diff --git a/HYPOTHESIS.md b/HYPOTHESIS.md index dae9caef..4fdd7ac2 100644 --- a/HYPOTHESIS.md +++ b/HYPOTHESIS.md @@ -137,9 +137,9 @@ With that caveat, **the interlingua and the certificate are one object seen twic Borrowed theorems are real; the framings are not — keep them separate. Some framings are nonetheless *corroborated* — independently reached from another field — a third grade, weaker than proof and noted last. -**Proven (citable).** Foster–Lyapunov drift ⇒ positive recurrence + $\mathbb{E}[\tau]\le V(s_0)/\varepsilon$ (Foster 1953; Meyn & Tweedie, *Markov Chains and Stochastic Stability*, 1993) — positive recurrence needs the usual irreducibility/petite-set hypotheses, while the absorbing-halt case used here needs only the weaker supermartingale optional-stopping hitting-time bound. The minimal $V$ is the expected hitting time, by first-step analysis + optional stopping (Norris, *Markov Chains*, 1997). For an absorbing chain that expected hitting time is the row sum of the fundamental matrix $N=\sum_{n\ge0}Q_{\mathrm{tr}}^{\,n}$ (Kemeny & Snell, *Finite Markov Chains*, 1960), with the general-state analogue the potential (Green) operator (Revuz, *Markov Chains*, 1984). Koopman's linear-operator view of nonlinear dynamics is classical (Koopman 1931), and Lyapunov functions can be assembled from its eigenfunctions when the spectrum is suitable (Mauroy & Mezić, 2016). You certify a candidate $\hat V$ by a *proven* drift inequality rather than by deriving $V^\star$, and estimate it empirically only where a proof is out of reach — the empirical drift checks, it does not certify (neural-Lyapunov: Chang, Roohi & Gao, *Neural Lyapunov Control*, NeurIPS 2019, arXiv:2005.00611). A classical monotone data-flow analysis gets its $V$ for free because a finite-height lattice is a well-founded descent (Kildall, POPL 1973). The gate-a-plant architecture itself is classical: supervisory control theory synthesizes a deterministic supervisor that disables controllable events of a plant it does not author, with the supremal controllable sublanguage as the largest admissible behavior (Ramadge & Wonham, SIAM J. Control and Optimization, 1987) — $\gamma$ is that supervisor, with a learned stochastic plant on general state spaces. The successor representation is Dayan (*Improving Generalization for Temporal Difference Learning: The Successor Representation*, Neural Computation 1993). Dialect-stack architecture: MLIR (Lattner et al., CGO 2021, arXiv:2002.11054); learned pass-ordering: MLGO (Trofin et al., arXiv:2101.04808). Single-pass low-depth expressivity: log-precision transformers are simulable by constant-depth logspace-uniform threshold circuits ($\mathsf{TC}^0$) (Merrill & Sabharwal, *The Parallelism Tradeoff: Limitations of Log-Precision Transformers*, TACL 2023) — fixed/constant precision is a stronger restriction, added autoregressive steps escape it (Merrill & Sabharwal, *The Expressive Power of Transformers with Chain of Thought*, ICLR 2024), and growing precision changes the picture, so the bound is suggestive for deployed models, not literal. +**Proven (citable).** Foster–Lyapunov drift ⇒ positive recurrence + $\mathbb{E}[\tau]\le V(s_0)/\varepsilon$ (Foster 1953; Meyn & Tweedie, *Markov Chains and Stochastic Stability*, 1993) — positive recurrence needs the usual irreducibility/petite-set hypotheses, while the absorbing-halt case used here needs only the weaker supermartingale optional-stopping hitting-time bound. The minimal $V$ is the expected hitting time, by first-step analysis + optional stopping (Norris, *Markov Chains*, 1997). For an absorbing chain that expected hitting time is the row sum of the fundamental matrix $N=\sum_{n\ge0}Q_{\mathrm{tr}}^{\,n}$ (Kemeny & Snell, *Finite Markov Chains*, 1960), with the general-state analogue the potential (Green) operator (Revuz, *Markov Chains*, 1984). Koopman's linear-operator view of nonlinear dynamics is classical (Koopman 1931), and Lyapunov functions can be assembled from its eigenfunctions when the spectrum is suitable (Mauroy & Mezić, 2016). You certify a candidate $\hat V$ by a *proven* drift inequality rather than by deriving $V^\star$, and estimate it empirically only where a proof is out of reach — the empirical drift checks, it does not certify (neural-Lyapunov: Chang, Roohi & Gao, *Neural Lyapunov Control*, NeurIPS 2019, arXiv:2005.00611). A classical monotone data-flow analysis gets its $V$ for free because a finite-height lattice is a well-founded descent (Kildall, POPL 1973). The gate-a-plant architecture itself is classical: supervisory control theory synthesizes a deterministic supervisor that disables controllable events of a plant it does not author, with the supremal controllable sublanguage as the largest admissible behavior (Ramadge & Wonham, SIAM J. Control and Optimization, 1987) — $\gamma$ is that supervisor, with a learned stochastic plant on general state spaces; the same theory's controllability condition (specifications must be closed under *uncontrollable* events) and its nonblocking requirement are the proven ancestors of gate-early-on-irreversibles and of the always-enabled escalation the appendix requires behind any learned veto. Covert-channel discipline — identify the channel, measure its bandwidth in bits, audit what cannot be closed — is the TCSEC lineage (*A Guide to Understanding Covert Channel Analysis of Trusted Systems*, NCSC-TG-030, 1993). The successor representation is Dayan (*Improving Generalization for Temporal Difference Learning: The Successor Representation*, Neural Computation 1993). Dialect-stack architecture: MLIR (Lattner et al., CGO 2021, arXiv:2002.11054); learned pass-ordering: MLGO (Trofin et al., arXiv:2101.04808). Single-pass low-depth expressivity: log-precision transformers are simulable by constant-depth logspace-uniform threshold circuits ($\mathsf{TC}^0$) (Merrill & Sabharwal, *The Parallelism Tradeoff: Limitations of Log-Precision Transformers*, TACL 2023) — fixed/constant precision is a stronger restriction, added autoregressive steps escape it (Merrill & Sabharwal, *The Expressive Power of Transformers with Chain of Thought*, ICLR 2024), and growing precision changes the picture, so the bound is suggestive for deployed models, not literal. -**Asserted (ours — not theorems).** That the harness is best modeled as nested stopped chains; that $V^\star$ is incompressible (no compression theorem); that "no lattice for $f(\cdot\,;W)$" means none is *known*, not that none exists; and everything under *Where this points* — including the Koopman/certificate co-determination, which is well-posed only under the spectral assumptions noted there, and the interlingua/certificate identification; and the design rules read off the objects rather than proven from them — the single-trusted-writer completion of the provenance partition, the narrow-only rule for learned checks, the composition law of the appendix. These organize the design; they are not results. +**Asserted (ours — not theorems).** That the harness is best modeled as nested stopped chains; that $V^\star$ is incompressible (no compression theorem); that "no lattice for $f(\cdot\,;W)$" means none is *known*, not that none exists; and everything under *Where this points* — including the Koopman/certificate co-determination, which is well-posed only under the spectral assumptions noted there, and the interlingua/certificate identification; and the design rules read off the objects rather than proven from them — the single-trusted-writer completion of the provenance partition, the narrow-only rule for learned checks and its influence-side twin (verdict payloads to the plant selected, never generated), the composition law of the appendix. These organize the design; they are not results. **Converged-upon (independently arrived at, from other framings).** The *Asserted* claims above are ours but not ours alone; several are reached independently, from starting points unconnected to this framing — which is the corroboration a definition earns: not a chorus of agreement (the systems below often disagree on method and goal), but that work approaching from capabilities, reinforcement learning, control theory, software architecture, and language-modeling theory each lands on a piece of the same object. That the **deterministic controller, not the model, carries the guarantee** is reached from four directions — capability and information-flow control (CaMeL: Debenedetti et al., *Defeating Prompt Injections by Design*, arXiv:2503.18813, securing the agent even when the underlying model is susceptible); reinforcement learning (shielding: Alshiekh et al., *Safe Reinforcement Learning via Shielding*, AAAI 2018, arXiv:1708.08611 — a deterministic reactive shield filtering a learned policy's actions against a temporal-logic specification); control theory (*Stable Agentic Control*, arXiv:2605.03034, enforcing finite action catalogs at the tool-output interface under a Lyapunov input-to-state-stability certificate against adversarial disturbance); and software architecture (the plan-then-execute / control-flow-integrity line, e.g. Beurer-Kellner et al., *Design Patterns for Securing LLM Agents against Prompt Injections*, arXiv:2506.08837). The **certified-vs-measured split** is reached from the construction side (CaMeL's provable security) and, independently, from the destruction side (guardrail-evasion results — *Bypassing Prompt Injection and Jailbreak Detection in LLM Guardrails*, arXiv:2504.11168, the v1 title — later versions retitle it; *No Free Lunch with Guardrails*, arXiv:2504.00441), with verification-oriented work stating it as the motivating gap (*Towards Verifiably Safe Tool Use for LLM Agents*, arXiv:2601.08012; VeriGuard, arXiv:2510.05156): a learned safeguard raises the odds of detection but cannot guarantee safety against a persistent attacker. The **inner readout as a composition of Markov kernels** is independently formalized in language-modeling theory — the autoregressive step as kernel composition in the category $\mathsf{Stoch}$ (*A Markov Categorical Framework for Language Modeling*, arXiv:2507.19247), and the broader "LLMs as Markov chains" line — though that work models the inner kernel alone and never closes it into an agentic loop, which is exactly the seam this definition adds. That **provenance shrinks the admissible adversary** is reached by datamarking / spotlighting (Hines et al., arXiv:2403.14720, 2024) and by CaMeL's data/control-flow separation; and a systematization of prompt injection against agentic coding assistants reaches the same verdict from the attack side — mitigation must be *architectural*, not model-level (*Prompt Injection Attacks on Agentic Coding Assistants*, arXiv:2601.17548); the sharper open problem this object is built to answer — formally specify the trust boundaries, then verify implementations respect them — is our phrasing of where that verdict points, not the paper's. Two convergences are weaker, and flagged. The **reach-avoid hitting-time certificate** is the independently developed reach-avoid supermartingale (RASM, arXiv:2210.05308, AAAI 2023) and stochastic Lyapunov–barrier apparatus, and its *hardness* is corroborated — expected-stopping-time problems for Markov chains are inter-reducible with the Positivity problem, a relative of the Skolem problem (Chatterjee & Doyen, *Stochastic Processes with Expected Stopping Time*, arXiv:2104.07278) — but this supports generic hardness only, not the specific incompressibility-at-$|W|$ conjecture, which remains ours and unproven. And **injection as an adversarial policy** is corroborated as a minimax game in the *detection* setting (DataSentinel: Liu et al., *A Game-Theoretic Detection of Prompt Injection Attacks*, arXiv:2504.11358) and as adversarial-disturbance robustness (*Stable Agentic Control*, above) — but no prior work assembles it as reach-avoid over the tool-output kernel with the gate as the irreversibility margin; here the relation is adjacency, not convergence. @@ -175,6 +175,8 @@ The same pressure lands on tooling from a second direction. The $\mathsf{action\ That last kind marks the seam where the gate stops being able to stay pure, and it is the same seam the rest of this document is built around. The *structural* slice of intent — does the action cohere with the plan in $s$ — is a deterministic predicate over $s$ and $y$, effect-free, and belongs in $\gamma$ without reservation. But whether an action matches what the user *actually meant*, in the full semantic sense, is exactly the thing the definition says cannot be checked: natural language is all undefined behavior, with no source-language standard to validate against. So a semantic intent check is a *learned* check, and an LLM judging "is this what they wanted" is a **stochastic kernel** — putting it inside $\gamma$ breaks the property the gate exists to hold, by the same move flagged for the fold-back verifier: a learned judge is a kernel, and belongs in $M_W$, not in a deterministic map. Semantic intent therefore does not live *in* the gate; it is a plant call — a separate authorize-the-proposal pass through $M_W$ whose output $\gamma$ then deterministically gates — or it is drift you measure, never a guarantee you hold. That nested call is not a new kind of thing: it is a mini-harness inside the gate's decision — a judge $M_W$, its own syntactic readout, its own deterministic gate — so its failure case answers itself, the inner gate fail-closing on an unparseable or low-confidence judgment exactly as the outer one does, because it *is* one. The object is **closed under this construction**: semantic gating is added by recursion, not by a new primitive. One constraint on the recursion is load-bearing enough to be a rule, because it is where this entry meets the provenance partition of the body: the judge's verdict is derived, through a learned kernel, from the very content an adversary may have bent, so folding it into authorization is exactly the fold the partition forbids — *unless the verdict can only cost capability*. **A learned check may narrow the deterministic admissible set; it must never widen it.** Judge-as-veto is safe by construction: attacker influence over the judge can at worst manufacture a denial, a liveness cost the certificate already prices. Judge-as-approver — a verdict granting what the deterministic checks alone would refuse, or standing in for the trusted principal's confirmation — lowers the certified floor to those deterministic checks alone; if avoiding $B$ depended on the deny the judge now withholds on the adversary's behalf, the certificate is gone. Only the trusted principal widens authorization; learned kernels only narrow it. (The recursion already obeys this: the mini-harness's inner gate fail-closes to $\bot$ — a deny — which is why the construction was safe to add at all.) The cost is real and worth stating — a judge pass is another full model call, with its latency and tokens — so it is a decision about *which* actions warrant it, not a free wrapper for all of them. The gate widens to every deterministic predicate over $s$ and $y$; it does not widen to the one predicate the document says is not deterministically checkable. +One more caveat keeps the veto's pricing honest, because a denial is free only in the *authority* lattice. In the dynamics it is an input like any other — folded into $s$, lowered into the next context, conditioning the plant's next proposal — so adversarial influence over a judge is influence over the *trajectory*: a selection channel (deny all but the path toward $B$, and the admissible set the plant experiences is a maze the adversary curated), and a targeted-liveness channel against load-bearing actions — the unstated dual of judge-as-approver: if avoiding $B$ depends on the action the judge now denies on the adversary's behalf, fail-closed's safe landing is an obligation the design earns per-state, not an axiom it inherits. The supervisory ancestry supplies the discipline: a learned veto requires a **nonblocking escape it cannot disable** — an always-enabled route to the trusted principal behind a bounded retry budget — or manufactured denials strand the run, or steer it. And whatever a verdict carries *back to the plant* is a second channel, wearing the judge's authority framing. Free prose there is *generative* influence — injected context, priced by the minimax descent, never by the veto's zero-widening — so the narrow-only rule has an influence-side twin: **a learned verdict's payload to the plant is selected, never generated** — controller-authored symbols, typed citations validated like any effect record, template text with no interpolated model prose — its per-verdict capacity a designed constant rather than a measured hope, and the residual selection pattern audited as the covert channel it is. The alphabet's bound is not a count but two thresholds: symbols become tokens when their semantics stop being controller-authored — the registry the trusted writer can actually audit is the real constant, and borrowed alphabets with upstream owners (a linter's rule registry) spend that budget well — and tokens become language when composition turns productive, arrangement carrying meaning the controller never wrote. Below both thresholds the alphabet may be as large as the audit budget affords. The strongest form dissolves the learned verdict into *scheduling*: the learned component chooses which deterministic checks to run — pass-ordering over verification passes — and the only verdicts that flow anywhere are what the oracles actually said, leaving attention misallocation, a liveness cost, as the entire attack surface. + But "before any invocation" has to be read as *before any effect*, which is sharper than it sounds — and the reason is the irreversibility point above: you validate before execution because execution is what you cannot take back, so the real invariant is **no effect crosses $\gamma$ unvalidated**. That catches three cases the naive reading misses. *Reads are not free*: a read-only call is still an injection vector (it pulls attacker-controlled content into context) or an exfiltration vector (a request whose URL is the payload), so the gate authorizes the *call* regardless of whether it mutates. *Validation must not act*: a "validator" that resolves a call by hitting an API, expanding a template that fires a webhook, or evaluating an argument that runs code has collapsed validation into invocation, and the effect has already happened *inside* $\gamma$ — so $\gamma$ itself must be **effect-free**, pure and total over the proposal and the current $s$, with no network and no execution; if deciding validity *requires* a side effect, that side effect is itself an action and must go through the gate, recursively. *The output is an action too*: the user-visible response and any logging are effects — for model-authored text, emitted either as an authorized action through $\gamma$ or only after an accepted halt (shell-templated status on any halt is the controller speaking, not the model) — streaming raw tokens to a sink before $\gamma$ has cleared them is the same bug from the other end. So the property, tightest: $\gamma$ is a **pure, effect-free authorization that every model-proposed action — tool call, read, write, or final output — must pass before any effect occurs**, with "before" enforced structurally by the gate being the only route from model text to $Q_E$. The two failure modes to design against are a path from model output to a sink that bypasses the gate, and a $\gamma$ that is not effect-free, so that "validating" a call already rang the bell. And the boundary, so the property does not overpromise: $\gamma$ guarantees *no unauthorized effect* — pure code ordering, fully in your control — but not that an *authorized* effect is safe or correct; that is the plant's problem, and the reason $\rho$ and the reach-avoid certificate exist. Fail-closed is the floor — nothing executes that did not pass the gate — not the ceiling.