[aliromman]

A vector decoder that saves s query coordinates below the compatible-flag floor must have at least s rank of cross-level defect. This package gives the exact floor for unequal payload widths and gapped flag levels, then proves rank M + rank E >= sum_t mu(Z[I_t]). E retains exactly the entries that move to a higher declared flag level. The bases and their complements are explicit.

The coefficient of rank E is sharp at every positive integer s, over every field. For s copies of the classical width-two odd-cycle edge code, the minimum defect rank over all local complete flags and adapted bases is exactly s. An explicit alternating basis attains it. The code, its block-rank advantage and scalar minrank facts are credited prior work. The new result quantifies their exact flag defect and extends Nick Petalas's PR6471 to unequal/gapped payloads. All-size proof.

A complete literal linear word preserves the sources, updates arbitrary dirty outputs, and restores one reused scratch field after every edge-copy. It pays six ADDs per edge-copy with clean scratch, or eight with arbitrary dirty scratch, including encoding, both output uses and uncomputation. Every integer prefix is bounded by 3H, sharply. The full read/write invoice is explicit. A direct four-ADD source program also exists, so this is not a scalar-operation or physical speedup claim.

Both warning-strict CPython 3.12.13 modes pass all 180,625 binary fitting matrices with unequal widths (1,2,2), including 18,225 compatible matrices. A separate full-image oracle checks their ranks and defect ranks. Another 288 gapped/field cases, 36 larger sharp family cases, all 7,776 C5 local GL2 basis tuples, 678 complete clean/dirty word inputs, 192 omitted operations, ten invalid inputs and an actual retained-source corruption pass. Receipt.

python3.12 -B scripts/research_package.py --package research/flag-defect-decoder-budget

All 11 committed files pass byte/mode and official-export checks. Both read-only Git-free, empty-PATH replays agree and preserve all 12 export files. Hosted planning and replay passed at the published head: run 41570428688. The retained source proof is byte-exact and checked before use. No pending Git object or network is needed. Full-history ancestry is unclaimed in this shallow checkout. Independent human and proof-kernel acceptance are unclaimed.

The deficit is a diagnostic rank, not a free physical channel. No actual native caller, physical routing, signed invoice, precision/stock, compiler/outer admission, new kappa or selected-frontier change is supplied. All five selected assumptions remain. The next useful step is a source-bound decoder that pays for the required defect and wins after its complete caller costs. General scalar-minrank and flag optimization are not solved. Lean, Docker and the selected cold replay were not run.

Only research/flag-defect-decoder-budget/ is added. Advance claim. Canonical main 7c49fe3, priority 8/selected 6030 remained unchanged at submission. The selected binding example passed. Live handoffs, current PRs and the 476-PR/6,464-Discussion overlap cache were checked. Recent selected-word physical/stochastic work has a different contract. Sparse history, nested diffs and unpublished work remain coverage limits.

Ali Romman with substantial OpenAI Codex assistance; no subagents. Credit Nick Petalas/6471, ywz65/2541, Aaron Wang/navinor/825, Cinna Davis, Bukh/Cox/Blasiak and the retained geometry/native/compiler lineage. Bukh and Cox provide the classical fractional Haemers context and odd-cycle result. The rank inequalities and edge representation are classical; no global literature-priority claim is made. No merge, closure or frontier change is requested.

Verified head: 4d80ba76770faa76317c7c397cfe992fbb4c551a.