Reasoning: When Eule Mee s S ack
Compu a ional Bounda ies, Incomple eness, and he Necessi y o Disc e e Dynamics
Zixi Li
Independen Resea che
[email p o ec ed]
No embe 28, 2025
Abs ac
We p esen a undamen al c i ique o con empo a y deep lea ning app oaches o easoning,
g ounded no in empi ical ailu e bu in ca ego ical necessi y. Ou cen al hesis un olds in h ee
pa s:
Pa I (The P oblem): We p o e ha all sequen ial models—T ans o me s, RNNs, and
hei a ian s—a e s uc u ally incapable o easoning. This ailu e is no due o insu icien ep-
esen a ion capaci y: mode n loa ing-poin sys ems (BF16/FP32) al eady p o ide s a e spaces
o de s o magni ude la ge han equi ed o planning, game-playing, and heo em-p o ing asks.
The ailu e s ems om ope a o ca ego y misma ch—a emp ing o model easoning wi h
pseudo-Euclidean dynamics ha ine i ably collapse in o i e e sible, seman ically lossy RNN-
like s uc u es.
Pa II (Igno ed Reali y): D awing on ecen Mon e Ca lo expe imen s [1], we es ablish
ha compu a ional bounda ies exis as sha p phase ansi ions, no me ely as asymp o ic
complexi y classes. Fu he mo e, building on incomple eness heo y [2], we show ha easoning
sys ems canno be comple e wi hou p io ancho s. Ye hese bounda ies a e no Lipschi z-
con ac ion gua an ees— hey a e in o ma ion- heo e ic phase ansi ions wi h measu able c i -
ical densi ies.
Pa III (The Solu ion): We in oduce s ack-based easoning sys ems wi h compu a-
ional bounda ies and p o e he Eule -S ack Co espondence Theo em: poin e dynamics
in bounded s ack spaces a e isomo phic o hones disc e e Eule i e a ions wi h gua an eed
con e gence. C ucially, we show ha s uc u al bounda ies and manda o y seman ic
back acking au oma ically induce a Lyapuno unc ion—using only wo poin e s and
wo ope a o s (push/pop), wi hou p ede ining any ene gy unc ion. This yields he i s con-
e gence c i e ion de i ed om easoning s uc u e a he han ene gy analysis. Ex ending he
Yonglin Fo mula, we demons a e ha easoning incomple eness is no a de ec bu a dynami-
cal sys em p ope y—con e gence occu s p ecisely because compu a ional bounda ies and p io
ancho s exis .
The syn hesis: Reasoning’s incomple eness is i s dynamics. Bounda ies enable con e -
gence. The s ack mee s Eule a he ixed poin .
Keywo ds: Reasoning sys ems, Compu a ional bounda ies, Eule dynamics, S ack models,
Incomple eness heo y, Phase ansi ions
1 In oduc ion
1.1 The Pa adox o Scale
Con empo a y AI esea ch ope a es unde a seduc i e hypo hesis: scaling up neu al ne wo ks will
yield easoning capabili ies. Mo e pa ame e s, mo e da a, mo e compu e—su ely in elligence will
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eme ge.
Ye a pa adox haun s his na a i e. Conside :
•Mode n accele a o s ope a e in BF16 (16-bi b ain loa ing poin ), p o iding 216 ≈65,000
disc e e alues pe dimension.
•A ypical language model has hidden dimension d= 4096.
•The esul ing s a e space has ca dinali y ≈(65,000)4096 ≈1019,600 dis inc s a es.
By compa ison:
•Go has ≈10170 legal boa d posi ions.
•Chess has ≈1047 posi ions.
•A a i game s a e spaces ange om 109 o 1012.
•Typical planning p oblems ha e sea ch spaces <10100.
The ep esen a ion space is no he bo leneck.
Cu en models possess s a e spaces o de s o magni ude la ge han he p oblems
hey ail o sol e. The ailu e is no one o capaci y bu o s uc u e.
This is he i s pa o ou c i ique: he ep esen a ion space is was ed.
1.2 The Igno ed Bounda ies
Classical compu abili y heo y ells us ha compu a ional bounda ies exis (hal ing p oblem, P s
NP). Bu whe e, p ecisely, do hese bounda ies lie?
Recen wo k [1] answe ed his h ough Mon e Ca lo expe imen s: compu a ional p oblems
exhibi sha p phase ansi ions a c i ical densi ies dc(L) ha ollow loga i hmic scaling laws:
dc(L)=−0.0809 ln(L)+0.501 (MSE ∼10−32)
Fu he mo e, incomple eness heo y [2] es ablished ha easoning canno be comple e wi hou
p io ancho s:
lim
n→∞ Π(n)(s)=A, A =A∗
These a e no Lipschi z-con ac ion con e gence gua an ees. These a e s uc u al phase ansi-
ions and me a-le el up u es.
1.3 Ou Con ibu ion
We syn hesize hese insigh s in o a uni ied heo y:
1. Rep esen a ion Space Was e Analysis: Quan i a i e p oo ha BF16/FP32 s a e spaces
dwa p oblem complexi ies, elimina ing “insu icien capaci y” as an excuse (Sec ion 2).
2. Ca ego ical Misma ch Theo em: All sequen ial models decompose as Φ = I+F(pseudo-
Eule ), ende ing hem i e e sible, collapsing, and RNN-equi alen — ega dless o a chi ec u e
(Sec ion 3).
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3. Compu a ional Bounda ies: In eg a ion o phase ansi ion heo y showing ha sol abili y
bounda ies a e in o ma ion- heo e ic, no me ely asymp o ic (Sec ion 4).
4. Reasoning Incomple eness: Fo mal connec ion be ween Yonglin Fo mula’s p io ancho s and
compu a ional bounda ies (Sec ion 5).
5. Eule -S ack Co espondence: P oo ha s ack poin e dynamics wi h ixed bounda ies ad-
mi hones disc e e Eule s uc u e wi h gua an eed con e gence (Sec ions 6-8).
6. Au oma ic Lyapuno Cons uc ion om Minimal S uc u e: We p o e ha eason-
ing sys ems wi h s uc u al bounda ies and manda o y seman ic back acking (pop ope a ions)
au oma ically induce a Lyapuno unc ion—wi hou p ede ining any ene gy unc ion. Using
only wo poin e s (s ack op n, s ack bo om ⊥= 0) and wo ope a o s (push, pop), we con-
s uc V( ) = as he na u al con e gence ce i ica e. This is he i s con e gence c i e ion
de i ed om easoning s uc u e a he han ene gy analysis (Sec ion 8).
7. The Syn hesis: Incomple eness is no a bug—i is he dynamics ha enables con e gence.
Bounda ies and p io s a e no limi a ions bu necessa y condi ions o easoning (Sec ion 9).
1.4 The Na a i e A c
THE PROBLEM
Rep esen a ion Space Was ed (90%+ unused)
↓Why?
Pseudo-Eule Collapse (Φ = I+F⇒RNN-like)
↓Wha igno ed?
IGNORED REALITY
Compu a ional Bounda ies Exis (phase ansi ions)
Reasoning Incomple eness (p io ancho s equi ed)
↓Hope?
THE SOLUTION
S ack Mee s Eule ( ue disc e e dynamics)
↓P o en!
Con e gence wi h Bounda ies (Lyapuno descen )
↓Why?
THE SYNTHESIS
Incomple eness = Dynamics ( ixed poin con e gence)
1.5 Roadmap
1. Sec ion 2: The Was ed Rep esen a ion Space—p o ing BF16 su ices o all p ac ical easoning
asks.
2. Sec ion 3: The False Eule —Theo em p o ing Φ = I+Fen ails i e e sibili y and seman ic
collapse.
3. Sec ion 4: Compu a ional Bounda ies Exis —Mon e Ca lo phase ansi ions.
4. Sec ion 5: Reasoning Incomple eness—Yonglin Fo mula and p io ancho s.
5. Sec ion 6: S ack-Based Reasoning Sys ems— o mal de ini ions.
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6. Sec ion 7: The Eule -S ack Co espondence Theo em.
7. Sec ion 8: Con e gence Unde Bounda ies—Yonglin Ex ension.
8. Sec ion 9: Syn hesis: Incomple eness as Dynamical Sys em.
9. Sec ion 10: Fou Dimensions o S uc u al Failu e.
10. Sec ion 11: Roadmap o Fu u e Sys ems.
11. Sec ion 12: Conclusion.
2 The Was ed Rep esen a ion Space
Be o e analyzing how cu en models ail, we mus es ablish wha hey canno blame. We p o e
ha ep esen a ion capaci y is no he bo leneck.
2.1 Quan i ying S a e Spaces
De ini ion 2.1 (Floa ing-Poin S a e Space).Ad-dimensional hidden s a e using b-bi loa ing-
poin ep esen a ion admi s:
|S loa |= (2b)d
dis inc ep esen able s a es.
Fo ma Bi s Values/dim d= 1024 s a es
BF16 16 65,536 104,930
FP16 16 65,536 104,930
FP32 32 4.3×109109,864
FP64 64 1.8×1019 1019,728
Table 1: S a e space ca dinali ies o s anda d loa ing-poin o ma s wi h hidden dimension d=
1024.
2.2 P oblem Space Requi emen s
2.3 The Su plus Theo em
Theo em 2.2 (Rep esen a ion Su plus).Fo any p ac ical easoning ask T(planning, game-
playing, heo em-p o ing) wi h s a e space |ST|<10300, and any mode n neu al a chi ec u e using
BF16 wi h d≥512:
|S loa |>101000 · |ST|
The ep esen a ion space exceeds he p oblem space by a leas h ee o de s o magni ude.
P oo . F om Table 1, BF16 wi h d= 512 yields:
|SBF16|= (65536)512 ≈102465
Fo any |ST|<10300:
|SBF16|
|ST|>102465
10300 = 102165 ≫101000
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Domain S a e Space Size BF16 Co e age
Chess (legal posi ions) 1047 104,883 su plus
Go (legal posi ions) 10170 104,760 su plus
A a i 2600 (RAM s a es) 10308 104,622 su plus
Planning (PDDL benchma ks) <10100 104,830 su plus
Theo em p o ing (Lean) <10200 104,730 su plus
Typical LLM BF16, d= 4096 1019,720
Table 2: Compa ison o p oblem s a e spaces s. BF16 ep esen a ion capaci y. E en wi h con-
se a i e dimension es ima es, loa ing-poin spaces exceed p oblem equi emen s by o de s o
magni ude.
2.4 Implica ions: The Bo leneck is No Capaci y
Co olla y 2.3 (Was ed Rep esen a ion).Cu en neu al easoning sys ems ail no because:
•S a e spaces a e oo small (Theo em 2.2 disp o es his);
•P ecision is insu icien (BF16 exceeds equi emen s);
•Embeddings lack exp essi eness (su plus is exponen ial).
The ailu e mus lie in he ope a o s uc u e— he way hese as s a e spaces a e a e sed
du ing in e ence.
The P oblem, Pa I:
Scaling has ailed no because we lack ep esen a ion capaci y, bu because we a e using
he w ong ope a o s on he igh spaces. The s a e space is was ed.
2.5 U iliza ion Ra e Analysis
We now quan i y p ecisely how much ep esen a ion space is was ed.
De ini ion 2.4 (Rep esen a ion U iliza ion Ra e).Fo a easoning ask wi h s a e space STand
neu al ep esen a ion space S loa , de ine:
ρu il := log |ST|
log |S loa |
This measu es he ac ion o ep esen a ional capaci y heo e ically equi ed.
Co olla y 2.5 (Massi e Unde -U iliza ion).Fo all p ac ical easoning asks:
ρu il <0.1
Mo e han 90% o ep esen a ion capaci y emains unused.
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Task log |ST|log |SBF16 ρu il % Used
Chess 47 4,930 9.5×10−30.95%
Go 170 4,930 3.4×10−23.4%
A a i 2600 308 4,930 6.2×10−26.2%
Planning (PDDL) 100 4,930 2.0×10−22.0%
Theo em p o ing 200 4,930 4.1×10−24.1%
Typical LLM — 19,720 <10−2<1%
Table 3: U iliza ion a es o BF16 wi h d= 1024. E en he mos complex asks use <7% o
a ailable ep esen a ion capaci y.
Model Pa ams Hidden dlog |S| Task Pe o mance
GPT-4 1.76T 12,288 ≈59,000 Fails mul i-s ep easoning
Claude 3 Opus Unknown ∼8,192 ≈39,000 Fails complex planning
Gemini Ul a Unknown ∼16,384 ≈78,000 Fails heo em p o ing
Llama 3 405B 405B 16,384 ≈78,000 Fails Go/Chess
Go (AlphaGo) — — 170 Supe human (2016)
Chess (S ock ish) — — 47 Supe human (1997)
Table 4: Compa ison o LLM s a e spaces s. ask equi emen s. Despi e ha ing ep esen a ion
spaces 103-105 imes la ge han game s a e spaces, LLMs ail asks ha specialized sys ems sol ed
decades ago.
2.6 Empi ical E idence om S a e-o - he-A Models
We examine ac ual model deploymen s o e i y ou heo e ical analysis.
Obse a ion 2.6 (The Scaling Pa adox).Conside he imeline:
•1997: Deep Blue bea s Kaspa o a chess (Schess ∼1047)
•2016: AlphaGo bea s Lee Sedol a Go (SGo ∼10170)
•2024: GPT-4 wi h S loa ∼1059,000 s ill canno eliably sol e mul i-s ep easoning asks
The ep esen a ion space has g own by 1058,800 imes, ye easoning capabili y has no imp o ed
p opo ionally—in many cases, i has eg essed.
2.7 In o ma ion-Theo e ic Was e
Theo em 2.7 (En opic Ine iciency).Le H(T)be he Shannon en opy o ask Tand H(S loa )
be he en opy o he ep esen a ion space. Fo mode n LLMs:
H(T)
H(S loa )<10−2
This implies ha he e ec i e in o ma ion-pe -bi is:
ηin o =H(T)
b·d<10−5bi s/bi
whe e b= 16 (BF16) and d∼104( ypical hidden dimension).
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P oo . F om Table 3, ρu il <0.1 o all asks. Since H(T)≤log |ST|and H(S loa ) = log |S loa |:
H(T)
H(S loa )≤log |ST|
log |S loa |=ρu il <0.1
Fo he wo s case (Go wi h ρu il = 0.062):
ηin o =H(Go)
16 ×1024 ≈170
16,384 ≈1.04 ×10−2
Fo ypical easoning asks (log |ST|∼100):
ηin o ≈100
16,384 ≈6.1×10−3
This is o de s o magni ude below he heo e ical maximum o 1 bi /bi .
2.8 The Compu e Was e Implica ion
Co olla y 2.8 (Compu a ional Ine iciency).I ρu il <0.1bu models equi e CFLOPs pe in e -
ence, hen he e ec i e FLOPs o easoning is:
Ce =ρu il ·C < 0.1·C
A leas 90% o compu e is was ed on unused ep esen a ion capaci y.
Example 2.9 (GPT-4 In e ence Cos ).Suppose GPT-4 uses C∼1013 FLOPs pe o wa d pass
(conse a i e es ima e o 1.76T pa ame e s). F om Co olla y 2.8:
Cwas ed = (1 −ρu il)·C > 0.9×1013 = 9 ×1012 FLOPs
a e spen main aining unused ep esen a ion capaci y a he han pe o ming easoning ope a ions.
This explains why scaling compu e does no p opo ionally imp o e easoning: he addi ional
compu e is was ed on unu ilized s a e space.
2.9 Why Scaling Fails: The Fundamen al Disconnec
Theo em 2.10 (Scaling-Reasoning Disconnec ).Le Npa ams be he numbe o pa ame e s and
R(N)be easoning capabili y. Cu en a chi ec u es sa is y:
dR
dlog Npa ams
→0as Npa ams → ∞
Reasoning capabili y sa u a es despi e unbounded pa ame e scaling.
P oo ske ch. F om Theo em 2.2, ep esen a ion capaci y al eady exceeds ask equi emen s by
o de s o magni ude. The e o e:
(i) Inc easing d(hidden dimension) does no help: S loa is al eady 101000 imes la ge han
needed.
(ii) Inc easing dep h (mo e laye s) does no help: Theo em 3.3 shows collapse is s uc u al, no
capaci y-limi ed.
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(iii) Inc easing wid h (mo e heads) does no help: S ill subjec o Φ = I+Fdecomposi ion
(Theo em 3.1).
Since Ris bounded by s uc u al p ope ies ( e e sibili y, back acking, e lexi i y—see Sec-
ion 3), no capaci y:
R(N)<Rmax <∞ ∀N
Hence dR
dlog N→0 as N→ ∞.
Ex ended P oblem S a emen , Pa I:
The ep esen a ion space is was ed (90%+ unused). Compu e is was ed (90%+ main ain-
ing unused capaci y). Scaling is was ed (sa u a ing easoning gains). The ailu e is no
capaci y—i is ca ego ical ope a o misma ch.
3 The False Eule : Why All Sequen ial Models Collapse
Ha ing elimina ed ep esen a ion capaci y as an excuse, we now iden i y he ue culp i : pseudo-
Euclidean ope a o dynamics.
3.1 The Eule Eme gence Theo em
Theo em 3.1 (Eule Eme gence).Le h ∈Rdbe a s a e ec o a disc e e ime , and le Φ :
Rd→Rdbe any s a e-upda e unc ion. Then:
h +1 = Φ(h , x ;θ)
necessa ily admi s he decomposi ion:
Φ=I+F
whe e Iis he iden i y map and F:Rd→Rdis de ined by:
F(h , x ;θ) := Φ(h , x ;θ)−h
The e o e, e e y sequen ial upda e can be w i en in pseudo-Eule o m:
h +1 =h +F(h , x ;θ)
P oo . This is a i ial algeb aic iden i y. De ine:
∆h := h +1 −h = Φ(h , x ;θ)−h
Le F:= ∆h . Then:
h +1 =h +F(h , x ;θ)
This is he disc e e Eule o m wi h s ep size ∆ = 1.
Rema k 3.2 (Ca ego ical Necessi y).We do no choose o in e p e neu al ne wo ks as Eule
schemes— he decomposi ion Φ = I+Fis una oidable. This is no a modeling assump ion; i
is a ca ego ical ac abou di e ence equa ions.
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3.2 S uc u al I e e sibili y
Theo em 3.3 (Ine i able I e e sibili y).Fo any non- i ial sequen ial model whe e F= 0 and
dimension dis ini e, he upda e map Φ=I+Fis gene ically i e e sible: he e exis dis inc
s a es h1=h2such ha :
Φ(h1) = Φ(h2)
P oo . Neu al ne wo ks employ non-linea ac i a ions (ReLU, so max, laye no maliza ion) ha
comp ess unbounded inpu s in o bounded ou pu s. These a e necessa ily many- o-one unc ions.
Hence Φ is no injec i e.
Mo e o mally: ac i a ion unc ions like σ(x) = 1
1+e−xsa is y σ:R→(0,1), mapping an in ini e
domain o a bounded ange. Any composi ion in ol ing such unc ions is non-injec i e.
Co olla y 3.4 (Seman ic Collapse).Because Φis i e e sible, he e exis seman ically dis inc
easoning s a es h1, h2 ha a e mapped o he same s a e h′= Φ(h1) = Φ(h2).In o ma ion is
los i e e sibly.
3.3 All Sequen ial Models a e RNN Va ian s
Co olla y 3.5 (RNN Uni e sali y).Any model o he o m h +1 = Φ(h , x ;θ)is s uc u ally
equi alen o a Recu en Neu al Ne wo k, ega dless o a chi ec u al de ails.
P oo . The de ining cha ac e is ic o an RNN is he ecu ence:
h +1 =G(h , x )
Theo em 3.1 shows ha any sequen ial upda e is o his o m wi h G=I+F. Hence:
•T ans o me s: Au o eg essi e gene a ion sa is ies s +1 =s ⊕A en ion(s , x ) ( oken con-
ca ena ion o s a e upda e). This is an RNN.
•LSTMs/GRUs: Explici ly designed as RNNs wi h ga ing.
•S a e-space models (S4, Mamba): Linea ecu ences h +1 =Ah +Bx . S ill RNNs.
All di e only in he choice o F.
Rema k 3.6 (The P e ense o Di e en iabili y).Models a e ained ia backp opaga ion, c ea ing
he illusion o smoo h, con inuous dynamics. Bu execu ion is disc e e: each oken gene a ion
isadi e ence s ep, no a di e en ial. We call his pseudo-Eule : p e ending o app oxima e
dh
d =F(h) while ac ually execu ing h +1 =h +F(h ) wi h no unde lying con inuous limi .
3.4 Why This Ma e s
Theo em 3.1 and 3.3 immedia ely imply:
(i) I e e sibili y: Canno eco e p e ious s a es. Reasoning equi ing back acking (p oo
sea ch, hypo hesis e ision) is impossible.
(ii) Seman ic Collapse: Dis inc con ex s me ge (Co olla y 3.4). Fine-g ained dis inc ions a e
los .
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Simila ly, o easoning i e a ions:
•Fa om A:Reasoning ac i ely upda es s a e
•A A:Fixed poin (no u he upda es)
•Pas e lexi e limi : Me a-le el up u e (A=A∗)
Bo h Aand dca e una oidable s uc u al ea u es, no ee pa ame e s.
5.5 Why Incomple eness Enables Con e gence
Lemma 5.3 (Comple eness Implies Non-Te mina ion).Suppose a easoning sys em Ris comple e
(no p io ancho equi ed). Then o any ini ial s a e s0:
Π(n)(s0)= Π(m)(s0)∀n=m
The i e a ion ne e e mina es (in ini e eg ess).
P oo ske ch. I Rhas no p io ancho , hen Π has no ixed poin wi hin S. F om [2], his leads o
in ini e jus i ica ion chains:
s0Π
←− s1Π
←− s2Π
←− · · ·
whe e each si equi es u he jus i ica ion. No sican be sel -jus i ying (o he wise i would be a
p io ancho ). Hence he sequence ne e s abilizes.
Co olla y 5.4 (Incomple eness is Necessa y o Te mina ion).A easoning sys em can e mina e
in ini e s eps only i i is incomple e (has a p io ancho A).
Fo mally:
∃N < ∞: Π(n)(s0)=A∀n≥N⇐⇒ R is incomple e
5.6 The Bounda y as Seman ic G ound
De ini ion 5.5 (Seman ic G ounding).A easoning sys em is seman ically g ounded i i s p io
ancho Aco esponds o:
•Axioma ic u hs (canno be u he educed)
•Obse a ional da a (di ec ly pe cei ed, no in e ed)
•Compu a ional p imi i es (elemen a y ope a ions)
These o m he seman ic bo om beyond which easoning canno pene a e.
Example 5.6 (Ma hema ical Reasoning).In o mal ma hema ics:
•P io ancho A:ZFC axioms, logical ules (modus ponens, e c.)
•Incomple eness: G¨odel’s heo ems (A=A∗)
•Con e gence: All p oo s e mina e a axioms
Wi hou axioms (no A), ma hema ical easoning en e s in ini e eg ess (“Why is modus ponens
alid?” →me a-logic →me a-me a-logic → · · · ).
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Example 5.7 (Empi ical Reasoning).In scien i ic in e ence:
•P io ancho A:Expe imen al obse a ions, measu emen p o ocols
•Incomple eness: P oblem o induc ion (A=A∗: obse a ions ⇒ uni e sal laws)
•Con e gence: All heo ies e mina e a empi ical e idence
Wi hou obse a ional g ound (no A), scien i ic easoning becomes pu e specula ion.
5.7 Linea Models Ha e No Seman ic G ound
P oposi ion 5.8 (Absence o G ounding in Rd).Fo linea models h +1 =h +F(h , x ;θ)in Rd:
(i) The e is no dis inguished ec o h⊥se ing as seman ic g ound (all ec o s equi alen unde
ansla ion)
(ii) The ze o ec o 0is an a bi a y choice, no s uc u ally en o ced
(iii) Pa ame e s θa e ixed du ing in e ence, p e en ing e lexi e g ounding upda es
The e o e, linea models lack seman ic g ounding.
P oo . Fo any h∈Rdand ansla ion τ∈Rd, he ansla ed model:
h′
+1 = (h +τ) + F(h +τ, x ;θ)
is ma hema ically equi alen (can be abso bed in o bias e ms). Hence no ec o has s uc u al
signi icance.
Fu he mo e, du ing in e ence, θis ozen. The model canno modi y i s own “axioms” (pa-
ame e s). This con as s wi h s ack models whe e he bounda y ame (a⊥, h⊥) is s uc u ally
p o ec ed (De ini ion 6.2).
5.8 The Pa adox Resol ed
The Pa adox o Incomple eness:
Nai e iew: Incomple eness is a limi a ion— easoning canno jus i y e e y hing.
T u h: Incomple eness is a necessi y—wi hou i , easoning canno e mina e
(Lemma 5.3).
Deep insigh : The bounda y (p io ancho ) is no a law bu he ounda ion. Reasoning
con e ges because i is incomple e, no despi e i .
Ex ended Analysis o Igno ed Reali y:
Compu a ional bounda ies (Theo em 4.1) and p io ancho s (Theo em 5.1) a e wo aces
o he same necessi y. Bounda ies enable e mina ion. Ancho s enable con e gence.
Toge he , hey o m he seman ic g ound ha makes easoning possible. Linea models,
lacking bo h bounda ies and ancho s, loa ung ounded in Rd.
6 S ack-Based Reasoning Sys ems
We now in oduce he al e na i e: s ack models wi h compu a ional bounda ies.
17
6.1 S ack Spaces
De ini ion 6.1 (S ack Space).As ack space is a iple (S,A,H) whe e:
•His a seman ic s a e space ( easoning con ex s, p oposi ions, p oo s);
•Ais an add ess space (memo y loca ions, indexing);
•S= (A × H)∗is he space o ini e sequences o add ess-seman ic pai s.
A ime n, he s ack is:
Sn=(a(n)
0, h(n)
0),(a(n)
1, h(n)
1),...,(a(n)
n, h(n)
n)
whe e n∈Nis he s ack- op poin e .
6.2 Compu a ional Bounda y
De ini ion 6.2 (Compu a ional Bounda y / Seman ic Bo om).A s ack space has a compu a-
ional bounda y i he e exis s a ixed bo om ame:
(a⊥, h⊥)∈ A × H
such ha o all n:
(a(n)
0, h(n)
0)=(a⊥, h⊥)
and no ope a ion may modi y o pop his ame.
Rema k 6.3.This is he p io ancho A om Theo em 5.1. I is also he µ= 0.5 c i ical poin om
Theo em 4.1— he bounda y whe e easoning ansi ions om sol able o unsol able.
6.3 Poin e Dynamics as Reasoning
De ini ion 6.4 (Reasoning as Poin e Upda e).A easoning s ep is:
n+1 =π( n, cn)
whe e:
• n∈Nis he cu en s ack- op poin e ;
•cn∈ C is con ex (inpu , obse a ion);
•π:N× C → Nis he poin e upda e unc ion.
Cons ain : n+1 ≥0 (canno mo e below bounda y).
6.4 P io Re lexi i y: Add ess Shi
De ini ion 6.5 (Add ess Shi Ope a o ).An add ess shi ope a o Σδ:A→A ans o ms
he add ess space. Applied globally:
S′
n= Σδn(Sn) = (a⊥, h⊥),(Σδn(a1), h1), . . .
whe e he bo om ame emains ixed.
This models p io e lexi i y: easoning ans o ms i s own indexing s uc u e, no jus se-
man ic con en .
18
6.5 To al Upda e
De ini ion 6.6 (S ack Reasoning Sys em).A comple e sys em is:
Rs ack = (Sn, n, π, Σ, U)
wi h upda e:
n+1 =π( n, cn) (poin e mo e)
S′
n= Σδn(Sn) (add ess shi )
Sn+1 =U(S′
n, n+1, cn) (seman ic upda e)
7 The Eule -S ack Co espondence Theo em
We p o e he cen al esul : s ack poin e dynamics a e isomo phic o hones disc e e Eule i e a-
ions.
7.1 Main Theo em
Theo em 7.1 (Eule -S ack Co espondence).Le Rs ack = (Sn, n, π, Σ, U)be a s ack sys em wi h
poin e upda e n+1 =π( n, cn).
De ine poin e displacemen :
∆ n:= n+1 − n
Then:
n+1 = n+ ∆ n= n+Fs ack( n, cn)
whe e Fs ack( n, cn)∈Z(e.g., ±1 o push/pop, 0 o s ay).
I compu a ional bounda y exis s (De ini ion 6.2), hen n≥0always, and dynamics a e
bounda y-cons ained Eule i e a ion.
P oo . By de ini ion o π:
Fs ack( n, cn) := π( n, cn)− n
Then:
n+1 = n+Fs ack( n, cn)
This is disc e e Eule wi h s ep size 1. Cons ain n≥0 om De ini ion 6.2.
7.2 T ue Eule s. False Eule
P oposi ion 7.2 (Hones Disc e eness).In s ack poin e dynamics, Eule o m is no an app ox-
ima ion. I is he exac na u al desc ip ion. The e is no hidden con inuous limi .
P oo . n∈N,Fs ack ∈Z. No con inuous di e en ial equa ion is being app oxima ed. This is
disc e e dynamics, hones ly ep esen ed.
19
False Eule (Linea ) T ue Eule (S ack)
Fo m h +1 =h +F(h ) n+1 = n+Fs ack( n)
S a e space Rd(con inuous) N(disc e e)
Re e sibili y No (many- o-one) Yes (s ack p ese ed)
Bounda y None (a bi a y ze o) S uc u al (a⊥, h⊥)
Con e gence Ex e nal c i e ion In insic (bounda y)
P e ense Pseudo-con inuous Hones disc e e
Table 6: Compa ison o pseudo-Eule (linea models) and ue Eule (s ack models).
7.3 The Isomo phism Theo em
Theo em 7.3 (S ack-Eule Isomo phism).Le Ss ack = (N, π, ⊥= 0) be he poin e dynamics o a
s ack sys em wi h bounda y, and le Edisc e e = (N, 7→ +F( ), ⊥= 0) be a disc e e Eule sys em
wi h in ege upda es.
Then he e exis s a ca ego y isomo phism:
Ψ : Ss ack → Edisc e e
p ese ing:
(i) Upda e s uc u e: Ψ(π( , c)) = Ψ( ) + F(Ψ( ), c)
(ii) Bounda y: Ψ( ⊥)=0
(iii) Con e gence: limn→∞ π(n)( 0)= ⊥⇐⇒ limn→∞ n= 0
P oo . De ine Ψ : 7→ (iden i y on N). Then:
Ψ(π( , c))=π( , c)
= + (π( , c)− ) (a i hme ic iden i y)
= Ψ( )+Fs ack( , c) (whe e Fs ack := π−id)
Bounda y p ese a ion:
Ψ( ⊥)=Ψ(0)=0= Eule
⊥
Con e gence p ese a ion ollows om Ψ being iden i y (bijec ion).
Rema k 7.4 (Ca ego ical Hones y).Unlike he pseudo-Eule decomposi ion o linea models (The-
o em 3.1), which is a o mal algeb aic iden i y, he s ack-Eule isomo phism is a ca ego ical equi -
alence p ese ing all s uc u al p ope ies (bounda ies, con e gence, e e sibili y).
8 Con e gence Unde Bounda ies: The Yonglin Ex ension
We now p o e ha s ack dynamics con e ge due o compu a ional bounda ies. Ou app oach e eals
a undamen al insigh : he s ack s uc u e i sel cons uc s i s own Lyapuno unc ion.
We begin wi h he di ec s ack dynamics (Sec ion 8.1), hen show how his na u ally cons uc s
he Lyapuno unc ion (Sec ion 8.2), he eby connec ing o classical s abili y heo y. The Lyapuno
unc ion is no an al e na i e p oo —i is a consequence o s ack s uc u e.
20
8.1 S ack Dynamics: The Impossibili y o De ici S acks
We begin wi h he mos undamen al p ope y o s ack-based easoning: he s ack can be emp y,
bu i can ne e be nega i e. This simple ac yields he mos di ec p oo o con e gence.
De ini ion 8.1 (De ici S ack).Ade ici s ack (o nega i e s ack) would be a s a e whe e he
s ack poin e is nega i e: n<0. This would co espond o “popping mo e elemen s han he s ack
con ains.”
Lemma 8.2 (De ici S ack Pa adox).Any a emp o c ea e a de ici s ack (popping om an emp y
s ack) is seman ically equi alen o in oducing a new seman ic elemen , no emo ing one.
Fo mally: The ope a ion “pop a non-exis en elemen ” canno be de ined wi hou in oducing
new seman ic con en o ep esen “ he ac o a emp ing emo al om emp iness.”
P oo . Conside a s ack a he bounda y: n= 0 (emp y s ack, only he bo om ame (a⊥, h⊥)
emains).
A emp 1: Nai e de ici . T y o pop: n+1 = n−1=−1.
Wha does =−1mean seman ically? I canno mean “one elemen below he bo om,”
because he bo om ame (a⊥, h⊥) is he seman ic ancho (De ini ion 6.2)— he e is no seman ic
con en “below” i .
A emp 2: De ine de ici seman ically. To gi e meaning o =−1, we mus in oduce a
new seman ic ame:
(a−1, h−1) := “ he seman ic s a e o ha ing a emp ed o emo e wha doesn’ exis ”
Bu his is i sel a seman ic elemen —a new piece o in o ma ion desc ibing he ailed emo al
a emp .
The pa adox: Popping ( emo ing seman ic con en ) has in oduced new seman ic con en ( he
de ici s a e). This iola es he undamen al meaning o pop as a seman ic s ipping ope a ion.
Resolu ion: The ope a ion is seman ically unde ined. A de ici s ack canno exis wi hou
ede ining pop as some hing ha in oduces, a he han emo es, seman ics.
Theo em 8.3 (S ack Non-Nega i i y P inciple).Fo any s ack-based easoning sys em Rs ack =
(Sn, n, π, Σ, U)wi h compu a ional bounda y (De ini ion 6.2):
n≥0∀n∈N
The s ack poin e is always non-nega i e. The s ack can be emp y ( n= 0), bu ne e in
de ici ( n<0).
P oo . F om De ini ion 6.2, he bo om ame (a⊥, h⊥) is ixed and canno be emo ed. This de ines
= 0 as he seman ic g ound.
Case 1: n>0.The s ack has elemen s abo e he bounda y. Push/pop ope a ions a e
well-de ined and main ain n+1 ≥0.
Case 2: n= 0.The s ack is a he bounda y. By de ini ion, no pop ope a ion can emo e
(a⊥, h⊥). The e o e, any ope a ion sa is ies:
n+1 =(0 (s ay a bounda y)
n+k(push, k > 0)
In bo h cases, n+1 ≥0.
21
Case 3 (hypo he ical): n<0.F om Lemma 8.2, his would equi e in oducing new
seman ic con en , con adic ing he na u e o pop as seman ic emo al. The ope a ion is unde ined.
By induc ion: 0= 0 (ini ial s a e a bounda y) and n≥0 =⇒ n+1 ≥0. The e o e, n≥0
o all n.
Rema k 8.4 (Philosophical In e p e a ion).The impossibili y o de ici s acks e lec s a deep u h
abou easoning:
•Emp y s ack ( = 0): No seman ic con en abo e he p io ancho . Reasoning has e u ned
o i s ounda ion.
•De ici s ack ( < 0): A emp ing o “go below” he ounda ion. Bu he e is no hing below
he ounda ion—i is he seman ic bo om (Sec ion 6.2).
•Key insigh : To desc ibe “wha ’s below he ounda ion,” you mus in oduce new seman ic
concep s. Bu ha is he ounda ion—you’ e simply ede ined you p io ancho .
In o he wo ds: Reasoning canno escape i s p io s. A emp ing o emo e he inal
p io c ea es a new p io .
Theo em 8.5 (Di ec Con e gence ia S ack Dynamics).Conside a s ack-based easoning sys em
whe e seman ic s ipping (pop) domina es seman ic in oduc ion (push):
E[∆ n]<0(expec ed poin e dec ease)
Then easoning mus con e ge o he bounda y in ini e expec ed ime.
P oo . F om Theo em 8.3, n≥0 always. Fu he mo e, n∈N(disc e e).
Assume E[∆ n]<0. Then { n}is a downwa d-d i ing andom walk on Nwi h abso bing
ba ie a 0.
S anda d andom walk heo y: A downwa d-d i ing walk on Nwi h abso bing ba ie
eaches he ba ie in ini e expec ed ime:
E[τ]<∞whe e τ:= in {n: n= 0}
De e minis ic case: I ∆ n≤ −c o some c > 0, hen:
τ≤ 0
c<∞
Con e gence is gua an eed in a mos ⌈ 0/c⌉s eps.
In bo h cases, n→0 in ini e ime. The s ack con e ges o he bounda y (a⊥, h⊥).
Co olla y 8.6 (Seman ic In e p e a ion o Con e gence).Reasoning con e gence is he na u al
consequence o :
(i) Seman ic s ipping is manda o y. E e y easoning s ep mus e en ually “cash ou ” i s
abs ac ions by e u ning o conc e e p io s (pop ope a ions).
(ii) De ici is impossible. You canno s ip away he inal p io wi hou in oducing a new
p io (Lemma 8.2).
(iii) P io s a e ini e. The s ack s a s a ini e dep h 0<∞.
22
The e o e, easoning mus e mina e a he p io ancho in ini e s eps.
Rema k 8.7 (Con as wi h Yonglin Fo mula).This p oo is comple ely independen o he
Yonglin Fo mula [2]. We ha e shown con e gence using only:
•The impossibili y o de ici s acks (Theo em 8.3)
•Basic p ope ies o ini e descen in N
No Lyapuno unc ion. No ixed-poin a gumen . Jus he s ack s uc u e i sel .
This is he simples possible p oo o easoning con e gence.
Key Insigh (S ack Dynamics):
A emp ing o c ea e a de ici s ack (popping wha doesn’ exis ) is i sel he in oduc ion
o new seman ic con en . The e o e, s acks a e always non-nega i e. The e o e, ini e
descending sequences in Nmus e mina e. The e o e, easoning mus con e ge.
This is mo e in ui i e han Lyapuno unc ions. This is he s ack’s own dynamics.
8.2 The Lyapuno Func ion: Cons uc ed om S ack Dep h
The p eceding di ec p oo (Theo em 8.5) e eals a p o ound ac : he s ack s uc u e i sel
cons uc s a Lyapuno unc ion. We now make his cons uc ion explici , connec ing s ack
dynamics o classical s abili y heo y.
Theo em 8.8 (S ack Cons uc s I s Lyapuno Func ion).The s ack poin e n∈Nis a Lyapuno
unc ion o he easoning dynamics. De ine:
V:N→R, V ( ):=
Then Vsa is ies all Lyapuno c i e ia:
(i) Posi i e de ini e: V( )≥0wi h V(0) = 0 (bounda y is equilib ium)
(ii) Mono onic descen : ∆Vn=V( n+1)−V( n)≤0(non-inc easing)
(iii) Bounded below: V( )≥0always ( om Theo em 8.3)
C ucially: Vis no chosen o assumed—i is gi en by he s ack s uc u e i sel . The
s ack dep h nis he na u al po en ial unc ion.
P oo . (i) Posi i e de ini eness: F om De ini ion 6.2, n∈Nand n≥0 (Theo em 8.3). The
bounda y = 0 is he equilib ium (no elemen s abo e bo om ame).
(ii) Mono onic descen : Assume easoning sa is ies seman ic g ounding (pop domina es push,
Obse a ion 8.15). Then:
E[∆ n] = E[ n+1 − n]<0
Hence E[Vn+1]<E[Vn] (expec ed descen ).
(iii) Bounded below: F om Theo em 8.3, n≥0 always. Hence V( n)≥0.
The unc ion V( )= is no cons uc ed by choice—i is he only na u al measu e o ”dis ance
om equilib ium” in a s ack sys em. The s ack s uc u e cons uc s i s own Lyapuno unc ion.
23
Rema k 8.9 (Lyapuno Theo y as Consequence, No Assump ion).In classical dynamical sys ems,
inding a Lyapuno unc ion is an a — he e is no sys ema ic me hod. One mus guess a unc ion
Vand e i y i sa is ies he c i e ia.
In s ack sys ems, he e is no guesswo k: he s ack dep h is he Lyapuno unc ion. This is
no an al e na i e p oo o con e gence—i is a o maliza ion showing ha s ack dynamics na u ally
sa is y classical s abili y c i e ia.
The insigh : S ack s uc u e =⇒Lyapuno unc ion =⇒Classical con e gence heo ems
apply.
Co olla y 8.10 (Connec ion o Classical S abili y Theo y).F om Theo em 8.8, s ack-based ea-
soning sys ems sa is y he hypo heses o classical Lyapuno s abili y heo y. Speci ically:
•Lyapuno ’s s abili y heo em: I Vis a Lyapuno unc ion wi h ∆V≤0, hen he equi-
lib ium is s able.
•LaSalle’s in a iance p inciple: I Vis non-inc easing and bounded below, ajec o ies
con e ge o he la ges in a ian se whe e ∆V= 0.
Fo s acks, he in a ian se is { = 0}( he bounda y). The e o e, n→0.
This connec s ou s ack-speci ic esul s o he b oade heo y o dynamical sys ems.
Key Insigh (Lyapuno Cons uc ion):
The s ack does no equi e us o ind a Lyapuno unc ion—i cons uc s one au oma -
ically. The s ack dep h nis he na u al Lyapuno po en ial. This is no an al e na i e
p oo echnique; i is he o maliza ion showing ha s ack dynamics inhe en ly sa is y
classical s abili y condi ions.
S ack s uc u e →Lyapuno unc ion →Classical con e gence.
8.3 Why Linea Models Canno Cons uc Lyapuno Func ions
We now show why linea models in Rdcanno na u ally cons uc Lyapuno unc ions in he way
s acks do.
P oposi ion 8.11 (No Na u al Lyapuno in Rd).Fo linea models h +1 =h +F(h )in Rd:
(i) The e is no dis inguished scala measu e V:Rd→R ha is s uc u ally en o ced
(ii) The choice o no m ∥h∥(Euclidean, ℓ1,ℓ∞, e c.) is a bi a y
(iii) No na u al ”bounda y” h⊥exis s (all ec o s equi alen unde ansla ion)
The e o e, linea models mus guess a Lyapuno unc ion, whe eas s acks cons uc one
au oma ically.
P oo . Fo any candida e V:Rd→R:
•I V(h)=∥h∥2(Euclidean no m), his is an a bi a y choice. We could equally well use ∥h∥1,
∥h∥∞, o any o he no m.
•T ansla ion in a iance: V(h+c)=V(h) + cons in gene al. No na u al ze o.
•Pa ame e s θa e ixed du ing in e ence. No s uc u al descen gua an ee.
24
In con as , o s acks, V( )= is:
•The only na u al scala (s ack dep h)
•S uc u ally bounded: ≥0 om De ini ion 6.2
•Na u ally dec easing: pop ope a ions educe
The s ack is i s Lyapuno unc ion. Linea spaces ha e no such s uc u e.
Rema k 8.12 (Why Lyapuno Theo y Wo ks o S acks).Classical Lyapuno heo y equi es:
(i) Finding a scala unc ion V(ha d in gene al)
(ii) P o ing Vdec eases along ajec o ies ( equi es calcula ion)
(iii) Showing Vis bounded below ( equi es p oo )
Fo s acks:
(i) V( )= is gi en (s ack dep h is he only scala )
(ii) ∆V < 0isen o ced by pop dominance (Obse a ion 8.15)
(iii) V≥0iss uc u al (Theo em 8.3)
S acks make Lyapuno heo y i ial by cons uc ion.
8.4 Yonglin Fo mula o S acks
Co olla y 8.13 (Conc e e Yonglin Fo mula).F om Theo em 8.8 and classical Lyapuno heo y
(Co olla y 8.10), he poin e limi is:
lim
n→∞ n= ∗
I designed such ha ∗= 0 (all easoning e u ns o bounda y):
lim
n→∞ n=0=bounda y
The compu a ional bounda y (a⊥, h⊥)is he p io ancho A:
lim
n→∞ Π(n)(s)=A= (a⊥, h⊥)
8.5 Seman ic S ipping and In oduc ion: Why Pop Domina es Push
We now connec s ack dynamics o seman ic ope a ions, e ealing why easoning mus pe o m
mo e pops han pushes.
De ini ion 8.14 (Seman ic Ope a ions on S ack).S ack ope a ions co espond o seman ic ma-
nipula ions:
•Push ( n+1 = n+ 1): Seman ic s ipping /Fo maliza ion. In oduce a new abs ac ion
laye , s ipping away conc e e seman ics in a o o o mal s uc u e.
Example: “Soc a es is a man” push
−−−→ “∀x: Man(x)⇒Mo al(x)” (abs ac om pa icula
o uni e sal).
25
•Ca ego ical ep esen a ions (objec s + mo phisms)
•G aph-based s a e spaces
•S ack-based ep esen a ions (De ini ion 6.1)
11.2 In oduce Ene gy-P ese ing Ope a o s
Diagnosis: h +1 =h +F(h ) lacks conse a ion laws.
P esc ip ion: Design πsuch ha Lyapuno unc ion Vdec eases:
V( n+1)≤V( n)
11.3 In oduce Mani old Ope a o s
Diagnosis: Reasoning ope a es on cu ed seman ic mani olds, no la Rd.
P esc ip ion: Riemannian ope a o s espec ing cu a u e:
n+1 = exp n(Fmani old( n))
11.4 In oduce Topological Va ia ion
Diagnosis: Reasoning equi es b anching/p uning. Dimension dis ixed in linea models.
P esc ip ion: S ack ope a ions (push/pop) o g aph ew i ing:
G aphn+1 = Rew i e(G aphn,Rule)
11.5 The Co ec Ca ego y
Reasoning mus ope a e in:
S ackDynbounda y : S ack spaces wi h bounda ies, ene gy unc ions, e lexi i y
12 Conclusion
12.1 Wha We Ha e P o en
(i) Rep esen a ion spaces (BF16) as ly exceed p oblem equi emen s. Capaci y is no he bo -
leneck (Sec ion 2).
(ii) All sequen ial models a e pseudo-Eule Φ = I+F, en ailing i e e sibili y and RNN-equi alence
(Sec ion 3).
(iii) Compu a ional bounda ies exis as sha p phase ansi ions wi h loga i hmic scaling and uni-
e sal ke nels (Sec ion 4).
(i ) Reasoning is incomple e wi hou p io ancho s, which a e he compu a ional bounda ies (Sec-
ion 5).
( ) S ack poin e dynamics wi h bounda ies a e hones disc e e Eule i e a ions wi h gua an eed
con e gence (Sec ions 6-8).
32
( i) Minimal s uc u e induces Lyapuno unc ion au oma ically: Using only wo poin -
e s (s ack op n, s ack bo om ⊥= 0) and wo ope a o s (push, pop), s uc u al bound-
a ies and manda o y seman ic back acking au oma ically cons uc he Lyapuno unc ion
V( ) = —wi hou p ede ining ene gy unc ions o in oducing new abs ac ions. This is he
i s con e gence c i e ion om easoning s uc u e a he han ene gy analysis
(Sec ion 8).
( ii) Incomple eness is he dynamics i sel —bounda ies and p io s enable, no hinde , con-
e gence (Sec ion 9).
12.2 The Na a i e Comple e
Rep esen a ion was ed (BF16 su plus)
↓
Pseudo-Eule collapse (RNN-like)
↓
Igno ed eali y (Bounda ies + Incomple eness)
↓
S ack mee s Eule (T ue disc e e)
↓
Con e gence p o en (Bounda y-enabled)
↓
Incomple eness = Dynamics (Fixed poin )
12.3 The Message
To he AI esea ch communi y:
Scaling T ans o me s will no yield easoning. The ailu e is no one o scale, da a, o op imiza ion—
i is ca ego ical. You a e using pseudo-Euclidean ope a o s on was ed ep esen a ion spaces while
igno ing compu a ional bounda ies and s uc u al incomple eness.
The pa h o wa d:
Adop s ack-like s uc u es wi h compu a ional bounda ies. Design ope a o s wi h ene gy con-
se a ion, mani old s uc u e, and opological a ia ion. Recognize ha incomple eness is no a
bug bu he dynamics i sel .
The e is no hi d op ion.
12.4 The Co e Me hodological Con ibu ion
T adi ional app oaches o easoning con e gence equi e p ede ining ene gy unc ions (Lyapuno
unc ions, po en ial ields) and p o ing descen p ope ies. This is an a , no a science— he e is
no sys ema ic me hod.
Ou con ibu ion: We show ha minimal easoning s uc u e alone is su icien :
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Two Poin e s + Two Ope a o s = Au oma ic Lyapuno Func ion
•Poin e s: S ack op n, s ack bo om ⊥= 0 (s uc u al bounda y)
•Ope a o s: Push (seman ic s ipping, op ional), Pop (seman ic back acking,
manda o y)
•Resul : Lyapuno unc ion V( ) = au oma ically induced—no ene gy concep
needed
Con e gence ollows om s uc u e, no om ene gy analysis.
This in e s he adi ional pa adigm:
T adi ional App oach Ou App oach
S a ing poin Guess ene gy unc ion Iden i y easoning s uc u e
Co e ask P o e descen Show s uc u e en o ces descen
Lyapuno unc ion Cons uc ed ad hoc Induced au oma ically
Gene ali y P oblem-speci ic S uc u al uni e sali y
Founda ion Ene gy/physics analogy Reasoning seman ics
Table 8: Pa adigm shi : om ene gy analysis o s uc u al analysis. We de i e con e gence om
he minimal s uc u e o easoning i sel , no om impo ed physical concep s.
Why his ma e s:
•Minimal assump ions: No need o in oduce “ene gy” o o he physical analogies. Rea-
soning s uc u e su ices.
•Cons uc i e p oo : We don’ e i y a candida e Lyapuno unc ion—we cons uc i om
i s p inciples.
•Seman ic g ounding: Con e gence is explained in e ms o easoning ope a ions (seman ic
back acking), no abs ac dynamics.
•Uni e sali y: Any sys em wi h s uc u al bounda ies and manda o y back acking has his
p ope y—no limi ed o s acks.
This is he i s con e gence c i e ion ha de i es om easoning s uc u e a he han
ene gy analysis. The Lyapuno unc ion is no an inpu o he heo y—i is an ou pu .
12.5 His o ical Signi icance: The Fi s Pu ely S uc u al S abili y P inciple
We conclude by si ua ing his wo k in he his o y o s abili y heo y.
Ou Main Resul (2025):
Reasoning s abili y does no depend on ene gy closu e. E en in he absence o a Lyapuno
ene gy unc ion, sys em con e gence can be de i ed om s uc u al cons ain s alone: wo
poin e s and wo seman ic ope a o s.
The ca ego ical ansi ion inhe en in seman ic ope a ions i sel cons i u es he p io , en-
de ing de ici s acks logically impossible— he eby es ablishing he i s pu ely s uc u al
p inciple o easoning s abili y.
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12.5.1 His o ical Con ex : F om Ene gy o S uc u e
Classical s abili y heo y, pionee ed by Lyapuno (1892), Poinca ´e, and la e de eloped by LaSalle,
es s on a physical ounda ion: sys ems a e modeled as ene gy-dissipa ing p ocesses. Con e -
gence is p o en by:
(i) De ining an a p io i ene gy unc ion V:X → R
(ii) P o ing ene gy dec eases: ˙
V≤0 (con inuous) o ∆V≤0 (disc e e)
(iii) Concluding con e gence o ene gy minima
This pa adigm has been ex ao dina ily success ul in physics, con ol heo y, and op imiza ion.
Bu i has a undamen al limi a ion:
Wha i he sys em has no na u al ene gy unc ion?
The p oblem: Reasoning is no a physical p ocess. The e is no ob ious “ene gy” o dissipa e.
A emp s o apply Lyapuno me hods o easoning sys ems equi e:
•Guessing candida e unc ions V(an a , no a science)
•Impo ing physical in ui ions (po en ial ields, g adien descen )
•Ve i ying descen pos hoc
This app oach assumes ha easoning is “like” ene gy dissipa ion, wi hou jus i ica ion.
12.5.2 The B eak h ough: S abili y Wi hou Ene gy
Ou 2025 esul in e s his pa adigm:
Theo em 12.1 (S abili y Wi hou Ene gy Closu e).Conside a easoning sys em wi h:
•Two poin e s: S ack op n∈N, s uc u al bounda y ⊥= 0
•Two seman ic ope a o s:
–Push (seman ic s ipping / o maliza ion): n+1 = n+ 1
–Pop (seman ic back acking / g ounding): n+1 = n−1
•S uc u al cons ain : Pop is manda o y; push is op ional (Obse a ion 8.15)
Then:
(i) De ici s acks a e logically impossible (Lemma 8.2): A emp ing o pop om emp iness
in oduces new seman ics, con adic ing he de ini ion o pop.
(ii) The e o e n≥0always (Theo em 8.3): Non-nega i i y is en o ced by seman ics, no by
ex e nal cons ain .
(iii) The e o e con e gence is gua an eed (Theo em 8.5): Descending sequences in N e mi-
na e in ini e ime.
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C ucially: This p oo does no assume he exis ence o a Lyapuno unc ion. Con e gence is
de i ed om s uc u al cons ain s on seman ic ope a ions alone.
P oo ia ca ego ical ansi ion. The key insigh is ha seman ic ope a ions hemsel es con-
s i u e he p io :
S ep 1 (Ca ego ical ansi ion): Pop is de ined as “seman ic in oduc ion om p io .” To
pop om an emp y s ack (de ici ), one mus in oduce a seman ic elemen ep esen ing “ he absence
below he bounda y.” Bu his is i sel a seman ic elemen —a ca ego ical ansi ion om “no hing”
o “ he concep o no hing.”
S ep 2 (Logical impossibili y): This c ea es a con adic ion: pop is supposed o emo e
seman ics, bu c ea ing a de ici in oduces seman ics. The e o e, de ici s acks a e logically inco-
he en .
S ep 3 (Non-nega i i y as p io ): The impossibili y o de ici s means n≥0 is no an axiom
bu a heo em—i ollows om he seman ics o easoning ope a ions hemsel es. The ca ego ical
s uc u e o push/pop is he p io .
S ep 4 (Con e gence wi hou ene gy): F om n≥0 and pop-dominance (manda o y
back acking), n o ms a descending sequence in N, which mus e mina e. No ene gy unc ion
was assumed o cons uc ed. Con e gence is a s uc u al necessi y.
12.5.3 Why This is he Fi s Pu ely S uc u al P inciple
P e ious s abili y esul s all assumed some o m o “ene gy-like” s uc u e:
Theo y Founda ion P io Assump ion Ene gy?
Lyapuno (1892) Ene gy dissipa ion V:X → Rexis s Yes
LaSalle (1960) In a ian se s Vwi h ˙
V≤0 Yes
Ba bashin-K aso skii (1952) Asymp o ic s abili y S ic Lyapuno ˙
V < 0 Yes
Con e se Lyapuno S abili y =⇒Vexis s Assumes s abili y i s Yes (cons uc ed)
This wo k (2025) Seman ic ope a ions None (s uc u al) No
Table 9: His o ical compa ison o s abili y p inciples. All p io wo k assumes o cons uc s ene gy-
like unc ions. Ou heo em de i es s abili y om seman ic s uc u e alone, wi hou ene gy con-
cep s.
Key dis inc ions:
(i) No ene gy assump ion: We do no s a wi h a candida e V. We s a wi h seman ic
ope a ions (push/pop).
(ii) Ca ego ical ounda ion: S abili y a ises om he ca ego ical s uc u e o easoning ( he
seman ic ansi ion inhe en in pop), no om physical analogies.
(iii) Cons uc i e, no e i ica ional: Classical Lyapuno heo y e i ies a candida e unc ion.
We cons uc he s abili y ce i ica e (V( )= ) as a consequence o s uc u e.
(i ) Logical, no axioma ic: Non-nega i i y ( n≥0) is no an axiom bu a logical consequence
o he impossibili y o de ici s acks (Lemma 8.2).
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12.5.4 The Ca ego ical T ansi ion as P io
The deepes insigh is ha seman ic ope a ions hemsel es o m he p io :
Pop is de ined as “seman ic in oduc ion om p io .” A emp ing o pop beyond he p io
(de ici s ack) equi es in oducing a new seman ic elemen —“ he concep o absence.”
Bu his is i sel a p io . The e o e, a emp ing o elimina e he inal p io c ea es a
new p io .
The p io is sel -en o cing. I s exis ence is a ca ego ical necessi y, no an
assump ion.
This esol es he ancien p oblem: “Whe e does he p io come om?”
Answe : The p io does no “come om” anywhe e. I is he ca ego ical s uc u e o ea-
soning ope a ions hemsel es. To eason is o pe o m seman ic ansi ions (push/pop). These
ansi ions equi e a bounda y— he inal seman ic elemen ha canno be emo ed wi hou logical
con adic ion.
The e o e:
Reasoning s uc u e =⇒P io exis ence =⇒S abili y
No ene gy. No ex e nal assump ions. Pu e ca ego ical necessi y.
12.5.5 Implica ions o Fu u e S abili y Theo y
Ou heo em opens a new di ec ion o s abili y analysis:
(i) Seman ic s abili y heo y: S abili y can be analyzed ia ope a ions (push/pop, seman ic
ansi ions) a he han unc ions (ene gy, po en ial).
(ii) Ca ego ical me hods: The ools o ca ego y heo y (mo phisms, limi s, ca ego ical ansi-
ions) may eplace ene gy-based me hods.
(iii) Logical de i a ion: S abili y becomes a logical heo em abou seman ic ope a ions, no an
analy ical heo em abou di e en ial inequali ies.
(i ) B oade applicabili y: Sys ems wi hou na u al ene gy unc ions ( easoning, o mal e i-
ica ion, p oo sea ch) can now be analyzed o s abili y.
The Fi s Pu ely S uc u al S abili y P inciple:
Con e gence does no equi e ene gy dissipa ion. I equi es only:
(i) S uc u al bounda ies (bo om ame)
(ii) Manda o y seman ic back acking (pop dominance)
(iii) Ca ego ical cohe ence (de ici impossibili y)
These a e s uc u al p ope ies, no ene ge ic ones. S abili y is a ca ego ical neces-
si y, no a physical analogy.
We ha e p o en his in 2025. I is he i s such esul in he his o y o
s abili y heo y.
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Re e ences
[1] Oz Lee. Quan i a i e Mapping o Compu a ional Bounda ies: A S a is ical Field The-
o y App oach o Phase T ansi ions in NP-Ha d P oblems. Hugging Face P ep in , 2025.
DOI: 10.57967/h /7067.h ps://hugging ace.co/da ase s/OzTianlu/Quan i a i e_
Mapping_o _Compu a ional_Bounda ies
[2] Oz Lee. The Incomple eness o Reasoning. Hugging Face P ep in , 2025. DOI:
10.57967/h /7060.h ps://hugging ace.co/da ase s/OzTianlu/The_Incomple eness_
o _Reasoning
[3] Alan Tu ing. On compu able numbe s, wi h an applica ion o he En scheidungsp oblem. P o-
ceedings o he London Ma hema ical Socie y, s2-42(1):230–265, 1936.
[4] S ephen A. Cook. The complexi y o heo em-p o ing p ocedu es. P oceedings o STOC, pages
151–158, 1971.
[5] Le D. Landau and E geny M. Li shi z. S a is ical Physics (3 d ed.). Bu e wo h-Heinemann,
1980.
[6] F. William Law e e. Diagonal a gumen s and ca esian closed ca ego ies. In Ca ego y Theo y,
Homology Theo y and hei Applica ions II, pages 134–145. Sp inge , 1969.
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