From 4baf6f81c329c3da2f4d0598dcdb4b1deb9ef3c4 Mon Sep 17 00:00:00 2001 From: Patrick Buckley Date: Fri, 26 Jun 2026 03:57:36 -0700 Subject: [PATCH] docs(hypothesis): clarity pass, GitHub-render fixes, consistency linter (#720) * docs(hypothesis): clarity pass, GitHub-render fixes, consistency linter Document (HYPOTHESIS.md): - split the dense "Formal" definition into labeled subsections - define the load-bearing terms: certificate (proven witness vs measured surrogate) and the controller / plant (= M_W) / shell triad - corrections: three-way drift split (+ r_env), scope the success/safety collapse to absorbing refusal, unify tau*->tau_H and drop the orphaned bare tau - calibrations: pin the incompressibility conjecture (still conjectural), mark the interlingua=certificate identity as figure, soften the two-walls trade - GitHub math rendering: brace command-subscripts (_\bot -> _{\bot}, etc.) so the markdown emphasis parser stops breaking inline math; replace R_\# with R_{\sharp} (\# unescapes to a raw # in GitHub math) Linter (lint_hypothesis.py): - deterministic consistency checks A-G; G adds an orphan/redundant-declaration scan that catches the bare-tau failure mode - residue guards so tau^star, unbraced _\cmd subscripts, and \# cannot return * fix(hypothesis): make lint_hypothesis.py pass ruff under py311 - precompute the inline-$ count so no backslash sits inside an f-string expression (backslashes in f-strings are 3.12+; the project targets 3.11) - split the one-line import (E401/I001); open HYPOTHESIS.md via a context manager (SIM115) --- HYPOTHESIS.md | 46 +++++---- lint_hypothesis.py | 246 +++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 274 insertions(+), 18 deletions(-) create mode 100644 lint_hypothesis.py diff --git a/HYPOTHESIS.md b/HYPOTHESIS.md index 9727d289..ef4d2a4c 100644 --- a/HYPOTHESIS.md +++ b/HYPOTHESIS.md @@ -10,17 +10,27 @@ Most descriptions of an agent framework are a feature list. This is an attempt a *Informal.* A harness is a **stopped, deterministically-controlled Markov process on task-state, closed around a stopped autoregressive process on context-space, driven by a learned model kernel** — a deterministic controller in closed loop with a stochastic learned plant. -*Formal.* A harness is a tuple $\mathcal{H} = (\mathcal{S}, \mathcal{C}, \mathcal{Y}, \mathcal{A}, \mathcal{E}, \pi, M_W, \gamma, Q_E, \rho, H, H_{\mathrm{ok}}, B)$ over **standard Borel** spaces: a deterministic lowering $\pi:\mathcal{S}\to\mathcal{C}$; a stochastic model-run kernel $M_W(c, dy)$ into a readout space $\mathcal{Y}$ (which includes the parse-failure $\bot$, so $M_W$ and $\gamma$ are total over it); a deterministic **authorization gate** $\gamma:\mathcal{S}\times\mathcal{Y}\to\mathcal{A}_\bot$ that parses/validates the model output into an authorized action in $\mathcal{A}$ or rejects it as $\bot$; a stochastic environment/tool kernel $Q_E:\mathcal{S}\times\mathcal{A}_\bot\rightsquigarrow\mathcal{E}$ on the authorized action (rejection included, with $Q_E(s,\bot,\cdot)=\delta_{e_0}$ for a distinguished no-op response $e_0\in\mathcal{E}$); and a deterministic verify-and-fold-back map $\rho:\mathcal{S}\times\mathcal{Y}\times\mathcal{A}_\bot\times\mathcal{E}\to\mathcal{S}$. The terminal set is an absorbing halt set $H\subseteq\mathcal{S}$ (the daemon "ready-state" recurrence of the note below is a separate, non-absorbing object) with accepting subset $H_{\mathrm{ok}}\subseteq H$; separately, a bad set $B\subseteq\mathcal{S}$ ($B\cap H_{\mathrm{ok}}=\varnothing$) marks the unsafe states for reach-avoid, possibly entered before any halt; hitting times are $\tau_A=\inf\{n\ge 0:s_n\in A\}$, and $\tau,\tau^\star$ are stopping times for the natural filtrations. The induced outer transition kernel, for $s\notin H$, is +*Formal — the objects.* A harness is a tuple $\mathcal{H} = (\mathcal{S}, \mathcal{C}, \mathcal{Y}, \mathcal{A}, \mathcal{E}, \pi, M_W, \gamma, Q_E, \rho, H, H_{\mathrm{ok}}, B)$ over **standard Borel** spaces: a deterministic lowering $\pi:\mathcal{S}\to\mathcal{C}$; a stochastic model-run kernel $M_W(c, dy)$ into a readout space $\mathcal{Y}$ (which includes the parse-failure $\bot$, so $M_W$ and $\gamma$ are total over it); a deterministic **authorization gate** $\gamma:\mathcal{S}\times\mathcal{Y}\to\mathcal{A}_{\bot}$ that parses/validates the model output into an authorized action in $\mathcal{A}$ or rejects it as $\bot$; a stochastic environment/tool kernel $Q_E:\mathcal{S}\times\mathcal{A}_{\bot}\rightsquigarrow\mathcal{E}$ on the authorized action (rejection included, with $Q_E(s,\bot,\cdot)=\delta_{e_0}$ for a distinguished no-op response $e_0\in\mathcal{E}$); and a deterministic verify-and-fold-back map $\rho:\mathcal{S}\times\mathcal{Y}\times\mathcal{A}_{\bot}\times\mathcal{E}\to\mathcal{S}$. + +*Terminal structure.* The terminal set is an absorbing halt set $H\subseteq\mathcal{S}$ (the daemon "ready-state" recurrence of the note below is a separate, non-absorbing object) with accepting subset $H_{\mathrm{ok}}\subseteq H$; separately, a bad set $B\subseteq\mathcal{S}$ ($B\cap H_{\mathrm{ok}}=\varnothing$) marks the unsafe states for reach-avoid, possibly entered before any halt; hitting times are $\tau_A=\inf\{n\ge 0:s_n\in A\}$, and $\tau_H$ is a stopping time for the natural filtration. + +*The outer kernel.* The induced outer transition kernel, for $s\notin H$, is $$T(s, A) = \int_{\mathcal{Y}}\!\int_{\mathcal{E}} \mathbf{1}_A\!\big(\rho(s, y, \gamma(s,y), e)\big)\; Q_E\big(s, \gamma(s,y), de\big)\; M_W(\pi(s), dy), \qquad T(s,A)=\mathbf{1}_A(s)\ \text{ for } s\in H,$$ -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. 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. +and the harness runs $s_{n+1} \sim T(s_n)$ from an initial $s_0 \sim \mu_0$ until $\tau_H = \inf\{n : s_n \in H\}$. 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 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.) + +*Fail-closed.* 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 $H_{\mathrm{ok}}$. + +*The harness invariants.* 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. + +*Beyond the stationary kernel.* 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 -$$M_W(c, \cdot) = \mathrm{Law}\big(R(z_\tau)\big), \quad z_t = (c_t, b_t, m_t), \quad v \sim K_W(c_t, \cdot), \quad K_W(c, v) = (U \circ \Phi_W \circ \mathrm{Emb})(c)[v], \quad c_{t+1} = \mathrm{suffix}_{\le L}(c_t\!\cdot\! v),\ \ b_{t+1} = b_t\!\cdot\! v,\ \ m_{t+1} = \mathsf{step}(m_t, v),\ \ \tau=\inf\{t:m_t\in\mathrm{Stop}\}.$$ +$$M_W(c, \cdot) = \mathrm{Law}\big(R(z_{\tau})\big), \quad z_t = (c_t, b_t, m_t), \quad v \sim K_W(c_t, \cdot), \quad K_W(c, v) = (U \circ \Phi_W \circ \mathrm{Emb})(c)[v], \quad c_{t+1} = \mathrm{suffix}_{\le L}(c_t\!\cdot\! v),\ \ b_{t+1} = b_t\!\cdot\! v,\ \ m_{t+1} = \mathsf{step}(m_t, v),\ \ \tau=\inf\{t:m_t\in\mathrm{Stop}\}.$$ -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 $m_t$ (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. +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 $M_W(c, \cdot) = R_{\sharp}\,\mathrm{Law}(z_{\tau})$, 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.) @@ -32,11 +42,11 @@ Two stopped processes, nested: **deterministic control over stochastic dynamics | $s \in \mathcal{S}$ | task-state: IR / dialect stack, tool results, plan, counters, **and every mutable interface variable** (model/tool versions, permissions, retrieved context) — only Markov *after* that augmentation | | $\pi : \mathcal{S} \to \mathcal{C}$ | **lowering** — prompt construction, dialect lowering, effective-program selection (deterministic) | | $M_W(c, dy)$ | the **model-run kernel** (inner solver) — a stopped autoregressive process; $\Phi_W$ is the residual-stream ("manifold") core in the transformer case | -| $Q_E(s, a, de)$ | the **environment/tool kernel** on the authorized action $a\in\mathcal{A}_\bot$ (with $Q_E(s,\bot,\cdot)=\delta_{e_0}$, the no-op $e_0$) — tool effects, API responses, the world (possibly adversarial) | -| $\gamma,\ \rho$ | 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 | +| $Q_E(s, a, de)$ | the **environment/tool kernel** on the authorized action $a\in\mathcal{A}_{\bot}$ (with $Q_E(s,\bot,\cdot)=\delta_{e_0}$, the no-op $e_0$) — tool effects, API responses, the world (possibly adversarial) | +| $\gamma,\ \rho$ | 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** $\rho:\mathcal{S}\times\mathcal{Y}\times\mathcal{A}_{\bot}\times\mathcal{E}\to\mathcal{S}$ | +| $H,\ \tau_H$ | 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. 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. +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 split — deterministic code around a stochastic oracle — wears two names. In control-theory terms it is **controller vs. plant**: the controller is those deterministic maps; the **plant** is the learned kernel $M_W$, *plant* in its exact sense — the element with its own dynamics you steer but do not author. In engineering terms it is **shell vs. plant**: the **shell** is the entire deterministic outer harness — the control logic *plus* the external memory and tools it administers (the files, databases, vector stores below) — of which the controller is just the control-logic slice. So *shell : plant :: the part you write : the part you don't*; $M_W$ is the only thing on the right, while the environment $Q_E$ is the world the actions meet — a disturbance into the loop, not the plant. 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 @@ -46,23 +56,23 @@ Classical software, inverted into latent geometry, then re-wrapped in classical ## The limit, stated honestly -**Raw halting is cheap; correct halting is not.** A deterministic budget — augment $s$ with a counter $k$ decremented each outer step, halting at $k=0$ — makes $V(s)=k$ a trivial Lyapunov certificate for *halting*, so the architecture does not lack a halting guarantee by construction. What it lacks for free is a certificate of *correct, safe, successful* halting under the learned dynamics. The un-budgeted halting object is still worth stating, since it shows where even the easy guarantee comes from: a certificate would be *sufficient* for almost-sure halting with bounded expected runtime — a $V \ge 0$ with +**Raw halting is cheap; correct halting is not.** A **certificate** is a *witness*: a checkable object — here a Lyapunov/drift function $V \ge 0$ — that *provably* satisfies a condition entailing the guarantee, through a standard supermartingale / optional-stopping theorem (the target picks the condition: drift toward $H$ for halting, a barrier for safety, reach-avoid for success). It is not the property, only an object cheap to check and hard to produce. One word then carries two senses, and the seam between them is what this section is about: the **proven** certificate, a $V$ whose bound actually holds; and the **measured** surrogate you fall back on when the architecture exhibits none — a candidate $\hat V$ with a sampled slack $\delta$, a *calibrated risk metric, not a certificate* until that bound is proven (or held to a high-confidence worst case). The gap between the two is the whole honest-limit argument. A deterministic budget — augment $s$ with a counter $k$ decremented each outer step, halting at $k=0$ — makes $V(s)=k$ a trivial Lyapunov certificate for *halting*, so the architecture does not lack a halting guarantee by construction. What it lacks for free is a certificate of *correct, safe, successful* halting under the learned dynamics. The un-budgeted halting object is still worth stating, since it shows where even the easy guarantee comes from: a certificate would be *sufficient* for almost-sure halting with bounded expected runtime — a $V \ge 0$ with $$\mathbb{E}[\,V(s_{n+1}) \mid s_n\,] \le V(s_n) - \varepsilon \quad\text{off the halt set}$$ -bounds $\mathbb{E}[\tau^\star] \le V(s_0)/\varepsilon$ under the usual integrability and optional-stopping conditions. Nothing in the harness hands you such a $V$ the way a compiler's structure does: a specific compiler analysis gets its $V$ for free where a finite-height lattice *is* a well-founded descent — termination by construction *for that analysis*, not for a whole compiler — and the harness has no analogous built-in descent for its model/environment loop. +bounds $\mathbb{E}[\tau_H] \le V(s_0)/\varepsilon$ under the usual integrability and optional-stopping conditions. Nothing in the harness hands you such a $V$ the way a compiler's structure does: a specific compiler analysis gets its $V$ for free where a finite-height lattice *is* a well-founded descent — termination by construction *for that analysis*, not for a whole compiler — and the harness has no analogous built-in descent for its model/environment loop. But the relevant $V$ is not *absent* — and this is the subtlety the blunt phrasing erased. The minimal certificate exists and is **forced**: it is the expected halting time itself, -$$V^\star(s) = \mathbb{E}[\,\tau^\star \mid s_0 = s\,],$$ +$$V^\star(s) = \mathbb{E}[\,\tau_H \mid s_0 = s\,],$$ -finite wherever $H$ is reached in finite expected time — the domain $\{s : \mathbb{E}_s[\tau_H] < \infty\}$ — though note this $V^\star$ certifies *halting* (reaching the terminal set $H$ at all), not *correct* halting; the stronger object, the expected time to an accepting $H_{\mathrm{ok}} \subseteq H$, is $V^\star_{\mathrm{ok}}$, taken up at the second wall below. So the honest claim splits in two: the architecture provides no certificate *for free*, and the one that exists is — **conjecturally, not as a theorem** — a functional of all of $W$ and the environment that does not compress below model scale. The compiler's certificate is structurally trivial; ours is *plausibly* as hard as the dynamics, though whether useful compressed certificates exist for structured sub-tasks is open. This is the quantitative form of *you can borrow how LLVM is built — not, in general, why it is correct.* +finite wherever $H$ is reached in finite expected time — the domain $\{s : \mathbb{E}_s[\tau_H] < \infty\}$ — though note this $V^\star$ certifies *halting* (reaching the terminal set $H$ at all), not *correct* halting; the stronger object, the expected time to an accepting $H_{\mathrm{ok}} \subseteq H$, is $V^\star_{\mathrm{ok}}$, taken up at the second wall below. So the honest claim splits in two: the architecture provides no certificate *for free*, and the one that exists is — **conjecturally, not as a theorem** — a functional of all of $W$ and the environment that does not compress below model scale — concretely, *no certificate of description length materially below $|W|$ satisfies the drift condition*, because any $V$ that genuinely certifies the dynamics must encode about as much about the world as the weights do (the dynamics it certifies *are* the weights). The compiler's certificate is structurally trivial; ours is *plausibly* as hard as the dynamics, though whether useful compressed certificates exist for structured sub-tasks is open. This is the quantitative form of *you can borrow how LLVM is built — not, in general, why it is correct.* So you never compute $V^\star$. You pick a candidate $\hat V$ and **estimate its drift slack** $$\delta = \sup_{s \notin H}\Big(\mathbb{E}[\,\hat V(s_{n+1}) \mid s_n\,] - \hat V(s_n) + \varepsilon\Big).$$ -The status of $\delta$ has to be stated carefully, because it is easy to oversell. If you can establish a *high-confidence upper bound* on the true worst-case slack and it is $\le 0$, optional stopping hands you a real, conservative certificate, $\mathbb{E}[\tau^\star] \le \hat V(s_0)/\varepsilon$. But an *empirical* $\delta$ estimated from sampled states is **not** a certificate: a measured $\delta > 0$ may mean the candidate $\hat V$ is poor, the sampled distribution missed rare failures, the supremum was never attained in-sample, the process is non-stationary, or the state abstraction is not Markov. So $\delta$ is **the number on the dashboard** — a *calibrated risk metric*, the evaluable surrogate for a guarantee the geometry will not give you, and a genuine bound only once it is statistically controlled against rare-event and adversarial tests. A weaker result is still useful: a true bound $\delta \le \bar\delta < \varepsilon$ (rather than $\le 0$) leaves descent intact with effective slack $\varepsilon - \bar\delta$ and $\mathbb{E}_s[\tau_H] \le \hat V(s)/(\varepsilon - \bar\delta)$. And the empirical quantity is distributional, not a supremum — write $\delta_\nu$ for drift averaged over a sampled $\nu$, reserving $\delta_{\sup}$ for the worst-case bound; only $\delta_{\sup}$ certifies. Its empirical noise floor and residual risk are driven by the measure $\mu(D)$ of the divergent region $D=\{s:\mathbb{E}_s[\tau_H]=\infty\}$ (states from which $H$ is not reached in finite expected time, under the reference/sampling measure $\mu$), the coverage of the sampled state distribution, and the hitting-time variance $\mathrm{Var}[\tau^\star]$ — properties of the trained weights, the environment, and the evaluation distribution, knowable only a posteriori. +The status of $\delta$ has to be stated carefully, because it is easy to oversell. If you can establish a *high-confidence upper bound* on the true worst-case slack and it is $\le 0$, optional stopping hands you a real, conservative certificate, $\mathbb{E}[\tau_H] \le \hat V(s_0)/\varepsilon$. But an *empirical* $\delta$ estimated from sampled states is **not** a certificate: a measured $\delta > 0$ may mean the candidate $\hat V$ is poor, the sampled distribution missed rare failures, the supremum was never attained in-sample, the process is non-stationary, or the state abstraction is not Markov. So $\delta$ is **the number on the dashboard** — a *calibrated risk metric*, the evaluable surrogate for a guarantee the geometry will not give you, and a genuine bound only once it is statistically controlled against rare-event and adversarial tests. A weaker result is still useful: a true bound $\delta \le \bar\delta < \varepsilon$ (rather than $\le 0$) leaves descent intact with effective slack $\varepsilon - \bar\delta$ and $\mathbb{E}_s[\tau_H] \le \hat V(s)/(\varepsilon - \bar\delta)$. And the empirical quantity is distributional, not a supremum — write $\delta_{\nu}$ for drift averaged over a sampled $\nu$, reserving $\delta_{\sup}$ for the worst-case bound; only $\delta_{\sup}$ certifies. Its empirical noise floor and residual risk are driven by the measure $\mu(D)$ of the divergent region $D=\{s:\mathbb{E}_s[\tau_H]=\infty\}$ (states from which $H$ is not reached in finite expected time, under the reference/sampling measure $\mu$), the coverage of the sampled state distribution, and the hitting-time variance $\mathrm{Var}[\tau_H]$ — properties of the trained weights, the environment, and the evaluation distribution, knowable only a posteriori. > 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. @@ -70,17 +80,17 @@ And the consolation rests in part on an assumption the world violates — though $$\sup_{\alpha \in \Pi}\ \int_{\mathcal{Y}}\!\int_{\mathcal{E}} V\big(\rho(s, y, \gamma(s,y), e)\big)\, Q_E^{\alpha(s,y)}\big(s, \gamma(s,y), de\big)\; M_W(\pi(s), dy) \;\le\; V(s) - \varepsilon,$$ -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 — 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 $s$.) +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 $p_{\mathrm{succ}}$ — but only under that absorbing-refusal assumption; once the spec retries (the non-terminal fail-closed of the definition), a run may refuse, restart, and still reach $H_{\mathrm{ok}}$ before $B$, and the middle form re-separates as a genuine third object. 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, 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). +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** — *directionally, not as a literal exchange rate*: parametric memory $|W|$ and working memory $L$ press on the same budget along the pretraining-vs-inference-scaling axis, with no clean unit-for-unit substitution of one for the other. 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 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. +- **$\pi$ 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, $r = r_{\text{shell}} + r_{\text{plant}} + r_{\text{env}}$ — presuming an additively separable $\hat V$, or a declared scheme attributing each step's drift to shell, plant, and environment 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 ($M_W$) is the irreducible residue, and the environment term ($Q_E$) is the one an adversary controls — the very quantity the minimax descent must bound, which the old two-way split folded out of sight. **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 the architecture does not hand it to you — you estimate it unless it is separately certified. +- **$\delta$, $\mu(D)$, $\mathrm{Var}[\tau_H]$ 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 @@ -99,7 +109,7 @@ Each claim is operational, not merely rhetorical: If $V^\star$ is incompressible only in *token* coordinates, the right change of coordinates might compress it — and that change of coordinates is a representation of meaning itself. Cost-to-go and representation co-determine each other: where the Koopman operator is diagonalizable — a point-spectrum idealization, since mixing dynamics carry continuous spectrum and admit no eigenbasis — the eigenbasis that linearizes the dynamics is also the one in which the certificate decomposes, and even then only for a $V$ in the span of those eigenfunctions; in reinforcement learning the discounted successor representation is the resolvent $(I-\beta P)^{-1}$ — discount $\beta$, not the gate $\gamma$ — with $V$ a *linear readout* of it — and in the undiscounted, absorbing case that actually matches a stopped harness the same role is played, in the finite setting — and countable settings where the Neumann series converges — by the **fundamental matrix** $N = \sum_{n \ge 0} Q_{\mathrm{tr}}^{\,n}$ (written $(I - Q_{\mathrm{tr}})^{-1}$ when the inverse exists), where $Q_{\mathrm{tr}}$ is the sub-stochastic kernel restricted to $H^c$ (transitions before absorption at $H$) and the row sums $N\mathbf{1}$ *are* $V^\star$ on the finite-mean hitting domain; on general state spaces the same series is read as the potential (Green) operator $G$, with $G\mathbf{1} = V^\star$ wherever it converges. Each of these is a clean identity only for a fixed, time-homogeneous kernel — under a nonstationary $Q_{E,n}$ the resolvent and fundamental matrix dissolve into a time-ordered product, and under an *adaptive* adversary into a controlled / game-value operator, so what is identity in the stationary regime is analogy beyond it. -With that caveat, **the interlingua and the certificate are one object seen twice** — and the reason neither can be written in closed form is the same "all undefined behavior": no canonical lowering of meaning, hence no finite header-file for either. The only representation of both is $W$ — a band-limited, lossy compression of a scale-free meaning-space, sharp where the record is thick and blurred where it thinned. That a finite object renders an infinite one *lossily but honestly* — declaring its resolution, and where it is unsure — is not a lie; it is the most an $f(\cdot\,;W)$ can do. **The search for $V$ and the search for the interlingua are not two programs. They are one** — and the day either is written in closed form, so is the other, or we will have proven why neither can be. +With that caveat, **the interlingua and the certificate are one object seen twice** — and the reason neither can be written in closed form is the same "all undefined behavior": no canonical lowering of meaning, hence no finite header-file for either. The only representation of both is $W$ — a band-limited, lossy compression of a scale-free meaning-space, sharp where the record is thick and blurred where it thinned. That a finite object renders an infinite one *lossily but honestly* — declaring its resolution, and where it is unsure — is not a lie; it is the most an $f(\cdot\,;W)$ can do. **The search for $V$ and the search for the interlingua are not two programs. They are one** — and the day either is written in closed form, so is the other, or we will have proven why neither can be. Read this as *figure*, not a lurking theorem: the only precise version would need the Koopman eigenbasis to fall on the very coordinates that lower meaning, and the mixing-spectrum caveat above already concedes that eigenbasis does not exist — which guts it. It is the least-defensible claim in this document, and it should announce that rather than imply a rigor it has not got. --- diff --git a/lint_hypothesis.py b/lint_hypothesis.py new file mode 100644 index 00000000..807ac194 --- /dev/null +++ b/lint_hypothesis.py @@ -0,0 +1,246 @@ +#!/usr/bin/env python3 +""" +Consistency linter for HYPOTHESIS.md. +Deterministic checks — no model, no confabulation: + A. delimiter / emphasis balance + B. residue regexes (things prior rounds fixed must not reappear) + C. single-capital-letter collision scan (one letter, two meanings) + D. definition check for the symbols recent rounds introduced + E. γ/ρ role-usage scan (gate=authorize/reject-proposal ; ρ=verify/fold-back/response) + F. display-only symbols (used in $$…$$ but nowhere in prose) + G. orphan / redundant-declaration scan (symbol used once; or two declaration sites) + +Path resolves to HYPOTHESIS.md beside this script, or argv[1] if given. +Known benign flags: E flags the γ,ρ symbol-table row; G2 flags τ_H (it legitimately +owns both a stopping-time/filtration statement and its = inf{…} formula). +""" + +import os +import re +import sys + +PATH = ( + sys.argv[1] + if len(sys.argv) > 1 + else os.path.join(os.path.dirname(os.path.abspath(__file__)), "HYPOTHESIS.md") +) +with open(PATH, encoding="utf-8") as _f: + T = _f.read() +LINES = T.splitlines() + + +def lineno(idx): # char index -> 1-based line + return T.count("\n", 0, idx) + 1 + + +def ctx(idx, w=55): + a = max(0, idx - w) + b = min(len(T), idx + w) + return T[a:b].replace("\n", " ") + + +# math spans (so we can scan symbols in math only) +math_spans = [] +for m in re.finditer(r"\$\$.*?\$\$", T, flags=re.S): + math_spans.append((m.start(), m.end())) +for m in re.finditer(r"(?3}: …{ctx(i, 38)}…") + +print("\n" + "=" * 70) +print("D. DEFINITION CHECK (symbols recent rounds introduced)") +print("=" * 70) +defs = { + "Π (adversary class)": r"the class \$\\Pi\$ of policies", + "D (divergent set)": r"D=\\\{s:\\mathbb\{E\}_s\[\\tau_H\]=\\infty\\\}", + "μ (measure)": r"reference/sampling measure \$\\mu\$", + "e_0 (no-op response)": r"no-op response \$e_0\\in\\mathcal\{E\}\$", + "Stop (stop set)": r"stop set \$\\mathrm\{Stop\}\$", + "p_succ": r"p_\{\\mathrm\{succ\}\}\(s\)", + "p_safe": r"p_\{\\mathrm\{safe\}\}\(s\)", + "β (RL discount)": r"discount \$\\beta\$", + "A_Y (pushforward set)": r"measurable \$A_Y", + "r (per-step drift)": r"per-step drift \$r\(s\)=", + "z_t triple": r"z_t = \(c_t, b_t, m_t\)", + "μ_0 (initial dist)": r"initial \$s_0 \\sim \\mu_0\$", + "certificate (2-sense)": r"A \*\*certificate\*\* is a \*witness\*", + "controller/plant/shell": r"\*shell : plant :: the part you write", + "r_env (3-way drift)": r"r_\{\\text\{env\}\}", +} +for lbl, rx in defs.items(): + found = bool(re.search(rx, T)) + print(f" {'OK ' if found else 'MISS'}{lbl}") + +print("\n" + "=" * 70) +print("E. γ / ρ ROLE SCAN") +print("=" * 70) +# γ should sit near authorize/gate/reject-proposal/capability/irreversible/before +# ρ should sit near verify/validate-response/fold-back/after +g_bad = re.compile(r"fold[- ]back|folds back", re.I) # γ doing ρ's job +r_bad = re.compile(r"rejects the proposal|authoriz|is the gate|gates ", re.I) # ρ doing γ's job + + +def scan(sym_rx, label, bad_rx): + flagged = 0 + for m in re.finditer(sym_rx, T): + if not in_math(m.start()): + continue + window = T[max(0, m.start() - 15) : m.start() + 70].replace("\n", " ") + if bad_rx.search(window): + flagged += 1 + print(f" FLAG {label} L{lineno(m.start())}: …{window}…") + if not flagged: + print(f" OK no {label} usages land in the wrong role-neighborhood") + + +scan(r"\\gamma", "γ", g_bad) +scan(r"\\rho", "ρ", r_bad) + +print("\n" + "=" * 70) +print("F. DISPLAY-ONLY SYMBOLS (in $$…$$, absent from prose)") +print("=" * 70) +disp = " ".join(T[a:b] for a, b in math_spans if T[a : a + 2] == "$$") +prose = re.sub(r"\$\$.*?\$\$", "", T, flags=re.S) +toks = set(re.findall(r"\\[A-Za-z]+(?:_\{[A-Za-z]+\})?|[A-Z]_[A-Za-z]|[A-Za-z]_\\[a-z]+", disp)) +suspicious = [] +for tk in sorted(toks): + base = tk.split("_")[0] + if base and base not in prose and tk not in prose and len(base) > 1: + suspicious.append(tk) +print(" (heuristic; review only) ", suspicious if suspicious else "none flagged") + +print("\n" + "=" * 70) +print("G. ORPHAN / REDUNDANT-DECLARATION SCAN (review only)") +print("=" * 70) +# G1 — a math symbol occurring exactly once is usually a rename residue or a typo +# (a unification can strip a symbol of all but one use). LaTeX operators and +# formatting commands are not symbols, so filter them out. Review, do not trust. +OPS = { + r"\Pr", + r"\sum", + r"\int", + r"\sup", + r"\inf", + r"\infty", + r"\in", + r"\notin", + r"\cap", + r"\cup", + r"\setminus", + r"\subseteq", + r"\subset", + r"\mid", + r"\ge", + r"\le", + r"\sim", + r"\circ", + r"\cdot", + r"\star", + r"\hat", + r"\bar", + r"\to", + r"\Rightarrow", + r"\rightsquigarrow", + r"\longrightarrow", + r"\quad", + r"\qquad", + r"\Big", + r"\big", + r"\mathbb", + r"\mathcal", + r"\mathrm", + r"\mathbf", + r"\text", +} +sym_rx = re.compile(r"\\[A-Za-z]+(?:_\{[^{}]*\}|_[A-Za-z0-9])?") +counts = {} +for a, b in math_spans: + for m in sym_rx.finditer(T[a:b]): + counts[m.group()] = counts.get(m.group(), 0) + 1 +singletons = sorted(s for s, c in counts.items() if c == 1 and s.split("_")[0] not in OPS) +print(" G1 singletons (occur once in math, operators filtered — orphan/typo candidates):") +print(" " + (", ".join(singletons) if singletons else "none")) + +# G2 — the bare-τ failure mode the τ-unification introduced: a stopping/hitting-time +# symbol carrying BOTH an enumeration declaration (a "…stopping/hitting time…" +# sentence) AND a separate "= \inf\{…}" formula on a *different* line — one of the +# two sites is usually redundant. A formula restated in adjacent prose is benign +# (same kind of site), and so is τ_H, which legitimately owns a filtration statement +# plus its formula. A *newly* enum+formula-split symbol is the smell. +decl_rx = re.compile( + r"hitting times? are|are stopping times|is a stopping time|stopping times? for the" +) +formula_tail = r"\s*=\s*\\inf\\\{" # "= \inf\{" — the hitting/stop-time def, not \infty +tau_syms = [r"\tau", r"\tau_A", r"\tau_H", r"\tau_B", r"\tau_F", r"\tau_{H_{\mathrm{ok}}}"] +print(" G2 stopping/hitting-time family (count | enum-decl lines | formula lines):") +for s in tau_syms: + pat = re.escape(s) + (r"(?![A-Za-z_^{])" if s == r"\tau" else r"(?![A-Za-z0-9])") + occ = list(re.finditer(pat, T)) + enum_lines, formula_lines = set(), set() + for m in occ: + ln = lineno(m.start()) + line = LINES[ln - 1] + if re.search(pat + formula_tail, line): + formula_lines.add(ln) + if decl_rx.search(line): + enum_lines.add(ln) + split = any(e != f for e in enum_lines for f in formula_lines) + note = " <-- enum + separate formula; eyeball (benign: τ_H)" if split else "" + print( + f" {s:24} count={len(occ):>2} enum={sorted(enum_lines)} formula={sorted(formula_lines)}{note}" + ) +print("\nDONE.")