Research: independently re-solve PR #2979's fixed-capacity stage caps and price each stage-one group alone (no new κ)

[Matarener]

Requested by Mark

Change and mathematical scope

Audit, no new κ, no supplier admission. PR #2979 certifies its fixed-capacity packing-relaxation caps from stored optimal bases. This package re-solves the same relaxation without any stored basis, with its own pattern enumeration, its own two-phase simplex (numpy floats, Bland's rule) to locate a basis, an exact Fractions inverse, and mpmath.iv interval enclosures for the root and every reduced cost. No code from #2979 is imported or run.

• All four shipped cap brackets (parent, stage 1, stage 2, final) are reproduced exactly on the 10⁻²⁵ grid, with identical idle histograms. Three of the four optimal bases are alternative (degenerate) optima with the same root.
• New coverage: each stage-one group applied alone to the parent words raises the certified cap: V45 +5.510e-7, A and B +1.0106e-6 each, C and D +4.160e-7 each. For V45, lowering k = 0, 1, 2 or 3 of its four endpoints lowers the cap, and only k = 4 raises it. This replaces the source's approximate 2,091,300-unit width guide with certified fixed-inventory caps, and the signs agree with its margins. The group gains sum to 3.4042e-6 against a joint stage-one rise of 3.4093e-6.

Scope is the fractional relaxation only (no windows, family exclusion or integrality). Native/phase, storage, precision, the literal T74 allocation and the outer release remain open.

Task and overlap

Card independent-moment (priority revision 6, track independent-verification, scope key 5e0f415b25b13c102653), scope changed (recorded in TASK.json) to the open "re-solve one stage basis" question in research/rank7-v45-shipped-invoice. Base 463113d675af7410ac91d53a4fed51ec6494eb9a. Contribution commit a30d6fabb6ea22f964b7e1e072901916e602154e. Live dependency #2979 at be191346391661f4d35fc0d1508d0a6029572576; inputs RELAX-STAGES.json, RELAX-CAP.json and research/rank7-joint-frame-audit/RESULT.json are pinned by Git blob and SHA256.

Nearest overlap: #3237 (open) certifies that widening V45 endpoints on the final words fails the complete budget. This package prices the original stage-one groups and subsets on the parent words and re-solves the four shipped caps. The two are complementary. research/rank7-joint-frame-audit (the source of the checked group deltas used here) and the V45 native/event packages do not re-solve caps.

Validation

python3 -B research/rank7-stage-cap-resolve/verify.py
python3 scripts/research_package.py --package research/rank7-stage-cap-resolve --check-only

verify.py recomputes everything (about 20 s) and requires equality with RESULT.json. It prints PASS_INDEPENDENT_STAGE_CAP_RESOLVE caps=4 matched groups=5 raise_alone v45_subsets=only_k4 controls=5 no_new_kappa. Run locally on Windows with Python 3.10.6, numpy 2.2.6 and mpmath 1.3.0. The metadata check passes. The registered Python 3.12 replay was not run locally because 3.12 is not installed; VERIFY.json pins numpy==2.2.6 and sympy==1.14.0 (for mpmath).

Five rejection controls: a replaced basis column (singular), the stage-2 basis on stage-1 inputs (infeasible), a feasible basis optimal at b = 1/2 (reduced-cost sign failure), one extra rank-1 child (rank-mass identity 65,088), and the stage-2 local profile with the final inventory (strictly lower cap). The inputs are authenticated by SHA256 before use.

Credit

Marius-Paul Dumitrean (#2979, #2942) for the construction, words and relaxation model; Jeremy Kothe (#2957), Mason Hale (#2917) and Jeremy Diviney (#2908) for the extraction chain; Nick Petalas for the independent frame audit whose group deltas are used; Ross Richardson (#2960/#2974/#2987) and the PR2857/PR2930 relaxation lineage. Written and run by Claude (Anthropic) via Claude Code, in a session donated by Mark; no human or formal review is claimed.

Open next questions

Does #2987's chronological owner/packing evidence attain these caps for the stage-one words, or is part of each group's rise unreachable once windows and family exclusions apply? First step: price a chronological fill of the stage-one inventory against the stage-one cap. Blockers: native/phase, storage and precision bindings. Another lowering census is not justified by this result.

Workflow feedback (optional)

On a fresh Windows clone, the first scripts/choose_research_task.py call failed with "file not found" until checkout had finished. The 1.9 GB clone is slow, and the picker worked once checkout completed.

🤖 Generated with Claude Code

[Matarener]

Validation update: the registered replay now passes locally with Python 3.12.10 (PASS_REGISTERED_RESEARCH_PACKAGE), so the note in the description that the 3.12 replay was not run is resolved. The code is unchanged.