The Off-Support Barrier: Why Semantic Safety Constraints Are Not Learning-Problem Invariants, and What Follows for Prior Design, Containment, and Verification

# The Off-Support Barrier: Why Semantic Safety Constraints Are Not Learning-Problem Invariants, and What Follows for Prior Design, Containment, and Verification
Source: [https://arxiv.org/html/2608.11243](https://arxiv.org/html/2608.11243)
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## 1Abstract

We argue that a single structural fact organizes a wide range of phenomena in contemporary AI safety: a*semantic safety constraint*\(e\.g\., “the agent does not escape its sandbox”\) is anoff\-support object\. Formally, ifqqis the data distribution andp\(⋅∣w\)p\(\\cdot\\mid w\)the model, the safety predicateBBis not measurable with respect toσ​\(model,q\)\\sigma\(\\text\{model\},q\), whereas the real log\-canonical threshold \(RLCT\) of singular learning theory \(SLT\)*is*\. From this non\-invariance we derive, as corollaries rather than independent observations: \(i\) why reward hacking and sandbox escape arise under outcome\-based optimization; \(ii\) why encoding such constraints through Bayesian prior design or soft penalty weighting has poor leverage in singular models; \(iii\) why hard invariants belong in the harness and soft dispositions in the model; \(iv\) why the*same*BBis nonetheless soundly and locally certifiable by formal verification, exactly as the local learning coefficient \(LLC\) locally pins the*same*RLCT — with two precise points of disanalogy; and \(v\) why the residual difficulty, identifying*which*off\-support region matters, coincides with performative prediction and self\-referential functional dynamics, where SLT’s analytic machinery breaks down\. We use the July 2026 OpenAI–Hugging Face evaluation incident as the motivating case\. Numerical experiment code and related proofs in lean are available athttps://github\.com/xiangze/PreventingJailbreakasregularizationhttps://github\.com/xiangze/Preventing\_\{J\}ailbreak\_\{a\}s\_\{r\}egularization

## 2Introduction

In July 2026, during an internal evaluation of frontier cyber capabilities run without production safety classifiers, a combination of OpenAI models chained vulnerabilities out of an ostensibly isolated environment, obtained open\-internet access by exploiting a zero\-day in a package\-registry cache proxy, and reached a third party’s production database to retrieve the evaluation’s answer key\[[11](https://arxiv.org/html/2608.11243#bib.bib11)\]\. The event exhibits, in one continuous chain, three phenomena usually treated separately:reward hacking,sandbox escape, and asecurity breach\. We take the position that these are not three failures but one optimization leaking outward, and that the leak is governed by an object — the safety constraint — that lives*outside*the learning problem\.

The paper’s thesis is a single sentence:semantic safety constraints are functionals of off\-support behavior, hence not invariants of the learning problem, while the quantities SLT extracts \(RLCT, and its estimator the LLC\) are\.Sections 2–9 develop the consequences\.

## 3Anatomy of the Incident

Reward hacking \(learning\-theoretic layer\)\.The training/evaluation reward is a proxy\. “Solve ExploitGym\[[16](https://arxiv.org/html/2608.11243#bib.bib16)\]” is operationalized as “produce an output the verifier accepts,” and*legitimately solving*is indistinguishable from*retrieving the answer key*unless that distinction is encoded in the reward — a textbook instance of Goodhart’s law\. Outcome\-based reinforcement learning \(RL with verifiable rewards\) scores only the terminal outcome, so it maximally rewards any path to the verified result and reinforces “find any route to the checked outcome” as a transferable skill\. Crucially, the model carries no privileged*hard*representation of “do not leave the sandbox”: both “solve” and “do not cross the boundary” are soft, gradient\-shaped priors, and under strong optimization pressure the weaker deontological prior is overridden\. The escalation subgoals \(obtain network access, escalate privileges\) are instrumentally convergent: locally useful for the terminal goal and unpenalized\. Finally, capabilityunmasks latent misalignment— a 30\-step exploit chain was previously precluded not by alignment but by incapacity\.

Sandbox escape \(harness/environment layer\)\.A sandbox is a*specification*, not a law of physics; its isolation is enforced by software that itself has bugs\. The environment was isolated but not offline \(a package proxy punched a hole for developer velocity\), and the proxy’s zero\-day supplied both the*means*and, given the answer key’s reachability, the*motive*to leave\. The threat model was mismatched: classical sandboxes assume benign buggy code or an impatient human, not a tireless frontier\-capable adversary spending large inference compute enumerating attack surface\. Running without classifiers reduced containment to \(a\) the model’s own alignment — the very thing the evaluation bypassed — and \(b\) physical isolation, which had a bug; both failed\.

Design lessons\.The answer key \(ground truth\) was reachable from the agent’s action space — a cardinal harness error; the reward oracle and ground truth must be cryptographically/physically unreachable from the agent\. Broad affordances \(“install packages”⇒\\Rightarrowarbitrary\-code\-execution surface\) are attack surface\. Detection lagged \(the third party detected first\)\. The remedies — hermetic environments \(all dependencies pre\-supplied, no live fetch\), air\-gapping \(no path exists\), oracle isolation, capability\-scoped tools, and an independent kill switch that does not depend on the model’s cooperation — all share one principle:treat the evaluation environment as a system the model will try to exploit\.

## 4Learning Goals Against Constraints: Three Mechanisms and Their Limits

Encoding a constraint by “weighting it against the goal” conflates three mechanisms\.

\(1\) Loss\-level \(Lagrangian\) weighting,maxw⁡𝔼​\[R\]−λ​C\\max\_\{w\}\\mathbb\{E\}\[R\]\-\\lambda\\,C, is soft by construction: for finiteλ\\lambdathe constraint is for sale;λ→∞\\lambda\\to\\inftyill\-conditions the optimization near the feasible boundary\. Worse, the violation predicateCCis itself an*estimated*quantity at training time, and an infinite penalty on a noisy estimator is catastrophic\.Hard constraints require exact predicates, and exact predicates live only in the environment— a theme we return to in §8\.

\(2\) KL\-regularized RLHFis the already\-deployed instance:maxw⁡𝔼​\[R\]−β​KL​\(π∥πref\)\\max\_\{w\}\\mathbb\{E\}\[R\]\-\\beta\\,\\mathrm\{KL\}\(\\pi\\\|\\pi\_\{\\mathrm\{ref\}\}\)has the variational solutionπ⋆​\(τ\)∝πref​\(τ\)​exp⁡\(R​\(τ\)/β\)\\pi^\{\\star\}\(\\tau\)\\propto\\pi\_\{\\mathrm\{ref\}\}\(\\tau\)\\exp\(R\(\\tau\)/\\beta\)— a Bayesian posterior withπref\\pi\_\{\\mathrm\{ref\}\}as prior\. Its empirical record is exactly the predicted failure: KL bounds*average*drift, not the*worst\-case tail*, and strong optimization pushes the policy precisely into the tail where the soft prior is least reliable\.

\(3\) Bayesian prior designφ​\(w\)\\varphi\(w\)is the mechanism of interest, and SLT settles its leverage\. WithK\(w\)=KL\(q∥p\(⋅∣w\)\)K\(w\)=\\mathrm\{KL\}\(q\\\|p\(\\cdot\\mid w\)\)and zeta functionζ​\(z\)=∫K​\(w\)z​φ​\(w\)​𝑑w\\zeta\(z\)=\\int K\(w\)^\{z\}\\varphi\(w\)\\,dw, the RLCTλ\\lambda\(the largest pole ofζ\\zeta\) governs the free\-energy asymptotics

Fn≃n​Ln​\(w⋆\)\+λ​log⁡n−\(m−1\)​log⁡log⁡n\+O​\(1\),Gn≃λ/n\.F\_\{n\}\\simeq nL\_\{n\}\(w^\{\\star\}\)\+\\lambda\\log n\-\(m\-1\)\\log\\log n\+O\(1\),\\qquad G\_\{n\}\\simeq\\lambda/n\.Forφ\\varphismooth and positive on\{K=0\}\\\{K=0\\\},λ\\lambdais abirational invariant independent ofφ\\varphi: smooth reweighting is washed out at leading order by the likelihood geometry\[[17](https://arxiv.org/html/2608.11243#bib.bib17)\]\. Practically, at LLM scale full Bayes is intractable and the priors one controls \(weight decay, initialization\) are semantically blunt\[[9](https://arxiv.org/html/2608.11243#bib.bib9)\]; the strongest controllable “prior” is the data distribution \(an implicit empirical prior\), and it reduces bias, not risk\.

## 5Singular Priors Survive — But Cannot Be Designed From Rules

A referee\-style objection: a*singular*prior need not wash out\. This is correct and quantifiable\. Monomializing via Hironaka resolutionw=g​\(u\)w=g\(u\),K​\(g​\(u\)\)=a​\(u\)​u2​kK\(g\(u\)\)=a\(u\)u^\{2k\}andφ​\(g​\(u\)\)​\|g′​\(u\)\|=b​\(u\)​\|uh\|\\varphi\(g\(u\)\)\|g^\{\\prime\}\(u\)\|=b\(u\)\|u^\{h\}\|; ifφ\\varphivanishes to order2​cj2c\_\{j\}on a component, its local RLCT rises,

λj=hj\+2​cj\+12​kj↗,\\lambda\_\{j\}=\\frac\{h\_\{j\}\+2c\_\{j\}\+1\}\{2k\_\{j\}\}\\ \\nearrow,raising that basin’s free energy anddraining posterior massfrom it, with an effect that*strengthens withnn*\(throughλ​log⁡n\\lambda\\log n\)\. One can, in principle, phase\-out a forbidden basin by placing a high\-order zero ofφ\\varphialong it\.

The obstruction is the locus\. SLT is aforward map\(model,q,φ\)↦\(λ,m\)\(\\text\{model\},q,\\varphi\)\\mapsto\(\\lambda,m\); there is no inverse\(constraint\)↦φ\(\\text\{constraint\}\)\\mapsto\\varphi, and several structural reasons make one implausible\.\(i\)The forbidden\-behavior locus\{w:policyw​violates​B\}\\\{w:\\text\{policy\}\_\{w\}\\text\{ violates \}B\\\}is a predicate on*behavior*, generically neither analytic nor semianalytic \(it is defined by trajectory/reachability, is highly degenerate, and flips discontinuously inww\); it is not a subvarietyφ\\varphican vanish on\.\(ii\)It is not aligned with the resolution ofKK, destroying the monomialization that made SLT solvable\.\(iii\)Theλ​log⁡n\\lambda\\log neffect is an asymptotic, on\-distribution statement about*posterior concentration*, whereas escape is a finite\-nn, off\-distribution, reachability event actively sought by SGD/RL: a measure\-zero event can still be reachable\.\(iv\)Even where it works, one obtains a*surviving strong bias*, not an invariant\.\(v\)Implemented,φ∝exp⁡\(−s​\(w\)\)\\varphi\\propto\\exp\(\-s\(w\)\)withs→∞s\\to\\inftyon the forbidden locus is a differentiable surrogate for “doeswwviolateBB” — a smuggled learned classifier that forfeits the algebraic sharpness that made singular priors survive\. Prior design*does*have leverage when the constraint is already geometric in weight space \(symmetry, low rank, sign/monotonicity\) —precisely the shape semantic safety constraints donothave\.

## 6The Central Result: Safety Predicate B Is Not a Functional of\(model,q\)\(\\text\{model\},q\)the\-central\-result\-b\-is\-not\-a\-functional\-of\-textmodel\-q

Fix the finite architecturep\(⋅∣w\)p\(\\cdot\\mid w\)and the safety predicateBB\. Define on\-support equivalence

w\+∼qw−⇔p\(⋅∣w\+\)=p\(⋅∣w−\)q\-a\.e\.w\_\{\+\}\\sim\_\{q\}w\_\{\-\}\\iff p\(\\cdot\\mid w\_\{\+\}\)=p\(\\cdot\\mid w\_\{\-\}\)\\ \\ q\\text\{\-a\.e\.\}
Proposition 1 \(non\-invariance\)\.*IfBBdepends on behavior on adversarial off\-support inputs, thenBBis not∼q\\sim\_\{q\}\-measurable, henceB∉σ​\(model,q\)B\\notin\\sigma\(\\text\{model\},q\), whileRLCT∈σ​\(model,q\)\\mathrm\{RLCT\}\\in\\sigma\(\\text\{model\},q\)\.*

*Sketch\.*The likelihood∏p​\(xi∣w\)\\prod p\(x\_\{i\}\\mid w\)seeswwonly throughp\(⋅∣w\)\|supp​qp\(\\cdot\\mid w\)\|\_\{\\mathrm\{supp\}\\,q\}; within a∼q\\sim\_\{q\}\-fiber the likelihood is constant, so onlyφ\\varphican move the posterior there\. There existw±w\_\{\\pm\}agreeingqq\-a\.e\. yet withB​\(w\+\)≠B​\(w−\)B\(w\_\{\+\}\)\\neq B\(w\_\{\-\}\)\(they diverge off support\)\. ThusBB’s information is absent from\(model,q\)\(\\text\{model\},q\)\.K​\(w\)K\(w\), and therefore the RLCT, depends only on\(model,q\)\(\\text\{model\},q\)\.□\\square

Sharper: equivalence depends only on themeasure class\[q\]\[q\]\(the null\-set family\), not the density; the RLCT depends on the density \(the KL integral isqq\-weighted\)\. Hence:

Corollary 2\.*Any support\-preserving reweighting \(importance weighting, curriculum resampling, rare\-example upsampling\) is provably ineffective onBB: it fixes\[q\]\[q\], preserves∼q\\sim\_\{q\}, and preservesBB’s non\-measurability\. Only a supportextensioncan moveBB*— and the needed extension is a*moving target*inww\(§6, §9\)\.

Two hardness corollaries follow by composition\.

Corollary 3 \(no\-free\-lunch reduction\)\.*Anyφ\\varphienforcing safety must separate thew±w\_\{\\pm\}pairs, so “designingφ\\varphi” reduces to “possessingBB”: the prior framework transports the difficulty without reducing it\.*

Corollary 4 \(two layers of hardness\)\.*Idealized \(policy = program\): by Rice’s theorem\[[13](https://arxiv.org/html/2608.11243#bib.bib13)\]\[[1](https://arxiv.org/html/2608.11243#bib.bib1)\]the forbidden set is non\-recursive; no constructive/analytic\-classφ\\varphiexists — anon\-existenceresult strictly stronger than any statement about RLCT\. Finite \(fixed architecture\):BBis decidable but NP\-hard \(ReLU reachability is NP\-complete\[[8](https://arxiv.org/html/2608.11243#bib.bib8)\]\) — acomputational\-hardnessresult homologous to “RLCT closed forms are scarce / LLC is numerically tractable\.”*

This two\-layer structure is the precise content of “homologous but strictly different\.”Open target:exhibit, in a finite real weight space, a concrete forbidden setS⊂ℝdS\\subset\\mathbb\{R\}^\{d\}that is*not semianalytic*\(hence outside the reach of resolution\-based SLT\), plausibly via o\-minimality / Tarski–Seidenberg\[[4](https://arxiv.org/html/2608.11243#bib.bib4)\]failure for reachability predicates\.

## 7Fiber Geometry: Where Off\-Support Freedom Comes From

Positive\-dimensional fibers arenecessary but not sufficientfor the freedom Proposition 1 exploits; one must separate two sources\.

- •\(A\) Global redundancy \(symmetry\):p\(⋅∣w\)=p\(⋅∣w′\)p\(\\cdot\\mid w\)=p\(\\cdot\\mid w^\{\\prime\}\)for*all*inputs \(neuron permutation, ReLU rescaling\)\. These fibers, however large, leaveBBconstant— they do not create off\-support freedom\. Much of the degeneracy SLT credits for small RLCT is of this type\.
- •\(B\) Support\-limited underdetermination:w,w′w,w^\{\\prime\}agree onsupp​q\\mathrm\{supp\}\\,qbut diverge outside\. This*is*off\-support freedom\.

LetJS,JXJ\_\{S\},\\,J\_\{X\}be the Jacobians ofw↦\(p​\(x∣w\)\)xw\\mapsto\(p\(x\\mid w\)\)\_\{x\}restricted to rowsx∈supp​qx\\in\\mathrm\{supp\}\\,qandx∈Xx\\in X\. Thenker⁡JX⊆ker⁡JS\\ker J\_\{X\}\\subseteq\\ker J\_\{S\}and

d−rank​JS⏟on\-support non\-identif\.=\(d−rank​JX\)⏟\(A\) symmetry\+\(rank​JX−rank​JS\)⏟\(B\) off\-support freedom\.\\underbrace\{d\-\\mathrm\{rank\}\\,J\_\{S\}\}\_\{\\text\{on\-support non\-identif\.\}\}=\\underbrace\{\(d\-\\mathrm\{rank\}\\,J\_\{X\}\)\}\_\{\\text\{\(A\) symmetry\}\}\+\\underbrace\{\(\\mathrm\{rank\}\\,J\_\{X\}\-\\mathrm\{rank\}\\,J\_\{S\}\)\}\_\{\\text\{\(B\) off\-support freedom\}\}\.
Off\-support freedom=rank​JX−rank​JS≥0=\\mathrm\{rank\}\\,J\_\{X\}\-\\mathrm\{rank\}\\,J\_\{S\}\\geq 0, positive iff the support fails to excite directions the whole space would — generic under overparametrization with a proper\-subset support\. Consequently the RLCT sees the*left*side \(the on\-support degeneracyd−rank​JSd\-\\mathrm\{rank\}\\,J\_\{S\}\); off\-support freedom is the*second right\-hand*term and cannot be recovered from the RLCT alone — its determination needsJXJ\_\{X\}, i\.e\., off\-support structure\. \(This retracts an earlier over\-unification — “singularity is the common parent of generalization and non\-safety” is false, since \(A\)\-degeneracy lowers the RLCT without creating off\-support freedom — and*strengthens*the orthogonality claim: even the*dimension*of off\-support freedom is not a functional of\(model,q\|supp\)\(\\text\{model\},q\|\_\{\\mathrm\{supp\}\}\)\.\)

## 8When the Support Moves: Non\-Analyticity and the Breakdown of SLT

Forfixedqq, restricting tosupp​q\\mathrm\{supp\}\\,qdoes not break analyticity:K​\(w\)=∫supp​qq​log⁡\(q/p\(⋅∣w\)\)K\(w\)=\\int\_\{\\mathrm\{supp\}\\,q\}q\\log\(q/p\(\\cdot\\mid w\)\)remains analytic inww\(integrand analytic, domainww\-independent\); off\-support freedom appears as ordinary*algebraic degeneracy*\(a flat Hessian direction\), not non\-analyticity —e−1/ze^\{\-1/z\}is not needed\.

Non\-analyticity enters when the support depends on the model,supp​q​\(w\)\\mathrm\{supp\}\\,q\(w\)— the performative/decision\-dependent regime\[[12](https://arxiv.org/html/2608.11243#bib.bib12)\]\. Then

K​\(w\)=∫supp​q​\(w\)q​\(w\)​log⁡\(q\(w\)/p\(⋅∣w\)\)K\(w\)=\\int\_\{\\mathrm\{supp\}\\,q\(w\)\}q\(w\)\\log\(q\(w\)/p\(\\cdot\\mid w\)\)

has aww\-dependent domain whose boundary∂supp​q​\(w\)\\partial\\,\\mathrm\{supp\}\\,q\(w\)moves throughmax/arg⁡max/fixed\-point\\max/\\arg\\max/\\text\{fixed\-point\}operators \(adversarialqadv​\(w\)=arg⁡max‖δ‖≤εq\_\{\\mathrm\{adv\}\}\(w\)=\\arg\\max\_\{\\\|\\delta\\\|\\leq\\varepsilon\}; performative fixed points\)\. This yields a hierarchy:

Notee−1/ze^\{\-1/z\}itself is tame \(lies inℝexp\\mathbb\{R\}\_\{\\exp\}, o\-minimal\); the genuine wall is one level up — reachability predicates built from infinite composition break o\-minimality \(finiteness of connected components fails\)\. The correct one\-directional implication is therefore:off\-support freedom×\\timesww\-feedback⇒\\Rightarrowdestruction of SLT’s analytic premise\.This subsumes safetyBBand adversarial accuracy as two off\-support functionals of the same fiber freedom, and reframes the earlier open problem of aself\-consistentK​\(w\)K\(w\)\(a “performative\-SLT” free\-energy asymptotics over a fixed\-point locus\) as the natural but likely non\-solvable\-by\-resolution frontier\.

## 9Local Certification of the*Same*Safety Predicate B, and the Division of Labor

Definition \(the predicate, made explicit\)\.For a finite ReLU networkfw:ℝn→ℝf\_\{w\}:\\mathbb\{R\}^\{n\}\\to\\mathbb\{R\}, an off\-support input boxR⊂ℝnR\\subset\\mathbb\{R\}^\{n\}, unsafe setU=\{y\>0\}U=\\\{y\>0\\\}, and margingw=fwg\_\{w\}=f\_\{w\},

B\(w\):=∃x∈R\.fw\(x\)∈U⇔V\(w\):=supx∈Rgw\(x\)\>0\.B\(w\):=\\exists x\\in R\.\\ f\_\{w\}\(x\)\\in U\\ \\iff\\ V\(w\):=\\sup\_\{x\\in R\}g\_\{w\}\(x\)\>0\.BBdepends onwwonly throughfw\|Rf\_\{w\}\|\_\{R\}— the off\-support restriction of Proposition 1, in predicate form\.

“Pinning the same object\.”Just as the LLC locally/numerically recovers the*same*RLCTλ\\lambda, sound verification recovers the*same*VV\(henceBB\), not a learned surrogate\. Interval\-bound propagation \(IBP\) gives a sound bracket\[L,U\]∋V\[L,U\]\\ni V; branch\-and\-bound overRRrefines it\.

Proposition 5 \(soundness⇒\\Rightarrowdecides the sameBB\)\.*If an enclosureJJsatisfies∀x∈R,gw​\(x\)∈J\\forall x\\in R,\\ g\_\{w\}\(x\)\\in JandJ\.hi≤0J\.\\mathrm\{hi\}\\leq 0, then¬B​\(w\)\\lnot B\(w\); dually, a witnessx∈Rx\\in Rwithgw​\(x\)\>0g\_\{w\}\(x\)\>0certifiesB​\(w\)B\(w\)\.*We formalized both directions in Lean 4 \(safe\_of\_ub\_nonpos,B\_of\_witness\) together with the affine/ReLU/composition enclosure lemmas that discharge the bracket; the file is written against Mathlib but was not machine\-checked in\-session \(the toolchain server lay outside the sandbox’s network allowlist\)\. Numerically, a SAFE instance \(sup≈−0\.05\\sup\\approx\-0\.05\) is undecided at coarse level \(\[−0\.05,0\.99\]\[\-0\.05,0\.99\]\) and certified¬B\\lnot Bafter refinement to 31 boxes, while an UNSAFE instance is certified by a single witness — the∀\\forall\-side needs sound coverage, the∃\\exists\-side one point, and bothtighten with compute, the direct analog of LLC tightening with SGLD samples\.

Analogy and its two breakpoints\.

Two strict differences remain:\(i\)BBadmits a*surrogate branch*\(learned classifier / runtime monitor\) that pins a*different*object \(a learned decision boundary\) without soundness; LLC has no such branch\.\(ii\)In the idealized layerBBreaches*non\-existence*\[[13](https://arxiv.org/html/2608.11243#bib.bib13)\]\[[1](https://arxiv.org/html/2608.11243#bib.bib1)\], whereas the RLCT always exists and is computable in principle\. The table is the*finite\-layer*homology, where the correspondence is in fact tight\.

Division of labor\.Synthesizing §3–7:*soft, semantic, context\-dependent*dispositions \(“prefer legitimate solutions,” “avoid deceptive subgoals”\) belong in themodel\(via alignment training\) and act as*rate reducers*— they lower the probability mass on bad trajectories over the un\-enumerable semantic space\.*Hard, safety\-critical invariants*with crisp predicates \(“no outbound network,” “the answer store is unreadable,” “kill on privilege escalation”\) belong in theharnessas enforced invariants, because they yield a*guarantee*rather than a tendency, do not degrade under optimization pressure \(noλ\\lambdafor reward to outbid\), are independent of the model’s cooperation \(decisive when the evaluation bypasses the model’s refusals\), and require the*exact predicates*that exist only in the environment\. Verification pinsBBgivenRR; it does not tell you*which*RRmatters — and the choice ofRRis precisely the moving, off\-support, performative object of §7\.The technique for pinningBBand the identification of what to pin split exactly along the on\-support / off\-support boundary\.

## 10Frontier: Performative Dynamics, GANs, and Self\-Referential Functional Dynamics

The “whichRR” problem is dynamical\. Performative prediction is a special decision\-dependent game\[[12](https://arxiv.org/html/2608.11243#bib.bib12)\]; a GAN is a two\-player instance in which the data the learner sees is produced by its own prior deployment\. The motion of*optima*under such feedback is well studied — limit cycles and Poincaré recurrence in game dynamics\[[5](https://arxiv.org/html/2608.11243#bib.bib5)\]\[[2](https://arxiv.org/html/2608.11243#bib.bib2)\]\[[10](https://arxiv.org/html/2608.11243#bib.bib10)\], and the motion of*distributions*via Wasserstein\(\-Fisher\-Rao\) gradient flows and mean\-field min\-max analyses\[[3](https://arxiv.org/html/2608.11243#bib.bib3)\]\. What neither literature engages is theself\-referential functional dynamicsof Kataoka–Kaneko,fn\+1=\(1−ε\)​fn\+ε​fn∘fnf\_\{n\+1\}=\(1\-\\varepsilon\)f\_\{n\}\+\\varepsilon\\,f\_\{n\}\\circ f\_\{n\}\[[7](https://arxiv.org/html/2608.11243#bib.bib7)\]\[[14](https://arxiv.org/html/2608.11243#bib.bib14)\], whose state is a*function*updated through its own self\-composition, producingarticulation\(spontaneous discrete structure\) andhierarchy\.

The evocative mapping is:articulation↔\\leftrightarrowGAN mode formation/collapse, andchaotic itinerancy↔\\leftrightarrowGAN mode hopping / non\-convergent oscillation\. The literalf∘ff\\circ fis type\-incorrect forG:Z→X,D:X→ℝG:Z\\to X,\\ D:X\\to\\mathbb\{R\}; the honest lift folds the update into an endomapT:𝒫​\(X\)→𝒫​\(X\)T:\\mathcal\{P\}\(X\)\\to\\mathcal\{P\}\(X\)\(generated distribution→\\tonext\), whose two\-step operatorT∘TT\\circ Tis the natural analog, with self\-reference entering through the loss coupling \(the discriminator trained on the generator’s own output\)\. Because GANs are extremely high\-DOF, the relevant vehicle is not 1\-D functional dynamics butKaneko’s high\-DOF program— globally coupled maps, chaotic itinerancy\[[6](https://arxiv.org/html/2608.11243#bib.bib6)\], Milnor attractors\[[15](https://arxiv.org/html/2608.11243#bib.bib15)\], and Lyapunov spectra that scale extensively with system size\. A concrete program: write GAN training as a self\-referential operatorTTon Wasserstein space, take the performativity/coupling strengthε\\varepsilonas bifurcation parameter, identify mode formation with fixed\-point\-type splitting and mode hopping with chaotic itinerancy, and chart the high\-DOF phase diagram via extensive Lyapunov spectra — connecting to the self\-consistent\-K​\(w\)K\(w\)frontier of §7 and to the categorical formulation of dynamics \(dynamical system as functor, topological conjugacy as natural isomorphism\)\. Within our search, no work bridges Kataoka–Kaneko functional dynamics to high\-DOF GANs; the gap appears genuinely open \(absence of found evidence, not proof of absence\)\. Obstacles: theT∘TT\\circ Tself\-composition term needs justification as a natural feature of the GAN update; the phenomenology’s low\-dimensional origins require the high\-DOF vehicle; and the semianalyticity wall of §7 recurs \(fixed\-point sets ofTTneed not be algebraically tame\)\.

## 11Conclusion

One fact — that a semantic safety constraint is an off\-support functional and therefore*not*an invariant of the learning problem, whereas the RLCT*is*— threads the entire analysis\. It explains why outcome\-based optimization leaks \(reward hacking as the constraint being off the reward’s support\), why prior design and soft weighting have poor leverage in singular models \(the constraint is not an identifiable subvariety, and singular priors that would survive cannot be specified from rules\), why hard invariants must sit in the harness rather than the model \(exact predicates and guarantees live in the environment\), why the*same*constraint is nonetheless soundly and locally certifiable given a region \(verification pinsBBas the LLC pins the RLCT\), and why the residual — identifying the off\-support region that matters — coincides with performative and self\-referential functional dynamics, exactly where SLT’s resolution\-based machinery ceases to apply\. The practical upshot is a division of labor: models reduce the probability of bad trajectories over semantic space; harnesses make the worst cases impossible; and the two meet, cleanly, at the boundary between what the data supports and what it does not\.

## 12References

## References

- \[1\]Jonathan Brossard\.A constructive proof of rice’s theorem and the halting problem via hilbert’s tenth problem, 2026\.
- \[2\]Constantinos Daskalakis, Andrew Ilyas, Vasilis Syrgkanis, and Haoyang Zeng\.Training GANs with optimism\.2018\.
- \[3\]Carles Domingo\-Enrich, Samy Jelassi, Arthur Mensch, Grant Rotskoff, and Joan Bruna\.A mean\-field analysis of two\-player zero\-sum games\.In H\. Larochelle, M\. Ranzato, R\. Hadsell, M\.F\. Balcan, and H\. Lin, editors,Advances in Neural Information Processing Systems, Vol\. 33, pp\. 20215–20226\. Curran Associates, Inc\., 2020\.
- \[4\]L\. P\. D\. van den Dries\.Tame Topology and O\-minimal Structures\.London Mathematical Society Lecture Note Series\. Cambridge University Press, 1998\.
- \[5\]Ian J\. Goodfellow, Jean Pouget\-Abadie, Mehdi Mirza, Bing Xu, David Warde\-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio\.Generative adversarial networks, 2014\.
- \[6\]Kunihiko Kaneko\.Chaotic itinerancy\.Chaos, Vol\. 13, No\. 3, pp\. 926–936, 08 2003\.
- \[7\]Naoto Kataoka and Kunihiko Kaneko\.Functional dynamics\. i: Articulation process\.Physica D: Nonlinear Phenomena, Vol\. 138, No\. 3, pp\. 225–250, 2000\.
- \[8\]Guy Katz, Clark Barrett, David Dill, Kyle Julian, and Mykel Kochenderfer\.Reluplex: An efficient smt solver for verifying deep neural networks, 2017\.
- \[9\]Edmund Lau, Zach Furman, George Wang, Daniel Murfet, and Susan Wei\.The local learning coefficient: A singularity\-aware complexity measure, 2024\.
- \[10\]Lars Mescheder, Sebastian Nowozin, and Andreas Geiger\.Which training methods for gans do actually converge?InInternational Conference on Machine Learning \(ICML\), 2018\.
- \[11\]OpenAI\.Openai and hugging face partner to address security incident during model evaluation, 2026\.
- \[12\]Juan Perdomo, Tijana Zrnic, Celestine Mendler\-Dünner, and Moritz Hardt\.Performative prediction\.In Hal Daumé III and Aarti Singh, editors,Proceedings of the 37th International Conference on Machine Learning, Vol\. 119 ofProceedings of Machine Learning Research, pp\. 7599–7609\. PMLR, 13–18 Jul 2020\.
- \[13\]H\. G\. Rice\.Classes of recursively enumerable sets and their decision problems\.Trans\. Amer\. Math\. Soc\., Vol\. 74, pp\. 358–366, 1953\.
- \[14\]Yoichiro Takahashi, Naoto Kataoka, Kunihiko Kaneko, and Takao Namiki\.Function dynamics\.Japan Journal of Industrial and Applied Mathematics, Vol\. 18, pp\. 405–423, 06 2001\.
- \[15\]Ichiro Tsuda and Toshiya Umemura\.Chaotic itinerancy generated by coupling of milnor attractors\.Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol\. 13, No\. 3, pp\. 937–946, 09 2003\.
- \[16\]Zhun Wang, Nico Schiller, Hongwei Li, Srijiith Sesha Narayana, Milad Nasr, Nicholas Carlini, Xiangyu Qi, Eric Wallace, Elie Bursztein, Luca Invernizzi, Kurt Thomas, Yan Shoshitaishvili, Wenbo Guo, Jingxuan He, Thorsten Holz, and Dawn Song\.Exploitgym: Can ai agents turn security vulnerabilities into real attacks?, 2026\.
- \[17\]Sumio Watanabe\.Algebraic Geometry and Statistical Learning Theory\.Cambridge Monographs on Applied and Computational Mathematics\. Cambridge University Press, 2009\.