[apoursh]

Change and mathematical scope

Proves an exact 38 H1 minimum for arbitrary rational ADD/SCALE refactorizations of the selected #5413 provider's twenty-field backward block [9044,9137). The retained block attains the minimum; no κ improvement is claimed.

This resolves the larger-boundary question left open by #5471. Four zero-frame helpers and higher-dimensional exits invalidate direct reuse of the eight-field argument. The new proof permits arbitrary scalar chronology, coefficients, temporary basis changes and intermediate binary subspaces, while fixing all twenty arbitrary entering fields, boundary values and frames, and monotone climbs. New warps, auxiliary fields, descents, chart-changing exchanges and changed boundaries are outside scope. Native chart semantics remain inherited.

All 154,618,822,656 monotone frame-path assignments are covered through exact reduction: 68 certified domain removals leave 448 assignments; 315 fail reachability and 132 fail rank cuts. The only surviving paths are the source paths. This establishes the paid floor, not uniqueness of the scalar program. A separately authored certificate checker verifies all removals and remaining coverage by direct edge deletion and exact minor ranks, without trusting the search's max-flow implementation.

Adds only research/helper300-twenty-field-floor/: proof, exact certificate, three separately authored checkers, source pins and registered replay metadata.

Task, source binding and overlap

• Canonical base: 32890bc5de034ab8fc93ac88ead7279a6a416d04, priority revision 7.
• Publication head: 2f586ad5253a0a8065bf444f33f728b93129b0ed.
• Task: complex-rank-shape-decoder, with changed scope and expansion decision recorded in TASK.json. The original card's 5% supplier-expansion gate is not claimed.
• Predecessors: Research (no κ claim): exact ten-H1 floor for an eight-field helper block #5471 at 77d29520c4f90a65787873d6becc205796630c23 and Research (no κ claim): reorder, re-frame and re-ranking ceilings on the #4610 productive-helper word #5465 at b123ff44a211b9fc96b9f8fd1538e9965a95b34e. Both are motivation/attribution references, not scientific replay dependencies.
• All live scientific inputs are pinned on canonical main. No pending-PR Git object or network access is needed.
• Canonical investigation and completed claim #5470. Targeted live and local overlap checks do not establish universal novelty.

Validation

CPython 3.12.14, macOS arm64, standard library only:

python3.12 -B scripts/research_package.py --package research/helper300-twenty-field-floor
python3.12 -B scripts/check_research_commit.py --package research/helper300-twenty-field-floor --check-source-history

Passed at the publication head:

• All six registered executions: each of three checkers normally and under -O.
• Exact 20×20 transfer, independent macro/basis-vector reconstruction, determinant one and two-sided inverse.
• All 68 domain removals, 315 missing-path witnesses, 132 rank-cut witnesses and complete residual coverage.
• Six mathematical controls, eleven source/semantic controls and three altered-certificate controls.
• Committed-byte/source-history preflight, committed export integrity, and all six direct checks in a Git-free, read-only export.

These finite checks support the written information-flow lemmas. They are not independent human review, formal-kernel acceptance, a fresh physical/phase replay or a full multiplication-theorem audit. No selected full-caller replay or exponent calculation was run; the selected construction is unchanged.

Credit and next questions

Credit Cao Yifan (#4610), Dillon Amburgey (#5402), Ali Romman's cached-centre relations, Chris R/Clear1nsight (#5413), GamingPuzzled (#5465), tomdif's historical #1286 basis-change screen, and the retained native/compiler lineage. Reachability and cut-rank bounds are standard arguments; the contribution is their exact source-bound application and reproducible certificate.

Prepared by apoursh, with substantial OpenAI Codex assistance from separate discovery, mathematical-audit and source-checking workers. Worker separation is not independent human review. Apache-2.0.

After two fixed-boundary floors, another construction round should change and fully price a transport or boundary premise. The README and TASK.json state concrete next questions and remaining producer/consumer, restoration, ownership and native-phase obligations. Merely enlarging this window is not the proposed next step.