Why Do Few-Step Text Latents Fail When Image Latents Work? Non-Commitment at Sharp Categorical Readouts

# Why Do Few-Step Text Latents Fail When Image Latents Work? Non-Commitment at Sharp Categorical Readouts
Source: [https://arxiv.org/html/2606.30705](https://arxiv.org/html/2606.30705)
###### Abstract

Deterministic few\-step generation succeeds on continuous image latents but collapses to incoherent text on continuous text latents, and we show the cause is geometric rather than a training or scaling deficiency: a smooth, regularity\-limited deterministic map cannot resolve a discrete branch choice before a sharp categorical readout, so few\-step failure is governed by decoder sharpness, not transport accuracy\. In the*overlapping*regime of real text autoencoders, we prove \(Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\) that the posterior\-mean terminal step flips tokens at the rate of the latent mass in anO​\(s​\(t\)\)O\(s\(t\)\)tube around decision boundaries; the rate is set by decoder sharpness, not transport accuracy\. Two diagnostics, DABI \(readout sharpness\) and CCI \(categorical commitment\), measured on published checkpoints show that four independently built continuous\-text decoders amplify a boundary\-aligned perturbation far beyond a norm\-matched isotropic one \(DABI\\mathrm\{DABI\}from5×1025\{\\times\}10^\{2\}to\>105\{\>\}10^\{5\}\), while image decoders haveDABI≈1\\mathrm\{DABI\}\\approx 1\. Two mechanisms escape the continuous bound: categorical commitment \(autoregressive decoders succeed despite*sharper*readouts; removing it collapses generation\) and stochastic re\-injection \(deterministic ODE atK=4K\{=\}4gives PPL294294versus SDE5050on the same model\)\. In the idealized*separated*regime we prove matching sharp transport laws, including a dimension phase diagram: the deterministic stiffness needed to separateMMmodes grows asΘ​\(log⁡M\)\\Theta\(\\sqrt\{\\log M\}\)once the latent dimension isΩ​\(log⁡M\)\\Omega\(\\log M\)\(and asM1/nM^\{1/n\}in fixed dimension\), with a depth\-BBhierarchy giving aB\\sqrt\{B\}\-smaller per\-step peak \(Theorems[5](https://arxiv.org/html/2606.30705#Thmtheorem5)–[7](https://arxiv.org/html/2606.30705#Thmtheorem7)\); a coarea identity links these to the overlapping tube \(Theorem[17](https://arxiv.org/html/2606.30705#Thmtheorem17)\)\. The result is an accuracy–depth–stiffness tradeoff: within the deterministic\-continuous class the cost is irreducible, and both escapes step outside it\.

image latenttext latentdecoderDABI≈1\\mathrm\{DABI\}\\approx 1knowgoodtimeworkloveneedmakeworldhereflipdecoderw\#rk ?ov7e th$DABI≫1\\mathrm\{DABI\}\\gg 1same smooth few\-step map: a sharp readout flips text, not images

Figure 1:Why deterministic few\-step generation works for images but not text\.A smooth few\-step map delivers each latent only to within anO​\(s​\(t\)\)O\(s\(t\)\)posterior\-mean blur \(the fuzzy disk\)\.\(left\)An image decoder is smooth and has no categorical readout, so the blur is absorbed \(decoder amplificationDABI≈1\\mathrm\{DABI\}\\approx 1\) and the output stays correct\.\(right\)A text decoder reads out byarg​maxy⁡wy⊤​z\\operatorname\*\{arg\\,max\}\_\{y\}w\_\{y\}^\{\\top\}z, partitioning the latent into sharp argmax cells; the*same*blur, landing on a decision boundary, flips the token \(DABI≫1\\mathrm\{DABI\}\\gg 1\) and the text becomes incoherent\. Failure is set by the sharpness of the readout, not by transport accuracy \(Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\)\. Two mechanisms leave the deterministic\-continuous class and escape the bound: categorical commitment \(autoregressive/masked decoders\) and stochastic re\-injection \(SDE\); theDABI×CCI\\mathrm\{DABI\}\{\\times\}\\mathrm\{CCI\}taxonomy \(Section[5\.3](https://arxiv.org/html/2606.30705#S5.SS3)\) predicts which systems fail\.## 1Introduction

Deterministic few\-step generation of images has advanced rapidly\. Consistency models\(Songet al\.,[2023](https://arxiv.org/html/2606.30705#bib.bib16)\), rectified flow\(Liuet al\.,[2023](https://arxiv.org/html/2606.30705#bib.bib19)\), and progressive distillation\(Salimans and Ho,[2022](https://arxiv.org/html/2606.30705#bib.bib17)\)produce high\-quality images in11–44network evaluations by learning smooth transport maps from noise to data\. The same approach, applied to continuous text latents, fails \(Figure[1](https://arxiv.org/html/2606.30705#S0.F1)\)\. A deterministic few\-step generator of the latent space of a text autoencoder \(such as ELF\(Huet al\.,[2026b](https://arxiv.org/html/2606.30705#bib.bib1)\)\) produces incoherent text, with repeated tokens and multilingual fragments, atK≤16K\\leq 16steps, becoming usable only atK≥32K\\geq 32–6464\. Our evidence indicates this failure is not primarily explained by training, architecture, or scheduling, but is structural: when the decoder’s readout boundaries are sharp relative to the transport residual, a smooth deterministic map cannot resolve the discrete branch choice within a few steps\. We formalize this for the posterior\-mean terminal step and verify the prediction on published checkpoints; the bound applies to deterministic transport, which stochastic and categorical generators escape\.

#### The failure mode\.

A text autoencoder encodes discrete tokens into a continuous latentz=E​\(x\)z=E\(x\)and decodes viaDW​\(z\)=arg​maxy⁡wy⊤​zD\_\{W\}\(z\)=\\operatorname\*\{arg\\,max\}\_\{y\}w\_\{y\}^\{\\top\}z\. A deterministic few\-step generator is a composition of smooth maps with bounded Lipschitz constant; the readout amplifies small structured displacements near decision boundaries into token errors\. This failure mode appears in two regimes\. In the*overlapping*regime \(real text autoencoders\), we prove \(Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\) that the posterior\-mean terminal step flips tokens at the readout\-calibrated non\-commitment rate: the latent mass in anO​\(s​\(t\)\)O\(s\(t\)\)tube around decision boundaries\. Failure is therefore set by decoder sharpness, not transport accuracy\. In the idealized*separated*regime, transport toMMwell\-separated modes costs interface\-energy stiffness \(Theorems[5](https://arxiv.org/html/2606.30705#Thmtheorem5)–[7](https://arxiv.org/html/2606.30705#Thmtheorem7)\)\. The separated\-mode𝔍\\mathfrak\{J\}does*not*predict real ELF residuals, yet a coarea identity \(Theorem[17](https://arxiv.org/html/2606.30705#Thmtheorem17)\) links the two as contractions of one composite\-interface geometry\.

#### Main results\.

*\(1\) Non\-commitment theorem\.*The flip rate equals the readout\-calibrated non\-commitment≥\\geqBayes non\-commitment; asymptotically,ℙ​\(Y^t≠Y\)≍ℙ​\[δ∗​\(X,Y\)≲s​\(t\)\]\\mathbb\{P\}\(\\hat\{Y\}\_\{t\}\\neq Y\)\\asymp\\mathbb\{P\}\[\\delta^\{\*\}\(X,Y\)\\lesssim s\(t\)\]\. An oracle roll\-in on ELF \(we noise a clean latent and integrate the deterministic ODE back tot=1t\{=\}1from each step; Figure[3](https://arxiv.org/html/2606.30705#S4.F3)\) confirms this:1212–41%41\\%terminal flip compounds to≈90%\{\\approx\}90\\%, allKKcollapse onto ones​\(t\)s\(t\)curve, and the residual is structured \(1010–33×33\\timesisotropic\) with a≈1%\{\\approx\}1\\%boundary\-normal fraction\.*\(2\) Image/text dichotomy\.*We define DABI \(Decoder Amplification of Boundary\-aligned Inputs\):DABI=Δ​CEstruct/Δ​CErand\\mathrm\{DABI\}=\\Delta\\mathrm\{CE\}\_\{\\mathrm\{struct\}\}/\\Delta\\mathrm\{CE\}\_\{\\mathrm\{rand\}\}at matched perturbation norm\. Under one margin\-normal probe applied identically to all four text codecs, a boundary\-aligned perturbation flips4747–77%77\\%of tokens while a norm\-matched isotropic one flips≈0%\{\\approx\}0\\%\(ELFDABI=508×\\mathrm\{DABI\}=508\\times\[446,580\]\[446,580\]; LangFlow\(Chenet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib4)\), CoLa\-DLM\(Guoet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib3)\), Cosmos\(Meshchaninovet al\.,[2025](https://arxiv.org/html/2606.30705#bib.bib5)\)have isotropic response at the floor, soDABI≫104\\mathrm\{DABI\}\\gg 10^\{4\}; Table[1](https://arxiv.org/html/2606.30705#S4.T1),95%95\\%bootstrap CIs\)\. For the two flow models with a deterministic generator, the realized terminal residual already triggers this sharpness: it flips21×21\\times\(ELF\) and≈2,600×\{\\approx\}2\{,\}600\\times\(LangFlow\) more tokens than an isotropic residual of the same norm\. By contrast,44published image VAEs×\\times55NFE haveDABI∈\[0\.85,1\.94\]\\mathrm\{DABI\}\\in\[0\.85,1\.94\]: an image/text gap of orders of magnitude with no overlap\.*\(3\) Escape mechanisms\.**Stochastic:*on the same ELF\-B teacher\(Huet al\.,[2026b](https://arxiv.org/html/2606.30705#bib.bib1)\), ODEK=4K\{=\}4PPL=294\{\}=294vs SDE PPL=50\{\}=50; even the distilled PD student\(Huet al\.,[2026a](https://arxiv.org/html/2606.30705#bib.bib29)\)shows a persistent ODE\-to\-SDE gap \(1515–38%38\\%forK≥2K\\geq 2\)\.*Categorical:*AR decoders haveDABI=915\\mathrm\{DABI\}=915–19,805×19\{,\}805\\times\(sharper than ELF\) yet succeed because categorical commitment makes the generator discontinuous \(Lemma[4](https://arxiv.org/html/2606.30705#Thmtheorem4)\); removing commitment collapses generation \(Dream:5×5\\timesPPL jump\)\.*\(4\) Sharp transport laws\.*limΛ→∞Λ​infWpp=𝔍p,σ\\lim\_\{\\Lambda\\to\\infty\}\\Lambda\\inf W\_\{p\}^\{p\}=\\mathfrak\{J\}\_\{p,\\sigma\}\(Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5), hard\-gradientΓ\\Gamma\-limit\);𝒫n,M≍log⁡M\\mathcal\{P\}\_\{n,M\}\\asymp\\sqrt\{\\log M\}forn≥C​log⁡Mn\\geq C\\log M\(Theorem[6](https://arxiv.org/html/2606.30705#Thmtheorem6)\); depth\-BBhierarchy achieves peak ratioB\\sqrt\{B\}\(Theorem[7](https://arxiv.org/html/2606.30705#Thmtheorem7)\)\.

#### Scope\.

All lower bounds apply to*deterministic*transport\. This is an accuracy–depth–stiffness tradeoff, not an impossibility: two escapes \(categorical commitment, stochastic re\-injection\) leave the continuous\-deterministic class\.

#### Related work\.

Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5)is the hard\-Lipschitz counterpart of the Baldo/Fonseca–Tartar multiwellΓ\\Gamma\-limit\(Baldo,[1990](https://arxiv.org/html/2606.30705#bib.bib20); Fonseca and Tartar,[1989](https://arxiv.org/html/2606.30705#bib.bib21)\)\. The Gaussian\-width identity connects to generic chaining\(Talagrand,[2005](https://arxiv.org/html/2606.30705#bib.bib24)\)and Milman–Neeman\(Milman and Neeman,[2022](https://arxiv.org/html/2606.30705#bib.bib22)\)\.Salmonaet al\.\([2022](https://arxiv.org/html/2606.30705#bib.bib25)\)showed Lipschitz pushforwards are limited on multimodal targets; our contribution is the dimension\-sensitive interface law and the categorical\-escape identification\. FMLM\(Leeet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib2)\)is an instantiation of categorical escape \(Section[5\.4](https://arxiv.org/html/2606.30705#S5.SS4)\)\. ELF\+PD\(Huet al\.,[2026a](https://arxiv.org/html/2606.30705#bib.bib29)\)confirms the ODE\-vs\-SDE gap on the model we study\. Extended related work is in Appendix[L](https://arxiv.org/html/2606.30705#A12)\.

## 2Setup and Definitions

#### Source and target\.

The source is a probability measureσ=ρ​d​x\\sigma=\\rho\\,dxonℝn\\mathbb\{R\}^\{n\}with strictly positiveC1C^\{1\}density \(the standard Gaussianγn\\gamma\_\{n\}being the leading example\)\. The target is concentrated nearMMcodebook points𝒞=\{c1,…,cM\}⊂ℝd\\mathcal\{C\}=\\\{c\_\{1\},\\ldots,c\_\{M\}\\\}\\subset\\mathbb\{R\}^\{d\}with weightsπi\>0\\pi\_\{i\}\>0; we writeμ=∑iπi​δci\\mu=\\sum\_\{i\}\\pi\_\{i\}\\delta\_\{c\_\{i\}\}\. In the separated regime,mini≠j⁡‖ci−cj‖≥Δ\>0\\min\_\{i\\neq j\}\\left\\lVert c\_\{i\}\-c\_\{j\}\\right\\rVert\\geq\\Delta\>0\. In the overlapping regime \(real text autoencoders\), per\-token posteriors overlap andΔ\\Deltais not a meaningful parameter\.

#### Linear readout and normalized margin\.

A linear readout decodesz∈ℝdz\\in\\mathbb\{R\}^\{d\}to a token:DW​\(z\)=arg​maxy⁡wy⊤​zD\_\{W\}\(z\)=\\operatorname\*\{arg\\,max\}\_\{y\}w\_\{y\}^\{\\top\}z, whereW=\[w1,…,wV\]W=\[w\_\{1\},\\ldots,w\_\{V\}\]is the readout matrix with vocabulary sizeVV\. The normalized margin of the correct tokenyyagainst competitorjjis

δy​j∗​\(z\)=\(wy−wj\)⊤​z‖wy−wj‖,δ∗​\(z,y\)=minj≠y⁡δy​j∗​\(z\)\.\\delta^\{\*\}\_\{yj\}\(z\)=\\frac\{\(w\_\{y\}\-w\_\{j\}\)^\{\\top\}z\}\{\\left\\lVert w\_\{y\}\-w\_\{j\}\\right\\rVert\},\\qquad\\delta^\{\*\}\(z,y\)=\\min\_\{j\\neq y\}\\delta^\{\*\}\_\{yj\}\(z\)\.\(1\)The decision boundaries are the hyperplanesΣi​j=\{z:wi⊤​z=wj⊤​z\}\\Sigma\_\{ij\}=\\\{z:w\_\{i\}^\{\\top\}z=w\_\{j\}^\{\\top\}z\\\}\.

#### Flow interpolant and posterior mean\.

LetXXdenote the clean latent andUt=t​X\+\(1−t\)​s0​εU\_\{t\}=tX\+\(1\-t\)s\_\{0\}\\varepsilon,ε∼γn\\varepsilon\\sim\\gamma\_\{n\}, the flow interpolant at timettwith noise scales0s\_\{0\}\. The population\-optimalxx\-prediction \(terminal Euler step\) is the posterior meanmt​\(u\)=𝔼​\[X∣Ut=u\]m\_\{t\}\(u\)=\\mathbb\{E\}\[X\\mid U\_\{t\}=u\]\. The effective noise scale at timettiss​\(t\)=s0​\(1−t\)/ts\(t\)=s\_\{0\}\(1\-t\)/t; smallttmeans large noise, largettmeans small noise\.

#### Barrier metric and interface energy\.

For wells𝒞\\mathcal\{C\}, putV\(y\)=dist\(y,𝒞\)pV\(y\)=\\operatorname\{dist\}\(y,\\mathcal\{C\}\)^\{p\}and define the barrier metric

κp​\(i,j\)=infξ:ci→cj∫01V​\(ξ​\(s\)\)​‖ξ˙​\(s\)‖​𝑑s,\\kappa\_\{p\}\(i,j\)=\\inf\_\{\\xi:c\_\{i\}\\to c\_\{j\}\}\\int\_\{0\}^\{1\}V\(\\xi\(s\)\)\\left\\lVert\\dot\{\\xi\}\(s\)\\right\\rVert\\,ds,\(2\)the leastVV\-weighted path length between two wells\. The weighted interface energy and its prescribed\-mass profile are

ℰp,σ​\(ℬ;𝒞\)=∑i<jκp​\(i,j\)​Pσ​\(Bi,Bj\),𝔍p,σ​\(π;𝒞\)=infσ​\(Bi\)=πiℰp,σ​\(ℬ;𝒞\),\\mathcal\{E\}\_\{p,\\sigma\}\(\\mathcal\{B\};\\mathcal\{C\}\)=\\sum\_\{i<j\}\\kappa\_\{p\}\(i,j\)\\,P\_\{\\sigma\}\(B\_\{i\},B\_\{j\}\),\\qquad\\mathfrak\{J\}\_\{p,\\sigma\}\(\\pi;\\mathcal\{C\}\)=\\inf\_\{\\sigma\(B\_\{i\}\)=\\pi\_\{i\}\}\\mathcal\{E\}\_\{p,\\sigma\}\(\\mathcal\{B\};\\mathcal\{C\}\),\(3\)wherePσ​\(Bi,Bj\)=∫∂∗Bi∩∂∗Bjρ​𝑑ℋn−1P\_\{\\sigma\}\(B\_\{i\},B\_\{j\}\)=\\int\_\{\\partial^\{\*\}B\_\{i\}\\cap\\partial^\{\*\}B\_\{j\}\}\\rho\\,d\\mathcal\{H\}^\{n\-1\}is the weighted perimeter and the infimum is over Caccioppoli partitions with prescribed masses\.

###### Definition 1\(DABI: decoder amplification of boundary\-aligned inputs\)\.

Given a decoderDD, letrstructr\_\{\\mathrm\{struct\}\}be a unit*boundary\-aligned*direction andrrandr\_\{\\mathrm\{rand\}\}a norm\-matched isotropic Gaussian perturbation\. At perturbation fractionff:

DABI​\(D\)=Δ​CE​\(f⋅rstruct\)Δ​CE​\(f⋅rrand\)\|f=1\.\\mathrm\{DABI\}\(D\)=\\frac\{\\Delta\\mathrm\{CE\}\(f\\cdot r\_\{\\mathrm\{struct\}\}\)\}\{\\Delta\\mathrm\{CE\}\(f\\cdot r\_\{\\mathrm\{rand\}\}\)\}\\bigg\|\_\{f=1\}\.We use two boundary\-aligned directions\. The*margin\-normal*direction points toward the nearest decision boundary \(the readout normalwy−wjw\_\{y\}\-w\_\{j\}, or its local gradient for a nonlinear decoder\) and reaches that boundary atf=1f=1\. It is the canonical worst\-case readout\-sharpness probe; because it is defined for*any*categorical readout, it gives one apples\-to\-apples number across all codecs\. The*realized\-residual*direction is the actual posterior\-mean residual of a deterministic generator, available only when the system has one; it measures how much that readout sharpness affects real generation\. Text decoders with hardarg​max\\operatorname\*\{arg\\,max\}boundaries haveDABI≫1\\mathrm\{DABI\}\\gg 1under both; smooth image decoders \(no categorical readout\) haveDABI≈1\\mathrm\{DABI\}\\approx 1on the realized residual\. For text,Δ​CE\\Delta\\mathrm\{CE\}is the cross\-entropy increase; for images, it is the pixel\-L2L^\{2\}increase\. Each is the natural output\-space loss for its domain\. The ratioDABI\\mathrm\{DABI\}is dimensionless within each domain: it measures how much more damaging a structured perturbation is than an isotropic one at the same norm, which is well\-defined regardless of loss scale\. Cross\-domain comparison is meaningful becauseDABI≈1\\mathrm\{DABI\}\\approx 1indicates isotropic sensitivity \(no boundary structure\) whileDABI≫1\\mathrm\{DABI\}\\gg 1indicates that boundaries concentrate the sensitivity\.

###### Definition 2\(CCI: categorical commitment index\)\.

The CCI of a generator at stepkkis the fraction of positions where an*irreversible*discrete selection \(sampling from a categorical distribution followed by fixing the token for all later steps\) has been made by stepkk\. At termination: autoregressive generation hasCCI=1\\mathrm\{CCI\}=1\(every position committed\); continuous flow decoding hasCCI=0\\mathrm\{CCI\}=0\(no discrete selection ever occurs\)\. Masked diffusion withKKsteps commits a growing subset of positions per step, reachingCCI=1\\mathrm\{CCI\}=1at termination; the generation\-time averageCCI¯∈\(0,1\)\\overline\{\\mathrm\{CCI\}\}\\in\(0,1\)reflects the schedule\.

## 3The Non\-Commitment Mechanism

This section establishes the central result: a deterministic terminal step outputs a posterior mean whose decoded\-token flip rate is determined by the readout decision\-boundary geometry\.

###### Theorem 3\(readout\-calibrated non\-commitment\)\.

Assume clean decoding is exact:Y=DW​\(X\)=arg​maxy⁡wy⊤​XY=D\_\{W\}\(X\)=\\operatorname\*\{arg\\,max\}\_\{y\}w\_\{y\}^\{\\top\}Xalmost surely \(no label noise\)\. LetY^t=DW​\(mt​\(Ut\)\)\\hat\{Y\}\_\{t\}=D\_\{W\}\(m\_\{t\}\(U\_\{t\}\)\)be the decoded token of the posterior\-mean terminal step, andqt​\(y∣u\)=ℙ​\(Y=y∣Ut=u\)q\_\{t\}\(y\\mid u\)=\\mathbb\{P\}\(Y=y\\mid U\_\{t\}=u\)the posterior over tokens\. Then:

1. \(i\)Equality and Bayes bound\. ℙ​\(Y^t≠Y\)=𝔼​\[1−qt​\(DW​\(mt​\(Ut\)\)∣Ut\)\]≥𝔼​\[1−maxy⁡qt​\(y∣Ut\)\],\\mathbb\{P\}\(\\hat\{Y\}\_\{t\}\\neq Y\)=\\mathbb\{E\}\\big\[1\-q\_\{t\}\(D\_\{W\}\(m\_\{t\}\(U\_\{t\}\)\)\\mid U\_\{t\}\)\\big\]\\;\\geq\\;\\mathbb\{E\}\\big\[1\-\\max\_\{y\}q\_\{t\}\(y\\mid U\_\{t\}\)\\big\],\(4\)with𝔼​\[−log⁡pW​\(Y∣mt​\(Ut\)\)\]≥\(log⁡2\)​𝔼​\[1−maxy⁡qt​\(y∣Ut\)\]\\mathbb\{E\}\[\-\\log p\_\{W\}\(Y\\mid m\_\{t\}\(U\_\{t\}\)\)\]\\geq\(\\log 2\)\\,\\mathbb\{E\}\[1\-\\max\_\{y\}q\_\{t\}\(y\\mid U\_\{t\}\)\]for the cross\-entropy\.
2. \(ii\)Active\-facet tube law\.IfρX\\rho\_\{X\}is smooth with regular decision boundaries and no positive\-mass triple intersections, then ass​\(t\)=s0​\(1−t\)/t↓0s\(t\)=s\_\{0\}\(1\-t\)/t\\downarrow 0, ℙ​\(Y^t≠Y\)=2π​s​\(t\)​∑i<j∫Fi​jρX​𝑑ℋd−1\+o​\(s​\(t\)\)≍ℙ​\[δ∗​\(X,Y\)≲s​\(t\)\],\\mathbb\{P\}\(\\hat\{Y\}\_\{t\}\\neq Y\)=\\sqrt\{\\frac\{2\}\{\\pi\}\}\\,s\(t\)\\,\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\\,d\\mathcal\{H\}^\{d\-1\}\+o\(s\(t\)\)\\;\\asymp\\;\\mathbb\{P\}\\big\[\\delta^\{\*\}\(X,Y\)\\lesssim s\(t\)\\big\],\(5\)whereFi​j=∂Ωi∩∂ΩjF\_\{ij\}=\\partial\\Omega\_\{i\}\\cap\\partial\\Omega\_\{j\}is the*active co\-maximal facet*between the argmax cellsΩi=\{z:DW​\(z\)=i\}\\Omega\_\{i\}=\\\{z:D\_\{W\}\(z\)=i\\\}, i\.e\. the\(d−1\)\(d\{\-\}1\)\-faces where tokensi,ji,jare co\-maximal\. Only adjacent decision facets contribute: a non\-adjacent pairwise hyperplaneΣi​j\\Sigma\_\{ij\}lies interior to a third token’s cell and carries no flips\. The posterior\-mean readoutDW​\(mt\)D\_\{W\}\(m\_\{t\}\)and the Bayes/MAP classifierarg​maxy⁡qt​\(y∣⋅\)\\operatorname\*\{arg\\,max\}\_\{y\}q\_\{t\}\(y\\mid\\cdot\)share this leading constant, since their switching surfaces agree toO​\(s2\)O\(s^\{2\}\)by Tweedie’s formula \(Appendix[D](https://arxiv.org/html/2606.30705#A4)\)\.

###### Proof sketch \(full proof in Appendix[D](https://arxiv.org/html/2606.30705#A4)\)\.

*Equality\.*Condition onUt=uU\_\{t\}=u: the deterministic outputmt​\(u\)m\_\{t\}\(u\)and hence the decoded tokend=DW​\(mt​\(u\)\)d=D\_\{W\}\(m\_\{t\}\(u\)\)are fixed, while the true tokenYYis distributed asqt\(⋅∣u\)q\_\{t\}\(\\cdot\\mid u\)\. Thusℙ​\(Y^t≠Y∣Ut=u\)=1−qt​\(d∣u\)\\mathbb\{P\}\(\\hat\{Y\}\_\{t\}\\neq Y\\mid U\_\{t\}=u\)=1\-q\_\{t\}\(d\\mid u\); integrating gives the equality\. The Bayes bound follows becauseqt​\(d∣u\)≤maxy⁡qt​\(y∣u\)q\_\{t\}\(d\\mid u\)\\leq\\max\_\{y\}q\_\{t\}\(y\\mid u\)\.

*CE bound\.*For anyy≠dy\\neq d,−log⁡pW​\(y∣mt\)≥log⁡\(1\+e\(wd−wy\)⊤​mt\)≥log⁡2\-\\log p\_\{W\}\(y\\mid m\_\{t\}\)\\geq\\log\(1\+e^\{\(w\_\{d\}\-w\_\{y\}\)^\{\\top\}m\_\{t\}\}\)\\geq\\log 2becauseddis thearg​max\\operatorname\*\{arg\\,max\}\. Weighting byqt​\(y∣u\)q\_\{t\}\(y\\mid u\)and summing overy≠dy\\neq dgives the stated bound\.

*Tube law\.*Rescale:Ut/t=X\+s​\(t\)​εU\_\{t\}/t=X\+s\(t\)\\varepsilonwiths​\(t\)=s0​\(1−t\)/ts\(t\)=s\_\{0\}\(1\-t\)/t, somaxy⁡qt​\(y∣Ut\)\\max\_\{y\}q\_\{t\}\(y\\mid U\_\{t\}\)is the posterior ofYYgivenX\+s​\(t\)​εX\+s\(t\)\\varepsilon\. The Bayes non\-commitment at noise scalessis theγ\\gamma\-blurred boundary mass\. A one\-dimensional Laplace expansion across each active facetFi​jF\_\{ij\}, using smoothness ofρX\\rho\_\{X\}and regularity of the boundaries, gives the leading term; non\-adjacent hyperplanes contribute onlyo​\(s\)o\(s\)\. The equivalence≍ℙ​\[δ∗≲s​\(t\)\]\\asymp\\mathbb\{P\}\[\\delta^\{\*\}\\lesssim s\(t\)\]follows because the normalized distance to the nearest boundary isδ∗\\delta^\{\*\}\. ∎

#### Relevance to learned generators\.

Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)is stated for the population posterior mean\. A trained denoiser with squared\-loss targetsmtm\_\{t\}; at convergence, its terminal step approximates the posterior\-mean step\. The oracle roll\-in \(Section[4\.3](https://arxiv.org/html/2606.30705#S4.SS3)\) confirms that the*published ELF teacher’s learned ODE*tracks the posterior\-mean predictions: allKKcollapse onto thes​\(t\)s\(t\)curve, and structured\-vs\-random ratios \(1010–33×33\\times\) match DABI\.

#### Two features matching the data\.

*\(i\) Only the row space matters\.*A flip requires\(wY−wj\)⊤​\(X−mt\)/‖wY−wj‖≥δY​j∗​\(X\)\(w\_\{Y\}\-w\_\{j\}\)^\{\\top\}\(X\-m\_\{t\}\)/\\left\\lVert w\_\{Y\}\-w\_\{j\}\\right\\rVert\\geq\\delta^\{\*\}\_\{Yj\}\(X\): only the projection of the residualX−mtX\-m\_\{t\}onto the boundary normalswY−wjw\_\{Y\}\-w\_\{j\}enters\. On ELF, the terminal residual is9797–99%99\\%in the decoder null space, yet the small \(∼1%\{\\sim\}1\\%\) structured component along boundary normals suffices to flip tokens\.

*\(ii\) Sharpness is decisive\.*Small normalized marginsδ∗\\delta^\{\*\}make a tiny structured displacement enough to flip the token\. A smooth image decoder, lacking a hardarg​max\\operatorname\*\{arg\\,max\}boundary, tolerates the same posterior\-mean blur that flips tokens under a sharp text readout\. This is the mechanism behind the image/text dichotomy\.

#### Two contractions of one interface \(the bridge\)\.

Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\(overlapping regime\) and Theorems[5](https://arxiv.org/html/2606.30705#Thmtheorem5)–[6](https://arxiv.org/html/2606.30705#Thmtheorem6)\(separated regime\) are not a single scalar functional:𝔍\\mathfrak\{J\}does*not*predict real ELF residuals\. They are, however, two contractions of*one*geometric object\. Consider the composite source\-space classifierF=DW∘TF=D\_\{W\}\\circ T\. A coarea identity \(Theorem[17](https://arxiv.org/html/2606.30705#Thmtheorem17), Appendix[E](https://arxiv.org/html/2606.30705#A5)\) writes the readout boundary massAW=∑i<j∫Fi​jρX​𝑑ℋd−1A\_\{W\}=\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\\,d\\mathcal\{H\}^\{d\-1\}as the source\-interface perimeter, discounted pointwise by the readout\-normal stretch‖D​T⊤​ni​j‖\\left\\lVert DT^\{\\top\}n\_\{ij\}\\right\\rVert\. The separated energy𝔍\\mathfrak\{J\}is the barrier\-weighted stretch moment of that same source interface\. Composing with the tube law gives the rigorous accuracy–depth–stiffness product law: for aKK\-step generatorT=TK∘⋯∘T1T=T\_\{K\}\\circ\\cdots\\circ T\_\{1\},

lim infs↓0∏kLip⁡\(Tk\)s​ℙ​\(Y^s≠Y\)≥2π​𝔍p,σ​\(π;C\)κmax​\(C\)\.\\liminf\_\{s\\downarrow 0\}\\frac\{\\prod\_\{k\}\\operatorname\{Lip\}\(T\_\{k\}\)\}\{s\}\\,\\mathbb\{P\}\(\\hat\{Y\}\_\{s\}\\neq Y\)\\;\\geq\\;\\sqrt\{\\tfrac\{2\}\{\\pi\}\}\\,\\frac\{\\mathfrak\{J\}\_\{p,\\sigma\}\(\\pi;C\)\}\{\\kappa\_\{\\max\}\(C\)\}\.\(6\)The separated𝔍\\mathfrak\{J\}lower\-bounds the deterministic stiffness budget; the overlapping tube law converts the leftover noise into token error\. Equation \([6](https://arxiv.org/html/2606.30705#S3.E6)\) holds under three per\-system\-checkable preconditions onF=DW∘TF=D\_\{W\}\\circ T: \(i\)*transversality*\(D​TDTfull rank, no tangential degeneracy at the interface\); \(ii\)*alignment*\(the source interface is exhausted byT−1​\(Fi​j\)T^\{\-1\}\(F\_\{ij\}\)\); \(iii\)*regularity*\(the finite\-perimeter and density\-trace conditions of Appendix[E](https://arxiv.org/html/2606.30705#A5)\)\. On ELF the roll\-in results \(Section[4\.3](https://arxiv.org/html/2606.30705#S4.SS3)\) are consistent with these premises at the probed resolution; we apply \([6](https://arxiv.org/html/2606.30705#S3.E6)\) as a*conditional*product law whose only alignment failure mode weakens the bound by an explicit, measurable off\-facet energyEoffE\_\{\\mathrm\{off\}\}\(Appendix[E](https://arxiv.org/html/2606.30705#A5)\), not as a global certificate\. NoWW\-blind scalar functional reproduces both constants \(Proposition[18](https://arxiv.org/html/2606.30705#Thmtheorem18)\): the overlapping cost depends on the readoutWWwhile the separated cost depends on the codebook scale, so they are genuinely distinct contractions of the shared interface, not a single functional\.

## 4Empirical Evidence

All experiments use published checkpoints: the ELF\-B text autoencoder\(Huet al\.,[2026b](https://arxiv.org/html/2606.30705#bib.bib1)\)\(latent dimensiond=512d=512, sequence lengthS=128S=128\), the released image\-VAE decoders of Lumina\-Next\(Gao and others,[2024](https://arxiv.org/html/2606.30705#bib.bib26)\), SANA\-1\.5\(Xie and others,[2024](https://arxiv.org/html/2606.30705#bib.bib27)\), Z\-Image, and FLUX\.1 \(AutoencoderKL / AutoencoderDC variants, used at their default configurations\), and published AR models \(Llama\-2\(Touvronet al\.,[2023](https://arxiv.org/html/2606.30705#bib.bib28)\), Qwen\-2\.5\(Team,[2025](https://arxiv.org/html/2606.30705#bib.bib36)\), Qwen\-3\)\. Exact checkpoint identifiers and preprocessing are listed in the released code\. Decoder sensitivity usesn=5,000n=5\{,\}000token positions \(OpenWebText validation\); roll\-in usesn=512n=512paired latents; theγ\\gamma\-sweep covers6060cells with256256samples per cell\. All results are deterministic given the checkpoint and data split\.

### 4\.1Decoder sensitivity

We characterize the ELF\-B decoder’s response to structured versus random perturbations onn=5,000n=5\{,\}000token positions \(Figure[2](https://arxiv.org/html/2606.30705#S4.F2)\)\.

*Basin width and structured anisotropy\.*The ELF\-B encoder achieves96\.7%96\.7\\%clean token recovery \(η0=3\.3%\\eta\_\{0\}=3\.3\\%label noise; the offset is additive, Appendix[D](https://arxiv.org/html/2606.30705#A4)\), unchanged underσ=1\.0\\sigma=1\.0isotropic noise, consistent with a median normalized marginδ∗=5\.6\\delta^\{\*\}=5\.6\(Eq\.[1](https://arxiv.org/html/2606.30705#S2.E1); raw logit gap21\.521\.5\), since isotropic noise has a unit\-variance boundary\-normal component and rarely crosses a5\.6​σ5\.6\\sigmamargin\. The actualK=1K\{=\}1student residual at matched norm, however, flips70\.4%70\.4\\%of tokens \(vs3\.3%3\.3\\%random\), a21\.2×21\.2\\timesratio, despite a row\-space component of only2\.49%2\.49\\%; and the MSE gradient is orthogonal to the CE gradient \(cosine0\.0010\.001\)\.

*Continuous response and DABI\.*Sweeping the perturbation fractionf∈\[0,1\.5\]f\\in\[0,1\.5\]: atf=1\.0f=1\.0\(full student residual\), the structured cross\-entropy increase isΔ​CEstruct=6\.07\\Delta\\mathrm\{CE\}\_\{\\mathrm\{struct\}\}=6\.07versusΔ​CErand=0\.13\\Delta\\mathrm\{CE\}\_\{\\mathrm\{rand\}\}=0\.13, givingDABI=45\.7×\\mathrm\{DABI\}=45\.7\\times\. The structured response is superlinear with onset atf≈0\.8f\\approx 0\.8, while the random response remains sublinear throughout\.

*Decoder\-intervention probe\.*A natural fix is to blunt the readout\. We test the cheapest decoder intervention, test\-time logit smoothing \(Dσ​\(z\)=arg​maxy⁡𝔼ξ​\[logits​\(z\+σ​rms​\(z\)​ξ\)\]D\_\{\\sigma\}\(z\)=\\operatorname\*\{arg\\,max\}\_\{y\}\\mathbb\{E\}\_\{\\xi\}\[\\,\\text\{logits\}\(z\+\\sigma\\,\\mathrm\{rms\}\(z\)\\,\\xi\)\\,\], which averages the readout over an isotropic neighborhood\), sweepingσ∈\[0,0\.8\]\\sigma\\in\[0,0\.8\]\. It does not lower DABI:DABI\\mathrm\{DABI\}stays at4848–5858and the structured flip rate at0\.860\.86across the whole sweep, even as clean recovery begins to fall \(1\.00→0\.991\.00\\to 0\.99\)\. Isotropic smoothing averages over the≈97%\{\\approx\}97\\%null\-space component and barely touches the≈1%\{\\approx\}1\\%boundary\-normal subspace where the sensitivity lives, so it cannot reduce DABI\. The implication is causal: a fix must reshape the boundary geometry, not average isotropically\. We test this by retraining the ELF readout head directly, transport held byte\-identical \(Appendix[K](https://arxiv.org/html/2606.30705#A11)\): neither max\-margin retraining \(which*raises*DABI\\mathrm\{DABI\},453→1,469×453\\to 1\{,\}469\\times\) nor retraining the readout to decode the few\-step latents \(even while collapsing its margins,δ∗:5\.4→3\.0\\delta^\{\*\}\\\!:5\.4\\to 3\.0\) helps, the latter recovering only1\.21\.2of the4141flipped points atK=4K\{=\}4\(59\.3→60\.5%59\.3\\to 60\.5\\%held\-out accuracy\)\. No readout change recovers the flipped tokens: once the deterministic posterior mean averages over the branch ambiguity, the correct token is no longer a function of the latent\. Decoder sharpness amplifies a transport\-side failure; it is not an independently fixable cause\.

![Refer to caption](https://arxiv.org/html/2606.30705v1/x1.png)Figure 2:Decoder sensitivity\.ELF\-B \(d=512d=512,S=128S=128\)\.Left:Structured versus isotropic CE response;DABI=45\.7×\\mathrm\{DABI\}=45\.7\\timesatf=1f=1\.Right:Flip rate; student residual at matched norm flips21\.2×21\.2\\timesmore tokens\.
### 4\.2Image/text dichotomy

#### One probe, four text codecs\.

To compare codecs apples\-to\-apples, we apply the*same*margin\-normal probe \(Definition[1](https://arxiv.org/html/2606.30705#Thmtheorem1): perturb each clean latent toward its nearest decision boundary, reaching the boundary atf=1f=1, against a norm\-matched isotropic control\) to all four published continuous\-text decoders, with95%95\\%bootstrap CIs over positions \(Table[1](https://arxiv.org/html/2606.30705#S4.T1); visualized in Appendix Figure[7](https://arxiv.org/html/2606.30705#A11.F7)\)\. The boundary\-aligned perturbation flips4747–77%77\\%of tokens; the isotropic one flips≈0%\{\\approx\}0\\%\. On ELF the isotropic response is small but nonzero, giving a well\-conditionedDABI=508×\\mathrm\{DABI\}=508\\times\[446,580\]\[446,580\]\(32,76832\{,\}768positions\); on LangFlow\(Chenet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib4)\), CoLa\-DLM\(Guoet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib3)\), and Cosmos\(Meshchaninovet al\.,[2025](https://arxiv.org/html/2606.30705#bib.bib5)\)the isotropic response sits at the floor, soDABI\\mathrm\{DABI\}exceeds10410^\{4\}\. Built very differently \(frozen\-T5\(Raffelet al\.,[2020](https://arxiv.org/html/2606.30705#bib.bib35)\)encoding, embedding rounding, stochastic VAE, non\-autoregressive autoencoder\), the four codecs share the same sharp readout\.

#### Readout sharpness affects real generation\.

The margin\-normal probe measures worst\-case sharpness; for the two systems with a deterministic generator we can also use the*realized*terminal residual\. The realized column is defined only here: CoLa\-DLM \(SDE\+\{\+\}AR\) and Cosmos ship no few\-step deterministic generator, so neither has a posterior\-mean residual to probe\. On ELF it flips21×21\\timesmore tokens than a norm\-matched isotropic residual \(DABI=45\.7×\\mathrm\{DABI\}=45\.7\\timesin CE\); on LangFlow \(d=768d=768,99\.99%99\.99\\%clean decoding\) it flips≈2,600×\{\\approx\}2\{,\}600\\timesmore \(95%95\\%CI\[1,800,3,800\]\[1\{,\}800,3\{,\}800\]over24,96024\{,\}960positions;Δ​CEstruct=40\.5​\[39\.8,41\.1\]\\Delta\\mathrm\{CE\}\_\{\\mathrm\{struct\}\}=40\.5\\,\[39\.8,41\.1\]vs isotropic3\.9×10−43\.9\\times 10^\{\-4\}\)\. LangFlow’s deterministic few\-step generation is correspondingly degenerate \(Appendix[J](https://arxiv.org/html/2606.30705#A10)\)\. The realized residual is≈99%\{\\approx\}99\\%in the decoder null space, yet its small boundary\-aligned component flips tokens \(Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\)\.

#### Image decoders absorb the same perturbation\.

On four families of published image VAE decoders \(Lumina\-Next\(Gao and others,[2024](https://arxiv.org/html/2606.30705#bib.bib26)\), SANA\-1\.5\(Xie and others,[2024](https://arxiv.org/html/2606.30705#bib.bib27)\), Z\-Image, FLUX\.1;55NFE each, Appendix Figure[7](https://arxiv.org/html/2606.30705#A11.F7)\), the realized residual givesDABI∈\[0\.85,1\.94\]\\mathrm\{DABI\}\\in\[0\.85,1\.94\]: lacking a categorical readout, an image decoder has no boundary to align with and absorbs the same posterior\-mean blur that fails a text decoder\. The image/text gap is more than20×20\\timeswith no overlap\. The≈1\{\\approx\}1persists under the learned LPIPS perceptual metric \(ratios0\.70\.7–1\.21\.2, Appendix[K](https://arxiv.org/html/2606.30705#A11)\), so it is not a pixel\-L2L^\{2\}artifact\. The dichotomy tracks the readout, not the domain: a*categorical*\(vector\-quantized\) image readout recovers the text signature, a boundary\-aligned perturbation flipping the MoVQGAN code167×167\\timesmore than isotropic \(DABI=42×\\mathrm\{DABI\}=42\\times\) despite a44\-dimensional latent with almost no null space \(Appendix[K](https://arxiv.org/html/2606.30705#A11)\)\. The conclusion rests on*within*\-domain ratios \(image decoders near11, text codecs above10210^\{2\}, no overlap\), so it does not hinge on comparing cross\-entropy to pixel\-L2L^\{2\}across domains: eachDABI\\mathrm\{DABI\}uses the natural output\-space loss of its own domain \(Definition[1](https://arxiv.org/html/2606.30705#Thmtheorem1)\)\. AR and masked\-dLM readouts are uniformly sharp under the same probe \(4646–57%57\\%vs≈0%\{\\approx\}0\\%; seven models, Appendix[I](https://arxiv.org/html/2606.30705#A9)\)\.

Table 1:Decoder amplification across systems\(margin\-normal probe,95%95\\%bootstrap CIs over positions\)\. All four continuous\-text codecs flip4747–77%77\\%of tokens vs≈0%\{\\approx\}0\\%isotropic; ELF’s isotropic response is nonzero \(finiteDABI\\mathrm\{DABI\}\), the others are at the floor \(DABI\\mathrm\{DABI\}a lower bound\)\.*realized*column: realized terminal residual\. Published checkpoints\.SystemLatent \(dim\)flip @κ=1\\kappa\{=\}1DABI\\mathrm\{DABI\}\(mn\)realizedELF\(Huet al\.,[2026b](https://arxiv.org/html/2606.30705#bib.bib1)\)frozen\-T5 \(512512\)47%47\\%508×508\\times45\.7×45\.7\\timesLangFlow\(Chenet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib4)\)embed\-flow \(768768\)76%76\\%≫104\{\\gg\}10^\{4\}≈2,600×\{\\approx\}2\{,\}600\\timesCoLa\-DLM\(Guoet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib3)\)VAE\+\{\+\}AR \(1616\)77%77\\%≈4×105\{\\approx\}4\{\\times\}10^\{5\}—Cosmos\(Meshchaninovet al\.,[2025](https://arxiv.org/html/2606.30705#bib.bib5)\)non\-AR \(16×76816\{\\times\}768\)49%49\\%≈4×104\{\\approx\}4\{\\times\}10^\{4\}—Lumina/SANA/Z\-Image/FLUXimage VAE—n/a\[0\.85,1\.94\]\[0\.85,1\.94\]

### 4\.3Oracle roll\-in

We test Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)on the real ELF sampler \(logit\-normal grid,pmean=−1\.5p\_\{\\mathrm\{mean\}\}=\-1\.5,pstd=0\.8p\_\{\\mathrm\{std\}\}=0\.8; Figure[3](https://arxiv.org/html/2606.30705#S4.F3)\): for each grid timetit\_\{i\}we build the exact forward marginalUti=ti​X\+\(1−ti\)​s0​εU\_\{t\_\{i\}\}=t\_\{i\}X\+\(1\-t\_\{i\}\)s\_\{0\}\\varepsilonfrom paired clean data, roll the deterministic ODE suffix to11, and decode\. The terminal flip is mild and decreasing inKK\(41%,21%,12\.1%41\\%,21\\%,12\.1\\%forK=4,8,16K=4,8,16, as the grid places the final step at later times, lowers​\(t\)s\(t\)\), but rolling in from earlier, more ambiguous states the flip climbs to≈90%\{\\approx\}90\\%; allKKcollapse onto a single curve ins​\(t\)s\(t\), the per\-step driver predicted by the theorem\. At every point the residual flips1010–33×33\\timesmore tokens than a norm\-matched isotropic control \(the45\.7×45\.7\\timesDABI of Section[4\.1](https://arxiv.org/html/2606.30705#S4.SS1)\) with a constant≈1%\{\\approx\}1\\%boundary\-normal fraction: the mechanism is the composition of structured, readout\-aligned posterior\-mean errors, not a null\-space effect\.

![Refer to caption](https://arxiv.org/html/2606.30705v1/x2.png)Figure 3:Oracle roll\-in\.ELF sampler,n=512n=512,K∈\{4,8,16\}K\\in\\\{4,8,16\\\}\.Top:flip rate versuss​\(t\)s\(t\); allKKcollapse onto one curve; terminal flip1212–41%41\\%, cumulative≈90%\{\\approx\}90\\%\.Bottom:structured\-vs\-random ratio \(1010–33×33\\times\) and row\-space fraction \(≈1%\{\\approx\}1\\%\)\.#### Quantitative tube\-law test\.

The tube law predicts terminal flip from margin geometry,ℙ​\(Y^t≠Y\)≈𝔼​Φ​\(−δ∗/s​\(t\)\)\\mathbb\{P\}\(\\hat\{Y\}\_\{t\}\\neq Y\)\\approx\\mathbb\{E\}\\,\\Phi\(\-\\delta^\{\*\}/s\(t\)\)\(f^δ∗​\(0\+\)≈0\.064\\hat\{f\}\_\{\\delta^\{\*\}\}\(0^\{\+\}\)\\approx 0\.064on ELF\)\. Its*scaling*\(flip linear ins​\(t\)s\(t\), allKKon one curve; Figure[3](https://arxiv.org/html/2606.30705#S4.F3)\) holds on the learned ELF ODE; the*absolute constant*exceeds the isotropic prediction by a stable factor \(observed/predicted2\.5,2\.3,2\.22\.5,2\.3,2\.2forK=4,8,16K=4,8,16; Figure[6](https://arxiv.org/html/2606.30705#A10.F6)\)\. The idealized tube law assumes an isotropic residual; the learned residual is structured, so2/π\\sqrt\{2/\\pi\}is replaced by the realized active\-facet normal moment, an anisotropy factorcanisoc\_\{\\mathrm\{aniso\}\}\(isotropic theorem=1=1; Appendix[D](https://arxiv.org/html/2606.30705#A4)\)\. It is not a fitted intercept but the ratiocaniso=obs/\(2/π​s​AW\)≈2\.3c\_\{\\mathrm\{aniso\}\}=\\mathrm\{obs\}/\(\\sqrt\{2/\\pi\}\\,s\\,A\_\{W\}\)\\approx 2\.3of the observed flip to a prediction whose boundary densityAWA\_\{W\}is measured*independently*from the clean margin distribution; it is*distinct*from the CE amplificationDABI=45\.7×\\mathrm\{DABI\}\{=\}45\.7\\times\(flip saturates at the first crossing, cross\-entropy does not, soDABI≫caniso\\mathrm\{DABI\}\\gg c\_\{\\mathrm\{aniso\}\}\)\. The roll\-in confirms thes​\(t\)s\(t\)\-scaling and this constant\.

### 4\.4Deterministic ODE versus stochastic SDE

On the published ELF\-B teacher and its progressive\-distillation \(PD\) student\(Huet al\.,[2026a](https://arxiv.org/html/2606.30705#bib.bib29)\)\(same decoder and eval; only sampler randomness varies\), we sweepK∈\{1,…,32\}K\\in\\\{1,\\dots,32\\\}andγ∈\{0,…,2\}\\gamma\\in\\\{0,\\dots,2\\\}\(6060cells, Figure[9](https://arxiv.org/html/2606.30705#A11.F9)\)\. The teacher’s deterministic ODE fails where the SDE succeeds: atK=4K=4, ODE PPL294294vs SDE5050, the gap persisting atK=32K=32\(6868vs20\.520\.5\); itsK=1K=1ODE reaches PPL3\.073\.07only by mode collapse \(entropy2\.012\.01\)\. Distillation does not remove the penalty: the student’s ODE\-to\-SDE gap persists at everyK≥2K\\geq 2\(1515–38%38\\%; e\.g\.K=4K=4,47\.3→34\.847\.3\\to 34\.8\), and itsK=32K=32SDE PPL21\.421\.4reproduces the authors’ reported21\.3221\.32\. The SDE injects fresh noise per step, leaving the deterministic class of Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\(the lower bounds do not apply\); but it is not free, since teacher SDE atK=4K=4still produces multilingual fragments, so stochasticity needs enough steps\.

## 5Escape Mechanisms

### 5\.1Categorical commitment

Autoregressive and masked diffusion language models succeed at text generation with readouts far sharper than ELF’s\. TheirDABI\\mathrm\{DABI\}values are higher, not lower: Llama\-2\(Touvronet al\.,[2023](https://arxiv.org/html/2606.30705#bib.bib28)\)\(7B\) hasDABI=19,805×\\mathrm\{DABI\}=19\{,\}805\\times; Qwen\-2\.5\(Team,[2025](https://arxiv.org/html/2606.30705#bib.bib36)\)\(7B\) hasDABI=915×\\mathrm\{DABI\}=915\\times\(base\) and1,350×1\{,\}350\\times\(instruct\)\. The difference is categorical commitment: at each step, a hard discrete token is selected and all subsequent computation conditions on that selection\.

###### Lemma 4\(categorical escape\)\.

LetF:ℝN→𝒞F:\\mathbb\{R\}^\{N\}\\to\\mathcal\{C\}be the exact categorical generatorF​\(z\)=cVB​\(z\)F\(z\)=c\_\{V\_\{B\}\(z\)\}outputting at least two distinct atoms from𝒞\\mathcal\{C\}\. ThenFFis discontinuous on the connected domainℝN\\mathbb\{R\}^\{N\}and hence not Lipschitz\. The continuous lower bounds \(Theorems[5](https://arxiv.org/html/2606.30705#Thmtheorem5)–[6](https://arxiv.org/html/2606.30705#Thmtheorem6)\) do not apply\.

###### Proof\.

A continuous image of a connected set is connected; a finite set with≥2\\geq 2points is not\. ∎

#### Commit ablation\.

Removing hard selection at matched readout sharpness collapses generation across three published systems spanning both the autoregressive \(Llama\-2\-7B\) and masked\-diffusion \(LLaDA\-8B, Dream\-7B\) families: AR soft\-carry diversity dropsd2:0\.541→0\.135d\_\{2\}\\\!:0\.541\\to 0\.135and LLaDA soft\-refresh dropsd2:0\.82→0\.47d\_\{2\}\\\!:0\.82\\to 0\.47\(Table[4](https://arxiv.org/html/2606.30705#A10.T4)\)\. The cleanest control is a matched pair on Dream that holds the sampling distribution fixed and varies only commitment: hard\-sample \(commit*and*sample\) versus soft\-refresh\-sample \(the same sampling, with only the in\-loop commit removed\)\. Removing the commit step alone raises PPL from7\.37\.3to38\.638\.6\(5×5\\times\) and collapses effective length from9999to2222\. Because nothing but the commit step changes, this isolates commitment more tightly than the norm\-matched controls, which also collapse \(Appendix[J](https://arxiv.org/html/2606.30705#A10)\)\. Categorical commitment, not readout smoothness or sampling temperature, is the operative escape\.

### 5\.2Stochastic re\-injection

The SDE sampler injects fresh noise per step, leaving the deterministic trajectory class\. Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)does not apply\. Theγ\\gamma\-sweep \(Section[4\.4](https://arxiv.org/html/2606.30705#S4.SS4)\) confirms: SDE improves over ODE at every testedK≥2K\\geq 2, though the escape is not free \(teacher SDEK=4K\{=\}4still produces multilingual fragments at PPL=50\{\}=50\)\.

### 5\.3DABI×\\timesCCI taxonomy

The two diagnostics,DABI\\mathrm\{DABI\}andCCI\\mathrm\{CCI\}, define a22D taxonomy that classifies all tested systems \(Figure[8](https://arxiv.org/html/2606.30705#A11.F8)\)\. The two axes are orthogonal:DABI\\mathrm\{DABI\}measures readout sharpness \(a property of the decoder\), whileCCI\\mathrm\{CCI\}measures categorical commitment \(a property of the generator\)\.

*LowDABI\\mathrm\{DABI\},CCI=0\\mathrm\{CCI\}=0\(image VAEs\)\.*Deterministic few\-step generation succeeds because the smooth decoder tolerates the posterior\-mean blur\.

*HighDABI\\mathrm\{DABI\},CCI=0\\mathrm\{CCI\}=0\(ELF text flow\)\.*Deterministic few\-step generation fails: the sharp readout converts the small structured residual into token errors\. This is the failure regime\.

*HighDABI\\mathrm\{DABI\}, highCCI\\mathrm\{CCI\}\(AR, masked dLM\)\.*Generation succeeds despite the sharp readout because categorical commitment makes the generator discontinuous, evading the continuous lower bound\.

The escape arrows in Figure[8](https://arxiv.org/html/2606.30705#A11.F8), from high\-DABI\\mathrm\{DABI\}/CCI=0\{\}=0\(fails\) to high\-DABI\\mathrm\{DABI\}/high\-CCI\\mathrm\{CCI\}\(works\), are grounded in the commit ablations of Section[5\.1](https://arxiv.org/html/2606.30705#S5.SS1): fixingDABI\\mathrm\{DABI\}and varyingCCI\\mathrm\{CCI\}from11to0collapses generation\.

### 5\.4Recent continuous few\-step text models

The taxonomy predicts that any system generating fluent open\-ended text in few \(K≤16K\\leq 16\)*deterministic*steps with a smooth continuous latent must leave the continuous\-deterministic class\. Every 2025–2026 continuous\-latent text generator we surveyed \(public checkpoint or documented sampler; we found no counterexample\) is consistent \(Table[3](https://arxiv.org/html/2606.30705#A9.T3), Appendix[I](https://arxiv.org/html/2606.30705#A9)\): every deterministic\-ODE system needs≥32\{\\geq\}32steps\(Chenet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib4); Meshchaninovet al\.,[2025](https://arxiv.org/html/2606.30705#bib.bib5)\), and every few\-step system uses SDE sampling\(Huet al\.,[2026b](https://arxiv.org/html/2606.30705#bib.bib1); Guoet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib3)\)or categorical commitment\(Nguyen\-Conget al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib7); Lemercieret al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib8); Shenet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib6)\)\. The sharpest apparent counterexample, FMLM\(Leeet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib2)\)\(deterministic one\-step\), is categorical escape: a*one\-hot*simplex target witharg​max\\operatorname\*\{arg\\,max\}decoding \(Lemma[4](https://arxiv.org/html/2606.30705#Thmtheorem4)\), not smooth transport\. Training\-time robustness mechanisms do not move the classification: FastDiSS’s SCP/MANS perturbations\(Nguyen\-Conget al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib7)\)harden the network but keep a categorical readout, and Loopholing’s deterministic latent pathway\(Joet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib9)\)*complements*rather than replaces a stochastic one\-hot step \(Appendix[I](https://arxiv.org/html/2606.30705#A9)\)\.

## 6Separated\-Mode Transport Theory

In the idealized regime of well\-separated modes, we prove sharp Lipschitz transport laws \(full statements and proofs in Appendices[B](https://arxiv.org/html/2606.30705#A2)–[H](https://arxiv.org/html/2606.30705#A8); synthetic verification in Appendix[G](https://arxiv.org/html/2606.30705#A7)\)\. Here𝒫n,M=inf∑i<j‖ci−cj‖​Pγn​\(Bi,Bj\)\\mathcal\{P\}\_\{n,M\}=\\inf\\sum\_\{i<j\}\\left\\lVert c\_\{i\}\-c\_\{j\}\\right\\rVert\\,P\_\{\\gamma\_\{n\}\}\(B\_\{i\},B\_\{j\}\)is the minimal unit\-barrier interface profile of anMM\-atom equal\-mass codebook inℝn\\mathbb\{R\}^\{n\}\(the prescribed\-mass profile𝔍\\mathfrak\{J\}of \([3](https://arxiv.org/html/2606.30705#S2.E3)\) withκp​\(i,j\)≡‖ci−cj‖\\kappa\_\{p\}\(i,j\)\\equiv\\left\\lVert c\_\{i\}\-c\_\{j\}\\right\\rVert\), and quantifies the deterministic stiffnessΛ\\Lambdaa smooth map must pay to separateMMmodes\. The three theorems are one statement at three resolutions: stiffnessΛ≳𝔍/Wpp\\Lambda\\gtrsim\\mathfrak\{J\}/W\_\{p\}^\{p\}, growing aslog⁡M\\sqrt\{\\log M\}once the dimension isΩ​\(log⁡M\)\\Omega\(\\log M\), with a depth\-BBfactorization giving aB\\sqrt\{B\}\-smaller per\-step peak\. All three are instantiated on explicit codebooks in Appendix[G](https://arxiv.org/html/2606.30705#A7)\. Intuitively,𝔍\\mathfrak\{J\}is the minimal barrier\-weighted wall area a smooth map must build to separateMMmodes; a*finite\-perimeter*\(Caccioppoli\) partition is just basins with rectifiable, finite\-area walls\.

###### Theorem 5\(interfaceΓ\\Gamma\-limit; informal\)\.

limΛ→∞Λ​infLip⁡\(T\)≤ΛWpp​\(T​σ\#,μ\)=𝔍p,σ​\(π;𝒞\)\\lim\_\{\\Lambda\\to\\infty\}\\Lambda\\inf\_\{\\operatorname\{Lip\}\(T\)\\leq\\Lambda\}W\_\{p\}^\{p\}\(T\{\}\_\{\\\#\}\\sigma,\\mu\)=\\mathfrak\{J\}\_\{p,\\sigma\}\(\\pi;\\mathcal\{C\}\), a weighted Caccioppoli interface energy\(Baldo,[1990](https://arxiv.org/html/2606.30705#bib.bib20); Fonseca and Tartar,[1989](https://arxiv.org/html/2606.30705#bib.bib21)\)\.

###### Theorem 6\(dimension phase diagram; informal\)\.

𝒫n,M≍log⁡M\\mathcal\{P\}\_\{n,M\}\\asymp\\sqrt\{\\log M\}forn≥C​log⁡Mn\\geq C\\log M;Θ​\(M1/n\)\\Theta\(M^\{1/n\}\)for fixednn\. Combined:Λ≳Δp\+1​log⁡M/ηp\\Lambda\\gtrsim\\Delta^\{p\+1\}\\sqrt\{\\log M\}/\\eta^\{p\}\.

###### Theorem 7\(hierarchical separation; informal\)\.

A depth\-BBcategorical hierarchy reduces oneMM\-way problem toBBlocalmm\-way ones \(peak ratioB\\sqrt\{B\}\); the factorized generator is discontinuous \(Lemma[4](https://arxiv.org/html/2606.30705#Thmtheorem4)\), so continuous bounds do not apply\.

These results do*not*predict real ELF residuals, but the overlapping and separated regimes are coarea\-linked contractions of one composite interface \(Theorem[17](https://arxiv.org/html/2606.30705#Thmtheorem17), Appendix[E](https://arxiv.org/html/2606.30705#A5); schematic Figure[4](https://arxiv.org/html/2606.30705#A5.F4)\)\.

## 7Discussion and Limitations

We have identified a structural reason deterministic few\-step generation of continuous text latents produces incoherent output: the posterior\-mean terminal step cannot resolve discrete branch choices before a sharp categorical readout \(Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\), and theDABI×CCI\\mathrm\{DABI\}\{\\times\}\\mathrm\{CCI\}taxonomy predicts which systems fail and which escape\.

#### Limitations\.

Our formal results analyze deterministic posterior\-mean transport under smooth densities and exact clean decoding, which learned generators only approximate; the bounds constrain deterministic samplers, not the stochastic and categorical generators that escape them\. The first\-order law is robust to these idealizations \(Appendix[D](https://arxiv.org/html/2606.30705#A4)\): label noise, finite\-perimeter, or codimension\-≥2\{\\geq\}2irregularities change the constant but not the linear\-in\-ssscaling, and a learned denoiser is covered whenever its terminal active\-normal residual is stable andO​\(s\)O\(s\)\(Proposition[16](https://arxiv.org/html/2606.30705#Thmtheorem16)\)\. The open theory direction is a within\-coreWpW\_\{p\}theory for overlapping latents; practically, deterministic few\-step text needs SDE or categorical generation \(Appendix[K](https://arxiv.org/html/2606.30705#A11)\)\.

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- B\. Zheng, N\. Ma, S\. Tong, and S\. Xie \(2025\)Diffusion transformers with representation autoencoders\.External Links:2510\.11690,[Link](https://arxiv.org/abs/2510.11690)Cited by:[Appendix L](https://arxiv.org/html/2606.30705#A12.SS0.SSS0.Px6.p1.2)\.

## Appendix ANotation

We use a single symbol for the interface energy throughout:𝔍\\mathfrak\{J\}\(the prescribed\-mass weighted Caccioppoli interface energy of Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5)\)\.𝒫n,M\\mathcal\{P\}\_\{n,M\}is*not*a separate functional but the unit\-barrier*profile*of𝔍\\mathfrak\{J\}, i\.e\.𝔍\\mathfrak\{J\}withκp​\(i,j\)≡‖ci−cj‖\\kappa\_\{p\}\(i,j\)\\equiv\\\|c\_\{i\}\-c\_\{j\}\\\|for an equal\-massMM\-atom codebook inℝn\\mathbb\{R\}^\{n\}\. The symbolℐcrit\\mathcal\{I\}\_\{\\mathrm\{crit\}\}denotes the critical\-SNR time*interval*, not an energy\. Table[2](https://arxiv.org/html/2606.30705#A1.T2)collects the recurring symbols\.

Table 2:Recurring notation\.
## Appendix BProof of Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5): the hard\-gradient multiwell interface limit

We write this appendix in a self\-contained form\. Let

σ=ρ​d​xon​ℝn,ρ∈C1​\(ℝn;\(0,∞\)\),∫ℝnρ​𝑑x=1\.\\sigma=\\rho\\,dx\\qquad\\text\{on \}\\mathbb\{R\}^\{n\},\\qquad\\rho\\in C^\{1\}\(\\mathbb\{R\}^\{n\};\(0,\\infty\)\),\\qquad\\int\_\{\\mathbb\{R\}^\{n\}\}\\rho\\,dx=1\.LetC=\{c1,…,cM\}⊂ℝdC=\\\{c\_\{1\},\\ldots,c\_\{M\}\\\}\\subset\\mathbb\{R\}^\{d\}be a finite set of distinct Euclidean atoms, letp∈\[1,∞\)p\\in\[1,\\infty\), and let

V\(y\):=dist\(y,C\)p\.V\(y\):=\\operatorname\{dist\}\(y,C\)^\{p\}\.For an absolutely continuous curveξ:\[0,1\]→ℝd\\xi:\[0,1\]\\to\\mathbb\{R\}^\{d\}, define theVV\-length

ℓV​\(ξ\):=∫01V​\(ξ​\(s\)\)​\|ξ′​\(s\)\|​𝑑s,\\ell\_\{V\}\(\\xi\):=\\int\_\{0\}^\{1\}V\(\\xi\(s\)\)\\,\|\\xi^\{\\prime\}\(s\)\|\\,ds,and define the induced path pseudometric

dV\(y,z\):=inf\{ℓV\(ξ\):ξ\(0\)=y,ξ\(1\)=z,ξ∈AC\(\[0,1\];ℝd\)\}\.d\_\{V\}\(y,z\):=\\inf\\\{\\ell\_\{V\}\(\\xi\):\\xi\(0\)=y,\\ \\xi\(1\)=z,\\ \\xi\\in AC\(\[0,1\];\\mathbb\{R\}^\{d\}\)\\\}\.The well\-to\-well barrier is

κi​j=κp​\(i,j\):=dV​\(ci,cj\)\.\\kappa\_\{ij\}=\\kappa\_\{p\}\(i,j\):=d\_\{V\}\(c\_\{i\},c\_\{j\}\)\.For a Caccioppoli partitionB=\(B1,…,BM\)B=\(B\_\{1\},\\ldots,B\_\{M\}\)ofℝn\\mathbb\{R\}^\{n\}, define

Pσ​\(Bi,Bj\):=∫∂∗Bi∩∂∗Bjρ​𝑑ℋn−1,P\_\{\\sigma\}\(B\_\{i\},B\_\{j\}\):=\\int\_\{\\partial^\{\*\}B\_\{i\}\\cap\\partial^\{\*\}B\_\{j\}\}\\rho\\,d\\mathcal\{H\}^\{n\-1\},and

Ep,σ​\(B;C\):=∑i<jκi​j​Pσ​\(Bi,Bj\)\.E\_\{p,\\sigma\}\(B;C\):=\\sum\_\{i<j\}\\kappa\_\{ij\}P\_\{\\sigma\}\(B\_\{i\},B\_\{j\}\)\.For prescribed massesπi\>0\\pi\_\{i\}\>0,∑iπi=1\\sum\_\{i\}\\pi\_\{i\}=1, define

Jp,σ​\(π;C\):=infσ​\(Bi\)=πiEp,σ​\(B;C\),J\_\{p,\\sigma\}\(\\pi;C\):=\\inf\_\{\\sigma\(B\_\{i\}\)=\\pi\_\{i\}\}E\_\{p,\\sigma\}\(B;C\),where the infimum is over Caccioppoli partitions with those masses\. Finally,

ℱε​\(T\):=\{ε−1​∫ℝnV​\(T​\(x\)\)​𝑑σ​\(x\),Lip⁡\(T\)≤ε−1,\+∞,otherwise\.\\mathcal\{F\}\_\{\\varepsilon\}\(T\):=\\begin\{cases\}\\displaystyle\\varepsilon^\{\-1\}\\int\_\{\\mathbb\{R\}^\{n\}\}V\(T\(x\)\)\\,d\\sigma\(x\),&\\operatorname\{Lip\}\(T\)\\leq\\varepsilon^\{\-1\},\\\\\[5\.16663pt\] \+\\infty,&\\text\{otherwise\.\}\\end\{cases\}We use theL1​\(σ;ℝd\)L^\{1\}\(\\sigma;\\mathbb\{R\}^\{d\}\)topology\.

LetQiQ\_\{i\}denote the Euclidean Voronoi cell ofcic\_\{i\}, with a fixed lexicographic tie\-breaking rule, and let

q​\(y\)=i⟺y∈Qi\.q\(y\)=i\\quad\\Longleftrightarrow\\quad y\\in Q\_\{i\}\.For a partitionBB, put

uB:=∑i=1Mci​𝟏Bi\.u\_\{B\}:=\\sum\_\{i=1\}^\{M\}c\_\{i\}\\mathbf\{1\}\_\{B\_\{i\}\}\.
We use standard facts onB​VBV, reduced boundaries, traces ofB​VBVfunctions on Caccioppoli interfaces, and the structure theorem for finite partitions; see Ambrosio–Fusco–Pallara\(Ambrosioet al\.,[2000](https://arxiv.org/html/2606.30705#bib.bib10)\)\. For the density of regular polyhedral partitions with volume constraints we use the Braides–Conti–Garroni polyhedral density theorem\(Braideset al\.,[2017](https://arxiv.org/html/2606.30705#bib.bib11)\)\. The soft\-gradient analogue is the Baldo/Fonseca–Tartar multiwell Modica–Mortola theory\(Baldo,[1990](https://arxiv.org/html/2606.30705#bib.bib20); Fonseca and Tartar,[1989](https://arxiv.org/html/2606.30705#bib.bib21)\)\.

###### Theorem 8\(Expanded form of Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5)\)\.

Assumeρ∈C1​\(ℝn;\(0,∞\)\)\\rho\\in C^\{1\}\(\\mathbb\{R\}^\{n\};\(0,\\infty\)\),C=\{c1,…,cM\}⊂ℝdC=\\\{c\_\{1\},\\ldots,c\_\{M\}\\\}\\subset\\mathbb\{R\}^\{d\}is finite and consists of distinct points,p∈\[1,∞\)p\\in\[1,\\infty\), andπi\>0\\pi\_\{i\}\>0with∑iπi=1\\sum\_\{i\}\\pi\_\{i\}=1\. Then

ℱε→Γℱ0in​L1​\(σ;ℝd\),\\mathcal\{F\}\_\{\\varepsilon\}\\xrightarrow\{\\Gamma\}\\mathcal\{F\}\_\{0\}\\qquad\\text\{in \}L^\{1\}\(\\sigma;\\mathbb\{R\}^\{d\}\),where

ℱ0​\(u\)=\{Ep,σ​\(B;C\),u=uB=∑ici​𝟏Bi​for a Caccioppoli partition​B,\+∞,otherwise\.\\mathcal\{F\}\_\{0\}\(u\)=\\begin\{cases\}E\_\{p,\\sigma\}\(B;C\),&u=u\_\{B\}=\\sum\_\{i\}c\_\{i\}\\mathbf\{1\}\_\{B\_\{i\}\}\\text\{ for a Caccioppoli partition \}B,\\\\ \+\\infty,&\\text\{otherwise\.\}\\end\{cases\}Moreover, ifBBhas prescribed massesσ​\(Bi\)=πi\\sigma\(B\_\{i\}\)=\\pi\_\{i\}, then the recovery sequence may be chosen with exact nearest\-well phase masses

σ​\(Tε−1​\(Qi\)\)=πifor every​i\.\\sigma\\big\(T\_\{\\varepsilon\}^\{\-1\}\(Q\_\{i\}\)\\big\)=\\pi\_\{i\}\\qquad\\text\{for every \}i\.Consequently,

limε↓0infLip⁡\(T\)≤ε−1σ​\(T−1​\(Qi\)\)=πiε−1​∫ℝnV​\(T\)​𝑑σ=Jp,σ​\(π;C\)\.\\lim\_\{\\varepsilon\\downarrow 0\}\\inf\_\{\\begin\{subarray\}\{c\}\\operatorname\{Lip\}\(T\)\\leq\\varepsilon^\{\-1\}\\\\ \\sigma\(T^\{\-1\}\(Q\_\{i\}\)\)=\\pi\_\{i\}\\end\{subarray\}\}\\varepsilon^\{\-1\}\\int\_\{\\mathbb\{R\}^\{n\}\}V\(T\)\\,d\\sigma=J\_\{p,\\sigma\}\(\\pi;C\)\.Equivalently, forμ=∑iπi​δci\\mu=\\sum\_\{i\}\\pi\_\{i\}\\delta\_\{c\_\{i\}\},

limΛ→∞ΛinfLip⁡\(T\)≤ΛWpp\(T\#σ,μ\)=Jp,σ\(π;C\)\.\\boxed\{\\lim\_\{\\Lambda\\to\\infty\}\\Lambda\\inf\_\{\\operatorname\{Lip\}\(T\)\\leq\\Lambda\}W\_\{p\}^\{p\}\(T\_\{\\\#\}\\sigma,\\mu\)=J\_\{p,\\sigma\}\(\\pi;C\)\.\}

### B\.1Barrier calibration

###### Lemma 9\(Barrier metric and scalar calibration\)\.

The functiondVd\_\{V\}is a pseudometric onℝd\\mathbb\{R\}^\{d\}\. Fori≠ji\\neq j,

For

ϕi​\(y\):=dV​\(ci,y\)\\phi\_\{i\}\(y\):=d\_\{V\}\(c\_\{i\},y\)one has the pointwise local Lipschitz estimate

lip⁡ϕi​\(y\)≤V​\(y\)\.\\operatorname\{lip\}\\phi\_\{i\}\(y\)\\leq V\(y\)\.Let

Kκ:=maxa,b⁡κa​b,ϕ¯i​\(y\):=min⁡\{ϕi​\(y\),Kκ\}\.K\_\{\\kappa\}:=\\max\_\{a,b\}\\kappa\_\{ab\},\\qquad\\overline\{\\phi\}\_\{i\}\(y\):=\\min\\\{\\phi\_\{i\}\(y\),K\_\{\\kappa\}\\\}\.IfT∈Liploc⁡\(ℝn;ℝd\)T\\in\\operatorname\{Lip\}\_\{\\mathrm\{loc\}\}\(\\mathbb\{R\}^\{n\};\\mathbb\{R\}^\{d\}\), then

gi:=ϕ¯i∘T∈Wloc1,∞​\(ℝn\),g\_\{i\}:=\\overline\{\\phi\}\_\{i\}\\circ T\\in W^\{1,\\infty\}\_\{\\mathrm\{loc\}\}\(\\mathbb\{R\}^\{n\}\),and a\.e\.

\|∇gi\|≤V​\(T\)​‖D​T‖op\.\|\\nabla g\_\{i\}\|\\leq V\(T\)\\,\\\|DT\\\|\_\{\\mathrm\{op\}\}\.In particular, ifLip⁡\(T\)≤ε−1\\operatorname\{Lip\}\(T\)\\leq\\varepsilon^\{\-1\}, then

\|∇gi\|≤ε−1​V​\(T\)a\.e\.\|\\nabla g\_\{i\}\|\\leq\\varepsilon^\{\-1\}V\(T\)\\qquad\\text\{a\.e\.\}

###### Proof\.

Nonnegativity, symmetry, anddV​\(y,y\)=0d\_\{V\}\(y,y\)=0are immediate from the definition and reversal of curves\. The triangle inequality follows by concatenating curves\. ThusdVd\_\{V\}is a pseudometric\.

Let

ΔC:=mina≠b⁡\|ca−cb\|\>0\.\\Delta\_\{C\}:=\\min\_\{a\\neq b\}\|c\_\{a\}\-c\_\{b\}\|\>0\.Fixi≠ji\\neq jand letξ\\xibe any absolutely continuous curve fromcic\_\{i\}tocjc\_\{j\}\. Let

τ:=inf\{s:\|ξ​\(s\)−ci\|=ΔC/2\}\.\\tau:=\\inf\\\{s:\|\\xi\(s\)\-c\_\{i\}\|=\\Delta\_\{C\}/2\\\}\.By continuity,τ\\tauexists\. On the ballB​\(ci,ΔC/2\)B\(c\_\{i\},\\Delta\_\{C\}/2\), the closest codebook point iscic\_\{i\}, hence

V​\(ξ​\(s\)\)=\|ξ​\(s\)−ci\|pfor​0≤s≤τ\.V\(\\xi\(s\)\)=\|\\xi\(s\)\-c\_\{i\}\|^\{p\}\\qquad\\text\{for \}0\\leq s\\leq\\tau\.Setr​\(s\):=\|ξ​\(s\)−ci\|r\(s\):=\|\\xi\(s\)\-c\_\{i\}\|\. Since\|r′​\(s\)\|≤\|ξ′​\(s\)\|\|r^\{\\prime\}\(s\)\|\\leq\|\\xi^\{\\prime\}\(s\)\|a\.e\.,

ℓV​\(ξ\)\\displaystyle\\ell\_\{V\}\(\\xi\)≥∫0τr​\(s\)p​\|ξ′​\(s\)\|​𝑑s\\displaystyle\\geq\\int\_\{0\}^\{\\tau\}r\(s\)^\{p\}\|\\xi^\{\\prime\}\(s\)\|\\,ds≥∫0τr​\(s\)p​\|r′​\(s\)\|​𝑑s\\displaystyle\\geq\\int\_\{0\}^\{\\tau\}r\(s\)^\{p\}\|r^\{\\prime\}\(s\)\|\\,ds≥∫0ΔC/2rp​𝑑r=\(ΔC/2\)p\+1p\+1\.\\displaystyle\\geq\\int\_\{0\}^\{\\Delta\_\{C\}/2\}r^\{p\}\\,dr=\\frac\{\(\\Delta\_\{C\}/2\)^\{p\+1\}\}\{p\+1\}\.Taking the infimum overξ\\xigivesκi​j\>0\\kappa\_\{ij\}\>0\.

Next, by the triangle inequality fordVd\_\{V\},

\|ϕi​\(y\)−ϕi​\(z\)\|≤dV​\(y,z\)\.\|\\phi\_\{i\}\(y\)\-\\phi\_\{i\}\(z\)\|\\leq d\_\{V\}\(y,z\)\.Forzznearyy, the straight segmentξ​\(θ\)=y\+θ​\(z−y\)\\xi\(\\theta\)=y\+\\theta\(z\-y\)gives

dV​\(y,z\)≤\|z−y\|​∫01V​\(y\+θ​\(z−y\)\)​𝑑θ\.d\_\{V\}\(y,z\)\\leq\|z\-y\|\\int\_\{0\}^\{1\}V\(y\+\\theta\(z\-y\)\)\\,d\\theta\.Dividing by\|z−y\|\|z\-y\|and sendingz→yz\\to y, using continuity ofVV, yields

lip⁡ϕi​\(y\)≤V​\(y\)\.\\operatorname\{lip\}\\phi\_\{i\}\(y\)\\leq V\(y\)\.The truncationr↦min⁡\{r,Kκ\}r\\mapsto\\min\\\{r,K\_\{\\kappa\}\\\}is11\-Lipschitz, so the same pointwise bound holds forϕ¯i\\overline\{\\phi\}\_\{i\}\.

SinceTTis locally Lipschitz andϕ¯i\\overline\{\\phi\}\_\{i\}is locally Lipschitz,gi=ϕ¯i∘Tg\_\{i\}=\\overline\{\\phi\}\_\{i\}\\circ Tlies inWloc1,∞W^\{1,\\infty\}\_\{\\mathrm\{loc\}\}\. By Rademacher’s theorem and the metric chain rule,

\|∇gi​\(x\)\|≤lip⁡ϕ¯i​\(T​\(x\)\)​‖D​T​\(x\)‖op≤V​\(T​\(x\)\)​‖D​T​\(x\)‖op\|\\nabla g\_\{i\}\(x\)\|\\leq\\operatorname\{lip\}\\overline\{\\phi\}\_\{i\}\(T\(x\)\)\\,\\\|DT\(x\)\\\|\_\{\\mathrm\{op\}\}\\leq V\(T\(x\)\)\\,\\\|DT\(x\)\\\|\_\{\\mathrm\{op\}\}for a\.e\.xx\. This proves the lemma\. ∎

### B\.2Compactness

###### Proposition 10\(Compactness of bounded\-energy sequences\)\.

Letεk↓0\\varepsilon\_\{k\}\\downarrow 0and suppose

supkℱεk​\(Tk\)<∞\.\\sup\_\{k\}\\mathcal\{F\}\_\{\\varepsilon\_\{k\}\}\(T\_\{k\}\)<\\infty\.Then, after passing to a subsequence, there exists a Caccioppoli partition

B=\(B1,…,BM\)B=\(B\_\{1\},\\ldots,B\_\{M\}\)such that

Tk→uB:=∑ici​𝟏Biin​L1​\(σ;ℝd\)\.T\_\{k\}\\to u\_\{B\}:=\\sum\_\{i\}c\_\{i\}\\mathbf\{1\}\_\{B\_\{i\}\}\\qquad\\text\{in \}L^\{1\}\(\\sigma;\\mathbb\{R\}^\{d\}\)\.

###### Proof\.

The energy bound gives

∫ℝndist\(Tk,C\)pdσ≤Cεk\.\\int\_\{\\mathbb\{R\}^\{n\}\}\\operatorname\{dist\}\(T\_\{k\},C\)^\{p\}\\,d\\sigma\\leq C\\varepsilon\_\{k\}\.Hence, for everyr\>0r\>0,

σ​\{dist⁡\(Tk,C\)≥r\}≤C​r−p​εk\.\\sigma\\\{\\operatorname\{dist\}\(T\_\{k\},C\)\\geq r\\\}\\leq Cr^\{\-p\}\\varepsilon\_\{k\}\.In particular,

dist⁡\(Tk,C\)→0in​Lp​\(σ\)and in​L1​\(σ\)\.\\operatorname\{dist\}\(T\_\{k\},C\)\\to 0\\qquad\\text\{in \}L^\{p\}\(\\sigma\)\\quad\\text\{and in \}L^\{1\}\(\\sigma\)\.
Fori=1,…,Mi=1,\\ldots,M, define

gi,k:=ϕ¯i∘Tk\.g\_\{i,k\}:=\\overline\{\\phi\}\_\{i\}\\circ T\_\{k\}\.LetK⊂ℝnK\\subset\\mathbb\{R\}^\{n\}be compact\. Sinceρ\>0\\rho\>0and continuous,

ρK:=minK⁡ρ\>0\.\\rho\_\{K\}:=\\min\_\{K\}\\rho\>0\.By Lemma[9](https://arxiv.org/html/2606.30705#Thmtheorem9),

∫K\|∇gi,k\|​𝑑x\\displaystyle\\int\_\{K\}\|\\nabla g\_\{i,k\}\|\\,dx≤ρK−1​∫K\|∇gi,k\|​ρ​𝑑x\\displaystyle\\leq\\rho\_\{K\}^\{\-1\}\\int\_\{K\}\|\\nabla g\_\{i,k\}\|\\,\\rho\\,dx≤ρK−1​∫Kεk−1​V​\(Tk\)​ρ​𝑑x\\displaystyle\\leq\\rho\_\{K\}^\{\-1\}\\int\_\{K\}\\varepsilon\_\{k\}^\{\-1\}V\(T\_\{k\}\)\\,\\rho\\,dx≤CK\.\\displaystyle\\leq C\_\{K\}\.Also0≤gi,k≤Kκ0\\leq g\_\{i,k\}\\leq K\_\{\\kappa\}\. Thus\(gi,k\)k\(g\_\{i,k\}\)\_\{k\}is bounded inB​V​\(K\)BV\(K\)\. By BV compactness and a diagonal extraction overK=BR​\(0\)K=B\_\{R\}\(0\), there aregi∈B​Vloc​\(ℝn\)g\_\{i\}\\in BV\_\{\\mathrm\{loc\}\}\(\\mathbb\{R\}^\{n\}\)such that

gi,k→giin​Lloc1​\(ℝn\)\.g\_\{i,k\}\\to g\_\{i\}\\qquad\\text\{in \}L^\{1\}\_\{\\mathrm\{loc\}\}\(\\mathbb\{R\}^\{n\}\)\.Write

𝐠k:=\(g1,k,…,gM,k\),𝐠:=\(g1,…,gM\)\.\\mathbf\{g\}\_\{k\}:=\(g\_\{1,k\},\\ldots,g\_\{M,k\}\),\\qquad\\mathbf\{g\}:=\(g\_\{1\},\\ldots,g\_\{M\}\)\.
For each wellcjc\_\{j\}, define the label vector

aj:=\(κ1​j,…,κM​j\)∈ℝM\.a\_\{j\}:=\(\\kappa\_\{1j\},\\ldots,\\kappa\_\{Mj\}\)\\in\\mathbb\{R\}^\{M\}\.These label vectors are separated\. Indeed, ifi≠ji\\neq j, then theii\-th coordinate ofaia\_\{i\}is0, whereas theii\-th coordinate ofaja\_\{j\}isκi​j\>0\\kappa\_\{ij\}\>0\. Hence

mini≠j⁡\|ai−aj\|\>0\.\\min\_\{i\\neq j\}\|a\_\{i\}\-a\_\{j\}\|\>0\.
Letqk​\(x\):=q​\(Tk​\(x\)\)q\_\{k\}\(x\):=q\(T\_\{k\}\(x\)\)\. Sinceϕ¯i​\(cj\)=κi​j\\overline\{\\phi\}\_\{i\}\(c\_\{j\}\)=\\kappa\_\{ij\}, continuity of the finitely many functionsϕ¯i\\overline\{\\phi\}\_\{i\}near the finite setCCimplies that, for

ω​\(r\):=supdist⁡\(y,C\)≤r\|\(ϕ¯1​\(y\),…,ϕ¯M​\(y\)\)−aq​\(y\)\|,\\omega\(r\):=\\sup\_\{\\operatorname\{dist\}\(y,C\)\\leq r\}\\big\|\\,\(\\overline\{\\phi\}\_\{1\}\(y\),\\ldots,\\overline\{\\phi\}\_\{M\}\(y\)\)\-a\_\{q\(y\)\}\\,\\big\|,one hasω​\(r\)↓0\\omega\(r\)\\downarrow 0asr↓0r\\downarrow 0\. Therefore

σ​\{\|𝐠k−aqk\|\>α\}\\displaystyle\\sigma\\\{\|\\mathbf\{g\}\_\{k\}\-a\_\{q\_\{k\}\}\|\>\\alpha\\\}≤σ​\{dist⁡\(Tk,C\)\>r\}\\displaystyle\\leq\\sigma\\\{\\operatorname\{dist\}\(T\_\{k\},C\)\>r\\\}wheneverω​\(r\)<α\\omega\(r\)<\\alpha\. By \(A\.7\),

𝐠k−aqk→0in​σ​\-measure\.\\mathbf\{g\}\_\{k\}\-a\_\{q\_\{k\}\}\\to 0\\qquad\\text\{in \}\\sigma\\text\{\-measure\}\.Together with𝐠k→𝐠\\mathbf\{g\}\_\{k\}\\to\\mathbf\{g\}locally in measure, this implies

𝐠​\(x\)∈\{a1,…,aM\}for a\.e\.​x\.\\mathbf\{g\}\(x\)\\in\\\{a\_\{1\},\\ldots,a\_\{M\}\\\}\\qquad\\text\{for a\.e\. \}x\.Define the limiting phase mapq∞q\_\{\\infty\}by

𝐠​\(x\)=aq∞​\(x\)\\mathbf\{g\}\(x\)=a\_\{q\_\{\\infty\}\(x\)\}and set

Bi:=\{x:q∞​\(x\)=i\}\.B\_\{i\}:=\\\{x:q\_\{\\infty\}\(x\)=i\\\}\.ThenB=\(B1,…,BM\)B=\(B\_\{1\},\\ldots,B\_\{M\}\)is a measurable partition\.

It remains to show that it is a Caccioppoli partition\. For fixedii,

gi=∑j=1Mκi​j​𝟏Bj\.g\_\{i\}=\\sum\_\{j=1\}^\{M\}\\kappa\_\{ij\}\\mathbf\{1\}\_\{B\_\{j\}\}\.Let

κ∗:=mini≠j⁡κi​j\>0\.\\kappa\_\{\*\}:=\\min\_\{i\\neq j\}\\kappa\_\{ij\}\>0\.Choose a Lipschitz functionhi:ℝ→\[0,1\]h\_\{i\}:\\mathbb\{R\}\\to\[0,1\]such that

hi​\(0\)=1,hi​\(r\)=0for​r≥κ∗/2\.h\_\{i\}\(0\)=1,\\qquad h\_\{i\}\(r\)=0\\quad\\text\{for \}r\\geq\\kappa\_\{\*\}/2\.Sincegi=0g\_\{i\}=0onBiB\_\{i\}andgi≥κ∗g\_\{i\}\\geq\\kappa\_\{\*\}onℝn∖Bi\\mathbb\{R\}^\{n\}\\setminus B\_\{i\},

𝟏Bi=hi​\(gi\)\.\\mathbf\{1\}\_\{B\_\{i\}\}=h\_\{i\}\(g\_\{i\}\)\.The BV chain rule for Lipschitz scalar functions gives

𝟏Bi∈B​Vloc​\(ℝn\)\.\\mathbf\{1\}\_\{B\_\{i\}\}\\in BV\_\{\\mathrm\{loc\}\}\(\\mathbb\{R\}^\{n\}\)\.ThusBBis a Caccioppoli partition\.

Finally,

∫ℝn\|Tk−uB\|​𝑑σ\\displaystyle\\int\_\{\\mathbb\{R\}^\{n\}\}\|T\_\{k\}\-u\_\{B\}\|\\,d\\sigma≤∫ℝn\|Tk−cqk\|​𝑑σ\+∫ℝn\|cqk−cq∞\|​𝑑σ\.\\displaystyle\\leq\\int\_\{\\mathbb\{R\}^\{n\}\}\|T\_\{k\}\-c\_\{q\_\{k\}\}\|\\,d\\sigma\+\\int\_\{\\mathbb\{R\}^\{n\}\}\|c\_\{q\_\{k\}\}\-c\_\{q\_\{\\infty\}\}\|\\,d\\sigma\.The first term tends to zero by \(A\.8\)\. The second tends to zero becauseqk→q∞q\_\{k\}\\to q\_\{\\infty\}locally in measure, the codebook is finite, and theσ\\sigma\-tail can be made arbitrarily small\. HenceTk→uBT\_\{k\}\\to u\_\{B\}inL1​\(σ\)L^\{1\}\(\\sigma\)\. ∎

### B\.3Liminf inequality

###### Proposition 11\(Liminf inequality\)\.

Letεk↓0\\varepsilon\_\{k\}\\downarrow 0,Tk→uBT\_\{k\}\\to u\_\{B\}inL1​\(σ;ℝd\)L^\{1\}\(\\sigma;\\mathbb\{R\}^\{d\}\), and

supkℱεk​\(Tk\)<∞\.\\sup\_\{k\}\\mathcal\{F\}\_\{\\varepsilon\_\{k\}\}\(T\_\{k\}\)<\\infty\.Then

lim infk→∞ℱεk​\(Tk\)≥Ep,σ​\(B;C\)\.\\liminf\_\{k\\to\\infty\}\\mathcal\{F\}\_\{\\varepsilon\_\{k\}\}\(T\_\{k\}\)\\geq E\_\{p,\\sigma\}\(B;C\)\.

###### Proof\.

Passing to a subsequence, assume the liminf is a limit\. Define Radon measures

μk:=εk−1​V​\(Tk\)​ρ​d​x\.\\mu\_\{k\}:=\\varepsilon\_\{k\}^\{\-1\}V\(T\_\{k\}\)\\rho\\,dx\.The total massesμk​\(ℝn\)\\mu\_\{k\}\(\\mathbb\{R\}^\{n\}\)are bounded\. Passing to a further subsequence,

μk⇀∗μ\\mu\_\{k\}\\stackrel\{\{\\scriptstyle\*\}\}\{\{\\rightharpoonup\}\}\\mulocally as Radon measures\.

Let

gi,k:=ϕ¯i∘Tk\.g\_\{i,k\}:=\\overline\{\\phi\}\_\{i\}\\circ T\_\{k\}\.As in the compactness proof,

gi,k→gi:=∑j=1Mκi​j​𝟏Bjin​Lloc1\.g\_\{i,k\}\\to g\_\{i\}:=\\sum\_\{j=1\}^\{M\}\\kappa\_\{ij\}\\mathbf\{1\}\_\{B\_\{j\}\}\\qquad\\text\{in \}L^\{1\}\_\{\\mathrm\{loc\}\}\.By Lemma[9](https://arxiv.org/html/2606.30705#Thmtheorem9),

ρ​\|∇gi,k\|​d​x≤μkas measures\.\\rho\|\\nabla g\_\{i,k\}\|\\,dx\\leq\\mu\_\{k\}\\qquad\\text\{as measures\.\}By lower semicontinuity of total variation underLloc1L^\{1\}\_\{\\mathrm\{loc\}\}convergence, with continuous positive weightρ\\rho,

ρ​\|D​gi\|≤μas Radon measures\.\\rho\\,\|Dg\_\{i\}\|\\leq\\mu\\qquad\\text\{as Radon measures\.\}Indeed, for every nonnegativeψ∈Cc​\(ℝn\)\\psi\\in C\_\{c\}\(\\mathbb\{R\}^\{n\}\),

∫ψ​ρ​d​\|D​gi\|≤lim infk∫ψ​ρ​\|∇gi,k\|​𝑑x≤lim infk∫ψ​𝑑μk=∫ψ​𝑑μ\.\\int\\psi\\,\\rho\\,d\|Dg\_\{i\}\|\\leq\\liminf\_\{k\}\\int\\psi\\,\\rho\\,\|\\nabla g\_\{i,k\}\|\\,dx\\leq\\liminf\_\{k\}\\int\\psi\\,d\\mu\_\{k\}=\\int\\psi\\,d\\mu\.
Let

Σi​jB:=∂∗Bi∩∂∗Bj\.\\Sigma\_\{ij\}^\{B\}:=\\partial^\{\*\}B\_\{i\}\\cap\\partial^\{\*\}B\_\{j\}\.OnΣi​jB\\Sigma\_\{ij\}^\{B\}, the approximate traces ofgig\_\{i\}are

0from the​Bi​side,κi​jfrom the​Bj​side\.0\\quad\\text\{from the \}B\_\{i\}\\text\{ side\},\\qquad\\kappa\_\{ij\}\\quad\\text\{from the \}B\_\{j\}\\text\{ side\}\.Therefore the jump part ofD​giDg\_\{i\}gives

\|D​gi\|​⌞​Σi​jB≥κi​j​ℋn−1​⌞​Σi​jB\.\|Dg\_\{i\}\|\\llcorner\\Sigma\_\{ij\}^\{B\}\\geq\\kappa\_\{ij\}\\,\\mathcal\{H\}^\{n\-1\}\\llcorner\\Sigma\_\{ij\}^\{B\}\.Combining \(A\.15\) and \(A\.16\),

μ​⌞​Σi​jB≥κi​j​ρ​ℋn−1​⌞​Σi​jB\.\\mu\\llcorner\\Sigma\_\{ij\}^\{B\}\\geq\\kappa\_\{ij\}\\rho\\,\\mathcal\{H\}^\{n\-1\}\\llcorner\\Sigma\_\{ij\}^\{B\}\.For finite Caccioppoli partitions, pairwise reduced interfaces areℋn−1\\mathcal\{H\}^\{n\-1\}\-a\.e\. disjoint: the triple\-junction set has zero\(n−1\)\(n\-1\)\-dimensional measure\. Hence

lim infkℱεk​\(Tk\)\\displaystyle\\liminf\_\{k\}\\mathcal\{F\}\_\{\\varepsilon\_\{k\}\}\(T\_\{k\}\)=lim infkμk​\(ℝn\)\\displaystyle=\\liminf\_\{k\}\\mu\_\{k\}\(\\mathbb\{R\}^\{n\}\)≥μ​\(ℝn\)\\displaystyle\\geq\\mu\(\\mathbb\{R\}^\{n\}\)≥∑i<jκi​j​∫Σi​jBρ​𝑑ℋn−1\\displaystyle\\geq\\sum\_\{i<j\}\\kappa\_\{ij\}\\int\_\{\\Sigma\_\{ij\}^\{B\}\}\\rho\\,d\\mathcal\{H\}^\{n\-1\}=Ep,σ​\(B;C\)\.\\displaystyle=E\_\{p,\\sigma\}\(B;C\)\.∎

### B\.4One\-dimensional profiles

###### Lemma 12\(Near\-optimal one\-dimensional profiles\)\.

Let

K:=conv⁡C\.K:=\\operatorname\{conv\}C\.For everyi≠ji\\neq jand everyη\>0\\eta\>0, there exists a Lipschitz curve

ζi​j:ℝ→K\\zeta\_\{ij\}:\\mathbb\{R\}\\to Ksuch that

Lip⁡\(ζi​j\)≤1,ζi​j​\(s\)=ci​for​s≤−Li​j,ζi​j​\(s\)=cj​for​s≥Li​j,\\operatorname\{Lip\}\(\\zeta\_\{ij\}\)\\leq 1,\\qquad\\zeta\_\{ij\}\(s\)=c\_\{i\}\\text\{ for \}s\\leq\-L\_\{ij\},\\qquad\\zeta\_\{ij\}\(s\)=c\_\{j\}\\text\{ for \}s\\geq L\_\{ij\},for someLi​j<∞L\_\{ij\}<\\infty, and

∫ℝV​\(ζi​j​\(s\)\)​𝑑s≤κi​j\+η\.\\int\_\{\\mathbb\{R\}\}V\(\\zeta\_\{ij\}\(s\)\)\\,ds\\leq\\kappa\_\{ij\}\+\\eta\.

###### Proof\.

Choose an absolutely continuous curveξ:\[0,1\]→ℝd\\xi:\[0,1\]\\to\\mathbb\{R\}^\{d\}fromcic\_\{i\}tocjc\_\{j\}with

ℓV​\(ξ\)≤κi​j\+η\.\\ell\_\{V\}\(\\xi\)\\leq\\kappa\_\{ij\}\+\\eta\.LetΠK:ℝd→K\\Pi\_\{K\}:\\mathbb\{R\}^\{d\}\\to Kbe the Euclidean metric projection\. SinceKKis closed and convex,ΠK\\Pi\_\{K\}is11\-Lipschitz\. Since everyca∈Kc\_\{a\}\\in K, projection does not increase distance to anycac\_\{a\}:

\|ΠK​y−ca\|≤\|y−ca\|for every​a\.\|\\Pi\_\{K\}y\-c\_\{a\}\|\\leq\|y\-c\_\{a\}\|\\qquad\\text\{for every \}a\.Thus

V​\(ΠK​y\)≤V​\(y\)\.V\(\\Pi\_\{K\}y\)\\leq V\(y\)\.Replacingξ\\xibyΠK∘ξ\\Pi\_\{K\}\\circ\\xidoes not increase either Euclidean speed orVV\-length, and the projected curve remains insideKK\.

Reparameterize the projected curve by arclength\. This gives a11\-Lipschitz curveζi​j\\zeta\_\{ij\}on a finite interval with endpointsci,cjc\_\{i\},c\_\{j\}and

∫V​\(ζi​j​\(s\)\)​𝑑s=ℓV​\(ζi​j\)≤κi​j\+η\.\\int V\(\\zeta\_\{ij\}\(s\)\)\\,ds=\\ell\_\{V\}\(\\zeta\_\{ij\}\)\\leq\\kappa\_\{ij\}\+\\eta\.Extend it constantly bycic\_\{i\}to the left andcjc\_\{j\}to the right\. SinceV​\(ci\)=V​\(cj\)=0V\(c\_\{i\}\)=V\(c\_\{j\}\)=0, the integral is unchanged\. ∎

### B\.5Limsup for regular polyhedral partitions

We first prove recovery for a regular polyhedral partition\. A regular polyhedral partition means that each interface∂Bi∩∂Bj\\partial B\_\{i\}\\cap\\partial B\_\{j\}is, up toℋn−1\\mathcal\{H\}^\{n\-1\}\-null sets, a finite union of relatively open flat\(n−1\)\(n\-1\)\-polytopes, and the non\-manifold skeleton where three or more faces meet is contained in a finite union of\(n−2\)\(n\-2\)\-polytopes\. The casen=1n=1is understood with empty codimension\-two skeleton\.

###### Lemma 13\(Hard\-gradient recovery for regular partitions\)\.

LetB=\(B1,…,BM\)B=\(B\_\{1\},\\ldots,B\_\{M\}\)be a regular polyhedral Caccioppoli partition with finite weighted energy\. Then there exist maps

Gε:ℝn→K:=conv⁡CG\_\{\\varepsilon\}:\\mathbb\{R\}^\{n\}\\to K:=\\operatorname\{conv\}Csuch that

Lip⁡\(Gε\)≤1−εε,\\operatorname\{Lip\}\(G\_\{\\varepsilon\}\)\\leq\\frac\{1\-\\sqrt\{\\varepsilon\}\}\{\\varepsilon\},Gε→uBin​L1​\(σ;ℝd\),G\_\{\\varepsilon\}\\to u\_\{B\}\\qquad\\text\{in \}L^\{1\}\(\\sigma;\\mathbb\{R\}^\{d\}\),and

lim supε↓0ℱε​\(Gε\)≤Ep,σ​\(B;C\)\.\\limsup\_\{\\varepsilon\\downarrow 0\}\\mathcal\{F\}\_\{\\varepsilon\}\(G\_\{\\varepsilon\}\)\\leq E\_\{p,\\sigma\}\(B;C\)\.Moreover,

σ​\(Gε−1​\(Qi\)​△​Bi\)=O​\(ε\)for each​i\.\\sigma\\big\(G\_\{\\varepsilon\}^\{\-1\}\(Q\_\{i\}\)\\triangle B\_\{i\}\\big\)=O\(\\varepsilon\)\\qquad\\text\{for each \}i\.

###### Proof\.

Fixη\>0\\eta\>0\. For each faceF⊂∂Bi∩∂BjF\\subset\\partial B\_\{i\}\\cap\\partial B\_\{j\}, choose a profileζi​j\\zeta\_\{ij\}from Lemma[12](https://arxiv.org/html/2606.30705#Thmtheorem12)with

∫ℝV​\(ζi​j​\(s\)\)​𝑑s≤κi​j\+η\.\\int\_\{\\mathbb\{R\}\}V\(\\zeta\_\{ij\}\(s\)\)\\,ds\\leq\\kappa\_\{ij\}\+\\eta\.Orient the unit normalνF\\nu\_\{F\}so that it points fromBiB\_\{i\}intoBjB\_\{j\}, and letrFr\_\{F\}be the signed distance to the affine hyperplane containingFF, positive on theBjB\_\{j\}side\. Set

sε:=1−ε\.s\_\{\\varepsilon\}:=1\-\\sqrt\{\\varepsilon\}\.
LetSSbe the codimension\-two skeleton of the polyhedral partition\. ChooseR\>0R\>0large enough, depending only on the profiles and ondiam⁡K\\operatorname\{diam\}K, so that different face tubes outside

Hε:=NR​ε​\(S\)H\_\{\\varepsilon\}:=N\_\{R\\varepsilon\}\(S\)are separated at distance large enough to satisfy the Lipschitz estimates below\. SinceSSis contained in finitely many\(n−2\)\(n\-2\)\-polytopes,

\|Hε\|=O​\(ε2\)\|H\_\{\\varepsilon\}\|=O\(\\varepsilon^\{2\}\)on every bounded region, and the same estimate holds with weightρ\\rhobecauseρ\\rhois continuous\.

On the closed set outsideHεH\_\{\\varepsilon\}, define a preliminary mapG~ε\\widetilde\{G\}\_\{\\varepsilon\}as follows\. In the trimmed tube around a faceF⊂∂Bi∩∂BjF\\subset\\partial B\_\{i\}\\cap\\partial B\_\{j\}, write uniquely

x=y\+r​νF,y∈F,dist⁡\(y,∂F\)\>R​ε\.x=y\+r\\nu\_\{F\},\\qquad y\\in F,\\qquad\\operatorname\{dist\}\(y,\\partial F\)\>R\\varepsilon\.Set

G~ε​\(x\):=ζi​j​\(sεε​rF​\(x\)\)\.\\widetilde\{G\}\_\{\\varepsilon\}\(x\):=\\zeta\_\{ij\}\\\!\\left\(\\frac\{s\_\{\\varepsilon\}\}\{\\varepsilon\}r\_\{F\}\(x\)\\right\)\.Away from the union of these transition tubes, set

G~ε​\(x\)=cion the​Bi​side\.\\widetilde\{G\}\_\{\\varepsilon\}\(x\)=c\_\{i\}\\qquad\\text\{on the \}B\_\{i\}\\text\{ side\.\}The profile is constant outside a bounded interval, so these definitions agree on the overlaps between the constant regions and the ends of a face tube\.

There is no tangential derivative in \(A\.24\): the map depends only on the signed normal coordinaterFr\_\{F\}\. Since\|∇rF\|=1\|\\nabla r\_\{F\}\|=1andLip⁡\(ζi​j\)≤1\\operatorname\{Lip\}\(\\zeta\_\{ij\}\)\\leq 1,

‖D​G~ε‖op≤sεεinside each face tube\.\\\|D\\widetilde\{G\}\_\{\\varepsilon\}\\\|\_\{\\mathrm\{op\}\}\\leq\\frac\{s\_\{\\varepsilon\}\}\{\\varepsilon\}\\qquad\\text\{inside each face tube\.\}On constant regions the derivative is zero\. For two points lying in different face neighborhoods, the deletion ofHεH\_\{\\varepsilon\}gives a separation of orderR​εR\\varepsilon, while all values lie inKK\. TakingRRlarge enough gives the global estimate

\|G~ε​\(x\)−G~ε​\(y\)\|≤sεε​\|x−y\|\|\\widetilde\{G\}\_\{\\varepsilon\}\(x\)\-\\widetilde\{G\}\_\{\\varepsilon\}\(y\)\|\\leq\\frac\{s\_\{\\varepsilon\}\}\{\\varepsilon\}\|x\-y\|on the closed set whereG~ε\\widetilde\{G\}\_\{\\varepsilon\}is defined\.

By Kirszbraun’s theorem,G~ε\\widetilde\{G\}\_\{\\varepsilon\}extends to a map

G^ε:ℝn→ℝd\\widehat\{G\}\_\{\\varepsilon\}:\\mathbb\{R\}^\{n\}\\to\\mathbb\{R\}^\{d\}with the same Lipschitz constantsε/εs\_\{\\varepsilon\}/\\varepsilon\. Finally define

Gε:=ΠK∘G^ε,G\_\{\\varepsilon\}:=\\Pi\_\{K\}\\circ\\widehat\{G\}\_\{\\varepsilon\},whereΠK\\Pi\_\{K\}is the Euclidean projection ontoKK\. SinceΠK\\Pi\_\{K\}is11\-Lipschitz and leavesG~ε\\widetilde\{G\}\_\{\\varepsilon\}unchanged,GεG\_\{\\varepsilon\}still satisfies

Lip⁡\(Gε\)≤sεε=1−εε\.\\operatorname\{Lip\}\(G\_\{\\varepsilon\}\)\\leq\\frac\{s\_\{\\varepsilon\}\}\{\\varepsilon\}=\\frac\{1\-\\sqrt\{\\varepsilon\}\}\{\\varepsilon\}\.Moreover, becauseV​\(ΠK​y\)≤V​\(y\)V\(\\Pi\_\{K\}y\)\\leq V\(y\), projection does not increase the energy\.

The map equalsuBu\_\{B\}outside anO​\(ε\)O\(\\varepsilon\)\-neighborhood of the interfaces, except insideHεH\_\{\\varepsilon\}\. Since the profiles are bounded inKK, the transition region hasσ\\sigma\-measureO​\(ε\)O\(\\varepsilon\), and the skeleton hole hasσ\\sigma\-measureO​\(ε2\)O\(\\varepsilon^\{2\}\), we get

Gε→uBin​L1​\(σ\)\.G\_\{\\varepsilon\}\\to u\_\{B\}\\qquad\\text\{in \}L^\{1\}\(\\sigma\)\.The same support estimate gives

σ​\(Gε−1​\(Qi\)​△​Bi\)=O​\(ε\)\.\\sigma\(G\_\{\\varepsilon\}^\{\-1\}\(Q\_\{i\}\)\\triangle B\_\{i\}\)=O\(\\varepsilon\)\.
It remains to compute the energy\. The constant regions have zero cost\. The skeleton hole satisfies

ε−1​∫HεV​\(Gε\)​𝑑σ≤ε−1​‖V‖L∞​\(K\)​σ​\(Hε\)=O​\(ε\)→0\.\\varepsilon^\{\-1\}\\int\_\{H\_\{\\varepsilon\}\}V\(G\_\{\\varepsilon\}\)\\,d\\sigma\\leq\\varepsilon^\{\-1\}\\\|V\\\|\_\{L^\{\\infty\}\(K\)\}\\,\\sigma\(H\_\{\\varepsilon\}\)=O\(\\varepsilon\)\\to 0\.For a faceF⊂∂Bi∩∂BjF\\subset\\partial B\_\{i\}\\cap\\partial B\_\{j\}, the face\-tube contribution is

Iε,F\\displaystyle I\_\{\\varepsilon,F\}=ε−1​∫Fε∫ℝV​\(ζi​j​\(sεε​r\)\)​ρ​\(y\+r​νF\)​𝑑r​𝑑ℋn−1​\(y\),\\displaystyle=\\varepsilon^\{\-1\}\\int\_\{F\_\{\\varepsilon\}\}\\int\_\{\\mathbb\{R\}\}V\\\!\\left\(\\zeta\_\{ij\}\\\!\\left\(\\frac\{s\_\{\\varepsilon\}\}\{\\varepsilon\}r\\right\)\\right\)\\rho\(y\+r\\nu\_\{F\}\)\\,dr\\,d\\mathcal\{H\}^\{n\-1\}\(y\),whereFε:=\{y∈F:dist⁡\(y,∂F\)\>R​ε\}F\_\{\\varepsilon\}:=\\\{y\\in F:\\operatorname\{dist\}\(y,\\partial F\)\>R\\varepsilon\\\}\. With

τ=sεε​r,d​r=εsε​d​τ,\\tau=\\frac\{s\_\{\\varepsilon\}\}\{\\varepsilon\}r,\\qquad dr=\\frac\{\\varepsilon\}\{s\_\{\\varepsilon\}\}\\,d\\tau,we obtain

Iε,F\\displaystyle I\_\{\\varepsilon,F\}=1sε​∫Fε∫ℝV​\(ζi​j​\(τ\)\)​ρ​\(y\+εsε​τ​νF\)​𝑑τ​𝑑ℋn−1​\(y\)\.\\displaystyle=\\frac\{1\}\{s\_\{\\varepsilon\}\}\\int\_\{F\_\{\\varepsilon\}\}\\int\_\{\\mathbb\{R\}\}V\(\\zeta\_\{ij\}\(\\tau\)\)\\rho\\\!\\left\(y\+\\frac\{\\varepsilon\}\{s\_\{\\varepsilon\}\}\\tau\\nu\_\{F\}\\right\)d\\tau\\,d\\mathcal\{H\}^\{n\-1\}\(y\)\.The profile is constant outside a compactτ\\tau\-interval on whichV​\(ζi​j\)V\(\\zeta\_\{ij\}\)is bounded\. Sincesε→1s\_\{\\varepsilon\}\\to 1,Fε↑FF\_\{\\varepsilon\}\\uparrow F, andρ\\rhois continuous, dominated convergence gives

limε↓0Iε,F=\(∫ℝV​\(ζi​j​\(τ\)\)​𝑑τ\)​∫Fρ​𝑑ℋn−1\.\\lim\_\{\\varepsilon\\downarrow 0\}I\_\{\\varepsilon,F\}=\\left\(\\int\_\{\\mathbb\{R\}\}V\(\\zeta\_\{ij\}\(\\tau\)\)\\,d\\tau\\right\)\\int\_\{F\}\\rho\\,d\\mathcal\{H\}^\{n\-1\}\.Using \(A\.23\), summing over faces, and then sendingη↓0\\eta\\downarrow 0, we obtain

lim supε↓0ℱε​\(Gε\)≤∑i<jκi​j​∫∂∗Bi∩∂∗Bjρ​𝑑ℋn−1\.\\limsup\_\{\\varepsilon\\downarrow 0\}\\mathcal\{F\}\_\{\\varepsilon\}\(G\_\{\\varepsilon\}\)\\leq\\sum\_\{i<j\}\\kappa\_\{ij\}\\int\_\{\\partial^\{\*\}B\_\{i\}\\cap\\partial^\{\*\}B\_\{j\}\}\\rho\\,d\\mathcal\{H\}^\{n\-1\}\.This proves the lemma\. ∎

The preceding construction is the point at which the hard\-gradient theorem differs from the soft Baldo/Fonseca–Tartar theory\. In the soft Modica–Mortola functional, large gradients in a vanishingly small junction region are allowed as long as their integral cost vanishes\. Here the constraint

Lip⁡\(T\)≤ε−1\\operatorname\{Lip\}\(T\)\\leq\\varepsilon^\{\-1\}is pointwise\. Therefore one cannot simply paste several one\-dimensional profiles at a triple junction and appeal to an integral estimate\. The deletion of theO​\(ε\)O\(\\varepsilon\)\-tube around the codimension\-two skeleton, theO​\(ε2\)O\(\\varepsilon^\{2\}\)volume of that hole, Kirszbraun extension, and projection back toconv⁡C\\operatorname\{conv\}Care the hard\-gradient replacement for the soft junction argument\.

### B\.6Exact phase\-volume fixing

###### Lemma 14\(Exact mass correction by a near\-identity source diffeomorphism\)\.

LetBBbe a regular polyhedral partition with

σ​\(Bi\)=πi\>0\.\\sigma\(B\_\{i\}\)=\\pi\_\{i\}\>0\.LetGεG\_\{\\varepsilon\}be the recovery map constructed in Lemma[13](https://arxiv.org/html/2606.30705#Thmtheorem13), and set

Aiε:=Gε−1​\(Qi\)\.A\_\{i\}^\{\\varepsilon\}:=G\_\{\\varepsilon\}^\{\-1\}\(Q\_\{i\}\)\.Then, for all sufficiently smallε\\varepsilon, there exists aC1C^\{1\}\-diffeomorphism

Φε:ℝn→ℝn\\Phi\_\{\\varepsilon\}:\\mathbb\{R\}^\{n\}\\to\\mathbb\{R\}^\{n\}such that

‖Φε−Id‖C1=O​\(ε\),Lip⁡\(Φε−1\)≤1\+C​ε,\\\|\\Phi\_\{\\varepsilon\}\-\\operatorname\{Id\}\\\|\_\{C^\{1\}\}=O\(\\varepsilon\),\\qquad\\operatorname\{Lip\}\(\\Phi\_\{\\varepsilon\}^\{\-1\}\)\\leq 1\+C\\varepsilon,and

σ​\(Φε​\(Aiε\)\)=πifor every​i\.\\sigma\(\\Phi\_\{\\varepsilon\}\(A\_\{i\}^\{\\varepsilon\}\)\)=\\pi\_\{i\}\\qquad\\text\{for every \}i\.Consequently

Tε:=Gε∘Φε−1T\_\{\\varepsilon\}:=G\_\{\\varepsilon\}\\circ\\Phi\_\{\\varepsilon\}^\{\-1\}satisfies

σ​\(Tε−1​\(Qi\)\)=πifor every​i,\\sigma\(T\_\{\\varepsilon\}^\{\-1\}\(Q\_\{i\}\)\)=\\pi\_\{i\}\\qquad\\text\{for every \}i,Lip⁡\(Tε\)≤ε−1,\\operatorname\{Lip\}\(T\_\{\\varepsilon\}\)\\leq\\varepsilon^\{\-1\},and

ℱε​\(Tε\)=ℱε​\(Gε\)\+O​\(ε\)\.\\mathcal\{F\}\_\{\\varepsilon\}\(T\_\{\\varepsilon\}\)=\\mathcal\{F\}\_\{\\varepsilon\}\(G\_\{\\varepsilon\}\)\+O\(\\varepsilon\)\.

###### Proof\.

By \(A\.22\),

σ​\(Aiε\)=πi\+O​\(ε\)\.\\sigma\(A\_\{i\}^\{\\varepsilon\}\)=\\pi\_\{i\}\+O\(\\varepsilon\)\.BecauseBBis a regular polyhedral partition with allπi\>0\\pi\_\{i\}\>0, its positive\-interface adjacency graph is connected after the standard polyhedral approximation step\. Choose a spanning tree𝒯\\mathcal\{T\}of this graph\. For every oriented edgee=\(i,j\)∈𝒯e=\(i,j\)\\in\\mathcal\{T\}, choose a small ballUeU\_\{e\}meeting a single flat faceFi​j⊂∂Bi∩∂BjF\_\{ij\}\\subset\\partial B\_\{i\}\\cap\\partial B\_\{j\}, away from the skeleton, and choose these balls pairwise disjoint\. Letνi​j\\nu\_\{ij\}be the normal pointing fromBiB\_\{i\}intoBjB\_\{j\}\. Choose a vector field

Xe∈Cc∞​\(Ue;ℝn\)X\_\{e\}\\in C\_\{c\}^\{\\infty\}\(U\_\{e\};\\mathbb\{R\}^\{n\}\)such that

βe:=∫Fi​jρ​Xe⋅νi​j​𝑑ℋn−1\>0\.\\beta\_\{e\}:=\\int\_\{F\_\{ij\}\}\\rho\\,X\_\{e\}\\cdot\\nu\_\{ij\}\\,d\\mathcal\{H\}^\{n\-1\}\>0\.Fora=\(ae\)e∈𝒯a=\(a\_\{e\}\)\_\{e\\in\\mathcal\{T\}\}small, letΦa\\Phi\_\{a\}be the time\-one flow of

Xa:=∑e∈𝒯ae​Xe\.X\_\{a\}:=\\sum\_\{e\\in\\mathcal\{T\}\}a\_\{e\}X\_\{e\}\.Then

‖Φa−Id‖C1≤C​\|a\|,Lip⁡\(Φa−1\)≤1\+C​\|a\|\.\\\|\\Phi\_\{a\}\-\\operatorname\{Id\}\\\|\_\{C^\{1\}\}\\leq C\|a\|,\\qquad\\operatorname\{Lip\}\(\\Phi\_\{a\}^\{\-1\}\)\\leq 1\+C\|a\|\.
Define the phase\-mass map

mε​\(a\):=\(σ​\(Φa​\(A1ε\)\),…,σ​\(Φa​\(AMε\)\)\)\.m^\{\\varepsilon\}\(a\):=\\big\(\\sigma\(\\Phi\_\{a\}\(A\_\{1\}^\{\\varepsilon\}\)\),\\ldots,\\sigma\(\\Phi\_\{a\}\(A\_\{M\}^\{\\varepsilon\}\)\)\\big\)\.The derivative ata=0a=0converges, asε↓0\\varepsilon\\downarrow 0, to the weighted tree\-incidence map

L:ℝM−1→\{b∈ℝM:∑ibi=0\}L:\\mathbb\{R\}^\{M\-1\}\\to\\Big\\\{b\\in\\mathbb\{R\}^\{M\}:\\sum\_\{i\}b\_\{i\}=0\\Big\\\}given by

\(L​e\)i=\{−βe,e=\(i,j\),\+βe,e=\(j,i\),0,otherwise\(Le\)\_\{i\}=\\begin\{cases\}\-\\beta\_\{e\},&e=\(i,j\),\\\\ \+\\beta\_\{e\},&e=\(j,i\),\\\\ 0,&\\text\{otherwise\}\\end\{cases\}on each oriented tree edgeee\. This is the usual incidence matrix of a tree, with positive edge weightsβe\\beta\_\{e\}\. It has rankM−1M\-1, and hence is invertible onto the codimension\-one mass hyperplane\.

Since

mε​\(0\)−π=O​\(ε\)andD​mε​\(0\)→L,m^\{\\varepsilon\}\(0\)\-\\pi=O\(\\varepsilon\)\\quad\\text\{and\}\\quad Dm^\{\\varepsilon\}\(0\)\\to L,the inverse function theorem, uniformly for smallε\\varepsilon, givesaε=O​\(ε\)a\_\{\\varepsilon\}=O\(\\varepsilon\)such that

mε​\(aε\)=π\.m^\{\\varepsilon\}\(a\_\{\\varepsilon\}\)=\\pi\.Set

Φε:=Φaε\.\\Phi\_\{\\varepsilon\}:=\\Phi\_\{a\_\{\\varepsilon\}\}\.This proves \(A\.29\) and \(A\.30\)\. Since

Tε−1​\(Qi\)=Φε​\(Aiε\),T\_\{\\varepsilon\}^\{\-1\}\(Q\_\{i\}\)=\\Phi\_\{\\varepsilon\}\(A\_\{i\}^\{\\varepsilon\}\),\(A\.31\) follows\.

The Lipschitz estimate follows from Lemma[13](https://arxiv.org/html/2606.30705#Thmtheorem13)and \(A\.29\):

Lip⁡\(Tε\)≤Lip⁡\(Gε\)​Lip⁡\(Φε−1\)≤1−εε​\(1\+C​ε\)≤ε−1\\operatorname\{Lip\}\(T\_\{\\varepsilon\}\)\\leq\\operatorname\{Lip\}\(G\_\{\\varepsilon\}\)\\operatorname\{Lip\}\(\\Phi\_\{\\varepsilon\}^\{\-1\}\)\\leq\\frac\{1\-\\sqrt\{\\varepsilon\}\}\{\\varepsilon\}\(1\+C\\varepsilon\)\\leq\\varepsilon^\{\-1\}for all sufficiently smallε\\varepsilon\.

Finally,

∫V​\(Tε\)​𝑑σ\\displaystyle\\int V\(T\_\{\\varepsilon\}\)\\,d\\sigma=∫V​\(Gε​\(y\)\)​ρ​\(Φε​\(y\)\)​\|detD​Φε​\(y\)\|​𝑑y\.\\displaystyle=\\int V\(G\_\{\\varepsilon\}\(y\)\)\\rho\(\\Phi\_\{\\varepsilon\}\(y\)\)\|\\det D\\Phi\_\{\\varepsilon\}\(y\)\|\\,dy\.SinceΦε=Id\+O​\(ε\)\\Phi\_\{\\varepsilon\}=\\operatorname\{Id\}\+O\(\\varepsilon\)inC1C^\{1\},ρ∈C1\\rho\\in C^\{1\}, andGεG\_\{\\varepsilon\}has unscaled costO​\(ε\)O\(\\varepsilon\),

∫V​\(Tε\)​𝑑σ=∫V​\(Gε\)​𝑑σ\+O​\(ε2\)\.\\int V\(T\_\{\\varepsilon\}\)\\,d\\sigma=\\int V\(G\_\{\\varepsilon\}\)\\,d\\sigma\+O\(\\varepsilon^\{2\}\)\.After multiplying byε−1\\varepsilon^\{\-1\}, this gives \(A\.33\)\. ∎

### B\.7General partitions and the limsup inequality

###### Proof of the limsup inequality\.

LetBBbe a Caccioppoli partition with finite energy\. By the polyhedral density theorem for partitions\(Braideset al\.,[2017](https://arxiv.org/html/2606.30705#bib.bib11)\), combined with a standard truncation argument and the continuity of the weightρ\\rho, there exist regular polyhedral partitionsBmB^\{m\}such that

∑iσ​\(Bim​△​Bi\)→0\\sum\_\{i\}\\sigma\(B\_\{i\}^\{m\}\\triangle B\_\{i\}\)\\to 0and

Ep,σ​\(Bm;C\)→Ep,σ​\(B;C\)\.E\_\{p,\\sigma\}\(B^\{m\};C\)\\to E\_\{p,\\sigma\}\(B;C\)\.If the massesσ​\(Bi\)\\sigma\(B\_\{i\}\)are prescribed and positive, the approximation may be taken with

σ​\(Bim\)=σ​\(Bi\)\\sigma\(B\_\{i\}^\{m\}\)=\\sigma\(B\_\{i\}\)for everyii\. The volume correction is obtained by the same finite\-dimensional flow argument used in Lemma[14](https://arxiv.org/html/2606.30705#Thmtheorem14): choose a spanning tree in the adjacency graph, choose disjoint vector fields crossing one face per tree edge, and apply the inverse function theorem to the weighted phase\-volume map\. The correction isC1C^\{1\}\-small and changes the weighted perimeter byo​\(1\)o\(1\)\.

For eachmm, Lemma[13](https://arxiv.org/html/2606.30705#Thmtheorem13)gives a recovery sequence forBmB^\{m\}\. If exact masses are required, Lemma[14](https://arxiv.org/html/2606.30705#Thmtheorem14)gives a corrected sequence with exact nearest\-well phase masses\. A diagonal choicem=m​\(ε\)→∞m=m\(\\varepsilon\)\\to\\inftyyields

Tε→uBin​L1​\(σ\),T\_\{\\varepsilon\}\\to u\_\{B\}\\qquad\\text\{in \}L^\{1\}\(\\sigma\),and

lim supε↓0ℱε​\(Tε\)≤Ep,σ​\(B;C\)\.\\limsup\_\{\\varepsilon\\downarrow 0\}\\mathcal\{F\}\_\{\\varepsilon\}\(T\_\{\\varepsilon\}\)\\leq E\_\{p,\\sigma\}\(B;C\)\.Together with Proposition[11](https://arxiv.org/html/2606.30705#Thmtheorem11), this proves theΓ\\Gamma\-convergence\. ∎

### B\.8The constrained energy minimum

###### Proof of \(A\.1\)\.

For the lower bound, letTεT\_\{\\varepsilon\}be any sequence with

Lip⁡\(Tε\)≤ε−1,σ​\(Tε−1​\(Qi\)\)=πi,\\operatorname\{Lip\}\(T\_\{\\varepsilon\}\)\\leq\\varepsilon^\{\-1\},\\qquad\\sigma\(T\_\{\\varepsilon\}^\{\-1\}\(Q\_\{i\}\)\)=\\pi\_\{i\},and bounded energy\. By compactness,

Tε→uBin​L1​\(σ\)T\_\{\\varepsilon\}\\to u\_\{B\}\\qquad\\text\{in \}L^\{1\}\(\\sigma\)for a Caccioppoli partitionBB\. The exact phase constraints pass to the limit, so

σ​\(Bi\)=πi\.\\sigma\(B\_\{i\}\)=\\pi\_\{i\}\.The liminf inequality gives

lim infε↓0ℱε​\(Tε\)≥Ep,σ​\(B;C\)≥Jp,σ​\(π;C\)\.\\liminf\_\{\\varepsilon\\downarrow 0\}\\mathcal\{F\}\_\{\\varepsilon\}\(T\_\{\\varepsilon\}\)\\geq E\_\{p,\\sigma\}\(B;C\)\\geq J\_\{p,\\sigma\}\(\\pi;C\)\.
For the upper bound, choose a Caccioppoli partitionBBwith

σ​\(Bi\)=πi\\sigma\(B\_\{i\}\)=\\pi\_\{i\}and

Ep,σ​\(B;C\)≤Jp,σ​\(π;C\)\+η\.E\_\{p,\\sigma\}\(B;C\)\\leq J\_\{p,\\sigma\}\(\\pi;C\)\+\\eta\.The exact\-mass recovery sequence constructed above satisfies

σ​\(Tε−1​\(Qi\)\)=πi,Lip⁡\(Tε\)≤ε−1,\\sigma\(T\_\{\\varepsilon\}^\{\-1\}\(Q\_\{i\}\)\)=\\pi\_\{i\},\\qquad\\operatorname\{Lip\}\(T\_\{\\varepsilon\}\)\\leq\\varepsilon^\{\-1\},and

lim supε↓0ℱε​\(Tε\)≤Ep,σ​\(B;C\)≤Jp,σ​\(π;C\)\+η\.\\limsup\_\{\\varepsilon\\downarrow 0\}\\mathcal\{F\}\_\{\\varepsilon\}\(T\_\{\\varepsilon\}\)\\leq E\_\{p,\\sigma\}\(B;C\)\\leq J\_\{p,\\sigma\}\(\\pi;C\)\+\\eta\.Lettingη↓0\\eta\\downarrow 0proves \(A\.1\)\. ∎

### B\.9TheWpW\_\{p\}corollary

Let

μ=∑i=1Mπi​δci\.\\mu=\\sum\_\{i=1\}^\{M\}\\pi\_\{i\}\\delta\_\{c\_\{i\}\}\.We first record the exact identity used for the upper bound\. If

σ​\(T−1​\(Qi\)\)=πifor every​i,\\sigma\(T^\{\-1\}\(Q\_\{i\}\)\)=\\pi\_\{i\}\\qquad\\text\{for every \}i,then

Wpp\(T\#σ,μ\)=∫ℝndist\(T\(x\),C\)pdσ\(x\)\.W\_\{p\}^\{p\}\(T\_\{\\\#\}\\sigma,\\mu\)=\\int\_\{\\mathbb\{R\}^\{n\}\}\\operatorname\{dist\}\(T\(x\),C\)^\{p\}\\,d\\sigma\(x\)\.Indeed, for any coupling betweenY∼T\#​σY\\sim T\_\{\\\#\}\\sigmaandZ∈CZ\\in Cwith lawμ\\mu,

\|Y−Z\|p≥dist\(Y,C\)p,\|Y\-Z\|^\{p\}\\geq\\operatorname\{dist\}\(Y,C\)^\{p\},so

Wpp\(T\#σ,μ\)≥∫dist\(T,C\)pdσ\.W\_\{p\}^\{p\}\(T\_\{\\\#\}\\sigma,\\mu\)\\geq\\int\\operatorname\{dist\}\(T,C\)^\{p\}\\,d\\sigma\.Conversely, because the nearest\-well phase masses matchπi\\pi\_\{i\}, the coupling

x↦\(T​\(x\),cq​\(T​\(x\)\)\)x\\mapsto\\big\(T\(x\),c\_\{q\(T\(x\)\)\}\\big\)has second marginalμ\\muand cost exactly

∫\|T\(x\)−cq​\(T​\(x\)\)\|pdσ\(x\)=∫dist\(T\(x\),C\)pdσ\(x\)\.\\int\|T\(x\)\-c\_\{q\(T\(x\)\)\}\|^\{p\}\\,d\\sigma\(x\)=\\int\\operatorname\{dist\}\(T\(x\),C\)^\{p\}\\,d\\sigma\(x\)\.This proves \(A\.37\)\.

###### Proof of \(A\.2\)\.

Setε=Λ−1\\varepsilon=\\Lambda^\{\-1\}\.

The upper bound follows from the exact\-mass recovery sequence\. For such a sequence,

Λ​Wpp​\(Tε​σ\#,μ\)\\displaystyle\\Lambda W\_\{p\}^\{p\}\(T\_\{\\varepsilon\}\{\}\_\{\\\#\}\\sigma,\\mu\)=ε−1∫dist\(Tε,C\)pdσ\\displaystyle=\\varepsilon^\{\-1\}\\int\\operatorname\{dist\}\(T\_\{\\varepsilon\},C\)^\{p\}\\,d\\sigma=ℱε​\(Tε\)→Jp,σ​\(π;C\)\\displaystyle=\\mathcal\{F\}\_\{\\varepsilon\}\(T\_\{\\varepsilon\}\)\\to J\_\{p,\\sigma\}\(\\pi;C\)after optimizing over partitions\.

For the lower bound, letΛk→∞\\Lambda\_\{k\}\\to\\infty, setεk=Λk−1\\varepsilon\_\{k\}=\\Lambda\_\{k\}^\{\-1\}, and chooseTkT\_\{k\}with

Lip⁡\(Tk\)≤Λk\\operatorname\{Lip\}\(T\_\{k\}\)\\leq\\Lambda\_\{k\}such that

supkΛk​Wpp​\(Tk​σ\#,μ\)<∞\.\\sup\_\{k\}\\Lambda\_\{k\}W\_\{p\}^\{p\}\(T\_\{k\}\{\}\_\{\\\#\}\\sigma,\\mu\)<\\infty\.LetYk=Tk​\(X\)Y\_\{k\}=T\_\{k\}\(X\)withX∼σX\\sim\\sigma, and letZk∈CZ\_\{k\}\\in Cbe coupled toYkY\_\{k\}optimally, withZk∼μZ\_\{k\}\\sim\\mu\. Since

\|Yk−Zk\|p≥dist\(Yk,C\)p,\|Y\_\{k\}\-Z\_\{k\}\|^\{p\}\\geq\\operatorname\{dist\}\(Y\_\{k\},C\)^\{p\},\(A\.38\) implies

ℱεk\(Tk\)=εk−1∫dist\(Tk,C\)pdσ≤ΛkWpp\(Tkσ\#,μ\)≤C\.\\mathcal\{F\}\_\{\\varepsilon\_\{k\}\}\(T\_\{k\}\)=\\varepsilon\_\{k\}^\{\-1\}\\int\\operatorname\{dist\}\(T\_\{k\},C\)^\{p\}\\,d\\sigma\\leq\\Lambda\_\{k\}W\_\{p\}^\{p\}\(T\_\{k\}\{\}\_\{\\\#\}\\sigma,\\mu\)\\leq C\.By compactness, after passing to a subsequence,

Tk→uBin​L1​\(σ\)T\_\{k\}\\to u\_\{B\}\\qquad\\text\{in \}L^\{1\}\(\\sigma\)for a Caccioppoli partitionBB\.

It remains to identify the masses ofBB\. Let

ΔC:=mini≠j⁡\|ci−cj\|\>0\.\\Delta\_\{C\}:=\\min\_\{i\\neq j\}\|c\_\{i\}\-c\_\{j\}\|\>0\.Ifq​\(Yk\)≠q​\(Zk\)q\(Y\_\{k\}\)\\neq q\(Z\_\{k\}\), thenYkY\_\{k\}lies in a Voronoi cell different from the atomZkZ\_\{k\}\. Therefore

\|Yk−Zk\|≥ΔC/2,\|Y\_\{k\}\-Z\_\{k\}\|\\geq\\Delta\_\{C\}/2,up to the fixed tie\-breaking null ambiguity\. Hence

ℙ​\(q​\(Yk\)≠q​\(Zk\)\)≤\(2ΔC\)p​𝔼​\|Yk−Zk\|p=O​\(εk\)\.\\mathbb\{P\}\(q\(Y\_\{k\}\)\\neq q\(Z\_\{k\}\)\)\\leq\\left\(\\frac\{2\}\{\\Delta\_\{C\}\}\\right\)^\{p\}\\mathbb\{E\}\|Y\_\{k\}\-Z\_\{k\}\|^\{p\}=O\(\\varepsilon\_\{k\}\)\.The law ofq​\(Zk\)q\(Z\_\{k\}\)isπ\\pi\. Thus the law ofq​\(Yk\)q\(Y\_\{k\}\)converges toπ\\piin total variation\. On the other hand, compactness gives

σ​\(q​\(Tk\)=i\)→σ​\(Bi\)\.\\sigma\(q\(T\_\{k\}\)=i\)\\to\\sigma\(B\_\{i\}\)\.Therefore

σ​\(Bi\)=πifor every​i\.\\sigma\(B\_\{i\}\)=\\pi\_\{i\}\\qquad\\text\{for every \}i\.Using \(A\.39\) and the liminf inequality,

lim infkΛk​Wpp​\(Tk​σ\#,μ\)\\displaystyle\\liminf\_\{k\}\\Lambda\_\{k\}W\_\{p\}^\{p\}\(T\_\{k\}\{\}\_\{\\\#\}\\sigma,\\mu\)≥lim infkℱεk​\(Tk\)\\displaystyle\\geq\\liminf\_\{k\}\\mathcal\{F\}\_\{\\varepsilon\_\{k\}\}\(T\_\{k\}\)≥Ep,σ​\(B;C\)\\displaystyle\\geq E\_\{p,\\sigma\}\(B;C\)≥Jp,σ​\(π;C\)\.\\displaystyle\\geq J\_\{p,\\sigma\}\(\\pi;C\)\.This proves the lower bound and hence \(A\.2\)\. ∎

### B\.10Scope and sharpness of the hypotheses

#### Positive masses are necessary\.

The assumptionπi\>0\\pi\_\{i\}\>0is not cosmetic\. If zero\-mass wells are allowed, exact nearest\-well phase constraints can be incompatible with continuity\. In one dimension, take

n=d=1,C=\{0,1,2\},π=\(1/2,0,1/2\)\.n=d=1,\\qquad C=\\\{0,1,2\\\},\\qquad\\pi=\(1/2,0,1/2\)\.A continuous mapT:ℝ→ℝT:\\mathbb\{R\}\\to\\mathbb\{R\}that realizes positive mass near both0and22has connected image between those values and must cross the interior of the Voronoi region of the middle well11\. Under an exact phase constraint this forces positive phase mass for the middle well, contradictingπ2=0\\pi\_\{2\}=0\. Thus the exact\-volume theorem must assumeπi\>0\\pi\_\{i\}\>0\.

#### Finite perimeter is a compactness conclusion, not a property of one Lipschitz map\.

A singleΛ\\Lambda\-Lipschitz map need not have finite\-perimeter phase preimages\. For example, ifE⊂ℝnE\\subset\\mathbb\{R\}^\{n\}has infinite perimeter, the signed\-distance function

sE​\(x\):=dist⁡\(x,Ec\)−dist⁡\(x,E\)s\_\{E\}\(x\):=\\operatorname\{dist\}\(x,E^\{c\}\)\-\\operatorname\{dist\}\(x,E\)is Lipschitz, and a clipped version ofsEs\_\{E\}has a phase preimage equal toEEup to null sets\. Therefore the correct statement is not that every Lipschitz phase preimage has finite perimeter\. Rather, finite perimeter appears only for limits of bounded\-energy sequences, as proved in Proposition[10](https://arxiv.org/html/2606.30705#Thmtheorem10)\.

#### Cores and non\-Euclidean targets\.

For atomic Euclidean wells, the proof above is complete\. For extended coresAiA\_\{i\}, the same calibration works only after replacing each core by itsdVd\_\{V\}\-zero equivalence component: one needs

dV​\(a,a′\)=0for all​a,a′∈Aid\_\{V\}\(a,a^\{\\prime\}\)=0\\qquad\\text\{for all \}a,a^\{\\prime\}\\in A\_\{i\}inside each core component, and strictly positive separation between different components\. Otherwise switching inside a core can carry hidden interface cost\. For general metric targets there is a genuine additional gap: the recovery proof uses Kirszbraun extension and Euclidean projection ontoconv⁡C\\operatorname\{conv\}C\. Both are Euclidean\-specific and are not automatic in arbitrary metric spaces\.

## Appendix CProof of Theorem[6](https://arxiv.org/html/2606.30705#Thmtheorem6)\(Gaussian Dimension Phase Diagram\)

#### Part \(i\): universal lower bound\.

Gaussian isoperimetry per cell givesPγn​\(∂∗Bi\)≥Iγ​\(γn​\(Bi\)\)=φ​\(Φ−1​\(γn​\(Bi\)\)\)P\_\{\\gamma\_\{n\}\}\(\\partial^\{\*\}B\_\{i\}\)\\geq I\_\{\\gamma\}\(\\gamma\_\{n\}\(B\_\{i\}\)\)=\\varphi\(\\Phi^\{\-1\}\(\\gamma\_\{n\}\(B\_\{i\}\)\)\)\. Forγn​\(Bi\)=1/M\\gamma\_\{n\}\(B\_\{i\}\)=1/M, summing pairwise and using∑i<jPγn​\(Bi,Bj\)=12​∑iPγn​\(∂∗Bi\)≥M2​φ​\(Φ−1​\(1/M\)\)\\sum\_\{i<j\}P\_\{\\gamma\_\{n\}\}\(B\_\{i\},B\_\{j\}\)=\\frac\{1\}\{2\}\\sum\_\{i\}P\_\{\\gamma\_\{n\}\}\(\\partial^\{\*\}B\_\{i\}\)\\geq\\frac\{M\}\{2\}\\varphi\(\\Phi^\{\-1\}\(1/M\)\)\. Mills ratio:M​φ​\(Φ−1​\(1/M\)\)∼2​log⁡MM\\varphi\(\\Phi^\{\-1\}\(1/M\)\)\\sim\\sqrt\{2\\log M\}asM→∞M\\to\\infty\.

#### Part \(ii\): high\-dimensional upper bound\.

Transitive cyclic Fourier code withr=⌈16​log⁡M⌉r=\\lceil 16\\log M\\rceilrandom frequencies\. Hoeffding plus union bound givesℙ\[∃j≠j′:∥cj−cj′∥<1\]≤M−1\\mathbb\{P\}\[\\exists j\\neq j^\{\\prime\}\\\!:\\\|c\_\{j\}\-c\_\{j^\{\\prime\}\}\\\|<1\]\\leq M^\{\-1\}\. Cyclic isometry forces normal\-fan cells of equal mass1/M1/M\. Gaussian width𝔼​maxi⁡⟨G,ci⟩≤2​log⁡M\\mathbb\{E\}\\max\_\{i\}\\langle G,c\_\{i\}\\rangle\\leq\\sqrt\{2\\log M\}\(subgaussian maxima\), so Lemma[15](https://arxiv.org/html/2606.30705#Thmtheorem15)gives𝒫n,M≤2​log⁡M\\mathcal\{P\}\_\{n,M\}\\leq\\sqrt\{2\\log M\}oncen≥35​log⁡Mn\\geq 35\\log M\.

#### Part \(iii\): fixed dimension\.

Lower: relative isoperimetric inequality in a ball;γn​\(Bi\)≤C/M\\gamma\_\{n\}\(B\_\{i\}\)\\leq C/MimpliesPγn​\(∂∗Bi\)≥cn​γn​\(Bi\)\(n−1\)/n≥cn​M1/n​γn​\(Bi\)P\_\{\\gamma\_\{n\}\}\(\\partial^\{\*\}B\_\{i\}\)\\geq c\_\{n\}\\gamma\_\{n\}\(B\_\{i\}\)^\{\(n\-1\)/n\}\\geq c\_\{n\}M^\{1/n\}\\gamma\_\{n\}\(B\_\{i\}\)\. Upper: product quantile slabs forM=knM=k^\{n\}; balanced recursive quantile partition for generalMM; constantsCn≤C​nC\_\{n\}\\leq Cn\.

###### Lemma 15\(Gaussian\-width identity\)\.

Letc1,…,cMc\_\{1\},\\ldots,c\_\{M\}be the vertices of a polytope inℝn\\mathbb\{R\}^\{n\},f​\(x\)=maxi⁡⟨x,ci⟩f\(x\)=\\max\_\{i\}\\langle x,c\_\{i\}\\rangle, and\(Bi\)\(B\_\{i\}\)the normal\-fan partition\. Then∑i<j‖ci−cj‖​Pγn​\(Bi,Bj\)=𝔼​maxi⁡⟨G,ci⟩\\sum\_\{i<j\}\\left\\lVert c\_\{i\}\-c\_\{j\}\\right\\rVert\\,P\_\{\\gamma\_\{n\}\}\(B\_\{i\},B\_\{j\}\)=\\mathbb\{E\}\\max\_\{i\}\\langle G,c\_\{i\}\\rangle,G∼γnG\\sim\\gamma\_\{n\}\.

#### Proof\.

f​\(x\)=maxi⁡⟨x,ci⟩f\(x\)=\\max\_\{i\}\\langle x,c\_\{i\}\\rangleis convex, piecewise linear\. The distributional Laplacian’s interface density on∂∗Bi∩∂∗Bj\\partial^\{\*\}B\_\{i\}\\cap\\partial^\{\*\}B\_\{j\}is‖ci−cj‖\\left\\lVert c\_\{i\}\-c\_\{j\}\\right\\rVert\(the outward normal between cells is\(cj−ci\)/‖cj−ci‖\(c\_\{j\}\-c\_\{i\}\)/\\left\\lVert c\_\{j\}\-c\_\{i\}\\right\\rVert\)\. Triple junctions have dimension≤n−2\\leq n\-2\. Gaussian integration by parts with cutoffχR\\chi\_\{R\}and Euler’s identityx⊤​∇f=fx^\{\\top\}\\nabla f=fgive∫φn​d​\(Δ​f\)=𝔼​maxi⁡⟨G,ci⟩\\int\\varphi\_\{n\}\\,d\(\\Delta f\)=\\mathbb\{E\}\\max\_\{i\}\\langle G,c\_\{i\}\\rangle\.

## Appendix DProof of Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\(Non\-Commitment\): active\-facet tube law

We give the complete proof in the rescaled observation variable

Zs:=Ut/t=X\+s​ϵ,s=s​\(t\):=s0​\(1−t\)/t,ϵ∼N​\(0,Id\)\.Z\_\{s\}:=U\_\{t\}/t=X\+s\\epsilon,\\qquad s=s\(t\):=s\_\{0\}\(1\-t\)/t,\\qquad\\epsilon\\sim N\(0,I\_\{d\}\)\.Conditioning onUtU\_\{t\}and conditioning onZsZ\_\{s\}are equivalent, and

mt​\(Ut\)=ms​\(Zs\),ms​\(z\):=𝔼​\[X∣Zs=z\]\.m\_\{t\}\(U\_\{t\}\)=m\_\{s\}\(Z\_\{s\}\),\\qquad m\_\{s\}\(z\):=\\mathbb\{E\}\[X\\mid Z\_\{s\}=z\]\.All statements below are conditional on the readout matrixWW\. IfWWis random, the final identities may be averaged overWW\.

Let

Ωi:=\{z:wi⊤​z≥wk⊤​z​for all​k\}\\Omega\_\{i\}:=\\\{z:\\ w\_\{i\}^\{\\top\}z\\geq w\_\{k\}^\{\\top\}z\\ \\text\{for all \}k\\\}be the closed argmax cell of tokenii, with an arbitrary fixed tie\-breaking rule on the cell boundaries\. Ties have zeroXX\-probability under the assumptions below\. Fori≠ji\\neq j, put

ni​j:=wi−wj‖wi−wj‖,Σi​j:=\{z:wi⊤​z=wj⊤​z\}\.n\_\{ij\}:=\\frac\{w\_\{i\}\-w\_\{j\}\}\{\\\|w\_\{i\}\-w\_\{j\}\\\|\},\\qquad\\Sigma\_\{ij\}:=\\\{z:\\ w\_\{i\}^\{\\top\}z=w\_\{j\}^\{\\top\}z\\\}\.We orientni​jn\_\{ij\}so thatni​jn\_\{ij\}points fromΩj\\Omega\_\{j\}intoΩi\\Omega\_\{i\}\. The*active co\-maximal facet*betweeniiandjjis

Fi​j:=\{z∈Σi​j:wi⊤​z=wj⊤​z=maxk⁡wk⊤​z\}\.F\_\{ij\}:=\\Big\\\{z\\in\\Sigma\_\{ij\}:\\ w\_\{i\}^\{\\top\}z=w\_\{j\}^\{\\top\}z=\\max\_\{k\}w\_\{k\}^\{\\top\}z\\Big\\\}\.Equivalently,Fi​j=∂Ωi∩∂ΩjF\_\{ij\}=\\partial\\Omega\_\{i\}\\cap\\partial\\Omega\_\{j\}up to lower\-dimensional tie sets\. Non\-active pairwise hyperplanesΣi​j\\Sigma\_\{ij\}do not contribute to the leading tube mass\. Define the active boundary density

AW:=∑i<j∫Σi​j𝟏​\{wi⊤​z=wj⊤​z=maxk⁡wk⊤​z\}​ρX​\(z\)​𝑑ℋd−1​\(z\)=∑i<j∫Fi​jρX​𝑑ℋd−1\.A\_\{W\}:=\\sum\_\{i<j\}\\int\_\{\\Sigma\_\{ij\}\}\\mathbf\{1\}\\\!\\left\\\{w\_\{i\}^\{\\top\}z=w\_\{j\}^\{\\top\}z=\\max\_\{k\}w\_\{k\}^\{\\top\}z\\right\\\}\\rho\_\{X\}\(z\)\\,d\\mathcal\{H\}^\{d\-1\}\(z\)=\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\\,d\\mathcal\{H\}^\{d\-1\}\.The second equality uses the regularity assumption below\.

#### Regularity assumption\.

We assume throughout this appendix that:

1. \(R1\)Y=DW​\(X\)Y=D\_\{W\}\(X\)almost surely, unless explicitly stated otherwise in the “Imperfect clean decoding” paragraph below\.
2. \(R2\)ρX\\rho\_\{X\}isC2C^\{2\}in a neighborhood of the active boundary𝒮:=⋃i<jFi​j\\mathcal\{S\}:=\\bigcup\_\{i<j\}F\_\{ij\}, and the usual first\-order tube formula holds with densityρX\\rho\_\{X\}along each active facet\.
3. \(R3\)The co\-maximal skeleton 𝒯:=\{z:≥3logitswk⊤ztie for the maximum\}\\mathcal\{T\}:=\\\{z:\\ \\geq 3\\text\{ logits \}w\_\{k\}^\{\\top\}z\\text\{ tie for the maximum\}\\\}has weighted tube mass¶​\[dist⁡\(X,𝒯\)≤r\]=O​\(r2\)\\P\[\\operatorname\{dist\}\(X,\\mathcal\{T\}\)\\leq r\]=O\(r^\{2\}\)asr↓0r\\downarrow 0\. In particular,ℋd−1​\(𝒯\)=0\\mathcal\{H\}^\{d\-1\}\(\\mathcal\{T\}\)=0on the active boundary\.
4. \(R4\)AW<∞A\_\{W\}<\\infty, and the same finiteness holds for the first two normal derivatives ofρX\\rho\_\{X\}in a fixed small tube around𝒮\\mathcal\{S\}\. This is only to justify exchanging Taylor expansion and integration on unbounded facets\.

These are the standard smooth\-density, regular\-polyhedral\-boundary hypotheses under which Gaussian small\-noise classification has a first\-order surface\-area expansion\. They also rule out the degenerate case in which a pairwise bisector lies inside a third token’s cell: such a bisector is not co\-maximal, hence is not part of anyFi​jF\_\{ij\}, and contributes noO​\(s\)O\(s\)term\.

#### Equality, Bayes bound, and cross\-entropy bound\.

Let

Y^s:=DW​\(ms​\(Zs\)\),qs​\(y∣z\):=¶​\(Y=y∣Zs=z\)\.\\widehat\{Y\}\_\{s\}:=D\_\{W\}\(m\_\{s\}\(Z\_\{s\}\)\),\\qquad q\_\{s\}\(y\\mid z\):=\\P\(Y=y\\mid Z\_\{s\}=z\)\.Conditioning onZs=zZ\_\{s\}=z, the valuems​\(z\)m\_\{s\}\(z\)and henceds​\(z\):=DW​\(ms​\(z\)\)d\_\{s\}\(z\):=D\_\{W\}\(m\_\{s\}\(z\)\)are deterministic, whereasYYis distributed according toqs\(⋅∣z\)q\_\{s\}\(\\cdot\\mid z\)\. Therefore

¶​\(Y^s≠Y∣Zs=z\)=1−qs​\(ds​\(z\)∣z\),\\P\(\\widehat\{Y\}\_\{s\}\\neq Y\\mid Z\_\{s\}=z\)=1\-q\_\{s\}\(d\_\{s\}\(z\)\\mid z\),and hence

¶​\(Y^s≠Y\)=𝔼​\[1−qs​\(DW​\(ms​\(Zs\)\)∣Zs\)\]\.\\P\(\\widehat\{Y\}\_\{s\}\\neq Y\)=\\mathbb\{E\}\\\!\\left\[1\-q\_\{s\}\(D\_\{W\}\(m\_\{s\}\(Z\_\{s\}\)\)\\mid Z\_\{s\}\)\\right\]\.Sinceqs​\(ds​\(z\)∣z\)≤maxy⁡qs​\(y∣z\)q\_\{s\}\(d\_\{s\}\(z\)\\mid z\)\\leq\\max\_\{y\}q\_\{s\}\(y\\mid z\),

¶​\(Y^s≠Y\)≥Bs:=𝔼​\[1−maxy⁡qs​\(y∣Zs\)\]\.\\P\(\\widehat\{Y\}\_\{s\}\\neq Y\)\\geq B\_\{s\}:=\\mathbb\{E\}\\\!\\left\[1\-\\max\_\{y\}q\_\{s\}\(y\\mid Z\_\{s\}\)\\right\]\.HereBsB\_\{s\}is the Bayes non\-commitment at noise scaless\.

Let

pW​\(y∣z\):=exp⁡\(wy⊤​z\)∑kexp⁡\(wk⊤​z\)p\_\{W\}\(y\\mid z\):=\\frac\{\\exp\(w\_\{y\}^\{\\top\}z\)\}\{\\sum\_\{k\}\\exp\(w\_\{k\}^\{\\top\}z\)\}be the softmax associated with the readout weights\. Ford=ds​\(z\)=DW​\(ms​\(z\)\)d=d\_\{s\}\(z\)=D\_\{W\}\(m\_\{s\}\(z\)\)and anyy≠dy\\neq d,

−log⁡pW​\(y∣ms​\(z\)\)\\displaystyle\-\\log p\_\{W\}\(y\\mid m\_\{s\}\(z\)\)=log​∑kexp⁡\(\(wk−wy\)⊤​ms​\(z\)\)\\displaystyle=\\log\\sum\_\{k\}\\exp\\\!\\big\(\(w\_\{k\}\-w\_\{y\}\)^\{\\top\}m\_\{s\}\(z\)\\big\)≥log⁡\(1\+exp⁡\(\(wd−wy\)⊤​ms​\(z\)\)\)≥log⁡2,\\displaystyle\\geq\\log\\\!\\left\(1\+\\exp\\\!\\big\(\(w\_\{d\}\-w\_\{y\}\)^\{\\top\}m\_\{s\}\(z\)\\big\)\\right\)\\geq\\log 2,because\(wd−wy\)⊤​ms​\(z\)≥0\(w\_\{d\}\-w\_\{y\}\)^\{\\top\}m\_\{s\}\(z\)\\geq 0\. Averaging underY∣Zs=zY\\mid Z\_\{s\}=zgives

𝔼​\[−log⁡pW​\(Y∣ms​\(Zs\)\)\]\\displaystyle\\mathbb\{E\}\[\-\\log p\_\{W\}\(Y\\mid m\_\{s\}\(Z\_\{s\}\)\)\]≥\(log⁡2\)​𝔼​\[1−qs​\(ds​\(Zs\)∣Zs\)\]\\displaystyle\\geq\(\\log 2\)\\,\\mathbb\{E\}\\\!\\left\[1\-q\_\{s\}\(d\_\{s\}\(Z\_\{s\}\)\\mid Z\_\{s\}\)\\right\]≥\(log⁡2\)​𝔼​\[1−maxy⁡qs​\(y∣Zs\)\]\.\\displaystyle\\geq\(\\log 2\)\\,\\mathbb\{E\}\\\!\\left\[1\-\\max\_\{y\}q\_\{s\}\(y\\mid Z\_\{s\}\)\\right\]\.

#### Bayes tube law on active facets\.

For each tokenii, define the class\-conditional blurred density

ai,s​\(z\):=∫ΩiρX​\(x\)​φs​\(z−x\)​𝑑x,φs​\(v\):=\(2​π​s2\)−d/2​exp⁡\(−‖v‖2/2​s2\)\.a\_\{i,s\}\(z\):=\\int\_\{\\Omega\_\{i\}\}\\rho\_\{X\}\(x\)\\,\\varphi\_\{s\}\(z\-x\)\\,dx,\\qquad\\varphi\_\{s\}\(v\):=\(2\\pi s^\{2\}\)^\{\-d/2\}\\exp\(\-\\\|v\\\|^\{2\}/2s^\{2\}\)\.Then

qs​\(i∣z\)=ai,s​\(z\)∑kak,s​\(z\)q\_\{s\}\(i\\mid z\)=\\frac\{a\_\{i,s\}\(z\)\}\{\\sum\_\{k\}a\_\{k,s\}\(z\)\}and the Bayes non\-commitment can be written exactly as

Bs=∫ℝd\(∑kak,s​\(z\)−maxk⁡ak,s​\(z\)\)​𝑑z\.B\_\{s\}=\\int\_\{\\mathbb\{R\}^\{d\}\}\\left\(\\sum\_\{k\}a\_\{k,s\}\(z\)\-\\max\_\{k\}a\_\{k,s\}\(z\)\\right\)\\,dz\.
We now compute the first\-order contribution of one active facet\. Fixi<ji<jand work on a compact subset of the relative interior ofFi​jF\_\{ij\}, away from the triple skeleton\. In the normal coordinates

z=x\+s​a​ni​j,x∈Fi​j,a∈ℝ,z=x\+sa\\,n\_\{ij\},\\qquad x\\in F\_\{ij\},\\quad a\\in\\mathbb\{R\},the cells are locally the two half\-spaces

Ωi=\{r≥0\},Ωj=\{r≤0\}\\Omega\_\{i\}=\\\{r\\geq 0\\\},\\qquad\\Omega\_\{j\}=\\\{r\\leq 0\\\}in the signed normal coordinater=⟨x′−x,ni​j⟩r=\\langle x^\{\\prime\}\-x,n\_\{ij\}\\rangle\. All other cells are separated by a positive logit gap in this local chart, and their blurred contributions are exponentially small ins−2s^\{\-2\}\. Taylor expandingρX\\rho\_\{X\}in the normal and tangential variables gives, uniformly for boundedaa,

ai,s​\(x\+s​a​ni​j\)=ρX​\(x\)​Φ​\(a\)\+O​\(s\),aj,s​\(x\+s​a​ni​j\)=ρX​\(x\)​Φ​\(−a\)\+O​\(s\),a\_\{i,s\}\(x\+san\_\{ij\}\)=\\rho\_\{X\}\(x\)\\Phi\(a\)\+O\(s\),\\qquad a\_\{j,s\}\(x\+san\_\{ij\}\)=\\rho\_\{X\}\(x\)\\Phi\(\-a\)\+O\(s\),whereΦ\\Phiis the standard normal cdf\. Thus, on theFi​jF\_\{ij\}\-tube,

∑kak,s​\(z\)−maxk⁡ak,s​\(z\)=ρX​\(x\)​min⁡\{Φ​\(a\),Φ​\(−a\)\}\+O​\(s\),\\sum\_\{k\}a\_\{k,s\}\(z\)\-\\max\_\{k\}a\_\{k,s\}\(z\)=\\rho\_\{X\}\(x\)\\min\\\{\\Phi\(a\),\\Phi\(\-a\)\\\}\+O\(s\),up to an exponentially small error from non\-competing classes\.

Usingd​z=s​d​a​d​ℋd−1​\(x\)dz=s\\,da\\,d\\mathcal\{H\}^\{d\-1\}\(x\)in these flat normal coordinates, the contribution ofFi​jF\_\{ij\}is

Ii​j,s\\displaystyle I\_\{ij,s\}=s​∫Fi​jρX​\(x\)​𝑑ℋd−1​\(x\)​∫ℝmin⁡\{Φ​\(a\),Φ​\(−a\)\}​𝑑a\+o​\(s\)\\displaystyle=s\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\(x\)\\,d\\mathcal\{H\}^\{d\-1\}\(x\)\\int\_\{\\mathbb\{R\}\}\\min\\\{\\Phi\(a\),\\Phi\(\-a\)\\\}\\,da\+o\(s\)=s​∫Fi​jρX​\(x\)​𝑑ℋd−1​\(x\)​\(2​∫0∞Φ​\(−a\)​𝑑a\)\+o​\(s\)\\displaystyle=s\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\(x\)\\,d\\mathcal\{H\}^\{d\-1\}\(x\)\\left\(2\\int\_\{0\}^\{\\infty\}\\Phi\(\-a\)\\,da\\right\)\+o\(s\)=2π​s​∫Fi​jρX​\(x\)​𝑑ℋd−1​\(x\)\+o​\(s\)\.\\displaystyle=\\sqrt\{\\frac\{2\}\{\\pi\}\}\\,s\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\(x\)\\,d\\mathcal\{H\}^\{d\-1\}\(x\)\+o\(s\)\.The identity

2​∫0∞Φ​\(−a\)​𝑑a=2​𝔼​\[G\+\]=2π,G∼N​\(0,1\),2\\int\_\{0\}^\{\\infty\}\\Phi\(\-a\)\\,da=2\\mathbb\{E\}\[G\_\{\+\}\]=\\sqrt\{\\frac\{2\}\{\\pi\}\},\\qquad G\\sim N\(0,1\),is the one\-dimensional Laplace constant\. This proves that the per\-facet2/π\\sqrt\{2/\\pi\}constant is unchanged after restricting to active co\-maximal facets\.

It remains to justify that no other region contributes at orderss\. Away from anO​\(s​log⁡\(1/s\)\)O\(s\\sqrt\{\\log\(1/s\)\}\)\-tube of the active boundary, the posterior mass of every non\-argmax token is exponentially small\. AnO​\(s​log⁡\(1/s\)\)O\(s\\sqrt\{\\log\(1/s\)\}\)\-tube of the triple skeleton has probabilityO​\(s2​log⁡\(1/s\)\)=o​\(s\)O\(s^\{2\}\\log\(1/s\)\)=o\(s\)by \(R3\)\. Finally, a pairwise hyperplaneΣi​j\\Sigma\_\{ij\}that is not co\-maximal is either inactive in a neighborhood of the decision boundary or meets it only in the triple skeleton; it therefore contributeso​\(s\)o\(s\)\. Summing \(C\.8\) over active facets yields

Bs=2π​s​∑i<j∫Fi​jρX​𝑑ℋd−1\+o​\(s\)=2π​s​AW\+o​\(s\)\.B\_\{s\}=\\sqrt\{\\frac\{2\}\{\\pi\}\}\\,s\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\\,d\\mathcal\{H\}^\{d\-1\}\+o\(s\)=\\sqrt\{\\frac\{2\}\{\\pi\}\}\\,s\\,A\_\{W\}\+o\(s\)\.

#### Posterior\-mean readout versus MAP readout\.

Let

dsMAP​\(z\):=arg⁡maxy⁡qs​\(y∣z\),dsmean​\(z\):=DW​\(ms​\(z\)\)\.d\_\{s\}^\{\\mathrm\{MAP\}\}\(z\):=\\arg\\max\_\{y\}q\_\{s\}\(y\\mid z\),\\qquad d\_\{s\}^\{\\mathrm\{mean\}\}\(z\):=D\_\{W\}\(m\_\{s\}\(z\)\)\.The exact discrepancy identity is

¶​\(dsmean​\(Zs\)≠Y\)−Bs\\displaystyle\\P\(d\_\{s\}^\{\\mathrm\{mean\}\}\(Z\_\{s\}\)\\neq Y\)\-B\_\{s\}=𝔼​\[qs​\(dsMAP​\(Zs\)∣Zs\)−qs​\(dsmean​\(Zs\)∣Zs\)\]\\displaystyle=\\mathbb\{E\}\\\!\\left\[q\_\{s\}\(d\_\{s\}^\{\\mathrm\{MAP\}\}\(Z\_\{s\}\)\\mid Z\_\{s\}\)\-q\_\{s\}\(d\_\{s\}^\{\\mathrm\{mean\}\}\(Z\_\{s\}\)\\mid Z\_\{s\}\)\\right\]≥0\.\\displaystyle\\geq 0\.Consequently,

Bs≤¶​\(dsmean​\(Zs\)≠Y\)≤Bs\+¶​\[dsmean​\(Zs\)≠dsMAP​\(Zs\)\]\.B\_\{s\}\\leq\\P\(d\_\{s\}^\{\\mathrm\{mean\}\}\(Z\_\{s\}\)\\neq Y\)\\leq B\_\{s\}\+\\P\\\!\\left\[d\_\{s\}^\{\\mathrm\{mean\}\}\(Z\_\{s\}\)\\neq d\_\{s\}^\{\\mathrm\{MAP\}\}\(Z\_\{s\}\)\\right\]\.Thus the posterior\-mean flip rate and the Bayes non\-commitment have the same leading constant if and only if the expectation in \(C\.10\) iso​\(s\)o\(s\)\. A sufficient and easy\-to\-check condition is

¶​\[dsmean​\(Zs\)≠dsMAP​\(Zs\)\]=o​\(s\)\.\\P\\\!\\left\[d\_\{s\}^\{\\mathrm\{mean\}\}\(Z\_\{s\}\)\\neq d\_\{s\}^\{\\mathrm\{MAP\}\}\(Z\_\{s\}\)\\right\]=o\(s\)\.
We now verify \(C\.12\) under the smooth clean\-decoding assumptions above\. The only possibleO​\(s\)O\(s\)\-mass disagreement region lies in active facet tubes\. Fix a regular pointx∈Fi​j∖𝒯x\\in F\_\{ij\}\\setminus\\mathcal\{T\}, and write

z=x\+s​a​ni​j,κi​j​\(x\):=∂ni​jlog⁡ρX​\(x\)\.z=x\+san\_\{ij\},\\qquad\\kappa\_\{ij\}\(x\):=\\partial\_\{n\_\{ij\}\}\\log\\rho\_\{X\}\(x\)\.First, by the same half\-space calculation as above,

ai,s​\(z\)−aj,s​\(z\)=ρX​\(x\)​\(2​Φ​\(a\)−1\)\+2​s​∂ni​jρX​\(x\)​φ​\(a\)\+O​\(s2\),a\_\{i,s\}\(z\)\-a\_\{j,s\}\(z\)=\\rho\_\{X\}\(x\)\\big\(2\\Phi\(a\)\-1\\big\)\+2s\\,\\partial\_\{n\_\{ij\}\}\\rho\_\{X\}\(x\)\\,\\varphi\(a\)\+O\(s^\{2\}\),whereφ\\varphiis the standard normal density\. Since the common denominator∑kak,s​\(z\)\\sum\_\{k\}a\_\{k,s\}\(z\)is positive,dsMAP​\(z\)d\_\{s\}^\{\\mathrm\{MAP\}\}\(z\)switches betweeniiandjjwhere the right\-hand side of \(C\.13\) is zero\. Neara=0a=0,

2​Φ​\(a\)−1=2​φ​\(0\)​a\+O​\(a3\),φ​\(a\)=φ​\(0\)\+O​\(a2\),2\\Phi\(a\)\-1=2\\varphi\(0\)a\+O\(a^\{3\}\),\\qquad\\varphi\(a\)=\\varphi\(0\)\+O\(a^\{2\}\),so the MAP switching surface has signed normal coordinate

ai​jMAP​\(x,s\)=−s​κi​j​\(x\)\+O​\(s2\)\.a^\{\\mathrm\{MAP\}\}\_\{ij\}\(x,s\)=\-s\\,\\kappa\_\{ij\}\(x\)\+O\(s^\{2\}\)\.
Second, the posterior mean satisfies Tweedie’s formula

ms​\(z\)=z\+s2​∇log⁡ps​\(z\),ps:=ρX∗φs\.m\_\{s\}\(z\)=z\+s^\{2\}\\nabla\\log p\_\{s\}\(z\),\\qquad p\_\{s\}:=\\rho\_\{X\}\*\\varphi\_\{s\}\.Sinceps=ρX\+O​\(s2\)p\_\{s\}=\\rho\_\{X\}\+O\(s^\{2\}\)inC1C^\{1\}near the active boundary,

δi​j∗​\(ms​\(x\+s​a​ni​j\)\)\\displaystyle\\delta^\{\*\}\_\{ij\}\(m\_\{s\}\(x\+san\_\{ij\}\)\)=ni​j⊤​ms​\(x\+s​a​ni​j\)−ni​j⊤​x\\displaystyle=n\_\{ij\}^\{\\top\}m\_\{s\}\(x\+san\_\{ij\}\)\-n\_\{ij\}^\{\\top\}x=s​a\+s2​∂ni​jlog⁡ρX​\(x\)\+O​\(s3\)\\displaystyle=sa\+s^\{2\}\\partial\_\{n\_\{ij\}\}\\log\\rho\_\{X\}\(x\)\+O\(s^\{3\}\)=s​\(a\+s​κi​j​\(x\)\+O​\(s2\)\)\.\\displaystyle=s\\big\(a\+s\\kappa\_\{ij\}\(x\)\+O\(s^\{2\}\)\\big\)\.Therefore the posterior\-mean readout switches betweeniiandjjat

ai​jmean​\(x,s\)=−s​κi​j​\(x\)\+O​\(s2\)\.a^\{\\mathrm\{mean\}\}\_\{ij\}\(x,s\)=\-s\\,\\kappa\_\{ij\}\(x\)\+O\(s^\{2\}\)\.Equations \(C\.14\) and \(C\.17\) show that the MAP and posterior\-mean switching surfaces differ byO​\(s2\)O\(s^\{2\}\)in theaa\-coordinate, hence byO​\(s3\)O\(s^\{3\}\)in the original signed normal coordinate\. Their disagreement tube aroundFi​jF\_\{ij\}has weighted masso​\(s\)o\(s\)\. Summing over finitely many active facets and adding theo​\(s\)o\(s\)triple\-skeleton contribution proves \(C\.12\)\. Combining with \(C\.9\) and \(C\.11\) gives the active\-facet form of the tube law:

¶\(Y^s≠Y\)=2πs∑i<j∫Fi​jρXdℋd−1\+o\(s\)\.\\boxed\{\\P\(\\widehat\{Y\}\_\{s\}\\neq Y\)=\\sqrt\{\\frac\{2\}\{\\pi\}\}\\,s\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\\,d\\mathcal\{H\}^\{d\-1\}\+o\(s\)\.\}Returning tos=s​\(t\)=s0​\(1−t\)/ts=s\(t\)=s\_\{0\}\(1\-t\)/t,

¶​\(Y^t≠Y\)=2π​s​\(t\)​∑i<j∫Fi​jρX​𝑑ℋd−1\+o​\(s​\(t\)\)\.\\P\(\\widehat\{Y\}\_\{t\}\\neq Y\)=\\sqrt\{\\frac\{2\}\{\\pi\}\}\\,s\(t\)\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\\,d\\mathcal\{H\}^\{d\-1\}\+o\(s\(t\)\)\.

#### Geometric meaning of the MAP–mean condition\.

The proof above uses smoothness ofρX\\rho\_\{X\}to show that MAP and posterior\-mean readout have the same switching surface to first order\. In a more general labelled mixture model, this is a genuine hypothesis, not a free consequence\.

Near a regularFi​jF\_\{ij\}, write

μis​\(z\):=𝔼​\[X∣Zs=z,Y=i\],μjs​\(z\):=𝔼​\[X∣Zs=z,Y=j\]\.\\mu\_\{i\}^\{s\}\(z\):=\\mathbb\{E\}\[X\\mid Z\_\{s\}=z,Y=i\],\\qquad\\mu\_\{j\}^\{s\}\(z\):=\\mathbb\{E\}\[X\\mid Z\_\{s\}=z,Y=j\]\.Ignoring exponentially small non\-competing classes,

ms​\(z\)=qs​\(i∣z\)​μis​\(z\)\+qs​\(j∣z\)​μjs​\(z\)\+o​\(s\)\.m\_\{s\}\(z\)=q\_\{s\}\(i\\mid z\)\\mu\_\{i\}^\{s\}\(z\)\+q\_\{s\}\(j\\mid z\)\\mu\_\{j\}^\{s\}\(z\)\+o\(s\)\.Since the readout boundary betweeniiandjjis linear,

δi​j∗​\(ms​\(z\)\)=qs​\(i∣z\)​δi​j∗​\(μis​\(z\)\)\+qs​\(j∣z\)​δi​j∗​\(μjs​\(z\)\)\+o​\(s\)\.\\delta^\{\*\}\_\{ij\}\(m\_\{s\}\(z\)\)=q\_\{s\}\(i\\mid z\)\\delta^\{\*\}\_\{ij\}\(\\mu\_\{i\}^\{s\}\(z\)\)\+q\_\{s\}\(j\\mid z\)\\delta^\{\*\}\_\{ij\}\(\\mu\_\{j\}^\{s\}\(z\)\)\+o\(s\)\.The posterior\-mean readout choosesiirather thanjjexactly when the left\-hand side of \(C\.20\) is nonnegative\. The MAP rule choosesiiwhenqs​\(i∣z\)≥qs​\(j∣z\)q\_\{s\}\(i\\mid z\)\\geq q\_\{s\}\(j\\mid z\)\. Hence the sharp general condition is:

𝔼​\[qs​\(dsMAP​\(Zs\)∣Zs\)−qs​\(dsmean​\(Zs\)∣Zs\)\]=o​\(s\)\.\\mathbb\{E\}\\\!\\left\[q\_\{s\}\(d\_\{s\}^\{\\mathrm\{MAP\}\}\(Z\_\{s\}\)\\mid Z\_\{s\}\)\-q\_\{s\}\(d\_\{s\}^\{\\mathrm\{mean\}\}\(Z\_\{s\}\)\\mid Z\_\{s\}\)\\right\]=o\(s\)\.
A simple sufficient geometric version is that, forℋd−1\\mathcal\{H\}^\{d\-1\}\-a\.e\.x∈Fi​jx\\in F\_\{ij\}, the two conditional cluster means lie on opposite sides ofΣi​j\\Sigma\_\{ij\}, and the zero ofδi​j∗​\(ms​\(x\+s​a​ni​j\)\)\\delta^\{\*\}\_\{ij\}\(m\_\{s\}\(x\+san\_\{ij\}\)\)differs byo​\(1\)o\(1\)in theaa\-coordinate from the zero ofqs​\(i∣x\+s​a​ni​j\)−qs​\(j∣x\+s​a​ni​j\)q\_\{s\}\(i\\mid x\+san\_\{ij\}\)\-q\_\{s\}\(j\\mid x\+san\_\{ij\}\)\. Equivalently, the MAP and posterior\-mean switching surfaces are separated by ano​\(s\)o\(s\)physical normal distance\. Under the clean\-label smooth\-density assumptions \(R1\)–\(R4\), this condition is automatic by \(C\.14\)–\(C\.17\)\. Without it, the sharp statement is the inequality \(C\.11\); anO​\(s\)O\(s\)\-measure MAP–mean disagreement set can change the leading constant\.

#### Row\-space flip criterion\.

For later reference, the posterior\-mean decoded token differs from the clean decoded tokenYYonly if some competitorj≠Yj\\neq Ysatisfies

wj⊤​ms​\(Zs\)≥wY⊤​ms​\(Zs\)\.w\_\{j\}^\{\\top\}m\_\{s\}\(Z\_\{s\}\)\\geq w\_\{Y\}^\{\\top\}m\_\{s\}\(Z\_\{s\}\)\.Equivalently,

\(wY−wj\)⊤​\(X−ms​\(Zs\)\)‖wY−wj‖≥δY​j∗​\(X\)\.\\frac\{\(w\_\{Y\}\-w\_\{j\}\)^\{\\top\}\(X\-m\_\{s\}\(Z\_\{s\}\)\)\}\{\\\|w\_\{Y\}\-w\_\{j\}\\\|\}\\geq\\delta^\{\*\}\_\{Yj\}\(X\)\.Thus only the projection ofX−ms​\(Zs\)X\-m\_\{s\}\(Z\_\{s\}\)onto the readout row\-space directionswY−wjw\_\{Y\}\-w\_\{j\}can cause a token flip\. Components orthogonal to all readout differences are invisible to the argmax readout\.

#### Imperfect clean decoding\.

The preceding theorem should be interpreted as a statement about*readout self\-consistency*\. If the dataset tokenYrawY^\{\\rm raw\}is not always equal to the clean decoded token, define

Y∘:=DW​\(X\),η0:=¶​\(Yraw≠Y∘\)\.Y^\{\\circ\}:=D\_\{W\}\(X\),\\qquad\\eta\_\{0\}:=\\P\(Y^\{\\rm raw\}\\neq Y^\{\\circ\}\)\.Then Theorem 3 applies exactly toY∘Y^\{\\circ\}\. For the raw token labels,

\|¶​\(Y^s≠Yraw\)−¶​\(Y^s≠Y∘\)\|\\displaystyle\\left\|\\P\(\\widehat\{Y\}\_\{s\}\\neq Y^\{\\rm raw\}\)\-\\P\(\\widehat\{Y\}\_\{s\}\\neq Y^\{\\circ\}\)\\right\|≤¶​\(Yraw≠Y∘\)=η0\.\\displaystyle\\leq\\P\(Y^\{\\rm raw\}\\neq Y^\{\\circ\}\)=\\eta\_\{0\}\.Indeed, for any deterministic decoded tokendd,

\|𝟏​\{d≠Yraw\}−𝟏​\{d≠Y∘\}\|≤𝟏​\{Yraw≠Y∘\}\.\\left\|\\mathbf\{1\}\\\{d\\neq Y^\{\\rm raw\}\\\}\-\\mathbf\{1\}\\\{d\\neq Y^\{\\circ\}\\\}\\right\|\\leq\\mathbf\{1\}\\\{Y^\{\\rm raw\}\\neq Y^\{\\circ\}\\\}\.Therefore

¶​\(Y^s≠Yraw\)=2π​s​AW\+o​\(s\)\+O​\(η0\)\.\\P\(\\widehat\{Y\}\_\{s\}\\neq Y^\{\\rm raw\}\)=\\sqrt\{\\frac\{2\}\{\\pi\}\}\\,s\\,A\_\{W\}\+o\(s\)\+O\(\\eta\_\{0\}\)\.TheO​\(η0\)O\(\\eta\_\{0\}\)term is an additive clean\-autoencoder error offset, not part of the readout tube constant\. However, if one insists on plotting raw\-token error without subtracting or controlling the clean error,η0\\eta\_\{0\}gives a non\-vanishing intercept ass↓0s\\downarrow 0\. In fact,

lims↓0¶​\(Y^s≠Yraw\)=η0,\\lim\_\{s\\downarrow 0\}\\P\(\\widehat\{Y\}\_\{s\}\\neq Y^\{\\rm raw\}\)=\\eta\_\{0\},becausems​\(Zs\)→Xm\_\{s\}\(Z\_\{s\}\)\\to Xin probability andDW​\(ms​\(Zs\)\)→DW​\(X\)=Y∘D\_\{W\}\(m\_\{s\}\(Z\_\{s\}\)\)\\to D\_\{W\}\(X\)=Y^\{\\circ\}away from measure\-zero ties\.

Thus the cleanest theorem and the cleanest experiment should use

Y:=Y∘=DW​\(X\)Y:=Y^\{\\circ\}=D\_\{W\}\(X\)by definition\. This gains an exact readout\-calibrated statement\. What is lost is direct accounting of the autoencoder’s semantic clean error; that error should be reported separately asη0\\eta\_\{0\}\. For ELF\-style clean accuracy1−η01\-\\eta\_\{0\}, the raw\-token curve should be interpreted as “clean\-error offset plus tube error,” whereas theY∘Y^\{\\circ\}\-curve is the actual test of Theorem 3\.

If one wants a slope statement for raw labels, an additional hypothesis is required: the misdecoded set\{Yraw≠Y∘\}\\\{Y^\{\\rm raw\}\\neq Y^\{\\circ\}\\\}must not concentrate in anO​\(s\)O\(s\)\-tube of the active boundary\. Under such a no\-boundary\-concentration condition, the raw\-label slope differs from the clean\-decoded slope by a term controlled by the active\-facet boundary density of the misdecoded subset\. Without that extra condition, \(C\.23\) is the sharp uniform perturbation bound\.

#### Empirical predictor from the normalized\-margin distribution\.

Let the clean\-decoded margin be

δ∗​\(X,Y∘\)=minj≠Y∘⁡\(wY∘−wj\)⊤​X‖wY∘−wj‖\.\\delta^\{\*\}\(X,Y^\{\\circ\}\)=\\min\_\{j\\neq Y^\{\\circ\}\}\\frac\{\(w\_\{Y^\{\\circ\}\}\-w\_\{j\}\)^\{\\top\}X\}\{\\\|w\_\{Y^\{\\circ\}\}\-w\_\{j\}\\\|\}\.Under \(R1\)–\(R4\),δ∗​\(X,Y∘\)≥0\\delta^\{\*\}\(X,Y^\{\\circ\}\)\\geq 0almost surely, and near zero it is the Euclidean distance to the nearest active readout facet\. Non\-active pairwise hyperplanes and triple intersections contribute onlyo​\(ε\)o\(\\varepsilon\)mass\. Hence

¶​\[0≤δ∗​\(X,Y∘\)≤ε\]\\displaystyle\\P\\\!\\left\[0\\leq\\delta^\{\*\}\(X,Y^\{\\circ\}\)\\leq\\varepsilon\\right\]=2​ε​∑i<j∫Fi​jρX​𝑑ℋd−1\+o​\(ε\)\\displaystyle=2\\varepsilon\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\\,d\\mathcal\{H\}^\{d\-1\}\+o\(\\varepsilon\)=2​ε​AW\+o​\(ε\)\.\\displaystyle=2\\varepsilon A\_\{W\}\+o\(\\varepsilon\)\.The factor22is important: the margin distribution sees both sides of each active facet, one fromΩi\\Omega\_\{i\}and one fromΩj\\Omega\_\{j\}\. Therefore the right\-density of the margin at zero is

fδ∗​\(0\+\):=limε↓0¶​\[0≤δ∗​\(X,Y∘\)≤ε\]ε=2​AW\.f\_\{\\delta^\{\*\}\}\(0\+\):=\\lim\_\{\\varepsilon\\downarrow 0\}\\frac\{\\P\[0\\leq\\delta^\{\*\}\(X,Y^\{\\circ\}\)\\leq\\varepsilon\]\}\{\\varepsilon\}=2A\_\{W\}\.Combining \(C\.18\) and \(C\.26\), the predicted posterior\-mean flip rate at roll\-in timetit\_\{i\}is

predicted​\_​flip\(ti\)=s​\(ti\)2​πfδ∗\(0\+\)\+o\(s\(ti\)\)\.\\boxed\{\\operatorname\{predicted\\\_flip\}\(t\_\{i\}\)=\\frac\{s\(t\_\{i\}\)\}\{\\sqrt\{2\\pi\}\}\\,f\_\{\\delta^\{\*\}\}\(0\+\)\+o\(s\(t\_\{i\}\)\)\.\}Equivalently,

predicted​\_​flip⁡\(ti\)=2π​s​\(ti\)​AW\+o​\(s​\(ti\)\)\.\\operatorname\{predicted\\\_flip\}\(t\_\{i\}\)=\\sqrt\{\\frac\{2\}\{\\pi\}\}\\,s\(t\_\{i\}\)\\,A\_\{W\}\+o\(s\(t\_\{i\}\)\)\.
Given clean latentsX1,…,XNX\_\{1\},\\ldots,X\_\{N\}, set

Yn∘:=DW​\(Xn\),δn:=δ∗​\(Xn,Yn∘\)\.Y\_\{n\}^\{\\circ\}:=D\_\{W\}\(X\_\{n\}\),\\qquad\\delta\_\{n\}:=\\delta^\{\*\}\(X\_\{n\},Y\_\{n\}^\{\\circ\}\)\.For a small bandwidthε\\varepsilonin the linear part of the empirical margin cdf, estimate

f^δ∗​\(0\+;ε\):=1N​ε​∑n=1N𝟏​\{0≤δn≤ε\},A^W​\(ε\):=12​f^δ∗​\(0\+;ε\)\.\\widehat\{f\}\_\{\\delta^\{\*\}\}\(0\+;\\varepsilon\):=\\frac\{1\}\{N\\varepsilon\}\\sum\_\{n=1\}^\{N\}\\mathbf\{1\}\\\{0\\leq\\delta\_\{n\}\\leq\\varepsilon\\\},\\qquad\\widehat\{A\}\_\{W\}\(\\varepsilon\):=\\frac\{1\}\{2\}\\,\\widehat\{f\}\_\{\\delta^\{\*\}\}\(0\+;\\varepsilon\)\.Then the plotted prediction should be

flip^​\(ti\)=s​\(ti\)2​π​f^δ∗​\(0\+;ε\)=2π​s​\(ti\)​A^W​\(ε\)\.\\widehat\{\\operatorname\{flip\}\}\(t\_\{i\}\)=\\frac\{s\(t\_\{i\}\)\}\{\\sqrt\{2\\pi\}\}\\,\\widehat\{f\}\_\{\\delta^\{\*\}\}\(0\+;\\varepsilon\)=\\sqrt\{\\frac\{2\}\{\\pi\}\}\\,s\(t\_\{i\}\)\\,\\widehat\{A\}\_\{W\}\(\\varepsilon\)\.The bandwidthε\\varepsilonshould be chosen by a stability plot of¶​\[0≤δ∗≤ε\]/ε\\P\[0\\leq\\delta^\{\*\}\\leq\\varepsilon\]/\\varepsilonover a small\-margin range; the theorem predicts a plateau asε↓0\\varepsilon\\downarrow 0\.

Equivalently, using an independentG∼N​\(0,1\)G\\sim N\(0,1\),

12​¶​\[δ∗​\(X,Y∘\)≤s​\|G\|\]=12​fδ∗​\(0\+\)​s​𝔼​\|G\|\+o​\(s\)=s2​π​fδ∗​\(0\+\)\+o​\(s\)\.\\frac\{1\}\{2\}\\,\\P\\\!\\left\[\\delta^\{\*\}\(X,Y^\{\\circ\}\)\\leq s\|G\|\\right\]=\\frac\{1\}\{2\}\\,f\_\{\\delta^\{\*\}\}\(0\+\)\\,s\\,\\mathbb\{E\}\|G\|\+o\(s\)=\\frac\{s\}\{\\sqrt\{2\\pi\}\}\\,f\_\{\\delta^\{\*\}\}\(0\+\)\+o\(s\)\.The factor1/21/2in \(C\.31\) corrects for the fact that the empirical margin cdf is two\-sided across each active facet, whereas the active\-facet surface densityAWA\_\{W\}counts each facet once\.

#### Anisotropic residual correction \(the realized constant\)\.

The predictor \(C\.27\)–\(C\.31\) assumes the terminal residual is the isotropic forward noise,X−ms⋆​\(Zs\)=s​εX\-m\_\{s\}^\{\\star\}\(Z\_\{s\}\)=s\\,\\varepsilonwithε∼N​\(0,I\)\\varepsilon\\sim N\(0,I\), wherems⋆=𝔼​\[X∣Zs\]m\_\{s\}^\{\\star\}=\\mathbb\{E\}\[X\\mid Z\_\{s\}\]is the exact posterior mean\. A learnedKK\-step ODE produces a terminal mapm~K,s\\widetilde\{m\}\_\{K,s\}and a*structured*residualRs:=X−m~K,s​\(Zs\)=s​ΞsR\_\{s\}:=X\-\\widetilde\{m\}\_\{K,s\}\(Z\_\{s\}\)=s\\,\\Xi\_\{s\}whose normalized lawΞs\\Xi\_\{s\}need not be isotropic\. Repeating the one\-dimensional Laplace expansion of the active\-facet tube withΞs\\Xi\_\{s\}in place ofε\\varepsilonreplaces the universal normal moment𝔼​\|G\|=2/π\\mathbb\{E\}\|G\|=\\sqrt\{2/\\pi\}by the realized active\-normal moment\. Flipping is a*one\-sided*boundary crossing, so with unit normaln^i​j=\(wi−wj\)/‖wi−wj‖\\widehat\{n\}\_\{ij\}=\(w\_\{i\}\-w\_\{j\}\)/\\\|w\_\{i\}\-w\_\{j\}\\\|oriented from celljjinto celliithe leading flip mass uses the one\-sided positive\-part moments

μi​j\(x\)=μi​j\+\(x\)\+μi​j−\(x\),μi​j±\(x\)=limη↓0𝔼\[\(±⟨Ξs,n^i​j⟩\)\+\|X=x±ηn^i​j\],\\mu\_\{ij\}\(x\)=\\mu\_\{ij\}^\{\+\}\(x\)\+\\mu\_\{ij\}^\{\-\}\(x\),\\qquad\\mu\_\{ij\}^\{\\pm\}\(x\)=\\lim\_\{\\eta\\downarrow 0\}\\mathbb\{E\}\\\!\\left\[\\bigl\(\\pm\\langle\\Xi\_\{s\},\\widehat\{n\}\_\{ij\}\\rangle\\bigr\)\_\{\+\}\\,\\middle\|\\,X=x\\pm\\eta\\widehat\{n\}\_\{ij\}\\right\],¶​\(Y^s≠Y∘\)=s​∑i<j∫Fi​jρX​\(x\)​μi​j​\(x\)​𝑑ℋd−1​\(x\)\+o​\(s\)=2π​s​AW​caniso\+o​\(s\),\\P\(\\widehat\{Y\}\_\{s\}\\neq Y^\{\\circ\}\)=s\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\(x\)\\,\\mu\_\{ij\}\(x\)\\,d\\mathcal\{H\}^\{d\-1\}\(x\)\+o\(s\)=\\sqrt\{\\tfrac\{2\}\{\\pi\}\}\\,s\\,A\_\{W\}\\,c\_\{\\rm aniso\}\+o\(s\),caniso:=∑i<j∫Fi​jρX​μi​j​𝑑ℋd−12/π​∑i<j∫Fi​jρX​𝑑ℋd−1\.c\_\{\\rm aniso\}:=\\frac\{\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\\,\\mu\_\{ij\}\\,d\\mathcal\{H\}^\{d\-1\}\}\{\\sqrt\{2/\\pi\}\\,\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\\,d\\mathcal\{H\}^\{d\-1\}\}\.When the residual law is continuous across the facet with no one\-sided sign asymmetry, this reduces to the symmetric\-trace formμi​j​\(x\)=𝔼​\[\|⟨Ξs,n^i​j⟩\|∣X=x\]\\mu\_\{ij\}\(x\)=\\mathbb\{E\}\[\\,\|\\langle\\Xi\_\{s\},\\widehat\{n\}\_\{ij\}\\rangle\|\\mid X=x\]\. The isotropic predictor \(C\.28\) is the further special caseΞs⇒N​\(0,I\)\\Xi\_\{s\}\\Rightarrow N\(0,I\), for whichμi​j≡2/π\\mu\_\{ij\}\\equiv\\sqrt\{2/\\pi\}andcaniso=1c\_\{\\rm aniso\}=1; equivalently the effective normal scale isseff=caniso​ss\_\{\\rm eff\}=c\_\{\\rm aniso\}\\,s\. Herecanisoc\_\{\\rm aniso\}is the active\-facet weighted normalL1L^\{1\}scale of the realized residual,*not*its total norm and*not*the row\-space energy fraction\.

#### Direct estimator ofcanisoc\_\{\\rm aniso\}\.

With signed boundary distancegi​j​\(x\)=\(ℓi​\(x\)−ℓj​\(x\)\)/‖wi−wj‖g\_\{ij\}\(x\)=\(\\ell\_\{i\}\(x\)\-\\ell\_\{j\}\(x\)\)/\\\|w\_\{i\}\-w\_\{j\}\\\|whereℓk=wk⊤​x\\ell\_\{k\}=w\_\{k\}^\{\\top\}x, the indicatorai​j​\(x\)a\_\{ij\}\(x\)thati,ji,jare the two largest logits atxx, and a kernelKδK\_\{\\delta\},

c^aniso​\(δ\)=∑n∑i<jai​j​\(Xn\)​Kδ​\(gi​j​\(Xn\)\)​\|⟨Rs,n,n^i​j⟩\|/s2/π​∑n∑i<jai​j​\(Xn\)​Kδ​\(gi​j​\(Xn\)\)\.\\widehat\{c\}\_\{\\rm aniso\}\(\\delta\)=\\frac\{\\sum\_\{n\}\\sum\_\{i<j\}a\_\{ij\}\(X\_\{n\}\)\\,K\_\{\\delta\}\(g\_\{ij\}\(X\_\{n\}\)\)\\,\\bigl\|\\langle R\_\{s,n\},\\widehat\{n\}\_\{ij\}\\rangle\\bigr\|/s\}\{\\sqrt\{2/\\pi\}\\,\\sum\_\{n\}\\sum\_\{i<j\}a\_\{ij\}\(X\_\{n\}\)\\,K\_\{\\delta\}\(g\_\{ij\}\(X\_\{n\}\)\)\}\.Equation \(C\.34\) is thes↓0s\\downarrow 0asymptotic estimator; at the finite effective noise of the roll\-in \(s≈2\.6s\\approx 2\.6–5\.25\.2\) the first\-moment identity is not yet tight, so we report the empirical anisotropy as the observed/predicted ratiocaniso=obs/\(2/π​s​AW\)≈2\.3c\_\{\\rm aniso\}=\\mathrm\{obs\}/\(\\sqrt\{2/\\pi\}\\,s\\,A\_\{W\}\)\\approx 2\.3, whose denominator uses the boundary densityAW=fδ∗​\(0\+\)/2A\_\{W\}=f\_\{\\delta^\{\*\}\}\(0^\{\+\}\)/2estimated*independently*from the clean margin distribution \(C\.29\)–\(C\.30\): the factor is a ratio of independent measurements, not a fitted intercept\. The collapse of theK=4,8,16K=4,8,16flip curves as functions ofs​\(t\)s\(t\)is exactly the statement that the normalized active\-normal residual law⟨RK,t,n^i​j⟩/s​\(t\)\\langle R\_\{K,t\},\\widehat\{n\}\_\{ij\}\\rangle/s\(t\)is approximatelyKK\-invariant: changingKKchanges the scalar scales​\(t\)s\(t\), not the anisotropic shape, so the three curves share the single multipliercanisoc\_\{\\rm aniso\}\.

#### The flip multiplier is not DABI\.

The constantcanisoc\_\{\\rm aniso\}is a boundary\-local, clipped,0/10/1crossing functional: to first order¶​\(flip\)\\P\(\\mathrm\{flip\}\)depends on𝔼​\[\(⟨Rs,n^i​j⟩\)\+\]\\mathbb\{E\}\[\(\\langle R\_\{s\},\\widehat\{n\}\_\{ij\}\\rangle\)\_\{\+\}\], which saturates once the prediction has crossed the facet\. The decoder\-amplification indexDABI\\mathrm\{DABI\}is a different functional of the same residual, the cross\-entropy increase𝔼​\[ℓCE​\(X−Rs,Y\)−ℓCE​\(X,Y\)\]\\mathbb\{E\}\[\\ell\_\{\\rm CE\}\(X\-R\_\{s\},Y\)\-\\ell\_\{\\rm CE\}\(X,Y\)\]relative to a norm\-matched isotropic perturbation\. It is not clipped at the first crossing \(wrong\-side logit gap keeps growing after a flip\), and the linear CE term of a symmetric isotropic baseline cancels in expectation, whereas a residual aligned with the CE gradient produces a first\-order increase\. HenceDABI\\mathrm\{DABI\}\(45\.7×45\.7\\timesin CE on ELF\) can be far larger than the flip multipliercanisoc\_\{\\rm aniso\}\(≈2\.3\\approx 2\.3\):DABI\\mathrm\{DABI\}certifies that the terminal residual is highly structured, but the flip\-rate tube constant iscanisoc\_\{\\rm aniso\}, notDABI\\mathrm\{DABI\}\.

#### Scope beyond the ideal assumptions\.

The clean coefficient2/π​AW\\sqrt\{2/\\pi\}\\,A\_\{W\}uses exact clean decoding, smooth density, regular boundaries, and exact posterior\-mean denoising; the first\-order law itself is more robust\.*\(i\) Label noise*is already handled in \(C\.23\)–\(C\.24\): it contributes anss\-independent floorη0\\eta\_\{0\}and can reweight the slope through boundary label gaps, but does not create the linear\-in\-ssterm; under a one\-sided boundary gapπDW​\(x\)​\(x\)−πj​\(x\)≥κ\>0\\pi\_\{D\_\{W\}\(x\)\}\(x\)\-\\pi\_\{j\}\(x\)\\geq\\kappa\>0the noisy slope is at leastκ\\kappatimes the clean slope\.*\(ii\) Non\-smooth density\.*Smoothness can be weakened to a weighted finite\-perimeter condition: theO​\(s\)O\(s\)law holds withFi​jF\_\{ij\}replaced by the reduced boundary∂∗Ωi∩∂∗Ωj\\partial^\{\*\}\\Omega\_\{i\}\\cap\\partial^\{\*\}\\Omega\_\{j\}andρX\\rho\_\{X\}by its approximate trace, provided∑i<j∫ρX∗​𝔼​\|⟨Ξ,n^i​j⟩\|​𝑑ℋd−1<∞\\sum\_\{i<j\}\\int\\rho\_\{X\}^\{\*\}\\,\\mathbb\{E\}\|\\langle\\Xi,\\widehat\{n\}\_\{ij\}\\rangle\|\\,d\\mathcal\{H\}^\{d\-1\}<\\infty; heavy tails away from the boundary are harmless\. The coefficient changes, and the rate can beo​\(s\)o\(s\)if the trace vanishes on the boundary, or have a different exponent under a non\-integrable boundary singularity\.*\(iii\) Singular strata\.*Codimension\-≥2\\geq 2nonsmooth strata and triple intersections havess\-tube volumeO​\(s2\)O\(s^\{2\}\), negligible against theO​\(s\)O\(s\)codimension\-one facets; only*positive\-mass*tie sets genuinely break the law and require an explicit tie\-break model\.*\(iv\) Approximate denoiser\.*The proof needs only first\-order active\-normal residual moments, not the exact posterior mean: ifRs/sR\_\{s\}/shas stable active\-facet normalL1L^\{1\}limits, \(C\.32\) applies with those limits\. Writinges=ms−𝔼​\[X∣Zs\]e\_\{s\}=m\_\{s\}\-\\mathbb\{E\}\[X\\mid Z\_\{s\}\], the posterior\-mean approximation is negligible when∑i<j∫Fi​jρX​𝔼​\[\|⟨es,n^i​j⟩\|/s\]→0\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\\,\\mathbb\{E\}\[\|\\langle e\_\{s\},\\widehat\{n\}\_\{ij\}\\rangle\|/s\]\\to 0; a finite nonzero limit merely modifiescanisoc\_\{\\rm aniso\}, while anω​\(s\)\\omega\(s\)normal error or a nonvanishing normal bias ats=0s=0breaks the linear law\. The observed collapse of theK=4,8,16K=4,8,16curves is direct evidence for the weaker condition the theorem actually needs: a stableO​\(s​\(t\)\)O\(s\(t\)\)active\-normal terminal residual\.

*Summary of scope\.*The lower\-bound mechanism is stable under label noise with a positive boundary label gap, weighted finite\-perimeter boundaries, codimension\-≥2\\geq 2singular strata, and approximate denoisers withO​\(s\)O\(s\)active\-normal error; these change the first\-order coefficient \(through label weights, density traces, or anisotropic moments\) but preserve the linear\-in\-sslaw\. The theorem no longer controls the leading order whenPXP\_\{X\}has atoms or positive mass on tie boundaries, when the boundary Minkowski content is not linear inss, or when the learned terminal residual has a normal component larger thanO​\(s\)O\(s\)\.

#### Observable remainder form \(finite\-sample readout\)\.

The asymptotic coefficient in \(C\.27\)–\(C\.31\) is controlled by a single observable: the failure of the empirical small\-margin cdf to be linear at the origin\. WithFδ∗​\(r\):=¶​\[0≤δ∗​\(X,Y∘\)≤r\]F\_\{\\delta^\{\*\}\}\(r\):=\\P\[0\\leq\\delta^\{\*\}\(X,Y^\{\\circ\}\)\\leq r\]and the linearization remainder

Rδ∗​\(r\):=Fδ∗​\(r\)−2​AW​r,R\_\{\\delta^\{\*\}\}\(r\):=F\_\{\\delta^\{\*\}\}\(r\)\-2A\_\{W\}\\,r,the isotropic active\-facet expansion \(C\.25\)–\(C\.31\) gives the exact one\-step identity¶​\(DW​\(ms⋆​\(Zs\)\)≠Y∘\)=12​𝔼​\[Fδ∗​\(s​\|G\|\)\]\+o​\(s\)\\P\(D\_\{W\}\(m\_\{s\}^\{\\star\}\(Z\_\{s\}\)\)\\neq Y^\{\\circ\}\)=\\tfrac\{1\}\{2\}\\,\\mathbb\{E\}\[F\_\{\\delta^\{\*\}\}\(s\|G\|\)\]\+o\(s\)withG∼N​\(0,1\)G\\sim N\(0,1\), and hence the explicit remainder bound

\|¶​\(DW​\(ms⋆​\(Zs\)\)≠Y∘\)−2π​s​AW\|≤12​𝔼​\[\|Rδ∗​\(s​\|G\|\)\|\]\+o​\(s\)\.\\Bigl\|\\,\\P\\\!\\left\(D\_\{W\}\(m\_\{s\}^\{\\star\}\(Z\_\{s\}\)\)\\neq Y^\{\\circ\}\\right\)\-\\sqrt\{\\tfrac\{2\}\{\\pi\}\}\\,sA\_\{W\}\\,\\Bigr\|\\leq\\tfrac\{1\}\{2\}\\,\\mathbb\{E\}\\\!\\left\[\\,\\bigl\|R\_\{\\delta^\{\*\}\}\(s\|G\|\)\\bigr\|\\,\\right\]\+o\(s\)\.The approximation error is thus*exactly*the non\-linearity of the measured margin cdf near zero: under \(R2\)Rδ∗​\(r\)=o​\(r\)R\_\{\\delta^\{\*\}\}\(r\)=o\(r\), and if the boundary tube content isO​\(r2\)O\(r^\{2\}\)the remainder in \(C\.36\) isO​\(s2\)O\(s^\{2\}\)\. In this sense the prediction is finite\-sample observable \(the estimatorA^W​\(ε\)\\widehat\{A\}\_\{W\}\(\\varepsilon\)and the empirical curvature ofF^δ∗\\widehat\{F\}\_\{\\delta^\{\*\}\}bound their own error\), and we report bootstrap intervals over sampled positions as a descriptive, not i\.i\.d\., summary, since text positions are correlated \(we deliberately avoid DKW/Hoeffding rates, which would import an i\.i\.d\. assumption the analysis does not need\)\. For raw labels the uniform bound\|¶​\(Y^s≠Yraw\)−¶​\(Y^s≠Y∘\)\|≤η0\|\\P\(\\widehat\{Y\}\_\{s\}\\neq Y^\{\\rm raw\}\)\-\\P\(\\widehat\{Y\}\_\{s\}\\neq Y^\{\\circ\}\)\|\\leq\\eta\_\{0\}withη0:=¶​\(Yraw≠Y∘\)\\eta\_\{0\}:=\\P\(Y^\{\\rm raw\}\\neq Y^\{\\circ\}\)is sharp without assumptions on where clean\-decoding errors lie\.

#### Approximate terminal denoisers \(learned generators\)\.

Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)is stated for the exact posterior meanms⋆=𝔼​\[X∣Zs\]m\_\{s\}^\{\\star\}=\\mathbb\{E\}\[X\\mid Z\_\{s\}\]; the following isolates the weaker hypothesis a*learned*terminal denoiser must satisfy, so the law is not tied to training convergence\.

###### Proposition 16\(approximate\-denoiser tube law\)\.

Letm~s\\widetilde\{m\}\_\{s\}be any deterministic terminal denoiser,R~s:=X−m~s​\(Zs\)\\widetilde\{R\}\_\{s\}:=X\-\\widetilde\{m\}\_\{s\}\(Z\_\{s\}\), and suppose the one\-sided active\-normal tracesμ~i​j±​\(x\)=lims↓0limη↓0𝔼​\[\(±⟨R~s,n^i​j⟩/s\)\+∣X=x±η​n^i​j\]\\widetilde\{\\mu\}\_\{ij\}^\{\\pm\}\(x\)=\\lim\_\{s\\downarrow 0\}\\lim\_\{\\eta\\downarrow 0\}\\mathbb\{E\}\[\(\\pm\\langle\\widetilde\{R\}\_\{s\},\\widehat\{n\}\_\{ij\}\\rangle/s\)\_\{\+\}\\mid X=x\\pm\\eta\\widehat\{n\}\_\{ij\}\]exist inL1​\(ρX​d​ℋd−1\)L^\{1\}\(\\rho\_\{X\}\\,d\\mathcal\{H\}^\{d\-1\}\)on the active facets\. Then

¶​\(DW​\(m~s​\(Zs\)\)≠Y∘\)=s​∑i<j∫Fi​jρX​\(x\)​\[μ~i​j\+​\(x\)\+μ~i​j−​\(x\)\]​𝑑ℋd−1​\(x\)\+o​\(s\)\.\\P\\\!\\left\(D\_\{W\}\(\\widetilde\{m\}\_\{s\}\(Z\_\{s\}\)\)\\neq Y^\{\\circ\}\\right\)=s\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\(x\)\\bigl\[\\widetilde\{\\mu\}\_\{ij\}^\{\+\}\(x\)\+\\widetilde\{\\mu\}\_\{ij\}^\{\-\}\(x\)\\bigr\]\\,d\\mathcal\{H\}^\{d\-1\}\(x\)\+o\(s\)\.Ifes:=m~s−ms⋆e\_\{s\}:=\\widetilde\{m\}\_\{s\}\-m\_\{s\}^\{\\star\}has vanishing active\-normal projection,∑i<j∫Fi​jρX​𝔼​\[\|⟨es,n^i​j⟩\|/s∣X→x\]​𝑑ℋd−1→0\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\\,\\mathbb\{E\}\[\|\\langle e\_\{s\},\\widehat\{n\}\_\{ij\}\\rangle\|/s\\mid X\\to x\]\\,d\\mathcal\{H\}^\{d\-1\}\\to 0, the exact coefficient2/π​AW\\sqrt\{2/\\pi\}\\,A\_\{W\}is unchanged; a finite nonzero limit leaves the exponent at one and rescales only the first\-order constant \(thecanisoc\_\{\\rm aniso\}of \(C\.33\)\); anω​\(s\)\\omega\(s\)normal error breaks the linear law\.

The proof is the Laplace expansion already used for \(C\.32\), withμ~±\\widetilde\{\\mu\}^\{\\pm\}replacing the isotropic moment\. The roll\-in collapse forK=4,8,16K=4,8,16\(Figure[3](https://arxiv.org/html/2606.30705#S4.F3)\) is therefore not invoked as a proof that training drivesm~K,s→ms⋆\\widetilde\{m\}\_\{K,s\}\\to m\_\{s\}^\{\\star\}; it is direct evidence for the weaker hypothesis Proposition[16](https://arxiv.org/html/2606.30705#Thmtheorem16)needs: that the normalized active\-normal residuals​\(t\)−1​⟨R~K,t,n^i​j⟩s\(t\)^\{\-1\}\\langle\\widetilde\{R\}\_\{K,t\},\\widehat\{n\}\_\{ij\}\\ranglehas a stable active\-facetL1L^\{1\}law acrossKK\. Convergence of the learned denoiser under realistic optimization, finite data, or distillation is outside our scope: the population law is posterior\-mean, and a learned system is covered exactly when its terminal active\-normal residual satisfies the displayed condition\.

#### Boundary\-concentrated label noise\.

Thess\-independent floorη0=¶​\(Yraw≠Y∘\)\\eta\_\{0\}=\\P\(Y^\{\\rm raw\}\\neq Y^\{\\circ\}\)bounds the gap between raw\-label and clean\-readout error uniformly, but it does not describe how label noise*at the boundary*reshapes the slope\. Letβi​j​\(x\)∈\[0,1\]\\beta\_\{ij\}\(x\)\\in\[0,1\]be the local cross\-facet probability that the raw label disagrees acrossFi​jF\_\{ij\}\. Under symmetric active\-normal residual traces, the raw\-label error keeps the linear\-in\-ssform with a reweighted slope,

ℰsraw=η0\+2π​s​∑i<j∫Fi​jρX​\(x\)​\(1−2​βi​j​\(x\)\)​𝑑ℋd−1​\(x\)\+o​\(s\)\.\\mathcal\{E\}\_\{s\}^\{\\rm raw\}=\\eta\_\{0\}\+\\sqrt\{\\tfrac\{2\}\{\\pi\}\}\\,s\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\(x\)\\,\\bigl\(1\-2\\beta\_\{ij\}\(x\)\\bigr\)\\,d\\mathcal\{H\}^\{d\-1\}\(x\)\+o\(s\)\.Boundary label noise thus attenuates the raw\-label slope by the measurable factor1−2​βi​j1\-2\\beta\_\{ij\}, and nulls it where labels are locally50/5050/50\(βi​j=1/2\\beta\_\{ij\}=1/2\); the clean\-readout flip slope2/π​AW\\sqrt\{2/\\pi\}\\,A\_\{W\}is unchanged\. If the raw\-label law has no finite boundary trace, or the mislabelledrr\-tube has non\-linear Minkowski content, the raw\-label curve can take a different exponent, while the self\-consistency theorem forY∘=DW​\(X\)Y^\{\\circ\}=D\_\{W\}\(X\)still applies\.

#### Non\-smooth readouts\.

The theorem does not requireDWD\_\{W\}to be smooth\. For a linear argmax readout,DWD\_\{W\}is discontinuous exactly on the codimension\-one facetsFi​jF\_\{ij\}, and the tube law is the first\-order measure of those discontinuity surfaces: regularity is asked of the boundary*measure*, not ofDWD\_\{W\}\. The same holds for nonlinear logitshkh\_\{k\}whose active surfacesFi​j=\{hi=hj=maxkhk,∥∇\(hi−hj\)∥\>0\}F\_\{ij\}=\\\{h\_\{i\}=h\_\{j\}=\\max\_\{k\}h\_\{k\},\\ \\\|\\nabla\(h\_\{i\}\-h\_\{j\}\)\\\|\>0\\\}are countablyC1C^\{1\}\(or form a locally finite\-perimeter partition with finite weighted Minkowski content\), on replacing the unit normaln^i​j=\(wi−wj\)/‖wi−wj‖\\widehat\{n\}\_\{ij\}=\(w\_\{i\}\-w\_\{j\}\)/\\\|w\_\{i\}\-w\_\{j\}\\\|byn^i​j​\(x\)=∇\(hi−hj\)⁡\(x\)/‖∇\(hi−hj\)⁡\(x\)‖\\widehat\{n\}\_\{ij\}\(x\)=\\nabla\(h\_\{i\}\-h\_\{j\}\)\(x\)/\\\|\\nabla\(h\_\{i\}\-h\_\{j\}\)\(x\)\\\|\. Only positive\-mass ties, non\-integrable boundary traces, or codimension\-one tangential contact∇\(hi−hj\)=0\\nabla\(h\_\{i\}\-h\_\{j\}\)=0fall outside the theorem\.

## Appendix EThe Bridge: Readout–Transport Interface Product Law \(Theorem[17](https://arxiv.org/html/2606.30705#Thmtheorem17)\)

Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\(overlapping regime\) and Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5)\(separated regime\) are not the same scalar functional\. This appendix shows they are two different*contractions of one geometric object*: the source\-space readout interface of the composite classifier

F=DW∘T:ℝn→\{1,…,V\},X=T​\(S\),S∼σ\.F=D\_\{W\}\\circ T:\\mathbb\{R\}^\{n\}\\to\\\{1,\\ldots,V\\\},\\qquad X=T\(S\),\\quad S\\sim\\sigma\.The readout boundary massAWA\_\{W\}of Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)is this interface measured with weight11; the separated interface energy𝔍\\mathfrak\{J\}of Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5)is a barrier\-weighted normal\-stretch moment of the same interface\. A coarea identity relates them, with an unavoidable*readout\-normal stretch*ηi​j​\(x\)=‖D​T​\(x\)⊤​ni​j‖\\eta\_\{ij\}\(x\)=\\left\\lVert DT\(x\)^\{\\top\}n\_\{ij\}\\right\\rVert\.

\(a\) overlapping: tube massAWA\_\{W\}Fi​jF\_\{ij\}ρX\\rho\_\{X\}mtm\_\{t\}n^i​j\\widehat\{n\}\_\{ij\}Pr⁡\(flip\)=2/π​s​AW,AW=∑i<j∫Fi​jρX​𝑑ℋd−1\\Pr\(\\mathrm\{flip\}\)=\\sqrt\{2/\\pi\}\\,s\\,A\_\{W\},\\quad A\_\{W\}=\\\!\\sum\_\{i<j\}\\\!\\int\_\{F\_\{ij\}\}\\\!\\rho\_\{X\}\\,d\\mathcal\{H\}^\{d\-1\}\(b\) separated: barrier energy𝔍\\mathfrak\{J\}cic\_\{i\}cjc\_\{j\}κi​j\\kappa\_\{ij\}𝔍=∑i<jκi​j​Pσ​\(Bi,Bj\),Pσ=∫Σ~i​jρ​𝑑ℋn−1\\mathfrak\{J\}=\\\!\\sum\_\{i<j\}\\\!\\kappa\_\{ij\}\\,P\_\{\\sigma\}\(B\_\{i\},B\_\{j\}\),\\quad P\_\{\\sigma\}=\\\!\\int\_\{\\widetilde\{\\Sigma\}\_\{ij\}\}\\\!\\rho\\,d\\mathcal\{H\}^\{n\-1\}Figure 4:One interface, two contractions\(the bridge, Theorem[17](https://arxiv.org/html/2606.30705#Thmtheorem17)\)\. The*same*source interfaceΣ~i​j=∂∗Bi∩∂∗Bj\\widetilde\{\\Sigma\}\_\{ij\}=\\partial^\{\*\}B\_\{i\}\\cap\\partial^\{\*\}B\_\{j\}is read two ways\.\(a\)Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)weights it by11: a posterior\-mean pointmtm\_\{t\}in theO​\(s\)O\(s\)tube around an active readout facetFi​jF\_\{ij\}flips to the wrong token with rate2/π​s​AW\\sqrt\{2/\\pi\}\\,sA\_\{W\}\.\(b\)Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5)weights the same interface by the barrierκi​j\\kappa\_\{ij\}and the normal stretch, giving the separation energy𝔍\\mathfrak\{J\}\. The coarea identity \([6](https://arxiv.org/html/2606.30705#A5.E6)\) links the two throughηi​j=‖D​T⊤​ni​j‖\\eta\_\{ij\}=\\left\\lVert DT^\{\\top\}n\_\{ij\}\\right\\rVert; no single readout\-blind scalar reproduces both \(Proposition[18](https://arxiv.org/html/2606.30705#Thmtheorem18)\)\.#### Setup and notation\.

Letσ=ρ​d​x\\sigma=\\rho\\,dxonℝn\\mathbb\{R\}^\{n\}withρ∈C1\\rho\\in C^\{1\},ρ\>0\\rho\>0\. LetT:ℝn→ℝdT:\\mathbb\{R\}^\{n\}\\to\\mathbb\{R\}^\{d\}beC1C^\{1\}in a neighborhood of the source\-space readout interfaces\. Write the argmax cellsΩi=\{z:DW​\(z\)=i\}\\Omega\_\{i\}=\\\{z:D\_\{W\}\(z\)=i\\\}, the active co\-maximal facetsFi​j=∂Ωi∩∂ΩjF\_\{ij\}=\\partial\\Omega\_\{i\}\\cap\\partial\\Omega\_\{j\}, and the unit target normalsni​j=\(wi−wj\)/‖wi−wj‖n\_\{ij\}=\(w\_\{i\}\-w\_\{j\}\)/\\left\\lVert w\_\{i\}\-w\_\{j\}\\right\\rVert\(oriented fromΩj\\Omega\_\{j\}intoΩi\\Omega\_\{i\}\)\. Define the composite phases and source interfaces

Bi:=T−1​\(Ωi\),Σ~i​j:=∂∗Bi∩∂∗Bj∩T−1​\(Fi​j\),ηi​j​\(x\):=‖D​T​\(x\)⊤​ni​j‖\.B\_\{i\}:=T^\{\-1\}\(\\Omega\_\{i\}\),\\qquad\\widetilde\{\\Sigma\}\_\{ij\}:=\\partial^\{\*\}B\_\{i\}\\cap\\partial^\{\*\}B\_\{j\}\\cap T^\{\-1\}\(F\_\{ij\}\),\\qquad\\eta\_\{ij\}\(x\):=\\left\\lVert DT\(x\)^\{\\top\}n\_\{ij\}\\right\\rVert\.LetAW=∑i<j∫Fi​jρX​𝑑ℋd−1A\_\{W\}=\\sum\_\{i<j\}\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\\,d\\mathcal\{H\}^\{d\-1\}be the readout boundary mass of Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\(hereρX\\rho\_\{X\}is the density ofT​σ\#T\{\}\_\{\\\#\}\\sigmanear the active boundary\), and letκmax​\(C\)=maxi<j⁡κp​\(i,j\)\\kappa\_\{\\max\}\(C\)=\\max\_\{i<j\}\\kappa\_\{p\}\(i,j\)\.

#### Assumptions\.

1. \(B1\)Alignment and finite\-energy partition\.The vocabulary is identified with the codebook \(V=MV=Mafter relabeling, each readout cellΩi\\Omega\_\{i\}carrying atomcic\_\{i\}\); the composite partitionB=\(Bi\)i=1MB=\(B\_\{i\}\)\_\{i=1\}^\{M\}withBi=T−1​\(Ωi\)B\_\{i\}=T^\{\-1\}\(\\Omega\_\{i\}\)is a Caccioppoli partition withσ​\(Bi\)=πi\>0\\sigma\(B\_\{i\}\)=\\pi\_\{i\}\>0and finite weighted interface energyEp,σ​\(B;C\)<∞E\_\{p,\\sigma\}\(B;C\)<\\infty\.
2. \(B2\)Nondegenerate pushforward and Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)regularity\.T∈C1T\\in C^\{1\}withrank⁡D​T​\(x\)=d\\operatorname\{rank\}DT\(x\)=dforσ\\sigma\-a\.e\.xxin a neighborhood of the active source interfaces \(in particularn≥dn\\geq d\), so thatT​σ\#=ρX​d​zT\{\}\_\{\\\#\}\\sigma=\\rho\_\{X\}\\,dzthere withρX\\rho\_\{X\}the density appearing in Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3); moreoverX=T​\(S\)X=T\(S\)andWWsatisfy all hypotheses of Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3),ρX\\rho\_\{X\}has finite normal traces on the active facets,∫Σ~i​jρ/ηi​j​𝑑ℋn−1<∞\\int\_\{\\widetilde\{\\Sigma\}\_\{ij\}\}\\rho/\\eta\_\{ij\}\\,d\\mathcal\{H\}^\{n\-1\}<\\infty, and for each active pair the normalized source mass of the active normal slab of half\-widthrroverΣ~i​j\\widetilde\{\\Sigma\}\_\{ij\}converges asr↓0r\\downarrow 0to∫Σ~i​jρ/ηi​j​𝑑ℋn−1\\int\_\{\\widetilde\{\\Sigma\}\_\{ij\}\}\\rho/\\eta\_\{ij\}\\,d\\mathcal\{H\}^\{n\-1\}\(every facet point is a Lebesgue point of the sliced density\)\.
3. \(B3\)Transversality\.0<ηi​j​\(x\)≤L⟂<∞0<\\eta\_\{ij\}\(x\)\\leq L\_\{\\perp\}<\\inftyforℋn−1\\mathcal\{H\}^\{n\-1\}\-a\.e\.x∈Σ~i​jx\\in\\widetilde\{\\Sigma\}\_\{ij\}, where L⟂=L⟂​\(T,W\):=maxi<j​ess​supx∈Σ~i​j⁡‖D​T​\(x\)⊤​ni​j‖≤Lip⁡\(T\);L\_\{\\perp\}=L\_\{\\perp\}\(T,W\):=\\max\_\{i<j\}\\operatorname\*\{ess\\,sup\}\_\{x\\in\\widetilde\{\\Sigma\}\_\{ij\}\}\\left\\lVert DT\(x\)^\{\\top\}n\_\{ij\}\\right\\rVert\\;\\leq\\;\\operatorname\{Lip\}\(T\);for theKK\-step form eachLip⁡\(Tk\)<∞\\operatorname\{Lip\}\(T\_\{k\}\)<\\infty\.
4. \(B4\)Interface exhaustion\.For every active pair,∂∗Bi∩∂∗Bj⊆T−1​\(Fi​j\)\\partial^\{\*\}B\_\{i\}\\cap\\partial^\{\*\}B\_\{j\}\\subseteq T^\{\-1\}\(F\_\{ij\}\)up toℋn−1\\mathcal\{H\}^\{n\-1\}\-null sets \(no off\-facet or inactive\-tie interface mass\), and the active triple skeleton carries noℋn−1\\mathcal\{H\}^\{n\-1\}interface mass\. The posterior\-mean/MAP switching agreement of Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)holds forXX\(Appendix[D](https://arxiv.org/html/2606.30705#A4), equations \(C\.12\)/\(C\.21\)\)\.

###### Theorem 17\(readout–transport interface product law\)\.

Under\(B1\)–\(B4\):

1. \(i\)Coarea transfer\.The readout boundary mass is the normal\-speed\-discounted source perimeter, AW=∑i<j∫Σ~i​jρ​\(x\)‖D​T​\(x\)⊤​ni​j‖​𝑑ℋn−1​\(x\)=∑i<j𝔪i​jT,W​\(Σ~i​j\),A\_\{W\}=\\sum\_\{i<j\}\\int\_\{\\widetilde\{\\Sigma\}\_\{ij\}\}\\frac\{\\rho\(x\)\}\{\\left\\lVert DT\(x\)^\{\\top\}n\_\{ij\}\\right\\rVert\}\\,d\\mathcal\{H\}^\{n\-1\}\(x\)=\\sum\_\{i<j\}\\mathfrak\{m\}\_\{ij\}^\{T,W\}\(\\widetilde\{\\Sigma\}\_\{ij\}\),\(6\)whered​𝔪i​jT,W:=ρ​‖D​T⊤​ni​j‖−1​d​ℋn−1d\\mathfrak\{m\}\_\{ij\}^\{T,W\}:=\\rho\\,\\left\\lVert DT^\{\\top\}n\_\{ij\}\\right\\rVert^\{\-1\}\\,d\\mathcal\{H\}^\{n\-1\}onΣ~i​j\\widetilde\{\\Sigma\}\_\{ij\}is the*normal\-stretch readout interface measure*\.
2. \(ii\)Same interface, two moments\.The source perimeter and the separated interface energy are weighted moments of the same measure, Pσ​\(Bi,Bj\)\\displaystyle P\_\{\\sigma\}\(B\_\{i\},B\_\{j\}\)=∫Σ~i​jρ​𝑑ℋn−1=∫Σ~i​j‖D​T⊤​ni​j‖​𝑑𝔪i​jT,W,\\displaystyle=\\int\_\{\\widetilde\{\\Sigma\}\_\{ij\}\}\\rho\\,d\\mathcal\{H\}^\{n\-1\}=\\int\_\{\\widetilde\{\\Sigma\}\_\{ij\}\}\\left\\lVert DT^\{\\top\}n\_\{ij\}\\right\\rVert\\,d\\mathfrak\{m\}\_\{ij\}^\{T,W\},\(7\)Ep,σ​\(B;C\)\\displaystyle E\_\{p,\\sigma\}\(B;C\)=∑i<jκi​j​Pσ​\(Bi,Bj\)\.\\displaystyle=\\sum\_\{i<j\}\\kappa\_\{ij\}P\_\{\\sigma\}\(B\_\{i\},B\_\{j\}\)\.Consequently 𝔍p,σ​\(π;C\)≤Ep,σ​\(B;C\)≤κmax​\(C\)​L⟂​AW,i\.e\.AW≥𝔍p,σ​\(π;C\)κmax​\(C\)​L⟂\.\\mathfrak\{J\}\_\{p,\\sigma\}\(\\pi;C\)\\;\\leq\\;E\_\{p,\\sigma\}\(B;C\)\\;\\leq\\;\\kappa\_\{\\max\}\(C\)\\,L\_\{\\perp\}\\,A\_\{W\},\\qquad\\text\{i\.e\.\}\\qquad A\_\{W\}\\;\\geq\\;\\frac\{\\mathfrak\{J\}\_\{p,\\sigma\}\(\\pi;C\)\}\{\\kappa\_\{\\max\}\(C\)\\,L\_\{\\perp\}\}\.\(8\)
3. \(iii\)Product law\.WithY^s=DW​\(ms​\(X\+s​ε\)\)\\widehat\{Y\}\_\{s\}=D\_\{W\}\(m\_\{s\}\(X\+s\\varepsilon\)\),Y=DW​\(X\)Y=D\_\{W\}\(X\), Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)givesℙ​\(Y^s≠Y\)=2/π​s​AW\+o​\(s\)\\mathbb\{P\}\(\\widehat\{Y\}\_\{s\}\\neq Y\)=\\sqrt\{2/\\pi\}\\,s\\,A\_\{W\}\+o\(s\), hence lim infs↓0L⟂s​ℙ​\(Y^s≠Y\)≥2π​𝔍p,σ​\(π;C\)κmax​\(C\)\.\\liminf\_\{s\\downarrow 0\}\\frac\{L\_\{\\perp\}\}\{s\}\\,\\mathbb\{P\}\(\\widehat\{Y\}\_\{s\}\\neq Y\)\\;\\geq\\;\\sqrt\{\\frac\{2\}\{\\pi\}\}\\,\\frac\{\\mathfrak\{J\}\_\{p,\\sigma\}\(\\pi;C\)\}\{\\kappa\_\{\\max\}\(C\)\}\.\(9\)For aKK\-step deterministic generatorT=TK∘⋯∘T1T=T\_\{K\}\\circ\\cdots\\circ T\_\{1\},L⟂≤Lip⁡\(T\)≤∏kLip⁡\(Tk\)L\_\{\\perp\}\\leq\\operatorname\{Lip\}\(T\)\\leq\\prod\_\{k\}\\operatorname\{Lip\}\(T\_\{k\}\), so lim infs↓0∏k=1KLip⁡\(Tk\)s​ℙ​\(Y^s≠Y\)≥2π​𝔍p,σ​\(π;C\)κmax​\(C\)\.\\liminf\_\{s\\downarrow 0\}\\frac\{\\prod\_\{k=1\}^\{K\}\\operatorname\{Lip\}\(T\_\{k\}\)\}\{s\}\\,\\mathbb\{P\}\(\\widehat\{Y\}\_\{s\}\\neq Y\)\\;\\geq\\;\\sqrt\{\\frac\{2\}\{\\pi\}\}\\,\\frac\{\\mathfrak\{J\}\_\{p,\\sigma\}\(\\pi;C\)\}\{\\kappa\_\{\\max\}\(C\)\}\.\(10\)

###### Proof\.

*Step 1 \(coarea through the readout facet\)\.*Fix an active facetFi​jF\_\{ij\}, written locally as the zero set of the affineϕi​j​\(z\)=ni​j⊤​z−bi​j\\phi\_\{ij\}\(z\)=n\_\{ij\}^\{\\top\}z\-b\_\{ij\}, and setgi​j​\(x\):=ϕi​j​\(T​\(x\)\)=ni​j⊤​T​\(x\)−bi​jg\_\{ij\}\(x\):=\\phi\_\{ij\}\(T\(x\)\)=n\_\{ij\}^\{\\top\}T\(x\)\-b\_\{ij\}\. Then∇gi​j=D​T⊤​ni​j\\nabla g\_\{ij\}=DT^\{\\top\}n\_\{ij\}, so by \(B3\)\|∇gi​j​\(x\)\|=ηi​j​\(x\)\>0\\left\\lvert\\nabla g\_\{ij\}\(x\)\\right\\rvert=\\eta\_\{ij\}\(x\)\>0onΣ~i​j\\widetilde\{\\Sigma\}\_\{ij\}\. Forr\>0r\>0let

Ui​j,r:=\{y\+t​ni​j:y∈Fi​j,dist⁡\(y,∂Fi​j\)\>r,\|t\|≤r\}⊂ℝdU\_\{ij,r\}:=\\\{\\,y\+t\\,n\_\{ij\}:\\ y\\in F\_\{ij\},\\ \\operatorname\{dist\}\(y,\\partial F\_\{ij\}\)\>r,\\ \\left\\lvert t\\right\\rvert\\leq r\\,\\\}\\subset\\mathbb\{R\}^\{d\}be thedd\-dimensional normal prism of half\-widthrrover the relative interior ofFi​jF\_\{ij\}\(away from the codimension\-two skeleton\); its preimageT−1​\(Ui​j,r\)T^\{\-1\}\(U\_\{ij,r\}\)is the*active*normal slab overΣ~i​j\\widetilde\{\\Sigma\}\_\{ij\}, a subset of\{\|gi​j\|≤r\}\\\{\\left\\lvert g\_\{ij\}\\right\\rvert\\leq r\\\}trimmed to the active facet\. By the pushforward identity \(B2\),

∫Ui​j,rρX​𝑑z=\(T​σ\#\)​\(Ui​j,r\)=σ​\(T−1​\(Ui​j,r\)\)=∫T−1​\(Ui​j,r\)ρ​𝑑x\.\\int\_\{U\_\{ij,r\}\}\\rho\_\{X\}\\,dz\\;=\\;\(T\{\}\_\{\\\#\}\\sigma\)\(U\_\{ij,r\}\)\\;=\\;\\sigma\\big\(T^\{\-1\}\(U\_\{ij,r\}\)\\big\)\\;=\\;\\int\_\{T^\{\-1\}\(U\_\{ij,r\}\)\}\\rho\\,dx\.Divide by2​r2rand letr↓0r\\downarrow 0\. The target\-side normal\-trace tube formula yields∫Fi​jρX​𝑑ℋd−1\\int\_\{F\_\{ij\}\}\\rho\_\{X\}\\,d\\mathcal\{H\}^\{d\-1\}; the source\-side active\-prism average converges to∫Σ~i​jρ/ηi​j​𝑑ℋn−1\\int\_\{\\widetilde\{\\Sigma\}\_\{ij\}\}\\rho/\\eta\_\{ij\}\\,d\\mathcal\{H\}^\{n\-1\}\(coarea forgi​jg\_\{ij\}on the trimmed prism, with\|∇gi​j\|=ηi​j\\left\\lvert\\nabla g\_\{ij\}\\right\\rvert=\\eta\_\{ij\}, the convergence and finiteness being part of \(B2\)\); the skeleton tube contributeso​\(1\)o\(1\)by \(B4\)\. Summing over active pairs gives \([6](https://arxiv.org/html/2606.30705#A5.E6)\)\.

*Step 2 \(compare to source perimeter\)\.*By \(B4\) the reduced interface∂∗Bi∩∂∗Bj\\partial^\{\*\}B\_\{i\}\\cap\\partial^\{\*\}B\_\{j\}agrees withΣ~i​j\\widetilde\{\\Sigma\}\_\{ij\}up toℋn−1\\mathcal\{H\}^\{n\-1\}\-null sets, soPσ​\(Bi,Bj\)=∫Σ~i​jρ​𝑑ℋn−1=∫Σ~i​jηi​j​𝑑𝔪i​jT,WP\_\{\\sigma\}\(B\_\{i\},B\_\{j\}\)=\\int\_\{\\widetilde\{\\Sigma\}\_\{ij\}\}\\rho\\,d\\mathcal\{H\}^\{n\-1\}=\\int\_\{\\widetilde\{\\Sigma\}\_\{ij\}\}\\eta\_\{ij\}\\,d\\mathfrak\{m\}\_\{ij\}^\{T,W\}, which is \([7](https://arxiv.org/html/2606.30705#A5.E7)\)\. Sinceηi​j≤L⟂\\eta\_\{ij\}\\leq L\_\{\\perp\},∑i<jPσ​\(Bi,Bj\)≤L⟂​AW\\sum\_\{i<j\}P\_\{\\sigma\}\(B\_\{i\},B\_\{j\}\)\\leq L\_\{\\perp\}A\_\{W\}\.

*Step 3 \(insert interface energy\)\.*Ep,σ​\(B;C\)=∑i<jκi​j​Pσ​\(Bi,Bj\)≤κmax​∑i<jPσ​\(Bi,Bj\)≤κmax​L⟂​AWE\_\{p,\\sigma\}\(B;C\)=\\sum\_\{i<j\}\\kappa\_\{ij\}P\_\{\\sigma\}\(B\_\{i\},B\_\{j\}\)\\leq\\kappa\_\{\\max\}\\sum\_\{i<j\}P\_\{\\sigma\}\(B\_\{i\},B\_\{j\}\)\\leq\\kappa\_\{\\max\}L\_\{\\perp\}A\_\{W\}\. Because𝔍\\mathfrak\{J\}is the infimum ofEp,σE\_\{p,\\sigma\}over Caccioppoli partitions with massesπ\\pi\(andBBis one such partition by \(B1\)\),𝔍≤Ep,σ​\(B;C\)\\mathfrak\{J\}\\leq E\_\{p,\\sigma\}\(B;C\), giving \([8](https://arxiv.org/html/2606.30705#A5.E8)\)\.

*Step 4 \(insert Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\)\.*The active\-facet tube law \(Appendix[D](https://arxiv.org/html/2606.30705#A4)\) givesℙ​\(Y^s≠Y\)=2/π​s​AW\+o​\(s\)\\mathbb\{P\}\(\\widehat\{Y\}\_\{s\}\\neq Y\)=\\sqrt\{2/\\pi\}\\,s\\,A\_\{W\}\+o\(s\), with the MAP/posterior\-mean agreement supplied by \(B4\)\. Combining with \([8](https://arxiv.org/html/2606.30705#A5.E8)\) and dividing bys/L⟂s/L\_\{\\perp\}gives \([9](https://arxiv.org/html/2606.30705#A5.E9)\)\.*Step 5*substitutesL⟂≤∏kLip⁡\(Tk\)L\_\{\\perp\}\\leq\\prod\_\{k\}\\operatorname\{Lip\}\(T\_\{k\}\)for aKK\-step composition, yielding \([10](https://arxiv.org/html/2606.30705#A5.E10)\)\. ∎

#### Reading the bridge\.

AWA\_\{W\}is*not*the source perimeter: it is the source perimeter discounted pointwise by the readout\-normal speedηi​j=‖D​T⊤​ni​j‖\\eta\_\{ij\}=\\left\\lVert DT^\{\\top\}n\_\{ij\}\\right\\rVert\. Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)reads this interface with weight11\(noise\-to\-error conversion\); Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5)reads the same interface with the barrier\-weighted stretchκi​j​ηi​j\\kappa\_\{ij\}\\eta\_\{ij\}\(source\-interface/stiffness law\)\. Equation \([10](https://arxiv.org/html/2606.30705#A5.E10)\) is the rigorous form of the accuracy–depth–stiffness tradeoff: to keep the terminal flip rate small, the deterministic stiffness budget∏kLip⁡\(Tk\)\\prod\_\{k\}\\operatorname\{Lip\}\(T\_\{k\}\)must grow at least like𝔍/κmax\\mathfrak\{J\}/\\kappa\_\{\\max\}, which by Theorem[6](https://arxiv.org/html/2606.30705#Thmtheorem6)is≳log⁡M\\gtrsim\\sqrt\{\\log M\}\.

#### Why the naive composition fails\.

A tempting shortcut identifiesAWA\_\{W\}with∑i<jPσ​\(Bi,Bj\)\\sum\_\{i<j\}P\_\{\\sigma\}\(B\_\{i\},B\_\{j\}\)and𝔍\\mathfrak\{J\}withAWA\_\{W\}up to constants\. Both are wrong: the first omits the stretch factorηi​j\\eta\_\{ij\}, and the second ignores that𝔍\\mathfrak\{J\}carries the barrier weightsκi​j\\kappa\_\{ij\}and the codebook scale\. The honest statement is one interface measure, two contractions\. This is made sharp by the following impossibility result, which also explains why no single*scalar*functional can serve\.

###### Proposition 18\(noWW\-blind scalar functional reproduces both regimes\)\.

\(a\)There is no functionalG​\(T\)G\(T\)of the transport map alone \(blind to the readoutWW\) that equalsAWA\_\{W\}for all readouts\.\(b\)There is no functionalH​\(T,W\)H\(T,W\)blind to the codebook scale that equals𝔍\\mathfrak\{J\}for all codebooks\. Hence the two constants are genuinely distinct scalar contractions of the shared interface, not one functional\.

###### Proof\.

*\(a\)*Taken=d=2n=d=2,T=IdT=\\mathrm\{Id\},X∼𝒩​\(0,diag⁡\(a2,b2\)\)X\\sim\\mathcal\{N\}\(0,\\operatorname\{diag\}\(a^\{2\},b^\{2\}\)\)witha≠ba\\neq b, and the two two\-token readouts with decision boundariesΣ\(1\)=\{z1=0\}\\Sigma^\{\(1\)\}=\\\{z\_\{1\}=0\\\}andΣ\(2\)=\{z2=0\}\\Sigma^\{\(2\)\}=\\\{z\_\{2\}=0\\\}\. Both give token masses\(12,12\)\(\\tfrac\{1\}\{2\},\\tfrac\{1\}\{2\}\), soTT,σ\\sigma,π\\piare identical\. But

AW\(1\)=∫\{z1=0\}ρX​𝑑ℋ1=12​π​a,AW\(2\)=∫\{z2=0\}ρX​𝑑ℋ1=12​π​b,A\_\{W^\{\(1\)\}\}=\\int\_\{\\\{z\_\{1\}=0\\\}\}\\rho\_\{X\}\\,d\\mathcal\{H\}^\{1\}=\\frac\{1\}\{\\sqrt\{2\\pi\}\\,a\},\\qquad A\_\{W^\{\(2\)\}\}=\\int\_\{\\\{z\_\{2\}=0\\\}\}\\rho\_\{X\}\\,d\\mathcal\{H\}^\{1\}=\\frac\{1\}\{\\sqrt\{2\\pi\}\\,b\},which differ sincea≠ba\\neq b\. NoWW\-blindG​\(T\)G\(T\)can equal both\.*\(b\)*FixT,W,πT,W,\\piand any source/codebook with𝔍p,σ​\(π;C\)\>0\\mathfrak\{J\}\_\{p,\\sigma\}\(\\pi;C\)\>0\(e\.g\.M≥2M\\geq 2distinct atoms,p≥1p\\geq 1,πi\>0\\pi\_\{i\}\>0, and a source whose weighted isoperimetric profile is bounded below\)\. Scale the codebook byr\>0r\>0: withV\(y\)=dist\(y,rC\)pV\(y\)=\\operatorname\{dist\}\(y,rC\)^\{p\}and the path reparametrizationξ↦r​ξ\\xi\\mapsto r\\xi,κp\(r\)​\(i,j\)=rp\+1​κp\(1\)​\(i,j\)\\kappa\_\{p\}^\{\(r\)\}\(i,j\)=r^\{p\+1\}\\kappa\_\{p\}^\{\(1\)\}\(i,j\), hence𝔍p,σ​\(π;r​C\)=rp\+1​𝔍p,σ​\(π;C\)\\mathfrak\{J\}\_\{p,\\sigma\}\(\\pi;rC\)=r^\{p\+1\}\\mathfrak\{J\}\_\{p,\\sigma\}\(\\pi;C\), whileAWA\_\{W\}is unchanged\. No codebook\-scale\-blindH​\(T,W\)H\(T,W\)can equal both\. ∎

#### Scope\.

Theorem[17](https://arxiv.org/html/2606.30705#Thmtheorem17)requires a single compositeTTthat is simultaneously \(B1\) aligned to the readout and \(B2\)–\(B3\) smooth and transverse near the active boundary with pushforward densityρX\\rho\_\{X\}\. A near\-optimal separated transport for Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5)need not satisfy this \(atomic outputs destroy the smoothρX\\rho\_\{X\}of \(B2\)\), and a smooth overlapping\-regimeTTsatisfying Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)need not minimize𝔍\\mathfrak\{J\}\. The bridge is therefore an*inequality*\([8](https://arxiv.org/html/2606.30705#A5.E8)\) on any aligned transversal composite, tight when the stretchηi​j\\eta\_\{ij\}is constant along the active interface and the barriersκi​j\\kappa\_\{ij\}are equal; it is not an identity betweenAWA\_\{W\}and𝔍\\mathfrak\{J\}\.

#### Approximate alignment \(relaxing \(B4\)\)\.

Interface exhaustion \(B4\) is the premise most likely to fail in practice: a learnedTTmay carry interface mass that does not map to an*active*readout facet\. Suppose this off\-facet interface has finite barrier\-weighted size

Eoff:=∑i<jκi​j​∫\(∂∗Bi∩∂∗Bj\)∖T−1​\(Fi​j\)ρ​𝑑ℋn−1\.E\_\{\\mathrm\{off\}\}:=\\sum\_\{i<j\}\\kappa\_\{ij\}\\int\_\{\(\\partial^\{\*\}B\_\{i\}\\cap\\partial^\{\*\}B\_\{j\}\)\\setminus T^\{\-1\}\(F\_\{ij\}\)\}\\rho\\,d\\mathcal\{H\}^\{n\-1\}\.Then Steps 2–3 of the proof bound only the*aligned*interface byκmax​L⟂​AW\\kappa\_\{\\max\}L\_\{\\perp\}A\_\{W\}, soEp,σ​\(B;C\)≤κmax​\(C\)​L⟂​AW\+EoffE\_\{p,\\sigma\}\(B;C\)\\leq\\kappa\_\{\\max\}\(C\)\\,L\_\{\\perp\}A\_\{W\}\+E\_\{\\mathrm\{off\}\}and, since𝔍p,σ​\(π;C\)≤Ep,σ​\(B;C\)\\mathfrak\{J\}\_\{p,\\sigma\}\(\\pi;C\)\\leq E\_\{p,\\sigma\}\(B;C\),

lim infs↓0L⟂s​ℙ​\(Y^s≠Y\)≥2π​\(𝔍p,σ​\(π;C\)−Eoff\)\+κmax​\(C\)\.\\liminf\_\{s\\downarrow 0\}\\frac\{L\_\{\\perp\}\}\{s\}\\,\\mathbb\{P\}\(\\widehat\{Y\}\_\{s\}\\neq Y\)\\;\\geq\\;\\sqrt\{\\frac\{2\}\{\\pi\}\}\\,\\frac\{\\bigl\(\\mathfrak\{J\}\_\{p,\\sigma\}\(\\pi;C\)\-E\_\{\\mathrm\{off\}\}\\bigr\)\_\{\+\}\}\{\\kappa\_\{\\max\}\(C\)\}\.Loss of alignment therefore weakens the product law only through the explicitly measurable off\-facet energyEoffE\_\{\\mathrm\{off\}\}, reducing to \([9](https://arxiv.org/html/2606.30705#A5.E9)\) whenEoff=0E\_\{\\mathrm\{off\}\}=0\. Likewise a non\-constant stretch enters only through the single scalarL⟂=ess​supi,j,x⁡‖D​T⊤​ni​j‖L\_\{\\perp\}=\\operatorname\*\{ess\\,sup\}\_\{i,j,x\}\\left\\lVert DT^\{\\top\}n\_\{ij\}\\right\\rVertalready isolated in \(B3\)\. We therefore apply \(B1\)–\(B4\) as a*conditional*product law whose failure mode is quantified rather than assumed away: on ELF the roll\-in diagnostics \(Appendix[D](https://arxiv.org/html/2606.30705#A4)\) are read as consistent with bounded transversality and small off\-facet energy at the probed resolution, not as a global certificate that \(B1\)–\(B4\) hold everywhere\.

#### What is measurable on an overlapping codec\.

On a real overlapping text autoencoder,AWA\_\{W\}andcanisoc\_\{\\rm aniso\}are intrinsic readout quantities, but𝔍p,σ​\(π;C\)\\mathfrak\{J\}\_\{p,\\sigma\}\(\\pi;C\)andκmax​\(C\)\\kappa\_\{\\max\}\(C\)are not: they are defined only after one fixes a separated atomic codebookCC, a barrierV\(y\)=dist\(y,C\)pV\(y\)=\\operatorname\{dist\}\(y,C\)^\{p\}, and prescribed massesπ\\pi\. We therefore do not estimate𝔍/κmax\\mathfrak\{J\}/\\kappa\_\{\\max\}on ELF and do not fit a numerical right\-hand side of \([9](https://arxiv.org/html/2606.30705#A5.E9)\) to an overlapping codec; a numerical𝔍/κmax\\mathfrak\{J\}/\\kappa\_\{\\max\}is meaningful only in the separated synthetic setting \(Appendix[G](https://arxiv.org/html/2606.30705#A7)\), whereCCandVVare specified\. What is intrinsic and measurable is the overlapping side together with the conditioning of the bridge map:A^W=12​f^δ∗​\(0\+\)≈0\.03\\widehat\{A\}\_\{W\}=\\tfrac\{1\}\{2\}\\widehat\{f\}\_\{\\delta^\{\*\}\}\(0^\{\+\}\)\\approx 0\.03on ELF, the realizedcanisoc\_\{\\rm aniso\}, and, in principle, the active\-facet transversalityηi​j=‖D​T⊤​n^i​j‖\\eta\_\{ij\}=\\left\\lVert DT^\{\\top\}\\widehat\{n\}\_\{ij\}\\right\\rVert\(a vector–Jacobian product of the few\-step map\), of whichL⟂L\_\{\\perp\}is the essential supremum\. Alignment can fail in two distinct ways: part of the source interface maps to non\-active facets \(the off\-facet termEoffE\_\{\\mathrm\{off\}\}above, itself defined only against a separated abstraction\), or the interface is tangential, soηi​j=0\\eta\_\{ij\}=0and1/ηi​j1/\\eta\_\{ij\}is non\-integrable \(a failure of transversality, not anEoffE\_\{\\mathrm\{off\}\}correction\)\. Equation \([9](https://arxiv.org/html/2606.30705#A5.E9)\) is thus a conditional inequality on any aligned separated abstraction, not an empirical identity to be fitted on an overlapping system\.

## Appendix FDynamics: Propositions and Empirical Confirmation

WithC∼μC\\sim\\mu,Z∼𝒩​\(0,Id\)Z\\sim\\mathcal\{N\}\(0,I\_\{d\}\),Xt=\(1−t\)​Z\+t​CX\_\{t\}=\(1\-t\)Z\+tC, the canonical independent\-coupling velocity isvt​\(x\)=𝔼​\[C−Z∣Xt=x\]=\(mt​\(x\)−x\)/\(1−t\)v\_\{t\}\(x\)=\\mathbb\{E\}\[C\-Z\\mid X\_\{t\}=x\]=\(m\_\{t\}\(x\)\-x\)/\(1\-t\)with posteriorwi∝exp⁡\(λt​⟨x,ci⟩\)w\_\{i\}\\propto\\exp\(\\lambda\_\{t\}\\langle x,c\_\{i\}\\rangle\)for equal radii,λt=t/\(1−t\)2\\lambda\_\{t\}=t/\(1\-t\)^\{2\}\.

###### Proposition 19\(critical acceleration\)\.

For aΔ\\Delta\-separated codebook in dimensiond=O​\(log⁡M\)d=O\(\\log M\), in the critical\-SNR window whereσt/t∈\[A​Δ/log⁡M,2​A​Δ/log⁡M\]\\sigma\_\{t\}/t\\in\[A\\Delta/\\sqrt\{\\log M\},2A\\Delta/\\sqrt\{\\log M\}\]\(A≥2A\\geq 2\),mmse⁡\(C∣Xt\)≥c​Δ2\\operatorname\{mmse\}\(C\\mid X\_\{t\}\)\\geq c\\Delta^\{2\}, hence𝔼​tr⁡Cov⁡\(C−Z∣Xt\)≳log⁡M/t2\\mathbb\{E\}\\operatorname\{tr\}\\operatorname\{Cov\}\(C\-Z\\mid X\_\{t\}\)\\gtrsim\\log M/t^\{2\}, and∫ℐcrit‖at‖L2​\(pt\)​𝑑t≳log⁡M\.\\int\_\{\\mathcal\{I\}\_\{\\mathrm\{crit\}\}\}\\left\\lVert a\_\{t\}\\right\\rVert\_\{L^\{2\}\(p\_\{t\}\)\}\\,dt\\gtrsim\\sqrt\{\\log M\}\.

###### Proposition 20\(Euler local barrier\)\.

W2​\(\(I\+h​vt\)​pt\#,pt\+h\)≳h2​log⁡M/ΔW\_\{2\}\(\(I\+hv\_\{t\}\)\{\}\_\{\\\#\}p\_\{t\},p\_\{t\+h\}\)\\gtrsim h^\{2\}\\log M/\\Deltaon the critical window\.

###### Proposition 21\(two\-step stiffness escape\)\.

For a centered, transitive, equal\-radius codebook, the two\-step scheme0→1−h→10\\to 1\-h\\to 1reaches the target ash↓0h\\downarrow 0withLip≳δ2/h\\operatorname\{Lip\}\\gtrsim\\delta^\{2\}/h\(divergent off\-path stiffness\)\.

###### Corollary 22\(accuracy–depth–stiffness tradeoff\)\.

No terminal lower bound onW2​\(p^K,μ\)W\_\{2\}\(\\hat\{p\}\_\{K\},\\mu\)depends on\(M,Δ,K\)\(M,\\Delta,K\)alone\. WithF=FK∘⋯∘F1F=F\_\{K\}\\circ\\cdots\\circ F\_\{1\},∏kLip⁡\(Fk\)≳Δp\+1​log⁡M/ηp\\prod\_\{k\}\\operatorname\{Lip\}\(F\_\{k\}\)\\gtrsim\\Delta^\{p\+1\}\\sqrt\{\\log M\}/\\eta^\{p\}, so a per\-step capSSforcesK≥log⁡\(c​Δp\+1​log⁡M/ηp\)/log⁡SK\\geq\\log\(c\\Delta^\{p\+1\}\\sqrt\{\\log M\}/\\eta^\{p\}\)/\\log S\.

## Appendix GSynthetic Verification of Theorems[5](https://arxiv.org/html/2606.30705#Thmtheorem5)–[6](https://arxiv.org/html/2606.30705#Thmtheorem6)

*InterfaceΓ\\Gamma\-limit \(Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5)\)\.*For an explicit equal\-mass point\-well construction inℝ1\\mathbb\{R\}^\{1\}, the optimalW22W\_\{2\}^\{2\}follows the1/Λ1/\\Lambdalaw: the log–log slope ofW22W\_\{2\}^\{2\}against the achieved Lipschitz constant is−0\.97,−0\.95,−0\.93,−0\.90\-0\.97,\-0\.95,\-0\.93,\-0\.90forM=4,8,16,32M=4,8,16,32\(Figure[5](https://arxiv.org/html/2606.30705#A7.F5)a\), approaching the predicted−1\-1asΛ\\Lambdagrows\. At the converged scale the renormalized ratioΛ⋅W22/\[3​\(2​ln⁡2−1\)​𝔍\]\\Lambda\\cdot W\_\{2\}^\{2\}/\[3\(2\\ln 2\-1\)\\mathfrak\{J\}\]matches the predicted logistic constant3​\(2​ln⁡2−1\)=1\.15893\(2\\ln 2\-1\)=1\.1589to within a few percent\.

*Dimension phase diagram \(Theorem[6](https://arxiv.org/html/2606.30705#Thmtheorem6)\)\.*Atn=256n=256, the interface profile𝒫n,M\\mathcal\{P\}\_\{n,M\}trackslog⁡M\\sqrt\{\\log M\}; at fixedn∈\{1,2,3,4\}n\\in\\\{1,2,3,4\\\}theM1/nM^\{1/n\}growth exponents are1\.14,0\.57,0\.38,0\.2851\.14,0\.57,0\.38,0\.285, matching the predicted1/n1/n\.

![Refer to caption](https://arxiv.org/html/2606.30705v1/x3.png)Figure 5:Synthetic verification of the separated\-regime theorems\.\(a\)The analytic equal\-mass construction realizes the1/Λ1/\\Lambdalaw of Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5)\(log–log slope−0\.90\-0\.90to−0\.97\-0\.97, approaching−1\-1\)\.\(b\)The normalized interface profileH=min⁡\(𝒫\)/log⁡MH=\\min\(\\mathcal\{P\}\)/\\sqrt\{\\log M\}across dimensionnnand mode countMM\.\(c\)At fixedn∈\{1,2,3,4\}n\\in\\\{1,2,3,4\\\}the profile grows asM1/nM^\{1/n\}\.\(d\)Atn=256n=256the profile trackslog⁡M\\sqrt\{\\log M\}, confirming Theorem[6](https://arxiv.org/html/2606.30705#Thmtheorem6)\.
## Appendix HAdmissible Hierarchy: Full Definition and Proof of Theorem[7](https://arxiv.org/html/2606.30705#Thmtheorem7)

###### Definition 23\(admissible categorical hierarchy\)\.

A depth\-BBadmissible representation ofμ=∑ℓπℓ​δcℓ\\mu=\\sum\_\{\\ell\}\\pi\_\{\\ell\}\\delta\_\{c\_\{\\ell\}\}consists of: \(1\) a tree disintegrationπu=πv​πu\|v\\pi\_\{u\}=\\pi\_\{v\}\\pi\_\{u\|v\},πv\>0\\pi\_\{v\}\>0; \(2\) independent fresh source blocksZb∼σbZ\_\{b\}\\sim\\sigma\_\{b\}; \(3\) semantic local transportTvT\_\{v\}with exact conditional phases, charged by Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5); \(4\) metric normalizationα​‖av,u−av,u′‖≤dist⁡\(Cu,Cu′\)≤β​‖av,u−av,u′‖\\alpha\\left\\lVert a\_\{v,u\}\-a\_\{v,u^\{\\prime\}\}\\right\\rVert\\leq\\operatorname\{dist\}\(C\_\{u\},C\_\{u^\{\\prime\}\}\)\\leq\\beta\\left\\lVert a\_\{v,u\}\-a\_\{v,u^\{\\prime\}\}\\right\\rVert; \(5\) immutable prefix commitment; \(6\) decoder with additive sensitivity\.

#### Path\-law induction\.

By induction on depthbb: atb=0b=0,V0=∅V\_\{0\}=\\varnothingandπ∅=1\\pi\_\{\\varnothing\}=1\. Assumingℙ​\(Vb−1=v\)=πv\\mathbb\{P\}\(V\_\{b\-1\}=v\)=\\pi\_\{v\}for allv∈𝒱b−1v\\in\\mathcal\{V\}\_\{b\-1\}, the event\{Vb=u\}\\\{V\_\{b\}=u\\\}requires\{Vb−1=v\}\\\{V\_\{b\-1\}=v\\\}\(wherevvis the parent ofuu\) andTv,Λv​\(Zb\)∈Qv,uT\_\{v,\\Lambda\_\{v\}\}\(Z\_\{b\}\)\\in Q\_\{v,u\}\. By independence ofZbZ\_\{b\}from the prefix and the exact phase constraint:

ℙ​\(Vb=u\)=ℙ​\(Vb−1=v\)⋅σb​\(Tv,Λv−1​\(Qv,u\)\)=πv⋅πu\|v=πu\.\\mathbb\{P\}\(V\_\{b\}=u\)=\\mathbb\{P\}\(V\_\{b\-1\}=v\)\\cdot\\sigma\_\{b\}\(T\_\{v,\\Lambda\_\{v\}\}^\{\-1\}\(Q\_\{v,u\}\)\)=\\pi\_\{v\}\\cdot\\pi\_\{u\|v\}=\\pi\_\{u\}\.HenceX∗=cVB∼μX^\{\*\}=c\_\{V\_\{B\}\}\\sim\\mu\.

#### Upper bound \(i\)\.

By decoder sensitivity \(6\):

‖D𝒗​\(Y1,…,YB\)−cVB‖≤∑bSvb−1​dist⁡\(Yb,𝒜vb−1\)\.\\\|D\_\{\\bm\{v\}\}\(Y\_\{1\},\\ldots,Y\_\{B\}\)\-c\_\{V\_\{B\}\}\\\|\\leq\\textstyle\\sum\_\{b\}S\_\{v\_\{b\-1\}\}\\operatorname\{dist\}\(Y\_\{b\},\\mathcal\{A\}\_\{v\_\{b\-1\}\}\)\.Minkowski’s inequality over rounds givesWp​\(μ^,μ\)≤∑b\(Ab/Λb\)1/pW\_\{p\}\(\\hat\{\\mu\},\\mu\)\\leq\\sum\_\{b\}\(A\_\{b\}/\\Lambda\_\{b\}\)^\{1/p\}whereAb=∑\|v\|=b−1πv​Svp​𝔍vA\_\{b\}=\\sum\_\{\|v\|=b\-1\}\\pi\_\{v\}S\_\{v\}^\{p\}\\mathfrak\{J\}\_\{v\}\.

#### Continuous lower bound \(iii\)\.

Any single continuousΛ\\Lambda\-LipschitzF:ℝN→ℝdF\\\!:\\\!\\mathbb\{R\}^\{N\}\\\!\\to\\\!\\mathbb\{R\}^\{d\}satisfiesΛ⋅Wpp​\(F​σ¯\#,μ\)≥𝔍global\\Lambda\\cdot W\_\{p\}^\{p\}\(F\{\}\_\{\\\#\}\\bar\{\\sigma\},\\mu\)\\geq\\mathfrak\{J\}\_\{\\mathrm\{global\}\}by Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5), and𝔍global≳Δp\+1​log⁡M\\mathfrak\{J\}\_\{\\mathrm\{global\}\}\\gtrsim\\Delta^\{p\+1\}\\sqrt\{\\log M\}by Theorem[6](https://arxiv.org/html/2606.30705#Thmtheorem6)\.

#### Discontinuity \(iv\)\.

By Lemma[4](https://arxiv.org/html/2606.30705#Thmtheorem4): the factorized generatorF​\(z1,…,zB\)=cVB​\(z\)F\(z\_\{1\},\\ldots,z\_\{B\}\)=c\_\{V\_\{B\}\(z\)\}outputs at least two atoms, hence is discontinuous on the connected domainℝn1\+⋯\+nB\\mathbb\{R\}^\{n\_\{1\}\+\\cdots\+n\_\{B\}\}, hence not Lipschitz\. The lower bound \(iii\) does not apply\.

## Appendix IRecent Continuous Few\-Step Text Models: Full Case Analysis

Throughout, a*continuous latent*means a smooth embedding \(nearby latents decode to related tokens, as in ELF\),*not*the categorical probability simplex whose vertices are one\-hot token codes\.

#### Case analysis: FMLM\.

Flow Map Language Models\(Leeet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib2)\)appear to refute us directly: a deterministic flow map that generates text in a single step\. They do not, because FMLM is an instance of our categorical escape rather than smooth continuous transport\. The flow interpolates between Gaussian noise and a*one\-hot*encoding on the vocabulary simplex, and the population\-optimal denoiser output is the posterior probability over tokens followed by anarg​max\\operatorname\*\{arg\\,max\}readout \(their Lemma 3\.1, cross\-entropy trained\)\. The terminal map is thus the discontinuous categorical generator of Lemma[4](https://arxiv.org/html/2606.30705#Thmtheorem4): “continuous denoising” toward one\-hot vertices witharg​max\\operatorname\*\{arg\\,max\}*is*categorical commitment \(CCI=1\\mathrm\{CCI\}=1\), not transport of a smooth latent\. Its reported open\-ended quality \(PPL129129at11step,7676at44steps on OpenWebText\) is consistent with this: fluency comes from the categorical target geometry\. This placement is not merely conceptual\.CCI=1\\mathrm\{CCI\}=1holds by construction \(thearg​max\\operatorname\*\{arg\\,max\}readout of their Lemma 3\.1*is*categorical commitment\), and we measure DABI directly on the official FMLM\-B\-OWT checkpoint\(Leeet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib2)\): under the same margin\-normal probe a boundary\-aligned perturbation flips≈50%\{\\approx\}50\\%of tokens \(\[42%,57%\]\[42\\%,57\\%\]\) versus≈0%\{\\approx\}0\\%for a norm\-matched isotropic one \(label\-DABI3,822×3\{,\}822\\times,95%95\\%CI\[1591,24837\]\[1591,24837\]; the isotropic response is at the measurement floor, soDABI≫104\\mathrm\{DABI\}\\gg 10^\{4\}\)\. This is the same sharp\-readout signature as the AR and masked decoders\. Because FMLM is an encoder\-less one\-hot\-simplex generator, there is no clean\-text reconstruction operating point as for the autoencoder codecs, so the probe is applied around the model’s own predictions; the readout sharpness is a property of the output projection regardless\. We use this only to place FMLM by readout sharpness andCCI\\mathrm\{CCI\},*not*as a reconstruction\-accuracy or realized\-residual test of Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\. The sharp margin\-normal readout coexists with successful one\-step generation precisely because the one\-hot target gives*large*margins \(δ∗=Θ​\(1\)\\delta^\{\*\}=\\Theta\(1\); the decoder\-geometry ablation of Appendix[J](https://arxiv.org/html/2606.30705#A10)\): the realized residual stays in\-cell, so categorical escape lives in the target geometry, not in a non\-sharp readout\.

#### The remaining field\.

Among genuinely smooth\-continuous latents, every deterministic\-ODE system needs*many*steps: LangFlow\(Chenet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib4)\)adopts a deterministic ODE for distillability and pays128128–10241024steps \(its authors note “stochasticity inherently resists flow\-based distillation”\); Cosmos\(Meshchaninovet al\.,[2025](https://arxiv.org/html/2606.30705#bib.bib5)\)runs a deterministic Euler ODE at200200steps with no few\-step results\. Every few\-step continuous system instead injects stochasticity or commits categorically: ELF\(Huet al\.,[2026b](https://arxiv.org/html/2606.30705#bib.bib1)\)reports SDE≫\\ggODE in the few\-step regime; CoLa\-DLM\(Guoet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib3)\)pairs an SDE latent prior with an autoregressive categorical decoder; FastDiSS\(Nguyen\-Conget al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib7)\)uses stochastic ancestral sampling on*conditional*seq2seq with a categorical readout; DiLaDiff\(Lemercieret al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib8)\)consistency\-distills only the latent prior, with text realized by a categorical decoder\. Independently, CoDAR\(Shenet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib6)\)names token rounding “the primary bottleneck” of continuous diffusion LMs and escapes via a categorical autoregressive decoder with a temperature knob, an external rediscovery of the readout\-sharpness mechanism\. Positioned by readout geometry rather than by sampler, these systems are uniform along the dimension the reviewer’s taxonomy cares about: all decode through a*sharp categorical*head \(nearest\-neighbor embedding rounding for Cosmos/LangFlow/FastDiSS, anarg​max\\operatorname\*\{arg\\,max\}/autoregressive head for CoLa\-DLM/DiLaDiff/CoDAR/FMLM\), so all sit at highDABI\\mathrm\{DABI\}and highCCI\\mathrm\{CCI\}\. They differ only in how they avoid the deterministic few\-step regime our bound forbids: many ODE steps \(Cosmos200200, LangFlow128128–10241024\), stochastic re\-injection \(ELF, FastDiSS\), or an autoregressive/categorical spine \(CoLa\-DLM, DiLaDiff, CoDAR, FMLM\)\. This is exactly the choice between the two escape columns of Table[3](https://arxiv.org/html/2606.30705#A9.T3)\.

#### Case analysis: training\-time mitigations \(FastDiSS\)\.

FastDiSS\(Nguyen\-Conget al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib7)\)is a continuous\-latent diffusion language model \(Gaussian diffusion over token embeddings\) with two*training\-time*regularizers: Self\-conditioning Perturbation \(SCP\), which noises the self\-conditioning signal during training so the network is robust to the noisier estimates it sees at inference, and Model\-aware Noise Scaling \(MANS\), which allocates more noise to tokens the model already reconstructs\. Neither changes the readout: discrete tokens are obtained by nearest\-neighbor rounding to the embedding table, i\.e\. categorical commitment\. SCP/MANS are therefore orthogonal to our taxonomy rather than counterexamples: they harden the network against the compounding self\-conditioning \(posterior\-mean\) error that is itself a symptom of the non\-commitment mechanism, while the final output still arrives through a categorical readout\. \(The paper does not state whether its*inference*sampler is deterministic or stochastic; we rely on neither\.\)

#### Case analysis: Loopholing\.

Loopholing Discrete Diffusion\(Joet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib9)\)is a discrete/categorical diffusion model whose subtitle \(“deterministic bypass of the sampling wall”\) invites reading as a deterministic counterexample\. It is not: each denoising step emits*both*a stochastic one\-hot sample*and*a deterministic continuous vector, and the deterministic latent pathway*complements*rather than replaces the categorical spine, carrying soft distributional information across steps \(a self\-conditioning\-style side\-channel\) so that information is not collapsed into one\-hot vectors between steps\. The categorical commitment \(escape \(i\)\) is fully retained; “deterministic” here refers to feature propagation, not to eliminating the categorical sample\. Loopholing thus sits inside the categorical\-escape column, augmented by a continuous side\-channel, consistent with rather than contradicting our analysis\.

Table 3:Recent continuous\-latent text generators by sampler and latent geometry\. The deterministic\-ODE×\\timessmooth\-continuous×\\timesfew\-NFE cell is empty across the surveyed literature\.

## Appendix JExtended Experiments

![Refer to caption](https://arxiv.org/html/2606.30705v1/x4.png)Figure 6:Tube\-law test\.\(a\)Predicted versus observed terminal flip atK∈\{4,8,16\}K\\in\\\{4,8,16\\\}; proportionality≈2\.3×\{\\approx\}2\.3\\times\.\(b\)Margin density at zerof^δ∗​\(0\+\)≈0\.064\\hat\{f\}\_\{\\delta^\{\*\}\}\(0^\{\+\}\)\\approx 0\.064\.#### LangFlow: second\-system replication\.

We repeat the decoder\-sensitivity and few\-step protocols on the published LangFlow OpenWebText checkpoint\(Chenet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib4)\)\(embedding\-space continuous diffusion,d=768d=768, GPT\-2\(Radfordet al\.,[2019](https://arxiv.org/html/2606.30705#bib.bib34)\)rounding readout\), using256256real Wikitext sequences of length128128\(24,96024\{,\}960positions\)\.*DABI\.*Clean decoding is99\.99%99\.99\\%\. At the posterior\-mean residual of the terminal step \(K=3K=3–44\), the structured residual flips5959–69%69\\%of tokens versus≈0\.02%\{\\approx\}0\.02\\%for a norm\-matched isotropic control, a flip ratio of2,600×2\{,\}600\\times\(95%95\\%bootstrap CI\[1,800,3,800\]\[1\{,\}800,3\{,\}800\]\);Δ​CEstruct=40\.5​\[39\.8,41\.1\]\\Delta\\mathrm\{CE\}\_\{\\mathrm\{struct\}\}=40\.5\\,\[39\.8,41\.1\]versus isotropic3\.9×10−43\.9\\times 10^\{\-4\}\. The response is superlinear in the perturbation fraction \(flip rate0\.02%→2\.3%→59%0\.02\\%\\to 2\.3\\%\\to 59\\%atf=0\.25,0\.5,1\.0f=0\.25,0\.5,1\.0forK=4K=4\), matching the ELF onset\.*Few\-step degeneracy\.*Deterministic Euler–EDM sampling is degenerate at low step counts:K=1K=1attains generated\-PPL10\.110\.1only through mode collapse \(entropy1\.71\.7bits, repeated tokens\),K=4K=4produces incoherent text \(PPL1,4011\{,\}401\), andK≤16K\\leq 16produces repetitive fragments; fluency requiresK≥64K\\geq 64\. This is an independent instance of the perplexity\-collapse hazard: perplexity alone is minimized by degenerate outputs, so it must be read alongside entropy and samples\. LangFlow is ODE\-native, so we do not transfer the ELF SDE\-versus\-ODE comparison to it; its escape is step count\.

#### Decoder geometry ablation\.

Fixing the ELF\-B decoder and varying the codebook geometry \(one\-hot FMLM\-style, VQ, smooth ELF\) isolates the role ofδ∗\\delta^\{\*\}distribution\. One\-hot readout: large margins \(δ∗=Θ​\(1\)\\delta^\{\*\}=\\Theta\(1\)\), so the realized residual rarely reaches a boundary and the*realized*\-residualDABI≈1\\mathrm\{DABI\}\\approx 1, even though the margin\-normal readout is itself sharp \(this large\-margin geometry is the categorical escape; cf\. FMLM, Appendix[I](https://arxiv.org/html/2606.30705#A9)\)\. Smooth ELF:δ∗\\delta^\{\*\}has a heavy left tail, so the realized residual concentrates near boundaries andDABI=45\.7×\\mathrm\{DABI\}=45\.7\\times\.

#### Dynamics on the ELF text encoder\.

Integrated acceleration:4\.2×4\.2\\timesisotropic\-Gaussian control, peak58×58\\timesatt≈0\.2t\\approx 0\.2\. Euler local error: peak att≈0\.2t\\approx 0\.2, confirming Proposition[20](https://arxiv.org/html/2606.30705#Thmtheorem20)\.

#### Teacher few\-step decoding\.

Incoherent output forK≤16K\\leq 16; usable atK≥32K\\geq 32–6464; teacher ceiling atK=256K=256\(PPL=36\.1=36\.1, entropy=3\.45=3\.45\)\.

#### Prefix conditioning\.

Conditioning on a clean prefix of up to96/12896/128latent tokens: the suffix acceleration peak is unchanged\. The multimodality is per\-token \(μ0⊗S\\mu\_\{0\}^\{\\otimes S\}\-product\); continuous latent factorization cannot escape\.

#### Autoregressive commit ablation: details\.

Llama\-2 \(7B\) on 200 prompts, max 128 tokens\. Hard\-arg​max\\operatorname\*\{arg\\,max\}: diversityd2=0\.541d\_\{2\}=0\.541, coherent text\. Soft\-carry \(logit distribution, no commitment\):d2=0\.135d\_\{2\}=0\.135, monotonic collapse into repetition\. Delayed commit \(commit at positionbb\):d2d\_\{2\}rises monotonically withbb; collapse onset matches the branching point predicted by Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\. Norm\-matched soft\-carry \(rescaling soft embeddings to match hard embedding norms\): still collapses, confirming the failure is discreteness, not norm\.

#### Masked\-diffusion commit ablation: details\.

LLaDA\-8B, 128 steps\. Hard \(standard\):d2=0\.82d\_\{2\}=0\.82, PPL=2\.4=2\.4\. Soft\-refresh \(carry distribution, no commit, with resampling\):d2=0\.47d\_\{2\}=0\.47\. No\-commit \(carry distribution, no resample\):d2=0\.01d\_\{2\}=0\.01\(immediate collapse\)\. Dream\-7B, 128 steps\. Hard\-sample: PPL=7\.3=7\.3, effective length=99=99, cross\-PPL=7\.3=7\.3\. Soft\-refresh\-sample \(remove only in\-loop commit, keep sampling\): PPL=38\.6=38\.6, effective length=22=22, cross\-PPL=38\.6=38\.6\. Matched\-pair ablation isolates the commit/no\-commit confound from the sample/greedy confound\.*Protocol \(Dream\)\.*Generation length128128;\{128,32,8\}\\\{128,32,8\\\}steps with timestepslinspace​\(1,ϵ,K\+1\)\\mathrm\{linspace\}\(1,\\epsilon,K\{\+\}1\)and globalnumber\_transfer; soft\-distribution top\-k=256k=256; no classifier\-free guidance\. Hard variants commit by greedyarg​max\\operatorname\*\{arg\\,max\}\(temperature0\) at each step;\*\-samplevariants draw from the*same*per\-step categorical distribution \(temperature11\), so the matched pair \(hard\-sample vs\. soft\-refresh\-sample\) holds the sampling law fixed and removes only the in\-loop commit; a norm\-matched soft variant additionally controls for embedding\-norm changes \(all collapse\)\. Cross\-PPL is scored by Qwen\-2\.5\-3B and sequences are truncated at the first EOS to keep pad tokens from contaminating the degeneracy metrics; the decoding path matches Dream’s officialgeneration\_utils\.py\(right\-shifted logits, global number transfer\)\. Table[4](https://arxiv.org/html/2606.30705#A10.T4)collects the three families\.

#### Interventions and metrics\.

The interventions hold the readout and the sampling distribution fixed and toggle only the in\-loop categorical step\.*Hard*and*hard\-sample*commit each step byarg​max\\operatorname\*\{arg\\,max\}\(temperature0\) or by a draw \(temperature11\);*soft\-refresh*carries the expected embedding𝔼p​\[e\]\\mathbb\{E\}\_\{p\}\[e\]instead of committing and recomputes it each step;*no\-commit*carries the soft embedding with neither resampling nor commitment; the*norm\-matched*soft variant rescales the soft embedding to the hard\-embedding norm\. Beyond perplexity and entropy we report three confound controls: the44\-gram repetition rate \(rep\-44\), distinct\-22diversity, and cross\-perplexity under an external Qwen\-2\.5\-3B scorer \(coherence\), together with the commitment fraction CCI\-AUC \(the path average of the hard\-committed share\)\. Removing commitment moves all of them together in the collapse direction: on LLaDA, hard→\{\\to\}soft\-refresh gives rep\-440\.08→0\.470\.08\{\\to\}0\.47, distinct\-220\.82→0\.460\.82\{\\to\}0\.46, cross\-PPL2\.4→18\.72\.4\{\\to\}18\.7; the Dream matched pair gives rep\-440\.01→0\.380\.01\{\\to\}0\.38, distinct\-220\.97→0\.760\.97\{\\to\}0\.76, cross\-PPL7\.3→38\.67\.3\{\\to\}38\.6, effective length99→2299\{\\to\}22, with CCI\-AUC0\.50→00\.50\{\\to\}0by construction\. Generation scripts and per\-condition metrics for all three families are released\.

Table 4:Commit ablations\.Holding the readout fixed and removing in\-loop categorical commitment collapses generation across autoregressive and masked\-diffusion families\. The cleanest control is the Dream matched pair, which removes*only*the in\-loop commit step while keeping the sampling distribution identical\.ModelDecodingCommit variantPPLd2d\_\{2\}eff\. lenverdictLlama\-2\-7BARhardarg​max\\operatorname\*\{arg\\,max\}—0\.5410\.541—coherentLlama\-2\-7BARsoft\-carry \(no commit\)—0\.1350\.135—collapseLlama\-2\-7BARsoft\-carry, norm\-matched—0\.140\.14—collapseLLaDA\-8Bmaskedhard \(standard\)2\.42\.40\.820\.82—coherentLLaDA\-8Bmaskedsoft\-refresh—0\.470\.47—degradesLLaDA\-8Bmaskedno\-commit—0\.010\.01—collapseDream\-7Bmaskedhard\-sample7\.37\.3—9999coherentDream\-7Bmaskedsoft\-refresh\-sample38\.638\.6—2222collapse

## Appendix KDecoder Causal Intervention

We test whether the few\-step failure can be fixed at the readout\. ELF’s flow \(transport\) head and decoder \(readout\) head are separate modules; we finetune*only*the readout \(the post\-transformer unembedding\{\\\{proj, unembed\}\\\},1717M parameters\) on the published checkpoint, holding clean token recovery near96\.7%96\.7\\%with a recovery cross\-entropy term\. Because the transport map is untouched, the*same*few\-step latent is decoded by the original and the retrained readout, isolating the readout’s contribution\. Per\-token unembedding columns are renormalized to their initial norms each step, so a margin objective cannot trivially rescale logits \(a no\-op for the normalized marginδ∗\\delta^\{\*\}and forarg​max\\operatorname\*\{arg\\,max\}decoding\)\. Metrics use a held\-out512512\-position subset, so the absoluteDABI\\mathrm\{DABI\}\(≈450\\approx 450\) is below the508×508\\timesheadline \(32,76832\{,\}768positions\); the*relative*changes are what the test turns on\.

Table 5:Decoder causal intervention\(ELF readout finetuned, transport frozen, clean recovery held\)\. Terminal recon accuracy decodes the controlled posterior\-mean roll\-in latent of Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)to the correct token \(baseline59\.3%=1−0\.4159\.3\\%=1\-0\.41terminal flip atK=4K\{=\}4, matching Figure[3](https://arxiv.org/html/2606.30705#S4.F3)\)\. No intervention recovers the flipped tokens\.*Max\-margin*maximizes the smooth\-max normalized margin\. It raises the median raw margin \(22→7022\\to 70\) but*increases*DABI\\mathrm\{DABI\}\(453→1,469×453\\to 1\{,\}469\\times\) and degrades generation at every step count \(K=4K\{=\}4PPL799→895799\\to 895\): a more confident readout has a*sharper*decision cliff, so a few\-step residual that crosses a boundary incurs a larger cross\-entropy jump, not a smaller one\.

*Residual\-targeted*training instead minimizes the cross\-entropy of decoding the realized few\-step latents to the correct tokens, the most direct “make the readout decode the few\-step latent” objective\. On the controlled terminal posterior\-mean step \(baseline flip41%41\\%\),4,0004\{,\}000steps*collapse*the readout margins \(δ∗:5\.4→3\.0\\delta^\{\*\}:5\.4\\to 3\.0, a much less sharp readout\) yet raise held\-out token accuracy by only1\.21\.2points \(59\.3→60\.5%59\.3\\to 60\.5\\%\)\. On uncoupled few\-step generation, perplexity moves only through mode collapse: the retrained readout’s outputs are degenerate repetitions \(entropy4\.4→3\.24\.4\\to 3\.2, e\.g\./s/s/s…\), so the apparent change is the perplexity\-collapse artifact, not recovery\.

*Nonlinear readout\.*The residual\-targeted test retrains a*linear*readout; to rule out a linearity artifact, we replace it with a22\-layer MLP \(GELU,10241024hidden,3434M parameters\) trained on the identical residual objective and the same frozen terminal hidden\. With strictly more capacity it overfits the training tokens to99\.6%99\.6\\%, yet its held\-out terminal recon is56\.8%56\.8\\%, no better than a linear probe on the same hidden \(57\.8%57\.8\\%\) or the model’s own readout \(59\.9%59\.9\\%\); the gap is the overfitting, not recovery\. Added nonlinearity recovers none of the flipped tokens, which is the information\-theoretic content of Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3): once the terminal step averages over the branch ambiguity, the correct branch is not a measurable function of the terminal latent, so no readout, linear or nonlinear, can recover it on held\-out data\.

The conclusion is consistent across all four interventions \(isotropic smoothing, max\-margin, and linear and nonlinear residual\-retraining\): the readout\-only interventions we tested do not recover the tokens flipped by the deterministic terminal step\. Under the posterior\-mean model, once the terminal step averages over the branch ambiguity and lands in theO​\(s​\(t\)\)O\(s\(t\)\)boundary tube, the branch identity is not present as a deterministic function of the terminal latent, so a readout\-only fix is not expected\. This places the failure on the transport side, with decoder sharpness as its faithful amplifier \(Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\)\.

#### Adversarial/margin training along boundary normals\.

Targeted adversarial or margin training that pushes apart the boundary normals is exactly the max\-margin intervention above: it widens the raw margins \(δ∗:5\.4→6\.3\\delta^\{\*\}\\\!:5\.4\\to 6\.3\) but*raises*DABI\\mathrm\{DABI\}\(453→1,469×453\\to 1\{,\}469\\times\) and degrades generation at every step count, because a more confident readout has a sharper decision cliff, so a boundary\-crossing residual incurs a larger jump\. Reshaping the boundary geometry faces the same wall as the nonlinear\-readout head: once the posterior mean averages over the branch ambiguity, the correct token is not a function of the latent, so no boundary reshaping recovers it without distorting clean decoding \(which collapses margins, the residual\-targeted arm\)\. Decoder\-side margin engineering thus does not reduce few\-step failure without sacrificing diversity or clean accuracy; the leverage is on the transport \(sampler\) side\.

#### Direction and norm sensitivity of DABI\.

The margin\-normal direction is the steepest\-descent direction of the active margin, hence the worst\-case boundary\-aligned perturbation;DABI\\mathrm\{DABI\}is therefore an upper envelope over locally adversarial directions rather than a knife\-edge artifact of one choice\. The response is also robust to the norm calibrationff: the structured curve is monotone with onset nearf≈0\.8f\\approx 0\.8and reaches a0\.860\.86flip rate byf=1f\{=\}1on ELF, while the isotropic curve stays at the floor throughout \(Figure[2](https://arxiv.org/html/2606.30705#S4.F2)\), soffslightly below or above11moves the absolute rate smoothly without changing the orders\-of\-magnitude gap\.

![Refer to caption](https://arxiv.org/html/2606.30705v1/x5.png)Figure 7:DABI image/text dichotomy\(visualizes Table[1](https://arxiv.org/html/2606.30705#S4.T1)\)\. Four continuous\-text codecs under the margin\-normal probe: ELF508×508\\times, Cosmos≈4×104\{\\approx\}4\{\\times\}10^\{4\}, CoLa\-DLM≈4×105\{\\approx\}4\{\\times\}10^\{5\}, LangFlow≫104\{\\gg\}10^\{4\}\(isotropic at floor\)\. Image VAEs \(Lumina\-Next, SANA\-1\.5, Z\-Image, FLUX\.1\), lacking a categorical readout, absorb their realized terminal residual atDABI≈1\\mathrm\{DABI\}\\approx 1\(\[0\.85,1\.94\]\[0\.85,1\.94\]\)\. The clusters are separated by orders of magnitude with no overlap\.
#### Image\-side robustness to the loss metric\.

The imageDABI≈1\\mathrm\{DABI\}\\approx 1is not an artifact of the pixel\-L2L^\{2\}readout loss\. With the learned LPIPS perceptual metric\(Zhanget al\.,[2018](https://arxiv.org/html/2606.30705#bib.bib40)\)\(AlexNet backbone, on the same decoded images\) in place of pixel\-L2L^\{2\}, the realized\-residual probe on Lumina\-Next, SANA\-1\.5, and FLUX\.1 gives structured/random ratios of0\.930\.93–1\.081\.08\(Lumina\),1\.151\.15–1\.251\.25\(SANA\), and0\.660\.66–0\.950\.95\(FLUX\) acrossK∈\{1,2,4,8\}K\\in\\\{1,2,4,8\\\}: all≈1\{\\approx\}1, the same regime as pixel\-L2L^\{2\}\(\[0\.85,1\.94\]\[0\.85,1\.94\]\), with no boundary amplification\. A semantically\-aware metric that up\-weights perceptually salient changes still finds no categorical boundary to align with: the cross\-domain gap is a property of the readout geometry, not of the choice of output\-space loss\.

#### Vector\-quantized image readout \(cross\-domain mechanism check\)\.

A vector\-quantized decoder reads out by nearest\-codebook lookup,code​\(z\)=arg​mink⁡‖z−ek‖2=arg​maxk⁡\(2​ek⊤​z−‖ek‖2\)\\mathrm\{code\}\(z\)=\\operatorname\*\{arg\\,min\}\_\{k\}\\\|z\-e\_\{k\}\\\|^\{2\}=\\operatorname\*\{arg\\,max\}\_\{k\}\(2e\_\{k\}^\{\\top\}z\-\\\|e\_\{k\}\\\|^\{2\}\), a linear\-argmax readout of the same form as a text decoder’sDW​\(z\)=arg​maxy⁡wy⊤​zD\_\{W\}\(z\)=\\operatorname\*\{arg\\,max\}\_\{y\}w\_\{y\}^\{\\top\}z\(herewk=2​ekw\_\{k\}=2e\_\{k\}\)\. We apply the margin\-normal probe directly to the published MoVQGAN codebook \(16,38416\{,\}384codes, latent dimensiond=4d=4\), sampling latents near codebook points\. A boundary\-aligned perturbation flips the assigned code in50\.3%50\.3\\%of cases atf=1f\{=\}1versus0\.3%0\.3\\%for a norm\-matched isotropic one \(flip ratio167×167\\times\), with code cross\-entropyDABI=42×\\mathrm\{DABI\}=42\\times: the same sharp\-readout signature as the text and AR decoders, now in an image model\. Because the VQ latent is only44\-dimensional, the isotropic null space is tiny, so the167×167\\timesgap cannot be attributed to null\-space avoidance; it is boundary\-normal alignment\. A smooth image VAE has no such categorical boundary \(DABI≈1\\mathrm\{DABI\}\\approx 1\); endowing an image decoder with one reproduces the failure signature, confirming the mechanism is the categorical readout, not the modality\.

#### Absolute flip rates \(not only ratios\)\.

The largeDABI\\mathrm\{DABI\}ratios are not ratio\-inflation artifacts of a tiny denominator: the absolute structured flip rates are themselves high while the isotropic ones sit at the floor\. Atκ=1\\kappa\{=\}1the boundary\-aligned perturbation flips85\.9%85\.9\\%of tokens on ELF \(isotropic3\.1%3\.1\\%\),100%100\\%on LLaDA\-8B and Dream\-7B \(isotropic0\.0%0\.0\\%\),≈50%\{\\approx\}50\\%on FMLM \(isotropic≈0%\{\\approx\}0\\%\), and5959–69%69\\%on LangFlow \(isotropic0\.02%0\.02\\%\)\. The conclusion rests on the absolute structured flips being near\-total and the isotropic flips being near\-zero; the ratio only summarizes a gap that is already evident in the raw rates\.

#### Row\-space control \(anisotropy beyond the null space\)\.

The realized residual is≈99%\{\\approx\}99\\%in the decoder null space, so the isotropic control of Figure[2](https://arxiv.org/html/2606.30705#S4.F2)spends most of its norm where the readout is blind\. To isolate anisotropy*within*the row space, we sketch the full cross\-position row spacerow​\(J\)\\mathrm\{row\}\(J\)of the logit JacobianJ=∂\(all logits\)/∂\(all latents\)J=\\partial\(\\text\{all logits\}\)/\\partial\(\\text\{all latents\}\)\(the active\-pair boundary normals at every content position pluskkrandom\-logit\-cotangent VJPs,k∈\{64,256\}k\\in\\\{64,256\\\}; per\-sample SVD basis\) and compare, at matched total norm, the structured margin\-normal push \(boundary\-aligned, lying inrow​\(J\)\\mathrm\{row\}\(J\)\) against a random direction drawn*within*row​\(J\)\\mathrm\{row\}\(J\)\. On4,0964\{,\}096ELF positions the boundary\-aligned push flips46\.9%46\.9\\%of tokens, a norm\-matched random row\-space direction flips3\.2%3\.2\\%, and an isotropic full\-space direction flips0\.9%0\.9\\%\. The structured\-vs\-isotropic ratio \(50×50\\times\) reproduces the realized\-residual DABI \(45\.7×45\.7\\times, a consistency check\); the structured\-vs\-random\-row ratio is14\.5×14\.5\\times\(15\.3×15\.3\\timesatk=64k\{=\}64, so stable in the sketch size\)\. Row\-space membership accounts for only a≈3\.5×\{\\approx\}3\.5\\timesfactor over isotropic; the remaining order of magnitude is specific boundary\-normal alignment, so the damage is anisotropy, not mere row\-space energy\. We work in the first\-order margin\-normal regime, where the row/null decomposition is valid \(the full realized residual is a large, partly nonlinear step\); code is released\.

![Refer to caption](https://arxiv.org/html/2606.30705v1/x6.png)Figure 8:DABI×CCI\\mathrm\{DABI\}\\times\\mathrm\{CCI\}taxonomy\.Image VAEs: lowDABI\\mathrm\{DABI\}, works\. ELF: highDABI\\mathrm\{DABI\},CCI=0\\mathrm\{CCI\}=0, fails\. AR/masked dLM: highDABI\\mathrm\{DABI\}, highCCI\\mathrm\{CCI\}, works\. Arrows: commit\-ablation collapse\.![Refer to caption](https://arxiv.org/html/2606.30705v1/x7.png)Figure 9:γ\\gamma\-sweep: ODE versus SDE\.ELF\-B teacher \(top\) and PD student \(bottom\)\.Left:PPL versusKK; ODE \(γ=0\\gamma=0\) stays high, SDE \(γ\>0\\gamma\>0\) escapes\.Right:Entropy; teacherK=1K=1ODE has PPL3\.073\.07but entropy2\.012\.01\(mode collapse\)\. An independent55\-seed rerun is stable: the ODE/SDE ordering and multilingual\-collapse signature hold across seeds \(per\-cell SD<10%\{<\}10\\%\)\.

## Appendix LExtended Related Work

#### Flow matching and rectified flow\.

Lipmanet al\.\([2023](https://arxiv.org/html/2606.30705#bib.bib18)\)andLiuet al\.\([2023](https://arxiv.org/html/2606.30705#bib.bib19)\)introduced flow matching and rectified flow for learning deterministic transport between distributions\. Consistency models\(Songet al\.,[2023](https://arxiv.org/html/2606.30705#bib.bib16)\)learn single\-step generators via consistency training\. Progressive distillation\(Salimans and Ho,[2022](https://arxiv.org/html/2606.30705#bib.bib17)\)reduces the step count of a pretrained diffusion model\. All achieve few\-step generation of images; none demonstrate few\-step generation of text latents without stochastic sampling\.

#### Lipschitz limitations for generative models\.

Salmonaet al\.\([2022](https://arxiv.org/html/2606.30705#bib.bib25)\)showed that Lipschitz pushforward measures are limited on multimodal targets and that per\-step noise injection can help\. Our Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5)provides the finer multiway, dimension\-sensitive interface law\. The Lipschitz\-constrained perspective also appears in Wasserstein GANs\(Arjovskyet al\.,[2017](https://arxiv.org/html/2606.30705#bib.bib32)\), where the constraint is on the discriminator; here it is on the generator\.

#### Adversarial robustness and margin geometry\.

DABI is, geometrically, a boundary\-aligned sensitivity analysis: it displaces a latent along the readout normalwi−wjw\_\{i\}\-w\_\{j\}and measures how few units flip the decoded token, precisely the small, boundary\-normal perturbations that define adversarial examples\(Szegedyet al\.,[2014](https://arxiv.org/html/2606.30705#bib.bib37); Goodfellowet al\.,[2015](https://arxiv.org/html/2606.30705#bib.bib38)\)and whose worst case PGD\(Madryet al\.,[2018](https://arxiv.org/html/2606.30705#bib.bib39)\)searches for\. That literature studies how to*shrink*this sensitivity \(large\-margin and certified training\); we use it as a diagnostic and in the opposite direction\. A sharp text readout has small normalized margins \(δ∗\\delta^\{\*\}of order the residual scale near the boundary\), so it is*maximally*susceptible to a structured, boundary\-aligned displacement, and the posterior\-mean transport error supplies exactly such a displacement for free, an “adversarial” perturbation produced by the generator rather than by an attacker \(Theorem[3](https://arxiv.org/html/2606.30705#Thmtheorem3)\)\. The image\-decoderDABI≈1\\mathrm\{DABI\}\\approx 1is the same statement read the other way: a smooth decoder with no categorical readout has no low\-margin boundary to attack\. Few\-step text failure is thus the generator inadvertently constructing a near\-worst\-case input for its own readout, which is why decoder\-side margin shaping \(Appendix[K](https://arxiv.org/html/2606.30705#A11)\) does not help: it moves the boundary the residual is already aligned to\.

#### Γ\\Gamma\-convergence and phase transitions\.

The Modica–MortolaΓ\\Gamma\-limit\(Modica and Mortola,[1977](https://arxiv.org/html/2606.30705#bib.bib30)\)and its multiwell generalizations\(Baldo,[1990](https://arxiv.org/html/2606.30705#bib.bib20); Fonseca and Tartar,[1989](https://arxiv.org/html/2606.30705#bib.bib21)\)are the soft\-gradient analogue of our Theorem[5](https://arxiv.org/html/2606.30705#Thmtheorem5)\. The difference is the constraint: we cap the pointwise operator norm of the gradient \(a Lipschitz condition\) rather than penalizing it with a double\-well potential\.

#### Gaussian isoperimetry and Gaussian widths\.

The Gaussian isoperimetric inequality\(Borell,[1975](https://arxiv.org/html/2606.30705#bib.bib23); Sudakov and Tsirelson,[1978](https://arxiv.org/html/2606.30705#bib.bib31)\)and generic chaining\(Talagrand,[2005](https://arxiv.org/html/2606.30705#bib.bib24)\)are the standard tools for our Theorem[6](https://arxiv.org/html/2606.30705#Thmtheorem6)\. The Milman–Neeman multi\-bubble theorem\(Milman and Neeman,[2022](https://arxiv.org/html/2606.30705#bib.bib22)\)gives the exact minimizer whenM≤n\+1M\\leq n\+1; our Fourier\-code construction complements this by handlingn≪Mn\\ll M\.

#### Text generation via continuous latents\.

ELF\(Huet al\.,[2026b](https://arxiv.org/html/2606.30705#bib.bib1)\)encodes text into a continuous latent and generates via flow matching\. FMLM\(Leeet al\.,[2026](https://arxiv.org/html/2606.30705#bib.bib2)\)uses one\-hot encoding witharg​max\\operatorname\*\{arg\\,max\}decoding, an instantiation of categorical escape\. RAE\(Zhenget al\.,[2025](https://arxiv.org/html/2606.30705#bib.bib12)\)uses a frozen semantic encoder with a trained decoder and high\-dimensional latent diffusion at5050NFE, consistent with our finding that many steps are needed for continuous text latents\. The concurrent ELF\+PD\(Huet al\.,[2026a](https://arxiv.org/html/2606.30705#bib.bib29)\)confirms the ODE\-vs\-SDE gap on the same model we study, with the authors explicitly noting that SDE corrects “deterministically amplifying imperfect trajectories\.”

#### Autoregressive and masked diffusion language models\.

Autoregressive models\(Brownet al\.,[2020](https://arxiv.org/html/2606.30705#bib.bib33); Touvronet al\.,[2023](https://arxiv.org/html/2606.30705#bib.bib28)\)generate tokens left\-to\-right with categorical commitment at each position\. Masked diffusion language models\(Sahooet al\.,[2024](https://arxiv.org/html/2606.30705#bib.bib13); Nieet al\.,[2025](https://arxiv.org/html/2606.30705#bib.bib14); Yeet al\.,[2025](https://arxiv.org/html/2606.30705#bib.bib15)\)generate by iteratively remasking and predicting discrete tokens\. Both escape the continuous lower bounds via categorical commitment \(Lemma[4](https://arxiv.org/html/2606.30705#Thmtheorem4)\)\.

#### Margin geometry and interface diagnostics\.

DABI is an interface audit of the decoder: it measures sensitivity along the readout\-normal direction, the same boundary\-aligned geometry studied by margin\-based robustness and Lipschitz\-margin bounds for classifiers\(Szegedyet al\.,[2014](https://arxiv.org/html/2606.30705#bib.bib37); Goodfellowet al\.,[2015](https://arxiv.org/html/2606.30705#bib.bib38); Madryet al\.,[2018](https://arxiv.org/html/2606.30705#bib.bib39)\), where a small normalized margin certifies vulnerability to a boundary\-normal perturbation\. Our contribution is to turn this diagnostic on the*generator*: the transport residual supplies the boundary\-aligned perturbation for free, so a sharp decoder \(small margins\) is exactly the regime a few\-step generator cannot satisfy\. The image/text dichotomy and the VQ control \(Appendix[K](https://arxiv.org/html/2606.30705#A11)\) read this interface across modalities; CCI complements it with the orthogonal commitment axis\.

#### Samplers that re\-inject categorical or endpoint uncertainty\.

Several continuous\-text samplers restore uncertainty during sampling rather than committing deterministically: predictor–corrector and bridge\-style samplers add a corrector or a stochastic bridge step that re\-injects endpoint or categorical noise\. In our taxonomy these are instances of the stochastic re\-injection escape \(Section[5\.2](https://arxiv.org/html/2606.30705#S5.SS2)\): leaving the deterministic transport class is precisely what defeats the lower bound, whether the injection is Gaussian \(SDE\), categorical \(commitment\), or a corrector that re\-randomizes the endpoint\. The bound constrains the deterministic\-continuous corner; every such sampler steps out of it by construction\.