Towa d a S uc u al Algeb a o F ac u es:
G o hendieckian Re lec ions o a Ma hema ical Founda ion o
Theo y F
An onio Be n´a dez Gumiel
Mad id, 9 July 2025
Abs ac
Theo y F p oposes a uni ica ion o undamen al physics by in e p e ing all known pa icles,
o ces, and symme ies as he esul o s uc u al ac u es wi hin a uni e sal inelas ic ield. This
a icle explo es he ma hema ical implica ions o such a ision, he ype o o malism i equi es,
and he limi a ions o cu en amewo ks. I e isi s he pa h aken by Eins ein in adap ing
di e en ial geome y o Gene al Rela i i y and p oposes ha G o hendieck’s ma hema ical
landscape—pa icula ly opos heo y, shea cohomology, and unc o iali y—o e s a p omising
di ec ion o s uc u ing he algeb a o ac u es. A unc o ial exp ession o he gene al ac u e
unc ion F, capable o de i ing ela i i y and quan um heo y om opological and cohomo-
logical da a, is ou lined. The documen concludes wi h an open p oposal o collabo a ion wi h
he ma hema ical communi y and a no e o g a i ude o P o esso Fe nando Zalamea o his
con ibu ions o he dissemina ion o G o hendieck’s legacy.
Resumen: La Teo ´ıa F p opone una uni icaci´on de la ´ısica undamen al in e p e ando odas
las pa ´ıculas, ue zas y sime ´ıas como esul ado de ac u as es uc u ales den o de un campo
inel´as ico uni e sal. Es e a ´ıculo explo a las implicaciones ma em´a icas de dicha isi´on, el
ipo de o malismo eque ido y las limi aciones de los ma cos exis en es. Se e isa el camino
omado po Eins ein al adap a la geome ´ıa di e encial pa a la ela i idad gene al, y se p opone
que el paisaje ma em´a ico de G o hendieck—pa icula men e la eo ´ıa de opos, la cohomolog´ıa
de haces y la unc o ialidad—o ece una di ecci´on p ome edo a pa a es uc u a un ´algeb a
de la ac u a. Se esboza una exp esi´on unc o ial de la unci´on gene al de ac u a F, capaz
de de i a ela i idad y eo ´ıa cu´an ica a pa i de da os opol´ogicos y cohomol´ogicos. El
documen o concluye con una p opues a abie a de colabo aci´on con la comunidad ma em´a ica
y un ag adecimien o al p o eso Fe nando Zalamea po su labo en la di ulgaci´on del legado de
G o hendieck.
Con en s
1 In oduc ion: A S uc u al Hypo hesis o F ac u e 2
2 P oblems and Ma hema ical Challenges in Theo y F 3
3 Wha Ma hema ics Would Theo y F Requi e? 4
4 The Legacy o G o hendieck and he Role o Topos Theo y in Theo y F 7
5 A F ac u e Func ion in he Language o G o hendieck 9
6 S eng hs, Risks, and Ma hema ical Al e na i es 11
7 Acknowledgemen s and Scien i ic In i a ion 13
Annex I: S uc u al Ske ch o he F ac u e Func ion F14
1
1. In oduc ion: A S uc u al Hypo hesis o F ac u e
The aspi a ion o uni y he ounda ions o physics has o en led o inc easingly complex ma hema -
ical cons uc ions, seeking e e deepe symme ies o highe -dimensional amewo ks. Theo y F, in
con as , depa s om a adical bu in ui i ely esonan p emise: ha he ab ic o physical eali y
is he esul o ac u es—no acciden al, bu s uc u ed—wi hin a uni e sal and inelas ic ield,
he e deno ed by T. In his ision, he known en i ies o physics—pa icles, o ces, cons an s, and
e en he obse able laws hemsel es—eme ge no as p ima y objec s, bu as cohe en esponses o
disc e e modes o ac u e, s uc u ally encoded in he geome y o T.
Each mode o ac u e, om longi udinal up u e o adial comp ession, de ines a dis inc i e
ype o s uc u al de o ma ion. These modes a e i e in numbe , unique and i educible, and
hei combina ions ep oduce he known ea u es o he physical wo ld, while also p edic ing new
possibili ies beyond cu en pa adigms.
Bu such a hypo hesis demands mo e han physical specula ion. I equi es a ma hema ical
language adequa e o he ask o exp essing discon inui y, hie a chy, ecombina ion, and eme gence.
A language capable o desc ibing:
•The in e nal logic o each ac u e mode,
•The s uc u e o hei mu ual in e ac ions,
•The bi h o opologies om hei composi ion,
•And he laws ha go e n hei empo al e olu ion wi hin he ield T.
This pape aims o ou line he pa h owa ds such a language.
In he wen ie h cen u y, Eins ein amously adap ed he ools o Riemannian geome y o ex-
p ess g a i a ion as cu a u e. Howe e , he hypo hesis o Theo y F poin s no owa d cu a u e
alone, bu owa d ac ali y, singula i y, and unc o ial ecombina ion. I sugges s ha he basic
on ological uni s o physics a e no poin s, no s ings, no ields in he classical sense, bu ac u e
s uc u es—disc e e ye p opaga i e, localized ye capable o building global cohe ence.
This is why we u n o he ma hema ical ision o Alexande G o hendieck. Few concep ual
a chi ec u es a e as sui ed o cap u e such s uc u al and gene a i e complexi y as he heo y o
schemes, shea es, and opoi de eloped unde his guidance. His no ion o a opos—a ca ego y
beha ing like a gene alized space, endowed wi h i s own logic—may p o ide p ecisely he kind o
dynamic in e nal geome y equi ed by Theo y F.
The cen al ques ion, hen, is his: Can G o hendieck’s ma hema ics, o some hing inspi ed by
i , se e as he ounda ion o a igo ous o mula ion o Theo y F?
To explo e his ques ion, we shall begin by examining he ma hema ical challenges ha Theo y
F poses, he exis ing ools a ailable, and he easons why cu en amewo ks may p o e insu icien .
Then, we shall en e he landscape o G o hendieck, acing i s key componen s and p oposing a
en a i e bu s uc u ed applica ion o his me hods o he unc ion F, he gene al exp ession o
ac u e in his new heo y.
This is no me ely a echnical unde aking. I is also a philosophical one. Fo i he uni e se is
indeed he esul o s uc u ed ac u e, hen ma hema ics i sel mus become a ool o s uc u al
e ela ion—a way o lis en, o deciphe , and o econs uc he silen g amma o physical becoming.
2
2. P oblems and Ma hema ical Challenges in Theo y F
While he physical in ui ion behind Theo y F is s iking in i s concep ual economy—de i ing all
s uc u e om combina ions o i e i educible ac u e modes—i quickly e eals an ex ao dina y
ma hema ical dep h when one seeks o o malize i igo ously. The heo e ical simplici y o i s
axioms conceals a o midable landscape o unsol ed challenges.
Le us enume a e and b ie ly desc ibe he p ima y ma hema ical p oblems cu en ly open in
Theo y F:
2.1. De ining he Algeb a o Modes
Each ac u e mode (I–V) is desc ibed geome ically—longi udinal up u e, angen ial shea , o -
sion, adial comp ession/expansion, and ans e se bi u ca ion. Howe e , no exis ing algeb a cap-
u es he in e ac ion ules, nonlinea i y, non-commu a i i y, and po en ial non-associa i i y be-
ween modes. Wha is needed is:
•A mode algeb a F, wi h basis elemen s co esponding o he i e modes,
•An ope a ion able de ining composi ion (e.g., MI·MIII =MIV ?),
•A sys em o s uc u al coe icien s encoding eme gen geome y, opology, and ene gy.
2.2. Modeling he Supe posi ion and In e ac ion o F ac u es
Unlike linea wa e in e ac ions o ield o e laps, ac u es in e ac nonlinea ly and o en p oduce
eme gen s uc u es ha do no co espond o a simple sum. Ma hema ically, his demands:
•Func o ial models o in e ac ion, whe e each mode may be seen as a unc o ac ing on local
con igu a ions,
•Homological acking o eme gen singula i ies,
•Possibly an ope adic o highe -ca ego ical s uc u e o accoun o ecombina ions.
2.3. Exp essing Local–Global Cohe ence
A key ea u e o Theo y F is ha local ac u es gene a e global ields—including g a i a ional,
elec omagne ic, o quan um ields. The ma hema ics o shea heo y and cohomology, especially
in G o hendieck’s o mula ion, seems essen ial he e. We need o:
•Associa e ac u e da a o open se s o a opological space,
•Ensu e gluing condi ions ha allow consis en global in e p e a ions,
•De i e opological in a ian s om ac u e con igu a ions.
2.4. Encoding S uc u al Time
Time in Theo y F is wo old:
• : he classical, s a is ical, measu able ime (as in ela i i y and quan um mechanics),
•T: a s uc u al ime, hidden in mass and ma e ial con igu a ion, which un olds h ough he
e olu ion o ac u es.
3
Ma hema ically, his sugges s ha Tis no a coo dina e bu an in e nal deg ee o eedom,
possibly modeled as:
•A unc o ial low in a opos,
•A g aded pa ame e in a de i ed ca ego y,
•O an e olu ion pa ame e o moduli o ac u e pa e ns.
2.5. Reco e ing Classical Theo ies as Limi s
Any accep able ma hema ical o mula ion o Theo y F mus ep oduce, as limi cases o s uc u al
p ojec ions:
•Gene al Rela i i y as cu a u e o space induced by mac o-cohe en ac u es,
•Quan um Field Theo y as in e ac ion o minimal ac u es o e opologically cons ained
acua,
•The S anda d Model’s symme ies and pa icles as combina ions o basic modes.
This challenge is no me ely symbolic: i equi es igo ous de i abili y, es ablishing b idges
be ween he ac u e unc ion Fand he ield equa ions o exis ing physics.
2.6. Building he F ac u e Func ion F
Ul ima ely, all physical mani es a ions in Theo y F de i e om a single abs ac unc ion F, which
de e mines:
•Which modes a e ac i e,
•How hey combine,
•Wha bounda y and ini ial condi ions igge hei ac i a ion,
•And wha obse able consequences (pa icles, o ces, cons an s) eme ge.
This unc ion is no ye o malized, bu i s ma hema ical o m mus be disco e ed, pe haps as
a sec ion o a shea o e a opos, o as a gene a ing mo phism in a highe ca ego y.
In summa y, he ask is monumen al. Theo y F demands he in en ion—o deep adap a ion—o
a new ma hema ical landscape. This is why he legacy o G o hendieck is so p omising: ew
ma hema icians ha e e e cons uc ed such an a chi ec u e o o mal hough , capable o hos ing
discon inui y, eme gence, locali y, and in e nal logic simul aneously.
3. Wha Ma hema ics Would Theo y F Requi e?
To b ing Theo y F in o ull ma hema ical o m, one canno ely solely on he es ablished pa adigms
o geome y and algeb a as inhe i ed om 20 h-cen u y physics. The heo y demands an exp essi e
logic o s uc u e, capable o modeling discon inuous eme gence, ecu si e gene a ion, and mul i-
modal in e ac ion in a undamen ally inelas ic con inuum. This sec ion ou lines he ma hema ical
ea u es ha a sui able o malism mus exhibi .
4
3.1. A Logic o Modes and S uc u e
Theo y F begins no wi h ields, pa icles, o me ics, bu wi h modes o ac u e—elemen al and
i educible gene a o s o s uc u e. Thus, he equi ed ma hema ics mus :
•Accommoda e gene a o s ha a e no con inuous ans o ma ions bu disc e e, geome ically
meaning ul up u e ac ions.
•Allow o non-associa i e composi ion laws, as combining Mode I wi h Mode II may yield
di e en esul s depending on o de ing and con ex .
•Be compa ible wi h in e nal hie a chies, such ha combina ions o modes gene a e highe -
o de s uc u es (e.g., pa icles, cons an s, ields).
This calls o a non-s anda d algeb a, possibly d awing om:
•Lie algeb as, o hei oo -s uc u e and symme y p ope ies,
•Cli o d algeb as, o encoding geome ic ans o ma ions,
•O en i ely new s uc u es, such as ac u e ope ads, mode la ices, o non-associa i e geome-
ies.
3.2. Func o ial Geome y o e Fixed Topologies
T adi ional physics models geome y o e smoo h mani olds. Bu in Theo y F, space i sel is he
esul o s uc u ed ac u e, and hus opologies mus eme ge om he algeb a o modes, no be
pos ula ed.
A G o hendieck-s yle esponse would in ol e:
•T ea ing each mode as a unc o om local da a o global cohe ence,
•Building shea ca ego ies o e p eshea es o mode in e ac ions,
•Allowing space o a ise as a colimi o gluing o mode ac ions.
This would mean ha geome y is no a backd op, bu a de i ed phenomenon, as i is in he
heo y o schemes and in he no ion o opos.
3.3. Time as In e nal G ading
Theo y F in oduces s uc u al ime T, a quan i y ela ed o mass and in e nal e olu ion o s uc-
u e. In classical physics, ime is an ex e nal pa ame e ; he e, i is immanen .
Ma hema ically, we may model Tas:
•A g ading o e a ca ego y o ac u es,
•A unc o ial low inside a opos, acking in e nal phase ansi ions,
•O a cohomological deg ee, measu ing he dep h o s uc u al en anglemen .
The idea is o dis inguish be ween he isible un olding o e en s ( ) and he in e nal p ocess o
s uc u al ealiza ion (T).
5
3.4. F om F ac u es o Physical Laws
Ul ima ely, he ma hema ics mus suppo a mapping om ac u e con igu a ions o obse able
phenomena. Tha is, om he o mal da a:
•Which modes a e p esen ,
•How hey in e ac ,
•Thei spa ial a angemen and his o y,
we mus de i e:
•The eme gence o pa icles ( e mions, bosons),
•The appea ance o o ces (g a i a ional, elec omagne ic. . . ),
•The alues o physical cons an s (c,ℏ,G),
•And he limi s o ansi ions be ween di e en egimes (classical, quan um, ela i is ic).
This sugges s a s uc u al unc o Φ : F → P, whe e Fis he ca ego y o ac u e con igu a ions
and P he ca ego y o physical s uc u es.
3.5. Compa ibili y wi h Rela i i y and Quan um Theo y
Theo y F does no ejec es ablished physics—i explains i as a limi ing case. The e o e, any
ma hema ics used mus admi :
•Cu ed space imes as mac o-limi s o dis ibu ed Mode I + IV ac u es,
•Quan um luc ua ions as high- equency, low-ampli ude Mode II + III in e ac ions,
•S anda d symme ies (SU(3), SU(2), U(1)) as eme ging om combina o ial mode la ices.
This equi es ha he new ma hema ics mus con ain, in sui able p ojec ions, bo h di e en ial
geome y and Hilbe -space quan um heo y, no as axioms bu as eme gen s uc u es.
3.6. Towa ds a New F ac u e Calculus
The mos specula i e—bu pe haps essen ial— equi emen is he de elopmen o a calculus o
ac u e:
•An in ini esimal s uc u e o e non-smoo h mani olds,
•Capable o acking s uc u al g adien s, discon inui ies, singula i y p opaga ion,
•And wi h an in e nal logic e lec ing he b anching, bi u ca ion, and coalescence o ac u es.
This could in ol e gene aliza ions o :
•Syn he ic di e en ial geome y,
•Noncommu a i e geome y,
•O de i ed algeb aic geome y.
In summa y, he ma hema ics equi ed by Theo y F is no me ely an ex ension o exis ing
amewo ks—i is a e- ounda ion, echoing G o hendieck’s p ojec o building a “new ma hema ical
uni e se” om oposes, ca ego ies, and in e nal logic.
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4. The Legacy o G o hendieck and he Role o Topos Theo y in
Theo y F
The ma hema ical ision o Alexande G o hendieck eshaped mode n ma hema ics no me ely by
ex ending i s s uc u es, bu by ans o ming i s e y epis emological ounda ions. His app oach
shi ed ocus om objec s o ela ionships, om spaces o mo phisms, om local manipula ions o
global logic.
In his sec ion, we explo e how he G o hendieckian amewo k—pa icula ly he heo y o
oposes, shea es, and unc o ial geome y—can p o ide he ma hema ical backbone needed o
o malize Theo y F.
4.1. Topos Theo y as a S uc u al Uni e se
Fo G o hendieck, a opos is mo e han a gene alized space; i is a uni e se o a iable se s, equipped
wi h i s own in e nal logic. This is pa icula ly powe ul o Theo y F, whe e:
•Local s uc u es ( ac u es) do no assemble o e a ixed space,
•Bu a he gene a e space i sel h ough hei in e ac ion pa e ns,
•And whe e logic mus adap o he non-classical, mul i-modal s uc u e o physical phenom-
ena.
In his con ex , a opos can se e as:
•A home o he algeb a o ac u e modes,
•A logic space whe e he beha io o ime Tis encoded,
•A geome ic en i onmen whe e ields and pa icles a ise as in e nal shea es o sec ions.
4.2. Shea es, Gluing, and Eme gen Geome y
One o G o hendieck’s deepes insigh s is ha local in o ma ion, when p ope ly o ganized and
“glued,” yields global s uc u e. In Theo y F:
•Each ac u e mode can be seen as gene a ing a local shea o beha io —a di ec ional, geo-
me ic, and ene ge ic con igu a ion.
•The in e ac ion o modes o e neighbo hoods (in e sec ions) de e mines cohe ence o dis up-
ion.
•The global “ ield” is hen a sec ion o he o al ac u e shea , sa is ying gluing condi ions
ac oss space.
This no only mi o s physical locali y/non-locali y bu o e s a p ecise mechanism o eme -
gence—a co ne s one in Theo y F.
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4.3. G o hendieck’s C i ique o Physics
In R´ecol es e Semailles, G o hendieck amously c i icized mode n physics—especially ela i -
i y— o being “banal” in ma hema ical s uc u e, compa ed o he adical e hinking equi ed
by quan um heo y. He w o e ha ela i i y was like mo ing be ween dialec s o F ench, while
quan um heo y was like swi ching o Chinese.
He ound he idea o a poin in quan um physics o be p o oundly ans o med—no longe s a ic,
bu laden wi h p obabili y, con ex , and opology. He sensed an analogy be ween his ede ini ion
o he poin and his own no ion o opos, in which poin s ca y en i e wo lds o local logic.
In Theo y F, his analogy is deepened: each ac u e poin is no a ze o-dimensional objec , bu
he mani es a ion o s uc u al ension, a node whe e he en i e ield Fcondenses empo a ily. I
is he e ha G o hendieck’s ision o poin s wi h in e nal opologies becomes essen ial.
4.4. F om Schemes o S uc u al Time
In his wo k on schemes, G o hendieck eplaced he classical no ion o space wi h a spec um o
algeb aic beha io —e e y poin being de ined no by coo dina es bu by local inged s uc u es.
This is highly esonan wi h Theo y F:
•The physical wo ld is no embedded in space ime bu eme ges om s uc u al logic,
•The passage o ime Tis no absolu e bu local and g aded,
•Each egion o mode can be seen as a ibe in a la ge shea , a ying wi h dep h, in ensi y,
and cohe ence.
The mo emen om poin -se opology o s uc u ed spec a an icipa es he mo e in Theo y F
om me ic space o ield o ac u es.
4.5. F om Descen o Uni ica ion
G o hendieck in oduced he concep o descen , whe e s uc u es de ined locally can be glued
oge he i ce ain compa ibili y condi ions a e sa is ied.
Theo y F’s ision o uni ying physics— ela i i y, quan um heo y, and cosmology— elies p e-
cisely on his mechanism:
•Local ac u e con igu a ions encode indi idual physics phenomena,
•Thei compa ibili y and in e sec ion ules de ine consis ency,
•Global laws (such as Eins ein’s ield equa ions o gauge symme ies) a e hen de i ed as
descen da a om he s uc u e o F.
In essence, G o hendieck’s ma hema ical e olu ion laid he concep ual g oundwo k o a heo y
like F, e en be o e i was imagined. His abs ac , s uc u al, and ca ego ical wo ld iew eplaces
geome y wi h ela ional logic, space wi h shea - heo e ic eme gence, and poin s wi h in e nal
dynamics. Theo y F—whose ounda ion lies in s uc u al up u e and ecombina ion—may be he
physical companion G o hendieck’s ma hema ics was wai ing o .
8
5. A F ac u e Func ion in he Language o G o hendieck
I G o hendieck we e o o malize Theo y F, he would no begin wi h pa icles, di e en ial equa-
ions, o space ime me ics. He would begin wi h he s uc u al idea o a unc ion, no me ely as
a map be ween se s, bu as a unc o ial mani es a ion o deepe logic, g ounded in ca ego ies, local
da a, and cohomological cohe ence.
This sec ion p esen s a concep ual and o mal ou line o wha he F ac u e Func ion Fcould
become when exp essed h ough G o hendieckian ma hema ics.
5.1. The Func ion as a Func o o S uc u e
Le Mdeno e he ca ego y o ac u e modes— he i e i educible s uc u al gene a o s:
•Mode I: Longi udinal cu a u e
•Mode II: Tangen ial shea
•Mode III: Helicoidal o sion
•Mode IV: Radial comp ession/expansion
•Mode V: Co- ac u al cohe ence (hypo he ical, s abilizing mode)
Each objec in Mco esponds o a mode, and mo phisms encode in e ac ions o ansi ions
be ween modes.
We de ine he F ac u e Func ion as a unc o :
F:Mop −→ Topos
which assigns o each mode Mi∈ M a opos TMi, ep esen ing he logic, space, and ield beha io
gene a ed by ha mode.
5.2. Shea es o Eme gen Geome y
Fo each opos TMi, we conside a si e o ac u e e en s, deno ed UMi, o ganized by hei local
cohe ence (in e ac ions wi hin a neighbo hood o s uc u al ension).
A shea SMio e UMi hen encodes:
•Local ield beha io induced by he mode,
•Phase ansi ions o bi u ca ions,
•Ene gy accumula ions and bounda y condi ions,
•The possible eme gence o pa icles as global sec ions.
These shea es a e no o e p ede ined geome ic spaces, bu a he o e spaces gene a ed by
he ac u e logic i sel .
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I.5. Towa d Physical Quan i ies
Le :
•E(F): S uc u al ene gy densi y
•Φ(F): Eme gen ield om mode in e ac ion
•m(F): S uc u al mass (measu e o ime ozen in o o m)
Hypo he ical co espondences:
E=ZΣ
|∇F|2dV,
Φ = Bounda y beha io o shea SX,
m=T−1.
These o mulas a e concep ual skele ons, pending ull speci ica ion ia algeb aic o ca ego ical
geome y.
I.6. Fu u e Ma hema ical Tasks
To igo ously de ine and compu e F, he ollowing esea ch a enues a e sugges ed:
•Cons uc he ac u e algeb a o M: gene a o s, ela ions, cohomology.
•Explo e non-associa i e ca ego ical amewo ks ha accommoda e in e ac ions wi h in e nal
cu a u e.
•De ine highe cohomological in a ian s o classi y pa icle amilies.
•Link global shea sec ions o s anda d quan um numbe s (cha ge, spin, e c.).
•Model s uc u al e olu ion ia in e nal logics and G o hendieck opologies on mode la ices.
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