[apoursh]

Change and mathematical scope

Proves an exact 10 H1 minimum for the dependency-closed eight-field backward helper-300 block at zero-based events [9044,9116) of the selected #5413 provider. The current block attains the minimum. This rules out the proposed H1 + H1 → H2 saving in this particular neighborhood; no κ improvement is claimed.

The proof permits arbitrary finite rational ADD/SCALE factorizations, temporary basis changes, scalar order and intermediate binary subspaces. It fixes the eight arbitrary entering fields and their entry/exit frames, and requires monotone climbs. Auxiliary fields, new warps, descents, chart-changing exchanges, cross-column conversions and changed boundary contracts are outside scope.

The obstruction is a transfer-rank cut: if either of the two zero-start helpers jumps directly to its terminal plane, only one carrier can transmit their two-dimensional entering information to other terminal planes, whereas the exact backward outputs require rank two. Both helpers therefore need two H1 climbs; the six line-start fields need one each, giving 2 + 2 + 6 = 10 H1.

Adds only research/helper300-eight-field-floor/: written proof, source-bound exact certificate, two separately authored checkers, and registered research metadata. Physical all-width interpretation remains conditional on the inherited native chart contract. This is neither a whole-provider lower bound nor a formal-kernel verification.

Task and overlap

• Base: canonical main 32890bc5de034ab8fc93ac88ead7279a6a416d04, priority revision 7; refreshed before submission.
• Publication head: 77d29520c4f90a65787873d6becc205796630c23.
• Task: complex-rank-shape-decoder, with changed scope recorded in TASK.json. The historical decoder card is narrowed to one possible paid-climb merge in the selected provider. Its 5% supplier-expansion gate is not claimed satisfied.
• Nearest predecessor: Research (no κ claim): reorder, re-frame and re-ranking ceilings on the #4610 productive-helper word #5465 at b123ff44a211b9fc96b9f8fd1538e9965a95b34e, R13 §§3–4. This addresses changed ADD factorizations rather than another fixed-word reorder or one-swap search. The note's illustrative forward x19 witness is not used for the backward conclusion.
• Source availability: all proof/replay inputs are pinned on the canonical base in SOURCES.json. The Research (no κ claim): reorder, re-frame and re-ranking ceilings on the #4610 productive-helper word #5465 note is an immutable motivation/overlap reference, not an unmerged Git-object dependency.
• Coordination: canonical investigation and completed claim #5470, linked from the handoff hub. Targeted overlap searches do not exclude unpublished work or hidden comments.

Validation

CPython 3.12.14 on macOS arm64, standard library only:

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

Passed at the publication head:

• All four registered executions: both checkers in ordinary and optimized Python.
• Primary checker: exact transfer and source binding, invertible prefix grammar, all 16 monotone frame-path assignments per orientation, rejection of all seven cost-saving assignments, five controls, protected continuation checkpoints, and equality with committed RESULT.json. The inherited forward window is an orientation control; the selected-provider claim is backward.
• Separately authored checker: identical backward matrix, boundary frames, field set, 28 retained events and 10-H1 invoice; determinant 1, basis-vector replay, and nine rejected semantic mutations.
• Committed-byte and source-history preflight; committed-package export integrity check; all four direct checks in a fresh read-only export with Git absent from PATH. Ordinary/optimized outputs agree.

Finite checks establish the source-bound hypotheses and support the written carrier lemma; they are not an enumeration of arbitrary programs. No complete provider/caller or absolute-phase replay, compiler-funding calculation, Lean audit, independent human review, or new exponent calculation was performed. The selected construction is unchanged.

Credit

Credit Cao Yifan's #4610 native word, Dillon Amburgey's #5402 coupled provider, Ali Romman's cached-centre relations, Chris R/Clear1nsight's #5413 completion and source validation, GamingPuzzled's #5465 bottleneck analysis, tomdif's historical #1286 helper-basis screen, and the retained native/compiler lineage. No priority claim is made for elementary cut-rank arguments.

Prepared by apoursh, with substantial OpenAI Codex assistance from a coordinator, synthesis worker, separate mathematical auditor and separately authored source checker. AI-worker separation is not independent human review. New work is Apache-2.0; predecessor notices are retained.

Open next questions

The package's README next questions and TASK.json identify the next bounded problem: extending the same seed through [9044,9137) closes over twenty fields. First derive its entry/exit geometry, rank cuts and complete paid profile; a larger interface could alter the one-carrier obstruction, but no profitable rewrite is supplied. Changed transport or auxiliary fields require an explicitly priced new contract. These follow-ups are outside this PR.

[apoursh]

See also #5478