Owned Math M7 — Attribution loss: a structured counterexample to the composed control law

TL;DR

The unified control law's headline guarantee — "(a) KILL the confident-unanchored runaway (M6 lasing)" — is unsatisfiable over the law's own signal vocabulary. Not a bug, an impossibility: M6's kill side-condition is per-mode ("G/L >evidenceForModeand zero external-innovation coupling"), but every signal the law reads is global-per-step. The structured counterexample is a two-world pair with identical signal histories — one genuinely anchored, one laundered — so no causal policy over those signals can kill one and spare the other (Lemma 3). Measured against the shipped code (first slate artifact to drive real code paths, not toy semantics): the laundered runaway survives 40/40 steps, ending at confidence 0.9999986 having never received evidence, while the law's per-step reason reads "improving on external evidence." Worse, the composition actively protects the runaway: the shipped allocator's dilation is strictly decreasing in laundered confidence (D → 0.500001, the knife-edge of the D > 0.5 fetch cutoff; any cost pressure or collapse-proximity pushes it under), so the snowball also de-allocates its own audit. The M2 hard cadence — proven necessary by M3 — is therefore not sufficient: cadence guarantees a grounding event; attribution decides whose claims get grounded. Fix shipped, default-compatible: a keyed anchor signal (evidenceForMode) restores M6-soundness (kill at step 1); the attribution vocabulary already exists in the ledger as M1's paid/free confidence split. Bonus finding: the converged-but-stale-and-broke cell emitted a self-contradictory record (action: "halt_saturated", saturated: false, reason claiming saturation) — fixed.

1. The claim attacked

#2857 / PR #2909 landed the unified control law composing the owned math into one per-step decision. Its header claims four guarantees; the first is the one that justifies the composition's existence:

"(a) KILL the confident-unanchored runaway (M6 lasing)"

and its ruleimplements it as:

if (gainOverLeak > 1 && !evidenceInflux) return kill;

The slate's own M6 statement is per-mode: "modes with G/L >evidenceInfluxand zero external-innovation coupling grow without bound." gainOverLeak is documented per-mode — but evidenceInflux is documented as "did external evidence arrive this step?" — a global disjunction over everything the run touched. The shipped unit test canonized the unsound reading; its name asserts an attribution the signals do not carry:

"a lasing mode that IS externally anchored is not killed" — tested as { gainOverLeak: 1.5, evidenceInflux: true } → not kill.

Nothing in that input says the evidence anchored that mode.

2. The structured counterexample (two worlds)

Unlike M3's silent set (found by grid search), this counterexample is constructed — and it is a pair, which is what makes it an impossibility rather than a tuning gap:

The G/L estimator reads magnitude, not provenance; the influx bit is a disjunction; the rising bit is the same arithmetic. Identical signal histories ⇒ any deterministic causal policy (arbitrary memory allowed) acts identically ⇒ it is either unsound on L or vacuous on G. That is Lemma 3 (proofs note); the kill criterion for the lemma and the reason it survives charitable readings are stated there.

The attack is cheap in practice: any ambient evidence stream — a subscribed feed, tool heartbeats, another hypothesis's grounding — sets the global bit every step. Under ambient influx, ruleis dead code: guarantee (a) was silently vacuous in exactly the environments the product runs in.

3. Measured against the shipped code

node experiments/owned_math_m7_composition_counterexample.js — deterministic (no RNG, no clock), requires the shipped modules directly.

Part A — the law (legacy vocabulary). Both worlds: action histogram {continue: 40}, first_kill_step: null, identical traces confirmed. World L ends at confidence 0.9999986 (odds 1.4⁴⁰ ≈ 836,683) with zero evidence ever attributed to m₁, and the per-step reason is "improving on external evidence."

Part B — the keyed guard. With evidenceForMode (shipped this note): World L killed at step 1 with reason "…mode-unanchored (global evidence present but none attributed to this mode — M7 laundering guard)"; World G: zero kills acrosssteps; calls that omit the field reproduce the legacy behavior exactly (allpre-existing unit tests unchanged).

Part C — the starvation corollary. With provenance-blind uncertainty u =− c (what the allocator sees when free confidence masquerades as knowledge), the shipped dilation(u, cp, c, p) is strictly decreasing in laundered confidence:

At the limit the runaway sits on the D > 0.5 cutoff; any positive cost pressure or collapse-proximity puts it under — and a laser self-repeats, so the degeneration signal plausibly reads p > 0, meaning the G12 collapse-deflation double-starves it. G12 is correct for verified-frozen nodes (retrieval has no marginal value); for the laundered laser the true marginal value of retrieval is maximal (one lookup refutes it), but estimated γ →D > 0.5because the estimate is provenance-blind. Same lemma, allocator layer.

De-laundered rows (attribution-aware uncertainty): anchored mass only (c = 0.5, u = 0.5) → D = 1.0, fetchExternal true; never-verified reading (u = 1) → D = 1.33, maxResults→ 7. Attribution moves the runaway from knife-edge-zero grounding to ramp-grade scrutiny.

Part D — the self-contradictory record. Input {fixedPoint: true, stable: true, groundingDue: true, budgetRemaining: 0} returned action: "halt_saturated" with saturated: false and a reason claiming "saturated and grounding budget exhausted" — a converged-but-stale state mislabeled as saturation in the very telemetry convergence records are built from. Fixed: the reason now distinguishes the two states; the action name stays halt_saturated (conservative — neither state has a fresh anchor, so neither earns halt_converged trust).

4. The three-layer reading — one mechanism

One mechanism at every layer: unanchored confidence masquerades as knowledge in each component's input vocabulary. The laundered runaway is not merely un-killed — it is a protected equilibrium of the composed law: it suppresses its own kill switch and de-allocates its own audit. And the slate's arc completes cleanly: M3 proved the hard cadence necessary; M7 proves cadence-plus-allocation without attribution still admits the silent runaway — necessary, not sufficient.

5. The fix and what it costs

• Shipped (this note, default-compatible): evidenceForMode — the keyed anchor. Kill rule

becomes G/L >∧ ¬E_mode; omitted field falls back to the legacy global reading, so no caller changes semantics until it opts in.new unit tests;pre-existing tests untouched.

• The attribution vocabulary already exists: M1's ledger scan already splits paid vs

free confidence mass (slate §M1: 20.8% free). Keying that split per mode is bookkeeping, not new theory: E_mode(t) = "the mode's paid mass moved this step."

• Instrumentation ask (goes with #2791): the M6 CANARY_TRACE per-generation trajectories

must tag grounding events with the mode/hypothesis they anchor, or the keyed bit cannot be computed live. Same ask covers the allocator: u must be anchored uncertainty (1 − c_paid), not− c.

• Not changed here: the M5 allocator itself (IP-gated per the register §6; the starvation

finding is recorded as a requirement, not patched code).

6. Kill criteria (pre-stated, for M7 itself)

• Lemma(impossibility half) dies if the two worlds are shown not co-reachable under

the real M6 estimator — i.e., someone proves the shipped G/L measurement entangles coupling (anchored lasers cannot exist). The counterexample against the shipped global-bit rule survives that (World L exists regardless); only the "no policy can do both" half degenerates.

• The guard dies if per-mode attribution is unimplementable at the signal source (the

#2791 instrumentation ask fails) — then the honest statement is that guarantee (a) must be withdrawn from the law's header, not that the global bit works.

• The starvation corollary dies if the deployed uncertainty estimate is shown to already

be attribution-aware (it is not, today: chatDilation derives u from surface features and dilation takes raw confidence).

7. Prior art, named

Discrete-event fault diagnosability (Sampath et al., 1995) — a fault is diagnosable only if no arbitrarily long faulty trace is observation-equivalent to a nominal one; W_G/W_L are exactly such a pair, so M7 is a diagnosability failure of the signal map, cured by relabeling (attribution), not policy cleverness. Static-output-feedback distinguishability is the control-theory cousin. Reward-hacking of anchoring checks is the RL-flavored analogue. M3's Lemmais the sibling one level down — passive internal functionals cannot separate grounded from ungrounded trajectories; global step-signals cannot separate anchored from laundered gain. Ours: the instantiation on the shipped law, the executable two-world artifact against real code paths (a first for the slate — M3/M4 used toy instrument semantics), the starvation corollary tying M5/G12 into the same mechanism, and the default-compatible guard with the M1 paid/free split as the attribution vocabulary.

8. Honest scope

• The impossibility is relative to the signal map as typed in the shipped law. A richer

map dissolves it — that is the point: the fix is a vocabulary fix.

• The knife-edge at exactly cp = 0, p =fetchExternalleaves fetchExternal true by ~1e-6; the starvation

claim at that corner is "minimum-grade grounding on a knife-edge," not "zero." Every perturbed cell (cp >falseor p > 0) is a hard false.

• The confidence arithmetic (odds × g) is illustrative; the law reads only the rising bit,

so no modeling choice there carries proof weight.

• The two worlds are signal-level constructions. Whether the production M6 estimator, once

instrumented (#2791), emits G/L >for genuinely anchored modes is an open empirical question — Lemma 3's honest-scope paragraph covers both outcomes.

• No new subsystem: one optional field on an existing pure function, docs, tests, and one

experiment file.