[petalas]

Summary

Prove the exact information retained by a rational-shear carry producer after reading its complete source low digit and before reading its target low digit. The interface forbids rereading the source and emitting information early, and charges all data-dependent controller/head state. Claim type: scoped written mathematical proof with finite corroboration.

g = gcd(A,q), d = q/g, e = Q/gcd(A/g,Q)
minimum cut states = 1                    if Q=1
minimum cut states = d * min(g,e)         if Q>1

An explicit equivalence-class label attains the bound. For a quarter shear over every odd q>=3 and high width w>=2, exactly q states are needed. The q=257 fixture requires nine retained binary bits although the final integer carry takes only eight values. Producing an exact integer packet and producing its high-modulus image have distinct state counts at width one.

This answers one serialized-access version of the low-cache packet-production question left open by Ali Romman's #5849 at 35ee8364ccfe54d6d5ec154fbf74993624b0a0d4. It does not audit or replay that construction. The new proof and verifier are produced by Nick Petalas. Credit Ali Romman #5849/#5841, JoyboyBrian, Tekalign and the retained ROW/native/tape lineage; the deterministic one-way state-equivalence method is classical.

Evidence and reproduction

Only research/quarter-carry-cut-states/ is added. Local CPython 3.12.15 on macOS arm64 passes:

• Warning-strict normal and optimized checks, separately timed at 13.300s and 13.450s. Both print PASS_QUARTER_CARRY_CUT_STATES with generated package-hash-bound receipts.
• 70,214 parameter systems and 16,603,686 source/target digit pairs; 192 quarter-family cases; all 32,896 source-pair separators at q=257; eight nonunit/collapse fixtures, fourteen invalid-input rejections and six semantic controls.
• A separate arithmetic probe checks 427 systems and 349,978 digit pairs using extended Euclid and direct full-modulus division. No predecessor verifier is imported or executed.
• Both registered VERIFY commands, metadata/provenance pins, exact committed bytes/modes and source-history checks; in-memory compilation and diff whitespace checks.
• Official committed-package export integrity and a read-only, empty-PATH, Git-free optimized replay, with every exported byte and mode unchanged.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/quarter-carry-cut-states/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/quarter-carry-cut-states/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/quarter-carry-cut-states
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/quarter-carry-cut-states --check-source-history

Any CPython 3.12 path can replace the local interpreter. Only the standard library is needed. SOURCES.json pins catalogue/selected-record provenance at canonical base de5ec2e7a6074068452f45b1a191b5c72832df4a; cited pending predecessors are scholarly context, with no executable or unmerged-history dependency.

Limitations and unrun obligations

The theorem counts information at one cut. It does not bound arbitrary interleaved, multipass or early-output algorithms, amortized joint packets, or an already prepared cache. Its attaining encoding permits later unrestricted computation and is not a literal tape implementation or a total-workspace bound. No claim identifies this scalar fixture with an actual released G20 port. Physical source access, cache layout, erasure, routing, native/caller integration, precision/stock and all-size implementation remain open and unrun. All five framework assumptions remain; no new kappa, supplier admission or frontier change is claimed. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Specify one actual low-cache layout that interleaves the two digit streams and charge its reads and erasure. This determines whether the serialized lower bound applies to a physical producer; the literal layout is missing.
2. Determine the joint cut-state count for two packets sharing one source digit, beginning with the quotient of the pair of response functions. A fixed joint output/access contract is needed before assigning separate storage charges.

Related open work

• Complete runtime G20 quarter-carry rotation with tight physical bounds #5849 supplies the original-owner rotation from a supplied packet and leaves its low-cache producer open; this PR prices information for one restricted producer access order.

• Lower all sparse G20 routes with exact whole-word q-adic carries #5841 supplies fused G20 carry algebra and potential correction sites; this PR counts distinguishable response functions for a scalar rational shear rather than route sites or carry-event density.

• Complete all G20 rational shear leaves with paid literal coefficient preparation #5846 implements rational-shear leaves with paid coefficient preparation; this PR supplies no replacement leaf or physical tariff.

• Complete original-owner execution of the full runtime affine shear #5842 installs runtime affine-shear results in the original owner; the state count here concerns packet preparation before that execution interface.

• Construct dyadic words for all 111 G20 port normalizers #5801 constructs dyadic port-normalizer words; this result does not resynthesize or realize those words.

• Bind all 7,548 G20 route markers to conditional ROW cover words #5805 binds actual markers to conditional ROW words and leaves cache/carry/commit details open; no physical marker binding is claimed here.

• Install the shorter G20 word in all 18 physical calls with fresh source records #5834 constructs the complete shorter global G20 caller while retaining cache/carry obligations; its endpoint, invoice and events are not replayed or changed.

• Halve G20 boundary work with equal-target bases and a complete physical caller #5840 simplifies global G20 boundary words while retaining cache/carry obligations; its synthesis and complete finite bill are not replayed or changed.

• Count exact nonzero carries for every reduced rational shear (no κ) #5915 counts low-input pairs with nonzero rational-shear carries; this PR counts distinguishable response functions after a source-before-target cut, a different quantity. No numerical comparison is made.

• Prove exact nearest-even shear history from binary code weight #5912 classifies retained history for nearest-even rounded shears; this PR uses exact modular arithmetic and does not compute those history costs.

CI notes

At head f0776668541d2828dd10e290c8e5adabe990f96d, run 40506305739 has plan failed; replay skipped. The only failed step is Onboarding helper tests. All ten failing cases belong to tests/test_research_current_assignment.py; the exact suite summary is FAILED (failures=8, errors=2, skipped=22). The two errors are missing follow_up_steps and catalog_first_step keys; the eight failures are fixture/reference expectation mismatches. This is the shared main de5ec2e7a onboarding failure, not a failure in this package. The same step, test file and summary were independently confirmed in PR #5911's run. No shared tests or other repository files were changed to address it.

The package replay job was skipped. Hosted replay has not certified this package; only the local mathematical/package checks reported above have run. Each registered local command completed below fourteen seconds with a 300-second timeout. No hosted scientific pass is claimed.

Merge danger

Reversible package-only addition. No shared code, frozen source, selected record or review ledger changes.