[lm1909]

Change and mathematical scope

Audit, no new κ. This independently checks three parts of dillona's #4623 (head fd5d3be12b9d6ee92c8fa089d95bd529c3236563) that #4625 did not replay. Nothing is imported from #4623 or #4625.

• Monotone extension: holds. I proved the separator lemma used to extend the result: if c(r−σ) ≥ 0, then d/dt[c(r/σ)^t] = c·ln(r/σ)(r/σ)^t ≥ 0. Its hypothesis does not depend on p, so J_p/σ^t is nondecreasing. Failure at C = .006/.007 therefore extends to every larger C for every p. The valid σ ranges are (5,7] complex and (7,8] BIT, and Construct border controllers without idles and bound recovery costs (no κ) #4623's σ = 6 and 15/2 lie inside them. I certified the bounds by a different method (exp/log series with explicit remainders, rounded outward to 2⁻¹²⁰). It reproduces A and B_J at .006/.007 to every printed digit.
• New sharpening. A < 0 and B_J < 0 already hold at C = .005687 (complex) and .006512 (BIT). A > 0 at .005685 / .006510, so this relaxation's sharp threshold lies between those values. This is a ceiling, not a supplier.
• Decoder: holds in the stated model. z = (I+J)X′ − Q⁻¹Jx = Q⁻¹Dx exactly for p = 2..7. This forces M = I+J and N = −J, so rank N = p−1. DECODER.md's cumulative base/slope table is reproduced and every entry is positive, so every concave moment fails.
• Controller sanity, p = 1..7. Three constructions agree coefficient by coefficient: a hand replay with idles deleted, DISCOVERY.md's closed-form U/V, and the literal source emitter complex_controller_factor.controller(p, 0). The border degree set is exactly {0..p−1}, det = Π D_s² and A(1) = I.

Not covered: the precision/H₀ lemma, the K≥2 repeated-evaluation argument, the determinant rank-mass invariant (taken as stated), physical/BIT/native admission, decoders outside the stated model, and p > 7 symbolically.

Task and overlap

• Card: independent-moment; scope refined to Construct border controllers without idles and bound recovery costs (no κ) #4623's uncovered items.
• Base: fbc9ea0249746acca4f504d4fda2b84e0e36cf0d, rechecked as current main before submission.
• Independently check #4623's idle-mass bookkeeping (177,408 - 141,568 = 35,840 = M_p) and its C=.006/.007 tangent bounds (no new κ) #4625 (pinterbanget) checked the mass bookkeeping and the tangent-bound values, and its verify.py includes a sign check of the separator premise. This package adds the derivation, the maximal σ ranges, a different certification method, the sharpened thresholds, the decoder and the controller.

Validation

python3.12 scripts/research_package.py --package research/pr4623-monotone-decoder-check (stdlib only, about 2 s) prints PASS_REGISTERED_RESEARCH_PACKAGE locally on Linux. #4623's BUDGET/DECODER/DISCOVERY are pinned with head: pins. Nine rejection controls all fail as expected: input tamper, a flipped backward sign, no thinning, reversed weights, a net-negative paid rank-9 term, non-certification at C = .005 complex / .006 BIT, a decoder without fresh queries, and a decoder without the raw-input term. These are finite certified checks plus a written proof of the lemma. There is no Lean.

Credit

dillona (#4623, #4611), pinterbanget (#4625), Matarener (#4602), Cinna Davis (#4470) and the controller lineage credited there. Prepared by lm1909 with Claude Code (Claude Opus 5.5) assistance; there has been no independent human review. Apache-2.0.

Open next questions

See the package README: the precision/H₀ lemma, and a symbolic proof that the idle-free controller identity holds for all p.

🤖 Generated with Claude Code