In o ma ion Conse a ion and he Eme gence o
Complex Quan um Ampli udes
Zenodo DOI: 10.5281/zenodo.17741392
Paul Cooney1, ∗
1Independen Resea che , Innis il, On a io, Canada
(Da ed: No embe 27, 2025)
We de i e he complex Hilbe -space s uc u e o quan um mechanics om
in o ma ion- heo e ic p inciples applied o coa se-g ained physical heo ies. S a -
ing om an on ology o mic ohis o ies— ully speci ied mic oscopic ajec o ies o -
de ed by some pa ame e —we de ine b anches as equi alence classes o ope a-
ionally indis inguishable mic ohis o ies. Fi e physically mo i a ed axioms go e n-
ing he s uc u e o b anch s a es— ini e in o ma ion capaci y, con ex mix u es,
e e sible in o ma ion-p ese ing dynamics, local omog aphy, and con inuous e-
e sibili y—uniquely de e mine complex p ojec i e space as he pu e-s a e mani old.
The Bo n ule eme ges om symme y cons ain s on b anch mul iplici ies combined
wi h Gleason’s heo em. No p io assump ions abou complex numbe s, supe po-
si ion, o quan um p obabili y a e made. The econs uc ion uni ies ope a ional,
in o ma ion-geome ic, and pa h-in eg al pe spec i es, demons a ing ha quan um
heo y is he unique e ec i e desc ip ion o any physical sys em admi ing a coa se-
g ained mic ohis o y s uc u e wi h he speci ied in o ma ion- heo e ic p ope ies.
I. INTRODUCTION
Quan um mechanics employs complex ec o spaces, uni a y e olu ion, and Bo n- ule
p obabili ies. While ex ao dina ily success ul empi ically, he physical o igin o hese ma h-
ema ical s uc u es emains concep ually opaque. Why complex a he han eal o qua e -
nionic ampli udes? Why uni a y dynamics? Why squa ed-ampli ude p obabili ies?
∗paul.co[email p o ec ed] o.ca
2
Se e al econs uc ion p og ams add ess hese ques ions om di e en s a ing poin s.
Ha dy’s ope a ional axioms [1], he Chi ibella–D’A iano–Pe ino i (CDP) pu i ica ion pos-
ula e [2], and he Masanes–M¨ulle causali y-based app oach [3] success ully eco e complex
Hilbe space om physically mo i a ed cons ain s on p epa a ions, ans o ma ions, and
measu emen s.
This wo k akes a complemen a y pe spec i e. We begin wi h an on ological assump ion:
he undamen al desc ip ion consis s o mic ohis o ies— ully speci ied mic oscopic con ig-
u a ions o de ed by some pa ame e λ. Physical obse a ions access only coa se-g ained
in o ma ion, pa i ioning mic ohis o ies in o b anches: equi alence classes yielding iden ical
p edic ions o all accessible measu emen s. We show ha i e in o ma ion- heo e ic axioms
go e ning b anch s uc u e uniquely o ce complex Hilbe space as he e ec i e s a e space.
The econs uc ion p oceeds in ou s eps:
1. Sec ions II–III: De ine mic ohis o ies, ope a ional obse ables, and b anch equi a-
lence classes.
2. Sec ion IV: S a e i e in o ma ion- heo e ic axioms cons aining b anch dynamics.
3. Sec ion V: P o e ha eal and qua e nionic Hilbe spaces iola e local omog aphy,
and show ha in o ma ion geome y o ces a unique K¨ahle s uc u e isomo phic o
complex p ojec i e space.
4. Sec ion VI: De i e he Bo n ule om b anch-mul iplici y symme y and Gleason’s
heo em.
Sec ion VII discusses concep ual implica ions and connec ions o o he econs uc ion
p og ams. The appendices p o ide echnical de ails on in o ma ion geome y and b anch
symme ies.
a. No a ion. We use na u al uni s ℏ=c= 1 excep whe e explici ly es o ed o
cla i y. Mic ohis o ies a e deno ed H, b anches [H]o B, he o de ing pa ame e λ, and he
e ec i e s a e space S.
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II. MICROHISTORIES AND OPERATIONAL OBSERVABLES
A. The On ological Laye
We posi ha he undamen al desc ip ion o a physical sys em consis s o mic ohis o-
ies: ully speci ied mic oscopic con igu a ions e ol ing along an o de ing pa ame e λ. Fo
conc e eness, λmay ep esen :
•physical ime in non- ela i is ic mechanics,
•p ope ime τalong wo ldlines in ela i i y,
•a disc e e upda e index in cellula au oma a,
•any mono onic pa ame e de ining a o al o de ing on mic oscopic s a es.
[Mic ohis o y] A mic ohis o y is a ajec o y
H={X(λ)}λ
λi,(1)
whe e X(λ) speci ies he comple e mic oscopic s a e a each alue o he o de ing pa ame e
λ∈[λi, λ ].
The mic oscopic on ology X(λ) is in en ionally le gene al: i may ep esen ield con ig-
u a ions, pa icle posi ions and momen a, spin con igu a ions, o any o he comple e spec-
i ica ion. We equi e only ha he se H([λi, λ ]) o dynamically admissible mic ohis o ies
is well-de ined o e e y in e al.
C ucially, no quan um o linea s uc u e is assumed a his s age. Mic ohis o ies
may e ol e de e minis ically, s ochas ically, o ia disc e e upda e ules. The dynamics need
no p ese e any no m, admi supe posi ion, o sa is y locali y.
B. Ope a ional Obse ables
Physical measu emen s access only coa se-g ained, mac oscopic obse ables, no ull mi-
c ohis o ies.
[Obse able] An obse able O∈ O is a unc ional
O:H([λi, λ ]) ×[λi, λ ]→R,(2)
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ex ac ing a eal numbe O[H, λ] om a mic ohis o y Ha o de ing alue λ.
The se Ois de e mined by expe imen al capabili ies. Two mic ohis o ies di e ing only
in deg ees o eedom inaccessible o a ailable de ec o s yield iden ical alues o all O∈ O.
In a quan um ield heo y, X(λ) migh speci y ield alues ϕ(x, λ) a all space ime poin s.
Obse ables O∈ O include only spa ially a e aged quan i ies, co ela ion unc ions a de-
ec o esolu ion, and in eg a ed ene gies—no poin alues ϕ(x0, λ0).
[Double-Sli Expe imen ] Conside a pa icle a e sing a double-sli appa a us. Mic o-
his o ies Hspeci y he comple e ajec o y h ough space, including which sli (i any) he
pa icle passes h ough. I no which-pa h de ec o is p esen , obse ables O∈ O include
only he inal sc een posi ion xsc een—no he ajec o y de ails. All mic ohis o ies ending
a he same sc een loca ion belong o he same b anch, ega dless o which sli hey passed
h ough. This na u ally explains in e e ence: he b anch s uc u e coa se-g ains o e pa h
de ails, and he e ec i e dynamics (quan um mechanics) mus accoun o con ibu ions
om all ajec o ies in he equi alence class.
This coa se-g aining is no a limi a ion bu a undamen al ea u e o eal expe imen s:
measu emen de ices ha e ini e esolu ion, ini e empo al esponse, and ini e sensi i i y.
III. OPERATIONAL EQUIVALENCE AND BRANCHES
A. Ope a ional Indis inguishabili y
Two mic ohis o ies a e ope a ionally equi alen i no accessible measu emen can dis in-
guish hem, ei he p esen ly o in he u u e.
[Ope a ional Equi alence] Two mic ohis o ies H, H′∈ H([λi, λ ]) a e ope a ionally equi -
alen a λ∗, w i en H∼λ∗H′, i :
1. Fo all obse ables O∈ O,
O[H, λ∗]=O[H′, λ∗],(3)
2. Fo all λ>λ∗, he p obabili y dis ibu ions o e u u e obse able alues—condi ioned
on he s a e a λ∗and any allowed in e en ions—a e iden ical o Hand H′.
Condi ion (1) ensu es p esen obse ables ag ee. Condi ion (2) ensu es u u e p edic ions
ag ee. Toge he , hey cap u e he ope a ional con en o a physical s a e: once Hand H′
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ag ee on all p esen ly measu able in o ma ion and gene a e iden ical condi ional p obabili ies
o all u u e measu emen s, hey ep esen he same physical si ua ion o all p ac ical
pu poses.
[Connec ion o Decohe ence] In sys ems wi h en i onmen al in e ac ions, condi ion (2) is
ypically sa is ied when mic oscopic his o ies ha e been en angled wi h la ge en i onmen al
deg ees o eedom. Decohe ence selec s p e e ed “poin e s a es” [4], and ope a ionally
equi alen mic ohis o ies co espond o hose lying wi hin he same poin e basis. Howe e ,
ou amewo k does no equi e decohe ence—ope a ional equi alence is de ined pu ely by
p edic i e indis inguishabili y.
B. B anches as Equi alence Classes
[B anch] A b anch B(λ∗) is an equi alence class o mic ohis o ies unde ∼λ∗:
B(λ∗)=[H] = {H′∈ H([λi, λ ]) : H′∼λ∗H}.(4)
The se o all b anches a λ∗is
B(λ∗)=H([λi, λ ])/∼λ∗.(5)
B anches a e he e ec i e s a es o he heo y. They e ain exac ly he ope a-
ionally ele an in o ma ion while disca ding inaccessible mic oscopic de ails. An obse e
who knows which b anch he sys em occupies a λ∗can p edic all u u e measu emen
ou comes wi h he same accu acy as an obse e wi h comple e mic ohis o y in o ma ion.
[Classical Phase Space] In classical mechanics, mic ohis o ies a e phase-space ajec o ies
(q( ), p( )). I measu emen p ecision is ini e, b anches a e small phase-space cells. Fo
pe ec p ecision, each b anch con ains a single ajec o y.
[Quan um Pa h In eg al] In he pa h-in eg al o mula ion, mic ohis o ies a e ield con-
igu a ions ϕ(x, ). B anches co espond o se s o pa hs con ibu ing o he same e ec i e
ampli ude a e en i onmen al acing. The b anch s uc u e depends on which obse ables
a e accessible.
[Spin Measu emen ] Conside a spin-1/2 pa icle p epa ed in s a e |ψ⟩=α|↑⟩ +β|↓⟩
along he z-axis, whe e |α|2+|β|2= 1. Be o e measu emen , he e exis s a single b anch
con aining all mic oscopic spin con igu a ions consis en wi h his p epa a ion.
Upon measu ing Sz, he sys em spli s in o wo b anches:
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•B anch B↑: all mic ohis o ies yielding ou come +ℏ/2,
•B anch B↓: all mic ohis o ies yielding ou come −ℏ/2.
Wi hin each pos -measu emen b anch, mic oscopic de ails may a y (en i onmen al de-
g ees o eedom, exac de ec o dynamics), bu all mic ohis o ies in B↑ag ee on u u e
p edic ions: subsequen measu emen s o Szwill yield +ℏ/2 wi h ce ain y. The b anches
a e ope a ionally dis inguishable because hey p edic di e en ou comes o u u e Szmea-
su emen s.
C ucially, i we ins ead measu e Sx, he sys em spli s in o di e en b anches B→and B←.
The b anch s uc u e is obse able-dependen — he e is no unique pa i ion o mic ohis o ies
in o b anches independen o measu emen con ex . This con ex uali y is a ea u e, no a
bug: i e lec s he ope a ional de ini ion o equi alence.
C. Why This F amewo k?
The mic ohis o y-b anch cons uc ion p o ides a physical in e p e a ion o e ec i e
quan um s a es: hey a e equi alence classes o mic oscopic con igu a ions indis inguishable
o obse e s wi h ini e esolu ion. This answe s he ques ion “wha is a quan um s a e?”
wi hou in oking wa e unc ion collapse, hidden a iables, o many wo lds.
Mo eo e , he amewo k is gene al: i applies o any heo y admi ing a mic oscopic
on ology and a coa se-g aining p ocedu e. The econs uc ion ha ollows will show ha i
b anches sa is y i e physically na u al in o ma ion- heo e ic cons ain s, hen he e ec i e
s a e space mus be a complex Hilbe space wi h uni a y dynamics and Bo n- ule p oba-
bili ies. Quan um mechanics eme ges no om undamen al quan um pos ula es bu om
uni e sal ea u es o in o ma ion comp ession.
IV. INFORMATION-THEORETIC AXIOMS
We now o malize cons ain s on he e ec i e s a e space So b anches. These axioms
desc ibe how in o ma ion s o ed in b anches can be mixed, ans o med, and composed.
Impo an ly, we do no assume linea i y, complex s uc u e, inne p oduc s, o
he Bo n ule. These will eme ge as consequences.
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A. Axiom 1: Fini e In o ma ion Capaci y
[Fini e In o ma ion] Fo any ini e spa ial egion and any ini e se o accessible obse -
ables, he numbe o ope a ionally dis inguishable b anches is ini e o coun ably in ini e.
The e ec i e s a e space Sadmi s a ini e-dimensional eal ec o -space embedding.
a. Physical Mo i a ion. No ini e appa a us can dis inguish in ini ely many mu ually
exclusi e mac oscopic s a es. Measu emen p ecision and de ec o sensi i i y impose unda-
men al limi s on in o ma ion capaci y. Axiom IV A encodes he Bekens ein–Hole o bound [5]
a a p e-quan um le el: ini e egions s o e ini e in o ma ion.
b. Ma hema ical Con en . Sis a compac con ex subse o RN o some ini e N.
This excludes pa hological in ini e-in o ma ion heo ies while pe mi ing classical p obabili y
heo y (S= p obabili y simplex) and quan um mechanics (S= densi y ope a o s).
B. Axiom 2: Con ex Mix u es
[Con exi y] I ω1, ω2∈ S a e p epa able s a es, hen o all p∈[0,1], he con ex combi-
na ion
ω=p ω1+(1−p)ω2(6)
ep esen s a s a e p epa able by selec ing ω1wi h p obabili y pand ω2wi h p obabili y
1−p.
a. Physical Mo i a ion. Classical unce ain y o e p epa a ion p ocedu es combines
linea ly. I an expe imen e p epa es ω1on Mondays and ω2on Tuesdays, he ensemble
s a e on a andomly chosen day is he mix u e (6). This is no quan um supe posi ion—i
is he ope a ional ac ha igno ance abou which p epa a ion occu ed is ep esen ed by a
weigh ed a e age.
b. Ma hema ical Con en . Sis a con ex se . Pu e s a es (ex emal p epa a ions) lie
on he bounda y ∂S; mixed s a es lie in he in e io .
C. Axiom 3: Re e sible In o ma ion-P ese ing Dynamics
Unde lying mic oscopic dynamics ypically conse e dis inguishabili y: de e minis ic e o-
lu ion p ese es phase-space olume (Liou ille’s heo em), and s ochas ic dynamics p ese e
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ensemble in o ma ion. A he coa se-g ained le el, his mani es s as no m p ese a ion.
[Re e sible Dynamics] The e exis s a one-pa ame e amily o ans o ma ions {T∆λ:
S → S} sa is ying:
1. T0= id,
2. T∆λ1+∆λ2=T∆λ2◦T∆λ1(g oup p ope y),
3. Each T∆λis bijec i e ( e e sibili y),
4. The e exis s a no m ∥·∥ on Ssuch ha
∥T∆λω∥=∥ω∥ ∀ω∈ S,∆λ. (7)
a. Physical Mo i a ion. Re e sible mic oscopic e olu ion conse es global dis inguisha-
bili y. Obse e s canno lose he abili y o di e en ia e ini ially dis inguishable s a es
h ough e e sible p ocesses. Axiom IV C ensu es ha he e ec i e heo y espec s his
conse a ion a he coa se-g ained le el.
b. Ma hema ical Con en . The no m ∥·∥ is no speci ied a p io i. In classical p ob-
abili y, ∥ρ∥=Pi|ρi|( o al a ia ion). In quan um mechanics, ∥ρ∥= T (ρ) ( ace no m).
The speci ic o m will be de e mined by Axioms IV D and IV E.
D. Axiom 4: Local Tomog aphy
To desc ibe composi e sys ems, we equi e ha join in o ma ion is eco e able om local
co ela ions.
[Local Tomog aphy] Fo subsys ems Aand Bwi h s a e spaces SAand SB, he composi e
sys em AB has s a e space SAB sa is ying:
1. Tenso p oduc s uc u e. Fo pu e s a es ωA∈ SA,ωB∈ SB, he p oduc s a e
ωA⊗ωB∈ SAB exis s.
2. Local de e mina ion. Join s a is ics o all locally accessible measu emen s on A
and Buniquely de e mine he global s a e in SAB.
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a. Physical Mo i a ion. I Alice measu es Aand Bob measu es B, hei join s a is ics
⟨OA⊗OB⟩con ain all ope a ionally accessible in o ma ion abou he composi e sys em.
Local omog aphy asse s ha no “hidden co ela ions” exis beyond wha is de ec able
h ough local and join ly local measu emen s.
b. Disc imina o y Powe . Local omog aphy is sa is ied by classical and complex-
quan um heo ies bu ails o eal and qua e nionic quan um mechanics [6, 7]. This axiom
he e o e plays a c ucial ole in selec ing complex Hilbe space uniquely.
E. Axiom 5: Con inuous Re e sibili y
To exclude disc e e classical heo ies ( ini e s a e spaces wi h pe mu a ion dynamics), we
impose con inui y o e e sible ans o ma ions.
[Con inuous Re e sibili y] The e e sible ans o ma ions {T∆λ}ac ansi i ely and con-
inuously on he pu e-s a e mani old P=∂S. Tha is:
1. Fo any pu e s a es ω1, ω2∈ P, he e exis s ∆λsuch ha T∆λω1=ω2.
2. The map (∆λ, ω)7→ T∆λωis con inuous in bo h a gumen s.
a. Physical Mo i a ion. Physical sys ems exhibi con inuous symme ies. Ro a ions o
spin s a es, ime ansla ions, and phase ans o ma ions all ac smoo hly on s a e space.
Axiom IV E ensu es ha Pis a connec ed, con inuously symme ic mani old, enabling
in o ma ion-geome ic analysis.
F. Summa y
The i e axioms collec i ely exp ess:
•Fini e in o ma ion esolu ion (Axiom IV A),
•Linea ep esen a ion o classical unce ain y (Axiom IV B),
•Conse a ion o dis inguishabili y (Axiom IV C),
•Absence o hidden co ela ions (Axiom IV D),
•Con inuous symme y (Axiom IV E).
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By Axiom IV B (con exi y), p obabili y assignmen s a e linea o e mix u es. By Ax-
iom IV E (con inuous e e sibili y), p obabili ies a y con inuously wi h he s a e. Combined
wi h he a ional-ampli ude esul (19), Gleason’s heo em uniquely ixes
p(Pk) = ⟨ψ|Pk|ψ⟩=|⟨k|ψ⟩|2.(22)
[Dimension Res ic ion] Gleason’s heo em equi es dim(H)≥3. Fo dim(H)=2
(qubi s), addi ional symme y a gumen s o appeal o composi e sys ems su ice o de i e
he Bo n ule [11].
E. Summa y
The Bo n ule eme ges om:
1. B anch-mul iplici y symme y: Equal ampli udes ⇒equal p obabili ies,
2. Ra ional ex ension: Ancilla cons uc ion o a ional ampli ude a ios,
3. Gleason’s heo em: Noncon ex uali y, addi i i y, and con inui y uniquely de e mine
p(k) = |⟨k|ψ⟩|2.
No sepa a e pos ula e is equi ed— he Bo n ule is a consequence o he in o ma ion-
heo e ic s uc u e imposed by Axioms IV A–IV E.
VII. DISCUSSION AND CONCEPTUAL IMPLICATIONS
A. Wha Has Been De i ed
We ha e shown ha i a physical heo y admi s:
•an on ology o mic ohis o ies,
•a coa se-g aining in o ope a ionally indis inguishable b anches,
• ini e in o ma ion capaci y, con ex mix u es, e e sible no m-p ese ing dynamics,
local omog aphy, and con inuous symme y,
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hen he e ec i e s a e space is necessa ily a complex Hilbe space wi h uni a y e olu ion
and Bo n- ule p obabili ies.
Quan um mechanics is no undamen al—i is he unique e ec i e desc ip ion o
any sys em sa is ying hese cons ain s.
B. In e p e a ion o Quan um S a es
E ec i e quan um s a es |ψ⟩a e in o ma ional summa ies o b anch equi alence classes.
The wa e unc ion does no ep esen a physical wa e bu encodes:
•which mac oscopic con igu a ions a e compa ible wi h obse a ions,
• he ela i e weigh s (mul iplici ies) o mic ohis o ies wi hin each con igu a ion,
•condi ional p obabili ies o u u e measu emen s.
This esol es he “wha is ψ?” ques ion wi hou in oking collapse, hidden a iables,
o many wo lds: ψis a comp essed ep esen a ion o ope a ionally ele an mic ohis o y
in o ma ion.
C. Rela ion o Decohe ence
The b anch amewo k na u ally accommoda es decohe ence-based in e p e a ions. When
a sys em in e ac s wi h a la ge en i onmen , in e e ence be ween mac oscopically dis inc
con igu a ions is dynamically supp essed. The esul ing poin e s a es co espond p ecisely
o ope a ionally dis inguishable b anches.
Howe e , ou econs uc ion does no equi e decohe ence. Ope a ional equi alence is
de ined by p edic i e indis inguishabili y, which may a ise h ough decohe ence, undamen al
coa se-g aining, o any o he mechanism ha pa i ions mic ohis o ies in o classes wi h
iden ical u u e s a is ics.
D. Rela ion o O he Recons uc ions
•Ha dy [1]: Begins wi h ope a ional heo ies o p epa a ions, ans o ma ions, and
e ec s. Ou mic ohis o y-b anch amewo k p o ides a physical on ology unde lying
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Ha dy’s abs ac ope a ions.
•CDP [2]: Uses pu i ica ion as a cen al pos ula e. Ou Axiom IV D implies pu i ica-
ion: any mixed s a e on Acan be ealized as a ma ginal o a pu e s a e on A⊗B.
•Masanes–M¨ulle [3]: Emphasizes causali y and in o ma ion- heo e ic consis ency.
Ou app oach complemen s his by g ounding ope a ional axioms in a mic ohis o y
on ology.
The p esen wo k uni ies hese pe spec i es: ope a ional axioms, in o ma ion geome y,
and mic ohis o y-based pa h in eg als all lead o he same conclusion—complex Hilbe
space is ine i able.
E. Wha is No De i ed
The econs uc ion es ablishes he ma hema ical s uc u e o quan um heo y bu does
no add ess:
•Speci ic Hamil onians: The gene a o Ho ime e olu ion mus come om addi-
ional physical inpu (e.g., symme ies, ield con en ).
•Planck’s cons an ℏ:The no maliza ion o he ac ion is no de e mined by in o ma ion-
heo e ic axioms alone.
•Measu emen p oblem: We cla i y he in o ma ion s uc u e o ou comes bu do
no selec a speci ic in e p e a ion (Copenhagen, E e e , GRW, e c.).
•Mic oscopic dynamics: The e olu ion o mic ohis o ies X(λ) emains model-
dependen .
Thus, he econs uc ion explains why quan um mechanics has i s o m bu no wha he
wo ld is made o o which Hamil onian go e ns i .
F. B oade Implica ions
a. Laye ed View o Physics. The de i a ion sugges s a wo- ie s uc u e:
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1. Mic oscopic on ology: Mic ohis o ies X(λ) e ol ing acco ding o some (po en ially
non-quan um) dynamics.
2. E ec i e heo y: B anches [H] obeying complex Hilbe -space ules due o in o -
ma ion comp ession.
Quan um mechanics esides a he e ec i e le el. I s ma hema ical o m is uni e sal—any
mic ohis o y heo y wi h he igh in o ma ion- heo e ic p ope ies yields he same e ec i e
desc ip ion.
b. Tes abili y. The econs uc ion i sel is no di ec ly es able (i is a s uc u al esul ),
bu i makes quan um mechanics mo e alsi iable: i u u e expe imen s e eal iola ions o
local omog aphy o e e sibili y, quan um heo y would need e ision—no as a undamen-
al pos ula e, bu as an eme gen e ec i e desc ip ion whose unde lying assump ions ha e
been iola ed.
c. Ex ensions. Na u al di ec ions o u u e wo k include:
•Gene aliza ion o inde ini e causal s uc u e [12],
•Applica ion o quan um ield heo y and holog aphic duali y,
•Explo a ion o al e na i e in o ma ion measu es (non-mono one me ics, non-con ex
s a e spaces),
•Connec ion o speci ic mic ohis o y models (e.g., la ice heo ies, disc e e causal se s,
cellula au oma a).
VIII. CONCLUSION
We ha e de i ed he complex Hilbe -space s uc u e o quan um mechanics—including
uni a y dynamics and he Bo n ule— om i e physically mo i a ed in o ma ion- heo e ic
axioms applied o coa se-g ained mic ohis o y ensembles. The econs uc ion shows ha
quan um heo y is no a undamen al pos ula e bu he unique e ec i e amewo k o
any physical sys em admi ing a b anch s uc u e sa is ying ini e in o ma ion capaci y,
con exi y, e e sible in o ma ion p ese a ion, local omog aphy, and con inuous symme y.
20
This p o ides a concep ually anspa en answe o ounda ional ques ions: quan um
s a es a e in o ma ional summa ies o ope a ionally indis inguishable mic ohis o ies; com-
plex ampli udes eme ge om K¨ahle geome y induced by in o ma ion conse a ion; and
Bo n- ule p obabili ies ollow om b anch-mul iplici y symme y. The esul uni ies ope a-
ional, in o ma ion-geome ic, and pa h-in eg al pe spec i es, o e ing a cohe en pic u e o
why na u e employs he ma hema ical s uc u es o quan um heo y.
ACKNOWLEDGMENTS
The au ho hanks he quan um ounda ions communi y o decades o wo k cla i ying
he ope a ional and in o ma ion- heo e ic unde pinnings o quan um mechanics.
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