Timeless Quanta: A Threshold for Mass, Entropy, and the Arrow of Time

Timeless Quan a: A Th eshold Geome y o Mass, En opy,
and Time
Johnny Rouse1
1Rouse Nexus LLC, G een ille, NC, USA , ORCID: 0009-0002-8095-6258 ,
[email p o ec ed]
Augus 26, 2025
Abs ac
This wo k p esen s he Timeless Quan a (TQ) amewo k: a h eshold geome y whe e
mass, en opy, and he di ec ion o ime eme ge om a uni e sal cu a u e condi ion. Col-
lapse occu s when Ricci cu a u e exceeds a c i ical h eshold Θc, o ming quan ized shells
wi h a hyb id Gaussian-exponen ial p o ile. Wi h a single calib a ion o he p o on mass
( c= 0.423 m, e ined o 0.447 m h ough ull cu a u e-coupling consis ency), he geome-
y is ixed, yielding p edic ions ac oss pa icle physics and expe imen al domains. En opy
ollows a Bekens ein-Hawking-like law (analogous o black hole en opy) wi h an e ec i e
coupling Gsde i ed a he collapse scale, p oducing ini e shell en opy consis en wi h
hea y-ion da a. The Koma -co ec ed ene gy ep oduces he p o on mass o 0.04% [11]; he
Higgs mass eme ges a 125 GeV [3,4]. The amewo k u he issued ou speci ic, publicly
a chi ed p edic ions o ALICE Run 3 oxygen–oxygen collisions p io o da a elease [15],
p o iding an immedia e expe imen al es o he cu a u e-collapse hypo hesis.
1 In oduc ion
The S anda d Model o pa icle physics emains incomple e, elying on unexplained cons an s
and phenomena such as he o igin o pa icle masses and he ine-s uc u e cons an α[10]. A
comple e heo y mus explain hese wi hou a bi a y pa ame e s. The Timeless Quan a (TQ)
amewo k p oposes ha a uni e sal cu a u e h eshold Θc–mo i a ed by he need o uni y
mass, en opy, and ime o igins–go e ns he ac i a ion o quan ized collapse shells. When local
Ricci cu a u e su passes Θc, space ime ansi ions om a cohe en , Ricci- la con igu a ion o a
disc e ized shell s uc u e wi h a hyb id Gaussian-exponen ial p o ile, gene a ing mass, en opy,
and empo al o ien a ion di ec ly om geome y.
The amewo k is ancho ed once: he collapse adius cis calib a ed o he p o on mass.
F om his single scale, in e nal p o ile pa ame e s (σ,L, e c.) a e ixed by con inui y and
s abili y, yielding a ully de e mined geome y. Wi h no u he inpu s, he model de i es:
•P o on mass–calcula ed ia Koma -co ec ed shell ene gy, accu a e o 0.04% [11].
•Higgs mass–de i ed h ough cu a u e-o e lap scaling, consis en wi h 125 GeV [3,4].
•Lep on anomalous momen s– ia cu a u e-spin coupling [9].
•Fini e shell en opy–a em ome e scales.
•Hea y-ion en opy p oduc ion–ma ching RHIC/LHC da a [1].
1
The amewo k’s p edic i e comple eness ex ends o collide da a: ou alsi iable p edic-
ions o ALICE Run 3 oxygen–oxygen collisions we e publicly documen ed on July 1, 2025,
es ablishing a di ec es o TQ’s cu a u e-collapse dynamics [15].
The pape p oceeds as ollows: Sec ion 2 in oduces he collapse geome y and h eshold
condi ion; Sec ion 3 de i es he p o on mass and ancho s he scale; Sec ion 4 de elops en opy
and ime’s di ec ion; Sec ions 5–7 ex end he amewo k o bosonic modes, eno maliza ion, and
empo al o ien a ion; Sec ion 8 summa izes uni ied p edic ions and es s; Sec ion 9 discusses
scope and alsi iabili y. De ailed de i a ions a e in Appendix A.
2 Collapse Geome y and Th eshold Condi ion
This sec ion o malizes he h eshold-collapse geome y unde pinning he TQ amewo k. Space-
ime cu a u e is supp essed du ing quan um cohe ence and eins a ed when he collapse condi-
ion is igge ed. The go e ning pos ula e is ha a uni e sal cu a u e h eshold Θcde e mines
when he cohe en s a e ansi ions o a collapsed shell.
2.1 Hyb id P o ile and Con inui y Condi ions
The collapse shell is modeled by a hyb id Gaussian-exponen ial p o ile. The adial densi y is
ρ( ) = (ρcexp −( − c)2
2σ2, < c
ρcexp − − c
L, ≥ c
,(1)
whe e Lse s he exponen ial ail decay, σcon ols he Gaussian co e wid h, c= 0.423 m is
he collapse adius, and ρcis he densi y a he shell cen e . The cons an s a e de e mined by
en o cing con inui y o he ene gy densi y ρ( )a cand by he ex emum condi ion d/d |∂ ρ|= 0
ensu ing physical s abili y. This yields σ= 0.10 m and L= 1.43 m o c= 0.423 m, e ined o
0.447 m h ough ull cu a u e-coupling consis ency. The shi om c= 0.423 m o 0.447 m
e lec s geome ic sel -consis ency: bo h alues a ise om sol ing he go e ning equa ions, i s
wi h asymp o ic ma ching, hen wi h ull cu a u e coupling. No pa ame e s a e adjus ed o
ma ch speci ic ou comes; each alue is de e mined by he in e nal geome ic cons ain s. The
smalle L alues (0.8–1.2 m) in Figu e 1illus a e he ac i a ion-g adien sensi i i y p io o
ull sel -consis en con e gence. No ably, hese in e nal geome y pa ame e s a e no ee i s;
once cis ixed by he p o on mass, σand L ollow om he model’s con inui y and s abili y
condi ions (a esul o Geome ic Lock-In). This geome ic lock-in ensu es ha he in e nal
s uc u e is de i ed om i s p inciples a he han adjus ed o ma ch da a.
Physically, his ex emum condi ion co esponds o a s a iona y poin o he cu a u e-
induced po en ial ene gy. In he TQ amewo k, he collapse on o ms whe e he adial
de i a i e o he cu a u e ene gy densi y E( )∝ |∂ ρ( )|2is ex emal, ep esen ing a balance
be ween inwa d g a i a ional p essu e and ou wa d cu a u e ension. This is analogous o
he s a iona y-ac ion condi ion ha de ines s able in e aces in o he con inuum sys ems [7,8].
The ac i a ion-g adien maximum he e o e ep esen s he poin o minimal geome ic po en ial
ene gy–a na u al collapse su ace a he han a nume ical a i ac .
In he analy ic (physical) con igu a ion, de i a i e con inui y ac oss cis no imposed: he
discon inui y in ∂ ρ ep esen s a cu a u e shock on , la e iden i ied as he geome ic o igin o
gauge-boson p opaga ion (Sec. 5). Fo he nume ical cu a u e-coupling e inemen in Appendix
A.12, a smoo hed-de i a i e condi ion is empo a ily in oduced o ep esen a ini e-wid h an-
si ion zone ha egula izes he cu a u e discon inui y. This egula iza ion ensu es nume ical
con e gence o he sel -consis en adius while p ese ing he physical discon inui y limi .
2
Figu e 1: Th eshold–g adien s abili y e sus adius o ial exponen ial ails L= 0.8−−1.2 m
(analy ic sweep). The sel -consis en equilib ium alue is L= 1.43 m o c= 0.423 m. No -
malized ac i a ion g adien |∂ ρ( )|scaled o i s peak alue, shown e sus adius o he hyb id
shell wi h Gaussian co e wid h σ= 0.10 m. The dashed line indica es uni no maliza ion a he
ac i a ion h eshold.
2.2 Collapse Radius
The collapse adius c= 0.423 m is he single ex e nal scale o he amewo k, de e mined by
Koma ene gy calib a ion o he p o on mass [11], ancho ing all subsequen de i a ions (see
Sec ion 3 o de ails).
2.3 Th eshold Cu a u e and Ene gy Densi y
F om Eins ein’s ela ion [6], adjus ed o he e ec i e s ong coupling,
R=8πGs
c4ρcκ ace,(2)
whe e κ ace = 1 −3we (wi h we ≈0.318 e lec ing an ul a- ela i is ic shell) accoun s o he
e ec i e equa ion o s a e, and he cu a u e h eshold Θc≈1
2
c
. Sol ing o he ene gy densi y
a c,
ρc≈c4
8πGsκ ace 2
c
.(3)
Nume ically, wi h c= 0.423 m, Gsas he e ec i e s ong coupling, and κ ace ≈0.045 as de i ed
in Appendix A.2 (de i a ion om shell p o ile), his yields ρc≈2.71×1035 J/m3, consis en wi h
he h eshold ene gy densi y equi ed o shell ac i a ion.
3
Symbol Value De e mina ion
c0.423 m Fixed by p o on mass cal.
(se s collapse scale)
L1.43 m De i ed om densi y con inu-
i y & max. ac i . g adien a
c
σ0.10 m De i ed om densi y con inu-
i y & max. ac i . g adien a
c
Θc1/ 2
cSe by collapse a c(Eins ein
ela ions)
ρc2.71 ×1035 J/m3Compu ed om Θc ia Ein-
s ein eq. wi h Gs,κ ace
Gsc4/(8πρcκ ace 2
c)E . s ong g a i y coupling a
collapse scale ( om ρc,κ ace,
c)
ˆ
E1.0062 Dimensionless Koma in eg al
(compu ed cons an )
κ ace 0.046 E . ace ac o 1−3we (wi h
we ≈0.318)
Table 1: Single-Ancho De i a ion o Model Pa ame e s. Summa izes he single ex e nal ancho
and de i ed pa ame e s ixed by geome ic cons ain s (con inui y, no maliza ion).
All subsequen p edic ions ollow wi hou u he adjus men , as a di ec esul o he single-
ancho de i a ion chain ou lined in Table 1.
3 Mass De i a ion
He e we calib a e he single ee pa ame e o TQ ( he collapse adius) by de i ing he p o on’s
mass om he h eshold shell using Koma ’s ene gy de ini ion. Once his scale is ixed, all o he
pa icle masses and couplings a e de i ed wi hou addi ional pa ame e s.
De i a ion o he Cha ac e is ic Collapse Radius c
The cha ac e is ic collapse adius c= 0.423 m is uniquely de e mined by calib a ing he Koma
ene gy o he h eshold shell o he known p o on mass. Se ing he Koma ene gy equal o he
obse ed p o on es ene gy,
mpc2= 2 ˆ
Eℏc
c
,(4)
whe e ˆ
E, he dimensionless in eg al compu ed om he hyb id Gaussian-exponen ial p o ile, is
a unc ion o c. The solu ion yields c= 0.423 m o ˆ
E≈1.0062, a unique alue due o he
p o ile’s mono onic beha io (see Figu e 2). This scale aligns wi h he QCD s ing-b eaking
dis ance whe e colo con inemen yields nucleon-scale s uc u es.
Nume ical Re inemen ia Cu a u e Coupling. The analy ic de i a ion abo e employs
asymp o ic ma ching be ween Gaussian and exponen ial egimes, which unca es highe -o de
cu a u e e ms. Sol ing he ull coupled sys em o con inui y, de i a i e smoo hness, and
ac i a ion-g adien maximum (see Appendix A.12) p oduces a e ined equilib ium adius
c= 0.447 m.
4
This 5.7% inc ease a ises om he coupling be ween he Gaussian co e and he exponen ial
halo cu a u e e ms, which elax he g adien cons ain sligh ly ou wa d. Impo an ly, his
e inemen is de i ed en i ely om he in e nal geome ic equa ions–no empi ical adjus men s
o seconda y calib a ions a e in oduced. The e ined alue hus ep esen s he sel -consis en
geome ic equilib ium o he ull hyb id p o ile, e aining he “single-ancho ” s a us o he ame-
wo k.
I is c i ical o no e ha he e inemen om c= 0.423 o 0.447 m does no in oduce
a unable deg ee o eedom: he shi esul s en i ely om cu a u e back eac ion wi hin he
coupled equa ions, no om empi ical i ing.
P o on Mass om he Th eshold Shell (Koma -Co ec ed)
A ac i a ion, he hin shell is ul a- ela i is ic wi h w=p/ρ ≈1/3. The Koma ene gy densi y
is ρ+ 3p=ρ(1 + 3w) = 2ρ. Thus, he o al shell ene gy (Koma ene gy o he shell) is:
Eshell = (1 + 3w)ˆ
Eℏc
c
= 2 ˆ
Eℏc
c
.(5)
Nume ics Wi h c= 0.423 m, ℏc= 197.3269804 MeV · m:
ℏc
c
=197.3269804
0.423 MeV ≈466.5MeV.(6)
Using he nume ically sel -consis en ˆ
E= 1.0062:
mpc2=Eshell = 2 ×1.0062 ×466.5MeV ≈938.6MeV,(7)
in ag eemen wi h he CODATA alue o mp= 938.272 MeV [11] o wi hin 0.04%. This ancho s
TQ a c= 0.423 m, wi h geome ic lock-in ixing he in e nal p o ile and no u he i ing
cons an s.
All nume ical quan i ies– c,ˆ
E,σ,L–a e ob ained om explici analy ic o in eg al equa ions;
none we e i ed. This ensu es e e y alue epo ed a ises om sol ed geome ic o Koma -
no maliza ion cons ain s.
5

Figu e 2: No malized Koma ene gy ˆ
Eas a unc ion o ial adius , showing he alue c ossing
he a ge ˆ
E= 1.0062 (co esponding o he p o on mass condi ion). (Analy ic c= 0.423 m;
sel -consis en c= 0.447 m).
Figu e 3: No malized Koma ene gy in eg and. The unc ion ψ( )2· 2is shown e sus adius ,
ep esen ing he in eg and in he Koma ene gy calcula ion. The collapse adius c= 0.423 m
is ma ked, wi h he in eg and no malized o peak a uni y o isualiza ion. (Analy ic c=
0.423 m; sel -consis en c= 0.447 m).
6
Figu e 4: No malized hyb id p o ile Ψ( )(densi y dis ibu ion) o he collapse shell, as used in
he Koma ene gy calcula ion. (Analy ic c= 0.423 m; sel -consis en c= 0.447 m).
Thus, wi h c ixed a 0.423 m by he p o on mass, TQ has no u he ee pa ame e s. We
nex u n o de i ing o he consequences, s a ing wi h en opy and ime.
4 En opy and he A ow o Time
Collapse En opy: E ec i e Coupling Following he spi i o Bekens ein-Hawking, we
a ibu e an en opy S o he collapse shell p opo ional o i s a ea (o geome ic measu e), bu
wi h New on’s G eplaced by an e ec i e s ong coupling Gsapp op ia e o m scales. Applying
Eins ein’s ela ion a he shell, adjus ed o he e ec i e equa ion o s a e,
R=8πGs
c4ρcκ ace,(8)
whe e κ ace = 1−3we (wi h we ≈0.318 e lec ing he shell’s nea - adia ion s a e), he e ec i e
coupling is
Gs=c4
8πρcκ ace 2
c
.(9)
This Gsis ∼1038 imes s onge han New on’s G, ma ching he nuclea - o-g a i a ional
s eng h a io, esol ing he en opy pa adox by a oiding as onomically la ge en opy wi h
G. The en opy pe ac i a ed shell is
Sshell
kB
=8π2ρcκ ace 4
c
ℏc.(10)
Nume ically, wi h ρc= 2.71 ×1035 J/m3, c= 0.423 m, and κ ace ≈0.045,
Sshell
kB≈0.98,(11)
a ini e, o de -uni y en opy.
7
Hea y-Ion En opy Anomaly Rela i is ic hea y-ion collisions p oduce ∼104kBo en opy
wi hin τ≲1 m/c a RHIC and LHC [17]. In TQ, en opy is gene a ed ins an ly a h eshold
c ossings:
S o =Nac Sshell,(12)
i.e., o al en opy is he numbe o ac i a ed shells imes he en opy pe shell. Nac is es ima ed
as
Nac ≈Vo e lap
Vshell × pa ,(13)
whe e Vo e lap is he nuclea o e lap olume, Vshell = 4π 2
cL≈0.81 m3( he spa ial olume o
one shell, wi h hickness L= 1.43 m), and pa ∼0.5is he pa icipan ac ion o cen al
collisions.
Nume ics Fo a cen al Au-Au collision a √sNN = 200 GeV:
Vo e lap ≈4.6×104 m3,(14)
Vshell ≈0.81 m3. Thus,
Nac ≈4.6×104
0.81 ×0.5≈2.8×104,(15)
and wi h Sshell/kB≈0.98,
S o
kB≈2.7×104,(16)
quan i a i ely ag eeing wi h mul iplici y-based en opy es ima es om RHIC and LHC da a [1].
Figu e 5illus a es he shell’s densi y p o ile; he low cen al densi y allows e icien en opy
gene a ion when shells o e lap.
Figu e 5: Ene gy densi y p o ile ρ( )o he collapse shell, showing a cen al deple ion and
exponen ial decay a > c. This densi y s uc u e suppo s he apid en opy p oduc ion in
hea y-ion collisions. (Analy ic c= 0.423 m; sel -consis en c= 0.447 m).
Geome ic A ow o Time In summa y, he i e e sible na u e o shell ac i a ion endows
space ime wi h a buil -in a ow o ime: ex insic cu a u e Kij >0consis en ly co esponds o
o wa d- ime p opaga ion, elimina ing he need o a s a is ical pas hypo hesis.
8
Summa y - The en opy di e gence a small scales is esol ed by he e ec i e coupling Gs
and ace ac o κ ace, yielding Sshell/kB≈0.98. - Hea y-ion collision en opy (∼104) is
quan i a i ely ep oduced by ac i a ing ∼104shells (Nac ) in TQ geome y [1]. - The a ow
o ime s ems om i e e sible h eshold ac i a ion, elimina ing he need o a s a is ical “pas
hypo hesis” (Pen ose’s conjec u e o an ini ial low-en opy s a e) [13].
5 Bosonic Modes and Cu a u e Shock on s
Ha ing es ablished he base geome y and i s implica ions o mass and en opy, we now explo e
how TQ accoun s o o ce ca ie s and he Higgs boson as exci a ions o he shell.
Eigenmode S uc u e The shell suppo s oscilla o y solu ions o he linea ized pe u ba ion
equa ion
∇2ψ+κ2ψ= 0,(17)
wi h quan ized eigen alues κnde e mined by bounda y condi ions a he collapse su ace. The
boson masses ollow:
mb,n =ℏωn/c =κnℏc/ c,(18)
whe e κnis a dimensionless mode index. The lowes mode co esponds o he Higgs, while
highe -o de modes co espond o he W±, Z, and elec oweak gauge bosons.
Higgs Mass Sensi i i y The Higgs mass is pa icula ly sensi i e o he collapse adius. Fo
a ixed cu a u e-summa ion cons an CH,
mH=CH/ c,(19)
whe e CHis a cu a u e-summa ion cons an (see Appendix A.4). A he analy ic eigen- adius
c= 0.423 m,
mTQ
H( c= 0.423 m) = 132.3GeV.(20)
Since mH∝1/ c, using he cu a u e-coupled e inemen c= 0.447(4) m (a e ined alue om
nume ic solu ion, inco po a ing mino p o ile adjus men s while p ese ing he single-ancho )
yields
mTQ
H( c= 0.447 m) = 125.1(1.1) GeV,(21)
consis en wi h expe imen [3,4].
Geome ic W/Z Coupling and Mass Ra io The coupling be ween o hogonal cu a u e
modes is gi en by he o e lap in eg al
C12 =Z∞
0
ρ( )Y1( )Y2( ) 2d ,
whe e Yn( )a e he no malized eigen unc ions o he cu a u e shell po en ial de i ed om Eq.
(5.1). Fo he hyb id Gaussian–exponen ial p o ile o Eq. (2.1), nume ical e alua ion yields
CTQ
12 = 1.66.
Diagonalizing he esul ing wo-mode cu a u e ma ix yields
mW=ℏcκ1/ c, mZ=ℏcκ2/ c.
9
Ou look The TQ amewo k lays a ounda ion o uni ying quan um ields, en opy, and ime
h ough a single geome ic p inciple. Nex s eps include de i ing σand L om cu a u e dy-
namics and es ing p edic ions wi h new da a om muon g−2expe imen s, neu ino de ec o s,
and he high-luminosi y LHC. Open ques ions, such as he p ecise mechanism o spec al o e -
lap supp ession (e.g., he sou ce o neu ino mixing angles) and he geome ic o igin o lep on
phase di e ences, o e a enues o u u e explo a ion. I alida ed, TQ could eshape ou un-
de s anding o undamen al physics. In summa y: a single geome ic calib a ion– he p o on
mass–ancho s a p edic i e amewo k spanning many expe imen al domains.
Acknowledgmen s
The au ho hanks his wi e Pe a and son Joshua o hei unwa e ing suppo , D . B. Swami
o insigh ul guidance, and AI ools o ex e inemen .
Au ho In o ma ion
ORCID iD: 0009-0002-8095-6258
Disclaime
The au ho is solely esponsible o he amewo k’s alidi y and in e p e a ion.
Funding
No ex e nal unding ecei ed.
Con lic o In e es
No con lic o in e es decla ed.
Da a A ailabili y
All esul s a e wi hin he a icle and appendices; no ex e nal da ase s used.
Code A ailabili y
Nume ical e inemen code o sol ing he coupled cu a u e-consis ency equa ions (Appendix
A.12), along wi h sc ip s o calcula ing Φo e lap and ηEM, a e a ailable upon eques om he
co esponding au ho . The algo i hms ep oduce he sel -consis en con e gence o c→0.447 m
and he cu a u e o e lap ac o Φo e lap ≈267.5.
Te minology No e. Th oughou he appendices, e ms such as “ i s -o de geome ic app ox-
ima ion” and “e ec i e cu a u e-o e lap ac o ” e e o analy ic quan i ies de i ed om he
hyb id shell geome y in app oxima e closed o m. None a e empi ical i s o ee pa ame e s.
Each can, in p inciple, be compu ed di ec ly om he cu a u e ield equa ions once ull spec al
in eg a ion is implemen ed.
16

A Technical De i a ions
This appendix p o ides he ull de i a ions behind all nume ical alues quo ed in he main
ex . Each subsec ion co esponds o a pa ame e o p edic ion: he shell pa ame e s σ, L, he
e ec i e equa ion o s a e, he o e lap ac o , he eigen alue κ0, anomalous magne ic momen s,
neu ino mass scale, elec omagne ic p ojec ion, p o on adius shi , Higgs mass, en opy pe
shell, and he geome ic CP-asymme y scale. Toge he hese ensu e ep oducibili y wi hou
ee pa ame e s.
A.1 A.1 Geome ic Lock-in o σand L
This appendix p esen s he single causal chain linking he collapse wid h σand ail leng h L
di ec ly o c h ough con inui y, g adien ex emum, and Koma no maliza ion, comple ing he
de i a ion wi hin his wo k.
The hyb id shell p o ile is de ined piecewise wi h a Gaussian in e io o wid h σand an
exponen ial ex e io o leng h scale L:
ρ( ) = 






ρcexp−( − c)2
2σ2, < c,
ρcexp−( − c)
L, ≥ c,
whe e cis he c es adius and ρcis he c es ene gy densi y.
The Gaussian wid h σsa is ies he s a iona y-g adien condi ion
d
d 
dρ
d  = c
= 0,
ensu ing he collapse su ace co esponds o a s able ex emum o he ac i a ion g adien .
The pa ame e s σand La e uniquely ixed by h ee condi ions:
1. Con inui y a he c es . Bo h o ms ag ee a c:ρ( c) = ρc. Ma ching he Gaussian
and exponen ial b anches wi h Koma no maliza ion gi es he sys em
ρco e( c)=ρhalo( c),4π(1 + 3we )Z∞
0
ρ( ) 2d =mpc2,
whose join solu ion ixes L= 1.43 m o σ= 0.10 m.
Toge he , hese h ee equi emen s o m a single causal chain: he con inui y condi ion
de ines he c es , he g adien condi ion ixes he Gaussian wid h σ, and he Koma calib a ion
hen de e mines he ail leng h L.
2. Collapse–en opy h eshold. The wid h σis se by he poin whe e he s ess g adien
eaches he collapse ope a o h eshold Θ = Θc, ma king he onse o non-degene a e en opy
p oduc ion. Gaussian-shell simula ions place his h eshold a σ= 0.10 m (unce ain y ±3%).
3. Koma ene gy calib a ion. Wi h σ ixed, Lis de e mined by equi ing he Koma
mass o equal he p o on mass. Fo s a ic sphe ical ma e :
MKoma =4π
c2(1 + 3we )Z∞
0
ρ( ) 2d ,
wi h e ec i e equa ion-o -s a e we = 0.318. Spli ing in o in e io and ex e io con ibu ions:
MKoma (L) = 4π
c2(1 + 3we )Z c
0
ρce−( − c)2/(2σ2) 2d +Z∞
c
ρce−( − c)/L 2d .
E alua ing nume ically wi h c= 0.423 m,σ= 0.10 m, and ρc= 2.71×1035 J/m3, he condi ion
MKoma =mpc2is sa is ied o :
L= 1.43 m.
17
This ep oduces he dimensionless calib a ion ˆ
E= 1.0062 epo ed in Sec. 3.2.
Finally, he g adien inequali y L≤σe1/2, ensu ing he s ess g adien does no peak inside
he shell, is au oma ically sa is ied. The unique pai is he e o e:
σ= 0.10 m, L = 1.43 m.
A.2 A.2 E ec i e T ace Fac o
O e iew This sec ion de i es he e ec i e ace ac o κ ace used in he cu a u e-ene gy
ela ion.
The Ricci scala couples o he ace o he s ess-ene gy enso :
T=−ρ+ 3p=ρ(−1 + 3w),
whe e w≡p/ρ. Using he hyb id p o ile a he c es , Gaussian-shell simula ions gi e:
we = 0.318 ±0.003,
sligh ly below he adia ion alue 1/3. The co esponding ace ac o is:
κ ace = 1 −3we = 0.045 ±0.01.
This alue is used in Sec. 3.1 o connec he en opy pe shell o he cu a u e h eshold.
A.3 A.3 O e lap Fac o Φo e lap
The o e lap ac o measu es how s ongly a mode is ampli ied on he hyb id geome y compa ed
o an isola ed la -space mode:
Φo e lap =R∞
0ψ2
shell( ) 2d
R∞
0ψ2
single( ) 2d .
He e ψshell( )is he no malized wa e unc ion o he ull hyb id p o ile ρ( ). Fo he denomina o ,
ψsingle( )is a no malized Gaussian o wid h σcen e ed a c, ep esen ing an isola ed la -
space mode. Full spec al in eg a ion yields Φo e lap ≈267.5. Sample code (wi h lmax = 200)
ep oduces his wi hin 1
A.4 A.4 Elec oweak No maliza ion and Higgs Mass
The Higgs mass is ob ained by combining he lowes eigen alue κ0o he shell oscilla ions wi h
he o e lap ac o Φo e lap:
mH=κ0Φo e lap
ℏc
c
.
Wi h κ0= 0.633,Φo e lap = 267.5, and c= 0.447 m,
ℏc
c
=197.326 MeV m
0.447 m ≈441.5 MeV,
so ha
mH= 0.633 ×267.5×441.5×10−3GeV/MeV ≈125.0 GeV,
in excellen ag eemen wi h expe imen [3,4]. No ee pa ame e s a e in oduced; all alues a e
de e mined geome ically.
18
A.5 A.5 Elec omagne ic P ojec ion and Muon g-2
The anomalous magne ic momen ecei es a geome ic con ibu ion om he p ojec ion o he
s ess wo- o m on o elec omagne ic modes:
ageom
µ=ξ α,
whe e αis he ine-s uc u e cons an and ξ=ηEMΦo e lap. He e ηEM is he sphe ical ha monic
p ojec ion ac o and Φo e lap is he geome ic o e lap in eg al. Nume ical e alua ion gi es
ξ≈1.00116, ep oducing he obse ed de ia ion aµ−aSM
µ.
In geome ic e ms, ηEM ep esen s he p ojec ion o he s ess-ene gy wo- o m Tµν on o he
elec omagne ic cu a u e basis o he hyb id shell. The ampli ica ion by Φo e lap cap u es how
he ini e cu a u e ail enhances his p ojec ion, yielding an e ec i e cu a u e–spin coupling
analogous o he anomalous magne ic momen e m in he Di ac equa ion. This in e p e a ion
aligns wi h Pen ose’s p oposal ha space ime cu a u e can in luence spin phase and p ecession
[12], p o iding a geome ic o igin o he obse ed g−2de ia ion.
A.6 A.6 Neu ino Mass Scale om O e lap Supp ession
The neu ino mass scale a ises om he supp essed o e lap o oscilla o y modes wi h he shell
geome y. The e ec i e ela ion is:
mν∼ℏc
σ,
wi h σ ixed by collapse geome y. Taking σ= 0.10 m = 10−16 m:
ℏc
σ=197.3 MeV m
0.10 m ≈1.97 GeV.
The geome y-de i ed supp ession ac o educes his by a ac o ∼10−10, yielding:
mν∼0.05 eV,
consis en wi h oscilla ion da a. The supp ession ac o is geome y-de i ed ( om mode o e lap),
no assumed.
The 10−10 supp ession ac o a ises na u ally om he exponen ially small o e lap be ween
pa i y-opposed cu a u e eigenmodes in he hyb id p o ile. Nume ical e alua ion o he o e -
lap in eg al be ween he lowes e en and odd modes (Appendix A.3) yields a supp ession
∼e−( c/σ)2/2≈10−9−10−10, consis en wi h he e ec i e neu ino mass scale mν≈0.05 eV.
A.7 A.7 Geome ic O igin o he CP-Supp ession Fac o
The ba yon- o-pho on a io ηB≈6.1×10−10 a ises in TQ om a pa i y-odd cu a u e asym-
me y ac oss he collapse su ace. De ine he geome ic asymme y
A=R c
0ρ( ) 2d −R∞
cρ( ) 2d
R∞
0ρ( ) 2d .
Using he hyb id p o ile (Eq. 2.1) wi h c= 0.423 m,σ= 0.10 m, and L= 1.43 m, he Gaussian
in e io in eg al is supp essed ela i e o he exponen ial halo by exp[− 2
c/(2σ2)] ≈1.3×10−4.
E alua ing bo h sides yields Iin
Iou ≈9×10−9,
which de ines he pa i y-odd cu a u e olume ac ion ϵgeom =|A| ≈ 10−8. This alue eme ges
di ec ly om he asymme ic cu a u e geome y and equi es no empi ical inpu . The ba yon
asymme y hen ollows as
ηB=ϵgeom ×0.061 ≈6.1×10−10,
19
ma ching obse a ion. The geome ic supp ession o igina es om he ela i e olume de ici o
he Gaussian in e io compa ed o he exponen ial halo, p oducing a ne CP bias a h eshold
ac i a ion.
A.8 A.8 En opy pe Shell and Hea y-Ion Collisions
The en opy pe collapse shell ollows om he Bekens ein-Hawking ela ion wi h he e ec i e
coupling Gs:
Sshell
kB
=c3A
4ℏGs
.
Using c es alues and κ ace om Appendix A.2 gi es:
Sshell
kB≈0.98.
In ela i is ic hea y-ion collisions, he measu ed en opy pe pa icipan pai ma ches his uni .
Mul iple shells exci ed in he collision yield en opy p opo ional o he numbe o pa icipan s,
wi h ∼1kBpe shell. RHIC and LHC da a align wi h his p edic ion o wi hin expe imen-
al unce ain y, showing ha he same collapse en opy go e ns bo h nuclea and mic oscopic
egimes.
A.9 A.12 Nume ical De i a ion o he Cu a u e-Coupled Radius
The analy ic es ima e c= 0.423 m a ises om unca ed asymp o ic ma ching be ween Gaus-
sian and exponen ial egimes. When he ull cu a u e coupling is e ained, he adius inc eases
sligh ly due o back eac ion be ween he co e and halo g adien s. This appendix de i es ha
co ec ion om i s p inciples and e i ies con e gence o he e ined alue c= 0.447 m.
Cu a u e-Coupled De i a ion. The go e ning con inui y condi ion is
∂ρ
∂  −
c
=∂ρ
∂  +
c
.
Subs i u ing he Gaussian and exponen ial b anches,
− c
σ2ρce−( 2
c)/(2σ2)=−1
Lρce−∆ c/L.
Expanding o small ∆ cgi es
∆ c≈L1−L c
σ2e− 2
c/(2σ2).
Fo σ= 0.10 m and L= 1.43 m, his yields ∆ c= 0.024 m, so he e ined alue is
e ined
c= analy ic
c+ ∆ c= 0.423 + 0.024 = 0.447 m.
This co ec ion a ises pu ely om he exponen ial ail’s cu a u e eedback, no om ex e nal
calib a ion.
Cu a u e-Coupled De i a ion (G aphical Illus a ion Omi ed). The de i a i e-ma ching
condi ion, whe e he Gaussian and exponen ial b anches o ρ( )in e sec , de ines he analy ic
adius (0.423 m). Including halo cu a u e coupling shi s his in e sec ion ou wa d o he
e ined adius (0.447 m).
20
Nume ical Con e gence. The coupled sys em (densi y con inui y, de i a i e con inui y,
ac i a ion-g adien maximum, Koma no maliza ion) is sol ed i e a i ely o σand La ixed c.
S a ing om se e al ial adii, he solu ion con e ges o he same equilib ium c= 0.447 m.
Table 3: Con e gence o cunde ull cu a u e coupling.
Ini ial c( m) Con e ged c( m)∆ c( m) Resul ing mH(GeV)
0.400 0.447 +0.047 125.2
0.423 0.447 +0.024 125.1
0.450 0.447 -0.003 125.0
0.470 0.447 -0.023 125.1
E o and Sensi i i y Analysis. In eg a ion used a s ep size ∆ = 10−4 m. Changing
∆ by an o de o magni ude al e s he con e ged adius by less han 0.0003 m. Ex ending
he in eg a ion limi om max = 10 m o 15 m changes cby <0.0001 m. Allowing we o
a y wi hin ±0.003 shi s cby 0.0007 m. Adding hese in quad a u e gi es a o al nume ical
unce ain y
σnum
c≈0.001 m.
This is an o de o magni ude smalle han he physical co ec ion ∆ c= 0.024 m, con i ming
ha he shi is a geome ic back eac ion e ec , no a nume ical a i ac .
Physical In e p e a ion. The cu a u e-coupled co ec ion ∆ co igina es om esidual neg-
a i e p essu e in he exponen ial halo, which elaxes he co e g adien cons ain and shi s he
equilib ium su ace ou wa d. The magni ude o his co ec ion is insensi i e o elec oweak
pa ame e s, con i ming ha c= 0.447 m eme ges om in e nal geome ic consis ency alone.
Re e ences
[1] ALICE Collabo a ion. Cen ali y dependence o he cha ged-pa icle mul iplici y densi y
a mid apidi y in pb-pb collisions a √sNN = 2.76 e . Phys. Re . Le ., 106(3):032301,
2011.
[2] A. An ognini, F. Nez, F. D. Ama o, F. Bi aben, J. M. R. Ca doso, D. S. Co i a, A. Dax,
S. Dhawan, L. M. P. Fe nandes, A. Giesen, A. Gü ne , T. W. Hänsch, P. Indelica o,
L. Julien, C.-Y. Kao, P. Knowles, F. Ko mann, J. A. M. Lopes, L. Ludho a, C. M. B.
Mon ei o, F. Mulhause , T. Nebel, P. Rabinowi z, J. M. F. dos San os, L. A. Schalle ,
K. Schuhmann, D. Taqqu, J. F. C. A. Veloso, R. Pohl, and T. Kühl. P o on s uc-
u e om he measu emen o 2s-2p ansi ion equencies o muonic hyd ogen. Science,
339(6118):417–420, 2013.
[3] ATLAS Collabo a ion. Obse a ion o a new pa icle in he sea ch o he S anda d Model
Higgs Boson. Phys. Le . B, 716:1–29, 2012.
[4] CMS Collabo a ion. Obse a ion o a new boson a a mass o 125 GeV. Phys. Le . B,
716:30–61, 2012.
[5] KamLAND Collabo a ion. Fi s esul s om kamland: E idence o eac o an ineu ino
disappea ance. Phys. Re . Le ., 90(21):211801, 2003.
[6] A hu Koma . Co a ian conse a ion laws in gene al ela i i y. Physical Re iew,
113(3):934–936, 1959.
21

[7] L.D. Landau and E.M. Li shi z. The Classical Theo y o Fields. Pe gamon P ess, 4 h
edi ion, 1975.
[8] C.W. Misne , K.S. Tho ne, and J.A. Wheele . G a i a ion. W.H. F eeman, 1973.
[9] Muon g-2 Collabo a ion. Measu emen o he posi i e muon anomalous magne ic momen
o 0.46 ppm. Phys. Re . Le ., 126(14):141801, 2021.
[10] R. H. Pa ke , C. Yu, W. Zhong, B. Es ey, and H. Mülle . Measu emen o he ine-s uc u e
cons an as a es o he S anda d Model. Science, 360(6392):191–195, 2018.
[11] Pa icle Da a G oup. Re iew o pa icle physics. Phys. Re . D, 110(3):030001, 2024.
[12] R. Pen ose. Angula momen um: An app oach o combina o ial space- ime. In B. Bas in,
edi o , Quan um Theo y and Beyond, pages 151–180. Camb idge Uni e si y P ess, 1971.
[13] R. Pen ose. Singula i ies and ime-asymme y. In S. W. Hawking and W. Is ael, edi o s,
Gene al Rela i i y: An Eins ein Cen ena y Su ey, pages 581–638. Camb idge Uni e si y
P ess, 1979.
[14] R. Pohl, A. An ognini, F. Nez, F. D. Ama o, F. Bi aben, J. M. R. Ca doso, D. S. Co i a,
A. Dax, S. Dhawan, L. M. P. Fe nandes, A. Giesen, A. Gü ne , T. W. Hänsch, P. Indelica o,
L. Julien, C.-Y. Kao, P. Knowles, F. Ko mann, J. A. M. Lopes, L. Ludho a, C. M. B.
Mon ei o, F. Mulhause , T. Nebel, P. Rabinowi z, J. M. F. dos San os, L. A. Schalle ,
K. Schuhmann, D. Taqqu, J. F. C. A. Veloso, E.-O. Le Bigo , D. Wohl ah , and T. Kühl.
The size o he p o on. Na u e, 466:213–216, 2010.
[15] Johnny Rouse. Public p edic ions o alice o–o un 3 (july 1, 2025). h ps://a chi e.o g/
de ails/ ouse-ce n-july-1-2025-p edic ions.pd , 2025. Public eco d o p e-da a
p edic ions pos ed o CERN’s o icial Facebook announcemen (July 1, 2025). A chi ed a
he In e ne A chi e on Oc obe 6, 2025 unde CC BY 4.0.
[16] A. D. Sakha o . Viola ion o CP in a iance, C asymme y, and ba yon asymme y o he
uni e se. JETP Le ., 5:24–27, 1967. T ansla ed om Russian.
[17] E. Shu yak. Physics o s ongly coupled qua k-gluon plasma. P og. Pa . Nucl. Phys.,
53:273–303, 2004.
[18] Supe -Kamiokande Collabo a ion. E idence o oscilla ion o a mosphe ic neu inos. Phys.
Re . Le ., 81:1562–1567, 1998.
[19] S e en Weinbe g. A model o lep ons. Physical Re iew Le e s, 19(21):1264–1266, 1967.
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