[petalas]

Summary

Exact finite certificate with a scoped proof: the three-dimensional mode shadow along the rank-five row flat generated by rows 0 through 4 has 103,333 lattice points, all attained by subset sums. Exactly 94,599 are interior and 8,734 are boundary points.

112 modes -> quotient by one rank-five flat -> Z^3
closed zonotope lattice points = all projected subset sums

This supplies one explicit case of Open next question 1 in Armin Pourshafeie / apoursh's #6829. Its facet/ridge results are cited, not replayed. The new quotient, proof and verifier were produced by Nick Petalas. Credit Dillon Amburgey #4933 for the unchanged matrix, apoursh #6320/#6333 for the finite reduction and retained radius bound, and Nick Petalas #6312 for the exact-mode contract. Classical elimination, zonotope support functions and subset-sum methods retain their credit.

Advance CLAIM. Canonical base e545235, priority revision 8. The uniform-transfer-audit card explicitly changes scope to this analytic-recovery side result. Its actual-caller gate remains unmet; selected #7000 is unchanged.

Evidence

Before: #6829 leaves shadows along rank-five flats uncomputed. After: this one flat has an exact quotient lattice, complete facet description and full closed/interior filling certificate.

Local CPython 3.12.15, standard library only, at head a2f593e:

• Check the rank-five basis, rank-three quotient map, exact 15-row closed flat, even projection and coordinate unit generators proving the quotient lattice is Z^3.
• Enumerate all 103,333 subset sums by both integer bitsets and independent tuple sets. Enumerate all 416,295 box points against 25 exact facet-normal pairs, and cross-check an independent interval scan. Every closed membership decision agrees.
• Five control predicates detect a removed interior subset sum in both scans, closed/interior holes in a small counterexample, and signed-shift encoding errors.
• Both registered warning-strict commands passed with identical computed hash-bound receipts. Individually timed direct executions took 4.924 seconds normally and 4.735 seconds optimized; the complete registered replay took 9.63 seconds. These are budget measurements, not speedup claims.
• Source-binding baseline, AST compilation, metadata/pins, whitespace, exact committed bytes/modes and declared source ancestry passed. Official export and declared-input integrity passed. Both direct modes also passed from a read-only Git-free export with empty PATH/PYTHONPATH; its eight files remained byte-identical. No predecessor scientific executable ran.

The receipt is generated from the current six package hashes and the computed subset bitset; it is not handwritten or committed.

Reproduce

From the repository root:

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

Substitute another CPython 3.12 interpreter path elsewhere. Expected markers are PASS_MODE_RANK_FIVE_SHADOW, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Materialize the new package and research/coded-native-caller/NORMALIZERS.json in sparse checkouts.

Limitations and unrun obligations

One rank-five flat only. No all-orbit classification, universal three-dimensional filling, full eight-dimensional filling or R=1 theorem. The retained radius interval remains 1 to 3/2. No physical rounder, native/caller realization, ownership, compiler, precision/tape or stock bill, supplier admission, new kappa or complete multiplication theorem is supplied. All five selected assumptions remain. No selected cold replay, predecessor scientific suite, Lean, Docker or separate type/lint analysis was run. Independent human and formal proof-kernel review are unclaimed.

Related open work

No open PR on exact three-dimensional mode-shadow filling surfaced in the live object/formula, current open-PR and Discussion searches. The cached full overlap search covered 503 open PR bodies and 7,185 Discussion records; sparse local-note coverage was incomplete, so main README and merged-PR searches were checked separately. Arbitrary historical diffs and unpublished work remain outside coverage.

• #7192 uses the same original mode data for a fixed rounded-inverse injectivity problem; this certificate concerns projected zonotope subset sums and computes no competing injectivity threshold.

• #7212 constructs balanced rank-one cycles for refined rounded-native recovery fibers using the same mode-data lineage; this PR concerns exact projected zonotope subset sums and computes no competing recovery-fiber optimum.

The completion refresh read this account's nine other open PRs created since the CLAIM: #7203, #7204, #7205, #7206, #7207, #7208, #7211, #7212 and #7213. Only #7212 adds a related mode-data question, as distinguished above. None computes this one-flat shadow count; no additional scientific comparison computation was run.

Open next questions

• Classify rank-five row-flat symmetry orbits and locate the displayed 15-row flat in that catalogue.
• Test another orbit with its exact quotient lattice and a complete membership scan. Larger boxes may require another finite representation.
• Connect any eventual full-dimensional filling theorem to a physical rounder and its complete caller costs. Shadow filling alone is insufficient.

CI notes

CI did not certify this package. At unchanged head a2f593e60346aaed903425c3bac84eb250b07972, run 43013336222, plan job 130498138797, failed in "Onboarding helper tests". I read the checks once and inspected the actual job log. It reports exactly one failure among 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found
FAILED (failures=1, skipped=24)

The assertion is at tests/test_research_current_status.py:58. Direct inspection of canonical base e54523517d0f03e58352e647c403a5af3e525a57 confirms that certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. The record, guide and test are byte-identical between that base and this PR head. This package did not cause the repository-level documentation mismatch.

The dependent package replay job was skipped because plan failed. Local normal/optimized, registered, source-history and portable-export checks remain the package's available execution evidence; hosted CI certified nothing for it.

I inspected all six comparison failed-step logs below and confirmed the same sole failing test, missing-command assertion and totals of 760 tests, one failure and 24 skipped:

• Run 42905094314.
• Run 42903313790.
• PR #7203, run 43012497455.
• PR #7204, run 43012524632.
• PR #7208, run 43013294091.
• PR #7211, run 43013361819.

No fix, retrigger commit, push or job rerun was made. Maintainers will repair main. The published head and six package files are unchanged.

Merge danger

Reversible isolated research package. Only six files under research/mode-rank-five-shadow are added. Shared implementation, frozen inputs and selected-frontier records are unchanged. Archival intake and scientific acceptance remain separate.