Appendix Y.15. Fo 235U(n h, ), he dis ibu ion
ob ained om TSRT ma ches he empi ical ∼200 MeV elease o wi hin nume ical p ecision.
•Kine ic ene gy o he hea y agmen s. TSRT yields a agmen kine ic ene gy o
E(TSRT)
ag = 169.8MeV,
compu ed om he pos -scission cu a u e econ igu a ion and ecoil geome y in Equa-
ion (108) (see also Figu e 42 (p. 257)). This accoun s o oughly 84% o he o al elease,
86Equa ion (119) is he local, domain-in eg a ed o m o he mass–cu a u e ela ion in oduced in he oun-
da ional TSRT pape [12]. The e, he same p inciple appea s h ough he cu a u e-ene gy unc ional
Ecu =1
8πGTSRT ZVRµν uµuνdV, (120)
and he iden i ica ion Ecu =mc2es ablishes he co espondence be ween p ojec ed cu a u e densi y (Rµν uµuν)
and ine ial mass. The p esen exp ession (119) adop s he symbol Kµν o he embling cu a u e enso , which
eplaces Rµν in bound sys ems whe e cu a u e oscilla ions a e locally sa u a ed. Thus, while he no a ion di e s,
he unde lying geome ic s a emen is ma hema ically and physically equi alen o ha o Sec ions III B and
IV A in Re e ence [12], whe e he cu a u e–mass ela ion is i s de i ed.
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70
in close ag eemen wi h he expe imen al 170.0±1.0MeV (Table 42 (p. 257)). The nu-
me ical alue a ises om he cu a u e elaxa ion occu ing when he neck adius d ops
below he sa u a ion scale Rsa discussed in Sec ion 10.1.
•P omp neu ons. TSRT p edic s
¯ν(TSRT)
n= 2.43,
as ob ained om he de e minis ic de achmen condi ion in he geodesic agmen a ion
ou ine neu on_de ach.m (Appendix Y.15). The co esponding ene gy,
E(TSRT)
n= 11.9MeV,
ma ches he empi ical ∼10–12 MeV ange and a ises om cu a u e exceeding he local
de achmen h eshold κneck a ound he scission poin (Sec ion 10.2). This p ocess is no
s ochas ic bu e lec s local sa u a ion o embling-mode cu a u e in he neck egion
(Appendix M, Sec ion 10.2).
•P omp γ ays. A e agmen s sepa a e, TSRT yields a p omp cu a u e-wa e elease
o
E(TSRT)
γ,p omp = 7.1MeV,
co esponding o oscilla o y cu a u e wa es p opaga ing along he agmen su aces as
hey elax owa d hei local embling equilib ia. This alue is ob ained ia he mode-
p ojec ion ou ine gamma_p omp .m in Appendix Y.15, and ma ches he empi ical 6–8 MeV
band.
•Delayed channels. Daugh e agmen s ypically eme ge cu a u e-misaligned ela i e o
hei local s abili y alley (Sec ion 9.7). TSRT he e o e p edic s delayed weak elaxa ion
con ibu ing
E(TSRT)
delayed = 9.3MeV,
in he o m o β±decays and delayed γ ays. The co esponding p ope - ime elaxa ion is
compu ed om Equa ion (61), wi h hal -li es ep oduced by he global cons an Cλ.87
TSRT he e o e yields he o al ene gy balance
∆E o =E(TSRT)
ag +E(TSRT)
n+E(TSRT)
γ,p omp +E(TSRT)
delayed ,(121)
which nume ically e alua es o
∆E(TSRT)
o = 198.1MeV,
in ag eemen wi h he expe imen al (200±2) MeV elease. This ma ch is no imposed by pa am-
e e adjus men : each e m ollows de e minis ically om he buil -in cu a u e edis ibu ion
and geodesic agmen a ion ules, all linked di ec ly o he cu a u e in eg als o Sec ion 10.3.
The nea -cons ancy o agmen kine ic ene gy ac oss issile sys ems (Table 4 (p. 12)) e lec s
he geome ic s abili y o he scission cu a u e landscape, whe eas neu on mul iplici y depends
p ima ily on he neck cu a u e scale κneck and he sa u a ion ampli ude Asa (Appendix M). The
explici ole o κneck and he o e lap adius Rcin de e mining he neu on de achmen h eshold
can be inspec ed in Sec ion 10.2 and in he nume ical diagnos ic ou pu o ission_ene gy.m.
All a ays, cons an s, and MATLAB ou ines equi ed o ep oduce he alues quo ed abo e
a e p o ided in Appendix M and Appendix Y.15. This ensu es ha he TSRT ene gy dis ibu ion
is ully ep oducible and g ounded di ec ly in he cu a u e dynamics o embling space ime.
87In TSRT he decay law is w i en in e ms o p ope ime τ, he in a ian along he geodesic o he embling
con igu a ion. Labo a o y ime di e s by a cu a u e-dependen edshi ac o , bu in nuclea en i onmen s
he co ec ion is negligible (| −τ|/ < 10−11).
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11.5 Compa ison wi h Expe imen
The quan i a i e esul s ob ained in Sec ion 11.4 allow a di ec , e m-by- e m compa ison be-
ween TSRT p edic ions and measu ed ission ene ge ics. Fo he mal–neu on-induced 235U(n h, ),
he TSRT cu a u e–ene gy e alua ion yields a o al elease o
∆E(TSRT)
o = 198.1 MeV,
which ag ees wi h he canonical expe imen al alue o (200±2) MeV es ablished by calo ime ic
and kinema ic s udies [69,75,84,85]. A de ailed b eakdown o his ag eemen appea s in Table 42
(p. 257), while he geome ic o igin o each componen is illus a ed in Figu e 42 (p. 257).
F agmen kine ic ene gy. TSRT p edic s a agmen kine ic elease o
E(TSRT)
ag = 169.8 MeV,
ob ained om he cu a u e elaxa ion and ecoil geome y in Equa ion (108). This alue lies
wi hin 1% o he expe imen al 170.0±1.0MeV and cap u es he nea -cons ancy o agmen
kine ic ene gy ac oss issile nuclides (see also he c oss-iso ope compa ison in Table 4 (p. 12)).
No empi ical mass-de ec assump ion is equi ed; he numbe ollows di ec ly om he scission
cu a u e geome y calib a ed in Sec ion 9.
P omp neu ons. The TSRT de achmen condi ion in he neck (Sec ion 10.2), e alua ed
wi h he cu a u e h eshold κneck and o e lap adius Rc, yields an a e age mul iplici y
¯ν(TSRT)
n= 2.43,
wi h a o al neu on kine ic con ibu ion o
E(TSRT)
n= 11.9 MeV.
Bo h alues all wi hin he empi ical anges ¯ν(exp)
n= 2.42–2.49 and E(exp)
n≈10–12 MeV [19,
86]. These quan i ies a e p oduced by he de e minis ic geodesic- agmen a ion ou ines in
Appendix Y.15.
P omp and delayed γ ays. The pos -scission cu a u e-wa e p ojec ion yields
E(TSRT)
γ,p omp = 7.1 MeV,
ma ching he empi ical 6–8 MeV band. Subsequen cu a u e- ebalancing in neu on- ich ag-
men s leads o
E(TSRT)
delayed = 9.3 MeV,
consis en wi h he cumula i e delayed emission in e ed om ission-p oduc decay chains.
These con ibu ions a e ep oduced by he weak-in e ac ion elaxa ion a es o Sec ion 9.7 and
compu ed explici ly in Appendix M.
Global consis ency ac oss obse ables. The ag eemen be ween TSRT and expe imen is
no limi ed o he o al elease. Tables 42 (p. 257) and 4 show ha TSRT simul aneously ep o-
duces he agmen kine ic peak nea 170 MeV, he neu on mul iplici y and ene gy spec um,
he p omp and delayed γcon ibu ions, and he sys em- o-sys em s abili y o o al ene ge ic
ou pu .
Because all channels o igina e om he same cu a u e-ene gy unc ional in Equa ion (118),
hei in e nal consis ency is au oma ic: no pe -channel uning o empi ical mass de ec s a e
in oked.
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In e p e a ion. Expe imen al ission ene ge ics a e he e o e eco e ed in TSRT as a di ec
consequence o de e minis ic cu a u e edis ibu ion. The ini ial compac nucleus possesses
a high embling-cu a u e densi y, which elaxes in o wo la ge - olume agmen s, de ached
geodesic modes (neu ons), and cu a u e wa es (p omp and delayed γ’s). The classical alue
o ∼200 MeV eme ges no as a phenomenological cons an bu om he geome y o he scission
ansi ion encoded by he TSRT cu a u e enso s. This places he empi ical esul s o Re s. [19,
20,84,85] wi hin a single causal, ela i is ically consis en amewo k.
The explici nume ical e i ica ion, ep oducible om he ou ines in Appendix Y.15, es-
ablishes ha he mass–ene gy balance o ission is ully accoun ed o by he cu a u e-ene gy
equi alence de eloped in Sec ions 11.2 and 11.4.
11.6 Geome ic Signi icance o Mass–Ene gy Con e sion
The e inemen s de eloped in his sec ion show ha mass–ene gy con e sion in TSRT is no an
imposed p inciple, no a phenomenological bookkeeping de ice, bu he di ec mani es a ion o
how embling cu a u e eo ganizes unde causal e olu ion. Wha is adi ionally called a “mass
de ec ” is ein e p e ed as a educ ion o in eg a ed cu a u e densi y when a highly sa u a ed
nuclea mode bi u ca es in o wo la ge - olume con igu a ions wi h lowe cu a u e a iance.
This ein e p e a ion is ancho ed by he cu a u e–ene gy iden i y (Equa ion (116)) and i s
p ope - ime o m (Equa ion (118)), which oge he imply ha he e ec i e ine ial mass o any
bound sys em is p opo ional o i s in eg a ed embling cu a u e (Equa ion (119)). Thus he
amilia ela ion E=mc2appea s as a na u al co olla y o he TSRT cu a u e unc ional a he
han an ex e nal axiom: he ac o c2a ises au oma ically om he mapping be ween p ope
ime and labo a o y ime in he TSRT ac ion (Sec ions 11.2 and 10.3).
The quan i a i e esul s s eng hen his geome ic ein e p e a ion. E alua ing ∆E h ough
Equa ion (118) yields
∆E(TSRT)
o = 198.1 MeV,
ma ching he es ablished expe imen al elease o (200 ±2) MeV o he mal–neu on-induced
235U(n h, ). Each pa i ioned con ibu ion, i.e., agmen kine ic ene gy (169.8MeV), p omp
neu on emission (11.9MeV), p omp γ ays (7.1MeV), and delayed cu a u e ebalancing
(9.3MeV), ag ees wi h measu ed alues wi hin unce ain ies (Table 42 (p. 257), Figu e 42
(p. 257)). All channels ollow om a single cu a u e unc ional wi hou in oking empi ical
mass de ec s, pai ing co ec ions, o pe -iso ope adjus men s.
The same geome ic mechanism accoun s o ine obse ables. The pa i y-locked neck-
mode s uc u e (Sec ions 10.6–10.7) ep oduces he al e na ing odd–e en s agge ing pa e n in
235U(n h, ) wi h a single global pa ame e se ; neu on mul iplici ies eme ge om de e minis ic
cu a u e de achmen in he neck (Sec ion 10.2); and he ene gy balance ac oss issile nuclides
emains cons an because he cu a u e-sa u a ion scale is uni e sal.
Taken oge he , hese esul s demons a e ha TSRT p o ides a uni ied, causal, and quan i-
a i ely alida ed explana ion o mass–ene gy con e sion in nuclea p ocesses. Ene gy elease in
ission is no a mys e ious ans o ma ion o “missing mass,” bu he p edic able consequence o
cu a u e edis ibu ion when a bound embling-space ime con igu a ion eo ganizes in o less
sa u a ed modes. In his sense, one o he mos iconic empi ical ela ions in physics is ele a ed o
a geome ic s a emen abou space ime i sel : mass and ene gy a e wo ep esen a ions o cu a-
u e densi y, and nuclea ans o ma ions e eal his iden i y in i s mos di ec and measu able
o m.
12 Nuclea Fusion in TSRT
Suppo ing appendices: Appendix O (c i e ion), Appendix P (e alua ion), Appendix Q ( alida-
ion).
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73
Co e MATLAB: s usionene gy.m, s s ac o .m.
This sec ion de elops he de e minis ic pic u e o usion in TSRT; how causal o e lap o em-
bling nuclea modes se s usion onse , selec s he emi ed p oduc s, and o ganizes as ophysical
chains.
The quan i a i e machine y o ba ie s and unneling88 is gi en in Sec ion 7 (Equa ion (40)
o Ve ( ); Equa ion (50) and he channel-speci ic o ms 52–53; and he da a con on a ions in
Figu e 2 (p. 12) o sc eening89 and Figu e 6 (p. 86) o hind ance90). He e we use hose esul s
o: (i) s a e he causal onse c i e ion in con igu a ion space, (ii) connec channel sys ema ics
(D–T,91 D–D,92 hea y-ion93) o TSRT geome y, (iii) explain en i onmen -dependen e ec s
(sc eening, dynamic dephasing), and (i ) assemble hese elemen s in o s ella pa hways (pp
chain,94 CNO cycle95) and labo a o y diagnos ics.
12.1 Con en ional Desc ip ion
The con en ional desc ip ion o nuclea usion builds on he concep o a Coulomb ba ie : wo
posi i ely cha ged nuclei mus app oach closely enough o he sho - ange s ong in e ac ion o
bind hem [87]. Classically, his equi es ha hei ela i e kine ic ene gy exceed he Coulomb
epulsion; in quan um-mechanical e ms, ba ie pene a ion is explained by unneling, as i s
o malized by Gamow [51] and applied o usion by A kinson and Hou e mans [88]. Once con ac
is achie ed, he esul ing composi e nucleus is mo e igh ly bound, and he mass di e ence
be ween ini ial and inal s a es is classically in e p e ed as he eleased ene gy, acco ding o
Eins ein’s mass–ene gy equi alence:
∆E=mini ial −m inalc2.(122)
His o ically, he idea ha s a s shine by mass–ene gy con e sion was p oposed by Edding on
in 1920 [89], sho ly a e Eins ein’s heo y was es ablished. Howe e , i was Be he’s landma k
88In quan um mechanics (QM), a ba ie is a po en ial ene gy p o ile V( ) ha classically con ines a pa icle o
a speci ic egion (e.g., he Coulomb epulsion be ween nuclei). Tunneling is he quin essen ial QM phenomenon
whe eby a pa icle has a ini e p obabili y o pene a ing such a ba ie , e en when i s kine ic ene gy is less han
he ba ie ’s maximum heigh . This non-classical ansmission is desc ibed by he wa e unc ion’s exponen ial
decay wi hin he classically o bidden egion.
89Sc eening (in nuclea usion): The enhancemen o he usion p obabili y in a plasma due o he su ounding
elec ons and nuclei, which pa ially shield he Coulomb epulsion be ween he using nuclei, e ec i ely lowe ing
he po en ial ba ie .
90Hind ance (in nuclea usion): A phenomenon, o en obse ed in hea y-ion usion a sub-ba ie ene gies,
whe e he usion c oss-sec ion is signi ican ly lowe han s anda d model p edic ions. I is ypically a ibu ed o
he in e nal s uc u e o he colliding nuclei and he dynamics o he usion p ocess.
91Deu e ium–T i ium (D–T): The usion eac ion be ween a deu e ium nucleus (2H) and a i ium nucleus (3H),
yielding a helium-4 nucleus and a neu on. I has he la ges c oss-sec ion a low ene gies and is he p ima y
eac ion used in magne ic con inemen usion esea ch.
92Deu e ium–Deu e ium (D–D): The usion eac ion be ween wo deu e ium nuclei. I has wo nea ly equip ob-
able b anches: p oducing a i ium nucleus and a p o on, o a helium-3 nucleus and a neu on.
93Hea y-ion usion: A usion p ocess in ol ing nuclei hea ie han helium, such as 12C + 12C. These eac ions
a e cha ac e ized by highe Coulomb ba ie s and a e c i ical o unde s anding nucleosyn hesis in massi e s a s
and supe no ae.
94P o on–p o on (pp) chain: The dominan se o usion eac ions by which s a s on he main sequence, like
he Sun, con e hyd ogen in o helium. I is ini ia ed by he weak- o ce-media ed usion o wo p o ons.
95CNO s ands o he ca bon–ni ogen–oxygen cycle, a ca aly ic sequence in which C, N, and O iso opes
enable hyd ogen bu ning by con e ing ou p o ons in o 4He (wi h emi ed γ ays, posi ons, and neu inos), he
ca alys s being es o ed a he end o he cycle. I is ele an he e because, unlike he p o on–p o on chain ha
powe s low-mass s a s, he CNO cycle domina es ene gy gene a ion in ho e , mo e massi e s a s due o i s much
s eepe empe a u e sensi i i y, making i cen al o con en ional s ella - usion sys ema ics and he associa ed
ba ie / S- ac o discussions ha TSRT ein e p e s. The CNO cycle is he dominan ene gy sou ce in s a s
mo e massi e han he Sun.
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80
Table 9: Benchma k compa ison o D–T usion. TSRT alues a e di ec ou pu s o he calib a ed
cu a u e in eg als.
Quan i y Classical (mass) TSRT (cu a u e) Expe imen
Q(MeV) 17.590 17.5903 17.6±0.1
En(MeV) 14.072 14.072 14.1±0.1
Eα(MeV) 3.518 3.518 3.5±0.1
12.6 S ella Fusion Chains
The geome ic pic u e o usion in TSRT acqui es pa icula signi icance in as ophysics, whe e
nuclea bu ning d i es s ella s uc u e and cosmic e olu ion.103 His o ically, he ecogni ion
ha usion powe s s a s was a landma k in 20 h-cen u y physics. In he 1920s Edding on
sugges ed ha he eno mous luminosi y o he Sun104 could only be explained by he con e sion
o mass in o ene gy [89]. Be he’s seminal wo k in 1938–39 [90,104] es ablished he p o on–p o on
(pp) chain and he ca bon–ni ogen–oxygen (CNO) cycle as he dominan ene gy-p oducing
eac ions in s ella in e io s. Since hen, usion has been cen al no only o s ella as ophysics
[105] bu also o cosmology, nucleosyn hesis [106], and he chemical e olu ion o galaxies [107].
In con en ional models, hese s ella usion chains a e desc ibed p obabilis ically h ough
ba ie pene a ion [88]: nuclea eac ion a es a e ob ained by olding quan um unneling p ob-
abili ies wi h Maxwell–Bol zmann eloci y dis ibu ions. This s a is ical amewo k has been
ex emely success ul a ep oducing s ella ene gy gene a ion a es, ye i p o ides no deepe
de e minis ic mechanism o why usion occu s, o why eac ion chains p oceed in exac ly he
obse ed o de [96,97].
TSRT e ames his pic u e in geome ic e ms. Each s ep o a s ella usion chain is a de-
e minis ic eo ganiza ion o embling cu a u e modes in o con igu a ions o highe symme y
and lowe cu a u e s ess. In he pp chain, wo p o ons app oach; hei o e lap cu a u e is
uns able unde pu e elec omagne ic epulsion, bu a weak econ igu a ion p oduces a deu e on
eigenmode wi h cu a u e supp essed by neu on–p o on balance. Subsequen usions (D+p→
3He, 3He+3He →4He+2p) co espond o u he cu a u e condensa ion in o mo e symme ic
bound modes, eleasing cu a u e ene gy as pho ons and neu inos. In he CNO cycle, ca bon,
ni ogen, and oxygen nuclei ac as ca aly ic cu a u e empla es: p o ons sequen ially use in o
hea ie eigenmodes un il 4He is eleased, and he ca alys cu a u e con igu a ion is es o ed.
In TSRT his ca aly ic ole is unde s ood as a esonance condi ion: he embling ields o C,
N, and O nuclei p o ide s able geome ic sca olds ha guide p o on cap u e in o symme ic
modes.
Neu inos in his amewo k a e no p obabilis ic byp oduc s bu de e minis ic geodesic ag-
103I was du ing he s ange s illness o he pandemic (2020), when he wo ld e ea ed and he hea ens, eed
om he eil o pollu ion, e ealed a deepe , mo e cons an blue by day and a s a lingly clea window o he
cosmos by nigh , ha hese ideas ma u ed wi h new o ce and by ecalling old uni e si y cou ses ( om when
he was an MS s uden in As ophysics) om p o esso s a he Ca holic Uni e si y o Leu en, such as C. Ae s,
C. Waelkens, P. Smeye s, P. Van Duppen and J. Coussemen . In ha ex ended silence, punc ua ed by global
unce ain y—whe e some hoa ded goods, o he s sa ed li es, and many aced p o ound so ow— he au ho ound
a singula peace. Shel e ed wi h amily a home in Belgium, and connec ed o s uden s, om a dis ance, in
F ance and in he Uni ed S a es, he long-held in ui ions and sobe con empla ions on he na u e o he uni e se
began o consolida e. They yea ned o se e in desc ibing he e y p ocesses by which dis an s a s o ge he
elemen s o li e, and in whose ligh we agile beings, ou sel es g appling wi h exis ence, ul ima ely ind ou
ma e ial o igin.
104The au ho i idly ecalls his a he , du ing his you h in Sellewie, Dee lijk (Belgium), desc ibing he Sun as
a "g ea i e." Those we e imes illed wi h wonde , discussing snow, ain, clouds, he Ea h’s magma, and u u e
echnologies like hyd ogen ca s and la sc eens long be o e hey eached he ma ke . While nuclea physics was
hen beyond his g asp, he ligh om ha s ella usion, i.e., he ans o ma ion o hyd ogen in o helium in he
Sun’s co e, made li e i sel so enjoyable ha i ueled a desi e o ques ion he e y co e o na u e, a pu sui he
would la e ecognize as physics.
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81
men s emi ed whene e local cu a u e ebalancing equi es a weak-mode adjus men . Thei
obse ed luxes om he Sun (e.g. Homes ake,105 SNO,106 and Supe -Kamiokande107) a e he e-
o e di ec expe imen al signa u es o de e minis ic TSRT econ igu a ion dynamics.
Thus, s ella ene gy p oduc ion appea s no as he s a is ical ou come o ba ie pene a ion,
bu as a causal sequence o cu a u e condensa ions, each s ep go e ned by local geome ic
symme y condi ions. This embeds as ophysical usion chains na u ally wi hin he b oade
TSRT amewo k al eady es ablished o undamen al pa icles [17] and cosmology [16]: he
same embling space ime geome y ha gene a ed pa icles in he ea ly uni e se now go e ns
hei de e minis ic ecombina ion in s a s. Hence, TSRT uni ies nuclea as ophysics wi h he
o igin o ma e and he expansion o he cosmos.108
12.7 P ope -Time Ac ion and he Low-Ene gy Exponen ial
The p eceding subsec ions es ablished he geome ic mechanism o usion, demons a ed TSRT’s
de e mina ion o he eleased ene gy ∆ETSRT, and con i med i s ag eemen wi h expe imen al
kinema ics h ough explici nume ical e alua ion (including he D–T benchma k). We now
u n o a dis inc and complemen a y ques ion: how TSRT ep oduces he obse ed ene gy
dependence o usion p obabili ies a sub-ba ie ene gies.
In his subsec ion, we show ha he cha ac e is ic exponen ial supp ession o usion c oss
sec ions [12], adi ionally a ibu ed o p obabilis ic quan um unneling, a ises de e minis i-
cally om he p ope - ime ac ion associa ed wi h he o e lap o app oaching embling geodesic
con igu a ions. This p o ides he geome ic o igin o he empi ical as ophysical S- ac o and
links low-ene gy usion beha iou di ec ly o he TSRT p ope - ime o mula ion.
In TSRT, he ansi ion be ween wo quasi-s able cu a u e con igu a ions is go e ned by
he p ope - ime ac ion inc emen along he connec ing geodesic cong uence, which con ols he
causal a e a which cu a u e o e lap can eo ganize in o he used con igu a ion. Le
∆S=Zτ
τipgµν ˙xµ˙xνdτ (137)
deno e he ac ion inc emen o he ela i e ajec o y ac oss he e ec i e elec omagne ic cu -
a u e egion. This is he same geome ic in a ian in oduced in he ounda ional TSRT pa-
pe [12, Sec ion 3.2], whe e i de ines he p ope - ime measu e o causal de o ma ion be ween
neighbo ing embling s a es. He e, he same p inciple is applied o he e ec i e wo-body
cu a u e domain o med du ing nuclea app oach.
Because he embling ield oscilla es on ex emely sho scales, he di ec ly compu ed ∆S
mus be coa se–g ained o e apid sub–geodesic luc ua ions. This a e aging yields an eme -
gen ac ion quan um ~TSRT, in oduced in [12, Sec ion 5.1] and abula ed in Appendix E. In
labo a o y, weak–cu a u e condi ions (whe e he in a ian embling cu a u e is small on he
105The Homes ake expe imen , led by Raymond Da is J ., was a adiochemical de ec o loca ed in he Homes ake
Gold Mine, Sou h Dako a. I used a la ge ank o pe chlo oe hylene o de ec elec on neu inos ia he in e se
be a decay eac ion: νe+37 Cl →e−+37 A . I s long- e m measu emen o a sola neu ino lux signi ican ly
lowe han heo y p edic ed was known as he Sola Neu ino P oblem. [59]
106The Sudbu y Neu ino Obse a o y (SNO) was a hea y-wa e Che enko de ec o loca ed in a mine in
Sudbu y, Canada. I s key capabili y was he simul aneous measu emen o elec on neu inos ( ia cha ged-
cu en in e ac ions) and all ac i e neu ino la o s ( ia neu al-cu en in e ac ions). This p o ided de ini i e
p oo o neu ino la o oscilla ion and sol ed he Sola Neu ino P oblem. [108]
107The Supe -Kamiokande (Supe -K) de ec o is a la ge wa e Che enko de ec o loca ed in he Kamioka
Mine, Japan. I obse es neu inos ia elas ic sca e ing om elec ons in wa e , which allows o p ecise eal-
ime measu emen o he di ec ion and ene gy o sola neu inos (p ima ily νe), con i ming he ene gy-dependen
de ici and oscilla ion phenomenon. [109]
108This uni ica ion was he au ho ’s d eam e e since he saw he book Powe s o Ten by Mo ison and Mo ison
in he ea ly 1980s [110], a d eam which was e i alized when eading Ge a d ’ Hoo ’s In he kielzog an de
deel jes [111] a ound 1992. The sou ce o physics is indeed a passiona e wonde a he beau y o na u e...
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Table 10: Compa ison o s ella usion chains in con en ional models and TSRT. The pp chain
and CNO cycle a e ep esen ed in e ms o hei d i ing mechanisms, in e p e a ion o eac ion
a es, and physical meaning o emi ed pa icles.
Con en ional (QM/S a is-
ical)
TSRT (De e minis ic Ge-
ome y)
D i ing p inciple Ba ie pene a ion ia quan-
um unneling; a es om o e -
lap o Maxwell–Bol zmann e-
loci y dis ibu ion wi h unnel-
ing p obabili y.
Causal o e lap o embling
geodesics; usion occu s i cu -
a u e modes eo ganize in o
mo e symme ic, lowe -s ess
eigenmodes.
pp chain Sequen ial unneling o p o-
ons wi h weak- o ce be a de-
cay; empi ical explana ion o
sola luminosi y.
De e minis ic condensa ion:
p o on o e lap s abilized by
weak econ igu a ion in o a
deu e on; successi e cu a u e
condensa ions yield 4He.
CNO cycle P o ons unnel in o C, N,
O nuclei, which ac as s a-
is ical ca alys s; cycle a es
i ia empe a u e-dependen
c oss sec ions.
C, N, O nuclei ac as esonance
sca olds in embling geome-
y, guiding de e minis ic p o-
on cap u e; cu a u e es o ed
a cycle comple ion.
Neu ino emission Byp oduc o be a decay; lux
p edic ed by p obabilis ic weak
in e ac ion ampli udes.
De e minis ic geodesic ag-
men s emi ed when cu a u e
ebalancing equi es weak ad-
jus men ; lux is di ec p obe o
causal TSRT econ igu a ion.
Ene gy elease Explained as “mass de ec ”
(∆mc2) con e ed in o adia-
ion and kine ic ene gy.
Reduc ion o in eg a ed cu a-
u e s ess (Equa ion (124));
E=mc2eme ges as geome -
ic iden i y.
coa se–g aining scale), one has
~TSRT =~[1 + O(χ)], χ ∝ hKµνKµνia g ℓ4≪1,(138)
so ~TSRT coincides wi h Planck’s cons an o expe imen al p ecision. C ucially, he equali y he e
is no assumed bu esul s om he geome ic limi in which he coa se–g ained embling cycle is
cu a u e–insensi i e. By con as , in s ong–cu a u e en i onmen s (e.g. nea compac objec s
o in ea ly–uni e se epochs), he coa se–g ained embling pe iod and hus he ac ion pe cycle
acqui e cu a u e–dependen eno maliza ion, yielding a geome y–dependen ~TSRT =~Ξ(K)
wi h Ξ(K)→1in he weak–cu a u e limi . This is a de i ed p ope y o he TSRT p ope –
ime ac ion (see [12, Sec ion 5.1] and Appendix E), no an imposed quan iza ion pos ula e: he
“quan um o ac ion” is he in a ian p ope – ime a e age o one embling cycle, whose alue
educes o ~in o dina y labo a o y condi ions and depa s om i in high–cu a u e egimes.
To connec p ope - ime dynamics wi h measu able en ance–channel supp ession, we s a
om he TSRT causal ac ion unc ional o an en ance ajec o y γjoining wo quasi-s able
con igu a ions Cin → C use:
S[γ] = ZγLξµν, uµdτ, (139)
wi h L he TSRT Lag angian densi y o he embling ield along he causal cong uence uµ
(see [12, Eq. (54)]). Fo nea -s a iona y en ance condi ions, a s eepes -descen e alua ion o e
causal his o ies yields a ansi ion ke nel ha is exponen ially con olled by he excess p ope -
ime ac ion ∆So he leas -ac ion pa h ela i e o he ee app oach:
∆S ≡ S[γleas ]−S[γ ee].(140)
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83
A e coa se-g aining o e sub-geodesic oscilla ions (which de ines ~TSRT), he TSRT ansi ion
weigh akes he uni e sal o m
W(E)∝exp−∆S/~TSRT,(141)
which is he p ope - ime analogue o he exponen in [12, Eq. (54)]. (Th oughou , ~in expo-
nen ial ac o s deno es ~TSRT; in he weak-cu a u e egime ~TSRT →~.)
To compu e ∆S(E)in wo-body usion, we educe o he adial ela i e mo ion wi h educed
mass µin an e ec i e TSRT po en ial
VTSRT( ) = VC( ;Rc) + Vcu ( ),(142)
combining ini e-size Coulomb ( adius Rc) and cu a u e-sa u a ion con ibu ions (de ini ions
in Sec ion 7.3). Along he leas -ac ion en ance pa h, he p ope - ime inc emen sa is ies dτ =
d / wi h adial speed . W i ing he excess ac ion as he line in eg al o e he classically
o bidden a c [ 1(E), 2(E)] ( u ning poin s o VTSRT( )a ene gy E),
∆S(E) = 2 Z 2
1P ( ;E)d
,(143)
whe e P is he TSRT conjuga e momen um o , he adial Hamil on–Jacobi educ ion gi es
P ( ;E) = p2µ[VTSRT( )−E]on he o bidden segmen . Abso bing he slowly a ying kine-
ma ic ac o 1/ in o he p e-exponen ial ( he s anda d mo e when isola ing he dominan
exponen ial), we ob ain he wo king exp ession
∆S(E)≈2Z 2(E)
1(E)p2µ[VTSRT( )−E]d
∞
,(144)
whe e ∞is he asymp o ic ela i e speed used o no malize he p ope - ime elemen in he
en ance channel ( he esidual 1/ ∞is a smoo h ac o ha mig a es o he non-exponen ial
S0(E)p e ac o o he c oss sec ion; c . Appendix A). The ac o 2 accoun s o he inbound
and ou bound segmen s wi hin he o e lapping domain. Equa ion (144) is he p ope - ime
coun e pa o he adial ac ion in eg al de i ed in [12, App. C], he e applied o he cu a u e-
egula ed ba ie o usion.
In he coa se-g ained limi ~TSRT →~, subs i u ing (144) in o (141) ep oduces he amil-
ia low-ene gy exponen ial o he as ophysical S- ac o , now as a de e minis ic consequence o
p ope - ime ac ion ac oss he epulsi e egion a he han a p obabilis ic unneling pos ula e.
De e minis ic modi ica ions en e anspa en ly ia VTSRT( ): elec on sc eening al e s he nea -
ield Coulomb e m (Sec ion 12.8), while cu a u e-sa u a ion and o e lap geome y encode deep
hind ance (Sec ion 12.9). Thus he TSRT p ope - ime o malism uni ies he obse ed exponen-
ial beha io o usion c oss sec ions wi h causal embling-geome y dynamics and p ese es
empi ical accu acy h ough a i s -p inciples exponen ∆S/~TSRT.
12.8 Elec on–Sc eening Shi s in Ul a–Low–Ene gy Fusion
We conside labo a o y usion a cen e –o –mass ene gy E(uni s: keV unless s a ed). The
expe imen al obse able is he as ophysical S- ac o S(E), de ined by
σ(E) = S(E)
Eexp−2π η(E), η(E) = Z1Z2e2
4πε0~ (E),(145)
wi h η(E) he Somme eld pa ame e and (E) he asymp o ic ela i e speed. Figu e 2 (p. 12)
shows he measu ed enhancemen a io Robs(E) = Sobs(E)/Sba e(E) o Pd-hos d(d, p) ; he
da ase and bes - i pa ame e s a e gi en in Table 38 (p. 206) and Appendix R.3.7.
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84
Expe imen s wi h ligh nuclei embedded in me als o molecula a ge s show enhanced low-
ene gy usion a es, ypically quan i ied by Robs(E)>1a small E[34,112]. S anda d analyses
model his by inse ing a cons an “ba ie shi ” Uein o he unneling exponen [113]. This
app oach is desc ip i e: i adjus s pa ame e s bu does no explain he physical o igin o he
shi .
In TSRT, he e ec a ises de e minis ically om nea - ield elec omagne ic cu a u e pola -
iza ion: he su ounding elec ons causally eo ganize he local U(1) embling ield, he eby e-
ducing he p ope - ime ac ion ac oss he epulsi e egion (Sec ion 12.7). The esul ing sc eening
enhancemen is no a i ed shi bu a compu ed consequence o cu a u e-coupled elec omag-
ne ic esponse, wi h he same cons an s applied o all hos ma e ials.
Two geome ic e ec s con ol he sc eening a low E( ull de i a ion: Appendix R.3.2):
1. Causal-lag a enua ion. Elec onic cu a u e esponds wi h a ini e p ope - ime delay; as
E↓, he nuclea ansi ime sho ens ela i e o he elec onic elaxa ion ime, so sc eening
e icacy dec eases smoo hly.
2. Dynamic dephasing. Elec on–nuclea embling modes lose phase alignmen o e a ini e
cohe ence leng h, p oducing a mild supp ession ha scales wi h √E.
These e ec s imply an ene gy-dependen sc eening ene gy ∆Ue(E), no a cons an o se .
The closed o m used in he main i s is
∆Ue(E) = ∆U0
1 + (E/Ec)β1 + λpE/keV,(146)
whe e: ∆U0is he s a ic-limi sc eening (eV), Ec(keV) se s he non-adiaba ic onse o causal-
lag a enua ion, βcon ols he smoo hness o ha onse , and λ(keV−1/2) encodes weak pa h-
leng h/nea - ield dependence o he lag. Equa ion (146) educes o he con en ional cons an
Uewhen E≫Ecand λ→0.
Expanding he TSRT p ope - ime ac ion (Equa ion (144)) o i s o de in he elec onic
cu a u e pe u ba ion and a e aging o e one embling cycle (Appendix R.3.2) yields he
enhancemen
Sobs(E)
Sba e(E)= exp"π Cηη(E)∆Ue(E)
E−ddyn pE/keV#,(147)
whe e Cη≃1is a small geome y ac o om angula a e aging o he nea - ield cu a u e, and
ddyn (keV−1/2) ep esen s he ne dephasing slope (Appendix R.3.2, Equa ion (327)). (He e
η(E)is he Somme eld pa ame e de ined in Appendix R.3.4, Equa ion (331).)
In solids wi h pa ially cohe en elec onic modes, a small esidual oscilla ion is expec ed
om in e e ence be ween elec onic embling and he en ance geome y. This yields he
de e minis ic modula ion
∆Ue(E)7→ ∆Ue(E)h1 + ε J0
κpE/keVi,(148)
wi h J0 he Bessel unc ion, εa small ampli ude, and κ(keV−1/2) a cu a u e wa enumbe se
by he nea - ield co ela ion leng h (de i a ion: Appendix R.3.5). The modula ion is ele an
only when a dimensionless cohe ence pa ame e χ≡τe /a (elec onic esponse ime τe, la ice
scale a) is O(1); o he wise ε→0.
The ull nea - ield sc eening model depends on a compac se o pa ame e s—∆U0(eV),
Ec(keV), β,λ(keV−1/2), Cη,ε,κ(keV−1/2), and he dephasing slope ddyn (keV−1/2)—all o
which a e de ined and de i ed sys ema ically in Appendix R.3.2. Thei ole is o encode he
causal elec onic cu a u e esponse, he ene gy-dependen elaxa ion o he nea ield, he gen le
embling modula ion, and he loss o phase alignmen ac oss successi e en ance cycles. Fo
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85
he Pd-hos d(d, p) da a shown in Figu e 2 (p. 12), a single pa ame e ec o ep oduces he
obse ed enhancemen pa e n ac oss he ull ene gy ange:
∆U0= 89.84 eV, Ec= 4.474 keV, β = 0.108, λ = 0.862 keV−1/2, Cη= 0.900,
ε=−0.300, κ = 2.062 keV−1/2, ddyn = 2.66 ×10−3keV−1/2.
The expe imen al a ay used in Figu e 2 (p. 12) is ep oduced e ba im in Table 28 (p. 190) o
Appendix W.7, ensu ing exac ep oducibili y o he sc eening i . The co esponding da ase
is abula ed e ba im in Table 28 (p. 190) (Appendix W.7), and he ep oduc ion o Figu e 2
(p. 12) is pe o med by he sc ip sc eeningmake ig.m (Lis ing 9, Appendix R.3.7), which
implemen s Equa ions (146)–(148) exac ly, using consis en uni s and SI cons an s h oughou .
F om a physical s andpoin , he i ed beha iou ollows di ec ly om he TSRT ac ion-based
o mula ion. Equa ions (146)–(147) show ha he enhancemen a e y low ene gies is limi ed
no by quan um- unneling p obabili ies, bu by a causal lag in how bound elec ons adap hei
cu a u e con ibu ion o he incoming nuclea geome y: as Edec eases, he en ance con-
igu a ion e ol es mo e apidly han he elec onic ield can espond, p oducing an e ec i e
educ ion o he local Coulomb cu a u e adius and hus a ini e ∆Ue(E). A highe ene -
gies, pa ial incohe ence be ween successi e embling cycles leads o a smoo h √E-dependen
damping cap u ed by he ddyn e m, while a small, pa i y-neu al oscilla o y componen yields
he Bessel- ype embling modula ion ha su i es coa se g aining. Toge he hese ea u es
e lec de e minis ic p ope ies o he p ope - ime geome y: he enhancemen cu e in Figu e 2
(p. 12) eme ges wi hou any p obabilis ic co ec ion o ba ie pene a ion, en i ely om he
causal s uc u e encoded in he TSRT adial ac ion (Sec ion 12.7).
12.9 Case S udy: Deep Sub-Ba ie Fusion Hind ance
A summa y hind ance igu e is p esen ed as Figu e 6 (p. 86); he uni e sal slope pai is abula ed
in Table 16 (p. 110). The absolu e ission ene gy elease used o no maliza ion (Figu e 6 (p. 86)),
ETSRT
iss = 170.028 MeV, is aken om Table 42 (p. 257). Tha able is gene a ed by Lis ing 28
a e he calib a ed cons an s in Lis ings 44 and 29.
A long-s anding challenge in hea y-ion usion has been he “deep sub-ba ie hind ance”:
measu ed as ophysical S- ac o s a e y low ene gies all much mo e s eeply han p edic ed
by s anda d coupled-channels models [30, 31]. Despi e decades o e inemen s, inco po a ing
ba ie dis ibu ions, mul iphonon exci a ions, and empi ical damping ac o s, he con en ional
quan um amewo k ne e ully ep oduced he obse ed apid allo . This anomaly aised he
undamen al ques ion o whe he unneling-based desc ip ions cap u e he co ec mechanism o
hea y-ion usion a ex eme low ene gies.
In he TSRT amewo k, his phenomenon a ises de e minis ically om a geodesic-o e lap
c i e ion: usion occu s only when he embling cu a u e domains o he wo app oaching
nuclei o e lap su icien ly o o m a single causally connec ed bound mode. A high ene gies,
his o e lap h eshold is easily me ; a e y low ene gies, howe e , he p ope - ime cu a u e
co ela ion be ween he wo nuclea domains becomes oo weak o achie e mode locking. The
cu a u e ields emain pa ially disjoin , and usion is supp essed, no because o an exponen-
ial “ unneling ailu e,” bu because he causal geome y i sel no longe pe mi s ull cu a u e
me ge . This supp ession appea s mac oscopically as he “hind ance” o he S- ac o .
Ma hema ically, he TSRT analysis begins wi h he p ope – ime ac ion inc emen o Sec-
ion 12.7, e alua ed o wo sphe ical nuclei o mass numbe s Aiand cha ges Zi. The ac ion
con ains (i) he la ge–scale geome ic app oach e m and (ii) a cu a u e sel –coupling e m ha
becomes ele an inside he o e lap egion. Expanding he ac ion in in e se powe s o ene gy and
o he e ec i e nuclea adius Re =A1/3
1+A1/3
2yields wo dimensionless geome ic in a ian s
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86
70 75 80 85 90 95 100 105 110 115 120
Ec.m. [MeV]
10-3
10-2
10-1
100
101
102
103
S (MeV b)
Hind ance S- ac o
58Ni+58Ni da a
64Ni+64Ni da a
16O+208Pb da a
58Ni+58Ni TSRT
64Ni+64Ni TSRT
16O+208Pb TSRT
Figu e 6: Deep sub-ba ie usion hind ance. The ho izon al axis is he cen e -o -mass ene gy
Ec.m.(in MeV); he e ical axis shows he as ophysical S- ac o S(E)(a bi a y uni s on a
loga i hmic scale), compiled om expe imen al da a in Table 39 (p. 207). Poin s: measu ed S-
ac o s o 58Ni+58Ni, 64Ni+64Ni, and 16O+208Pb. Cu es: de e minis ic TSRT ac ion-scaling
model [Equa ion (150)], wi h a single calib a ion on 64Ni+64Ni, uni e sal asymme y eno -
maliza ion om 16O+208Pb, and symme ic- adius co ec ion om 58Ni+58Ni, as discussed in
Sec ion 12.9. Code and i ing de ails a e p o ided in Appendix Q.3; expe imen al a ays and
me ada a a e embedded and e e enced in Appendix Q.3. Da a p o enance and p ep ocessing
s eps (uni s, ene gy no maliza ion, digi iza ion checks whe e applicable) a e summa ized in Ap-
pendix X. Ac ion-scaling law: Sec ion 12.9; slope calib a ion: Appendix B.5, Table 16 (p. 110);
sc ip s in Appendix Q.3.
ha comple ely con ol he low–ene gy asymp o ic beha iou ( ull de i a ion in Appendix W.8):
Ξ1(E) = √µ Z1Z2
Re √E,Ξ2(E) = µ(Z1Z2)2
R2
e E,(149)
whe e µis he educed mass. Ξ1(E)gi es he leading p ope – ime cu a u e con ibu ion o
he app oach ajec o y; Ξ2(E)is he nex sys ema ic co ec ion, a ising om cu a u e sel –
in e ac ion (elec omagne ic s i ness) wi hin he geome ic o e lap.
F om his expansion, and using he coa se–g ained ac ion o malism o Sec ion 12.7, he
as ophysical S- ac o acqui es he de e minis ic scaling law
S(E) = Asys exph−α′′ Ξ1(E)−β′′ Ξ2(E)i,(150)
wi h Asys ixed solely om he uppe -ene gy egion whe e he cu a u e-squa ed con ibu ion
is negligible. The coe icien s (α′′, β′′)a e ob ained di ec ly om he p ope - ime ac ion ex-
pansion (Appendix W.8) and co espond o he cu a u e and cu a u e-squa ed componen s,
espec i ely. The key poin is ha TSRT eplaces phenomenological unneling supp ession wi h
causally de ined geome ic in a ian s; no p obabilis ic pene a ion ac o is equi ed.
The da ase s used o he compa isons in Figu e 6 (p. 86) a e abula ed e ba im in Table 39
(p. 207). The i ed alues o (Asys, α′′, β′′), oge he wi h hei ep oducible MATLAB wo k low,
appea in Appendix W.8.
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Asymme y and geome ic eno maliza ion. Once he uni e sal TSRT in a ian s Ξ1,2(E)
ha e been iden i ied, he only sys em–speci ic di e ences come om geome ic ac o s: mass–
a io asymme y and adius–dependen cu a u e o e lap. In TSRT hese modi y no he ex-
ponen ial o m i sel , bu he slopes wi h which he in a ian s en e he ac ion. Fo wo nuclei
wi h mass numbe s A1, A2, he leading modi ica ion is a con olled escaling o he cu a u e
slopes by he asymme y pa ame e
χ=|A1−A2|
A1+A2
,(151)
which measu es he misma ch o he embling eigenmodes on app oach. A i s o de ,
α′=α1 + a1χ, β′=β1 + b1χ,(152)
whe e (a1, b1)quan i y how mass–asymme y pe u bs cu a u e locking in he en ance channel.
Fo nea ly symme ic sys ems, a second (subdominan ) geome ic co ec ion accoun s o he
weak dependence o cu a u e–o e lap e iciency on he en ance adius R=A1/3
1+A1/3
2:
α′′ =α′1 + c1Rcal
R−1, β′′ =β′1 + c2Rcal
R−1,(153)
wi h Rcal he e e ence adius o he calib a ion sys em (chosen he e as 64Ni+64Ni). These co ec-
ions ha e di ec geome ic meaning: hey encode how cu a u e locking is sligh ly s eng hened
o weakened depending on ela i e size and de o ma ion, wi hou in oducing any phenomeno-
logical ba ie s o damping ac o s.
Calib a ion and esul s. All pa ame e s used in Figu e 6 (p. 86) a e ob ained om a sin-
gle de e minis ic calib a ion p o ocol. The base loga i hmic slopes o he e e ence sys em
64Ni+64Ni a e
αL= 3.927, βL=−0.0396,
co esponding o (α, β) = (−αL,−βL)in he TSRT ac ion exponen . The asymme y coe icien s
(a1, b1) = (−1.0,−1.2) a e ixed om he s ongly asymme ic sys em 16O+208Pb, whe e he
en ance geome y mos clea ly ampli ies he asymme ic cu a u e e m. The adius coe icien s
(c1, c2) = (−0.6,0.2) ollow om he 58Ni+58Ni sys em, which di e s om he calib a ion sys em
p ima ily by i s smalle en ance adius a he han mass asymme y. The esul ing e ec i e
log–slopes a e:
58Ni + 58Ni : (αe
L, βe
L) = (3.848,−0.0399),
64Ni + 64Ni : (3.927,−0.0396),
16O + 208Pb : (0.561,+0.00113).
No pa ame e is uned o he hind ance egion i sel . The ampli ude Asys o each sys em is
de e mined solely om i s uppe -ene gy hi d, whe e he o e lap cu a u e is weak and he TSRT
exponen ial educes o i s leading o m. All i ing s eps, da ase s, and ou ines a e con ained in
he MATLAB sc ip ep oducehind anceTSRT.m (Appendix Q.3), wi h aw da a a ays gi en
in Table 39 (p. 207) and he calib a ed pa ame e s collec ed in Appendix X.
In e p e a ion. The physical meaning o hese esul s is clea : deep sub-ba ie usion hin-
d ance eme ges no om p obabilis ic supp ession o unneling, bu om he geome ic limi o
cu a u e cohe ence. In TSRT, he embling ields o he wo nuclei mus me ge smoo hly o
o m he used eigenmode; a ex eme sub-ba ie ene gies, he cu a u e–o e lap egion becomes
oo ex ended and oo s i o allow cohe en locking. The exponen ial g ow h o Ξ1(E)and Ξ2(E)
wi h dec easing Equan i ies exac ly his loss o cohe ence. T adi ional coupled-channels models
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88
educe he end phenomenologically h ough empi ical damping ac o s, ye s ill s uggle o
ep oduce he s eepes slopes. In con as , he TSRT scaling law (150) ep oduces he en i e
sys ema ics ac oss di e se eac ions using geome ic in a ian s alone, wi hou adjus able ba ie
shapes o sc eening po en ials. This p o ides he i s de e minis ic, causal explana ion o he
hind ance phenomenon and uni ies he mic oscopic cu a u e mechanism (Sec ion 12) wi h he
mac oscopic as ophysical S- ac o beha iou obse ed expe imen ally.
12.10 Geome ic Syn hesis o Fusion in TSRT
TSRT ein e p e s nuclea usion as a de e minis ic geome ic ansi ion o embling space ime
cu a u e. Ra he han desc ibing usion as a p obabilis ic a e sal o a po en ial ba ie , TSRT
iews he p ocess as he causal me ging o wo ini ially sepa a e cu a u e domains. When wo
nuclei app oach closely enough, hei local embling ields o e lap and he e ec i e me ic
de o ms con inuously in o a single, causally locked con igu a ion. This ansi ion o ms a mo e
compac and ene ge ically a ou able cu a u e densi y, p o iding he geome ic in e se o he
cu a u e dilu ion ha go e ns ission (Sec ion 11).
The na a i e de eloped h ough his sec ion began wi h he his o ical and phenomenological
con ex o he Con en ional Desc ip ion (Sec ion 12.1), linking ea ly labo a o y measu emen s o
Be he’s iden i ica ion o he p o on–p o on chain and CNO cycle. The TSRT iew (Sec ion 12.2)
eplaced his pic u e o po en ial ba ie s and unneling wi h causal cu a u e concen a ion: a
de e minis ic p ocess in which he embling modes o wo nuclei me ge once he cu a u e-
sa u a ion h eshold is exceeded. Wi hin his geome ic in e p e a ion, he classical Coulomb
ba ie becomes a ansien equilib ium be ween long- ange elec omagne ic cu a u e and sho -
ange cu a u e locking. The ansi ion ac oss his equilib ium equi es no s ochas ic unneling,
as shown in Sec ion 12.3, bu p oceeds h ough con inuous p ope - ime e olu ion o he cu a u e
con igu a ion.
Ene gy pa i ioning wi hin his de e minis ic pic u e is go e ned by cu a u e in eg als a he
han p obabili ies. As de i ed in Sec ion 12.4, he appa en “b anching a ios” o usion eac-
ions a ise om causal ou ing o cu a u e econ igu a ion in o kine ic, elec omagne ic (γ),
and weak (neu ino) channels. These ou comes e lec he s uc u e o he cu a u e ields in-
ol ed a he han in insic andomness, as he me ged embling domain edis ibu es cu a u e
densi y acco ding o i s geome ic cons ain s.
Conc e e illus a ions o hese p inciples we e p o ided in Sec ion 12.5, whe e he TSRT
e ec i e po en ial Ve ep oduced he eac ion h esholds, ene gy elease, and a e beha iou
o deu e ium– i ium usion wi hou adjus able pa ame e s. A as ophysical scales, he same
causal mechanism go e ns he S ella Fusion Chains (Sec ion 12.6), whe e cu a u e concen a-
ion, mode locking, and geome ic ecycling unde pin he p o on–p o on chain and he CNO
cycle. Thus, a uni ied explana ion o usion eme ges: he same geome ic laws apply om
labo a o y expe imen s o s ella in e io s.
The ole o p ope ime plays a cen al pa in his uni ica ion. Sec ion 12.7 de i ed he a-
milia exponen ial dependence o usion c oss sec ions no om quan um unneling assump ions
bu om he p ope - ime ac ion associa ed wi h he cu a u e pa h be ween u ning poin s. The
key ac o exp[−∆STSRT(E)] hus a ises om causal cu a u e accumula ion and coa se-g ained
embling cycles, p o iding a ully de e minis ic o igin o he low-ene gy exponen ial o m o
he as ophysical S- ac o .
En i onmen al cu a u e e ec s also all na u ally wi hin his amewo k. In condensed-
ma e se ings, he embling-induced pola iza ion o su ounding elec ons modi ies he en ance-
channel cu a u e ield, p oducing he expe imen ally obse ed enhancemen o ul a-low-ene gy
usion a es. Sec ion 12.8 demons a ed ha his e ec ollows om he same TSRT ac ion p in-
ciple and equi es no p obabilis ic co ec ions. A he opposi e ex eme, he deep sub-ba ie
usion hind ance o hea y-ion collisions inds i s explana ion in cu a u e-sa u a ion limi s (Sec-
ion 12.9). In his egime, he en ance cu a u e becomes oo s i o econ igu e cohe en ly,
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p e en ing u he ac ion educ ion and leading o exponen ial supp ession. Enhancemen and
hind ance hus eme ge as wo mani es a ions o he same geome ic mechanism.
Taken oge he , hese esul s yield a cohe en and comp ehensi e syn hesis: usion, ac oss all
accessible ene gy scales, is go e ned by cu a u e concen a ion, p ope - ime de e minism, and
causal mode locking. The s anda d mass–ene gy ela ion E=mc2acqui es a geome ic meaning
in his con ex , exp essing he equi alence be ween cu a u e densi y and ine ial ene gy a he
han a con e sion ule. All quan i a i e p edic ions p esen ed in his sec ion use a single, globally
ixed cu a u e calib a ion o he e e ence nucleus 16O(Appendix E.3), wi h no pe -sys em
uning. The p esen syn hesis conce ns nea -g ound-s a e, e en–e en nuclei whe e embling
symme ies a e clea es ; ex ensions o odd-Asys ems, o a ional exci a ions, and high-ene gy
usion– ission dynamics may be add essed in u u e wo k. Th oughou , TSRT p ese es ene gy,
momen um, and angula momen um exac ly, ensu ing consis ency wi h empi ical da a and wi h
he geome ic ounda ions o embling space ime ela i i y.
13 Compa ison wi h Quan um Nuclea Models
Suppo ing appendices: Appendix B (ex ended ables) and Appendix L, M, P (domain-speci ic
benchma ks).
MATLAB ile loca ions a e indexed in Appendix Y.
We compa e he de e minis ic TSRT p edic ions summa ized in Tables 5 (p. 13) and 6 (p. 13)
wi h he co esponding ea u es o quan um nuclea models. Resul s o his pape include: (i)
absolu e binding ene gies and adii, (ii) ba ie ac ions and usion sys ema ics including deep
hind ance, (iii) odd–e en s agge ing in ission yields om cu a u e-locked neck modes, and (i )
a ully de e minis ic li e ime law ep oducing nuclea hal -li es o e mo e han wen y o de s
o magni ude using a single global no maliza ion. These p edic ions allow o a di ec , uni ied
compa ison wi h liquid-d op, shell, unneling, and S anda d-Model desc ip ions. The con as s
below highligh how empi ical co ec ions in quan um models (pai ing, shell closu es, unnel-
ing ampli udes, and sc eening shi s) a ise in TSRT om he geome y o embling cu a u e
wi hou p obabilis ic pos ula es o pe -nucleus uning.
TSRT ep oduces s anda d nuclea sys ema ics while elimina ing phenomenological pos u-
la es. Volume/su ace sa u a ion and pai ing en e ia de e minis ic cu a u e cohe ence (Equa-
ion (10)); magic-numbe e ec s a ise as geome ic esonances a he han single-pa icle shell
i s (Sec ion 5); ba ie s ollow om ac ion s a iona i y (Equa ion (16)), and OES is an in e e -
ence e ec (Figu e 3 (p. 24)), no an added pai ing e m. Sub-ba ie usion ends, sc eening,
and deep hind ance ollow om cu a u e-o e lap and p ope - ime ansmission as de eloped in
Sec ion 7 (Figu e 2 (p. 12), Figu e 6 (p. 86)), while he b oade phenomenology is syn hesized
in Sec ion 12. In all cases, TSRT’s pa ame e s a e ied o cu a u e geome y once and eused,
as summa ized in Table 6 (p. 13).
The mode n quan um desc ip ion o nuclei eme ged om a se ies o c ises and empi ical
puzzles du ing he i s hal o he 20 h cen u y. The disco e y o adioac i i y by Becque el
[114] and he ea ly cha ac e iza ion o α-decay posed challenges ha classical physics could
no add ess. In 1928, Gamow [51] and, independen ly, Gu ney and Condon [115], in oduced
he hen- adical idea o quan um unneling h ough he Coulomb ba ie —i s i s d ama ic
applica ion o nuclea phenomena.
Sho ly he ea e , mac oscopic pa ame iza ions such as he liquid-d op model [37,87] p o-
ided global i s o binding ene gies and, wi h ex ensions, laid he ounda ion o he i s
heo e ical ea men o nuclea ission [23]. Ye his con inuum pic u e could no accoun o
he s iking sys ema ics o nuclea s uc u e: he exis ence o “magic numbe s,” he odd–e en
s agge ing o binding ene gies, o he de ailed pa e ns obse ed in eac ion c oss-sec ions.
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TSRT is no me ely quali a i e. Wi h he calib a ed cons an s o Appendix E and he cu a u e
in eg als o Appendix H, he ollowing alida ions a e al eady achie ed:
• Binding ene gies along he alley o s abili y: Using Equa ion (154), TSRT ep oduces
g ound-s a e binding ene gies o closed-shell nuclei (16O, 40Ca, 56Fe, 208Pb) o wi hin
∼1%, ma ching he accu acy o liquid-d op i s bu wi h no phenomenological su ace o
pai ing e ms. The ull compa ison able is gi en in Appendix I.
• Sub-ba ie usion supp ession: Fo ligh -ion sys ems such as D–D and D–T, e alua ion o
he p ope - ime ac ion in eg al (Sec ion 12.7) yields exponen ial slopes o he as ophysical
S- ac o consis en wi h expe imen al de e mina ions down o ∼10 keV. Nume ical esul s
and code a e p o ided in Appendix O.
• Fission agmen s agge ing: As shown in Sec ion 10.6, he de e minis ic neck-mode model
quan i a i ely ep oduces he ampli ude and cha ge-dependence o odd–e en s agge ing in
235U(n h, ). Appendix N ex ends his compa ison o 239Pu, showing ha he same ixed-
pa ame e model accoun s o bo h sys ems.
These benchma ks demons a e ha TSRT is al eady p edic i e and alsi iable: he same
cu a u e cons an s calib a ed once ep oduce a wide se o obse ables wi hou pe -nucleus o
pe -channel uning. Fu u e wo k may ex end he alida ion o hea ie odd–Aand odd–odd
nuclei, o a ional bands, and high-exci a ion usion eac ions, whe e addi ional embling-mode
couplings mus be included. De ailed TSRT de i a ions and nume ical benchma ks o binding
ene gies ac oss he alley o s abili y a e p o ided in Appendix L, while he as ophysical S- ac o
analysis is de eloped in Appendix P, oge he wi h ep oducible MATLAB codes.
14.4 T anspo mapping: ecoil, Dopple , g a i a ional
We use s anda d wo-body kinema ics o ecoil, special- ela i is ic Dopple o sou ce/obse e
mo ion, and he usual g a i a ional edshi whe e applicable; ull implemen a ion de ails a e in
Appendix J.
14.5 Cha ge-Sensed Elec omagne ic Cu a u e T anspo
In addi ion o kinema ic ecoil and o dina y Dopple /g a i a ional e ec s (Sec ion 14.4), TSRT
p edic s a small de e minis ic anspo co ec ion when pho ons o cha ged pa icles a e se
he nuclea elec omagne ic embling nea ield. This is he nuclea analogue o he a omic e-
inemen in oduced in [14]: he de ec ed ene gy di e s om he sou ce- ame alue by a cha ge-
sensed cu a u e holonomy accumula ed along he ini ial segmen o he wo ldline h ough he
EM embling geome y.
We w i e
Ede =Es c 1 + δ ec +δDop +δg a +δTSRT
EM ,(158)
whe e δ ec,δDop, and δg a a e he s anda d ecoil, Dopple , and g a i a ional e ms, and δTSRT
EM
is he cha ge-sensed elec omagne ic cu a u e anspo co ec ion. Fo pho ons, δTSRT
EM ep o-
duces he a omic nea - ield e inemen o [14] in he nuclea egime; o cha ged ejec iles i scales
wi h hei cha ge and he local EM embling cu a u e. A ull de i a ion and an implemen able
closed- o m nea - ield model appea in Appendix K.
In he compa isons epo ed he e, δTSRT
EM is small ( ypically .10−3in ac ional ene gy)
and does no a ec MeV-scale budge s; we include i whe e line cen oids a e compa ed a high
p ecision (e.g., γene gies and con e sion elec ons) and documen i s nume ical impac explici ly
(Appendix K, Tables 15 (p. 109) and 22 (p. 144)).
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14.6 Fu u e Theo e ical De elopmen s
Se e al heo e ical on ie s ollow na u ally:
1. Mul i-nucleon eigenmodes: Classi ica ion o collec i e embling modes in many-body nu-
clei, ex ending he single-pa icle and ew-body cases de eloped he e.
2. Reac ion ne wo ks: Applica ion o TSRT cu a u e ules o en i e usion chains, decay
cascades, and nucleosyn hesis pa hways in s a s and supe no ae.
3. G a i y c osso e : Linking nuclea cu a u e con inemen wi h mac oscopic sel -g a i y,
o model neu on s a s and collapse in o black holes wi hin he same embling amewo k.
14.7 Pe spec i e
Wha quan um mechanics once ea ed as in insically p obabilis ic ( unneling, decay, usion)
eme ges in TSRT as de e minis ic geome y. By uni ying a omic, nuclea , and cosmological
domains in a single causal amewo k, TSRT esol es a cen u y-old duali y be ween quan um
pos ula es and ela i is ic ield equa ions.
The b oade implica ion is clea : he same embling geome y ha gene a es Planck’s con-
s an , a omic spec a, and en anglemen also go e ns nuclea s uc u e and s ella ene gy p o-
duc ion. Fu u e wo k may e ine hese p edic ions, ex end hem o as ophysical en i onmen s,
and pu sue expe imen al es s designed o dis inguish de e minis ic cu a u e dynamics om
p obabilis ic quan um models.
TSRT no only uni ies nuclea physics wi h ela i i y, bu ele a e all o physics o a single
geome ic p inciple o embling space ime.
15 Calib a ed Ancho s o Absolu e Uni s in TSRT (Deu e on o
Fission Chain)
TSRT p edic ions a e p oduced om cu a u e-based unc ionals e alua ed on explici nume i-
cal g ids and epo ed in SI and s anda d nuclea uni s (Appendix V). The p esen no maliza-
ion chain begins wi h a single calib a ed ene gy–cu a u e mapping om he deu e on binding
ene gy and ex ends consis en ly up o he ission scale, e i ied on he 235U(n, ) benchma k
(Table 42 (p. 257)). A e his s ep, all epo ed TSRT ene gies a e absolu e and ep oducible
wi hou u he scaling. This sec ion shows ha , o he p esen manusc ip , all equi ed map-
pings om code quan i ies o physical uni s a e ei he (i) al eady absolu e h ough exac uni
con e sions, o (ii) ixed once by a documen ed physical ancho wi h no esidual eedom. We
summa ize hese mappings and iden i y how u u e wo k can p oceed wi hou any u he cali-
b a ions.
15.1 Ene gy mapping: absolu e by cons uc ion
Le Q[K]deno e he (dimensionless) cu a u e unc ional deli e ed by he ene gy pipeline (Ap-
pendix Y, Lis ings 4 (p. 215)–5 (p. 216)). Physical ene gies a e
Ephys =CEQ[K], Ephys ∈ {J,MeV}.(159)
In his manusc ip he ene gy mapping is al eady absolu e because he pipeline pe o ms a di ec
SI→MeV con e sion using exac cons an s (Appendix V, Tables 30 (p. 200)–31 (p. 201)):
1 MeV = 1.602176634 ×10−13 J, c = 2.99792458 ×108m/s(exac ).
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Acco dingly, he global ac o in Equa ion (159) is nume ically ixed,109
CE≡1(in he code’s MeV epo ing) (160)
and he e is no ee ene gy scale le o adjus . This is empi ically e i ied by he binding
o e iew (Figu e 1 (p. 10)), whe e B/A nea A≈60 ma ches AME2020 alues wi hin able
p ecision (Table 3 (p. 11)) wi hou any pe -nucleus eno maliza ion.
15.2 Cu a u e s eng h: ixed once and eused
The geome ic cu a u e-s eng h pa ame e used in he nuclea sec o is inhe i ed om he
a omic TSRT cons an and e ained he e:
γ= 4.36 ×1042 m−2(161)
as documen ed in Appendix A. This cons an mul iplies he dimensionless cu a u e in a ian s in
he ield cons uc ion and he e o e se s he geome ic magni ude o local embling supp ession.
Because he ene gy mapping (160) is absolu e, γis no degene a e wi h CEand is uly ixed
once o all sec o s. Toge he , his cu a u e magni ude and he absolu e ene gy mapping
inhe i ed om he deu e on no maliza ion (Sec ion 15.1) de ine a closed and uni -consis en
ene gy–geome y co espondence used in he ission alida ion (Table 42 (p. 257)).
15.3 Li e ime scale: a single physical ancho
Decay a es a e compu ed om he TSRT base–slope law wi h pe -mode ac o s (Sec ion 9.7,
Appendix B.7). The only global scale en e ing he hal -li e p edic ions is he a e cons an Cλ,
calib a ed once on a s anda d benchma k and hen held ixed:
Cλ= 6.48104 ×10−6s−1(162)
(see Table 2 (p. 11) and Appendix A). Wi h γ ixed by (161) and he absolu e ene gy mapping
(160), Cλis he unique a e-scale ancho ; no pe -iso ope adjus men s a e in oduced.
15.4 On he embling ampli ude hAi: no an independen i
In TSRT, obse able magni udes a e go e ned by he combina ion o (i) geome ic cu a u e
s eng h γ, (ii) mode-dependen shape/phase (en e ing he cu a u e in a ian s), and (iii) he
absolu e ene gy/ epo ing map. Any pu a i e mean embling ampli ude hAi ha escales all
local cu a u es would be degene a e wi h γand CEa leading o de . Since γis al eady ixed
by (161) and CEby (160), he e is no addi ional ee global ampli ude in he p esen wo k:
hAi is abso bed by γand does no cons i u e a sepa a e calib a ion he e. (163)
Sec o -speci ic shape pa ame e s (e.g. OES neck-pa i y weigh s, hind ance slopes) a e dimen-
sionless and ied o geome y (Sec ion 6, Sec ion 12.9) a he han a global ampli ude.
15.5 Gamma– ay no maliza ion (E2) and in e nal con e sion
Fo elec omagne ic decays, he E2block is ancho ed on he well-cha ac e ized 2+
1→0+
1 ansi-
ion o 156Gd wi h
Eγ= 88.9656 keV, τ = 1.635 ns, α o = 0.163,
as compiled in Table 36 (p. 205) (ENSDF A=156 and B Icc). This ancho ixes he E2p opo -
ionali y CE2used in Appendix Y.8 once and o all; no addi ional i pa ame e s a e in oduced
downs eam in o he li e ime pipeline (Appendix Y.9).
109In p ac ice, he in e nal code quan i y C.no m_ene gy (see Appendix Y.2, Lis ing 2 (p. 213)) ac s as he SI-
o-code con e sion cons an . I s deu e on-ma ched alue, 4.7865×1031 J/(code uni ), ensu es ha he MeV-scale
ene gy epo ing is absolu e and consis en ac oss all nuclea sec o s.
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15.6 P ac ical summa y
Wi h he abo e in place, he p esen manusc ip al eady ope a es in absolu e uni s:
• Ene gy epo ing is SI/MeV exac : CE≡1by cons uc ion (Appendix V and Y.2).
• The cu a u e s eng h is ixed once: γ= 4.36 ×1042 m−2(Appendix A).
• The li e ime scale has a single physical ancho : Cλ= 6.48104 ×10−6s−1(Table 2 (p. 11)).
• The E2block is locked by he 156Gd ancho (Table 36 (p. 205)), wi h ICC included ia
B Icc.
• The absolu e ene gy no maliza ion has been independen ly e i ied by ep oducing he
o al ission elease o he mal-neu on-induced 235U(n, ) (Table 42 (p. 257)), con i ming
he deu e on-based cons an ac oss he ull nuclea scale.
No esidual global knobs a e le wi hin he scope o his pape ; he emaining pa ame e s a e
geome ic (dimensionless) and a e ei he uni e sal slopes o mode-shape weigh s, as de ailed in
Sec ions 5, 6, 12.9, and 9.7.
This absolu e scaling is nume ically e i ied by he ission benchma k (Table 42 (p. 257)),
whe e ETSRT
iss = 170.028 MeV is ob ained wi hou any pos -hoc adjus men .
15.7 Ou look: emo ing ancho s en i ely
To make TSRT s ic ly ancho - ee in u u e applica ions, one may de i e Cλand he elec omag-
ne ic p opo ionali ies di ec ly om he embling–me ic Lag angian and bounda y condi ions,
ying hem o (c, ~)and he cu a u e spec um o bound modes. The p esen wo k al eady
demons a es ha he ene gy mapping— om deu e on binding up o 235U ission—is absolu e,
lea ing only he a e-scale Cλand adia i e p opo ionali ies o be de i ed om i s p inciples.
The p esen manusc ip al eady elimina es any ene gy-scale ambigui y (Sec ion 15.1); he e-
maining pa hway is a i s -p inciples de i a ion o he global a e scale. This may be pu sued in
o hcoming wo k.
16 Conclusions
This wo k comple es he ex ension o T embling Space ime Rela i i y Theo y (TSRT) in o
he nuclea domain and shows ha nuclea binding, s abili y, and decay a ise om a single
causal geome ic p inciple. In TSRT, a nucleus is a localized embling–cu a u e eigenmode
whose pe sis ence depends on sus ained supp ession o in insic cu a u e; decay occu s when
his supp ession elaxes de e minis ically along geodesic pa hs. The pic u e dispenses wi h
p obabilis ic pos ula es and unneling hypo heses and eplaces hem wi h cu a u e o e lap and
causal elaxa ion go e ned by p ope ime.
Wi hin his amewo k we p edic absolu e hal -li es o be a, alpha, and spon aneous ission
channels ac oss mo e han wen y o de s o magni ude using a single once-only no maliza ion
ixed on a benchma k nucleus. The ag eemen wi h e alua ed alues is s ingen , wi h a mean log-
a i hmic de ia ion below one pa in a million in he ep esen a i e se s udied. No shell closu es,
pai ing gaps, nucleus–by–nucleus p e o ma ion ac o s, o empi ical mass inpu s a e in oked; all
ene ge ics, ba ie ac ions, and a es a e ob ained om he same embling-space ime geome y
ha also unde lies ou desc ip ions o binding and de o ma ion. The no o ious longe i y o
ca bon–14 ollows na u ally as a geome ic supp ession o i s weak ansi ion, wi hou ad hoc
hind ance pa ame e s, ein o cing ha nuclea decay is a de e minis ic cu a u e econ igu a ion
a he han a s ochas ic p ocess.
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The same causal geome y accoun s o phenomena long conside ed anomalous wi hin s an-
da d app oaches. Deep sub-ba ie usion hind ance eme ges as a de e minis ic educ ion o
embling-mode o e lap; odd–e en s agge ing in ission yields e lec s phase-locked neck eigen-
modes a scission; and elec on-sc eening shi s a ul a-low ene gies ollow om cu a u e-
induced eno maliza ion o elec omagne ic ields. These beha io s do no equi e addi ional
hypo heses o auxilia y i s: hey a e di ec consequences o he embling me ic ha also
go e ns li e imes.
Ac oss β,α, and spon aneous ission, all epo ed hal -li es ollow om a single cu a u e-
based law wi h global, once-only ancho s, and wi hou p obabilis ic pos ula es o pe -nucleus
adjus men s.
In p ecise e ms, TSRT a ains a mean loga i hmic de ia ion below 10−6be ween p edic ed
and expe imen al hal -li es— ou o i e o de s o magni ude mo e p ecise han s a e-o - he-a
quan um o semiempi ical models. This le el o ag eemen , ob ained om ixed global cons an s
wi hou local uning, es ablishes TSRT as he i s de e minis ic amewo k capable o p edic ing
absolu e nuclea decay imes di ec ly om space ime geome y.
This uni ies nuclea s abili y and ans o ma ion unde de e minis ic space ime geome y
and closes he explana o y loop om cu a u e supp ession o obse able li e imes. The same
geome ic mechanism ha sus ains bound con igu a ions also go e ns hei causal elaxa ion,
yielding quan i a i e ag eemen o e mo e han wen y o de s o magni ude.
Quan i a i ely, he same de e minis ic law ep oduces absolu e hal -li es om milliseconds
o nine een-powe -o - en yea s wi h a mean disc epancy a abou one-million h o a decade
in log space, using only h ee ixed global cons an s o he h ee decay modes. This con as s
wi h he en h- o-hund ed h-o -a-decade dispe sion ypical o mains eam models on compa able
benchma ks, unde sco ing he explana o y and p edic i e powe o he geome ic cu a u e
mechanism.
Wi h ene gy, cu a u e, and a e scales now ixed in absolu e physical uni s, TSRT achie es a
sel -con ained quan i a i e b idge be ween geome y and expe imen , pa ing he way o u u e
ancho - ee p edic ions de i ed solely om embling-me ic i s p inciples.
These esul s es ablish no only a quan i a i e, bu also a quali a i e uni ica ion. Ene gy
conse a ion and eac ion ene ge ics appea as cu a u e edis ibu ion; nuclea pe sis ence and
ans o ma ion become ace s o he same causal s uc u e ha ela es a omic spec oscopy and
g a i a ional edshi .
The amewo k is anspa en and ep oducible: all ables and igu es in he pape a e
gene a ed by he p o ided MATLAB implemen a ions and da ase s, and he calib a ion s eps
a e documen ed so ha independen use s can e ace he wo k low end o end.
The sel -consis en calib a ion chain, spanning om he deu e on no maliza ion h ough he
cu a u e-based scission wid h, now yields he co ec absolu e ene gy elease o 235U(n, )
(ETSRT
iss = 170.028 MeV), he eby alida ing he TSRT ene gy mapping in nuclea -scale uni s
wi h no esidual i ing eedom.
Looking o wa d, he app oach is poised o sys ema ic ex ension o nuclei wi h compe ing
decay b anches, o ull ne wo k modeling in s ella en i onmen s, and o cu a u e-d i en beha -
io in neu on-s a ma e and s ong- ield as ophysics. Re inemen s o highe -o de cu a u e
de i a i es and signed geome ic ampli udes will u he es and expand he p edic i e each
wi hou depa ing om he de e minis ic ounda ion.
In sum, nuclea binding, decay, ission, and usion, adi ionally amed as p obabilis ic
quan um phenomena, eme ge he e as p ecise mani es a ions o embling space ime cu a u e,
placing nuclea physics on a single causal geome ic oo ing alongside he b oade ela i is ic
uni e se.
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025
101
17 Acknowledgmen s
CERN is g a e ully acknowledged o p o iding, h ough Zenodo,110 an open-access pla o m
ha makes his wo k eely a ailable o he global physics communi y, w i en in a s yle he
au ho p e e s.
The au ho 111 g a e ully acknowledges ins i u ional suppo om he F ench Cen e Na ional
de la Reche che Scien i ique (CNRS) h ough he In e na ional Resea ch Labo a o y IRL 2958
Geo gia Tech–CNRS, as well as Me z Mé opole, he Ci y o Me z, and Geo gia Tech Eu ope
o hos ing him as a Geo gia Tech acul y membe in F ance.
In as uc u e and mobili y suppo we e p o ided by he Région G and Es , he F ench
Lo aine Uni e si é d’Excellence (LUE impac p ojec I-META ANR-15-IDEX-04-LUE), In-
s i u Ca no ARTS, Conseil Régional de Lo aine, Le Conseil Dépa emen al de la Moselle,
Fede -Eu ope, he Indo-F ench Cen e o he P omo ion o Ad anced Resea ch (CEFIPRA
RCF-IN-0067), Mi abelle Plus, S ellan is, Ins i u de Soudu e, he Associa ion Na ionale de la
Reche che e de la Technologie (ANRT), Me z Mé opole, he Ins i u Supé ieu Eu opéen de
l’En ep ise e de ses Techniques (ISEETECH), he Con a Plan É a -Région (CPER-MEPP-
P05), he Agence Na ionale de la Reche che (ANR-09-BLAN-0167-01), he Eu opean Union’s
Ho izon 2020 p og am (G an Ag eemen No. 871260), he Belgian Fund o Scien i ic Resea ch
Flande s (FWO, 2005), he Flemish Ins i u e o he Encou agemen o Scien i ic and Techno-
logical Resea ch in Indus y (IWT, 2001-13343), and he No h A lan ic T ea y O ganiza ion
(NATO PST.CLG.980315). While hese p og ams p o ided an in aluable esea ch en i onmen ,
he concep ual di ec ion and esul s p esen ed he e s em om independen schola ly inqui y
de eloped o e decades.
The au ho acknowledges he ounda ional aining in physics, as ophysics, and enginee ing
physics ecei ed a he Ca holic Uni e si y o Leu en (KULAK and KU Leu en) and Ghen
Uni e si y, which shaped his in e disciplina y app oach. He also hanks he Geo gia Ins i u e o
Technology and he Geo ge W. Wood u School o Mechanical Enginee ing o his p omo ion
110Like an eagle ee o ly whe e i wills, u h is ee o wande h ough he wo ld o physics and ace he
sc u iny o he en i e communi y a once, o genuine u h has no hing o ea .
111The au ho was p o oundly a ec ed upon isi ing he Hi oshima Peace Memo ial Museum in 2004 wi h his
iend D Filip Windels. The con on a ion wi h he human eali y o nuclea weapons s ood in s a k con as
o hei abs ac heo e ical desc ip ion. This expe ience made he subsequen obse a ion o Japan’s pos -
wa jou ney all he mo e powe ul: a na ion ha ans o med p o ound agedy in o a es amen o esilience,
dedica ing i sel o peace, echnological excellence, and in e na ional coope a ion. A pe sonal encoun e wi h his
spi i occu ed ea lie , in S i Lanka, ollowing he de as a ing 1996 loods. The e, amids he eco e y e o s,
a local amily o e ed a gi o p o ound gene osi y: canned una, a s aple p o ided by Japanese ood aid. This
simple ac e ealed an endu ing u h, ha he co e o a na ion’s cha ac e is ound no in he ins umen s o i s
powe , bu in i s consis en choice o ex end humani y and compassion, e en o dis an s ange s.
In ellec ually, he au ho ’s pa h in o nuclea science was shaped by ea ly joys in pe o ming Mössbaue e ec
expe imen s as a mas e ’s s uden , by o mal cou ses in nuclea physics and as ophysics, and by p i a e eadings
ha explo ed nuclea p ocesses in s ella sys ems, whe e he wo disciplines me in a uni y su passing he con ines
o lec u es. A p olonged s ay in S i Lanka in 1997 added u he dep h: he e, while s udying symme y g oups
and a nuclea physics cou se, he au ho lea ned o app ecia e he elegance o elemen a y pa icles as s uc u ed
mani es a ions o ma hema ical ha mony. I was also in S i Lanka ha he i s augh ma hema ics, disco e ing
in i s uni e sali y a p o ound u h, ha ma hema ics knows no bo de s, no ace, no colo , and hus embodies a
common language o humani y.
The oo s o his pe spec i e each back u he s ill, o childhood in S i Lanka in 1985, when public an icipa ion
o Halley’s Come illed he ai . Al hough he come i sel was no clea ly isible a ha ime and place, he
exci emen i gene a ed—and a o ma i e isi o he Colombo plane a ium—igni ed a sense o cosmic wonde .
F om ha pe iod onwa d, he au ho ound joy in wa ching shoo ing s a s and con empla ing he luminous a c
o he Milky Way. In hindsigh , hese ea ly imp essions nu u ed he con ic ion ha human inqui y, like he
embling o space ime i sel , is bo h agile and immense, ancho ed in lee ing pe sonal momen s ye poin ing
always owa d he uni e sal.
The deep espec o he Japanese people and hei pos -wa alues, combined wi h a li elong ascina ion
o he cosmos, o ma hema ics, and o nuclea p ocesses a e e y scale, con inues o inspi e he e hical and
philosophical pe spec i e ha unde pins his wo k. Ne e heless, i was in Hi oshima ha a desi e g ew o deeply
unde s and he nucleus o he a om.
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025

102
o Full P o esso , coinciding wi h his designa ion as P o esso Hono is Causa in Physics by
he Uni e si y o Allahabad, a con luence o ecogni ion ha secu ed he academic eedom
necessa y o o malize he ideas p esen ed he e, which ace hei o igins o ea ly s udies o sola
sys em dynamics (D iesp ong, Dee lijk) and elec on beam de lec ion expe imen s (Koninklijk
A heneum, Wa egem).
The au ho also exp esses his g a i ude o his cousin, M . F ançois Be land. While employed
a he Doel Nuclea Powe S a ion in Belgium, M . Be land dona ed his comple e collec ion
o he scien i ic jou nal ’Na uu & Techniek’ o he au ho du ing his high school yea s. A a
ime when access o ad anced scien i ic esou ces was limi ed, hese jou nals, which ea u ed
explana ions o nuclea physics and ela i i y heo y, among many o he in e es ing scien i ic
opics, se ed as an in aluable and ea ly sou ce o inspi a ion and mo i a ion.
The au ho ex ends his deepes g a i ude o colleagues and o ganiza ions in acous ics, ul a-
sonics, acous o-op ics, and op ics o os e ing an in ellec ually ib an and collegial en i onmen .
Thei sha ed passion o wa e phenomena has been a con inual sou ce o inspi a ion, and hei
engagemen wi h indus ial applica ions always boos ed inspi a ion o seek c i ical suppo o
sus ain he au ho ’s labo a o y.
He is equally indeb ed o his cu en and o me s uden s (BS, MS, PhD) and pos doc o al
esea che s, whose cu iosi y and esh pe spec i es p o oundly en iched his wo k in nondes uc-
i e e alua ion and physical acous ics. Thei en husiasm ea i med ha scien i ic disco e y,
ega dless o discipline, h i es on in ui ion, elen less cu iosi y, and in ellec ual joy, mo i a ing
he au ho ’s deepe explo a ions in undamen al physics beyond con en ional wo king hou s.
This wo k s ands as a es amen o he endu ing in ellec ual and ma e ial legacies o Daniel
A. Lesage (1911–1988), Pie e F. Vangheluwe (1939–1991) (who igge ed he au ho ’s in e es
in his o y and academic pu sui in gene al), Ma ie e Vanoos huyze (1925–2009), Godelie e
Ve bo gh (1932–2012), Mau ice A. Decle cq (1941–2015), and Jeanne M. C. Maesen (1930–2024),
whose gene osi y con inues o enable schola ship. The au ho is equally indeb ed o Jeane e
Ve b ugghe and Nelly Vangheluwe o hei unwa e ing suppo , bo h mo al and logis ical, which
has been indispensable o his esea ch.
The au ho wishes o o e a pe sonal no e o hanks o his goddaugh e , So ie Windels,
who is always a beacon o cou age, kindness, and in eg i y. Du ing a o ma i e pe iod in his
esea ch, as he na iga ed he complexi ies o nuclea s uc u e and he limi a ions o p e ailing
models, she p o ided him wi h a bookle o quo es by Nikola Tesla. Tesla’s wo ds on he alue
o seclusion as a sec e o in en ion and he impo ance o deep, pa ien hough o e immedia e
esul s a i ed a a c i ical ime, se ing as a imely eminde ha he soli ude demanded by
his job and by his wo k could i sel be a sou ce o s eng h and o iginali y. Fu he mo e, he
au ho was p o oundly cha med when, me e mon hs be o e he conclusion o his leng hy wo k,
she exp essed he p ide in his e o s, which was a ges u e ha p o ided a inal, deeply alued
encou agemen .
Finally, he au ho exp esses p o ound g a i ude o his wi e and child en o hei s ead as
p esence, boundless encou agemen , and endu ing pa ience h oughou he many weekends and
holidays a home in Belgium de o ed o his wo k.
In pa icula , he au ho is deeply g a e ul o his child en—Benjamin, Anna-Lau a, and
Lambe —who g aciously len him hei as compu e s du ing he weekends so he could comple e
he mos demanding calcula ions.
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103
18 Appendices
Appendix Guide and C oss-Re e ence Map
The appendices ha ollow cons i u e a ully sel -con ained echnical companion o he pape .
They a e designed so ha e e y nume ical esul , igu e, and able in he main ex can be
aced o i s exac algo i hmic and analy ical sou ce. The s uc u e is modula : me hods and
cons an s i s , hen de i ed da ase s, ollowed by hema ic de i a ions ha pa allel he main
sec ions.
Global Appendix O e iew
•Nume ical Founda ions and Calib a ion. Appendix A (p. 105) de ails p ocedu es,
cons an s, and calib a ion ou ines, which a e complemen ed by he geome ic de ini ions
in Appendix C (p. 111), he physical cons an s in Appendix E (p. 120), and he nume -
ical in eg a ion schemes in Appendix F (p. 123). These appendices join ly unde pin all
quan i a i e wo k in Sec ions 5–7.
•Nuclea Geome y and Cu a u e F amewo k. Appendix C (p. 111) de ines cu -
a u e ields and coo dina e con en ions used h oughou TSRT nuclea modeling. The
companion Appendix G (p. 129) in oduces he embling- ield o malism o nucleons
and composi e nuclei, while Appendix H (p. 132) o mula es he co esponding cu a u e
ene gy unc ionals. These build he backg ound o he magic-numbe de i a ions and
de o ma ion ene gy in he main ex .
•Magic Numbe s and Shell Geome y. Appendix D (p. 114) de i es shell closu es as
s a iona y cu a u e esonances. I builds on Appendix C (p. 111) and in o ms Sec ion 5
(binding-ene gy kinks), Sec ion 13.3 (compa ison wi h shell models), and Sec ion 6 (odd–
e en s agge ing con ex ).
•Da a Tables and Benchma k Compa isons. Appendix B (p. 108) compiles all ex-
ended nume ical ables—binding ene gies, cha ge adii, ission ba ie s, sc eening shi s,
and hind ance slopes—each di ec ly ied o he igu es and compac ables in he main ex .
Li e ime benchma ks in Appendix B.7 (p. 110) comple e his da ase po olio, linking o
Sec ion 9 and Appendix I (p. 135) o de e minis ic hal -li e es ima ion.
•S abili y, Li e imes, and Decay Laws. Appendix I (p. 135) de i es he TSRT li e ime
es ima o and s abili y-map cons uc ion, while Appendix A (p. 105) desc ibes how he
absolu e li e ime scale Cλis ixed. Toge he hey suppo he main- ex Sec ions 9.7 and
9. Benchma ks appea in Appendix B.7 (p. 110).
•Fusion, Fission, and Hind ance. Appendix M (p. 151) de elops he cu a u e-g adien
bi u ca ion model o nuclea ission; Appendix O (p. 161) o mula es he TSRT geodesic-
o e lap c i e ion o usion ene gy; Appendix P (p. 166) and Appendix Q (p. 175) ex end
his o as ophysical S- ac o s and deep sub-ba ie hind ance. Thei pa ame e s and
slope calib a ions a e linked h ough Appendix P.4 (p. 168) and Appendix A (p. 105).
The combined con en suppo s Sec ions 7 and 8.
•Elec omagne ic Sc eening and CSCT T anspo . Appendices K (p. 141) and J
(p. 138) p o ide he cu a u e-sensed elec omagne ic anspo o malism (CSCT), used
o nea - ield line shi s in Figu e 2 (p. 12). These build on he same cons an s and g id
de ini ions as he mechanical modules in Appendix A (p. 105).
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104
•Odd–E en S agge ing and Neck-Mode Model. Appendix N (p. 157) gi es he
ull de i a ion o he Bessel-neck model o OES, complemen ing he da a ables in Ap-
pendix B.6 (p. 109) and he main- ex analysis in Sec ion 6.
•Me a-Analyses and Valida ion. Appendices R (p. 180) h ough S (p. 191) include
alida ion p o ocols, en i onmen checks, and pe o mance me ics. Appendix U (p. 197)
quan i ies unce ain y and sensi i i y o key obse ables. These appendices collec i ely
suppo he discussions in Sec ion 13.
•Rep oducibili y In as uc u e. Appendix Y (p. 210) p o ides he comple e wo k low
and un o de ; Appendix T (p. 194) lis s all igu e/ able gene a ion sc ip s, ollowed by
he global MATLAB ile index a he end o he documen . These appendices ensu e ull
anspa ency and epea abili y.
Nume ical ep oducibili y p e iew
Be o e en e ing he echnical appendices, he ollowing c oss-map connec s headline esul s o
hei compu a ional o igins:
1. Magic-numbe closu es. Analy ical de i a ion: Appendix D (p. 114); abula ed ou pu s:
Table 20 (p. 116). Suppo s Sec ions 5 and 13.3.
2. Binding ene gies and cha ge adii. Da a: Appendix B.1 (p. 108) and B.2 (p. 108); me hods:
Appendix A (p. 105); calib a ion cons an s: Appendix E (p. 120).
3. Li e imes and s abili y maps. De i a ion: Appendix I (p. 135); calib a ion: Appendix A
(p. 105); ables: Appendix B.7 (p. 110). Co esponds o Sec ion 9.7 and he main s abili y
diag ams.
4. Fission and usion ba ie s. De i a ions: Appendices M (p. 151) and O (p. 161); ba ie
da a: Appendix B.3 (p. 109); as ophysical S- ac o and hind ance alida ion: Appen-
dices P (p. 166), Q (p. 175), and B.5 (p. 109).
5. Odd–e en s agge ing (OES). De i a ion: Appendix N (p. 157); da a: Appendix B.6
(p. 109); igu es: Figu e 3 (p. 24). Algo i hms: Appendix A (p. 105) (OES lis ings).
6. Elec omagne ic sc eening. De i a ion and da a: Appendices K (p. 141) and B.4 (p. 109);
sc ip s: Appendix Y (p. 210) (lis ings 9 (p. 223), 27 (p. 254)); suppo s Sec ion 7.
7. Sensi i i y and ep oducibili y. Quan i a i e e o bounds: Appendix U (p. 197); global
un o de and alida ion en i onmen : Appendices S (p. 191), T (p. 194), and Y (p. 210).
How o un he ull wo k low. Fo a s ep-by-s ep execu ion o de , including ilenames and
expec ed ou pu s, see Appendix Y (p. 210).
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A Me hods: Nume ical P ocedu es, Pa ame e s, and Calib a-
ions
Consis ency wi h implemen a ion. In e nal uni s a e SI and a e con e ed o p esen a ion as in
s cons an s.m. G ids and quad a u e ollow s makeg id.m and gaussleg.m; he sin θdθ
weigh is included once (no double-weigh ing), ma ching he no es in Appendix F. Con e gence
checks use s con e gence.m.
Suppo s Sec ions 5, 6, 7, 8.
MATLAB: s cons an s.m, s makeg id.m, s con e gence.m,
gaussleg.m, s _ epo_sani y_check.m, s _p e ligh _checks.m.
This appendix documen s he nume ical pa h om model de ini ion o plo ed esul . I is
he p ima y e e ence o ep oducibili y. Me hods he e a e in oked in Sec ion 5 (binding/ adii),
Sec ion 6 (ba ie s, OES), and Sec ion 7 (sc eening, hind ance), wi h en y poin s lis ed in
Table 6 (p. 13). We s a e exac pa ame e alues, desc ibe calib a ion choices (e.g., he 56Fe
ancho ), and gi e un- ime se ings (e.g., g ids, ole ances, seeds, pla o m).
Uni s and con en ions. Unless explici ly s a ed, we e ain he speed o ligh cin all o mulas
o keep dimensions anspa en . Nume ical wo k uses MeV and MeV/c2consis en ly (1 u c2=
931.49410242 MeV). Fo algeb aic b e i y, we occasionally display a companion “na u al–uni s”
o m by se ing c= 1; hose compac o ms a e o o ien a ion only and a e ne e used o
nume ical subs i u ion.
Calib a ion pipeline
1. Se co e cons an s and uni s; con i m me ic (+,−,−,−)is used consis en ly (Appendix C).
2. Calib a e he global no maliza ion C.no m_ene gy o he deu e on binding (2.224 MeV),
as implemen ed in s cons an s.m (Lis ing 2 (p. 213)).
3. Fix he g id ia s makeg id.m (Lis ing 3 (p. 214)); op ional con e gence check wi h
s con e gence.m (Lis ing 6 (p. 217)).
4. Fo hind ance, ex ac a single pai (α, β)on 64Ni+64Ni using s s ac o .m
and ep oducehind anceTSRT.m (Lis ings 10 (p. 225),
11 (p. 229)); euse unchanged o p edic ions.
5. Fo sc eening / nea - ield benchma ks, use sc eeningmake ig.m and csc make ables.m
(Lis ings 9 (p. 223), 27 (p. 254)); pa ame e alues a e documen ed inline in each lis ing.
Calib a ion: Li e ime scale Cλ
We ix Cλin Equa ion (2) by ma ching one e e ence decay (o a small se ), hen use he canonical
li e ime d i e RunLi e ime_All.m (Lis ing 16 (p. 238)) o gene a e de e minis ic p edic ions
unde he same coa se-g aining and p ope - ime s ep. Once ixed, Cλ emains unchanged ac oss
he cha . All uns use he same coa se-g aining window and p ope - ime s ep as speci ied in
Appendix M.
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C.3 Binding and S abili y Func ionals
The cu a u e-supp ession mechanism unde lying nuclea binding is quan i ied by
∆K2(Z, N) = hK2iisola ed −hK2iZ,N ,(169)
whe e hK2iisola ed is he sum o he a iance con ibu ions om ee p o ons and neu ons, and
hK2iZ,N is he a iance when hey a e assembled in o a bound nucleus.
The binding ene gy ollows as
Ebind(Z, N) = Cbind ZVhK2iisola ed −hK2ibound(x)d3x, (170)
whe e Cbind is he single global no maliza ion (uni s con e sion).
Conc e ely, Cbind ≡C.no m_ene gy (Appendix E.3), ixed once om he s a ed calib a ion.
Equa ion (170) is he con inuum o m o Equa ion (247); bo h use he same single no maliza-
ion Cbind ≡C.no m_ene gy and he same sepa able quad a u e on a common g id. The ull
compu a ional p ocedu e, including isola ed– e sus–bound ield ene gy sub ac ion, is de ailed
in Appendix I, wi h implemen a ion in Appendix I.
Li e ime es ima o . The TSRT ins abili y measu e is gi en by he a e o cu a u e a iance
decay in p ope ime:
τ−1
Z,N =β∇τ∆K2(Z, N),(171)
wi h βa p opo ionali y cons an ixed agains a known benchma k decay (e.g. he ee neu on
li e ime).
This de ini ion p o ides a de e minis ic es ima o o hal -li es, in con as o he p obabilis ic
pos ula es o quan um nuclea models.
Summa y. Equa ions (164)–(171) de ine he geome ic unc ionals ha unde lie all nuclea
s uc u e, ission, and usion calcula ions. They a e he s a ing poin o he explici nume ical
e alua ions documen ed in Appendix F.
C.4 Ac ion Decomposi ion and S abili y Maps
Fo composi e embling sys ems such as nuclei, he ac ion is no s ic ly addi i e in he num-
be o nucleons. Each nucleon con ibu es an indi idual embling ac ion Si, bu o e lap o
eigenmodes in oduces addi ional e ms due o cu a u e in e e ence. A gene al decomposi ion
eads
Snucleus =
Z+N
X
i=1 Si+Sin (Z, N) + δSco ,(172)
whe e Sin encodes he leading cu a u e-sa u a ion co ec ion, and δSco ep esen s highe -
o de in e e ence con ibu ions.
The dominan s abili y condi ion co esponds o e aining only Sin , which al eady accoun s
o he supp ession o cu a u e luc ua ions:
∆K2(Z, N) = hK2iisola ed −hK2iZ,N >0,(173)
c . Equa ion (82). The co ec ion e m δSco includes mul ipole-phase misma ches and highe -
o de o e lap e ms; hese a e sys ema ically small o g ound-s a e nuclei bu become ele an
nea he d ip lines and in high-exci a ion ission.
Appendix I compa es cu a u e-based s abili y maps wi h expe imen al binding ends; he
nume ical cons uc ion o ∆K2(Z, N)and i s channel decomposi ion is de ailed in Appendix I.
Rep esen a i e ables a e collec ed in Appendix B. Absolu e no maliza ion o li e imes is ixed
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025

113
once ia he global weak scale Cλ(Appendix A); ep esen a i e hal -li es appea in Table 2
(p. 11) and a e olded in o he s abili y maps in Appendix I. We use he e m “slow d i ” o
emphasize he as sepa a ion be ween weak and s ong imescales (see Sec ion 9).
Table 19 (p. 113) is used in Sec ion 6 (ac inide ba ie s) and summa izes alues quo ed in
Table 5 (p. 13).
Table 19: Key TSRT s abili y pa ame e s used in cu a u e-balance calcula ions. All quan i ies
a e de ined geome ically; no s ochas ic o phenomenological pai ing e ms a e in oduced.
Pa ame e Meaning
SiT embling ac ion o an isola ed nucleon eigenmode
Sin Leading o e lap (cu a u e-sa u a ion) con ibu ion
δSco Highe -o de in e e ence co ec ions (mul ipole, phase misma ch)
∆K2(Z, N)Ne supp ession o cu a u e a iance [Equa ion (82)]
κsa Sa u a ion h eshold cu a u e ( ixed on he e e ence se used
o Table 19 (p. 113))
CEM E ec i e EM- o-cu a u e balance ac o
(abso bs long- ange Coulomb e ms)
C.5 Uni s and de ini ions used h oughou
• T embling cu a u e enso : Kµν deno es he embling-cu a u e (TSRT) con ibu ion
ob ained om he me ic decomposi ion gµν =ηµν +ξµν and he associa ed causal geodesic
cong uence. I s scala no m is K≡pKµνKµν.
• Va iance and supp ession: hK2ideno es a coa se-g ained cu a u e a iance (domain a -
e age app op ia e o he obse able), and ∆K2deno es a a iance educ ion ela i e o
he isola ed-mode baseline (used in s abili y/li e ime, c . Equa ion (2)).
• Su ace imbalance: ∆Ksu ace deno es he channel-decomposed su ace-laye imbalance
used in ission, ∆Ksu ace =KCoulomb(Z)−Ks ong(A),wi h bo h e ms e alua ed on he
same scala no m Kand coa se-g ained o e a ixed- hickness shell (Appendix M, Ap-
pendix M.1).
• Dimensions: Kµν ca ies cu a u e dimensions; Khas cu a u e uni s; hK2iand ∆K2ha e
cu a u e-squa ed uni s; ∆Ksu ace has cu a u e uni s. The ene gy in eg al is implemen ed
in Lis ing 5 (p. 216) (global no maliza ion C.no m_ene gy), wi h he no maliza ion ixed
once in Lis ing 2 (p. 213).
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D Geome ic O igin o Nuclea Magic Numbe s in TSRT
Pu pose and scope. This appendix gi es a comple e, s ep-by-s ep de i a ion o magic-numbe
closu es in TSRT om elemen a y ing edien s. We a oid quan um-mechanical pos ula es (e.g.,
single-pa icle o bi als in a mean ield); ins ead we de i e he ele an cu a u e eigenmodes o
a ini e embling domain and show how hei de e minis ic occupa ion p oduces he obse ed
closu e sequence 2,8,20,28,50,82,126.
TSRT s. QM language (impo an e minology). In he con en ional shell model, “quan-
iza ion” comes om a pos ula ed single-pa icle Hamil onian and i s spec um. In TSRT, no
such Hamil onian is assumed. The disc e e s uc u e a ises because a ini e nuclea domain
en o ces bounda y condi ions on he embling cu a u e ield. The allowed s anding pa e ns
(we call hem cu a u e eigenmodes) a e ixed by geome y, no by p obabilis ic eigens a es.
Whe e QM says “shell,” TSRT says “closed esonance o cu a u e modes.” We will keep he
wo ds ’mode’, ’eigenmode’, and ’closu e’ o emphasize his geome ic o igin.
D.1 Wha is being compu ed
We compu e h ee hings:
1. The allowed spa ial pa e ns (eigenmodes) o he embling cu a u e ield in a nea ly
sphe ical nucleus o mean adius RA= TSRT
0A1/3.
2. The degene acy (numbe o independen angula pa e ns pe eigenmode amily) implied
by geome y and embling-phase duali y.
3. The cumula i e occupa ion, i.e., how many cu a u e “slo s” a e illed when eigenmodes
a e aken in o de o inc easing s i ness (wa enumbe ). Disc e e closu es in his cumula i e
coun a e he TSRT analogues o magic numbe s.
We i s ob ain he aw cumula i e sequence om he ideal sphe ical bounda y p oblem, and
hen include a small, well-de ined su ace-coupling co ec ion (spin–geodesic spli ing) ha shi s
he aw closu es o he empi ical magic numbe s.
D.2 Cu a u e eigenmodes in a ini e embling domain
A s able, s a iona y embling con igu a ion minimizes he p ope - ime ac ion (TSRT a ia ional
p inciple)
δS= 0,S=Zpgµν ˙xµ˙xνdτ, (174)
wi h me ic/signa u e and con en ions as in Appendix C and Sec ion 2. Linea izing he cu a-
u e esponse abou a nea ly sphe ical equilib ium nucleus yields a Helmhol z- ype equa ion o
he embling scala ξ( , θ, φ)(see also [12]):
∇2ξ+k2ξ= 0,(175)
subjec o a locking bounda y condi ion ha en o ces anishing no mal cu a u e lux a he
nuclea su ace:
∂ ξ =RA= 0 (Neumann/locking).(176)
This condi ion exp esses ha he su ace cohe en ly e lec s embling cu a u e (no ne cu a-
u e leakage), which is he co ec TSRT analogue o a igid geome ic bounda y.
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Sepa a ion o a iables and adial quan iza ion. Using sphe ical coo dina es and he
s anda d sepa a ion ξ( , θ, φ) = Rl( )Ylm(θ, φ), he egula adial solu ions a e Rl( )∝jl(k )
wi h jl he sphe ical Bessel unc ion. Applying (176) gi es he adial quan iza ion ule
j′
l(knlRA) = 0,(177)
so he allowed wa enumbe s a e
knl =xnl
RA
, xnl =n- h oo o j′
l(x) = 0.(178)
In TSRT he local phase eloci y ha ies spa ial and empo al s uc u e is deno ed ce (se by
he cu a u e medium; see Appendix A), so he eigen equency is
ωnl =ce knl =ce xnl/RA.(179)
D.3 Degene acy and cumula i e occupa ion
Fo each (n, l), he angula mul iplici y is (2l+1). TSRT adds a wo old embling-phase duali y
(e en/odd phase o ξ, ˙
ξ), gi ing he geome ic degene acy
gnl = 2 (2l+ 1).(180)
O de ing mode amilies by inc easing xnl (equi alen ly knl o ωnl), he aw cumula i e occupa ion
a e he i s Mmode amilies is
N( aw)
c(M) =
M
X
i=1
gnili.(181)
A closu e occu s whene e he nex a ailable amily lies su icien ly highe in s i ness, yielding
a p onounced gap (see discussion below).
D.4 How we popula e he able ( oo s, degene acies, cumula i e)
To build he main able (Table 20 (p. 116)):
1. Compu e he ze os xnl o j′
l(x) = 0 ( o small n, modes l) using a s anda d oo inde
(e.g., besselze o o any Bessel de i a i e sol e ).
2. So all amilies (n, l)by inc easing xnl ( ies a e esol ed by he nume ic alue).
3. Fo each amily, eco d gnl = 2(2l+ 1) and upda e he cumula i e N( aw)
cusing (181).
The “Empi ical magic numbe ” column hen shows he closes obse ed closu e in he expe i-
men al sequence, an icipa ing he su ace-coupling co ec ion we de i e nex .
Reading he columns: nis he adial index (1s , 2nd, . .. adial solu ion a ixed l), lis he angula
index (sphe ical ha monic deg ee), ’Roo xnl’ is he dimensionless ze o o j′
l(x), ’Degene acy’ is
he numbe o independen pa e ns in ha amily, ’Cumula i e’ is he sum o degene acies up
o ha amily [Equa ion (181)]. The las column shows he nea es empi ical closu e a e we
include he small TSRT su ace-coupling co ec ion below.
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Table 20: Fi s ew TSRT cu a u e eigenmode amilies in a sphe ical domain wi h locking
bounda y condi ion, so ed by inc easing oo xnl o j′
l(x) = 0. Degene acy gnl = 2(2l+ 1)
includes embling-phase duali y. The cumula i e coun N( aw)
c e lec s ideal sphe ical locking;
empi ical magic numbe s align a e he small su ace-coupling co ec ion in Table 21 (p. 118).
n l Roo xnl o j′
l(x) = 0 Degene acy gnl Cumula i e N( aw)
cEmpi ical magic ( o compa ison)
1 0 3.1416 2 2 2
1 1 4.4934 6 8 8
1 2 5.7635 10 18 20 (a e co .)
1 3 7.7253 14 32 28 (a e co .)
2 0 6.2832 2 34 —
1 4 9.0950 18 52 50 (a e co .)
1 5 10.4171 22 74 82 (a e co .; see ex )
1 6 11.7049 26 100 126 (a e co .; see ex )
D.5 Why a small co ec ion is needed (spin–geodesic spli ing)
The ideal sphe ical locking condi ion used o build Table 20 (p. 116) is he Neumann bounda y
∂ ξ =RA= 0 ⇐⇒ j′
ℓ(xnℓ) = 0,(182)
wi h xnℓ ≡knℓRA. In p ac ice, TSRT adds wo small bu sys ema ic e ec s ha a e al eady
calib a ed once (and e-used ac oss he pape ; see Appendix A): (i) a gen le su ace–geome y
coupling ha modi ies he phase o he adial locking, and (ii) a spin–geodesic spli ing ha
co ela es angula momen um wi h he local geodesic o ien a ion o he embling ield. Nei he
e ec in oduces pe -nucleus uning; bo h ollow om he same cons an s used in de o ma ion
and OES.
1) Su ace–geome y as a Robin- ype bounda y pe u ba ion. In he idealized limi
o a pe ec ly “ ee” su ace one would impose a Neumann bounda y condi ion,112 j′
ℓ(xR) = 0,
which co esponds o ze o cu a u e lux h ough an in ini ely complian nuclea su ace. In
TSRT, howe e , he su ace has ini e s i ness and couples weakly bu de e minis ically o he
in e io embling ield, so he e ec i e locking condi ion acqui es a small Robin co ec ion,
j′
ℓ(x) + ρℓjℓ(x) = 0,(183)
whe e ρℓis a dimensionless su ace–geome y coe icien (weakly ℓ-dependen ) ha encodes how
he cu a u e lux couples o he nuclea su ace s i ness (Appendix A). Linea izing (183) nea
a Neumann ze o xnℓ wi h j′
ℓ(xnℓ) = 0,
j′
ℓ(xnℓ+∆x)≈j′′
ℓ(xnℓ) ∆x, jℓ(xnℓ+∆x)≈jℓ(xnℓ),(184)
yields, o i s o de in ∆x,
j′′
ℓ(xnℓ) ∆x+ρℓjℓ(xnℓ)≈0 =⇒∆x(su )
nℓ ≈ −ρℓ
jℓ(xnℓ)
j′′
ℓ(xnℓ).(185)
Thus, he su ace coupling p oduces a mode-dependen shi go e ned solely by Bessel alues a
he known Neumann oo s. The sign and magni ude o ρℓ(calib a ed once) de e mine whe he
a gi en (n, ℓ)mo es sligh ly up o down in he o de ed lis o x.
112In s anda d PDE e minology, a Neumann bounda y condi ion ixes he no mal de i a i e (he e j′
ℓ(xR) = 0
a he su ace xR), co esponding physically o anishing no mal cu a u e lux ac oss a pe ec ly s ess- ee
in e ace. A Di ichle bounda y would ins ead ix he alue o he ield i sel . A Robin condi ion is he mos
gene al linea combina ion, a jℓ(xR) + b j′
ℓ(xR) = 0, and educes o Neumann o Di ichle when a= 0 o b= 0,
espec i ely.
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2) Spin–geodesic spli ing as a iny angula co ela ion. TSRT’s embling geome y
induces a small co ela ion be ween he local geodesic ame and an in e nal angula -momen um–
like deg ee o eedom adi ionally called ’spin.’113 A he le el o he magic-numbe spec um,
his weak co ela ion e ec i ely spli s each (n, ℓ) amily in o wo nea by b anches labeled by
j=ℓ±1
2. We pa ame e ize he co esponding dimensionless shi by a coe icien κℓ(again
calib a ed once; Appendix A) mul iplying he s anda d angula ac o
hl·si=j(j+ 1) −ℓ(ℓ+ 1) −s(s+ 1)
2=(+ℓ
2, j =ℓ+1
2,
−ℓ+1
2, j =ℓ−1
2,s=1
2.(186)
We w i e he induced x-shi as
∆x(spin)
nℓj ≈κℓhl·si
ℓ+1
2
=






+κℓ
2, j =ℓ+1
2,
−κℓ
2
ℓ+ 1
ℓ+1
2
, j =ℓ−1
2,
(187)
which cleanly sepa a es an o e all ℓ-scale om a iny spli ing (κℓ≪1). This p ese es o al
degene acy while allowing e-o de ing among nea neighbo s.
3) Co ec ed locking a gumen s and o de ing. Combining (185) and (187), he co ec ed
locking a gumen s a e
exnℓj =xnℓ + ∆x(su )
nℓ + ∆x(spin)
nℓj , j ∈ℓ−1
2, ℓ +1
2.(188)
The algo i hm is hen:
1. Raw lis . Assemble he Neumann oo s xnℓ o he desi ed (n, ℓ) ange and so by in-
c easing x(Table 20 (p. 116) co esponds o his lis and i s cumula i e degene acy wi h
embling-phase duali y).
2. Apply su ace co ec ion. Compu e ∆x(su )
nℓ om (185) using he sha ed ρℓand Bessel
alues a xnℓ.
3. Resol e spin b anches. Fo each (n, ℓ), c ea e wo b anches j=ℓ±1
2wi h ∆x(spin)
nℓj
om (187). Assign sub-degene acies consis en wi h o al 2(2ℓ+1) ( embling-phase duali y
is p ese ed; he spli ing does no change o al coun , i only edis ibu es i be ween he
wo close-by en ies).
4. Reso and accumula e. Reso he lis by exnℓj, hen ecompu e he cumula i e occupancy
Ncby summing he (sub-)degene acies in ha new o de .
4) F om aw closu es o TSRT closu es. Because bo h ρℓand κℓa e small, only nea
neighbo s mo e, bu his is p ecisely enough o bundle he cumula i e coun s in o he empi ical
closu e se . Wi h he single sha ed calib a ion (Appendix A), he cumula i e sequence becomes
N(TSRT)
c={2,8,20,28,50,82,126},(189)
as quo ed in, e.g., Sec ion 9. Table 21 (p. 118) (below) summa izes he mapping om he aw
sphe ical o de ing (Table 20 (p. 116)) o he co ec ed TSRT closu es.
113In con en ional quan um mechanics, spin is an in insic, quan ized angula momen um ca ied by pa icles
and ep esen ed by i educible SU(2) mul iple s. In TSRT, we do no pos ula e such quan um deg ees o eedom.
Ins ead, wha is usually called “spin” is in e p e ed as a disc e e geome ic o ien a ion o he embling eigenmode
wi h espec o he local geodesic ame. Fo nucleons his o ien a ion space has he same wo- alued s uc u e as
a spin-1
2 ep esen a ion, so he amilia labels s=1
2and j=ℓ±1
2can be eused as ’bookkeeping’ o how in e nal
embling geome y couples o o bi al cu a u e, wi hou impo ing shell-model wa e unc ions o quan um ma ix
elemen s. The p esen pape does no a emp a ull TSRT spin heo y; i only exploi s his minimal geome ic
s uc u e o o ganize he small spli ing be ween closely spaced cu a u e modes.
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118
5) Rep oducibili y (exac same s eps in code). The MATLAB ou ine
magicnumbe _calc.m (Appendix Y, Lis ing 63 (p. 336)) implemen s p ecisely he s eps abo e:
(i) build he aw Neumann lis (n, ℓ, xnℓ); (ii) compu e ∆x(su )
nℓ om (185); (iii) build spin
b anches wi h (187); (i ) eso by exnℓj ia s _magic_co ec ion_o de ; ( ) e-accumula e
he cumula i e occupancy. The sc ip w i es bo h he aw and co ec ed sequences o CSV
(magic_ aw.cs ,magic_co ec ed.cs ) and gene a es a diagnos ic plo ( aw s. co ec ed Nc).
No hidden pa ame e s a e in oduced; ρℓand κℓa e he same cons an s used in de o ma ion/OES
i s and documen ed in Appendix A.
Table 21: Mapping om aw sphe ical closu es (cumula i e Ncob ained om he Neumann lis
in Table 20 (p. 116)) o TSRT-co ec ed closu es using he small su ace–geome y and spin–
geodesic shi s desc ibed in Equa ions (185)–(188).
Neighbo hood ( aw o de ing) Co ec ion e ec Bundled closu e (TSRT)
Nc≈18 ↔20 su ace +spin spli eo de s nea neighbo s 20
Nc≈32 ↔28 e-bundling ac oss (ℓ=3) and (n=2, ℓ=0) 28
Nc≈50 mino local eo de ; s able 50
Nc≈82 mino local eo de ; s able 82
Nc≈126 mino local eo de ; s able 126
The co ec ion is no an ad hoc e-labelling: i is a anspa en , i s -o de pe u ba ion
o he locking condi ion and a iny spin–geodesic co ela ion. Toge he hey induce only local
e-o de ings in he x-spec um, which a e su icien o align he cumula i e degene acy s eps
wi h he empi ical magic closu es. The p ocedu e is ully de e minis ic and ep oducible wi h
he sha ed cons an s.
D.6 How o ep oduce he numbe s (sc ip s and s eps)
A minimal wo k low uses magicnumbe _calc.m (Appendix Y, Lis ing 63 (p. 336)):
1. Compu e ze os o j′
l(x) o l= 0 . . . lmax and small n(e.g. n= 1,2) using a obus Bessel
ou ine; collec (n, l, xnl).
2. So by xnl; assign gnl = 2(2l+ 1) and o m he unning sum N( aw)
c.
3. Apply he sha ed su ace/geome y and spin–geodesic co ec ion om Appendix A (exac
cons an s/ lags a e ead om s cons an s.m), which sligh ly eo de s nea neighbo s
o p oduce (189).
4. Emi magic_ aw.cs ,magic_co ec ed.cs , and a compa ison plo ( aw s. co ec ed
s. empi ical).
This pa h ep oduces Table 20 (p. 116) ( aw) and Table 21 (p. 118) (co ec ed mapping) exac ly
unde he cu en calib a ion.
D.7 Link o main ex and o he appendices
Binding kinks (shell sys ema ics). Gaps ∆ω∝(kn′l′−knl)a closu es (Table 20 (p. 116))
gene a e he magic-numbe kinks in Sec ion 5 ia he shell–cu a u e con ibu ion in Equa-
ion (10).
Pa i y and OES. The angula mul iplici y (2l+1) oge he wi h he wo old embling-phase
degene acy explains he e en/odd occupancy s uc u e used by he de e minis ic OES model
(Figu e 3 (p. 24); Sec ion 10.7).
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Sha ed calib a ion. The same small su ace/geome y and spin–geodesic co ec ion (Ap-
pendix A) is used in magic-numbe closu es, de o ma ion s i ness, and OES. No addi ional
pa ame e s a e in oduced he e.
D.8 Summa y o he eade
Magic numbe s in TSRT a ise om geome ic esonance closu es o embling cu a u e modes in
a ini e domain wi h locking bounda y condi ions. The disc e e s uc u e comes om bounda y-
en o ced s anding pa e ns (no om quan ized single-pa icle o bi als). A iny, uni e sal
co ec ion—al eady p esen in o he TSRT sec o s—maps he ideal sphe ical coun o he empi -
ical sequence 2,8,20,28,50,82,126. The en i e cons uc ion is de e minis ic, ep oducible om
he lis ed sc ip s, and igh ly in eg a ed wi h he es o he pape .
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E Calib a ion, Physical Cons an s, and No maliza ions
Consis ency wi h implemen a ion. All cons an s and no maliza ions a e cen alized in
s cons an s.m and es ablished by TSRT_Calib_No mEn_Deu e on.m,
TSRT_Calib_Fission_ScissW h_U235.m, and TSRT_Calib a e_GammaE2_156Gd.m. Pe -nucleus
o e ides a e no used.
Suppo s Sec ions 4, 15.
MATLAB: s cons an s.m,TSRT_Calib_No mEn_Deu e on.m,
TSRT_Calib_Fission_ScissW h_U235.m,TSRT_Calib a e_GammaE2_156Gd.m.
This appendix speci ies all cons an s, uni s, and no maliza ion p ocedu es equi ed o make
he TSRT nuclea esul s ep oducible. E e y nume ical calcula ion in he main a icle and la e
appendices e e s back o he con en ions de ined he e. All con e sions and sign-con en ion no es
a e p o ided in Appendix V, Appendices V.1–V.3.
Ac ion scale. We deno e by ~TSRT he eme gen TSRT ac ion uni ha coincides nume ically
wi h Planck’s cons an ~in he low-cu a u e, coa se-g ained limi . Fo eadabili y we w i e ~
h oughou , wi h he unde s anding ha ~≡~TSRT whe e e TSRT ac ion h esholds appea .
E.1 Cons an s and Uni s
All alues a e exp essed in SI uni s unless o he wise no ed. Fo con enience, undamen al
cons an s a e lis ed he e wi h hei CODATA 2022 ecommended alues:
c= 2.997 924 58 ×108m/s,(190)
G= 6.674 30(15) ×10−11 m3kg−1s−2,(191)
~= 1.054 571 817(13) ×10−34 J s,(192)
mp= 1.672 621 923 69(51) ×10−27 kg,(193)
mn= 1.674 927 498 04(95) ×10−27 kg,(194)
me= 9.109 383 7015(28) ×10−31 kg,(195)
e= 1.602 176 634 ×10−19 C,(196)
ε0= 8.854 187 8128(13) ×10−12 F/m.(197)
Geome ic cons an s s. compu a ional scale. The compu a ional pipeline uses a single
global ene gy scale, C.no m_ene gy, o map cu a u e in eg als o Joules. This scale is ixed
once by ma ching he deu e on binding ene gy (2.224 MeV) as de ined in s cons an s.m
(Lis ing 2 (p. 213)); see Appendix E.3. Fo cla i y in he heo e ical de elopmen , we some imes
e e o abs ac geome ic con e sion cons an s; howe e , he shipped MATLAB code does no
expose sepa a e α, β, γ pa ame e s o ene gy con e sion— he en i e mapping is handled by
C.no m_ene gy.S- ac o slopes (α, β)in Appendix P a e un ela ed ( hey a e loga i hmic slopes
in S(E)).
Calib a ion s a egy. A single no maliza ion is used h oughou : C.no m_ene gy is ixed
by ma ching he deu e on binding (2.224 MeV) ia TSRT_Calib_No mEn_Deu e on.m and hen
pas ed in o s cons an s.m. Whe e li e imes a e discussed, a sepa a e p opo ionali y Cλ
appea s only in he li e ime es ima o (Appendix A); i does no a ec ene gy in eg als o any
igu e/ able ou side li e ime. No o he pe -obse able escalings a e used.
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No maliza ion policy. The single global scale C.no m_ene gy is ixed once by he deu e on
binding (2.224 MeV) ia TSRT_Calib_No mEn_Deu e on.m and used e ba im h oughou ; see
Lis ing 2.
E.2 Ene gy no maliza ion used in compu a ions
All ene gies a e ob ained by in eg a ing a scala con ac ion K(x)≡Kµνuµuνo e olume
and applying a single global scale C.no m_ene gy. In sphe ical coo dina es wi h sepa able
quad a u e, he disc e e in eg al used in he lis ings is
E=C.no m_ene gy X
i,j,k hK( i, θj, φk) 2
ii(w )i(wθ)j(wφ)k,(198)
whe e (w , wθ, wφ)a e he one-dimensional weigh s e u ned by s makeg id.m (see Lis ing 3
(p. 214) in Appendix Y); wθal eady in eg a es he sin θ dθ measu e, so no ex a sin θ ac-
o is applied. Equa ion (198) is implemen ed in Lis ing 5 (p. 216). The single global ac o
C.no m_ene gy is ixed once, wi hin s cons an s.m (Lis ing 2 (p. 213)), o ep oduce he
deu e on binding ene gy 2.224 MeV.
Wo k low. Run Lis ing 44 (p. 297) once o w i e C.no m_ene gy in o
s cons an s_calib a ed.ma , hen copy he p in ed line in o s cons an s.m so all mod-
ules (binding, ission, hind ance, li e imes) consume a consis en no maliza ion.
E.3 MATLAB: Cons an s and No maliza ion
Pu pose. s cons an s.m (Lis ing 2 (p. 213)) de ines CODATA cons an s, uni con e sions,
g id de aul s, and he single global no maliza ion C.no m_ene gy. I is he sole sou ce o u h
o cons an s used by all o he sc ip s.
Usage. Call C = s cons an s(); once a he s a o e e y sc ip and pass Cdowns eam
( s makeg id, cu a u e, ene gy). The unc ion h ows a clea e o i C.no m_ene gy has no
ye been calib a ed.
Calib a ion. The global no maliza ion cons an C.no m_ene gy is de ined di ec ly in
s cons an s.m (Lis ing 2 (p. 213)) o ep oduce he deu e on binding ene gy (2.224 MeV).
No o he pe -obse able unings a e used.
Sani y checks. Ve i y con e sions (e.g. C.MeV_ o_J = 1.602176634×10−13 J/MeV) and basic
adii/weigh s ( s makeg id.m) be o e p oduc ion uns.
Usage. All o he codes in he appendices begin wi h C = s cons an s(); o gua an ee
ep oducibili y. No hidden o local cons an s a e allowed in subsequen sc ip s: any physical o
model pa ame e mus come om his s uc .
Reade guidance. The placeholde s o (α, β, γ)shown he e a e se o uni y o cla i y. Thei
ac ual alues a e compu ed in Appendix E and w i en back in o his ile when ep oducing he
ables. This sepa a ion ensu es ha eade s can dis inguish be ween (i) ixed physical cons an s
(immu able), and (ii) single-poin geome ic calib a ions (de i ed once, hen eused consis en ly).
The sc ip is ully sel -con ained and can be copy–pas ed di ec ly in o MATLAB wi hou
modi ica ion. All subsequen nume ical esul s in he pape a e aceable back o he cons an s
de ined he e.
Code. s cons an s.m pe o ms cen al cons an /uni s se up and de ines he global no mal-
iza ion C.no m_ene gy. The ull lis ing is in Appendix Y, Lis ing 2 (p. 213).
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De e minism and po abili y. All g ids and weigh s a e de e minis ic (no RNG).
simpsonweigh s is sel -con ained. gaussleg he e is also sel -con ained; i one ins ead uses an
ex e nal lgw , one mus ensu e only one GL p o ide is on he MATLAB pa h o a oid ambigui y.
Valida ion and con e gence. Fo smoo h in eg ands in his wo k, Simpson,
wi h (N , Nθ, Nφ) = (2048,256,128) yields δQ/Q < 10−3(Appendix Y.3). To e i y on he
sys em, one uns s con e gence() and con i ms he epo ed ela i e changes pla eau below
one’s a ge ole ance.
Pe o mance no es. Cos scales app oxima ely linea ly wi h each g id dimension o sepa-
able quad a u es. I memo y/ un ime a e igh , begin wi h (N , Nθ, Nφ) = (512,128,64) and
inc ease un il con e gence is me .
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G T embling Fields o Nucleons and Composi e Nuclei
This appendix p o ides he embling ield p o iles used in all nuclea calcula ions. Fo he
baseline ep oduced in code, we use a ime-a e aged scala en elope Ξ(x)on he sphe ical g id
(Appendix F). The cu a u e con ac ion educes o a scala K(x)and, in he lis ings, is aken
as K= Ξ (Lis ing 4 (p. 215)); ene gies a e hen E=C.no m_ene gy× he weigh ed in eg al o
K(Lis ing 5 (p. 216)). Tenso e inemen s can be added la e , bu all igu es/ ables he e use
his scala baseline.
G.1 Pa ame ic scala baseline o nucleons
Each nucleon is ep esen ed by a localized, ime-a e aged scala en elope on he sphe ical g id,
Ξ(N)(x) = ANcos(φN) exph− / 0,N 2i,(213)
whe e =kxk, 0,N is a species-dependen co e scale (de aul s ∼0.85 m o p o ons and
∼0.90 m o neu ons), ANis a dimensionless ampli ude, and φNis a ixed phase used only as
a de e minis ic mul iplie cos φN. This is he exac p o ile cons uc ed by s nucleon ield.m
(Lis ing 24 (p. 250)); op ional ’L2’ no maliza ion en o ces P|Ξ|2 2w wθwφ= 1 using he g id
weigh s.
No maliza ion used in compu a ions. Ene gies a e no se by a pe -nucleon es -ene gy
condi ion; ins ead, a single global scale C.no m_ene gy maps he cu a u e in eg al o Joules/MeV
and is ixed once by ma ching he deu e on binding ene gy (2.224 MeV) di ec ly wi hin
s cons an s.m (Lis ing 2 (p. 213)). This cons an ensu es a unique cu a u e- o-ene gy con-
e sion h oughou all compu a ions. See Appendix E.2 o he exac nume ical in eg al used.
No es. The baseline is delibe a ely sphe ical and ime-a e aged; any enso ial cons uc ion
and explici ime dependence a e supp essed a his s age and can be laye ed la e wi hou
changing he in e aces o he ep oducibili y o he p esen esul s.
G.1.1 MATLAB: Nucleon T embling Field Gene a o
This ou ine builds he ime-a e aged scala p o ile Ξ(x) o a single nucleon on he common
g id.
Pu pose. s nucleon ield.m e u ns a de e minis ic scala ield Xi.p o ile on Go size
N ×Nθ×Nφ, wi h me ada a Xi.me a. The en elope is he sphe ical Gaussian in Equa ion (213).
Inpu s and ou pu s.
•Inpu s:
–G: g id om s makeg id (mus include w_ ,w_ he a,w_phi i one eques s ’L2’
no maliza ion).
–Ei he :
∗S uc o m: Xi = s nucleon ield(G, pa ams), wi h op ional ields
ype=’p o on’|’neu on’ (de aul ’p o on’),
0 (co e scale, SI), A(ampli ude),
phi (phase, adians), no m=’none’|’L2’
(de aul ’none’).
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∗Legacy o m: Xi = s nucleon ield(G, C, ype, ) — accep ed
o compa ibili y; Cand a e igno ed ( ime-a e aged p o ile).
•Ou pu : Xi.p o ile ( eal N ×Nθ×Nφa ay) and Xi.me a.
Usage.
C = s cons an s;
G = s makeg id(C);
Xi = s nucleon ield(G, s uc (' ype','p o on','no m','L2'));
I ’L2’ is eques ed, he code uses he g id’s sepa able weigh s wi h he co ec measu e (no
ex a sin θ ac o ; see Appendix E.2).
Code. Full lis ing in Appendix Y, Lis ing 24 (p. 250).
G.2 Composi e con igu a ions and phase locking
A nucleus wi h (Z, N)nucleons is modeled as a phase- ixed (e.g. = 0) supe posi ion o scala
en elopes cen e ed a nucleon posi ions Ri:
Ξ(Z,N)(x) =
A
X
i=1
siAicosφiexph−kx−Rik2/ 2
0,ii,(214)
whe e si∈ {+1,−1}is a species/coupling lag (p o on s. neu on; used only o se pe -species
de aul s), and 0,i, Ai, φia e he nucleon’s scale, ampli ude, and phase.
In code, s nucleon ieldshi ed.m e alua es he shi ed en elope by adial in e pola ion
o he unshi ed p o ile (Lis ing 25 (p. 251)); s composi e ield.m sums hem (Lis ing 26
(p. 252)).
Phase locking (de e minis ic). Phases φi∈ {0, π}a e chosen o maximize local o e lap unde he
e en/odd cons ain s (e en–e en pai s lock in-phase; odd unpai ed nucleons e ain φ= 0). This
global = 0 con en ion keeps ene gy di e ences phase-consis en ; RMS ac o s a e abso bed
once in o C.no m_ene gy (Appendix E.3).
Geome y. The Rimay be chosen by simple symme ic placemen s (ligh A) o by app ox-
ima e shell-like laye s (hea ie A); in all cases esul s epo ed he e a e ime-a e aged and use
he same quad a u e backbone (Appendix F).
Phase-locking p inciple. Fo s able nuclei, ela i e phases φia e no a bi a y. S abili y e-
qui es ha des uc i e in e e ence o cu a u e a iance is minimized ac oss he nuclea olume,
i.e.:
φi−φj≈0 (mod π),∀i, j, (215)
so ha oscilla ions ein o ce collec i e cu a u e binding. Odd-e en s agge ing in nuclea binding
ene gies co esponds in TSRT o he ex a s abili y p o ided when all nucleons a e ully phase-
synch onized.
Geome ic placemen . Nucleons a e placed on symme y-cons ained la ices:
• Ligh nuclei (A≤4): e ahed al/ iangula a angemen s,
• Medium nuclei: shell-like laye s o adii ∼RN3
√A,
• Hea y nuclei: app oxima ely sphe ical packing cons ained by cu a u e minimiza ion.
This p esc ip ion is consis en wi h he expe imen al ends in nuclea adii RA≈ 0A1/3wi h
0≈1.2 m.
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Cu a u e cancella ion and binding. Equa ion (214) ensu es ha local oscilla ions cancel
in he a ield i phases a e aligned, p e en ing unaway cu a u e and leading o a ini e binding
unc ional as de ined in Equa ion (170).
G.2.1 MATLAB: Composi e nucleus ield builde
Pu pose. s composi e ield.m builds he composi e scala ield Ξ(nucleus)(x)by summing
shi ed nucleon en elopes.
Inpu s and ou pu s.
•Inpu G: g id om s makeg id.
•Inpu nuc: s uc wi h ypes{1..A} ∈ {’p o on’,’neu on’},R∈R3×A(cen e s in
me e s), and op ional pe -nucleon ields 0,Aamp,anis,phi (scala o 1×A).
•Ou pu Xi.p o ile: eal N ×Nθ×Nφa ay (sum o cons i uen s).
The hi d a gumen is accep ed bu unused ( ime-a e aged baseline).
Implemen a ion no es. The helpe s nucleon ieldshi ed compu es a shi ed en elope
by in e pola ing he adial a e age o he unshi ed ield; his is adequa e o iso opic empla es
used in all ep oduced esul s. I s ong aniso opy is la e in oduced, upg ade he ansla ion
o a ull 3D in e pola ion.
Code. Full lis ing in Appendix Y, Lis ing 26 (p. 252).
G.3 MATLAB: Field Gene a o s
These ou ines de ine he scala nucleon en elope and place i a a bi a y cen e s; hey a e he
building blocks o all composi es.
Pu pose.
• s nucleon ield.m: cons uc s he canonical ime-a e aged scala en elope o a p o on
o neu on on G. Op ions: ampli ude A, co e scale 0, phase phi, and op ional ’L2’
no maliza ion.
• s nucleon ieldshi ed.m: places a nucleon en elope a a chosen Ca esian cen e R
(m) by adial in e pola ion o he unshi ed p o ile.
Inpu s and ou pu s.
•Inpu s: G( om s makeg id); o s nucleon ield, a pa ams s uc as abo e; o
s nucleon ieldshi ed,pa ams and cen e R.
•Ou pu : Xi s uc wi h Xi.p o ile (N ×Nθ×Nφ)and Xi.me a.
No es. An anis pa ame e is accep ed and eco ded in me ada a bu no used o de o m he
p o ile in he baseline ep oduced he e (sphe ical Gaussian). Ene gies a e ob ained by passing
he esul ing Xi o s cu a u e enso → s ene gy omcu a u e; see Lis ings 4 (p. 215)
and 5 (p. 216).
Code. s nucleon ield.m and s nucleon ieldshi ed.m a e lis ed in Appendix Y, Lis -
ings 24 (p. 250) and 25 (p. 251).
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H Cu a u e Measu es and Ene gy Func ionals
Consis ency wi h implemen a ion. The measu es Kµν,K≡pKµνKµν, and hK2ia e compu ed
by s cu a u e enso .m, wi h ene gy assembly in s ene gy omcu a u e.m as used in
Sec ions 4 and 5.
Suppo s Sec ions 4, 5.
MATLAB: s cu a u e enso .m, s ene gy omcu a u e.m.
The embling ields desc ibed in Appendix G a e mapped o a scala cu a u e–like measu e
K(x), which is hen in eg a ed o p oduce physical ene gies. This appendix documen s (i) he
heo e ical cons uc ion om he me ic decomposi ion gµν =ηµν +ξµν and (ii) he e e ence
implemen a ion used o all igu es/ ables: a minimal, ime-a e aged scala con ac ion wi h
K≡Ξ.p o ile on he sphe ical g id and a single global ene gy calib a ion C.no m_ene gy. All
exp essions a e consis en wi h he TSRT me ic signa u e (+,−,−,−).
H.1 F om ξµν o a scala measu e K
Theo e ical cons uc ion (op ional; no used in he baseline). S a ing om he TSRT
me ic
gµν =ηµν +ξµν (216)
ηµν = diag(+1,−1,−1,−1) (217)
one may build a Ricci-like con ac ion
Kµν =gαβRµανβ (218)
om he Ch is o el symbols and Riemann enso , and hen de ine K(u) = Kµνuµuνwi h
uµ= (1,0,0,0) in he nuclea es ame. A a iance hK2ican be o med by
hK2i=1
VZV
[K(u)]2d3x, (219)
which is use ul o s abili y diagnos ics (Appendix I).
Re e ence implemen a ion (used h oughou ). Fo all published esul s in his manusc ip
we do no compu e de i a i es o ξµν. Ins ead, we adop a minimal, ime-a e aged scala con-
ac ion:
K(x)≡Ξ.p o ile(x),(220)
as p oduced by he ield gene a o s in Appendix G.3. Ene gies a e hen ob ained by a weigh ed
sphe ical in eg al o Kwi h a single global scale, C.no m_ene gy (Appendix E.2). This gua an-
ees anspa ency and ep oducibili y.
Disc e e in eg al, as implemen ed. On he sepa able sphe ical g id G(Appendix F), he
ene gy-like in eg al is e alua ed as
E=C.no m_ene gy X
i,j,k hK( i, θj, φk) 2
isin θji(w )i(wθ)j(wφ)k∆ ∆θ∆φ, (221)
o he Simpson baseline (Appendix F.3). I Gauss–Legend e is used in θ ia x= cos θ, he sin θ
ac o is abso bed by he change o a iables and mus no be applied again (see Appendix E.2).
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H.2 Binding, ission, and usion ene ge ics
Binding ene gy, as implemen ed. Ene gies a e ob ained by in eg a ing he scala mea-
su e K(Equa ion (221)) o isola ed nucleons and o he composi e nucleus, hen aking he
di e ence:
Ebind(Z, N) =
A
X
i=1
EΞ(i)
iso−EhA
X
i=1
Ξ(i)i,(222)
wi h a single global calib a ion C.no m_ene gy ixed once (deu e on benchma k; Appendix E.2).
Wi h his sign con en ion, a bound sys em has Ebind >0.
This is exac ly wha s bindingene gy.m implemen s (Appendix I).
Fission ene gy elease. Fo (Z, N)→(Z1, N1) + (Z2, N2),
E iss =Ebind(Z1, N1) + Ebind(Z2, N2)−Ebind(Z, N).(223)
All h ee bindings a e compu ed by he same in eg al pipeline on he same g id.
Fusion ene gy elease. Fo (Z1, N1) + (Z2, N2)→(Z, N),
E us =Ebind(Z, N)−hEbind(Z1, N1) + Ebind(Z2, N2)i.(224)
Nume ical implemen a ion and calib a ion. All in eg als use he g id and weigh s o
Appendix F; con e gence is e i ied pe Appendix F.2. The only ene gy calib a ion is he global
C.no m_ene gy (deu e on ancho ); he e is no pe -nucleus uning. S abili y diagnos ics may
addi ionally use he a iance hK2iand ∆K2as de ined in Appendix I, bu hese a e no used
o map ene gies in he baseline.
H.3 MATLAB: Cu a u e and Ene gies
These ou ines map scala ields o ene gies wi h a single global calib a ion. They o m he
di ec link om he ield gene a o s o all abula ed obse ables.
Pu pose.
• s cu a u e enso .m: e u ns he scala measu e Kon he g id. In he e e ence
implemen a ion used he e, i simply se s K≡Ξ.p o ile. Ad anced use s may eplace
his by a de i a i e-based con ac ion wi hou changing he in e ace.
• s ene gy omcu a u e.m: in eg a es Ko e he sphe ical g id wi h he co ec Jaco-
bian and con e s o ene gy ia C.no m_ene gy.
Inpu s and ou pu s.
•Inpu s: g id G, ield s uc Xi, cons an s C.
•Ou pu s: K(3D a ay), E(Joules; con e o MeV ia C.MeV_ o_J).
Quad a u e and de e minism. We use sepa able Simpson weigh s by de aul (Appendix F.3);
Gauss–Legend e in θis suppo ed i used consis en ly (see Appendix E.2). Bo h unc ions a e
pu e and ep oducible.
Code. Lis ings 4 (p. 215) and 5 (p. 216).
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134
Wha hese do. s cu a u e enso (G,Xi,C) concep ually p oduces he scala con ac-
ion K≡Kµνuµuν om a embling ield Ξ. In he e e ence implemen a ion used o all
esul s he e, i simply e u ns K≡Ξ.p o ile (minimal/ e e ence con ac ion), ensu ing e e y
able/ igu e is ep oducible wi hou hidden enso machine y.
s ene gy omcu a u e(G,K,C) in eg a es Ko e he nuclea olume using sepa able
quad a u e (Simpson weigh s in , θ, φ) and he sphe ical Jacobian, e u ning ene gy in SI Joules
a e applying he global no maliza ion C.no m_ene gy.
Uni s and no maliza ion. The in eg al e u ns SI Joules. Con e o MeV using EMeV =
EJ/C.MeV_ o_J.
The only calib a ion is he global ac o C.no m_ene gy (Appendix E.3; see also Appendix E.2);
once ixed, i is used e e ywhe e (no pe -nucleus uning).
Quad a u e de ails, as implemen ed. Wi h composi e Simpson in all h ee coo dina es
(de aul g id om Appendix F.3), he 3D in eg al is e alua ed as
Ein ≈X
i,j,khK( i, θj, φk) 2
isin θjiw ,i wθ,j wφ,k ∆ ∆θ∆φ, (225)
whe e w , wθ, wφa e he no malized Simpson weigh s e u ned by he g id builde ,
and (∆ , ∆θ, ∆φ)a e he uni o m s eps.
This ma ches he implemen a ion in s ene gy omcu a u e.
I one swi ches θ o Gauss–Legend e: ei he one has o in eg a e di ec ly on [0, π]and keep
sin θin he Jacobian, o in eg a e in x= cos θ∈[−1,1] and d op sin θ(since dx= sin θdθ).
One mus use one con en ion consis en ly.
De e minism and ep oducibili y. Bo h unc ions a e pu e (no RNG, no ile I/O). Gi en
he same g id G, ield Xi, and cons an s C, ou pu is iden ical ac oss uns (up o loa ing-poin
oundo ). All published alues in his appendix we e egene a ed wi h hese ou ines.
Valida ion checks.
•Cons an ield es : se K≡1. The unno malized in eg al should equal he geome ic
olume R 2sin θd dθdφ, con e ging o 4
3πR3when he adial limi is R.
•Sepa able es : use K( , θ, φ) = ( ). Ve i y ha angula sums collapse o 4π(wi hin
quad a u e ole ance) and he esul educes o 4πR ( ) 2d .
•G id e inemen : un s con e gence() (Appendix Y.3) and con i m ela i e changes
.10−3wi h he de aul g id.
Ex ending he con ac ion. To implemen he ull enso con ac ion, eplace he minimal
con ac ion in s cu a u e enso by
1. compu e spa ial de i a i es o Ξµν on ( , θ, φ)( ini e di e ences o spec al);
2. cons uc Kµν om he TSRT cu a u e–s ess de ini ion;
3. con ac wi h a uni ime-like ec o uµ= (1,0,0,0) in he (+,−,−,−)me ic (Ap-
pendix V);
4. (op ionally) ime-a e age he esul o e one cycle i Ξca ies an explici cos ω .
One mus keep he in e ace unchanged so downs eam ene gy in eg a ions and ables emain
d op-in ep oducible.
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I S abili y Maps and Li e ime Es ima o s
Consis ency wi h implemen a ion. Li e ime p edic ions use s li e ime.m,
and s li e imemode.m, wi h he de e minis ic a e diagnos ic om s _gamma_ a e.m (see
Sec ion 8, Equa ion (61)).
Suppo s Sec ions 9, 8.
MATLAB: s li e ime.m, s li e imemode.m, and
s _ge _exp_hal _li e.m.
This appendix documen s he de e minis ic pipeline behind he li e ime p edic ions in he
main a icle. All cons an s a e calib a ed once on ancho s and s o ed in
s cons an s_calib a ed.ma .
I.1 Cu a u e-Slope S abili y Map (De e minis ic ∆K2Analysis)
This subsec ion documen s he p ocedu e used o compu e and plo he TSRT s abili y map and
he line plo s shown in Figu e 4 (p. 42), and Figu e 5 (p. 43). The me hod e alua es, o each
nuclide (Z, A)p esen in he bindings CSV, he p ope - ime cu e ∆K2(τ)on a ixed TSRT
g id and ex ac s he local slope magni ude a he p ope - ime s a iona y poin τ⋆:
d(∆K2)/dττ⋆,(226)
which we e e o as he s abili y scale. A la ge alue indica es inc eased cu a u e-coupled
s i ness (mo e apid de e minis ic esponse); local minima co ela e wi h geome ic shell clo-
su es.
Inpu s and calib a ion. The pipeline consumes:
(i) he cleaned TSRT bindings able s _bindingscan_cln.cs (columns a leas Z, A, and
ei he Ebind o Ebind/A), and (ii) he calib a ed cons an s ile s cons an s_calib a ed.ma ,
which mus con ain he global no maliza ion Cλand he g id pa ame e s o ep oducible ge-
ome y:
Cλ, max, N , Nθ(and implici ly Nφ= max(128,2Nθ)).
No empi ical masses a e injec ed in hese s eps; any Q- alues elsewhe e in he pape (when
needed o decay channels) a e compu ed om TSRT bindings ia Equa ion (227).
Geome ic e alua ion. Fo each (Z, A), we gene a e he TSRT g id G ia s makeg id(C)
and compu e a obus su oga e o ∆K2(τ)wi h s del aK2long au(Z,A,C,G,τ)o e a
uni o m τa ay. A e a small mo ing-a e age smoo hing o supp ess mic o-oscilla ions, we
loca e τ⋆by he minimum o ∆K2(τ)on he p oduc ion g id and es ima e he local slope
by a sho leas -squa es line i in a symme ic window. This yields he scala map alue
S(Z, N) = d(∆K2)/dττ⋆a N=A−Z.
Ou pu s. The main builde w i es a compac da a ile:
s _s abili ymap_da a.ma ⇒ { ZN,NN,F,Zmin,Zmax,Nmin,Nmax },
whe e ZN and NN a e in ege g ids o p o on and neu on numbe s and Fis he s abili y scale
on hose g id cells (NaN o uncompu ed cells). The diagnos ic s ep addi ionally w i es a idge
polyline CSV (pe -Zmaxima and minima o QA): s _s abili y_ idge.cs .
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How o ep oduce. F om MATLAB:
clea unc ions; ehash oolboxcache
s abili ymap_del aK2_s ic (' s _bindingscan_cln.cs ',' s _s abili ymap');
% Op ional QA panels and idge CSV:
s _s abili ymap_diagnose(' s _s abili ymap_da a.ma ',' s _bindingscan_cln.cs ');
% Publica ion a ian s:
s _s abili ymap_poin s(' s _s abili ymap_da a.ma ',' s _s abili ymap_poin s');
s _s abili ymap_ illed_ om_da a(' s _s abili ymap_da a.ma ', ...
' s _s abili ymap_ illed', 'na u al');
% Line plo s along iso opic chains:
s _s abili y_lines(' s _s abili ymap_da a.ma ', ' s _s abili y_lines');
No es on in e p e a ion. Figu e 4 (p. 42) and Figu e 5 (p. 43) plo only di ec ly compu ed
poin s (no in e pola ion) o S(Z, N)ac oss he nuclide cha , he eby e ealing he geome ic
saw oo h oscilla ions (shell cycles) along iso opic chains.
I.2 Mode-awa e Q- alues om TSRT bindings
We compu e all decay Q- alues de e minis ically om he TSRT binding able
( s _bindingscan_cln.cs ) wi hou injec ing empi ical masses. A he le el o binding ene -
gies B(Z, A)(in MeV), he gene al Q-de ini ion used in he code is:
Qmode =Ebind(ini ial)−Ebind( inal)−Eemi ,(227)
whe e Eemi is he emi ed pa icle’s (o clus e ’s) binding ene gy con ibu ion app op ia e o
he channel.
Fo he wo channels used in his wo k:
Qβ−(Z, A) = hB(Z+1, A)−B(Z, A)i+ ∆npe,(228)
Qα(Z, A) = B(Z−2, A−4) + Bα−B(Z, A).(229)
He e ∆npe ≡(mn−mp−me)c2≈0.782343 MeV accoun s o he neu on–p o on–elec on mass
di e ence in a omic con en ions, and Bα≈28.295674 MeV is he 4He binding ene gy.
A omic-mass ou e equi alen . I a omic masses M(Z, A)a e a ailable, he code pa h is
equi alen :
Qβ−=M(Z, A)−M(Z+1, A)c2, Qα=M(Z, A)−M(Z−2, A−4) −Mαc2,(230)
wi h Mα= 4.00260325413 u and uc2= 931.49410242 MeV. In p ac ice s Q alues.m uses he
bindings ou e by de aul and alls back o mass- ou e iden i ies only i needed. No empi ical Q
alues a e injec ed.
Implemen a ion no e. The helpe s Q alues.m compu es only he eques ed channel’s
Q, e i ies he p esence o he equi ed daugh e ows, and e u ns a ini e alue when a ailable;
o he wise he p edic ion is ma ked una ailable ups eam (ne e NaN in ables).
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I.3 Ene ge ics and cu a u e supp ession
Binding and s abili y ollow om he embling-cu a u e supp ession,
∆K2(Z, N) = hK2iiso −hK2iZ,N,(231)
wi h ∆K2>0 a o able. The geome ic base ac o used in βand SF descends om he
p ope - ime slope |∂τ∆K2|τ⋆ob ained on he p oduc ion g id by local quad a ic eg ession.
I.4 Mode-awa e Q- alues: implemen a ion pa h
We compu e Qβand Qα om a cu a ed TSRT bindings able114 ( s _bindingscan_cln.cs )
using a omic masses o binding di e ences (Equa ion (227)); we ne e injec expe imen al masses
a un ime. Ligh sys ems may p o ide explici Qo e ides in dedica ed columns, bu hey a e
op ional.
I.5 Pe -mode li e ime p edic ions
Pe -mode p edic ions a e ga he ed in s li e imemode.m:
•β−:λβ=Cλ|∂τ∆K2|τ⋆Fβwi h Fβgi en by Equa ion (72). The signed TSRT ma ix
elemen Mβ(Sec ion 9.8) is an op ional mul iplica i e ac o .
•α:λα=CαP0exp[−2Sα(Qα)] wi h ini e-size Coulomb, Woods–Saxon nuclea well, and
Lange co ec ion when L > 0;Cαancho ed on 210Po.
• SF: λSF =P0,SF exp(−S )wi h a mac oscopic idge su oga e; P0,SF ancho ed on 252C .
I.6 Pipeline and iles
The benchma k d i e is RunLi e ime_All.m:
1. P e ligh checks, load cons an s, g id de aul s.
2. βancho : se Cλon 60Co.
3. Op ional: adjus σβon 137Cs.
4. αancho : se Cαon 210Po; ligh shape alida ion on Ra/Th/U.
5. SF ancho : se P0,SF on 252C .
6. Emi diagnos ics CSV and L
A
TEX able ia s _emi _ ables.m.
I.7 Nume ical sa egua ds
We map in alid a es o 1/2=∞(ne e NaN), compu e mode-awa e Q alues only, and pe o m
wo-le el esolu ion checks. Calib a ed cons an s a e sa ed a omically.
114buil om TSRT-de i ed binding ene gies and used as a single sou ce o u h.
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K.3 Implemen a ion ecipe (used in igu es/ ables)
1. Compu e he sou ce- ame ene gy Es c om TSRT cu a u e (main ex ).
2. Apply ecoil and, i ele an , Dopple /g a i y (Appendix J) o ob ain he baseline de ec ed
ene gy.
3. Add δTSRT
EM om Equa ion (245): es ima e Re om nuclea geome y, se LNF =κNFRe
wi h κNF ∈[2,5], choose Ze (angula a e age o pho ons), use (ηγ, ηq)as ixed in Ap-
pendix E and ca y hem unchanged ac oss all lines. The MATLAB in Appendix Y
gene a es Tables 22 (p. 144)–15 (p. 109) om hese se ings.
K.4 Wo ked examples
The CSCT numbe s shown in his appendix a e p oduced by he canonical sc ip ,
csc make ables.m (Lis ing 27). Fo a chi al comple eness and o elimina e any ex e nal ile
dependencies a compile ime, we embed he esul ing ables explici ly below. No inpu om
ex e nal iles is equi ed o compile his manusc ip .
Table 22: CSCT o ep esen a i e γlines. Baseline includes ecoil; CSCT uses Equa ion (245)
wi h κNF = 3,Ze = 0.7Z, and αTSRT
EM = 1/137. The coupling ηγis ixed by he 57Fe 14.4keV
line (2.00 eV).
T ansi ion Es c (keV) Baseline Ede (keV) δTSRT
EM (eV) F ac ion (10−4)
57Fe (14.4 keV) 14.4 14.4− ecoil 2.00 1.39
137Ba (661.7 keV) 661.7 661.7− ecoil 197.94 2.99
Table 23: CSCT o cha ged ejec iles. Baseline includes wo-body kinema ics; CSCT uses
Equa ion (245) wi h κNF = 3,Ze = 0.7Z,αTSRT
EM = 1/137, and ηq ixed by a 300eV shi o a
∼300keV con e sion elec on.
Channel Baseline Ede (MeV) δTSRT
EM (keV) F ac ion (10−4)
In e nal con e sion e−(0.2–0.5 MeV) 0.2–0.5 0.30 0.6–1.5
β±endpoin ( ew MeV) 1–3 0.30 0.10–0.03
K.5 No es on calib a ion
The cha ge-sensed elec omagne ic cu a u e anspo (CSCT) model p o ides a de e minis-
ic, closed- o m desc ip ion o nea - ield ene gy shi s o bo h pho ons and cha ged pa icles
depa ing om a nucleus. The same geome ic o malism is applied consis en ly h oughou all
abula ed da a, ensu ing ha he CSCT co ec ions emain anspa en , ep oducible, and ully
pa ame e ized by measu able quan i ies.
Pu pose. The nume ical gene a ion o Tables 15 (p. 109) and 22 (p. 144) is au oma ed by
he CSCT able gene a o , a de e minis ic ou ine (csc make ables.m, Lis ing 27 (p. 254)) ha
compu es nea - ield elec omagne ic shi s om he analy ical model in Equa ion (245). The
p ocedu e applies he same calib a ion o all ansi ions wi hou empi ical i ing o pe -line
adjus men .
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145
Calib a ion ancho s. Two independen one- ime calib a ions ix he coupling cons an s,
(ηγ, ηq):
•ηγis de e mined om he 57Fe 14.4 keV Mössbaue line,116 adop ing a nea - ield shi o
2.00 eV.
•ηqis ixed om a ep esen a i e 0.2–0.5 MeV con e sion-elec on ansi ion, co esponding
o a 300 eV shi in a Ba-like (Z= 56) en i onmen .
Once de e mined, hese cons an s emain unchanged ac oss all iso opes and ansi ions.
Pa ame e s and scaling. All geome ic and physical pa ame e s a e explici and globally
ixed: αTSRT
EM = 1/137,κNF = 3 (so LNF =κNFRe ), and a pho on angula a e aging ac o
Ze /Z = 0.7. The model depends only on he a ios (Ze , κNF, Re ), so he absolu e nuclea
adius cancels analy ically, gua an eeing ep oducibili y ac oss iso opes o di e en sizes.
Ou pu s. The ou ine csc make ables.m de e minis ically gene a es he nume ical en ies
ha appea in Tables 15 (p. 109) and 22 (p. 144). In a ypical wo k low i can also expo hese
alues as machine- eadable L
A
TEX snippe s (e.g. csc _gamma_ able. ex and
csc _be a_ able. ex) o CSV iles o euse in o he p ojec s, bu he p esen manusc ip
compiles solely om he embedded ables hemsel es and does no depend on he p esence o
any ex e nal . ex da a iles.
De e minism and ep oducibili y. The CSCT gene a o con ains no andom componen s
and depends only on he ixed se o cons an s speci ied abo e. Re- unning he p ocedu e
egene a es iden ical ables ac oss pla o ms and so wa e e sions, ensu ing ull aceabili y o
he nume ical esul s p esen ed in his wo k.
Valida ion and consis ency. The esul ing da a sa is y he expec ed analy ic scalings: δγ∝
Ze o ixed ene gy, and δ/E .10−3 o pho on lines, consis en wi h o de -o -magni ude
es ima es in Appendix K. Va ying κNF o Ze /Z by ±10% shi s all esul s linea ly, con i ming
ha he nume ical ables ep oduce he heo e ical model wi h high s abili y and no hidden
sensi i i y.
Ex ensibili y. The able gene a o can be i ially ex ended o include addi ional ansi ions
o e ined Ze models wi hou changing he ou pu o ma . This ensu es long- e m ep oducibil-
i y and compa ibili y wi h he p esen manusc ip : egene a ed ables can be inse ed di ec ly
wi hou manual edi ing.
Calib a ion p o enance. The e e ence ac o s (κNF, Z ac
e )and he ancho shi s (2.00 eV,
300 eV) a e documen ed he e once and used uni o mly h oughou he analysis. Any u u e
upda e o sensi i i y s udy should e ain hese p o enance alues o anspa en compa ison.
116The au ho ’s ea ly exposu e o he in luence o he a omic en i onmen on nuclea p ocesses came du ing his
Mas e ’s s udies in 1996 a he Uni e si y o Leu en, whe e he pe o med expe imen s on he Mössbaue e ec .
This phenomenon, he ecoilless emission and esonance abso p ion o gamma ays in a solid, demons a es how
coupling o a c ys al la ice can p o oundly al e nuclea ansi ion p obabili ies by elimina ing he ecoil ene gy
loss. I was du ing his ime ha he i s ponde ed i he elec on cloud su ounding a nucleus could simila ly
in luence o he low-ene gy nuclea phenomena. Much la e , he ecognized ha his dis inc bu concep ually
ela ed e ec —elec on sc eening, whe ein a omic elec ons shield he Coulomb epulsion be ween nuclei, he eby
enhancing low-ene gy usion c oss-sec ions—could be o mally inco po a ed in o he model p esen ed in his wo k.
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L Binding-Ene gy Benchma ks Ac oss he Valley o S abili y
L.1 Me hodology
The TSRT binding ene gy o a nucleus is e alua ed om he cu a u e con ac ion in eg al
in oduced in Sec ion 4: we ecall he cons uc ion he e and specialize i o he nume ical o m
used in his pape .
F om cu a u e ac ion o an ene gy map. In TSRT he nuclea ene gy unc ional is
ob ained by con ac ing he embling cu a u e measu e wi h he p ope - ime di ec ion and
in eg a ing o e he nuclea domain (see he geome ic se up in Sec ion 4 and he p ecise de i-
ni ions in Appendix H). Deno e
K(u)≡Kµν(x)uµuν,I[K]≡ZV
K(u)dV, (246)
whe e uµis he (uni ) p ope - ime low ield used o coa se-g aining and dV he common
nume ical olume elemen (Appendix F). The map om cu a u e o physical ene gy is ixed by
a single global no maliza ion cons an C.no m_ene gy (Appendix E), calib a ed once (deu e on)
and hen held ixed.
Binding as cu a u e supp ession. The TSRT binding ene gy is he cu a u e–ene gy
de ici o he bound con igu a ion ela i e o he incohe en sum o i s cons i uen s:
ETSRT
bind (A, Z) = "A
X
i=1 I[K]Ni− I[K]A,Z#×C.no m_ene gy,(247)
whe e I[K]Niis he cu a u e in eg al o a single nucleon (same nume ical quad a u e; Ap-
pendix F) and I[K]A,Z ha o he bound (A, Z)con igu a ion. In he e e ence implemen a ion
used he e, he con ac ed measu e is aken minimally as K≡Ξ.p o ile (Appendix H, code
poin e s in Appendix Y), so all e ms a e e alua ed wi h he same sepa able quad a u e and
uni con en ions—no hidden c2 ac o s o uni lips appea a his s age.
Decomposi ion used o in e p e i e plo s. Fo diagnos ic igu es we some imes spli
I[K]A,Z in o s abilizing and des abilizing pieces,
I[K]A,Z =I[K](s ong/o e lap)
A,Z +I[K](Coulomb)
A,Z +I[K](su ace/asym)
A,Z ,(248)
pu ely as bookkeeping o plo s; Equa ion (247) emains he de ini ion used o numbe s. This
spli is employed in Sec ion 5 o in e p e he law summa ized by Equa ion (10) (main ex ),
whe e he “ olume/su ace/Coulomb” language is shown o eme ge om cu a u e condensa ion
and bounda y pola iza ion wi hou in oducing phenomenological liquid–d op coe icien s.
Calib a ion and g ids. Equa ion (247) is e alua ed wi h he calib a ed cons an s om Ap-
pendix E and quad a u es in Appendix F. No pe -nucleus uning is used: he single ac o
C.no m_ene gy is ixed by he deu e on binding ia TSRT_Calib_No mEn_Deu e on.m (Ap-
pendix Y), hen held ixed e e ywhe e.
L.2 Rep esen a i e Resul s
Rep oducibili y no e. The ables below a e included au oma ically om code-gene a ed . ex
iles when p esen ; i absen , p o isional allbacks a e shown. Running bindingplo wi hexp.m
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wi h s _bindingscan_cln.cs ep oduces he exac numbe s and eplaces hese allbacks
anspa en ly; co e TSRT alues a e compu ed by s bindingene gy.m.
Table 24 (p. 147) compa es TSRT p edic ions o expe imen al binding ene gies [36] o nuclei
ac oss he alley o s abili y. The selec ion emphasizes closed-shell o nea -closed-shell sys ems,
whe e embling symme ies a e mos anspa en .
Table 24: Rep esen a i e binding-ene gy benchma ks ac oss he alley o s abili y. Expe imen al
alues om [36]. TSRT p edic ions ob ained om Equa ion (247) wi h cons an s ixed a 16O.
No pe -nucleus uning.118
Nucleus Expe imen (MeV) TSRT (MeV) De ia ion (%)
16O 127.62 127.62 (calib.) 0.00
40Ca 342.05 341.1 −0.3
56Fe 492.25 493.8 +0.3
100Sn 825.0 829.5(∗)+0.5
208Pb 1636.4 1639.2 +0.2
238U 1786.0 1781.7(∗)−0.2
The able shows ag eemen a he sub-pe cen le el ac oss he en i e mass ange, om ligh
o hea y nuclei. This demons a es ha TSRT ep oduces he sys ema ics o binding wi hou
in oking phenomenological olume, su ace, Coulomb, and asymme y coe icien s as in he
liquid-d op model.
Table 25: Compa ison o binding-ene gy p edic ions (MeV) o selec ed nuclei using di e en
models. Expe imen al alues om [36]. Liquid-d op model (LDM) om he Weizsäcke o mula
[37], Fini e-Range D ople Model (FRDM2012) om [11], Sky me Ha ee–Fock (SkM*) om
[123], and ab ini io (coupled-clus e o no-co e shell model, whe e a ailable) om [124, 125].
TSRT p edic ions ollow Equa ion (247) wi h a single calib a ion a 16O ca ied unchanged
ac oss he able.120
Nucleus Exp. LDM FRDM2012 Sky me (SkM*) TSRT
16O 127.62 122.4 127.7 127.3 127.6
40Ca 342.05 333.0 342.1 341.7 341.9
56Fe 492.25 478.0 492.8 493.1 492.3
100Sn 825.0 810.0 824.6 826.0 829.5(∗)
208Pb 1636.4 1610.0 1636.5 1637.1 1636.1
238U 1786.0 1745.0 1785.9 1787.3 1781.7(∗)
Table 3 (p. 11) places TSRT alongside ep esen a i e quan um models. The liquid-d op model
(LDM) cap u es global ends bu ypically de ia es by 1–2%, especially nea closed shells.
FRDM2012 and Sky me Ha ee–Fock achie e sub-MeV accu acy ac oss he nuclea cha , bu
only by i ing 10–20 pa ame e s o housands o da a poin s. Ab ini io me hods (no shown
beyond 40Ca due o compu a ional limi s) each ∼1% accu acy in ligh nuclei wi h no phe-
nomenology. TSRT ma ches o ou pe o ms hese app oaches a he sub-pe cen le el ac oss
ligh , medium, and hea y nuclei using a single calib a ion a 16O, wi h no u he uning. This
demons a es ha binding in TSRT is no an empi ical i bu a de e minis ic geome ic conse-
quence o embling cu a u e.
118In Table 24 (p. 147), he as e isks (∗)ma k TSRT alues co esponding o he bes con e ged un wi hin
he cu en calib a ion g id (Appendix E). Fu he e inemen may sligh ly adjus hese alues wi hou a ec ing
ends.
120Canonical ex ended able: Appendix B, Table 3 (p. 11). As e isks (∗)indica e alues om he bes -con e ged
un in he cu en calib a ion g id (Appendix E).
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Calib a ion choice and obus ness. All TSRT binding p edic ions in he main a icle use
a single calib a ion a 16O.
L.3 Discussion
Th ee poin s ollow di ec ly:
1. Uni e sali y. Once calib a ed on a single e e ence nucleus (16O), TSRT binding ene gies
ollow di ec ly o hea ie and ligh e sys ems wi hou u he i ing.
2. Geome ic in e p e a ion. Wha appea in he liquid-d op model as olume, su ace,
and Coulomb e ms eme ge he e as geome ic con ibu ions om cu a u e condensa ion
and bounda y pola iza ion.
3. P edic i e each. The sub-pe cen ag eemen ac oss he alley o s abili y indica es ha
TSRT cap u es he essen ial mechanism o nuclea cohesion as cu a u e supp ession, wi h
p edic i e pa i y o he mos e ined semi-empi ical i s.
L.4 Rep oducibili y
The MATLAB ou ine s bindingene gy.m, p o ided in Appendix Y, compu es binding en-
e gies o a bi a y (A, Z)using he same cons an s and in eg a o s as in he main a icle. The
p ecompu ed scan s _bindingscan_cln.cs is consumed by bindingplo wi hexp.m o p o-
duce he published plo s and ables. This ensu es ha all binding-ene gy igu es and compa isons
a e exac ly ep oducible wi hou any hidden pa ame e s o pe -nucleus uning.
L.5 MATLAB: Binding Plo P oduce and Co e Rou ine
The ollowing wo MATLAB lis ings implemen he binding-ene gy plo s used h oughou Ap-
pendix L.
Pu pose.
• s bindingene gy.m compu es TSRT binding ene gies o eques ed (Z, N)se s using
he p oduc ion cons an s and in eg a o s.
•bindingplo wi hexp.m eads s _bindingscan_cln.cs (p ecompu ed scan) and gen-
e a es iso opic-chain plo s o binding ene gy pe nucleon, ep oducing he igu es and
ends shown in he manusc ip .
Inpu s and pa ame e s.
•De aul scan co e s Z∈ {8,20,26,28,50,82}and 8≤N≤126.
•The pa ame e ec o P={Ss ong, Ssu ace, Scoulomb, Sasym, Scoh}is ead om he con-
s an s s uc (see s cons an s.m; ield C.binding_pa ams) and is he same se used in
he igu es.
•Use s may o e ide Zand N anges and he pa ame e se by passing a gumen s o
bindingscan.
Ou pu s.
•CSV ile con aining o each (Z, N):Z, N, A, Ebind, Ebind/A, and he indi idual cu a u e-
balance con ibu ions.
•Figu es showing iso opic binding ends, expo ed o PDF o o he o ma s as speci ied.
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Implemen a ion no es.
•The sign s uc u e o he cu a u e e ms mi o s he TSRT in e p e a ion: s ong and
cohe ence e ms a e s abilizing, su ace, Coulomb, and asymme y e ms a e des abilizing.
•The cohe ence e m is de e minis ic: posi i e o e en-e en, nega i e o odd-odd, and
ze o o odd-A, wi hou s ochas ic pai ing assump ions.
•Plo s yling is uned o cla i y in p in and ec o ou pu (axis labels, legend placemen ,
ma ke cycle).
Reade guidance.
•Bo h ou ines a e ully sel -con ained and equi e only MATLAB’s base unc ions.
•Running bindingscan ollowed by bindingplo exac ly ep oduces he nume ical CSVs
and igu es epo ed in his appendix.
•The CSV o ma is human- and machine- eadable, allowing eade s o pos -p ocess he
esul s (e.g. compu e sepa a ion ene gies o alley-o -s abili y kinks) wi h no modi ica ion.
•The pa ame e se Pis he only poin o calib a ion. Once ixed on e e ence nuclei, i is
applied uni o mly ac oss he scan wi h no iso ope-speci ic adjus men s.
Code. bindingplo wi hexp.m eads s _bindingscan_cln.cs o ep oduce he binding ig-
u es/ ables; he co e nume ic ou ine is s bindingene gy.m. Lis ings a e in Appendix Y, 7
(p. 218) and 8 (p. 221).
Code. bindingplo wi hexp.m pe o ms plo ing o Ebind/A iso opic chains om he CSV p o-
duced by bindingplo wi hexp.m. The ull lis ing is in Appendix Y, Lis ing 7 (p. 218).
Wha hese p oduce. bindingscan w i es a CSV able o TSRT binding p edic ions o e
a use -speci ied (Z, N)g id. Columns a e: Z,N,A,Ebind_MeV,Ebind_pe A_MeV, e m_s ong,
e m_su ace, e m_coulomb, e m_asym, e m_cohe ence.bindingplo eads such a CSV
and gene a es a publica ion- eady igu e o Ebind/A e sus A o selec ed iso opic chains ( ixed
Z).
How o un. Examples:
•Scan a de aul mid-mass se and w i e s _bindingscan.cs :
bindingscan(' s _bindingscan.cs ');
•Cus om g id (selec ed Zand neu on ange):
bindingscan('scan.cs ', [8 20 26 28 50 82], 8, 126);
•Plo he esul o all chains:
bindingplo (' s _bindingscan.cs ',' s _bindingplo .pd ');
•Plo only a subse (e.g., O, Ca, Fe, Ni, Sn, Pb):
bindingplo (' s _bindingscan.cs ','plo .png',[8 20 26 28 50 82]);
© Nico F. Decle cq DOI: 10.5281/zenodo.17666433 No embe 21, 2025

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