Coupling the harness to a neural learner

Parent: recursive-harness

safe-recursion-theorem treats the executor as a black-box ε-fallible solver. But the executor is a NEURAL LEARNER, and three of its specific properties are load-bearing — ignoring them leaves the theory a shell.

1. ε is not a constant — it's a landscape

A neural learner's error is wildly non-uniform: small near its training distribution, exploding off it. This is the real reason decomposition works at all. If ε were uniform, splitting would just trade one big error for n small ones at the same rate. Decomposition helps because it is a change of coordinates that projects a strange task back onto the model's high-density manifold — each piece lands in the low-ε region (boilerplate-sized, idiom-shaped leaves). Consequences: the leaf criterion is not "few tokens" but "familiar enough"; the measure P×C is really P×C relative to the model's competence landscape; and a split that produces unfamiliar pieces has made things WORSE even if each piece is smaller.

2. The repair loop is pseudo-SGD — so k = feedback legibility

A gate's failure signal is the learner's gradient: error signal → adjustment → re-evaluate is exactly the SGD shape. Hence the contraction factor k of inexact-contraction is NOT a property of the model alone but of the pair (model × feedback legibility): a check that only says "FAIL" yields k ≈ 1 (the learner gropes blindly); a check that localizes the error (which assertion, which input, expected vs got) yields k ≪ 1. Design principle that falls out: checks should emit gradients, not verdicts — write error messages FOR the learner. This refines recursion-convergence-contraction's "verification bandwidth = k": bandwidth includes how actionable the signal is, not just whether it fires.

3. The leaf criterion is PAC: a child's spec must be learnable from its injection

The stage injection (goal + context + playbook, stage-anatomy) is a sample; the context window is the sample budget m; the child's behavior is the hypothesis the learner fits. By the logic of pac / vc-dimension: if the sub-task's spec is too complex relative to what m tokens of injection can pin down, the child fills the gap with its priors — a hallucinated spec, the failure mode of tacit-spec-as-spec-compression. So the precise stopping rule for decomposition: split until each child's spec is learnable from the injection you can afford. Too little context for the spec's complexity = an unlearnable leaf = guaranteed tacit drift.

Two minor couplings

• Decorrelation: same model = correlated α (honest-caveats); the ensemble remedy — vary temperature, prompts, model families across redundant checks — buys back some independence for the union bound.
• Self-consistency as a cheap pre-gate: sample the learner n times and measure agreement; low agreement = high variance = route to a stronger gate or a further split, before spending real verification budget.

The harness is not scaffolding around a black box — it is a control loop wrapped around a LEARNER, and its three couplings (competence landscape, gradient-legible feedback, learnable leaves) are where the theory stops being air.