[CubeJerry]

Under the exact label-mask hypothesis of #6846, prove that the full P104 system needs at least 9k dimensions over every field. The predecessor's original fourteen ports have the smaller exact minimum ceil(17k/2); replacing port 57 by 61 gives a sharp 9k subsystem. The full mask attains 9k in characteristic two.

The same certificates prove exact fixed-length binary broadcast lengths, even for nonlinear deterministic encoders: ceil(17k/2) on the original subsystem and 9k on the replacement and full graph. An explicit entropy reduction and attaining parity broadcasts are included.

This closes #6846's intermediate block-dimension question by transferring the established Petersen mechanism from merged #1744. No new general Petersen theorem is claimed. Concurrent #6854 excludes exactly 8k for every sufficiently large E8 subset and leaves this fractional interval outside its scope.

Base: 7c49fe3.
#6846 head: 0165179.
Input: research/p104-designer-toolkit-v2/words/5561.json.gz, Git blob c367d865a154ffd2ae0b6f6053469580a139f487. Complete source and citation pins are retained.

Reproduce (Python 3.12, standard library):

python3.12 -B scripts/research_package.py --package research/p104-sharp-block-hall-dimension

Expected: PASS_P104_SHARP_BLOCK_HALL_DIMENSION, PASS_P104_PETERSEN_EXACT_RANK_PROOF and PASS_REGISTERED_RESEARCH_PACKAGE.

Executed: all four normal/optimized registered checks; separate set-based and bitmask coefficient implementations; the 17-row hand-proof expansion and 43-row certificate; coordinate witnesses at k=1..12; all 10,816 binary Gram entries. Ten primary and four independent controls reject corrupted sources, parity/premise changes, invalid coefficients and broken witnesses. Committed bytes/modes, official export integrity and all four fresh Git-free, empty-PATH replays pass. The complete remote tree equals the tested tree and adds only eight package files.

No new κ. The mask remains an explicit hypothesis; no necessity theorem for every legal supplier, physical caller or complete paid invoice is supplied. Full-mask attainment outside characteristic two remains open. All five framework assumptions and the selected frontier remain. The universal results follow from written proofs and exact axiom certificates, not finite sampling; human and proof-kernel acceptance are unclaimed.

By @CubeJerry with substantial OpenAI GPT-6 Astra Pro assistance and separate AI proof/implementation checks. Credit DreamingOfClouds (#6846), Thiago/Tuafo (#1744), Blasiak–Kleinberg–Lubetzky, Bukh–Cox, and the unchanged #5561 source lineage. Apache-2.0.

Proof, source pins, limitations and open questions:
https://github.com/CubeJerry/integer-mult-bounds/tree/e1e6f9de0610b44072ff864cba8bb0d46062c02d/research/p104-sharp-block-hall-dimension