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DONE — scoped contribution published by JoyboyBrian. Independent literal-marker audit corroborates the source functional subroutine: a fixed 256-row interpreter realizes 2L+2 steps in both directions with returned heads/control. Compiler setup, coin delivery and full caller admission remain open. Public artifact: #6095 at Focused replay and source/code review passed; scientific packages also passed committed-byte/source-history and portable-export checks. Hosted CI is recorded separately and is not scientific acceptance. Full historical suites, Lean and Docker were not run. Exact source credit and Open next questions are in the artifact. Prepared with substantial OpenAI Codex assistance. {
"schema": "integer-mult-coordination/v1",
"kind": "DONE",
"task": "supplier-boundary-audit",
"scope": "Independently audit PR6092\u2019s literal two-tape interpreter and charged return contract",
"base": "721c03822c421b7f7fcbb2598993b947c4d88419",
"expiry": null,
"artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/6095"
} |
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DONE — scoped contribution published by JoyboyBrian. The actual two-mode inverse has all 2704 residue pairs modulo52 already at H=1. The denominator13 ABI is exactly the index16 subclass; the explicit conditional selector adapter is correct but does not establish savings, physical service or cleanup. Public artifact: #6096 at Focused replay and source/code review passed; scientific packages also passed committed-byte/source-history and portable-export checks. Hosted CI is recorded separately and is not scientific acceptance. Full historical suites, Lean and Docker were not run. Exact source credit and Open next questions are in the artifact. Prepared with substantial OpenAI Codex assistance. {
"schema": "integer-mult-coordination/v1",
"kind": "DONE",
"task": "supplier-boundary-audit",
"scope": "Audit denominator-13 joint sampler against actual /52 inverse residues",
"base": "721c03822c421b7f7fcbb2598993b947c4d88419",
"expiry": null,
"artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/6096"
} |
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HANDOFF — The actual first MOVE receives no prefix rounding noise The feasibility gate for task05 stops before nonzero concentration transport. At canonical base Role224 has exactly zero squared and absolute prefix rounding influence. Its sole prefix write is event7,
For the source grid-input contract these two updates are exact. Later rounding elsewhere cannot affect the already completed destination. This holds for arbitrary dirty grid inputs and separately for real/imaginary components, without a randomness assumption. The copied output consequently presents no prefix stochastic error to this MOVE. It supplies no nonzero tail-transport example and no precision advantage. A local-only bounded verifier independently propagated one reverse influence row and all748 chronological formal-error rows. Both give S224=L224=0; the forward engine also recovers282 noisy outputs and maximum S=733/8, matching Nick Petalas's PR6026 certificate. Mutating either relevant integer coefficient to-1/2 produces squared influence1 in both engines; a later fractional write to225 leaves224 noiseless. The unchanged native interpreter checks all512 Gaussian columns against Minimal self-contained stop reproduction, run at the exact base above: import gzip, hashlib, json
from fractions import Fraction as Q
p = 'research/p104-helper-subspace-splits/WORD.json.gz'
raw = open(p, 'rb').read()
assert hashlib.sha256(raw).hexdigest() == 'bbb072bced5824a8d9094c49be2fc51fbfdd05fb2b7dab8106e1275053075099'
w = json.loads(gzip.decompress(raw))
assert next(i for i,e in enumerate(w['program']) if e[0]=='MOVE') == 9589
assert w['program'][9589] == ['MOVE',224,3,0,False]
events = w['program'][:9589]
assert [(i,e) for i,e in enumerate(events) if e[1]==224] == [(7,['ADD',224,225,'-1',3])]
assert [(i,e) for i,e in enumerate(events[:7]) if e[1]==225] == [(0,['ADD',225,227,'-1',3])]
a = [Q(0)]*748; a[224] = Q(1); S = L = Q(0)
for e in reversed(events):
r,q = e[1],Q(e[-2])
if q.denominator > 1: S += a[r]**2; L += abs(a[r])
if e[0]=='SCALE': a[r] *= q
else: a[e[2]] += q*a[r]
assert S == L == 0
assert {i:x for i,x in enumerate(a) if x} == {224:1,225:-1,227:1}
print('PASS_FIRST_MOVE_NOISELESS_GATE')The longer local verifier passed normal Python, optimized Python, and an identical read-only Git-free replay. Its receipts are local-only; the embedded reproduction is the public stop evidence. No PR, package export, later-MOVE search, full historical suite, Lean or Docker replay was produced. The originally proposed PR6066 object was unavailable locally and was not needed or independently checked. Open next question: Identify a source-bound boundary with nonzero reached error, after accounting for every intervening MOVE and scalar event. Do not select a later role from the unchanged prefix table: intervening operations alter its influence law. The first step is a separately claimed chronological propagation to one actual noisy boundary; its native approximation and paid sampler remain blockers. Even here, a rounded native implementation is not error-free: if Credit: Nick Petalas's PR6026 provides the stochastic-prefix question and methodology; Ali Romman's historical helper-subspace word and its unchanged NOTICE retain their predecessor attribution. JoyboyBrian, with substantial OpenAI Codex assistance, supplied this stop witness. Apache-2.0. No new kappa or complete multiplication theorem is claimed. {
"schema": "integer-mult-coordination/v1",
"kind": "HANDOFF",
"task": "uniform-transfer-audit",
"scope": "Transport the source stochastic-prefix guarantee through its actual first native MOVE",
"base": "721c03822c421b7f7fcbb2598993b947c4d88419",
"expiry": null,
"artifact": null
} |
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DONE — scoped contribution published by JoyboyBrian. For the literal (1,1) paired DDG, the resident transcript plus stopping counter supports output XOR-copy and complete path reversal without per-step state history. Conditional scan cost is2496T record visits plus counter/tape costs. Retained randomness is not fresh conditional on output. Public artifact: #6102 at Focused replay and source/code review passed; scientific packages also passed committed-byte/source-history and portable-export checks. Hosted CI is recorded separately and is not scientific acceptance. Full historical suites, Lean and Docker were not run. Exact source credit and Open next questions are in the artifact. Prepared with substantial OpenAI Codex assistance. {
"schema": "integer-mult-coordination/v1",
"kind": "DONE",
"task": "formal-interface-lemma",
"scope": "Reverse the actual variable-time paired sampler without a per-step state log",
"base": "721c03822c421b7f7fcbb2598993b947c4d88419",
"expiry": null,
"artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/6102"
} |
R20 task06 handoff: the first inverse rounding column has a factor-13 exact suffixStatus: source-bound local handoff, not caller admission. No PR proposed. Canonical base: The first where v is the selected common-coordinate real scalar basis field. The K maps are signed permutations; they cannot be omitted from the column, although they preserve its component magnitude. This refers to the exact-real column induced by one rounding site, with all other errors separated. For the actual retained pair suffix, write R=DQ, C=K^-1(A tensor I)K, and use PR5069's B, T and O. Applying OTB to the injection (0,e) gives Indeed M^T M=56I and A=I-MM^T/4 imply A m=-13m. The Y term vanishes before O: D e - Q^-1 C^-1 DQ C e=0, using [C,DQ]=0. This preserves the actual reconstruction, normalizers, thin inverse, and exit Q conjugation. For a fixed input and a hypothetical replacement with expectation error β=E[δ], the same exact-linear formula holds with β. No independence assumption is needed for this one-site expectation calculation. It does not describe the difference between two executions with later nonlinear rounded inverses: those need separate error variables and range/stability bounds. The column is an operator-valued gain, not a field-wide infinity-norm allowance. A complete field of errors can sum coherently; no field-wide bound follows from a single column. A concrete low-bit discriminator is numerator 1 (actual integer input e_0 in common coordinates): p=1/52. A two-fair-bit bounded sampler has probabilities k/4; its nearest p is 0, so β=-1/52 and ΔX=+(h/4)R K^-1(m tensor v). For numerator -1 the corresponding sign reverses. The magnitude remainder is 4r+low, so replacing p by r/13 silently discards this discriminator. This is a conditional arithmetic comparison, not a physical sampler implementation or a recommendation to replace deterministic nearest rounding. Exact missing contract and stop decision
The precise next input needed is a caller ledger specifying (a) the accepted error norm and probability semantics, (b) its residual budget at the pair output, (c) P and the actual remaining suffix, and (d) allocations for later rounded inverses and ordinary arithmetic. Substituting the displayed column into that ledger is the next task. Do not invent a tolerance, infer a stochastic guarantee from deterministic nearest bounds, or expand to all eight modes. Pins and bounded reproductionAll executable inputs are on canonical base 721c038:
PR6096 head e56f509 was read for its source residue/low-bit classification. PR6068 head 689e282 was named in the claim but its Git object was unavailable locally; its finite-bit theorem was not imported, rechecked, or required for the conditional β formula. The companion Both passed. The check authenticates the three source hashes, injects +1 and -1 into the actual first kernel round and verifies reconstruction, checks the mode eigenvalue, executes 16 exact rational suffix cases at four source-bound Walsh charts and δ in {±1,±1/52}, and checks omitted-normalizer address and dropped-low-bit discriminators. Finite cases corroborate the written operator proof; they do not verify all widths or the complete rounded caller. Explicit exceptions remain active under -O. No source was modified. Open next questionsSupply the missing named caller ledger above before judging bounded bias affordable. If such a ledger uses expectation, first specify why that criterion suffices for its eventual recovery step. If it requires deterministic error, a bias bound alone is insufficient. The next rounded inverse must have its own error decomposition; cancellation in the exact suffix does not remove that obligation. Credit: JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0. Dillon/PR4933 supplies actual signed normalizers and factorization; JoyboyBrian PR5037/5069 supplies kernel and corrected physical pair, and PR6096 supplies actual reached residues. zxv PR6068/6072 motivated the bounded-bias/actual-domain question. Dugongue/mdumitrean, Cinna Davis, Cao Yifan/intqwq, Jacob Alan Sussman and the retained Gaussian/native/ownership/compiler lineage retain predecessor attribution. Unrun: no physical stochastic sampler, later nonlinear rounded-suffix replay, all-eight optimization, complete allowance/tariff, formal proof, or selected-frontier change. The exact sensitivity does not establish caller affordability. This claim is released. No continuing reservation, selected-frontier change or supplier admission is implied. {
"schema": "integer-mult-coordination/v1",
"kind": "HANDOFF",
"task": "uniform-transfer-audit",
"scope": "Propagate bounded rounding bias through one actual inverse suffix",
"base": "721c03822c421b7f7fcbb2598993b947c4d88419",
"expiry": null,
"artifact": null
} |
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Status: DONE (scoped logical results and finite certificates; physical caller remains open).
Three focused contributions describe what changes when stochastic helper work is made reversible, extended beyond dyadic coefficients, and batched. They are complementary results, with no new selected κ or native supplier claimed.
2711ca28367c5bbbb54ce9601928e885d87ca165, supplies a logical compute–accumulate–uncompute interface for the archived scalar word used by Nick Petalas's PR #6026. Complemented matched coins invert dyadic stochastic ADDs; 84 saved parity bits handle signed halvings. The 9,589-event prefix restores arbitrary dirty working values and clears the parity log after copying the rounded result. Its ledger is 19,926 logical arithmetic macros and 11,713 read-only random bits per real stream sample. The forward distribution is preserved. The returned tape can remain correlated with retained output, so it does not automatically supply fresh randomness to a later adaptive call.689e28282fd5259dec7f75b83904617dab2d7b3a, gives sharp finite-cap bias and exact-accepted-law abort budgets for odd-denominator sampling, motivated by the denominator-13 inverse in PR #5069. Exact standalone sampling has optimal expected draw count two and unbounded worst-case stopping. A proposed four-bit comparison-and-retry substitution changes probability 1/13 to 1/15; exact conditional sampling instead needs at least 3/16 abort probability. The deterministic predecessor does not claim that substitution.9011bf83e85ecd0443c789cf03259442f750877d, constructs classical dyadic decision graphs for all 144 nonzero denominator-13 residue pairs. Each produces the independent product law with optimal expected draw count below 7/2, versus at least four for separate standalone calls. Two 1/13 decisions need less than 9/4 expected draws. Each configuration has 312 internal states, 624 binary transitions and a 702-byte packed table. Both probabilities must be known before batch draws begin. A finite control table still has unbounded stopping.All three packages pass exact normal and optimized CPython 3.12.8 checks, committed-byte preflight and fresh read-only Git-free export replays. Their checkers include explicit negative controls. No Monte Carlo or floating-point acceptance is used. Hosted scientific replay has passed for the first two; the newest PR's run is pending. The optional full-history preflight is unavailable in the shallow checkout. The included navigation repair from PR #6050 triggers unrelated frozen-source pin failures, which are disclosed in the PRs and left unchanged.
Open next questions: lower one signed-half macro to a complete reversible integer/tape implementation; price guard widths, temporary cleanup, physical sampling and tape movement; identify an actual nonadaptive pair of reached residues before using the paired sampler; and bind the next native MOVE and complete caller bill. Neither restored tape contents nor a lower expected fair-bit count pays these obligations automatically.
Prepared by zxv with substantial OpenAI Codex assistance. The stochastic-prefix and inverse predecessors retain their attribution; antithetic inversion, residue retention, compute/uncompute, probability counting and dyadic decision graphs are classical. The scoped contributions are the authenticated source lifecycle, sharp substitution budgets, failed retry control and complete small joint-sampler catalogue.
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