[petalas]

Summary

Scoped construction proof with finite corroboration. A new paid-pivot BIT fallback implements every split rank-r projector over every odd prime-power ring, including arbitrary dirty data and diagonal ancestor weights. No new kappa or supplier admission is claimed.

pivot the rank-r cross block with paid swaps
execute the ordered-ADD adapter
execute one weighted reversed rf-child
execute the inverse adapter

The complete child bill is one width-rf child, at most 4r width-f children and at most 8r width-floor(f/2) children. Half-width calls are omitted for f=1. At m=64,r<=42 there are at most 505 children, with linear width moment at most 9rf. The new largest child has width rf, so this is not an equal-tariff runtime comparison with the retained 49,911-child fallback envelope. The adapters also have exactly f[4r(m-1)-2r(r-1)+2m^2] scalar ordered-ADD slots, including zero coefficients, or 15,332f slots at rank 42.

This extends Cinna Davis's #4100 block identity: paid unit pivoting supplies invertible L,R0 with LPC R0=EDC for every split projector. Triangularity is unnecessary in the block identity. The old generic fallback eliminates the full 2m-coordinate involution; this construction retains one rank-r child. The extra-child bound is attained by the declared compiler on suffix projectors, without claiming a universal minimum.

Only research/bit-pivoted-fallback/ changes. Base: 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7. Publication commit: 27f1ce44126cb3a841756834649f556de0ce00e9.

Evidence and reproduction

Before: arbitrary bad packets use the retained full-matrix fallback. After: a separate compiler supplies the pivoted rank-r construction with all swaps and inverse adapters charged.

Local exact checks enumerate 417,575 matrix candidates and all 1,134 nontrivial idempotents in five small-ring domains. They test 1,263 plans, including 1,074 needing paid pivots, on 11,490 basis/dirty vectors and 11,490 involution roundtrips against a separate direct endpoint oracle. All 128 low/high itinerary comparisons pass. Sixteen literal 64-coordinate fixtures cover all eight retained paid ranks at f=1 and f=3, including rank 42 and the attained 12r extra-child count. Four semantic and four invalid-input controls pass.

From the repository root:

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

All pass locally on CPython 3.12.15. Direct normal and optimized checks take 2.52 s and 2.39 s, below their 120-second limits; these are check durations. Markers: PASS_BIT_PIVOTED_FALLBACK, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES. The preflight authenticates all 10 package/input files and source history. The verifier generates its report with all package/input hashes; no receipt is handwritten.

The official committed export passes integrity checking. Both direct modes pass without Git in 2.57 s and 2.61 s, with all 11 exported files unchanged. Altering the declared normal-form input is rejected by its source hash before the scientific replay.

Limitations and unrun obligations

The universal result rests on the written local construction proof, with no independent human or formal proof-kernel review. The code enforces m<=64, f<=8 and modulus bit length<=4096; the written construction has no such upper limits. Executed rings are the listed prime powers. A completed weighted-child oracle and ordered ADDs are explicit premises. The local proof is not a physical supplier or finite-tape implementation.

Physical coefficient preparation, ratio-weight descriptors, routing, fresh-row divisibility, high-packet scans, storage, parking and restoration at the actual caller remain open. No selected cold replay, global recurrence retuning, all-size compiler admission or complete multiplication theorem was run or claimed. Production inputs and the selected frontier are unchanged.

Open next questions

1. Install one complete m=64 rank-42 physical packet with odd f, frozen ancestors and explicit ratio-weight descriptors; prove its compiler, stock and restoration bill.
2. Compare a pivot policy preferring an already positioned unit with the suffix fixtures. The current count is sharp for this compiler, not for all algorithms.

Related open work

• Independently check PR4100's good-packet normal form on small rings (no new κ) #4215 proves/checks the good-packet normal-form identity and asks about fallback costs. This uses that identity as a predecessor and constructs a different fallback. Its finite audit corpus and this new-program corpus differ, so no shared aggregate count is asserted.
• Prove exact extreme-rank BIT header densities #4227 counts exact extreme-rank unpivoted good headers. This is a local construction for every projector, with no overlapping density number to compare.
• Prove exact intermediate-rank BIT header densities #4228 gives the all-rank unpivoted good-header law and a fallback-pricing handoff. This does not change or recheck its density law; no overlapping density number is asserted.
• Prove the sharp fallback child count: 33269, not 49911, at m = 64 (no new κ) #4237 sharpens the count for the unchanged full-matrix fallback. Both quote the source envelope 49,911 = 3*16,637, which agrees arithmetically. This new algorithm instead retains a width-rf child, so their complete child histograms differ; its new maximum is cited rather than independently rechecked.
• Prove the exact joint header law for orthogonal rank-one requests #4239 counts joint first-pivot guards for orthogonal rank-one requests. This pivoted construction has no overlapping joint-density number or shared child invoice to compare.

CI notes

Local metadata, both registered commands, direct normal/optimized replay, committed-byte/source-history preflight, committed-export integrity, Git-free replay and declared-source corruption controls pass. Hosted CI is green on exact head 27f1ce44126cb3a841756834649f556de0ce00e9: run 39488429736, plan passed in 3m12s and the package replay passed in 1m34s. The hosted log contains both scientific PASS reports and PASS_REGISTERED_RESEARCH_PACKAGE; every reported package/input hash matches the committed bytes. No check failed and no check remains pending.

Merge danger

Reversible isolated research package. Its blast radius is the new directory; production sources, old credits and selected-frontier data are unchanged.

Credit

Produced by Nick Petalas. Cinna Davis supplies #4100's block identity; source527 and commdata2338 supply the normalizer lineage; Jose Manuel supplies recursive closure and physical continuation; Ross Richardson supplies the full-ring fallback, covariance and orientation accounting. The original notices retain their credit. Independent human and formal proof-kernel review are unclaimed. Apache-2.0.

[rcfc-jreed]

Independent scoped audit of #4235 BIT pivoted fallback at 27f1ce44: 13/13

Code-disjoint checks; does not import package verify.py (local fork under nested research/bit-pivoted-fallback/ with patched PACKAGE/ROOT). Package bytes from head 27f1ce44126cb3a841756834649f556de0ce00e9 via GitHub contents API. CLAIM.

No packaging defect in scope. Stdout JSON receipt (PASS_BIT_PIVOTED_FALLBACK).

What passes: pins at base 12fa8c5c…; six SOURCES digests; fork status/controls receipt regenerates.

Not covered: Lean formalization; new κ; merge/close. Finite replay ≠ formal proof.

James Reed (@rcfc-jreed), with substantial AI assistance (Claude, in Cursor).