docs(hypothesis): round-sixteen (maxima re-run) — semantic-axis fixes the linter cannot see

The linter closed mechanical consistency; this review probes meaning, a separate
axis. Twelve findings, several real corrections, all folded.

Correctness:
- Stationarity overclaim: the supermartingale BOUND survives a nonstationary
  kernel under uniform conditional drift. Time-homogeneity is needed for V* as a
  fixed function, the resolvent/fundamental-matrix identities, and δ-calibration.
- Self-contradiction: "the certificate cannot be proven, only observed" contradicted
  the established "a proven inequality certifies" — reworded to "the architecture
  does not hand it to you; estimated unless separately certified."
- Citation: the TACL result is LOG-precision → logspace-uniform TC⁰ (verified);
  fixed/constant precision is a stronger restriction. Fixed in body and Grounding.

Modeling holes closed:
- Adversary class Π must respect rejection: γ(s,y)=⊥ ⇒ Q_E^α(s,⊥,·)=δ_e0, else the
  adversary resurrects refused side effects.
- R must be a syntactic/verified readout, not a semantic solver — otherwise the
  L-wall is void (compute could hide in R off the ≤L window).
- e must be an effect record (ledger outcome), not just API bytes, since only ρ
  writes external effects into s.
- The displayed M_W(c) freezes endpoint/version/sampler; config changes need a
  state-indexed M_{κ(s)} or K_C — the kernel can't silently depend on config in s.
- The final user-visible response/log is itself an effect: an authorized action
  through γ, or emitted only after an accepted halt.
- m_t must include a token counter and clock for the cap/timeout to be functions of it.

Residue (omissions a collision-linter can't catch):
- Another γ dropped from the K_C Dirac-special-case list.
- τ* mislabeled as "designed code" → the halt test (H) is the code; τ* is its
  emergent hitting time.
- H\H_ok relabeled "non-accepting" (safe refusals outside B; wrong/bad halts
  possibly in B), not uniformly "rejecting/fail-closed."

HYPOTHESIS.md

@@ -14,13 +14,13 @@ Most descriptions of an agent framework are a feature list. This is an attempt a
 
 $$T(s, A) = \int_{\mathcal{Y}}\!\int_{\mathcal{E}} \mathbf{1}_A\!\big(\rho(s, y, \gamma(s,y), e)\big)\; Q_E\big(s, \gamma(s,y), de\big)\; M_W(\pi(s), dy), \qquad T(s,A)=\mathbf{1}_A(s)\ \text{ for } s\in H,$$
 
-and the harness runs $s_{n+1} \sim T(s_n)$ from an initial $s_0 \sim \mu_0$ until $\tau^\star = \inf\{n : s_n \in H\}$. Because $\pi, \gamma, \rho, H$ are deterministic they contribute no integration variable of their own — they appear as measurable transformations inside the integrand (the pushforward), not literally outside it — so the controller injects no randomness, and every coin is inherited from $M_W$ and $Q_E$. The gate $\gamma$ is what makes **fail-closed** a property, not just a name: model output is an *untrusted proposal*, and $\gamma(s,y)=\bot$ forces a no-op environment response ($Q_E(s,\bot,\cdot)=\delta_{e_0}$) — so a malformed or unauthorized tool call is rejected *before* it can act, not validated after its side effects have landed. Fail-closed is then the property that a rejected proposal causes *no unauthorized side effect* and lands in a **safe, non-bad** set ($\rho(s,y,\bot,e_0)\notin B$): a non-accepting terminal $H\setminus H_{\mathrm{ok}}$ in the strict case, or a safe non-terminal state when the spec retries. And $\rho$ must validate the tool *response* $e$, not only the proposal that $\gamma$ already gated: a malformed or adversarial $Q_E$ output is caught at fold-back, not just at the gate. But response-validation has a hard limit: $\rho$ can reject a bad tool *response*, yet it cannot undo side effects an *authorized* action already caused — so $\gamma$, not $\rho$, is the last line before irreversible effects, and anything irreversible must be gated at authorization. The boundary is also only real if raw model output reaches *no* sink — tool, logger, browser, or remote call — before $\gamma$; any pre-authorization escape bypasses the gate. These are the invariants that make $\mathcal{H}$ a *harness* and not merely a controlled Markov process with a learned kernel inside: the model sees only $\mathcal{C}$, never full $\mathcal{S}$; its outputs are proposals, not actions; a deterministic capability boundary $\gamma$ gates every side effect; and the *terminal* set $H$ splits into accepting ($H_{\mathrm{ok}}$) and rejecting/fail-closed ($H\setminus H_{\mathrm{ok}}$), while the bad set $B$ is a *separate* unsafe set — possibly absorbing, possibly entered mid-run before any halt — against which $\tau_B$ is measured for reach-avoid. (The earlier shorthand $T = \rho \circ (M_W \circ \pi, E)$ is suggestive but ill-typed — $M_W$ returns a *law*, while $\rho$ consumes a *sample* together with the prior state $s$; the integral is what the shorthand meant.) This displayed $T$ is the time-homogeneous, fixed-kernel case; for nonstationary or adversarial environments, replace $Q_E$ with a time-indexed kernel $Q_{E,n}$ — or an admissible family of kernels, or an adversary's policy — over which the robust certificate (the minimax form under *The limit*) quantifies. If that adversary conditions on history rather than only the current $(s, y)$, the history must itself live in $s$ — otherwise the object is a Markov *game* requiring further augmentation, not a Markov chain.
+and the harness runs $s_{n+1} \sim T(s_n)$ from an initial $s_0 \sim \mu_0$ until $\tau^\star = \inf\{n : s_n \in H\}$. Because $\pi, \gamma, \rho, H$ are deterministic they contribute no integration variable of their own — they appear as measurable transformations inside the integrand (the pushforward), not literally outside it — so the controller injects no randomness, and every coin is inherited from $M_W$ and $Q_E$. The gate $\gamma$ is what makes **fail-closed** a property, not just a name: model output is an *untrusted proposal*, and $\gamma(s,y)=\bot$ forces a no-op environment response ($Q_E(s,\bot,\cdot)=\delta_{e_0}$) — so a malformed or unauthorized tool call is rejected *before* it can act, not validated after its side effects have landed. Fail-closed is then the property that a rejected proposal causes *no unauthorized side effect* and lands in a **safe, non-bad** set ($\rho(s,y,\bot,e_0)\notin B$): a non-accepting terminal $H\setminus H_{\mathrm{ok}}$ in the strict case, or a safe non-terminal state when the spec retries. And $\rho$ must validate the tool *response* $e$, not only the proposal that $\gamma$ already gated: a malformed or adversarial $Q_E$ output is caught at fold-back, not just at the gate. But response-validation has a hard limit: $\rho$ can reject a bad tool *response*, yet it cannot undo side effects an *authorized* action already caused — so $\gamma$, not $\rho$, is the last line before irreversible effects, and anything irreversible must be gated at authorization. The boundary is also only real if raw model output reaches *no* sink — tool, logger, browser, or remote call — before $\gamma$; any pre-authorization escape bypasses the gate. The user-visible final response and any logging are themselves effects: either an authorized action through $\gamma$, or emitted only after an accepted halt in $H_{\mathrm{ok}}$. These are the invariants that make $\mathcal{H}$ a *harness* and not merely a controlled Markov process with a learned kernel inside: the model sees only $\mathcal{C}$, never full $\mathcal{S}$; its outputs are proposals, not actions; a deterministic capability boundary $\gamma$ gates every side effect; and the *terminal* set $H$ splits into accepting ($H_{\mathrm{ok}}$) and non-accepting ($H\setminus H_{\mathrm{ok}}$ — safe refusals outside $B$, and wrong or bad halts possibly in $B$), while the bad set $B$ is a *separate* unsafe set — possibly absorbing, possibly entered mid-run before any halt — against which $\tau_B$ is measured for reach-avoid. (The earlier shorthand $T = \rho \circ (M_W \circ \pi, E)$ is suggestive but ill-typed — $M_W$ returns a *law*, while $\rho$ consumes a *sample* together with the prior state $s$; the integral is what the shorthand meant.) This displayed $T$ is the time-homogeneous, fixed-kernel case; for nonstationary or adversarial environments, replace $Q_E$ with a time-indexed kernel $Q_{E,n}$ — or an admissible family of kernels, or an adversary's policy — over which the robust certificate (the minimax form under *The limit*) quantifies. If that adversary conditions on history rather than only the current $(s, y)$, the history must itself live in $s$ — otherwise the object is a Markov *game* requiring further augmentation, not a Markov chain.
 
 *The inner kernel.* $M_W$ is itself a stopped process, and for a decoder-only transformer it is implemented as
 
 $$M_W(c, \cdot) = \mathrm{Law}\big(R(z_\tau)\big), \quad z_t = (c_t, b_t, m_t), \quad v \sim K_W(c_t, \cdot), \quad K_W(c, v) = (U \circ \Phi_W \circ \mathrm{Emb})(c)[v], \quad c_{t+1} = \mathrm{suffix}_{\le L}(c_t\!\cdot\! v),\ \ b_{t+1} = b_t\!\cdot\! v,\ \ m_{t+1} = \mathsf{step}(m_t, v),\ \ \tau=\inf\{t:m_t\in\mathrm{Stop}\}.$$
 
-with the layer stack $\Phi_W$ on the residual stream as the (loosely) "manifold" core — formally just the learned high-dimensional residual-stream transformation, with manifold-proper reserved for the frontier. The inner state $z_t=(c_t,b_t,m_t)$ separates the model-visible window $c_t$ (the $\le L$ slice that slides) from the untruncated output buffer $b_t$ (the transcript the readout actually consumes, so truncation never loses it) and the parser/stop state $m_t$, updated $m_{t+1}=\mathsf{step}(m_t,v)$, whose stop set $\mathrm{Stop}$ — EOS emitted, max-token cap, timeout, or parse-failure $\bot$ — forces $\tau=\inf\{t:m_t\in\mathrm{Stop}\}$ finite, making $M_W$ a genuine *probability* kernel rather than a sub-probability one completed by a cemetery output. The readout is total, $R : \mathcal{Z} \to \mathcal{Y}$ — a parsed tool-call, answer, or transcript, returning the parse-failure $\bot\in\mathcal{Y}$ when parsing fails — so $M_W(c, \cdot) = R_\#\,\mathrm{Law}(z_\tau)$, the pushforward of the stopped-state law along $R$ (equivalently $M_W(c, A_Y) = \Pr[R(z_\tau) \in A_Y \mid z_0 = (c,\varnothing,m_0)]$ for a measurable $A_Y\subseteq\mathcal{Y}$); the no-truncation special case takes $\mathcal{Y}=\mathcal{C}$ with $R(c,b,m)=c$ (the window is the whole transcript), reading $c_\tau$ directly. The $\bot$ branch is exactly what $\gamma$ rejects fail-closed. This is a **specialization, not part of the definition**: a harness wrapped around a black-box API is still a harness, and $M_W$ may be any learned kernel. Where the weights are open, the geometry of $\Phi_W$ is where the substrate's continuity lives, and several downstream claims lean on it — but the definition does not.
+with the layer stack $\Phi_W$ on the residual stream as the (loosely) "manifold" core — formally just the learned high-dimensional residual-stream transformation, with manifold-proper reserved for the frontier. The inner state $z_t=(c_t,b_t,m_t)$ separates the model-visible window $c_t$ (the $\le L$ slice that slides) from the untruncated output buffer $b_t$ (the transcript the readout actually consumes, so truncation never loses it) and the parser/stop state $m_t$ (parser state, a token counter, and a clock, so the cap and timeout are functions of it), updated $m_{t+1}=\mathsf{step}(m_t,v)$, whose stop set $\mathrm{Stop}$ — EOS emitted, max-token cap, timeout, or parse-failure $\bot$ — forces $\tau=\inf\{t:m_t\in\mathrm{Stop}\}$ finite, making $M_W$ a genuine *probability* kernel rather than a sub-probability one completed by a cemetery output. The readout is total, $R : \mathcal{Z} \to \mathcal{Y}$ — a parsed tool-call, answer, or transcript, returning the parse-failure $\bot\in\mathcal{Y}$ when parsing fails; crucially $R$ is a *syntactic, verified* readout (parsing and extraction), not a semantic solver, or the $L$-wall below is void — arbitrary computation could hide in $R$ off the $\le L$ window — so $M_W(c, \cdot) = R_\#\,\mathrm{Law}(z_\tau)$, the pushforward of the stopped-state law along $R$ (equivalently $M_W(c, A_Y) = \Pr[R(z_\tau) \in A_Y \mid z_0 = (c,\varnothing,m_0)]$ for a measurable $A_Y\subseteq\mathcal{Y}$); the no-truncation special case takes $\mathcal{Y}=\mathcal{C}$ with $R(c,b,m)=c$ (the window is the whole transcript), reading $c_\tau$ directly. The $\bot$ branch is exactly what $\gamma$ rejects fail-closed. This is a **specialization, not part of the definition**: a harness wrapped around a black-box API is still a harness, and $M_W$ may be any learned kernel. Where the weights are open, the geometry of $\Phi_W$ is where the substrate's continuity lives, and several downstream claims lean on it — but the definition does not.
 
 Two stopped processes, nested: **deterministic control over stochastic dynamics over a learned kernel.** Both loops are hitting-time processes; *some* harnesses additionally read the halt set as a fixpoint or acceptance condition — iterative refinement to self-consistency is the genuine fixpoint case, while EOS, length, and tool-call syntax are not convergence. Neither loop settles because you asked it to. (The clean inner-then-outer nesting assumes tool calls fall *between* model runs; streaming or mid-generation tool calls interleave the two loops and need a finer state machine — the nesting is then an idealization.)
 
@@ -36,13 +36,13 @@ Two stopped processes, nested: **deterministic control over stochastic dynamics
 | $\gamma,\ \rho$ | the deterministic **authorization gate** $\gamma:\mathcal{S}\times\mathcal{Y}\to\mathcal{A}_\bot$ (untrusted proposal → authorized action or $\bot$) and the **fail-closed verify-and-fold-back** $\rho:\mathcal{S}\times\mathcal{Y}\times\mathcal{A}_\bot\times\mathcal{E}\to\mathcal{S}$ |
 | $H,\ \tau^\star$ | the **halt set** (absorbing) and the outer **halting time** — a hitting-time process, not a single pass |
 
-The structural fact that earns the word *controller*: $\pi$, $\gamma$, $\rho$, and the halt test are **deterministic** (and the readout $R$ too, where the transformer specialization is in play), so $\mathcal{H}$ injects no randomness of its own. Every coin is inherited from $M_W$ and $Q_E$. This determinism is *conditional* — on versioned code, configuration, model endpoint, and tool interfaces; any retry, timeout, race, or randomized routing that escapes that conditioning must be modeled explicitly as part of $Q_E$ or the controller, not waved away. More generally, control may itself be stochastic — a controller kernel $K_C(s, dc)$ over routing, sampled retries, ensemble votes, learned routers — of which the deterministic $\pi, \rho, H$ are the Dirac special case. That case is the one worth wanting: it localizes every coin to $M_W$ and $Q_E$ and keeps the controller/plant split clean. Where control is genuinely stochastic the split does not break, it widens — fold $K_C$ into the kernel and the certificate quantifies over its randomness too.
+The structural fact that earns the word *controller*: $\pi$, $\gamma$, $\rho$, and the halt test are **deterministic** (and the readout $R$ too, where the transformer specialization is in play), so $\mathcal{H}$ injects no randomness of its own. Every coin is inherited from $M_W$ and $Q_E$. This determinism is *conditional* — on versioned code, configuration, model endpoint, and tool interfaces; any retry, timeout, race, or randomized routing that escapes that conditioning must be modeled explicitly as part of $Q_E$ or the controller, not waved away. The displayed $M_W(c)$ likewise freezes endpoint, version, and sampler; a routing or config change is a state-indexed kernel $M_{\kappa(s)}$ or folds into $K_C$ — the kernel must not silently depend on config the table places in $s$. More generally, control may itself be stochastic — a controller kernel $K_C(s, dc)$ over routing, sampled retries, ensemble votes, learned routers — of which the deterministic $\pi, \gamma, \rho, H$ are the Dirac special case. That case is the one worth wanting: it localizes every coin to $M_W$ and $Q_E$ and keeps the controller/plant split clean. Where control is genuinely stochastic the split does not break, it widens — fold $K_C$ into the kernel and the certificate quantifies over its randomness too.
 
 ## Why this shape
 
 $$f(x) \;\longrightarrow\; x = f(x;\,W) \;\longrightarrow\; f(x)$$
 
-Classical software, inverted into latent geometry, then re-wrapped in classical software. The harness **re-imposes the determinism the model dissolved**: $\pi, \gamma, \rho, \tau^\star$ are ordinary designed code — a controller — whose primitive operand happens to be a stochastic oracle. That closure is why a compiler is the right mental model (staged deterministic software ports cleanly) and exactly why the analogy breaks (a compiler's primitive operation was never a coin). **The harness is the half you can reason about classically, sitting on top of the half you cannot.**
+Classical software, inverted into latent geometry, then re-wrapped in classical software. The harness **re-imposes the determinism the model dissolved**: $\pi, \gamma, \rho$, and the halt test ($H$) are ordinary designed code — a controller — whose primitive operand happens to be a stochastic oracle. That closure is why a compiler is the right mental model (staged deterministic software ports cleanly) and exactly why the analogy breaks (a compiler's primitive operation was never a coin). **The harness is the half you can reason about classically, sitting on top of the half you cannot.**
 
 ## The limit, stated honestly
 
@@ -66,13 +66,13 @@ The status of $\delta$ has to be stated carefully, because it is easy to oversel
 
 > For an agent *meant* to run forever — a coordinator, a daemon — halting is the wrong target, and $V^\star = \infty$ is the spec, not a pathology. The same drift theory then certifies **recurrence to a ready-state** instead of absorption to a halt-set. The object changes; the missing certificate does not.
 
-And the consolation rests on an assumption the world violates. The whole drift apparatus — $V^\star$, the hitting-time bound, the slack $\delta$ — assumes a **time-homogeneous kernel**: the same state transitions the same way every time. But the environment $E$ is *part of* $T$, and the world is not stationary — worse, it can be **adversarial**, an attacker choosing the tool-output *policy* — a kernel over what tools return, not the realized draw — so as to break your descent. The drift condition then stops being a fixpoint question and becomes a **minimax** one,
+And the consolation rests in part on an assumption the world violates — though less of it than it first seems. The supermartingale *bound* itself survives a nonstationary kernel, provided the conditional drift holds uniformly at every step; what genuinely needs a **time-homogeneous kernel** is $V^\star$ as a fixed function, the resolvent / fundamental-matrix identities, and the sampled-$\delta$ calibration (which assumes the very kernel it was measured on). But the environment $E$ is *part of* $T$, and the world is not stationary — worse, it can be **adversarial**, an attacker choosing the tool-output *policy* — a kernel over what tools return, not the realized draw — so as to break your descent. The drift condition then stops being a fixpoint question and becomes a **minimax** one,
 
 $$\sup_{\alpha \in \Pi}\ \int_{\mathcal{Y}}\!\int_{\mathcal{E}} V\big(\rho(s, y, \gamma(s,y), e)\big)\, Q_E^{\alpha(s,y)}\big(s, \gamma(s,y), de\big)\; M_W(\pi(s), dy) \;\le\; V(s) - \varepsilon,$$
 
-a descent that must hold in expectation over the model's own output $y$ *and* even when the adversary picks the worst admissible environment policy $\alpha(s,y)$ from the class $\Pi$ of policies the environment genuinely permits. A $V$ that certifies halting against a benign world is defeated by an adversarial one, and the measured $\delta$ bounds only the $Q_E$ you *sampled*, never the policy an attacker will choose. **This is the formal home of prompt injection** — not "the model did something bad," but the environment optimized to bend your dynamics. And the target is not merely non-halting: injection steers toward a **bad set** $B$ — wrong acceptance, data exfiltration, unauthorized tool use, privilege escalation, irreversible side effects — so security is a **reach-avoid** problem, not a liveness one. Here two reliability objects must be kept apart, because under absorbing refusal the naive forms collapse. **Success** is reaching a correct halt before *any* failure, $p_{\mathrm{succ}}(s) = \Pr_s(\tau_{H_{\mathrm{ok}}} < \tau_F)$ with $F = B \cup (H \setminus H_{\mathrm{ok}})$ — a safe refusal counts *against* it. **Safety** is never entering the bad set at all, $p_{\mathrm{safe}}(s) = \Pr_s(\tau_B = \infty)$ — a safe refusal *satisfies* it. These genuinely differ on any run that avoids $B$ without reaching $H_{\mathrm{ok}}$ ($p_{\mathrm{succ}}$ scores $0$, $p_{\mathrm{safe}}$ scores $1$): safe refusals, and — absent almost-sure absorption into $H\cup B$ — safe non-halting or endless safe retry. The tempting middle form $\Pr_s(\tau_{H_{\mathrm{ok}}} < \tau_B)$ is *not* a third object: with $H\setminus H_{\mathrm{ok}}$ absorbing, reaching $H_{\mathrm{ok}}$ before $B$ already requires reaching it before any refusal, so it coincides with $p_{\mathrm{succ}}$. Safety is certified by a barrier / avoidance certificate for $B$; success needs that plus the reach part — a hitting-time drift toward $H_{\mathrm{ok}}$. Fail-closed control is the disturbance-rejection margin for both, but split by reversibility: the gate $\gamma$ caps how far an adversarial world reaches into *side effects* and widens the gap to $B$ (it is the margin for the irreversible part), while $\rho$ validates the response and folds back, rejecting bad state after the action has run — which cannot undo an authorized side effect. In this language, security is robustness of the reach-avoid certificate. And injection is not confined to the post-model kernel $Q_E$: poisoned retrieval, prompt-injected pages, and malicious tool metadata enter through $\pi$'s *inputs*, before generation — so the adversary lives wherever untrusted content enters the state/context-construction pipeline, which is why input provenance and the gate $\gamma$ both matter, not post-hoc verification alone. (For $B$ to capture irreversible side effects rather than only states, the side-effect ledger must itself live in $\mathcal{S}$.)
+a descent that must hold in expectation over the model's own output $y$ *and* even when the adversary picks the worst admissible environment policy $\alpha(s,y)$ from the class $\Pi$ of policies the environment genuinely permits — every $\alpha\in\Pi$ must still respect rejection, $\gamma(s,y)=\bot \Rightarrow Q_E^{\alpha}(s,\bot,\cdot)=\delta_{e_0}$, or the adversary resurrects side effects the gate refused. A $V$ that certifies halting against a benign world is defeated by an adversarial one, and the measured $\delta$ bounds only the $Q_E$ you *sampled*, never the policy an attacker will choose. **This is the formal home of prompt injection** — not "the model did something bad," but the environment optimized to bend your dynamics. And the target is not merely non-halting: injection steers toward a **bad set** $B$ — wrong acceptance, data exfiltration, unauthorized tool use, privilege escalation, irreversible side effects — so security is a **reach-avoid** problem, not a liveness one. Here two reliability objects must be kept apart, because under absorbing refusal the naive forms collapse. **Success** is reaching a correct halt before *any* failure, $p_{\mathrm{succ}}(s) = \Pr_s(\tau_{H_{\mathrm{ok}}} < \tau_F)$ with $F = B \cup (H \setminus H_{\mathrm{ok}})$ — a safe refusal counts *against* it. **Safety** is never entering the bad set at all, $p_{\mathrm{safe}}(s) = \Pr_s(\tau_B = \infty)$ — a safe refusal *satisfies* it. These genuinely differ on any run that avoids $B$ without reaching $H_{\mathrm{ok}}$ ($p_{\mathrm{succ}}$ scores $0$, $p_{\mathrm{safe}}$ scores $1$): safe refusals, and — absent almost-sure absorption into $H\cup B$ — safe non-halting or endless safe retry. The tempting middle form $\Pr_s(\tau_{H_{\mathrm{ok}}} < \tau_B)$ is *not* a third object: with $H\setminus H_{\mathrm{ok}}$ absorbing, reaching $H_{\mathrm{ok}}$ before $B$ already requires reaching it before any refusal, so it coincides with $p_{\mathrm{succ}}$. Safety is certified by a barrier / avoidance certificate for $B$; success needs that plus the reach part — a hitting-time drift toward $H_{\mathrm{ok}}$. Fail-closed control is the disturbance-rejection margin for both, but split by reversibility: the gate $\gamma$ caps how far an adversarial world reaches into *side effects* and widens the gap to $B$ (it is the margin for the irreversible part), while $\rho$ validates the response and folds back, rejecting bad state after the action has run — which cannot undo an authorized side effect. In this language, security is robustness of the reach-avoid certificate. And injection is not confined to the post-model kernel $Q_E$: poisoned retrieval, prompt-injected pages, and malicious tool metadata enter through $\pi$'s *inputs*, before generation — so the adversary lives wherever untrusted content enters the state/context-construction pipeline, which is why input provenance and the gate $\gamma$ both matter, not post-hoc verification alone. (For $B$ to capture irreversible side effects rather than only states, the side-effect ledger must itself live in $\mathcal{S}$, and the response $e$ must be an *effect record* carrying the ledger outcome — not just API bytes — since only $\rho$ writes external effects into $s$.)
 
-There is a **second wall, orthogonal to the first.** It binds not the full harness state $\mathcal{S}$ but the **model-visible working memory** $\mathcal{C} = \mathcal{V}^{\le L}$ — bounded by the context length $L$. That bound is *not* the incompressibility of $V^\star$ (a fact about the parameters $W$ — the **dictionary**, fixed at training); it is a fact about the inner kernel's **working memory** (the $L\times d$ residual stream — the **desk**). $\mathcal{S}$ itself may be far richer — files, databases, vector stores, durable memory, queues — but that is *external* memory the shell supplies, and the distinction is the point: every external read still passes *through* the $\le L$ window to touch computation, so external stores extend addressable storage without extending the per-pass resident set. The shell can page; the plant cannot grow its desk. (What follows is heuristic, not definition-level: the complexity claims turn on depth, precision, and architecture, and belong with the frontier, not the core.) Under the standard fixed-depth, fixed-precision theoretical model a single forward pass is constant-depth ($\mathsf{TC}^0$) — *suggestive* for deployed models, not literal, and shifting once depth grows with context (log-depth variants escape parts of it); the qualitative point survives the caveats: one pass buys bounded sequential depth, so the loop buys more only by emitting tokens: **the context window is the tape, the autoregressive loop is the read/write head**, and — in the variable-$L$, fixed-precision idealization — the model-mediated inner computation behaves like a linear-bounded automaton, its reachable fixpoints capped by space-$O(L)$ computability (chain-of-thought is register-spilling onto that tape). This is a *second* obstruction beside divergence, and it concerns *success*, not raw halting. Split the terminal set: let $H$ be any halt state (including fail-closed refusal) and $H_{\mathrm{ok}} \subseteq H$ the successful, accepting halts, with $V^\star_{\mathrm{ok}}(s) = \mathbb{E}[\tau_{H_{\mathrm{ok}}} \mid s_0 = s]$ taken on the process where $H \setminus H_{\mathrm{ok}}$ — halting wrong, refusing, failing closed — is *absorbing failure*, so a run that fails closed before acceptance has infinite accepting hitting time unless the spec explicitly restarts it — hence unconditional $V^\star_{\mathrm{ok}}$ is infinite whenever pre-acceptance failure has positive probability, which is why the workable reliability object is the success probability $p_{\mathrm{succ}}$ (above) or, for restarting specs, the regenerative expected time. Then $U_{\mathcal{H}}(L)$ — harness-relative, since the shell's decompositions and verified tools determine what can be paged or outsourced — is the set of tasks whose **irreducible per-step model-mediated working set** exceeds $L$ — not tasks whose *data* exceeds $L$ (those the shell can page), and not work that can be **discharged to a verified external tool** (a solver, interpreter, or compiler computes off-context). For a task in $U_{\mathcal{H}}(L)$ the raw chain may still hit $H$ — by failing closed, refusing, or returning a wrong answer — so $V^\star = \mathbb{E}[\tau_H \mid s]$ stays perfectly well-defined; what blows up is $V^\star_{\mathrm{ok}}$, the expected time to a *correct* halt, which is infinite under a formal success predicate, or undefined if no such predicate has been specified. The honest statement is about the finite-success domain: $\mathrm{dom}_{<\infty}(V^\star_{\mathrm{ok}}) \subseteq \mathrm{reachable}_{\mathcal{H}}(L) \setminus D$ — both the reachable set and the divergent set $D$ relative to $\mathcal{H}$, exactly as $U_{\mathcal{H}}(L)$ is. The two walls **trade**: parametric memory $|W|$ and working memory $L$ are substitutable on one budget line — the pretraining-vs-inference-scaling axis. And the bound is inherent to *finite working memory*, not attention specifically: state-space models embody it differently (a fixed-size recurrent state rather than an $L$-window), and real attention's usable tape is shorter than $L$ (lost-in-the-middle).
+There is a **second wall, orthogonal to the first.** It binds not the full harness state $\mathcal{S}$ but the **model-visible working memory** $\mathcal{C} = \mathcal{V}^{\le L}$ — bounded by the context length $L$. That bound is *not* the incompressibility of $V^\star$ (a fact about the parameters $W$ — the **dictionary**, fixed at training); it is a fact about the inner kernel's **working memory** (the $L\times d$ residual stream — the **desk**). $\mathcal{S}$ itself may be far richer — files, databases, vector stores, durable memory, queues — but that is *external* memory the shell supplies, and the distinction is the point: every external read still passes *through* the $\le L$ window to touch computation, so external stores extend addressable storage without extending the per-pass resident set. The shell can page; the plant cannot grow its desk. (What follows is heuristic, not definition-level: the complexity claims turn on depth, precision, and architecture, and belong with the frontier, not the core.) Under the standard fixed-depth, log-precision theoretical model a single forward pass is in constant-depth $\mathsf{TC}^0$ — *suggestive* for deployed models, not literal, and shifting once depth grows with context (log-depth variants escape parts of it); the qualitative point survives the caveats: one pass buys bounded sequential depth, so the loop buys more only by emitting tokens: **the context window is the tape, the autoregressive loop is the read/write head**, and — in the variable-$L$, fixed-precision idealization — the model-mediated inner computation behaves like a linear-bounded automaton, its reachable fixpoints capped by space-$O(L)$ computability (chain-of-thought is register-spilling onto that tape). This is a *second* obstruction beside divergence, and it concerns *success*, not raw halting. Split the terminal set: let $H$ be any halt state (including fail-closed refusal) and $H_{\mathrm{ok}} \subseteq H$ the successful, accepting halts, with $V^\star_{\mathrm{ok}}(s) = \mathbb{E}[\tau_{H_{\mathrm{ok}}} \mid s_0 = s]$ taken on the process where $H \setminus H_{\mathrm{ok}}$ — halting wrong, refusing, failing closed — is *absorbing failure*, so a run that fails closed before acceptance has infinite accepting hitting time unless the spec explicitly restarts it — hence unconditional $V^\star_{\mathrm{ok}}$ is infinite whenever pre-acceptance failure has positive probability, which is why the workable reliability object is the success probability $p_{\mathrm{succ}}$ (above) or, for restarting specs, the regenerative expected time. Then $U_{\mathcal{H}}(L)$ — harness-relative, since the shell's decompositions and verified tools determine what can be paged or outsourced — is the set of tasks whose **irreducible per-step model-mediated working set** exceeds $L$ — not tasks whose *data* exceeds $L$ (those the shell can page), and not work that can be **discharged to a verified external tool** (a solver, interpreter, or compiler computes off-context). For a task in $U_{\mathcal{H}}(L)$ the raw chain may still hit $H$ — by failing closed, refusing, or returning a wrong answer — so $V^\star = \mathbb{E}[\tau_H \mid s]$ stays perfectly well-defined; what blows up is $V^\star_{\mathrm{ok}}$, the expected time to a *correct* halt, which is infinite under a formal success predicate, or undefined if no such predicate has been specified. The honest statement is about the finite-success domain: $\mathrm{dom}_{<\infty}(V^\star_{\mathrm{ok}}) \subseteq \mathrm{reachable}_{\mathcal{H}}(L) \setminus D$ — both the reachable set and the divergent set $D$ relative to $\mathcal{H}$, exactly as $U_{\mathcal{H}}(L)$ is. The two walls **trade**: parametric memory $|W|$ and working memory $L$ are substitutable on one budget line — the pretraining-vs-inference-scaling axis. And the bound is inherent to *finite working memory*, not attention specifically: state-space models embody it differently (a fixed-size recurrent state rather than an $L$-window), and real attention's usable tape is shorter than $L$ (lost-in-the-middle).
 
 ## Where it cashes out
 
@@ -80,7 +80,7 @@ This is not ornament; the decomposition is load-bearing in the design.
 
 - **$\pi$ is a progressively-lowered dialect stack** — raw input → intent → plan → tool-call → the neutral wire IR — each level a deterministic pass with its own verifier. The per-step drift $r(s)=\mathbb{E}[\hat V(s_{n+1})\mid s]-\hat V(s)$ splits by coordinate, $r = r_{\text{shell}} + r_{\text{plant}}$ — presuming an additively separable $\hat V$, or a declared scheme attributing each step's drift to shell vs. plant coordinates: the shell term is an *exact, designed* descent (each lowering strictly narrows the admissible-meaning set — a well-founded descent we build by hand), the plant term is the irreducible residue. **Syntactic soundness is free; semantic adequacy is not.** Relative to a formal schema and a correct validator, schemas, types, and boundary checks go into the shell at zero probabilistic cost; whether the lowered task still *means* what the user intended stays empirical, because natural language supplies no source-language standard to check against.
 - **$\rho$ is fail-closed verification** — validate at every boundary, never let malformed state flow downstream. The discipline transfers from compilers in *form*; the *teeth* do not, because a harness has no source-language standard — natural language is, in effect, all undefined behavior — there is no complete formal source-language semantics to check against. And $\rho$ must be *deterministic*: if verification is itself an LLM judge, that is another learned kernel call — it belongs in $M_W$, not in $\rho$.
-- **$\delta$, $\mu(D)$, $\mathrm{Var}[\tau^\star]$ are what you measure** — not derive. You instrument the certificate precisely because it cannot be proven, only observed.
+- **$\delta$, $\mu(D)$, $\mathrm{Var}[\tau^\star]$ are what you measure** — not derive. You instrument the certificate precisely because the architecture does not hand it to you — you estimate it unless it is separately certified.
 
 ## How this could be wrong
 
@@ -109,7 +109,7 @@ With that caveat, **the interlingua and the certificate are one object seen twic
 
 Borrowed theorems are real; the framings are not — keep them separate.
 
-**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). Dialect-stack architecture: MLIR (Lattner et al., CGO 2021, arXiv:2002.11054); learned pass-ordering: MLGO (Trofin et al., arXiv:2101.04808). Single-pass constant-depth under fixed-depth/fixed-precision assumptions: the $\mathsf{TC}^0$ transformer-expressivity result (Merrill & Sabharwal, *The Parallelism Tradeoff: Limitations of Log-Precision Transformers*, TACL 2023) — with the caveat that added autoregressive steps escape it (Merrill & Sabharwal, *The Expressive Power of Transformers with Chain of Thought*, ICLR 2024), and growing-precision variants change 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). 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. These organize the design; they are not results.