Beyond Smooth Baselines: A Falsifiable Search for Universal Patterns

Beyond Smoo h Baselines: A Falsi iable Sea ch o
Uni e sal Pa e ns
Ma hew J. Hall∗1
1Independen Resea che , Wilming on, DE, USA
Sep embe 17, 2025
Abs ac
Backg ound: A e s anda d baselines a e emo ed, ain pa e ns some imes linge ac oss
e y di e en se ings — om quan um in e e ence and p ecision QED shi s (Casimi ,
Lamb) o cosmological spec a such as he CMB and he s ochas ic g a i a ional-wa e
backg ound (SGWB). These ecu ing hin s aise a simple ques ion: a e hey aces o
eal physical s uc u e, o jus s a is ical noise?
Me hods: To add ess his, we ou line a model-agnos ic amewo k o sea ching such
pa e ns. Candida e signals a e es ed wi h Lomb–Sca gle pe iodog ams and Whi le like-
lihood me hods, combined wi h explici ials co ec ion and model-selec ion c i e ia. The
design emphasizes ep oducibili y, po abili y ac oss da ase s, and p e- egis a ion o make
ou comes unambiguous.
Resul s: We ske ch a compac able- op expe imen using ime-domain double-sli in-
e e ome y as a di ec quan um p obe, and include a oy e-analysis on mock da a. The
oy exe cise shows ha he me hod e u ns clean nulls when no s uc u e is p esen , ye
eliably eco e s weak injec ed pa e ns. Exis ing public da ase s — Casimi , Lamb, CMB,
and SGWB — o e immedia e oppo uni ies o apply he app oach mo e b oadly.
Conclusion: The amewo k is buil o be alsi iable. Consis en nulls would ein o ce
communi y baselines, while he eco e y o common pa ame e s ac oss independen domains
would call o deepe in es iga ion o scale-in a ian phenomena. Ei he way, he ou come
mo es us owa d clea e limi s on wha cu en models can (and canno ) explain.
1 In oduc ion
Sub le s uc u es ha emain a e p ecision measu emen s a e compa ed agains well-es ablished
baselines ha e o en p o ided he i s hin s o o e looked physics. F om labo a o y in e e ence
expe imen s o cosmological su eys, hese “le o e ” pa e ns a e now being examined mo e
ca e ully as possible indica o s o hidden s uc u e.
Classic elec on and pho on double-sli expe imen s demons a ed in e e ence as a di ec man-
i es a ion o quan um cohe ence [1–3], while a osecond “ ime-sli ” expe imen s showed ha
cohe ence can also be con olled in he empo al domain [4,5]. These sys ems se clean base-
lines: any unexplained modula ion is immedia ely es able agains well-unde s ood p edic ions.
Vacuum luc ua ion obse ables such as he Casimi e ec and Lamb shi ex end his logic o
p ecision QED. Bo h ha e been measu ed wi h ex ao dina y accu acy, om Casimi ’s o iginal
p oposal [6] o mode n o sion-balance and mic o esona o s udies [7], and om he i s Lamb
shi obse a ions in hyd ogen [8,9] o ad anced QED calcula ions [10]. He e oo, he baselines
a e so well cha ac e ized ha e en ain depa u es can be quan i a i ely isola ed.
∗ORCID: 0009-0001-7066-2558
1
On he la ges scales, he cosmic mic owa e backg ound (CMB) has been mapped o sub-
pe cen p ecision [11–13], while pulsa iming a ays and g ound-based in e e ome e s p obe
he s ochas ic g a i a ional-wa e backg ound (SGWB) ac oss complemen a y equency bands
[14,15]. Nex -gene a ion obse a o ies such as LISA and DECIGO [16,17] p omise o push hese
baselines e en u he . Collec i ely, hese da ase s ep esen he mos s ingen communi y-wide
es s o scale-in a ian s uc u e.
Sea ching o ain , epea ing pa e ns in such da a equi es obus s a is ical me hods. Lomb–
Sca gle pe iodog ams [18,19], Whi le likelihood es ima o s [20], and in o ma ion c i e ia such
as AIC and BIC [21,22] p o ide well- es ed ools o iden i ying weak signals in noisy con ex s.
A he same ime, sa egua ds such as ials co ec ion and p e- egis a ion o analysis s eps a e
essen ial o a oid alse posi i es [23–25].
In his wo k, we p esen a model-agnos ic amewo k o es whe he shallow log-pe iodic pa -
e ns appea consis en ly ac oss independen domains. The guiding hypo hesis is ha any
genuine s uc u e would eme ge as epea ing ea u es in ln Xwi h a common spacing pa ame-
e β, ega dless o whe he Xdeno es delay ime, pla e sepa a ion, equency, o wa enumbe .
By b inging oge he labo a o y p obes ( ime-domain double-sli ), p ecision QED measu emen s
(Casimi and Lamb), and cosmological da ase s (CMB and SGWB), he amewo k de ines a de-
libe a ely alsi iable es : a consis en βac oss sys ems would mo i a e new heo e ical inqui y,
while obus nulls would sha pen cu en baselines.
2 Scope, Posi ioning, and Falsi iabili y
We emphasize om he ou se ha he p esen amewo k is no a p oposal o new pa icles
o ields, bu a phenomenological esidual empla e o be es ed agains accep ed baselines.
Speci ically, we conside shallow oscilla ions in ln Xwi h spacing β,
R(X)∼cosβln(X/X0),
as a gene ic unc ional o m ha can be applied ac oss domains whe e smoo h heo e ical
expec a ions a e well es ablished: quan um in e e ence, acuum luc ua ion obse ables, and
cosmological backg ounds.
The posi ioning o his s udy is me hodological. We consolida e s a is ical ools al eady s anda d
in as onomy and physics—Lomb–Sca gle pe iodog ams o une en sampling [18,19], Whi le
likelihoods o s a iona y ime se ies [20], and in o ma ion c i e ia such as AIC and BIC o
model selec ion [21,22]. Ou con ibu ion is o p e- egis e he use o hese me hods o sea ch
o a common βac oss o he wise un ela ed physical con ex s.
Falsi iabili y is buil in o he amewo k. A success ul ou come equi es a consis en , globally
signi ican βac oss independen domains, obus unde da a spli s and null es s. Failu e
occu s unde any o h ee condi ions: (i) no peaks emain a e co ec ing o ials ac o s
and look-elsewhe e penal ies, (ii) inconsis en alues o βeme ge be ween domains beyond
quo ed unce ain ies, o (iii) he signal is uns able unde jackkni e es s o disappea s unde
andomized con ols. We emphasize he need o p e- egis a ion o sea ch anges, h esholds,
and signi icance c i e ia o a oid hindsigh bias, in line wi h bes p ac ices o mode n da a
analysis [25,26].
2.1 Quan um In e e ence and Time-Domain Double-Sli
In e e ence phenomena emain one o he mos di ec illus a ions o quan um cohe ence. Elec-
on double-sli expe imen s, beginning wi h Tonomu a’s classic isualiza ion o single-elec on
build-up [1], ha e es ablished ha ma e wa es in e e e wi h hemsel es e en when pa icles
a e se he appa a us one a a ime. Analogous esul s wi h pho ons and neu ons ein o ce
he uni e sali y o he p inciple [2,3].
2
Recen ad ances ha e ex ended he concep om spa ial o empo al domains. In a osecond
physics, “ ime-domain double-sli ” expe imen s gene a e wo well-sepa a ed ioniza ion bu s s
by con olling he elec ic ield o a d i ing lase [4,5]. The esul ing pho oelec on spec a
display in e e ence inges whose spacing encodes he empo al sepa a ion o he sli s. Such
expe imen s ha e opened a new window o p obing cohe ence in he em osecond- o-a osecond
egime.
The heo e ical modeling o isibili y in bo h spa ial and empo al sli s ypically in okes co-
he en supe posi ion o wo ampli udes wi h ela i e phase de e mined by pa h (o delay) di -
e ence [27]. Decohe ence sou ces— ini e cohe ence leng h, en i onmen al noise, and spec al
bandwid h—en e as en elope unc ions ha educe inge con as wi hou al e ing he unda-
men al sinusoidal o m. These models p o ide a eliable baseline agains which any addi ional
s uc u e in esiduals mus be assessed.
2.2 Vacuum Fluc ua ions: Casimi and Lamb
Quan um elec odynamics p edic s measu able consequences o acuum luc ua ions, wo o he
bes known being he Casimi e ec and he Lamb shi . The Casimi o ce be ween pa allel
conduc ing pla es was i s de i ed as a consequence o al e ed ze o-poin modes in con ined
geome ies, yielding an a ac i e o ce in e sely p opo ional o he ou h powe o sepa a ion
L[6]. Mode n p ecision expe imen s, including o sion-pendulum and mic o esona o se ups,
ha e e i ied his scaling o sub-pe cen accu acy while also quan i ying co ec ions om ini e
conduc i i y, empe a u e, and su ace oughness [7,28].
The Lamb shi , o iginally obse ed as a small spli ing be ween he 2S1/2and 2P1/2le els o
hyd ogen, p o ided one o he ea lies con i ma ions o acuum luc ua ions [8,9]. I is now
calculable in QED wi h ex ao dina y p ecision, inco po a ing adia i e, ecoil, and acuum
pola iza ion e ec s [10]. These wo cases p o ide well-es ablished baselines whe e esiduals
beyond he known heo e ical co ec ions can be clea ly de ined and quan i a i ely es ed.
2.3 CMB and S ochas ic G a i a ional-Wa e Backg ound
On cosmological scales, acuum luc ua ions du ing in la ion a e belie ed o seed he empe -
a u e aniso opies obse ed in he cosmic mic owa e backg ound (CMB). The Planck 2018
analysis emains he e e ence baseline, o e ing sub-pe cen le el cha ac e iza ion o he CMB
angula powe spec um [11]. G ound-based ins umen s such as ACT and SPT ha e ex ended
hese measu emen s o smalle angula scales, p o iding independen c oss-checks and high-ℓ
esidual analyses [12,13].
In he g a i a ional-wa e sec o , pulsa iming a ay (PTA) collabo a ions ha e ecen ly e-
po ed common-spec um signals consis en wi h a s ochas ic g a i a ional-wa e backg ound
(SGWB) [14]. A highe equencies, LIGO, Vi go, and KAGRA ha e placed s ingen iso opic
SGWB uppe limi s ac oss he 20–100 Hz band [15]. Fu u e obse a o ies such as LISA and
DECIGO aim o b idge he equency gap, p obing mHz o deci-Hz windows wi h o de s-o -
magni ude g ea e sensi i i y [16,17]. Toge he , hese da ase s o m he cu en communi y
baselines o cosmological and g a i a ional-wa e luc ua ions.
2.4 S a is ical Tooling Used by he Communi y
The sea ch o weak, s uc u ed signals in noisy da ase s elies on well-es ablished s a is ical
me hods. Lomb–Sca gle pe iodog ams p o ide a s anda d app oach o une enly sampled da a,
widely used in as onomy and physics [18,19]. Fo s a iona y ime se ies, he Whi le likelihood
u nishes an e icien app oxima e likelihood in he equency domain [20]. Model selec ion and
e alua ion ypically employ in o ma ion c i e ia such as he Akaike In o ma ion C i e ion (AIC)
[21] and he Bayesian In o ma ion C i e ion (BIC) [22].
In all such sea ches, ca e mus be aken o co ec o he “look-elsewhe e” e ec and o ap-
3
ply ials ac o s, ensu ing ha nominal signi icances e lec he ue p obabili y o spu ious
de ec ions. These p ac ices cons i u e he me hodological baseline agains which any claims o
pe iodic esiduals mus be judged.
3 Me hods
The p oposed amewo k is s uc u ed o allow ep oducible sea ches o shallow log-pe iodic
esiduals ac oss labo a o y and cosmological da ase s. We desc ibe below he expe imen al
p o ocols, e-analysis p ocedu es, and s a is ical ools ha o m he me hodological basis o
his wo k.
3.1 Labo a o y P o ocols
Two complemen a y able- op implemen a ions p o ide di ec expe imen al access o esiduals
in he ime domain. The i s is a pho on-based ime-domain double-sli , whe e wo elec o-
op ic o acous o-op ic modula o s de ine ansmission windows sepa a ed by a a iable delay
∆ . In e e ence isibili y V(∆ ) is measu ed in he a ield using SPAD o sCMOS de ec o s,
and esiduals a e compu ed ela i e o a calib a ed baseline V0. The second is a dis ance-
dependen double-sli geome y, in which sli sepa a ion is a ied in con olled s eps, enabling
es s o loga i hmic pe iodici y in spa ial in e e ence. T ade-o s be ween pho on and elec on
implemen a ions a e summa ized in Table 1, and a schema ic o he ime-domain appa a us is
shown in Fig. 3.
3.2 Re-Analysis o Exis ing Da a
The same esidual empla e can be applied di ec ly o es ablished high-p ecision da ase s: (i)
Casimi o ce measu emen s as a unc ion o pla e sepa a ion L, (ii) Lamb-shi compila ions
ac oss p incipal quan um numbe s, (iii) CMB powe spec um esiduals as a unc ion o mul i-
pole ℓo wa enumbe k, and (i ) SGWB pos e io s as a unc ion o equency . In each case, he
accep ed smoo h baseline B(X) (QED heo y, ΛCDM cosmology, o in e e ome e noise model)
is used o no malize he obse able Y(X), and esiduals a e de ined as R(X)=Y(X)/B(X)−1.
Nega i e esul s in any domain a e epo ed as uppe limi s on he ampli ude a1(β).
3.3 S a is ical F amewo k
Residuals a e epa ame e ized on o a loga i hmic g id x= ln(X/X0), and spec al sea ches a e
pe o med o oscilla o y modula ions o he o m cos(βx +ϕ). Fo une en sampling, Lomb–
Sca gle pe iodog ams [18,19] a e employed; o s a iona y se ies, Whi le likelihood me hods
[20] a e used. In o ma ion c i e ia (AIC, BIC) guide model selec ion [21,22], and global p- alues
a e compu ed ia simula ions o con ol he look-elsewhe e e ec [23]. Boo s ap and jackkni e
esampling p o ide empi ical co a iance es ima es, while injec ion– eco e y es s quan i y sen-
si i i y o shallow modula ions. The o e all analysis pipeline is summa ized in Fig. 4.
3.4 P e-Regis a ion and Con ols
To a oid hindsigh bias, analysis anges, binning choices, and signi icance h esholds a e spec-
i ied in ad ance. Labo a o y p o ocols include single-sli con ols, ga e-o de e e sal, and
bandwid h a ia ion o isola e sys ema ics. Fo a chi al da ase s, null es s a e pe o med by
phase-sc ambling o segmen ing da a, ensu ing ha any signi ican βde ec ion is obus o
su oga e analyses. A simula ed eco e y o an injec ed modula ion is shown in Fig. 5.
4 Residual Templa e and Join -Consis ency P inciple
Fo each da ase , we conside an obse ed spec um o measu emen Y(X) and an accep ed
smoo h baseline B(X) de i ed om communi y-s anda d heo y o calib a ion. The no malized
4
esidual is de ined as
R(X)≡Y(X)
B(X)−1, x ≡ln(X/X0),(1)
whe e X0is an a bi a y e e ence scale. This epa ame e iza ion maps mul iplica i e scalings
in X o addi i e ansla ions in x, acili a ing he sea ch o pe iodic modula ions in log-space.
The esiduals a e modeled as
R(x) =
M
X
m=1
amcosm βx +ϕm+ϵ(x),(2)
whe e amand ϕma e he ampli ude and phase o he m h ha monic, and ϵ(x) deno es noise
con ibu ions modeled by an empi ically calib a ed co a iance C[29]. This o m co esponds
o shallow log-pe iodic modula ions supe imposed on o he wise smoo h spec a.
A cen al p edic ion o his amewo k is he exis ence o a common spacing pa ame e βac oss
independen domains, such as quan um in e e ence, acuum luc ua ion obse ables, and cos-
mological backg ounds. While ampli udes and phases may a y due o ins umen esponse and
physical con ex , he eco e ed alue o βshould be s a is ically consis en i he unde lying
phenomenon is eal. Inconsis ency o βac oss domains, o i s disappea ance unde null and
jackkni e es s, cons i u es a decisi e alsi ica ion.
This app oach echoes log-pe iodic esidual sea ches al eady used in c i ical phenomena and
ea hquake p ecu so s [30], as well as ha monic analysis in cosmology whe e oscilla o y ea-
u es a e sough in he p imo dial powe spec um [31]. By o malizing a c oss-domain join -
consis ency equi emen , we p o ide a alsi iable c i e ion ha sepa a es uni e sal s uc u e
om domain-speci ic sys ema ics.
5 P e-Regis e ed P edic ions (P1–P4)
To ensu e cla i y and alsi iabili y, we s a e explici p e- egis e ed p edic ions (P1–P4). Each
p edic ion speci ies he obse able, he expec ed o m o esidual s uc u e, and he co espond-
ing null condi ion.
•P1 (In e e ence): In a ime-domain double-sli expe imen , he no malized inge is-
ibili y V(∆ ), a e di iding by a smoo h en elope unc ion, is analyzed as a unc ion o
loga i hmic delay ln ∆ . I he esidual empla e is co ec , shallow pe iodic modula ions
wi h spacing βshould be eco e ed. The expe imen al baseline is ha isibili y decays
mono onically wi h inc easing ∆ due o ini e cohe ence leng h [4,5]. Any log-pe iodic
s uc u e mus he e o e appea abo e his well-es ablished end.
•P2 (Vacuum): A e sub ac ion o s anda d QED p edic ions o he Casimi o ce
and Lamb shi [7,10], esiduals plo ed agains ln L(pla e sepa a ion) o ln n(p incipal
quan um numbe in hyd ogenic s a es) should exhibi he same βas in P1. I no consis en
βis p esen , he phenomenology is dis a o ed.
•P3 (CMB ×SGWB): In cosmology, he esidual powe spec um a e sub ac ion o
bes - i ΛCDM baselines [11] can be examined o oscilla o y ea u es in ln k. Likewise,
s ochas ic g a i a ional-wa e backg ound analyses yield pos e io s o s ain powe as a
unc ion o ln [14,15]. A decisi e p edic ion is ha bo h domains eco e consis en β
alues wi hin join unce ain ies. Disag eemen cons i u es a alsi ica ion.
•P4 (Nulls and Con ols): Any genuine signal mus su i e obus ness es s. Phase-
sc ambled o su oga e da ase s should elimina e he pe iodici y, con i ming i is no an
a i ac o spec al leakage o binning. Con e sely, jackkni e esampling and spli -sample
5

analyses should p ese e he eco e ed βi i e lec s unde lying physics. Failu e unde
hese con ols in alida es he claim.
Toge he , hese p e- egis e ed p edic ions ensu e ha he amewo k can be decisi ely es ed.
Posi i e de ec ion equi es c oss-domain consis ency, while nega i e ou comes a e equally in o -
ma i e as hey ule ou hidden log-pe iodic s uc u e a he sensi i i y achie ed.
6 Expe imen al P o ocol: Time-Domain Double-Sli (Table-Top)
6.1 Appa a us
Two complemen a y implemen a ions a e conside ed, bo h based on s anda d in e e ome y
componen s. In he i s , a na owband pho on (o elec on) sou ce is di ec ed h ough wo
as empo al ga es (elec o-op ic o acous o-op ic modula o s o pho ons; pulsed ga ing o
elec ons). These de ine wo empo al ape u es sepa a ed by a con ollable delay ∆ . The
spa ial pa h is held ixed, and a - ield in e e ence is eco ded by a single-pho on a alanche
diode (SPAD) a ay o equi alen de ec o .
In he second app oach, he appa a us is a con en ional spa ial double-sli in e e ome e wi h
a cohe en pho on sou ce (e.g. a enua ed lase o single-pho on emi e ). The no el y lies in
moun ing he de ec ion plane on a calib a ed ansla ion ail, allowing he sli –sc een dis ance L
o be a ied sys ema ically om he nea - ield o he a - ield egime. This e ames he sc een
as a unable p obe o empo al e olu ion, con e ing successi e posi ions in o e ec i e “ ames”
o pho on dynamics [32].
6.2 Measu emen P ocedu e
1. Tempo al ga ing a ian : Calib a e a e e ence isibili y V0a ∆ = ∆ 0. Sweep ∆
on a loga i hmic g id (e.g. mul iplica i e s eps ×1.1) spanning se e al decades. Compu e
esidual isibili y R(∆ ) = V(∆ )/V0−1, se x= ln(∆ /∆ 0), and pe o m Lomb–Sca gle
and Whi le-likelihood scans in βwi h p e- egis e ed ials con ol.
2. Dis ance-dependen a ian : Reco d in e e ence inges a mul iple, e enly spaced
de ec o posi ions L. Ex ac inge isibili y V(L) as a unc ion o Land no malize o a
sho -dis ance e e ence. Classical op ics p edic s mono onic decay o V(L) wi h L, while
he s uc u ed- ime hypo hesis p edic s non-mono onic b ea hing and e i als wi h pe iod
∆LT[32].
6.3 Con ols and Sys ema ics
Robus null es s a e c i ical. Fo he empo al-ga ing me hod: ope a e wi h only one ga e open
(single-sli con ol), e e se ga e o de , a y sou ce bandwid h, spli he da ase by de ec o
segmen o acquisi ion ime, and injec syn he ic signals o con i m eco e y.
Fo he dis ance-dependen me hod: eplica e wi h mul iple pho on wa eleng hs, a y sli sep-
a a ion, and pe o m uns unde di e en en i onmen al condi ions (ai s acuum). S anda d
op ics p edic s ha wa eleng h o medium only a ec s inge spacing, no isibili y e i als.
Obse a ion o dis ance-pe iodic b ea hing independen o wa eleng h would be dis inc i e.
Toge he , hese p o ocols p o ide wo independen p obes— empo al ga ing and dis ance-
dependen de ec ion— ha sha e he same alsi iable c i e ion: mono onic decay suppo s con-
en ional op ics, while ep oducible log-pe iodic e i als suppo s uc u ed empo al dynamics.
7 Re-Analysis Playbook o Exis ing Da a
We ou line a ep oducible wo k low o applying he esidual empla e o ou well-s udied
domains: (i) Casimi o ce s. pla e sepa a ion L, (ii) Lamb-shi compila ions s. p incipal
6
quan um numbe , (iii) CMB powe -spec um esiduals s. ln k, and (i ) SGWB pos e io s s.
ln . The goal is o enable anspa en e-analyses ha ei he de ec a common spacing βo se
uppe limi s on he leading ampli ude a1(β). Nega i e (null) esul s a e equally aluable and
should be epo ed as cons ain s.
7.1 Common Wo k low
1. Da a inges ion and baseline: Load measu emen s Y(X) and adop he communi y
baseline B(X) (QED/Casimi heo y; Planck/ACT/SPT bes - i ΛCDM; PTA/LIGO
SGWB baselines).[7,10–15,28]
2. Residual cons uc ion: Fo m no malized esiduals R(X)=Y(X)/B(X)−1; de ine
x= ln(X/X0).
3. Spec al sea ch: Compu e Lomb–Sca gle pe iodog ams o une en xg ids and/o i
he cosine empla e wi h Whi le likelihood.[18–20]
4. T ials con ol: P e- egis e he βscan ange and ha monic con en m≤M. Con e
local o global signi icance ia Mon e Ca lo o G oss–Vi ells app oxima ions o look-
elsewhe e e ec s.[21–23]
5. Join consis ency: I one domain yields a candida e ˆ
β, es consis ency ac oss o he
domains by a join likelihood o me a-analysis on βwi h nuisance ampli udes/phases.
6. Robus ness: Apply jackkni e/spli es s ( ime/segmen /ins umen ), null su oga es
(phase-sc amble, boo s ap), and injec ion– eco e y o alida e sensi i i y.
7. Repo ing: P o ide ei he (a) a globally signi ican common βwi h unce ain ies and
goodness-o - i , o (b) uppe limi s on a1(β) ac oss he scan, including da a and code o
ep oducibili y.
7.2 (i) Casimi Fo ce s. L
Da ase : P ecision pla e–pla e o ce measu emen s wi h quan i ied co ec ions ( ini e conduc-
i i y, empe a u e, oughness).[7,28]
Baseline: Theo e ical B(L) om QED wi h accep ed co ec ions.
Analysis: Build R(L) and e alua e R s. x= ln(L/L0). Sea ch o a cosine a β; check ha
de ending (choice o B) does no abso b pe iodic s uc u e ( a y smoo hness/spline ension as
a con ol).
Null epo ing: Quo e 95% CL limi s a1(β) o e he scanned ange; p o ide sensi i i y om
injec ion– eco e y.
7.3 (ii) Lamb-Shi Compila ions
Da ase : Hyd ogenic le el shi s ac oss p incipal quan um numbe s wi h mode n QED i s.[8–
10]
Baseline: S a e-o - he-a QED p edic ions B(n) including adia i e/ ecoil/ acuum e ms.
Analysis: Cons uc esiduals s. x= ln n(o ln(∆E/∆E0)). Sea ch o shallow pe iodici y
wi h βconsis en wi h (i).
Null epo ing: Limi s on a1(β); show s abili y unde al e na i e compila ions and e o
in la ion es s.
7.4 (iii) CMB Powe Spec um Residuals
Da ase : Planck 2018 empe a u e/pola iza ion spec a and g ound-based ACT/SPT high-ℓ
da a.[11–13]
Baseline: Bes - i ΛCDM ans e unc ions; esiduals Rℓmapped o R(ln k) ia s anda d ℓ↔k
7
app oxima ions. Apply beam/ o eg ound ma ginaliza ions as in o icial likelihoods.
Analysis: Pe iodog am/likelihood sea ch o βin R(ln k); accoun o mul ipole windowing
and co ela ed e o s (use published co a iance). Con ol he la ge ials space (b oad k ange,
mul iple spec a TT/TE/EE) wi h global p- alues.[18–20,23]
Null epo ing: Tabula e a1(β) limi s o TT, TE, EE and combined; con i m ha phase-
sc ambles e ase any appa en peaks.
7.5 (i ) SGWB Pos e io s s. ln
Da ase : PTA common-spec um pos e io s (e.g., NANOG a 15-yea ) and LIGO/Vi go/KAGRA
iso opic limi s.[14,15]
Baseline: Powe -law o b oken-powe -law s ain spec a B( ) used by he collabo a ions.
Analysis: Build R( ) agains ln ; include band-limi ed co a iances and window unc ions.
Sea ch o βconsis en wi h (iii). Fo PTAs, espec Hellings–Downs spa ial co ela ions as
handled in eleased pos e io s; o LIGO, use he s ochas ic c oss-co ela ion pipelines’ co a i-
ances.
Null epo ing: F equency-dependen limi s a1(β) and combined CMB×SGWB join con-
s ain s; publish analysis no ebooks o eplica ion.
7.6 Uppe Limi s and Global Signi icance
Fo a scan o e β∈[βmin, βmax], epo :
a95%
1(β) : he smalles ampli ude excluded a 95% CL om he likelihood p o ile, (3)
pglobal : he global alse-ala m p obabili y a e ials co ec ion (G oss–Vi ells o MC),
(4)
∆AIC,∆BIC : model-selec ion penal ies when adding he cosine e m(s). (5)
A claim equi es a globally signi ican peak and c oss-domain consis ency o βwi hin quo ed
unce ain ies; o he wise, esul s should be amed as uppe limi s wi h comple e obus ness
checks.
8 S a is ical Me hods
8.1 Baseline Fi ing and De ending
The choice o baseline B(X) is c i ical, as o e - lexible models can abso b he e y esidual
s uc u e o in e es . In p ac ice, B(X) may be aken as a powe -law, spline, o pa ame ic
o m mo i a ed by he physics o he sys em (e.g. QED co ec ions o Casimi /Lamb da a;
ΛCDM ans e unc ions o CMB).[11,28] Robus ness checks include a ying he smoo hing
scale o spline ension, and e i ying ha candida e pe iodici ies pe sis .
Apodiza ion and binning choices can in oduce leakage in o he esidual spec um. We he e o e
ecommend mul iple windowing unc ions (Hann, Tukey, Kaise ) and bin-wid h scans o es
o s abili y. This is s anda d p ac ice in spec al analysis o physical da a [29].
8.2 Pe iodog am and Likelihood
Fo i egula ly spaced x alues (e.g. CMB mul ipoles, Casimi sepa a ions), we employ Lomb–
Sca gle pe iodog ams [18,19]. Fo uni o m x, as Fou ie ans o ms can be used o e iciency.
A complemen a y likelihood-based app oach uses he Whi le likelihood o s a iona y esiduals:
−2 ln L(θ) =  − θ⊤C−1 − θ+ cons ,(6)
whe e θ={β, am, ϕm}, a e he obse ed esiduals, θ he model p edic ion, and C he
co a iance ma ix es ima ed om noise modeling o spli -sample me hods.[20]
8
Unce ain ies on βcan be ob ained om he Fishe in o ma ion ma ix o om Mon e Ca lo
simula ions o syn he ic da ase s. Boo s ap esampling p o ides a u he check agains non-
Gaussian noise.
The o e all analysis pipeline, om baseline i ing o join -consis ency es s, is illus a ed in
Fig. 4.
8.3 Model Selec ion and T ials
Model compa ison employs in o ma ion c i e ia such as he Akaike In o ma ion C i e ion (AIC)
and Bayesian In o ma ion C i e ion (BIC), which penalize he addi ion o cosine e ms cos(βx)
ela i e o smoo h baselines.[21,22]
Because βcan span a wide ange and mul iple ha monics mmay be es ed, ials ac o s ( he
“look-elsewhe e” e ec ) mus be included when quo ing signi icances. This can be add essed
ei he ia Mon e Ca lo simula ions o using analy ic app oxima ions such as he G oss–Vi ells
me hod [23]. Fo mul iple da ase s, alse-disco e y- a e (FDR) con ol can be applied o main-
ain o e all e o a es [24].
A de ec ion claim he e o e equi es: (i) global p- alues below con en ional h esholds a e
ials co ec ion, (ii) imp o emen in AIC/BIC o e baselines, and (iii) consis ency o βac oss
independen domains.
9 Resul s
The amewo k was alida ed h ough simula ed da ase s and applica ion o ep esen a i e
analysis pipelines. Resul s a e p esen ed in wo ca ego ies: a che ypal eco e y es s using
injec ed signals, and null o exclusion scena ios ha de ine alsi iabili y.
9.1 A che ypal Signal Reco e y
Syn he ic esiduals wi h injec ed log-pe iodic modula ions we e gene a ed o es he sensi i i y
o he analysis pipeline. Figu e 5illus a es eco e y o an injec ed cosine wi h ampli ude
a1= 0.02 a β= 4.5 in he p esence o Gaussian noise. A p ominen peak is eco e ed in he
Lomb–Sca gle pe iodog am a he injec ed equency, wi h signi icance su i ing global ials
co ec ion. These examples con i m ha he pipeline co ec ly iden i ies shallow modula ions
when p esen , and quan i ies hei global s a is ical signi icance.
9.2 Null and Exclusion Ou comes
In he absence o an injec ed signal, he pipeline p oduces no spu ious peaks abo e he 95%
global con idence h eshold. Phase-sc ambled and jackkni e esampling con i m he s abili y o
null ou comes, ensu ing ha signi ican peaks canno a ise om s anda d baseline luc ua ions
o s a is ical a i ac s. Such null es s o m he basis o publishing uppe limi s on a1(β) in eal
da ase s.
9.3 Falsi iabili y Condi ions
To guide in e p e a ion, a concise alsi iabili y checklis was cons uc ed. The empla e is uled
ou i : (i) no globally signi ican βpeak is ound in any domain a e p ope ials co ec ion;
(ii) inconsis en β alues eme ge ac oss independen domains; o (iii) appa en signals ail unde
spli s, jackkni es, o su oga e null es s. This s uc u e ensu es ha esul s, whe he disco e y
o null, a e decisi e and ep oducible.
10 Resul s A che ypes (Simula ed Examples)
To illus a e he expec ed ou comes o he p oposed amewo k, we p esen simula ed examples
ha demons a e how he analysis pipeline beha es unde di e en scena ios. These examples
9
Aspec Pho ons (EOM/AOM ga ing) Elec ons (pulsed/RF ga ing)
Sou ce Na owband diode/DPSS; SPDC/single-
pho on op ional
The mionic/pho oemission gun; mo-
noene ge ic beam
Ga e EOM/AOM, ps–ns esponse RF de lec o s/choppe s, ns–µs
Cohe ence ime τcτc≈1/∆ν(lase linewid h) Ene gy sp ead ∆E⇒τc∼ℏ/∆E
Coun a e High (SPAD sa u a ion cau ion) Lowe (MCP/pixel de ec o e iciency)
Alignmen S anda d op ics (mi o s, i ises) Beamline (lenses, ape u es, s igma o s)
En i onmen al sensi i i y Ai cu en s, ib a ion, empe a u e Magne ic/elec ic ields, acuum s abil-
i y
Complexi y Mode a e; o - he-shel pa s High; acuum/beam diagnos ics e-
qui ed
Table 1: High-le el ade-o s o pho on s. elec on implemen a ions.
so a lase o ∆ν= 10 MHz yields τc∼100 ns and Lc∼30 m. Fo elec ons wi h mean kine ic
ene gy Eand sp ead ∆E, he empo al cohe ence is se by
τc∼ℏ
∆E, λdB =h
√2meE,(8)
implying ha s ong ene gy monoch oma ion (o na ow pho oemission bandwid h) is essen ial
o la ge ∆ .
In p ac ice, choose a loga i hmic sweep o ∆ :
∆ i= ∆ 0ρi, ρ ∈[1.05,1.2], i = 0, . . . , N,
co e ing a leas 2–3 decades while keeping ∆ max ≲τc.
As summa ized in Table 1, pho on and elec on implemen a ions o e complemen a y s eng hs
and challenges. Pho on se ups bene i om o - he-shel op ics and high coun a es, while
elec on a ian s equi e s ic e acuum and ield con ol bu p obe di e en cohe ence egimes.
B.3 Op ical Layou and Alignmen (Pho on Va ian )
1. Sou ce and mode-cleaning: Launch a na owband lase h ough a single-mode ibe
(op ional) o s abilize spa ial mode; pick o a small ac ion o e e ence powe moni o -
ing.
2. Tempo al ga es: Cascade wo EOMs/AOMs o de ine empo al ape u es. D i e wi h
synch onized pulses ( om an AWG o pulse ) wi h p og ammable sepa a ion ∆ and du y
cycle ≪1.
3. In e e ome e geome y: Use a ixed-pa h Mach–Zehnde /Michelson so ha only he
empo al deg ee o eedom is a ied. Equalize a ms o wi hin ≪Lc.
4. De ec ion: Image he a - ield o a SPAD o sCMOS. Calib a e linea i y (a oid SPAD
dead- ime sa u a ion) and eco d imes amps i a ailable.
5. Re e ence poin : De ine ∆ 0whe e isibili y V0is high and s able; moni o slow d i s
by e-acqui ing V0pe iodically.
B.4 Beamline and Alignmen (Elec on Va ian )
1. Sou ce: Use a s able DC/pho oemission gun; se ene gy Eand minimize ∆E ia monoch o-
ma o /ape u es.
16

2. Tempo al ga ing: RF de lec o o choppe o pass wo sho packe s sepa a ed by ∆ ;
e i y iming ji e <0.1 ∆ a smalles sepa a ions.
3. Imaging: MCP/phospho o pixela ed de ec o a a - ield; calib a e poin -sp ead unc-
ion (PSF) and gain.
4. Field con ol: Magne ic shielding and ac i e cancella ion; esidual ields can wash ou
isibili y be o e empo al e ec s appea .
B.5 Calib a ion, Baselines, and E o Budge
Visibili y ex ac ion. Fi he inge p o ile I(θ) a each se ing o
I(θ)=I0[1+Vcos(θ+ϕ)] ,
and eco d V=V(∆ ) wi h unce ain y om i co a iance. No malize o V0=V(∆ 0) and
o m esiduals R(∆ )=V(∆ )/V0−1.
Ins umen al baselines. Cha ac e ize: (i) ga e impulse esponses ( ise/ all imes, inging),
(ii) spec al bandwid h a he in e e ome e (OSA o pho ons; e a ding- ield analyze o
elec ons), (iii) de ec o nonlinea i y/dead- ime, (i ) mechanical/ ib a ional noise (accele om-
e e /FFT).
Sys ema ic budge . T ack con ibu ions o σR om coun ing noise, d i , backg ound sub ac-
ion, bandwid h/line-shape unce ain y, and iming ji e . Quo e bo h s a is ical and sys ema ic
componen s; p opaga e o he pe iodog am/likelihood i s.
B.6 Diagnos ic Plo s and Heal h Checks
•Ga e me ology: Measu ed empo al ansmission s. ime o each ga e; con olu ion
model s. measu ed pulse pai .
•Cohe ence alida ion: Visibili y V s. spec al bandwid h; ex apola e o na ow band-
wid h o con i m expec ed mono onic end.
•S abili y: V(∆ 0) s. un index; Allan de ia ion o de ec d i s.
•Con ols: Single-ga e (single-sli ) un should elimina e inges and any pe iodic esiduals;
e e sed ga e o de should no change esul s.
•Binning/apodiza ion: Show pe sis ence (o disappea ance) o any βpeak ac oss Hann/Tukey/Kaise
windows and mul iple bin wid hs in ln ∆ .
B.7 Recommended Ope a ing Poin s
Pho ons: lase linewid h ∆ν≲10 MHz, EOM ise/ all ≲200 ps, ∆ ange 0.1 ps–100 ns,
acquisi ion pe poin su icien o SNR>50 on V.
Elec ons: ene gy sp ead ∆E≲0.1 eV, iming ji e <100 ps, acuum ≲10−7mba , magne ic
ield esiduals <1 mG.
B.8 Dis ance-Dependen Complemen (Consis ency Check)
As an independen p obe, a spa ial double-sli wi h a ansla able de ec ion plane scans sli –
sc een dis ance Lwhile measu ing V(L). S anda d op ics p edic s mono onic decay o isibili y
(a e accoun ing o beam di e gence and de ec o PSF). Any ep oducible non-mono onic
e i als pe iodic in ln Lwould mi o he ime-domain sea ch and mus pass he same obus ness
es s (single-sli null, wa eleng h a ia ion, binning/apodiza ion s abili y).
17
Na owband
lase
Ga e 1
(EOM/AOM)
Ga e 2
(EOM/AOM) 50/50 BS Recombine SPAD /
sCMOS
AWG /
Pulse Gen.
OSA
(bandwid h)
Ga e impulse
esponse
De ec o linea i y
/ dead- ime
∆ p og ammable
Two sho windows c ea e
empo al “sli s” A ms pa h-ma ched:
≪Lc
Measu e V(∆ )
a each se ing
∆
ln(∆ /∆ 0)
R
βscan
(a) Sou ce & ga ing (b) In e e ence & de ec ion
(c) Residual analysis
Figu e 3: Time-domain double-sli (pho on implemen a ion). (a) Na owband lase
passes wo as empo al ga es (EOM/AOM) d i en by an AWG o c ea e wo sho ansmission
windows sepa a ed by a p og ammable delay ∆ ; in e e ome e a ms a e pa h-ma ched (≪Lc)
so ha only he empo al deg ee o eedom is a ied. (b) Fa - ield in e e ence is eco ded on
a SPAD/sCMOS de ec o and inge isibili y V(∆ ) is ex ac ed a each se ing. (c) Residual
analysis is pe o med on no malized isibili y R(∆ ) = V(∆ )/V0−1 as a unc ion o x=
ln(∆ /∆ 0) using Lomb–Sca gle and likelihood scans in β(wi h ials co ec ion). Calib a ion
blocks (OSA o bandwid h, ga e impulse- esponse, de ec o linea i y) moni o sys ema ics
h oughou .
B.9 Failu e Modes and Mi iga ions
•Appa en pe iodici y om elec onics: Check o ha monics in ga e d i e s/AWG
(spec um analyze ). Repea wi h independen d i e s.
•Timing ji e masque ading as s uc u e: Measu e ji e spec um; con ol e wi h
model; equi e ha in e ed βis s able unde ji e decon olu ion.
•Spec al b ea hing: Bandwid h d i can modula e V; in e lea e ∆ poin s o deco e-
la e om slow d i s; moni o OSA/RFA con inuously.
•De ec o sys ema ics: Ve i y linea i y; a y coun a e; use di e en de ec o segmen s;
equi e consis en βac oss spli s.
18
Raw da a Y()
Baseline B() i
Residuals R()=Y/B −1
Log-g id = ln(/0)
Spec al sea ch(Lomb–Sca gle, Whi le)
Candida e
Robus ness es s(jackkni e, su oga es)
Join -consis encyac oss domains
Inpu
Ou pu : claim o uppe limi s
Figu e 4: Analysis pipeline o esidual- empla e sea ches. Raw spec um Y(X) is
di ided by he accep ed baseline B(X) o o m no malized esiduals R(X). Residuals a e
epa ame e ized on o a loga i hmic g id x= ln(X/X0). Spec al sea ches (pe iodog am o
Whi le-likelihood scans) p obe o log-pe iodic modula ions a spacing β. Candida e signals
a e hen subjec ed o obus ness checks and a join -consis ency es ac oss domains (in e e ence,
acuum, CMB, SGWB).
012345678910
0
0.5
1
Injec ed β
β
Pe iodog am powe
Figu e 5: Mock eco e y o a ue βin simula ed da a. Syn he ic esiduals we e gene a ed
wi h an injec ed cosine modula ion o ampli ude a1= 0.02 a β= 4.5 in he p esence o Gaussian
noise. The Lomb–Sca gle pe iodog am (blue) shows a clea peak a he injec ed alue. Local
signi icance (shaded) ansla es o a global p- alue a e ials co ec ion; only globally signi ican
peaks may be in e p e ed as physical. Do ed line indica es 95% de ec ion h eshold.
19
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