[petalas]

Summary

An explicit rational centre-basis shear reduces the actual rank-30 Gaussian reader's four-shear coefficient tariff by eight ADDs:

- 2 * (208 source + 9968 reader coefficients) = 20352
+ 2 * (208 source + 9964 reader coefficients) = 20344

After Ali Romman's #5646 basis, use D5 += (3/40) D4 and J4 -= (3/40) J5. The source/reader product is unchanged, every arbitrary dirty centre is restored, and the word still uses four shears and thirty borrowed fields. This is a finite explicit rational-kernel construction with a scoped algebraic proof. No new kappa or selected-frontier change.

The common denominator changes from 960 to 1200, adding a second factor of five. The maximum coefficient remains one, and the maximum absolute row sums remain 9/4 and 1036/15. This improves the declared coefficient-support tariff; physical rational arithmetic may have a different price.

Only research/rational-reader-basis is added. The first strict improvement ends the search; no optimality is claimed. The result was produced by Nick Petalas. Credit Ali Romman #5646 for the actual factors, published basis and open question, David Ganz / Dugongue #5634 for the reader identity, #5640 for the grouped tariff, and the retained centre-code-services lineage and source notices.

Evidence

• Before: reader row five has 892 coefficients; the complete bill is 20,352.
• After: it has 888 coefficients; the complete bill is 20,344.
• The new replay compares all 11,208 nonzero coefficients of the affected two-term kernel product, reverses the exact basis and checks all untouched factors.
• A literal 512-address execution counts every instruction in both words, compares every data output and restores all thirty arbitrary dirty fields. Five controls detect losing coefficients, an omitted inverse source update and a corrupted reader coefficient. The written algebra covers arbitrary payloads; the one finite payload is an instruction-order control.
• Local CPython 3.12.15 warning-strict normal and optimized checks pass, each taking about 1.84 seconds. These times establish timeout margin, not a speedup claim.
• Registered metadata and both replay commands pass. Exact committed bytes/modes and full source history pass at dfba21817680dbbd1ca0455cae5e5e93c97710fb. Official export/input coverage, a read-only Git-free optimized replay and source-corruption rejection pass.

From the repository root:

~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/rational-reader-basis/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/rational-reader-basis/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/rational-reader-basis
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/rational-reader-basis --check-source-history

Expected statuses are PASS_RATIONAL_READER_BASIS, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The verifier emits a generated hash-bound JSON receipt. No predecessor verifier, rank/inverse audit, local-optimum replay or historical suite is run.

Limitations and unrun obligations

The supplied boundary DJ=0 and factor construction are cited from the predecessors. The new proof establishes preservation and the new support count; it does not independently validate those predecessors. Rational arithmetic interiors, rounding, routing, tape/storage, a complete recursive invoice, native/outer admission and a full multiplication theorem remain open. No proof assistant or selected cold replay was run. All five framework assumptions remain. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Does the eight-instruction saving survive the extra factor of five in one concrete rational-arithmetic caller? Start by pricing these exact coefficients, rounding, routing and storage.
2. Can a broader basis reduce support and the complete physical bill further? Start from these explicit factors and a priced candidate; this contribution claims no optimum or follow-up reservation.

Related open work

Refreshed this account's open PRs created after the advance CLAIM: #5920, #5921, #5922, #5923, #5924, #5925, #5926, #5927 and #5928. Their titles and result scopes were read; none studies the actual rank-30 Gaussian-reader coefficient bill. No additional computation was run to compare other contributions.

No open PR studying this same reader was found in the earlier live searches for 5634, 5646 and Gaussian reader. Merged #5646, #5634 and #5640 are credited predecessors. The other rank-30 search reference concerns a historical moment profile, not this reader. These searches do not cover unpublished work.

CI notes

At dfba21817680dbbd1ca0455cae5e5e93c97710fb, run 40513266276 finished with plan failed; replay skipped. The only failed step is Onboarding helper tests. The unchanged tests/test_research_current_assignment.py ends with FAILED (failures=8, errors=2, skipped=22). Its two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight failures concern current-assignment card/receipt and missing/sparse-reference expectations.

The same step, test file and failure summary were directly confirmed for PR #5918 at d3d1e2155620e2a491523c759e1569587678b922 in run 40506994991. This package adds only its seven scoped files and changes none of those shared tests or helpers. Maintainers own the shared failure; no repair or rerun was attempted here.

Hosted replay has not certified this package. Only the local scientific and publication checks listed above have run. CI did not execute the package replay.

Merge danger

Door: two-way. Blast radius: isolated research package. Existing sources, shared tooling and the selected frontier are unchanged.