[petalas]

Summary

Finite explicit factor construction with a scoped written proof. One new rational shear after the translated Gaussian reader basis of #6677 reduces the complete four-shear bill to 20,312 coefficient ADDs. Every translation is charged. The axis-translation invoice is 91,852 units, the maximum absolute coefficient is 9/8, and the scatter/gather absolute row-sum bounds remain 43/16 and 72. The common denominator increases to 4,800.

New zero-based factor update:
L0 -= (1/5) L1
J1 += (1/5) J0

Four-shear response: x -> (I+JL)x
All thirty arbitrary dirty borrowed fields return exactly.

This answers one further part of Ali Romman's #5646 broader-basis question. Credit Ali Romman for the supplied factors, constant dyadic basis and question; Nick Petalas #6677 for the translated input basis and adapted reconstruction/word code; Nick Petalas #5919 for the earlier rational escape; David Ganz / Dugongue #5634 for the reader substitution and four-shear identity; Ali Romman #5640 for the borrowed-space contract. The retained native/numerical/compiler contributors are credited in the README. The new shear, proof and checker were produced by Nick Petalas. Predecessor scientific suites are cited rather than audited or replayed.

Only research/translated-reader-step/ is added. Base main is e54523517d0f03e58352e647c403a5af3e525a57, priority revision 8. The complex-new-helper-inventory card changes scope to an ordinary reader-interface construction. No new productive-helper or complete physical-invoice gate is claimed. Advance claim. Discovery stopped at the first new verified step.

Evidence

Before: the cited translated factors use 220 scatter terms and 9,940 gather terms, hence 20,320 coefficient ADDs and 91,924 axis-translation units at denominator 960.

After: the new factors use 228 scatter terms and 9,928 gather terms. Each factor occurs twice, so twelve removed gather terms and eight added scatter terms save eight ADDs. The complete translation invoice saves 72 declared axis units. The denominator cost increases from 960 to 4,800; no physical benefit is inferred from the support counts alone.

Local CPython 3.12.15, macOS arm64, standard library only, at head 5ff6f378515e15d9268e1cf4d1f7b9ebef4142b1:

• Exact changed-pair group-algebra equality, with all 14,320 nonzero baseline coefficients and possible new positions; exact inverse basis and every untouched factor vector pass.
• A separate literal complete four-shear executor compares all 57,344 data coordinates and restores all 15,360 dirty-bank coordinates on a dense 512-address payload. The written proof covers arbitrary inputs and every representation of the commuting translations.
• Eight algebraic/input rejection controls pass. All coefficient, translation, denominator, magnitude and norm metrics are enumerated.
• Both registered normal/optimized commands pass. Timed direct warning-strict runs took 3.06 s and 3.07 s, below each 120 s registered timeout; their generated hash-bound receipts are byte-identical. The registered two-command run took 6.57 s total.
• Metadata/source pins, AST compilation, function-size inspection, whitespace and committed-byte/source-history checks pass. The last check reports 12 files and base ancestry e54523517d0f03e58352e647c403a5af3e525a57.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/translated-reader-step
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/translated-reader-step/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/translated-reader-step/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/translated-reader-step --check-source-history

Expected markers are PASS_REGISTERED_RESEARCH_PACKAGE, PASS_TRANSLATED_READER_STEP and PASS_RESEARCH_COMMIT_BYTES. Another CPython 3.12 executable may replace the local path. Direct replay writes only its generated JSON receipt to stdout.

Limitations

This is a finite explicit coefficient-factor improvement with a scoped proof, not a new kappa, physical supplier, admission or complete multiplication theorem. No local or global basis optimum is asserted. Source null-product/native identities are cited premises. The denominator gains one factor of five; rational multiplication, primitive-interior precision, physical routing/tape, the section's own bank, storage/ownership, stock, uniform compiler closure and outer application remain separate obligations. All five selected assumptions and the selected #7000 frontier remain unchanged. Predecessor scientific suites, selected cold replay, Lean, Docker, independent human review and formal proof-kernel review are unrun or unclaimed.

Open next questions

1. Does this eight-ADD saving survive a complete arithmetic and physical routing bill? Start by implementing both factor pairs in the same priced kernel, charging coefficient multiplication, rounding, movement, retained storage and returned ownership. The arithmetic backend and full caller are missing.
2. Can another basis improve a fixed complete physical objective? Define that objective before another cancellation search. This witness proves no optimum, and a consumer contract remains missing.

The README and TASK.json record relevance, first steps and blockers.

Related open work

No open PR studying the actual rank-30 Gaussian reader's coefficient or translation bill surfaced in the live overlap checks. Merged #5646, #5919 and #6677 are the direct constructive predecessors. Open #7185 studies generic three-block square-basis signatures, a different quantity; it supplies no competing Gaussian-kernel count. The live checks were refreshed before committing. Sparse cached local-note coverage, unpublished work and arbitrary historical diffs remain coverage limits.

The completion check examined this account's three other open PRs created since the CLAIM: #7191 classifies polynomial two-gap insertions, #7192 studies rounded inverse injectivity from owner residues, and #7193 bounds five address-twist contributions. None studies the actual Gaussian reader's coefficient or translation bill or reports a competing value for it. No additional scientific computations were run for this comparison.

CI notes

CI did not certify this package. At exact head 5ff6f378515e15d9268e1cf4d1f7b9ebef4142b1, run 43000211746, plan job 130454190330, failed in Onboarding helper tests. I read the check summary once and confirmed the following exact failure from the job log:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found
Ran 760 tests
FAILED (failures=1, skipped=24)

The log's CI verification document still describes #6030 and make frontier6030-verify. Direct comparison with base main e54523517d0f03e58352e647c403a5af3e525a57 confirms that its selected record already names make frontier7000-verify. The test, CI document and selected record are byte-identical between that base and this PR. The failure predates this package.

The same exact failing test, missing command, 760-test total, one failure and 24 skips were independently confirmed from the failed-step logs of all five comparison runs:

• Other contributors: 42905094314 and 42903313790.
• Prove sharp multiplicative-shell growth for every rank alphabet (no κ) #7178: 42992088329.
• Prove five A2 targets need five rank-three calls (no κ) #7179: 42992340683.
• Prove the sharp half-shift Gaussian inverse lower envelope (no κ) #7180: 42992579989.

The package replay job 130456102593 was skipped because planning failed. Local normal/optimized, registered-package and committed-byte/source-history checks above remain the available automated package evidence. No fix, retrigger commit, push or job rerun was made for this repository-level failure. Maintainers will repair main. Mathematical acceptance remains separate from replay.

Merge Danger

Door: two-way. Blast radius: isolated. Eight new package files; shared implementation, pinned evidence and selected-frontier files are unchanged.