Bound sparse feedback volume by conditioning and the native entropy jump

[Clear1nsight]

The selected coupled boundary requires (C_f x+B_f y/2,y). PR4009's full-array pass bound does not by itself charge sparse updates. This package proves a separate arithmetic-volume/conditioning tradeoff using an Ailon-style preconditioned quasi-entropy:

8 K S + Re(Delta_Q) >= fV/2.

Here S is the total number of updated coordinates, K bounds every full formal machine prefix condition number, and Delta_Q is the explicit entropy jump across the sole native call. Arbitrary sparse feedback, variable mask sizes, restored arbitrary dirty auxiliaries and clean returned auxiliaries are allowed. Monomial routing/scaling is granted free for the lower bound; its actual physical invoice remains extra.

When the target child runs before arithmetic, preceded only by payload-preserving monomials, its jump is zero: S >= fV/(16K). Thus uniformly bounded-condition child-first sparse adapters still require order fV arithmetic. A general interleaved call retains its actual jump; unrestricted conditioning, general sparse algorithms and complete supplier admission remain open. No new kappa or selected frontier change is claimed.

The scoped contribution is the analytical block preconditioner J_xy=C_f^-1 B_f, its exact endpoint value, the dimension-free complex coordinate-shear bound, and explicit native-cut accounting. Ordinary determinant and unpreconditioned quasi-entropy miss the triangular correction. The proof includes clean auxiliaries without imposing their off-domain nonzero-input behavior. This adapts the established potential method, not a new general lower-bound technique.

Validation at 8721aa7738db8094a14cb417c5d4ed614b1a2f48: registered normal and optimized replay pass; exact Gaussian-rational endpoint fixtures,32 complex pair-column invariants, nonunit monomials, noncommuting BA/AB fixture, returned-clean auxiliary fixture, explicit interleaved jump 1+i, and five failure controls. Two same-team AI proof reviews found no issue. Exact committed-byte/source-history preflight passes. Official export check passes; ordinary and optimized replay from a read-only, Git-free export passes with all8 file hashes unchanged. The written all-dimension proof is the theorem evidence; finite controls are not a quantified proof or formal-kernel certification.

Scientific base20ecf289, priority6; canonical main refreshed unchanged before commit. Entry overlap covered733 open PR bodies including drafts plus latest1166/1901 coordination; broader discussion coverage remains partial. PR4023 fixed-support unary queries, PR4027 H6 U9 splice, and the adjacent PR3954 paired-owner context use different contracts. The sibling signed-pair sparse construction supplies a complementary upper bound and is not a runtime dependency.

Credit Nir Ailon (1403.1307) and Nir Ailon/Gal Yehuda (1604.02557v4, Section1.1/Lemma1.1), plus #4009/#3989, Gonzalo Mier/#3975 and their retained selected/native lineage. Prepared by Clear1nsight with substantial OpenAI Codex assistance in research, proof, implementation, independent agent review and packaging. Apache-2.0. Routing, precision, host placement and native/compiler admission are not discharged.

Final hosted confirmation: run39243065063 succeeds at the unchanged published head 8721aa7738db8094a14cb417c5d4ed614b1a2f48. Both normal and optimized five-control replays, registered-package success and unchanged-input verification pass. Final feedback sweep found no comments, reviews or inline threads. Human/formal mathematical acceptance and the stated general-interleaving gap remain open.