[petalas]

Summary

Prove a uniform total-rank floor 13q for five distinct compatible, pairwise commuting full stages in a shortest n+1-query root on nq independent inputs, n even and characteristic different from two. This is a scoped coefficient-space proof using the cited inequalities of thu-nmrc #6717 and CubeJerry #6773.

- Four-stage averaging alone: five-stage total >= 25q/2
+ Directed leakage aggregation: five-stage total >= 13q
+ Equality requires endpoint/odd ranks (2q,3q,3q,3q,2q)

An even interior gives 14q; no even interior and zero or one endpoint gives 40q/3. The displayed linear relaxation attains 13, but no attaining projector tuple or finite search exclusion is claimed. The contribution is the new five-stage aggregation and equality necessity, produced by Nick Petalas. Attainment remains open.

Credit Ali Romman #6398, thu-nmrc #6717, CubeJerry #6773/#6781, ToikY #6721/#6734, Cinna Davis and the retained query/controller lineage. Ali Romman's earlier scalar six-field floor 14 and CubeJerry's sharp four-field four-stage value are stronger in their narrower domains and are explicitly retained as predecessor results. Their proofs are cited, not replayed.

Only research/p104-five-stage-rank-budget is added. Base and refreshed canonical main are e545235, priority revision 8.

Evidence

The written proof covers all allowed n,q by site type. Exact rational dual sums pass for all 1,897 five-site placements at n=4,6,8,10,12. All 84 abstract relaxation witnesses satisfy the listed rows. Six false-certificate controls reject. Source-byte and receipt corruptions also reject in an exported copy.

CPython 3.12.15, standard library only. Direct normal and optimized checks each took under 0.1 seconds locally; the registered checks have 60-second limits. These times establish timeout headroom, not a speedup. Metadata/pins, both registered commands, committed bytes/modes, source-history ancestry, official export/input coverage, and read-only Git-free normal/optimized replay with PATH=/nonexistent all pass. Exported bytes are unchanged after replay and restored corruption controls.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/p104-five-stage-rank-budget/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/p104-five-stage-rank-budget/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-five-stage-rank-budget
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-five-stage-rank-budget --check-source-history --check-modes

Another CPython 3.12 may replace the absolute path. Expected markers are PASS_FIVE_STAGE_RANK_BUDGET, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. RESULT.json is generated and binds the proof inputs and package files by SHA256.

Limitations

The universal claim is the written aggregation conditional on the cited lemmas. Finite checks do not establish the optimum of actual projector tuples or independently certify predecessors. No exact five-stage construction, physical/native supplier, paid invoice, new κ, frontier change or complete multiplication theorem is supplied. Native routing, ownership, compiler, precision/tape and analytic recovery are unrun and outside this result. Independent human and formal proof-kernel review are unclaimed.

The delivered scope is narrower than the advance CLAIM: the progress finding leaves attainment open.

Open next questions

• Is 13q attainable at an even n >= 8, with arbitrary coupled width? Start with the forced equality ranks and impose the actual-prefix equations exactly.
• Can a structural argument raise the uniform floor? The displayed linear inequality system has optimum 13 at a two-endpoint placement, so it needs an additional implication.

Related open work

The pre-commit refresh checked 467 open PR titles/bodies, this account's open PRs and the latest 35 comments in #1166. No open PR was found computing this five-stage total under the shortest coupled-root contract. Sparse cached notes, search-index omissions, arbitrary diffs and unpublished work remain coverage limits.

• #7126 bounds thin rank for every-stage paired-star gaps, a different schedule and rank quantity; its result is not extended or recomputed here.

Completion refresh checked all 12 other open PRs on this account created after the CLAIM at 2026-11-10T22:13:28Z: #7132, #7133, #7134, #7135, #7167, #7168, #7169, #7170, #7172, #7174, #7175 and #7176. Their titles and bodies concern stock fit, dyadic windows, rectangle residual rank, net splitting tax, E8 hosting/COPY/Gram geometry, tensor Gram stabilization, helper-split invoice ceilings and six-source chain windows. None studies the shortest coupled-root five-stage total proved here. No shared numerical result requires comparison, and no additional same-object line is warranted.

Merged #6717 and #6773 supply the exact live proof inputs; #6398 and #6781 are provenance-pinned narrower predecessors.

CI notes

At head a30cb01ae877386d0ede459491100fc55cc28c83, run 42984180909, plan job 130401334725, failed in Onboarding helper tests. I read the completed checks once and inspected the job log. It reports 760 tests, exactly one failure and 24 skipped. The sole failing test is:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record

Its AssertionError says that `make frontier7000-verify` was not found in the CI verification document. At canonical main e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json names make frontier7000-verify, while the current section of docs/ci-verification.md still describes #6030 and make frontier6030-verify. I compared both files between main and this head; their bytes are unchanged. This is a repository-level navigation mismatch outside this package.

I also inspected the failed plan logs of run 42905094314, run 42903313790, #7167 run 42983769381 and #7172 run 42984145008. Each shows the same sole failing test, missing command, and 760-test summary with one failure and 24 skipped. The earlier inspected #7134 run 42914997118 and #7135 run 42914936342 show the same failure.

The package replay job 130403220838 was skipped after plan failed. CI did not certify this package. The scientific evidence is the local normal/optimized checks, registered replay, source-history/byte/mode checks and unchanged read-only export replay described above. No job was rerun, no retrigger commit was pushed, and no shared file was edited.

Merge Danger

Door: two-way. The isolated research package can be removed.

Blast radius: package. Selected records, shared implementation and predecessor evidence are unchanged.