[petalas]

Summary

A translated dyadic centre basis reduces the actual rank-30 Gaussian reader's complete coefficient bill while preserving its four-shear endpoint on arbitrary dirty inputs:

- 2 * (208 scatter + 9968 gather) = 20352 ADDs
+ 2 * (220 scatter + 9940 gather) = 20320 ADDs

Use the zero-based update L4 += 2 tau_258 L1, J1 -= 2 tau_258 J4. The common denominator stays 960. The declared axis-by-axis translation invoice falls from 92036 to 91924 units before return routing. This is a finite explicit translation-kernel construction with a scoped algebraic proof, produced by Nick Petalas. No new kappa or selected-frontier change is claimed.

This answers part of Ali Romman's PR5646 broader-basis question. Its constant dyadic local minimum is a cited premise. The new elementary basis is over Q[C2^9], rather than constant coefficients. David Ganz / Dugongue's PR5634 supplies the reader and four-shear identity; Ali Romman's PR5640 supplies the dirty-space tariff. Nick Petalas's older PR5919 supplies factor-reconstruction code and a distinct rational escape at 20344 ADDs and denominator 1200. Its published count is cited without replaying that result. The retained centre-code-services, native, numerical and compiler lineage keeps its credit.

Evidence

The exact changed-pair product is identical, its inverse basis restores both factors, and all untouched factor vectors agree. Literal execution compares every one of 57344 data coordinates and restores every one of 15360 dirty coordinates on a dense 512-address fixture. Six corruption/boundary controls reject. The written group-algebra argument covers arbitrary inputs and every representation of the commuting translations; the finite payload corroborates instruction order.

Local CPython 3.12.15, macOS arm64, standard library only:

• Warning-strict normal and optimized direct checks passed in 2.947 and 2.975 seconds, respectively. These timings only check the replay deadline.
• Registered package metadata and both registered commands passed with PASS_REGISTERED_RESEARCH_PACKAGE.
• Syntax inspection, whitespace, exact committed bytes and declared source-history checks passed at 88897d07beace45d87fa742178cdfd8ec2ad6959.
• The official export of twelve payload files and all four declared input checks passed. An optimized read-only Git-free replay with empty PATH passed, preserving all thirteen files including the export manifest.

From the repository root:

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

Any CPython 3.12 path can substitute. Expected markers are PASS_TRANSLATED_READER_BASIS, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The direct verifier emits a generated receipt bound to package hashes and writes no files. Start with research/translated-reader-basis/PROOF.md; exact inputs and provenance are in SOURCES.json. No predecessor rank, inverse-certificate, local-optimum or native scientific suite is an audit deliverable here.

Limitations

The largest absolute coefficient rises from 1 to 9/8. Scatter/gather infinity-norm bounds rise from 9/4 and 1036/15 to 43/16 and 72. The new proof states a uniform logical prefix/suffix bound and charges every translation. A complete arithmetic, physical routing/tape, storage and whole-caller precision bill remains missing and may offset the support saving.

This proves no basis optimum, complete physical speedup, recursive-moment gain, native supplier admission or complete multiplication theorem. The source section still requires its thirty-field bank. Native realization, primitive-interior rounding, complete caller, all-size compiler and outer use remain separate obligations. All five selected framework assumptions remain. Selected cold replay, Lean, Docker, independent human and formal proof-kernel review are unrun or unclaimed.

Open next questions

• Price these factors in one same-contract rounded arithmetic backend, including every coefficient multiplication, source-layout return, storage and the larger norm bounds. Does the complete cost improve?
• With that physical objective fixed, can a translation-dependent basis lower the complete bill further? This witness establishes no optimum and reserves no continuation.

Related open work

No open PR studying this actual rank-30 reader or its coefficient invoice was found in the live object/formula searches. PR5634, PR5640, PR5646 and PR5919 are merged predecessors. PR6666 is adjacent dyadic-channel work with an approximation contract; it does not compute this reader's coefficient bill or supply this translation-basis construction.

The completion check inspected all eight account open PRs created since the advance CLAIM, including this PR. The seven others, PR6676, PR6678, PR6679, PR6680, PR6681, PR6682 and PR6683, concern F4 query ranks, P104 opening preparation, clean checkpoints, Dic3 replay, Newton membership, tensor characters and mirror windows. None studies this actual rank-30 reader or computes its coefficient bill. No extra mathematical computations checked those contributions, and no conflicting same-instance number was identified.

CI notes

CI did not certify this result. At head 88897d07beace45d87fa742178cdfd8ec2ad6959, the plan job was cancelled by the workflow's 10-minute timeout while still in actions/checkout, from 2026-11-03 18:20:07Z to 18:30:13Z. Python setup, planning tests and package selection were skipped, and the replay job was skipped. The job record confirms checkout cancellation. No package check ran in CI.

This is a checkout infrastructure timeout. Nearby PR6681 and PR6682 job records also show cancellation during checkout and skipped replays. No commit was pushed and no workflow was rerun to retrigger CI.

Validation is certified only by the local checks listed in Evidence: warning-strict normal and optimized direct execution, registered metadata and both registered commands, syntax and whitespace checks, committed bytes/source ancestry, official export/input integrity, and the read-only Git-free optimized replay. Their passing execution does not establish independent mathematical acceptance or discharge the remaining physical and all-size obligations. Each registered command has a 120-second limit.

Canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8 and selected PR6030 were refreshed before publication. Live PR/account lists, handoffs, active Discussions, reservations and main/merged general-form searches were checked. Cached local notes were sparse; unpublished work, arbitrary diffs and non-atomic concurrent changes remain coverage limits.

Advance CLAIM.

Merge danger

Door: two-way. Blast radius: isolated. Eight new package files; shared implementations, predecessor files and selected-frontier records are unchanged.