[aliromman]

This scoped negative result proves that a common-D controller cannot gain a favorable raw power moment in the stated characteristic-two FULL/THIN alphabet, even when it reuses productive H8 helpers and mixes scalar coordinates across ports. It closes a proposed way to remove idle calls. It does not improve the complete kappa calculation or change the selected frontier.

The proof, exact receipt, source pins and validation record define the result and its limits.

For n actual fields, f FULL calls with v main pairs each, s reused full-rank helper occurrences and h paid thin H1 calls, coefficient comparison and the classical rank inequality give M=2vf+s>=n and h>=2n-2vf-2s. The complete compact BIT invoice then has rank surplus at most -63f. Retaining its four H4 helper credits gives the exact bound Delta_t<=-47f*8^(t-1)<0 for every 0<=t<=1. The helper counts may be unequal and need not be multiples of 111. The favorable proposed family n/v=2p, f=p+2, h/v=p-1 needs p<=3 for its endpoint and p>=66 for its relaxed rank price; these conditions cannot both hold.

The alphabet requires four independent coefficients in F[q,d]/(q^2,d^2), one shared D, the explicit FULL support/defect ranks, funded thin ranks and ordinary constant scalar changes of basis. It excludes distinct fresh D variables, productive thin outputs, different native calls and operator-valued free codes. The existing compact source controller uses distinct fresh D blocks. Its retained positive rank surplus 753 is checked as a control and is not ruled out.

All consumed files are present at the publication base 32890bc5de034ab8fc93ac88ead7279a6a416d04. That checkpoint's selected conditional kappa is 0.003027288235 under its existing assumptions. The compact source receipt supplies the native local profile. The checker independently joins it to the complete saved 18-FULL invoice; it does not rerun the source native construction.

Python 3.12 or later and the standard library suffice:

python3 -B scripts/research_package.py --package research/reused-d-helper-rank-bound
python3 -I -O -B research/reused-d-helper-rank-bound/verify.py

Expected token: PASS_SHARED_D_HELPER_REUSE_RANK_BOUND. Normal and optimized checks passed, including 13 exact four-coefficient attaining examples, four rejected endpoint mutations, the separate missing-helper-debit calculation, 75 integer relaxations and the shared signed-invoice comparison. A Git-free read-only export with empty PATH passed both modes with all input hashes unchanged. Committed-byte and export-integrity checks passed for 13 files. The strict source-history check stopped because the checkout is shallow; full-history preflight remains open. No selected cold replay, physical supplier admission, compiler, Lean proof, hosted CI or complete multiplication theorem check ran.

The source credit notice credits ai-nanachi's compact436 word from PR #4326, its partial productive interface, Cinna Davis's query-thinned controllers in PR #4440 and PR #4470, and the retained zxv, Jacob Alan Sussman, PR #589, geometry, framing, COPY, compiler and analytic lineage. Those interfaces are inherited. The shared invoice checker supplies the existing concave-order comparison.

The campaign's BIT worker supplied the first projector-factor proof; its branching worker supplied the productive-helper extension. This package gives their common field-count formulation and complete partial-helper power bound. Two other campaign workers reviewed the algebra; one also ran an independent 448-field, cross-port fixture with helper counts 1 and 3. These are author-side AI checks. Prepared for Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. No independent human acceptance or proof-assistant verification is claimed.

Related PR #5507 extends PR #5456 to two-owner multigraphs under independent stage-axis assumptions. The present proof has one shared D and arbitrary constant scalar coupling; it does not impose a two-owner coordinate support graph. These are different restricted alphabets.