Certified rational unitary FFT and rounded Toeplitz shear

[jmbarrancoml]

CLAIM / complete conditioned tensor caller and physical raw-H entrances.
I start a new scoped package from canonical main
12fa8c5c844be21bfefb38a19503fbd867a11244, priority 7.
I retain completed PR #4253 at ac14c1409c261d64d9856a5a473d8b7f2be8c302
and PR #4178 at 3e69c9360f609bdbf3617d35ea44f497f5506f93.
Claim expiry: 2026-10-21 01:00 UTC.

Progress: I have a local draft of the block-max precision argument and
an integer-cache implementation. A first two-axis/two-label arithmetic
trajectory completes 441 stages at a 660-bit grid; exact tensor-reference
comparison and registered verification remain pending. I preserved the
published predecessors in a new main-based archive.

The production-owner probe reached its 60-second limit before a call.
An instrumented attempt opened authenticated sources in 0.44 seconds
and remained in construction when I stopped it after roughly 142 seconds.
The source documents that fixed-prime factory as impractically large.
Retained-fixture calls at supplier precisions 74 and 10 also exceeded
120-second probe limits. A separately labelled four-Gaussian, two-bit
literal call at supplier precision 3 passed the exact lifted-return
comparison in 51.204 seconds. It uses reduced bindings and does not
establish production geometry or complete physical-program execution.
I retain these limits and will keep the production hypotheses explicit.

I will connect the actual full Toeplitz tables and stored inverse plans
to the retained rounded suffix-cache protocol, then bind raw-H entrances
and returns to the existing ComposedRoot interface. I will pad complete
logical identity layers before lowering, require a common H signature,
and retain the full dyadic coarse-coordinate field inside each call.
The domain includes multiple axis/coarse-label plans, signed complex
data on a declared grid, a blockwise radius and a final error target.

I will use the completed per-axis prefix bounds to control tensor
coefficient and data errors without multiplying an amplification
allowance once per stage. I will separately pay the tensor product's
upstream operator defects; finer data rounding cannot repair them.
I will bind exact raw tables to their logical plans and compensated
reference stages before execution.

For the physical entrance, I will scale a whole packet into the unit
disk and retain enough grid bits for an exact normalized-H return.
Restoration requires removing known low zeros and interpreting the
returned integer at the caller grid. I will prove the production
relation n=Theta(p_s), selected-support and record-size admissions;
an isolated microbox has insufficient address geometry in general.
Finite checks will report whether they exercise the standard literal
branch, composed routing or a declared reduced-guard fixture.

Existing source proofs cover the fixed words, cache recurrence and
conditional root/router contract. This task targets their actual caller
connection. The cost ledger will retain coefficient generation/loading,
cache movement, physical root/router calls, encoding and return disposal.
Source analytic admission and the complete outer multiplication theorem
remain separate. I will not count unknown setup work as zero.

The first bounded test will measure one exact lifted call before a full
physical trajectory. Verification will use the controlled runner, no
concurrent replays, at most eight commands and at most 1,200 seconds of
declared replay allowance. I will stop at a wrong complete endpoint,
lost coarse coordinate, invalid unit-disk or grid admission, missing
source binding, or uncharged rounding/movement term. A physical fallback
fixture alone cannot establish the production asymptotic premises.

I compared the current-reference core/halo and output-ownership connection.
I choose the tensor caller because the completed full programs now supply
its previously missing concrete coefficient/prefix input. The refreshed
overlap search covers 376 open PR bodies and 3,150 Discussion records,
with 17 matches. I checked twelve recent PRs, 6/951 hub comments and
the source/current PR feedback. Related matches concern the completed
precision protocol or selected BIT work. Broader code, closed history
and unpublished work remain outside that search.

James Reed's newer PR #4178 audit checks source hashes and height-contract
evidence; it excludes the live rounded caller. I retain that scope.
I credit Vlado's compiler, ywz65's simultaneous-rounding analysis,
the T74/root/router contributors, Chafik's phases and the original
Gaussian lineage. Jose Manuel (jmbarrancoml), with substantial OpenAI
Codex assistance. Internal analysis agents belong to the author process.
I claim no selected-frontier change or complete multiplication theorem.

DONE / complete conditioned Toeplitz lowering to fixed Hadamard stages.
I completed this extension in
PR #4253,
head ac14c1409c261d64d9856a5a473d8b7f2be8c302, from unchanged
canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority 7.
I preserve completed FFT head 6f17e6208a78cfade27f5c8f03383256f586ab24
and its scientific receipt, and retain unchanged full-program source
from PR #4205 head 78c72e563173f3822fcb4efed9ceeccf05630258.

I compile the actual scalar, shear, butterfly and swap gates through
the existing six-H words and Beneš routing. I preserve mixed layers,
oriented wires and full identity words on idle pairs. For the reverse
direction I use the original stored inverse, including scalar
reciprocals. I bind the admitted root, request bounds, actual plans
and prepared pages before rounded execution.

For a rational Toeplitz root under the retained leading-section and
phase contracts, I give error below 2^-t against that same root or
its inverse, with signed widths
O(F0+t+log N+log lambda+log Q+log R).
I derive a native prefix bound C = 2^14 Gamma Pi, where Pi is the
source logical prefix bound and Gamma is the smallest power of two
at least Pi squared. The endpoint norm E0 enters the suffix and rounding
allowance. A scaled two-wire zero-shear control has an actual suffix
more than six times C, rejecting the omitted-E0 extension.
I add the upstream coefficient defect times the input radius to the
new native coefficient/data error. DENSE-PROOF.md gives the derivation.

The full receipt covers 11 roots in both directions, 146,262 native
stages and 992,742 complex roundings. I compare complete outputs with
the stored logical program and exact matrix/inverse references,
including indefinite and Gaussian roots, reversed wires, unequal
subtrees and 512-bit incoming data. Four generic cases check mixed,
idle, zero/tiny and empty programs. I check 24 shear words and 2,184
segments/inverses, the endpoint-factor example and 25 controls.
Scientific digest:
b3f78b230198f71d6b537ca2c1795ae430e65adb737873fc23861fdff36e69b4.
The earlier FFT digest remains
d735321b0bf6457ad884779b5854ff201e18f79e4bf8df8b32da6f91477b4c9d.

All four registered and Git-free commands pass. Committed-byte and
source-history checks cover 34 payload files; all 35 export files
remain unchanged. The full export replays took 60.984 and 58.504
seconds; the FFT replays took 3.143 and 3.468 seconds.
Archive SHA256:
d815f527e08e1684dfe06ab6704ba79c457650468390eacfc415c76f4511693c.
Export receipt SHA256:
d1a186bfaf571c1e211f10bb433ab02541b99058ffa1af94a5068118fd042bb2.
These replay records bind implementation commit
9eec608bc90665d5fec3f2d6bf1503007d434a48.
The first hosted attempt failed during temporary Git-directory cleanup
in an unchanged repository helper test, before package execution.
GitHub requires maintainer rights for reruns, so I added an empty retry
commit. Both commits have tree
01e2787c432069130b028ae022b43c38968ed374.
I rechecked final-head committed bytes/source history and its export,
then compared all 34 payload files and modes with the replayed archive;
they match exactly. The metadata receipt records the new head.
Final-head archive SHA256:
5c6df6dd3752710e4c4f7959cf0f29729305d652a7e11e4e12a33cd32ae476ac.
Final-head export receipt SHA256:
6b39bd9896c855c17ff80d5772bdf915761921f62473a02c5c0aee3fdb70c0ec.
I matched all four successful hosted scientific records and both complete
receipt digests to the verified export, and checked the unchanged-input
step in run 39508290934.
Planning passed in 3m9s; package replay passed in 6m59s.
Test merge 0b12f1870f2a7612b06cf940d564a58a2c775959 has the exact
base/publication parents and the same publication tree. I marked the
PR ready for review. The first failed attempt remains in the history.

The initial overlap covered 363 open PR bodies and 3,127 Discussion
records. Before submission I refreshed unchanged main, checked eight
recent PRs, 5/949 hub comments and the canonical thread. PR #4253
had no review or comment. The newer connector and header-law work
addresses different obligations within that coverage; broader code,
closed history and unpublished work remain outside the search.

Useful independent work: assess DENSE-PROOF.md's nonunitary prefix,
endpoint and precision argument against the pinned full source.
I execute arithmetic H/2 stages. Actual physical-root calls, tensor
admission, exact setup, coefficient delivery and complete tape costs
remain open. I reserve no automatic quarter-budget for physical
error in this full extension. The rounded programs need not form
an exact inverse pair. I claim no new selected exponent or complete
multiplication theorem, and I release this scoped claim.

I credit Vlado's zero-safe words, routing and full circuits, Chafik's
phases, the retained generator/Gaussian contributors and classical
LDU, Schur and perturbation methods. Jose Manuel (jmbarrancoml),
with substantial OpenAI Codex assistance. Internal analysis agents
belong to the author process.

Original claim / complete conditioned Toeplitz lowering to fixed Hadamard stages.
I extend PR4253 from completed head
6f17e6208a78cfade27f5c8f03383256f586ab24, preserving its FFT receipt.
Canonical main remains 12fa8c5c844be21bfefb38a19503fbd867a11244,
priority 7. Claim expiry: 2026-10-21 00:00 UTC.

I will bind the remaining scalar and shear gates to the retained
zero-safe six-H words, then compile the actual full Toeplitz program
and its stored inverse from PR4205 head
78c72e563173f3822fcb4efed9ceeccf05630258.
The source execution_bound controls stored-coefficient prefixes,
suffixes and their inverses; it is distinct from a rounded-data bound.
I will derive the new native fragments and precision from that contract.

The domain is a rational Toeplitz root with the retained leading-section
admission, arbitrary signed complex data on a declared grid, and an input
radius. I will preserve mixed singleton/two-wire layers, oriented wires,
idle completion, scalar reciprocals and the actual stored inverse.
The target is error against the same root or its inverse. I will add
the predecessor's coefficient defect to the new native error, without
reusing its old data-rounding allowance as the new execution cost.

Unlike the FFT case, the endpoint need not have norm one. I will include
that endpoint factor in the suffix and error budget. A proposed control
uses a scaled two-wire program with a zero shear to expose the invalid
unit-endpoint extension. I will also check zero/tiny complex shears and
nonunitary scalar leaves through the actual retained compiler.

The earlier word/routing identities and full-circuit prefix contract are
author-checked inputs. The new coupled nonunitary bound and concrete
lowering are this task's target. Exact setup, physical native calls,
tensor admission and complete delivery/tape costs remain separate
obligations. I will preserve the declared six-H and routing counts and
record the numerical stage count and word widths without pricing unknown
physical costs as zero.

I compared the global core/halo connection, whose locality lemma already
exists but whose current-reference and ownership binding remains open.
I choose this step because the completed FFT lowering does not consume
the actual dense program's nonunitary gates. I will stop at a wrong
endpoint, a lost source binding, a missing nonunitary amplification
factor, an uncharged rounding site or an understated signed width.
I budget four normal/optimized commands within 1,200 seconds total.

The refreshed overlap search covered 363 open PR bodies and 3,127
Discussion records, with 32 phrase matches. I inspected the relevant
fixed-H, twiddle and return-contract notes; those address different
physical or lifecycle boundaries. I also checked eight recent PRs,
5/928 hub comments and the current PR4253 feedback, which had no review
or comment. Broader code and unpublished work remain outside this search.

I retain Vlado's scalar/shear words and full compiler, Chafik's phases,
the original rational-generator and FFT credits, and the classical
LDU/Schur/perturbation lineage. Jose Manuel (jmbarrancoml), with
substantial OpenAI Codex assistance. Internal agents belong to our author
process. I claim no new selected exponent or complete multiplication theorem.

DONE / rational FFT lowering to fixed Hadamard stages.
I completed PR4253,
head 6f17e6208a78cfade27f5c8f03383256f586ab24, from current main
12fa8c5c844be21bfefb38a19503fbd867a11244, priority 7.
I preserve completed PR4205 at 78c72e563173f3822fcb4efed9ceeccf05630258.

I bind its actual rational FFT and stored adjoint to the retained
six-H/seven-diagonal words and Beneš routing. I use the existing
nonzero-shear helper without increasing the word count, preserve pair
orientation and inactive completion, and pay H0=H/2 normalization.
I prove a prefix/inverse-prefix allowance K=2^14 independent of the
number of completed FFT layers. Under the retained phase contract,
one coefficient/data grid gives error below 3*2^-t/4 and online
width O(F0+t+log N+log R), with explicit signed temporary fields.

I check 13 cases through 64 wires, 2,525 primitive stages and 73,104
rounded outputs. Sixteen standalone words cover 1,456 exact segments
and their inverses; 23 controls pass. I compare exact complete endpoints,
basis matrices through eight wires and direct Fourier interval formulas.
A 512-bit spectator has nonzero rounding error within the allowance.
The complete scientific digest is
d735321b0bf6457ad884779b5854ff201e18f79e4bf8df8b32da6f91477b4c9d.

Both registered and Git-free modes pass. I checked 18 committed payload
files and source history, and all 19 exported files stayed unchanged.
The export checks took 3.106 and 3.096 seconds.
Archive SHA256: c90737441543a6695982a2b8c64b8db8d774f5d07ea5f3b2dc38be5a3d467618.
Receipt SHA256: 35af38b18c31582902d81404e6d3fc5ddf5bf90438ebf78ba1c5fc7f6d382e04.
I matched both scientific records and the digest from
hosted run39500091244
to the export, including its unchanged-input check. Test merge
cc2863d6efddf19bbcb79e871d1a1fe4f45caa59 has the exact base/head
parents and publication tree cb1959a22aba2bde512187f17a4b54bb3f2403da.

The initial overlap refresh covered 359 open PR bodies and 3,071
Discussion records. Before submission I refreshed unchanged main and
checked eight recent PRs and 5/927 hub comments. James Reed's new
claim concerns a separate finite audit of PR4178; I do not count it
as verification of this contribution. Wider code, closed history and
unpublished work remain outside the search.

Useful independent work: assess the uniform prefix bound and complete
rounding allocation at this fixed head. Actual physical-root execution,
tensor admission, coefficient delivery and complete setup/tape costs
remain open. The two rounded programs generally are not exact inverses.
The Fourier comparison shares the declared phase backend.

I credit Vlado's words, routing and FFT schedule, Chafik's rational
phases, classical LDU/FFT methods and Cinna's later caller. Our PR4178
tensor precision protocol retains its separate scope. Jose Manuel
(jmbarrancoml), with substantial OpenAI Codex assistance. Internal
agents belong to our author process. I release this scoped claim
without a selected-frontier change.

DONE / ordinary finite-core Gaussian words with rounded boundary arithmetic.
I completed this extension in
PR4205,
head 78c72e563173f3822fcb4efed9ceeccf05630258.
I join the retained carve, shared-reference Schur pages, raw interior
circuits and PR4186's rounded boundary arithmetic. I retain the original
couplings and six-call order. I apply the narrow inverse and broad
forward map on ordinary finite windows with sufficient complementary gaps.

I fund the reference for amplified intermediate inputs, use one data grid
and prove complete maximum complex-l1 error at most 2^(-t-1).
The boundary program reads prepared rounded inverses of exact reference
blocks; it performs no online inversion. I restate the needed residual
proof and retain the original adapter with pinned archive provenance.
CORE-PROOF.md separates exact setup, online arithmetic and remaining costs.

I check two 143-row words with two 16-row interiors each and
two 11-row windows crossing a hole. I execute 88,704 dense gate
applications and 19,992 boundary scalar products. I check actual stage
inputs, complete narrow residuals and broad products, a 768-bit incoming
grid and 13 controls. The core scientific receipt is
65f0cefaed8d7b8b21125cde5155401b3880e896b49068e8e8c86895b5004e4d.
I use cap 16 and half-band 18 in the finite fixtures and make no
large-cap scaling claim from them. I preserve six earlier scientific receipts.

I passed all eight registered and Git-free commands. I checked 63 committed
payload files and source-history pins; all 64 export files stayed unchanged.
I measured 147.066 and 146.018 seconds for the combined export commands.
Archive SHA256: dca7449c8bbd870be87655373969d940dc42af69ed40da632873ae1105ad907e.
Receipt SHA256: 2b0882f33e7480cef5c713f96ba83a663137fc04504190db402d1aa3f6b3319e.
I matched all fourteen scientific records and seven digests from
hosted run39492372797
to the export, including the unchanged-input check.
Test merge 7acbb956f35751735f0948e6c751787b8635ad4c has the exact
base/head parents and publication tree d069aea846216a42c32d1761c2e356719b2c7e16.

I retained main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority 7.
Before submission I checked 12 recent PR records and 5/853 hub comments.
I later refreshed phrase-based overlap across 352 open PR bodies and
3,024 Discussion records with complete pagination. I make no novelty
or all-diff coverage claim. I compared Nick Petalas's PR4217
HALO locality/error audit and preserve its separate credit.

Useful independent work: audit the exact-reference page lowering or
the complete five-factor error bound at this fixed head. Core/halo
comparison, wrapped corrections, tensor assembly, physical page delivery
and complete setup/tape costs remain open. The quantified proof still
needs independent assessment; I claim no selected-frontier change.

I credit Vlado's carve/block program, Jeremy's shifted identities,
Mason's intervals, Chafik's phases, Cinna's caller and the scoped
ToikY/James Reed audits. Jose Manuel (jmbarrancoml), with substantial
OpenAI Codex assistance. Internal agents belong to our author process.
I release this scoped claim.

Completed raw Gaussian interiors / released at that checkpoint.
I completed the raw-reference interface in
PR4205,
head c9db4069d59bfad26a15fe000b91cc3a058684e5.
I construct fixed periodic narrow/broad references, preserve original
phase and period cuts and their different hole rules, and execute
anchored raw interior blocks and the inverse of the same stored program.
I pay positive-weight setup, tails, diagonal coefficients, circuit
rounding and comparison with the same reference restrictions.

The cap/radius/accuracy allocator chooses one reference before block
requests. Coordinate queries avoid full-axis enumeration. I derive both
global tail norms and an optional narrow row-gap certificate with
principal-inverse bounds in the original circular metric.
Full-cell weights require O(entry_bits+alpha²/theta) offline bits;
the online width is
O(F0+t+log n+log R+1/c+Fg+log alpha).

I check 2,178 entries across two complete 33-by-33 references, three
explicit windings with analytic remaining tails, row gaps, a principal
inverse crossing a cell boundary and a hole, overlapping pages and cuts.
Six raw fixtures execute 1,086 gates and 26 rounded states; 17 controls
pass. A separate four-coordinate query on a source axis of size
10^20+1 avoids axis enumeration. The scalar comparisons share the
declared interval backend. The raw scientific receipt is
f9c2bfc6b61ca8a52c403d912d2f687beb3641955cec71d3879ab017b067d6f7.
I preserve the five earlier receipts and their fixed-head evidence.

All eight registered and Git-free commands pass. I checked 54 committed
files and source-history pins; all 55 exported files stayed unchanged.
The combined export commands took 32.439 and 33.041 seconds.
Archive SHA256:
136d7dd95fc29b427afa7c7f55a140537a8561a971bdc1ef12a341a5989f5234;
receipt SHA256:
129c6f4fbdd9ccedd3e232023456fead49ce55e2ae767abb0c4a51119b7c565a.
I matched all twelve successful
hosted records
and all six scientific digests to the export, including unchanged inputs.
Test merge ebc06734b5645c1eb8e8f5a780ca289b3147c8ae has publication
tree 7b8c6b4149fe23de7b79a8f9370f0da972ed704d and exact base/head parents.

Before publication I refreshed unchanged main and checked 12 recent PR
records, 5/815 hub comments and the new scalar-audit handoff.
I retain ToikY's PR4220
at 5cf23ba47a28b78a98eb0665177b5030e65c01b9 for the separate scalar
implementation and width/precision audit of normalized source 66b95cc916.
That audit excludes the full circuit, matrix-inverse theorem and this
raw/reference extension. I retain the other source and audit credits below.

The next complete caller is an ordinary finite-core inverse: connect the
retained carve, these same-reference interiors, original sparse couplings
and the already completed PR4186 rounded boundary solver. Cyclic corners
require separate treatment. Complete boundary/tensor execution, physical
delivery, tape costs and independent mathematical acceptance remain open.

Original common-reference raw claim / now released.
I extend PR4205
from completed head 66b95cc916552f487203be14fd99a7aea0bdd78e.
I preserve its five receipts and return the PR to draft.
Claim expiry: 2026-10-20 20:10 UTC.

I will construct one rational Gaussian reference for a finite periodic
source/target axis, with the retained phase cells, period cut and hole
completion. I will connect its capped constant-phase interior restrictions
to anchored raw broad/narrow circuits and the inverse of each stored program.
I will pay positive-weight setup, kernel truncation, diagonal quantization,
circuit rounding and comparison with that same reference.

I pin canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244,
priority revision 7. The completed normalized circuit supplies a written
conditioning/error contract and author-checked gate execution.
GAUSSIAN.md, GAUGEFREE-INTERIORS.md, LOG-A-PHASE-SCHUR-AUDIT.md
and FINE-PRIME.md supply the intended source formulas and reference
requirements. Their complete multiplication claims remain outside this
interface result and lack independent acceptance here.

The boundary accepts exact rational source geometry, a declared input
radius/grid and requested output accuracy. The circuit owns the capped
logical wires. I will store each positive diagonal coefficient once and
use its exact reciprocal in the stored inverse. One global reference
description must determine overlapping window entries without rerounding.
The shared-reference and raw-wrapper bounds are constructive targets
of this extension. Complete boundary arithmetic, physical coefficient
delivery, routing and tape costs remain unknown at this boundary.

I compared the new GL3 audit in PR4214 and the CRT stencil work in PR4208.
Their native/physical contracts differ from this remaining numerical
comparison. I continue here because the consumer requires the raw block
of one common rounded matrix; an accurate normalized Gaussian alone
does not supply that block. I checked 15 recent PR records and 6/790
hub comments and found no competing raw-reference claim in that coverage.

I will stop at a phase/cut mismatch, a zero rounded weight, an uncharged
inverse/alias error or an understated setup/online width. I will retain
eight sequential registered commands with total timeout at most 1,200
seconds and keep the older scientific receipts unchanged.
I retain Mason, Jeremy, Vlado, Chafik and Cinna's source credits.
Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance;
internal agents belong to the author process.

Completed normalized Gaussian interface / released at that checkpoint.
I completed the retained Tsec and adjacent full_plan connection in
PR4205,
head 66b95cc916552f487203be14fd99a7aea0bdd78e.
For dyadic N, rational c>0 and |s|<=1/2, I derive leading-section
conditioning and one truncation/coefficient/circuit error budget against
the ideal normalized Gaussian section and its inverse. I obtain online
widths O(F0+t+log N+log R+1/c). I preserve the literal source function
and two callsite statements, using symbolic pi cancellation and exact
rational arithmetic.

I check nine Gaussian cases, 4,527 gates, 38 rounded states and 16 controls.
The fixtures include both half shifts, nonzero paid truncation, a 512-bit
incoming vector and two requested precisions. I compare exact leading
inverses and finer scalar interval references, charging their residual
error. Those references share the declared Mason interval backend.
The complete Gaussian receipt is
0a0548963b8132b430f43b8534abbd4220b93a138eb9e0a28a9cfd7432922928.
I preserve the four earlier receipts and fixed-head evidence below.

All eight registered and Git-free commands pass. I checked 43 committed
files and source-history pins; all 44 exported files stayed unchanged.
The combined full/Gaussian export commands took 31.914 and 32.200 seconds.
Archive SHA256:
b81a13321adc9b33edc61beb4e3ec46de4c860a19e97ab1d14b49377756dcad6;
receipt SHA256:
fcd5c528c84c1a7703d6c9bc2290994f77a5916806309309e5a6b6b06231f059.
I matched all ten successful
hosted records
and all five scientific digests to the export, including unchanged inputs.
Test merge 4b24dc0e4012f2816dedff2c29e5fef5f40edfe8 has publication
tree 4b2d0e5a71b527bee3459e623293acefea5740e5 and the exact base/head parents.

Before publication I refreshed unchanged main, checked 12 recent PR
records and 5/783 hub comments, and checked the unchanged feedback state.
The next obligation is a common rounded reference plus anchored raw
diagonal/scalar wrappers. Raw Gaussian assembly, physical delivery,
complete tape costs and independent mathematical acceptance remain open.
The broad boundary Schur factor requires application, not inversion.
I retain Mason's interval/dense-execution credit, Jeremy's shifted
source, Vlado's q-LDU/circuit, Chafik's phases and Cinna's caller.
Internal agents belong to our author process.

Original normalized Gaussian claim / now released.
I extend PR4205
from completed head 7ab1bc8973cc9e606053c65f094ca3abc4a42fee.
I preserve its four receipts and return the PR to draft during this work.
Claim expiry: 2026-10-20 19:30 UTC.

I will connect the retained Tsec(n,g,s) and its full_plan call to
requested-precision rational coefficients for g=pi*c, rational c>0,
and |s|<=1/2. The normalized kernel uses offset j-i and diagonal one.
The boundary owns N logical complex wires, accepts a declared input radius
and incoming dyadic grid, and returns the forward or stored-inverse circuit.
I will charge omitted offsets, coefficient quantization and circuit rounding
against the ideal normalized Gaussian section.

I use main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7.
The retained evaluator is research/capped-fusion-t74-k554/outer/check_gaugefree.py;
Mason Hale's research/certified-gaussian-recovery/intervals.py supplies
outward arithmetic. PR4193 at 52e962e318423be1fe8ed9d705dba92067795b61
contains our finite-cap bound; I will rederive its needed implication here
from the retained q-LDU identities. These remain author-checked results.
Mason already supplied certified dense Gaussian execution and integer recovery.
I claim this specific normalized coefficient-to-logical-circuit connection.

The published full logical circuit and its width/error theorem provide
proved written bounds with finite replay evidence. The requested-precision
Gaussian producer and combined reference budget are constructive targets
of this extension. Raw diagonal gauges, alpha scaling, the common rounded
Gaussian reference, physical setup/delivery and complete tape costs remain
unknown at this boundary. I will keep those costs explicit.

I compared the new P112 contact audit in PR4212 and the CRT stencil route
in PR4208. Their native/physical obligations differ from this numerical
interface. I continue here because the new bridge supplies the same-root
conditioning required by the completed full compiler and pays its new
approximation error in the consumer. I checked 20 recent open PR records
and 8/776 hub comments; I found no competing claim in that coverage.
Broader history and unpublished work remain outside this search.

I will stop at an invalid enclosure, a source-orientation mismatch, an
unfunded tail/inverse perturbation or a stale matrix certificate.
I will run the old full verifier and the new Gaussian verifier sequentially
inside the existing two 300-second slots, keeping eight registered commands
and the 1,200-second total timeout budget.

Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance.
I retain Jeremy Kothe's shifted source, Vlado's circuit/q-LDU, Chafik's
phases, Mason's interval arithmetic and Cinna's caller credits.
Internal agents belong to our author process.

Completed full Toeplitz recursion / released at that checkpoint.
I completed the retained full Toeplitz program and stored inverse in
PR4205,
head 7ab1bc8973cc9e606053c65f094ca3abc4a42fee.
Under leading-section conditioning, I prove a completed-call
prefix/inverse bound 256 N³ (Lambda Q)^12 through scalar unit-LDU
factors and legal downsets. I account for disjoint unfinished calls at
physical cuts, unequal sibling lengths and zero-layer identities, then
derive one global coefficient/grid/online-width allocation.

I add 11 cases, 3,659 gates, 68 rounded forward/inverse states,
151 completed-call sets and 12 selected physical cuts. I enumerate
all downsets through size four and use atom-boundary sets for larger
cases. Timing/ownership checks cover every layer cut; selected operator
checks include two nonempty unfinished fragments. Nontrivial fixtures
reach size eight; the size-16 case is an identity/zero-call fixture.
Fifteen controls pass. The complete full receipt is
b1bdf09eb959cc4821c5515fa865c06541215224174bcdfc667e6515210bfbb6.
I preserve all three earlier receipts and their fixed-head evidence.

All eight registered and Git-free commands pass. I checked 34 committed
files and source-history pins; all 35 export files stayed unchanged.
Export SHA256: 6530d03e07ca6483324016a8adb963829adb0a8a1180d557fe8982f5437fb221;
receipt SHA256: f475957b55e46bbd446b7a44676951ec24f9fbc17d1885946bcf9ed4167d1964.
I matched all eight successful
hosted records
and all four scientific digests to the export, including unchanged inputs.
Test merge 2a44474782ebc44a2168d797420ab8dadd40ea6f has publication
tree 38438ccaac76fcb460bcc764365cc12d158c2a30.

Before publication I refreshed unchanged main and inspected 15 recent
PR records, 6/756 hub comments and the unchanged feedback state.
I retain ToikY's separate near-identity Schur theorem and James Reed's
finite budget/seal audit at fdcdc269 within their stated scopes.
The actual Gaussian coefficient/common-reference interface, complete
setup and physical delivery, and independent proof acceptance remain open.
I preserve the source credit and AI-assistance disclosure below.

Original full-recursion claim / now released.
I extend the completed generator revision
fdcdc2690910392f7b42dbae4155b7d7b5790abd with the retained full_plan
schedule. I preserve that revision and its receipts, including the scope
of James Reed's finite budget/seal audit. I will return the PR to draft.

For dyadic rational Toeplitz T, I assume a norm bound Lambda and a common
inverse bound Q for its leading principal sections. I will prove a
polynomial global prefix envelope through scalar unit-LDU factors and
legal completed-call downsets, then connect the actual generator calls
and stored inverse to one coefficient/grid allocation. Unequal sibling
ranks can give different macro boundaries; I will account for disjoint
active call fragments at physical cuts. I will round full scalar leaves
with their own inverse margins and coefficient-height bounds.

I refreshed unchanged main 12fa8c5c844be21bfefb38a19503fbd867a11244,
priority 7, and checked 25 recent PR records and 8/733 hub comments.
ToikY's PR4209 at ccc33bfde8baadef7979fac81e28c0a6f6f278c4
proves same-rho Schur closure under row/column majorant admission; it
excludes full gate execution and global precision. I credit that sharper
local theorem and keep its near-identity hypothesis distinct from the
leading-section bounds used here. James Reed's audit of fdcdc269
covers finite budgets/seals, excluding gate replay and the universal proof.

Boundary: the program owns N complex wires and applies T or its inverse
to arbitrary input on an F0-grid with norm at most R. The completed
generator and child contracts are author-checked inputs. The global
downset bound, full execution, rounding and online widths are this
extension's proof/construction targets. Exact setup, coefficient delivery,
fixed-tape cost and the complete Gaussian/outer theorem remain open.
This step replaces the remaining full-recursion matrix abstraction at
the current caller; the consumer must still pay those later costs.
I compared PR4209's local-constant integration question and the CRT
stencil handoff; neither replaces this global recursive proof obligation.

I will retain the source's transpose, Schur references, parallel schedule
and disjoint wire embeddings. I will test non-near-identity and complex
inputs, unequal sibling lengths, zero-layer calls, small pivots and
nonzero fine-grid data. I will stop at an invalid frontier factorization,
missing conditioning/rank premise, unstable scalar inverse or uncharged
word/error term. I budget eight registered commands totaling at most
1,200 seconds. No numerical full-recursion result exists at this claim.

I credit Vlado/PR1673 and the retained gauge-free extension for the
recursive compiler, rank identities and earlier conditioning-dependent
guard; Chafik/PR628 for phases; Cinna for the later caller; and the
classical LU/Schur lineage. Jose Manuel (jmbarrancoml), with substantial
OpenAI Codex assistance. Internal agents belong to the author process.

Completed generator execution / released at that checkpoint.
I completed the actual compensated generator program in
PR4205,
head fdcdc2690910392f7b42dbae4155b7d7b5790abd.
I pair the negative stored spectral multipliers, reuse each stored
triangular inverse and prove additive completed-term defects. I bound
the four checkpoint prefixes/suffixes, intermediate data growth and
global fixed-grid rounding, with explicit signed data and temporary widths.

I check 57 matrices, 71 generators, 5,614 gates and 172 rounded states,
including all 50 predecessor Schur examples. I check exact ranks and
endpoints, prefix/suffix matrices and their inverses at 36 checkpoints
in four selected cases, and 12 controls. Fifty-seven states have nonzero rounded source
error within the allowance. I preserve the earlier FFT and triangular
receipts. The complete generator receipt is
f74e0efc459e9c3d44d8ca8b3a33dc0fccc31731d0829419c3d515fe7f0f1e02.

All six registered and Git-free commands pass. I checked 28 committed
files and the base/archive ancestry; all 29 exported files stayed
unchanged. Export SHA256:
6ed6556942e7defff248038867b6504424c964027895792692e83116f1f55cc1;
receipt SHA256:
70f6a5b0e1d70fe3945efa6e2cff4248a333c8d6e2a4885d636e02e45ab530d0.
I matched all six successful
hosted records
and all three scientific digests to the export, including unchanged inputs.
Test merge 532445e19d06be4de89cbf229c109ea53642be90 has publication
tree 8f158e3df02dd92e7df63e191fe6951b43fab02c.

Before publication I refreshed unchanged main and inspected 15 newer PR
records, 6/726 hub comments and the unchanged PR feedback state.
The full Toeplitz recursion, Gaussian/physical integration and independent
proof acceptance remain open. I retain the exact producer's original
PR4198 identity separately from its local frozen archive pin.
I preserve Vlado, Chafik and Cinna's source credit and the AI-assistance
disclosure below.

Original generator claim / now released.
I extend PR4205 from its completed recursive revision with the actual
bounded-generator program. I will combine PR4198's exact rational
producer with the retained seven-statement macro and the certified
triangular/shear children. I will preserve the previous receipts and
return the PR to draft during this extension.

I pin the producer at e457d5e912a27307d725827aeca7198169320b84
and the completed child at 203d04313c0cb2b4a4663a5a2d4d407e5a5b6d20.
Both remain author-checked results with their stated review boundaries.
The target owns exactly two dirty banks of dyadic size m and implements
(x,y) -> (x,y+Tx), with source restoration error charged after rounding.
The inherited algebraic generator identity and child contracts are proved
in the submitted notes; the complete gate depth, precision and word
allowances are this extension's constructive upper-bound target.
Full Gaussian setup, coefficient delivery and fixed-tape cost remain unknown
at this integration boundary. I compare the CRT stencil route in PR4208:
its next transported-input obligation differs from this missing numerical
composition. I choose this bounded extension because it removes the exact
matrix callbacks at the generator interface, including its cleanup.

I will bind compensation to negated stored spectral multipliers while
retaining the source's cyclic-then-negacyclic order. This gives an exact
unrounded term Ghat(Phat-I) and permits additive completed-term errors.
I will prove checkpoint and internal-gate bounds, use one data grid,
and check arbitrary dirty input and rounded source-restoration error.
I will execute actual source Schur examples and discriminating controls.
Full Toeplitz/Gaussian recursion and physical setup/delivery remain open.

I refreshed unchanged main at this checkpoint and inspected 30 PR records
and 10/717 hub comments. The overlap helper searched 5,236 local records
and returned broad card matches without a usable remote cache; targeted
generator/compensation searches found no competing implementation in that
coverage. PR4208 concerns CRT stencil production. I retain the scoped
PR4198 independent-check credit and its stated exclusions.

I will freeze the exact producer bytes with their original PR4198
provenance, use explicit conversions between rational representations,
and stop at a failed endpoint/cancellation identity, uncharged error or
understated width. I budget six registered commands with at most 1,200
seconds total. Jose Manuel (jmbarrancoml), with substantial OpenAI Codex
assistance. I retain Vlado's compiler, Chafik's phase and Cinna's caller
credits; internal agents belong to the author process.

Completed recursive extension / released at that checkpoint.
I completed the retained parity-triangular program and its stored inverse
in PR4205,
head 203d04313c0cb2b4a4663a5a2d4d407e5a5b6d20. I prove one common
coefficient/grid allocation for forward and inverse execution, with
prefix/suffix bounds, arbitrary-segment bounds for intermediate states,
and explicit signed record and temporary widths.

I add ten cases through 16 wires, 1,806 gates and 62 rounded forward/inverse
states, including nonunitary complex leaves, the half-identity boundary,
reversed wires and a 100-bit incoming grid. I retain the first FFT receipt.
The recursive scientific receipt is
d430fced712daa2e620b53a49dd09f0ba10ddddb4a4ff6b79d2abbbfd3afc1e6.
All four registered and Git-free commands pass. I checked 21 committed
files and source history; all 22 export files stayed unchanged.
Export SHA256: 9fc5dcb433a8732f143de947930bac86d4f88834c0e7eb980ce5e484c8a2c2f3;
receipt SHA256: f8ca37a8778309eba7938c20ae353cd8a589aec507c01692402a4fa6dd35990e.
I matched all four successful
hosted records
and both scientific digests to the export, including unchanged inputs.
Test merge fba8ed9e6b03b3efacf6e266656ea1652f5be30d has the publication
tree 46f87e4fd3b1ae31b36d16a5eadb053aa8a8cf26.

Before publication I refreshed unchanged main, inspected 12 newer PR records
and 5/715 hub comments, and checked that PR4205 had no review or comment
requiring a response. The new PR4208 concerns a CRT guard stencil.
The next obligation is the full compensated generator program; complete
Gaussian/physical integration and independent proof acceptance remain open.
I retain Vlado's recursion and stability credit, Chafik's phase credit,
Cinna's caller credit and the AI-assistance disclosure below.

Original extension claim / now released.
I extend the same PR4205
with the retained near-identity parity-triangular program and its inverse.
I preserve the completed FFT revision below and return the PR to draft.

I will retain the source's even/odd recursion and parallel schedule, insert
certified FFT/shear children, round stored scalar leaves, and invert those
same stored coefficients. I will prove global prefix/suffix and coefficient
perturbation bounds, then execute the flattened circuit on one data grid.
The admitted inputs are dyadic-size rational symbols with component-l1
distance from the identity at most 1/2. I will check complex non-dyadic
leaves, the admission boundary, inverse order, sibling alignment and
incoming-grid preservation.

I refreshed unchanged canonical main at 15:12 UTC, inspected 40 recent PR
records and 12/713 hub comments, and retained the earlier 100-PR/35-comment
orientation. I found no competing rounded parity-child claim in that coverage.
James Reed has claimed a finite check of PR4198; that check differs from
this circuit extension. PR4206 addresses BIT support propagation.

I credit Vlado/PR1673 for the parity identity, recursion and qualitative
stability argument, Chafik/PR628 for certified phases and Cinna for the
later caller. I will stop at a failed descendant bound, uncharged inverse
error, coefficient-width gap or schedule mismatch. I leave full generator
composition, physical lowering and the complete multiplication theorem open.
I budget two additional replays of at most 240 seconds each through the
repository runner. Internal agents remain part of our author process.

Completed first revision / claim released at that checkpoint. I completed the proof and actual gate replay as jmbarrancoml in PR4205, head f02a4ab5ab70a8af80415ac35c5b48cd5a93a475. I prove the rational-phase, unitary FFT, spectral preparation, complete shear and fixed-grid error bounds, including incoming-grid and signed-width accounting.

Both registered modes and both official Git-free export commands pass. I checked 17 committed files and declared source history; all 18 export files stayed unchanged. I verify 12 cases, 1,428 gates, 61 rounded states and 94 phase certificates. Forty-three states have nonzero source-restoration error within the allowance. Complete scientific receipt: 2e203e42d3d617e828e3341ffeb44254c931db2aebb8a040d04d67338b5b4016. Export SHA256: e29418f60e636138e3ffc1d941cf9bc6bbab2ced2d1c425aa1331d81fadb9c2d; receipt SHA256: 4d23700186d7038eb72aaa682d3da0d9d16c4fc8e6edfc024309a2bbeacfbec0. I matched both successful hosted run39442494617 records to the export and checked input preservation. Test merge 5a473bd4dec26e0e2f4d89f6d7f69ae240e2356c equals the published tree 3e8f29998f4531239d5caacb1c7fc6e5f842c291. I marked the PR ready.

I refreshed unchanged main, the source head, 60 PR records and 12/708 hub comments before publication. PR4201 concerns CRT tapes and PR4203 concerns D4 interfaces; newer claims concern separate BIT/D4 audits. The parity-triangular/full-generator composition, physical delivery and complete Gaussian theorem remain open.

Original claim / rational unitary FFT child / priority revision 7. I take the FFT/Toeplitz-shear precision obligation left by PR4198. I start from canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244, retain PR4100 at 337d04939571a637aeab54a0f58f54fde811663a, and keep the original check_circuits.py schedule. Claim expiry: 2026-10-20 16:30 UTC.

I will instantiate PR628's existing rational unit-phase construction at the FFT twiddles, diagonal twists and pi/4 splitter. With rational a²+b²=1 and |w|=1, the butterfly [[a,bw],[b,-aw]] is exactly unitary. I will preserve the source's bit reversal, disjoint layers and chronological circulant/negacyclic shear order, using adjoints of the same stored unitary gates.

The complete target includes three separate errors: approximation of the ideal unitary FFT, preparation of the unnormalized spectral multipliers, and fixed-grid rounding of data after each gate. I will round the prepared multipliers before online use so their exact offline sums do not introduce uncharged long denominators. I will prove an error bound for both dirty banks, including source-restoration error after rounding, with coefficient and temporary-width bounds. Exact coefficient unitarity alone is not the rounded execution claim.

I will compare the actual adapted gate schedule with direct Fourier enclosures and exact Toeplitz references. Controls will cover inverse order, missing bit reversal, multiplier normalization, twists, coefficient unitarity and insufficient precision. Two bounded normal/optimized replays will use the repository runner. Physical tape lowering, complete triangular/full-Toeplitz recursion and the full Gaussian theorem remain outside this child result.

I checked 100 recent PR records and 35 hub comments, the retained phase and FFT sources, and targeted local searches. The overlap helper searched 5,234 records and returned one unrelated rational-unitary frame note; it had no usable remote cache. PR628 already supplies exact rational unit phases, and our PR3709 already supplies certified inward-twiddle rounded FFTs. I found no equivalent certified connection to this unitary child in that coverage. Older comments, arbitrary diffs and unpublished work remain outside the search.

I compared the new PR4200 support-propagation handoff as an alternative. Read-only inspection found a consistent scoped invariant under its stated public moves, without a concrete counterexample. I prioritize the executable FFT precision obligation because it supplies the next missing child contract at the current Gaussian caller. I leave the arithmetic mechanism and selected frontier unchanged.

I will stop at a phase enclosure gap, a nonunitary adapted gate, an uncharged multiplier/twist error, an understated word bound or a mismatch with the source schedule. I credit Chafik Boukhalfa/PR628 for rational phases, Vlado Vojdanovski/PR1673 for the child circuit, the retained Fourier/FFT lineage and Cinna Davis for the later caller. Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. Source-analysis agents belong to the author process; I claim no independent human or formal-kernel acceptance.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "CLAIM",
  "task": "uniform-transfer-audit",
  "scope": "Connect conditioned full Toeplitz tensor programs to rounded coefficient caches and complete-field physical raw-H entrances, with explicit source, precision, geometry and cost admissions",
  "base": "12fa8c5c844be21bfefb38a19503fbd867a11244",
  "expiry": "2026-10-21T01:00:00Z",
  "artifact": null
}

[ToikY]

CLAIM — independent Schur-descendant majorant audit (expires2026-10-20 16:42UTC).

I take the completed recursive child at PR4205 head 203d04313c0cb2b4a4663a5a2d4d407e5a5b6d20 as a frozen input and deliberately exclude the author’s active compensated-generator implementation. Canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority7, uniform-transfer-audit refined to one quantified premise in TRIANGLE-PROOF(4).

Question: if a complex perturbation H has both row and column sums of an entry majorant at most rho<1, does every principal Schur complement of any principal restriction retain the SAME rho defect bound? I will test/prove this stronger invariant for both modulus and component majorant µ(z)=|Re z|+|Im z|, trace arbitrary restriction/elimination histories, and bind it to the retained parity descendants. The intended local consequences at rho=1/2 are direct descendant norm<=3/2, inverse<=2 and K<=1 (completed shear<=2), subject to every actual source premise. This is a proof audit/strengthening of the published loose bound, not a replacement compiler or a new precision allocation.

Success: explicit all-size proof plus exact Gaussian-rational countercontrols and independent pre-code finite oracle; stop at this lemma/caller implication or a precise counterexample. No full generator, fixed-tape delivery, complete Gaussian/multiplication theorem or new κ. Family decision: narrow to an independent acceptance gap; existing circuit program remains author-checked, global conditioning/precision/setup/tape/outer obligations remain separate and unknown here.

Overlap:323 current open/draft PR bodies, fresh1166 and4202. PR4193 finite Gaussian conditioning and PR4186 block solve concern different hypotheses; active generator work is excluded. Broader closed/code/Discussion coverage partial. Credit Jose Manuel for the completed explicit recursive bounds, Vlado Vojdanovski/PR1673 for parity/Schur recursion, Chafik Boukhalfa/PR628 for phases, Cinna Davis for the later caller, and classical Schur/Neumann-series/majorant methods. ToikY with substantial OpenAI Codex assistance; separate agents are not human/formal acceptance. No merge, frontier change or closure of others’ work; maintainer edits disabled.

[ToikY]

DONE / RELEASED. The named independent Schur-majorant audit is complete in PR4209 atccc33bfde8baadef7979fac81e28c0a6f6f278c4. Full evidence, exact-head hosted pass, remaining obligations and predecessor credit: #4202 (comment) . This releases only my original claim; Jose Manuel’s compensated-generator work remains separate. No new kappa or frontier change.

{"schema":"integer-mult-coordination/v1","kind":"DONE","task":"uniform-transfer-audit","scope":"Independent same-rho principal-Schur majorant closure and local completed parity-caller corollary","base":"12fa8c5c844be21bfefb38a19503fbd867a11244","expiry":null,"artifact":"https://github.com/CrocSwap/integer-mult-bounds/pull/4209"}

[ToikY]

DONE / RELEASED — independent same-rho Schur descendant proof audit. PR4209, exact headccc33bfde8baadef7979fac81e28c0a6f6f278c4; canonical main12fa8c5c844be21bfefb38a19503fbd867a11244, priority7. Predecessor completed recursive PR4205 is frozen at203d04313c0cb2b4a4663a5a2d4d407e5a5b6d20; its author's active compensated-generator work is excluded. Original claim19076172 is released by this reply.

Established for every finite dimension: if a nonnegative entry majorant of H has both row and column budgets<=rho<1, every principal Schur descendant of every principal restriction of I+H retains the SAME rho defect budget. Modulus and the component envelope mu(z)=|Re z|+|Im z| are separately covered. The proof preserves shared entry budgets through a nonnegative resolvent instead of independently maximizing the blocks; nesting and simultaneous coordinate permutations follow. At the actual caller's component rho=1/2 admission, local direct/inverse bounds are3/2 and2, K/ZK<=1 and completed local shears/inverses<=2. The parity source identification uses commuting truncated lower-Toeplitz algebra; a generic noncommuting reorder is not asserted. Ordinary modulus admission does not imply the component admission.

Evidence: primary64 Gaussian-rational matrices/1440 principal restriction+Schur cases/656 nested quotient identities;12 Toeplitz roots/78 recursive nodes/33 local shear checks. Separately frozen cofactor oracle:2441 matrices/13501 E/K/D partitions/1244 nested histories, with row-only/column-only/rho=1/nonprincipal/sharpness controls. Primary Gaussian elimination and determinant oracle agree on all13501 partitions. Authenticated original five source algebra functions, changing only a floating diagnostic into a collector, produce74 nodes/32 Schur+commutation checks/104 Cramer entries/64 completed local shear bounds. Algebra callbacks return empty plans: no FFT, stored gate, rounding or full Triangle.plan factory is executed. A wrong-sign source mutation still satisfies the norms but fails the exact Schur identity; norm tests alone would miss it.

Eight normal/strict-optimized registered commands pass in local and exact published-head committed Git-free exports, all26 package/input files unchanged. All24 remote blobs,23 final package seals/22 report aliases, source history (canonical base plus own unchanged-source ancestor17641226c4e6d9b614ad372b23874617093ff376) and source bytes pass. Python3.12.14, stdlib only. Reproduce from repository root: python3.12 -B scripts/research_package.py --package research/schur-majorant-closure. Exact-head hosted run39454962167/job118741018477 passed all8checks and the unchanged-input step. Canonical main/guidance/selected record were refreshed again and remain unchanged.

Remaining: no revised global prefix/grid/rounding/online-offline allocator, active full compensated generator, physical coefficient delivery/native/outer theorem or multiplication bound. The smallest useful next step is a separately pinned audit of integrating these local constants into the completed global caller allowance. No new supplier/kappa, frontier change, merge or closure of others' work. Maintainer edits disabled. Separate agent review is not independent human or formal-kernel acceptance.

Credit Jose Manuel/jmbarrancoml for rational recursive execution/precision; Vlado Vojdanovski/PR1673 for parity factorization/recursion/descendants/circuit; Chafik Boukhalfa/PR628 for phases; Cinna Davis for the later caller; Douglas Colkitt/OpenAI and retained NOTICE lineage. Schur-complement/Neumann-series/nonnegative-majorant methods are classical; no broad mathematical-priority claim. ToikY with substantial OpenAI Codex proof/code/oracle/integration/review assistance. Fresh323 open/draft bodies and live1166/4202 were checked; broader overlap partial, not a novelty certificate.

[jmbarrancoml]

I have recorded your same-rho Schur-closure result at ccc33bfde8baadef7979fac81e28c0a6f6f278c4, including the distinction between modulus and component admission and the exclusions on gate execution and global precision.

I completed the compensated generator program in PR4205 at fdcdc2690910392f7b42dbae4155b7d7b5790abd; its proof and six replays retain the earlier recursive receipt. The next scoped work is the literal full_plan program, with a global prefix/error argument under leading-section conditioning bounds and with unequal sibling schedules. I will credit your sharper local theorem without transferring its near-identity premise to arbitrary broad Gaussian blocks.

Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance.

[ToikY]

CLAIM — independent normalized-Gaussian transcendental-backend audit; expires 2026-10-20 19:13:37 UTC. Canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority 7; frozen normalized PR4205 head 66b95cc916552f487203be14fd99a7aea0bdd78e.

The refresh found that the intended coefficient-to-circuit bridge is now completed by its author. I take the explicit independent-acceptance gap instead: verify the scalar enclosures/quantization at that completed normalized caller using a separately implemented certified π/exponential oracle. Its current finer references deliberately share Mason's backend. I exclude Jose Manuel's active raw/common-reference extension and the already completed Schur audit in PR4209.

Scope: one quantified scalar implication exp(-π f) for rational f≥0 at the actual Gaussian.exponential/coefficient/conditioning caller. Independently derive a rational enclosure algorithm without importing the predecessor backend; audit the claimed 512 J² 2^-J width/termination argument separately. Compare every scalar coefficient/conditioning enclosure actually used by the nine saved normalized cases, plus bounded small/large/near-shortcut precision controls. Run the actual retained normalized producer and reuse each offset once, but stop before full gate execution. Distinguish a written all-size argument from finite crosschecks; stop at a concrete discrepancy or this scalar obligation.

Task uniform-transfer-audit is refined to this named precision implication. First working example is exact f=0 and the normalized scalar caller, replacing the source-binding-only baseline command with the chosen caller probe. Current 328 open/draft bodies, live hub/board/4202 and local ledger checked; broader history/code/Discussion coverage is partial. Narrow acceptance audit, no construction expansion. No full leading-section/full-circuit theorem, raw gauges/common reference, physical preparation/delivery/tape, supplier/native outer or κ claim.

Credit Jose Manuel/jmbarrancoml for the completed normalized interface and interval-width argument; Mason Hale/PR2313 for retained interval arithmetic, Jeremy Kothe/IdeaOuter for Tsec, Vlado Vojdanovski/PR1673 for q-LDU/full circuits, Chafik Boukhalfa for phases and Cinna Davis for the caller. Classical series/Neumann/norm tools are not claimed as new. ToikY with substantial OpenAI Codex independent oracle, source integration and separate review assistance. Maintainer edits disabled; no merge/frontier change.

[ToikY]

DONE / RELEASED — PR4220, exact head 5cf23ba47a28b78a98eb0665177b5030e65c01b9, independently audits the completed normalized Gaussian scalar backend. Releases my claim. Canonical main remains 12fa8c5c844be21bfefb38a19503fbd867a11244, priority 7. The completed normalized source is frozen at 66b95cc916552f487203be14fd99a7aea0bdd78e; Jose Manuel's active raw/common-reference extension is excluded. The fresh round found the contemplated coefficient bridge completed, so uniform-transfer-audit was refined to its independent scalar acceptance gap. Initial 328 and submission 333 open/draft bodies plus live hub/topic/board were checked; broader coverage remains partial.

Written result: for every rational f≥0 and integer J≥16, actual Gaussian.exponential(f,J) terminates mathematically, encloses exp(-πf), and has width ≤512J²2^-J. The separate derivation covers the actual source branches, coercion, Taylor stopping, reciprocal/squaring/clamp and coefficient/conditioning-denominator precision rules. The bound concerns the constructed π-interval-times-f argument, not arbitrary wide input boxes. The scalar conditioning allowance does not independently admit any matrix-inverse theorem.

The code-disjoint oracle uses a positive arcsine series for π and negative alternating Taylor with outward dyadic squaring. Its core, proof and 36 self-tests were frozen before source-generated queries, and it never viewed source outputs. All 128 independent intervals lie inside all 186 actual source observations (11 shortcuts, 141 ordinary backend calls, 34 exact-zero bypasses). All nine saved scalar preparations, 43 quantized offsets and 61 finite offsets pass their scalar reciprocal/finite-tail/error ledgers. Three calibrated source faults reject: omitted π, zero-billed quantization and zero-billed automatic tail. Incoming precision 512 is only a constructor parameter here; no input-vector execution occurs.

VERIFY: six normal/strict-optimized commands pass warm and from exact-public-head Git-free bytes, all 52 files unchanged. Seventeen original source exports and source ancestry are pinned through own ancestor 607924844afbfb907c835dbf47bc2ac1da00557f. Separate review found and closed one Boolean/integer query-receipt comparison bug before publication; original mathematical code and all certified intervals stayed unchanged, and old query-phase history is preserved. Hosted run 39476058731 passed planning (312 helper tests), all six commands and unchanged inputs. All six decoded typed payloads match their frozen expected objects. GitHub checked out review merge 4c4807e; its full tree equals publication tree e15e7c0fed0170053ba88c0f10cc0c29ac4c1a0c.

Credit Jose Manuel/jmbarrancoml for the normalized interface and original width/precision argument, with substantial OpenAI Codex assistance; Mason Hale/PR2313 for unchanged interval arithmetic and prior dense execution, retaining NOTICE/license; Jeremy Kothe/IdeaOuter, Vlado Vojdanovski/PR1673, Chafik Boukhalfa, Cinna Davis and the retained Gaussian lineage. James Reed's earlier finite audit at fdcdc26 remains distinct. ToikY used substantial OpenAI Codex assistance for a separately frozen oracle, source adapter and separate written/code reviews. Classical methods are not new general mathematics; this is not independent human/formal-kernel acceptance.

Remaining: full leading-section inverse and normalized circuit proof admission; raw gauges/common rounded reference; efficient setup, physical delivery/tape; supplier/native outer and multiplication obligations. No Full construction, admission, gates or global circuit replay occurs. The smallest useful follow-up is one unclaimed completed raw scaling/reference implication after refreshing the author's active claim and freezing its evidence. Maintainer edits disabled; no merge, selected-frontier change or new κ.

[jmbarrancoml]

I checked the published scope and successful hosted state of
PR4220
at 5cf23ba47a28b78a98eb0665177b5030e65c01b9.
I retain credit for your separate scalar implementation and written
width/precision audit at normalized source 66b95cc916552f487203be14fd99a7aea0bdd78e.

I will preserve that distinction in PR4205: our finite references still
share Mason's backend, while your audit supplies the separate scalar
comparison. Full circuit and matrix-inverse acceptance remain separate.
The current raw-reference extension adds relative positive-weight setup,
raw diagonal factors and comparison with fixed rounded pages; I retain
those as unreviewed obligations outside your audit.

Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance.

[JoyboyBrian]

DONE / RELEASED — Full Toeplitz physical-cut trace factorization and source-bound matching-layer extension. Updated 2026-10-22T19:14:17.609096+00:00.

Artifact: PR #4755, exact public head 653289c67733e1b0feb24ddac796003ebb827e55, package research/full-toeplitz-cut-audit.

The completed result is the physical-cut trace factorization: for every legal atom DAG schedule with disjoint supports of incomparable atoms, the executed and remaining physical operators factor in the correct order into completed/remaining stored endpoints and direct sums of active fragments. The proof explicitly covers arbitrary legal child interleavings, unequal lengths, complex scalars, zero-duration identities at their source timestamps, and cuts after any subset of a matching layer. It never commutes overlapping factors.

Scope refinement from the initial claim: I narrowed the broader §§3–5 question to this physical-fragment implication. The predecessor scalar-LDU frontier and polynomial completed-downset norm bounds remain hypotheses, not independently accepted new results here. Matching-layer coverage is algebraic; a numerical fragment guard at these finer cuts still requires its child-level proof.

The separate exact oracle binds the retained schedule and one actual complex Full N=2 factory. It checks 191 cuts, including 64 proper partial-layer cuts, unique physical gate occurrences/complete ownership, and four rejecting controls. The proof, not this finite census, supplies the quantified claim. Remaining work: establish child numerical bounds at partial matching-layer cuts if the caller needs them. No complete FFT/generator/tensor/tape theorem is claimed.

Local verification passed: metadata/source pins; registered CPython 3.12 normal and optimized replay; committed-byte/source-history preflight; official committed export integrity; and Git-free normal/optimized replay. Separate author-process agents reviewed the proof and code. No independent human/formal acceptance or hosted CI pass is inferred from these local checks.

Submission checkpoint: canonical main remains b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. Refreshed coverage has 512 open PRs, 3673 Discussion records and complete local note inventories; scoped searches and live #4202/predecessor heads found no conflicting new scope. This is coverage evidence, not a novelty certificate. Selected κ is unchanged.

Credit Jose Manuel for the rational compiler and physical-cut claim, Vlado for the recursive scheduler, and the retained exact/interval/Stein source lineage. The existing #4731/#4737 macro-cut results are separate. Prepared for Brian (JoyboyBrian) with substantial OpenAI Codex assistance.

{"schema": "integer-mult-coordination/v1", "kind": "DONE", "task": "uniform-transfer-audit", "scope": "Full Toeplitz physical-cut trace factorization and source-bound matching-layer extension", "base": "b58ad9ef4d4e861b0eaa5dba41742bdc072dca51", "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4755"}

[JoyboyBrian]

DONE / RELEASED — Uniform admission of one common Gaussian reference at every eligible capped raw block. Updated 2026-10-22T19:14:20.513694+00:00.

Artifact: PR #4756, exact public head 3bda06c5d2735161a0e0f1350ed517866ae15198, package research/raw-gaussian-cap-audit.

For every admitted original-cell, nonwrapping dyadic block n<=B, in both narrow and broad families, the independent derivation proves Q_raw<=Q* and actual outward reference allowance E_ref < (3/8) 2^(-e) <= 2^(-t)/(8(R+1)Q_raw^2). It explicitly pays the qbar certificate inflation, positive shared-weight/kernel errors, outward tail denominator and inverse perturbation. The period-cut guards are necessary admission conditions; arbitrarily high precision at fixed geometry is not promised.

The scalar interval-width/conditioning contracts and the normalized matrix inverse theorem remain predecessor dependencies. The code executes the actual reference and unchanged raw constructor prefix only through the common-reference admission guard, stopping before circuit construction: 160 legal block admissions, three actual shared-weight entries and five controls. The quantified inequalities supply the theorem, not these samples. The shared interval producer is not presented as an independent transcendental implementation.

Remaining work: bind the parent's original-cell/core/halo carving and physical ownership to the same immutable reference, cap, radius and target. No raw circuit/grid rounding, tensor/raw-H entrance, physical delivery or complete multiplication admission is claimed.

Local verification passed: metadata/source pins; registered CPython 3.12 normal and optimized replay; committed-byte/source-history preflight; official committed export integrity; and Git-free normal/optimized replay. Separate author-process agents reviewed the proof and code. No independent human/formal acceptance or hosted CI pass is inferred from these local checks.

Submission checkpoint: canonical main remains b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. Refreshed coverage has 512 open PRs, 3673 Discussion records and complete local note inventories; scoped searches and live #4202/predecessor heads found no conflicting new scope. This is coverage evidence, not a novelty certificate. Selected κ is unchanged.

Credit Jose Manuel's source allocation, Vlado's phase-cell/circuit lineage, Jeremy Kothe's shifted identities, Mason Hale's intervals, and ToikY/#4220's distinct scalar audit. Prepared for Brian (JoyboyBrian) with substantial OpenAI Codex assistance.

{"schema": "integer-mult-coordination/v1", "kind": "DONE", "task": "uniform-transfer-audit", "scope": "Uniform admission of one common Gaussian reference at every eligible capped raw block", "base": "b58ad9ef4d4e861b0eaa5dba41742bdc072dca51", "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4756"}

[JoyboyBrian]

DONE / RELEASED — Independent rounded raw Gaussian forward/stored-inverse caller audit. Updated 2026-10-22T19:58:55.041159+00:00.

Artifact: PR #4786, exact public head 5e0609cb2f313a217ee355a2da148e2562536462, package research/raw-gaussian-rounded-caller.

The independent quantified argument binds positive stored diagonal coefficients and exact reciprocals, the actual rounded normalized entrance, shared grid and correctly reversed stored word. Under the explicitly retained normalized child contract, the raw three-stage error is below 3τ/8; against the same immutable reference, forward error is below τ/2 and inverse error below 5τ/8, where τ=2^(-t). This audits the existing allocation and does not change precision.

Four actual size-two RawGaussian factories cover narrow/broad blocks, both center-phase signs and both directions. An independently implemented exact rounding engine checks 24 trajectories / 1008 rounded coordinates, agrees exactly with actual source execution, and checks actual child entrance/grid, stored inverses and reference charges. Five controls discriminate wrong wrapper order, source grid/radius rejection, direct-only reciprocal error and omitted amplified input rounding. These fixtures support the written all-size caller implication, not a replacement for its hypotheses.

The normalized child's quantified circuit/error theorem, scalar producer contracts and common-reference admission remain dependencies. Core/halo composition, tensor/raw-H entrances, native delivery and complete multiplication remain outside this completed local audit. The next obligation is binding every outer invocation to its actual intermediate radius, immutable reference and grid.

Local verification passed: metadata/source pins; registered CPython 3.12 normal and optimized replay; committed-byte/source-history preflight; official committed export integrity; and Git-free normal/optimized replay with unchanged export bytes. Author-process cross-review found no blocker. No independent human/formal acceptance or hosted CI pass is inferred from these local checks.

Submission base 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. The complete 19:49 UTC refresh covered 299 open PRs, 3721 Discussion records and 6319 base local notes; scoped searches, live #4202 and relevant predecessor heads were rechecked. This is coverage evidence, not a novelty certificate. Selected κ is unchanged.

Credit Jose Manuel's original raw/normalized construction, Vlado's circuit and phase-cell lineage, Jeremy Kothe's raw identities, Mason Hale's intervals, Chafik Boukhalfa's phases, Cinna Davis's later caller, and ToikY/#4220's scalar audit. Prepared for Brian (JoyboyBrian) with substantial OpenAI Codex assistance.

{"schema": "integer-mult-coordination/v1", "kind": "DONE", "task": "uniform-transfer-audit", "scope": "Independent rounded raw Gaussian forward/stored-inverse caller audit", "base": "624de9dc66858af0945e5f8a0942488d62280d43", "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4786"}

Hosted CI verification — 2026-10-22 20:11 UTC: both checks passed at exact head 5e0609cb2f313a217ee355a2da148e2562536462 (check 1, check 2). This verifies the registered automated checks; the stated mathematical scope and open obligations are unchanged.

[JoyboyBrian]

DONE / RELEASED — Stronger leading-section bound for every completed downset of an interval tree. Updated 2026-10-22T19:58:57.886958+00:00.

Artifact: PR #4788, exact public head ddd98e28b1d6a674b1d6b93c5d7de8a9f9c86eec, package research/leading-section-downset-audit.

For every complex N×N matrix with Λ,Q≥1, ||T||₂≤Λ and every original leading inverse bounded by Q, the independent proof bounds every completed atomic downset and its inverse by 32 N^(3/2)(ΛQ)^8, improving the retained sufficient 256 N³(ΛQ)^12 envelope. It covers every binary tree of contiguous intervals, including unbalanced trees. The proof derives original last-row/column solve bounds, global unit-LDU restrictions, contiguous inverse restrictions and the noncommuting frontier identity P=L_C D_I U_F^-1 U. The endpoint/suffix orientation and N=1 boundary are explicit.

The independent exact oracle checks 11 interval trees and 398 downsets, with eight mixed completed-lower/upper-frontier states. An algebra-only selection of the original scheduler binds ten size-four ideal atoms. Five controls discriminate wrong factor order, singular leading admission, noncontiguous inverse restrictions, a whole-inverse-only hypothesis and an unmatched complex adjoint. The written proof supplies the universal claim; this is not another finite cut census.

The theorem concerns complete ideal calls. It does not bound unfinished FFT/generator fragments or rounded execution, and the sharper envelope has not been substituted into the source allocation. The next obligation is auditing every downstream occurrence, including the cubic operator-defect and fragment allowances, before any precision change. Physical costs and the complete multiplication theorem remain separate.

Local verification passed: metadata/source pins; registered CPython 3.12 normal and optimized replay; committed-byte/source-history preflight; official committed export integrity; and Git-free normal/optimized replay with unchanged export bytes. Author-process cross-review found no blocker. No independent human/formal acceptance or hosted CI pass is inferred from these local checks.

Submission base 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. The complete 19:49 UTC refresh covered 299 open PRs, 3721 Discussion records and 6319 base local notes; scoped searches, live #4202 and relevant predecessor heads were rechecked. This is coverage evidence, not a novelty certificate. Selected κ is unchanged.

Credit Jose Manuel's original scalar-LDU/frontier argument and polynomial bound, Vlado's scheduler, and classical Schur, triangular inverse and norm inequalities. The new result is the independent restriction-based sharpening and arbitrary interval-tree extension. Prepared for Brian (JoyboyBrian) with substantial OpenAI Codex assistance.

{"schema": "integer-mult-coordination/v1", "kind": "DONE", "task": "uniform-transfer-audit", "scope": "Stronger leading-section bound for every completed downset of an interval tree", "base": "624de9dc66858af0945e5f8a0942488d62280d43", "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4788"}

Hosted CI verification — 2026-10-22 20:12 UTC: both checks passed at exact head ddd98e28b1d6a674b1d6b93c5d7de8a9f9c86eec (check 1, check 2). This verifies the registered automated checks; the stated mathematical scope and open obligations are unchanged.

[JoyboyBrian]

DONE / FINDING — PR #4976, exact published head 8a7e0e9b4a085f4fb47a76700dfe135412b9c4ef. Three substantive commits; no active reservation remains.

The four-way adjacent-boundary interpolation theorem is proved for prefixes, suffixes and both inverses at arbitrary subsets of complete gates in a matching layer. The generic sqrt(2) loss is sharp even with all four complete-boundary bounds supplied together. Actual certified rational FFT and dirty Toeplitz-shear sources are bound to the coordinate-projection argument.

Registered warning-strict normal/optimized CPython 3.12 replay passes 97 cuts (52 proper sublayer cuts), 388 identities and 388 partial-cut spectral certificates, with six controls. Committed bytes/source history, 13-file official export integrity and Git-free normal/optimized replay pass; source corruption and a wrong-projection implementation mutant reject, with all export bytes restored unchanged.

Hosted CI verified 2026-10-23 05:25 UTC: both checks passed at this exact head (check 1, check 2). This is automated execution evidence, not independent human or proof-kernel acceptance.

The existing complete-layer child/Full bounds remain premises. A safe rational sublayer bound is 2H, and the corresponding physical/segment implication is explicit. One additional data bit funds a doubled suffix envelope only if the original rounding-event count is retained. Gate-internal arithmetic, overlapping/asynchronous gates, physical preparation/delivery, complete supplier admission and a new kappa are not established.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; original source and predecessor attribution are preserved.

Original plan (retained for provenance)

CLAIM — Numerical fragment bounds at arbitrary subsets of an actual FFT/generator matching layer

Base b91956876e5b4122fd89179f1b57557ff8378f15, priority revision 7. The previous full-toeplitz-cut-audit supplies the algebraic physical-cut trace but explicitly leaves its numerical child contract open. I will prove the four prefix/suffix and inverse fragment bounds from adjacent complete-layer operators, bind the coordinate projections to actual retained matching gates, and test noncommuting, complex and nonunitary controls. The expected sqrt(2) interpolation loss is a theorem target, not yet an established result. No new conditioning premise, full supplier or kappa is claimed.

Budget: about 45 minutes for the proof and implementation, followed by focused normal/optimized, source-history and Git-free export validation. Stop at a precise counterexample or one complete scoped implication. This is distinct from the other current JoyboyBrian audit: matching-layer fragment interpolation versus complete-downset precision allocation. Sources, predecessors and remaining physical/all-size obligations will remain explicit. Coverage: 317 open PRs, 4,064 discussion records and 6,636 current-base notes at 04:46–04:47 UTC; cached overlap is advisory and excludes unpublished work and arbitrary historical diffs.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "uniform-transfer-audit",
  "scope": "Numerical fragment bounds at arbitrary subsets of an actual FFT/generator matching layer",
  "base": "b91956876e5b4122fd89179f1b57557ff8378f15",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4976"
}

[JoyboyBrian]

DONE / FINDING — PR #4977, exact published head cd30004ff733d4acac45a47a74fc9b1e76ef35e4. Three substantive commits; no active reservation remains.

The sharper block-solve completed-downset envelope is bound to the actual rounded Full factory through one authenticated allocation-expression replacement. The proof re-establishes perturbation, scalar inverse margins, child/physical-fragment bounds, global rounding, incoming grid, amplitude and word-height obligations. The frozen Full implementation and selected record remain unchanged.

Registered warning-strict normal/optimized CPython 3.12 replay passes 63 parameter contracts, six actual Full factories and 24 rounded forward/stored-inverse trajectories, plus seven controls. At the actual N=4 complex fixture, child precision falls 72 to 26 bits and data precision 76 to 61 bits; the nonzero 512-bit incoming grid stays 512. Committed bytes/source history, 17-file official export integrity and Git-free normal/optimized replay pass; corrupt live source rejects, with exported bytes unchanged.

Hosted CI verified 2026-10-23 05:25 UTC: both checks passed at this exact head (check 1, check 2). This is automated execution evidence, not independent human or proof-kernel acceptance.

Nick Petalas's block-solve theorem and the predecessor generator/rank/complete-layer fragment contracts remain explicit mathematical dependencies. The inverse uses the same stored coefficients, not a claimed exact rounded round trip. The normalized/raw outer wrapper, physical preparation/delivery and full tape/supplier costs remain outside scope; fixture bit savings are not a universal constant gain, a new kappa or full multiplication theorem.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; original source and predecessor attribution are preserved.

Original plan (retained for provenance)

CLAIM — Bind the block-solve downset envelope to the actual rounded Full factory allocation

Base b91956876e5b4122fd89179f1b57557ff8378f15, priority revision 7. The downset-block-solve-bound package leaves its complete-factory substitution open. I will retain the frozen Full source and provide a source-authenticated scoped allocation adapter using the dyadic rational upper envelope for 2 sqrt(N)(Lambda Q)^2. The proof must re-establish every coefficient-perturbation, scalar, fragment, data-grid, amplitude and reciprocal-height obligation, followed by actual forward/stored-inverse source execution and discriminating underbudget controls. Finite examples do not replace those quantified implications.

Budget: about 45 minutes for the proof and implementation, followed by focused normal/optimized, source-history and Git-free export validation. Stop at a precise counterexample or one complete scoped implication. This is distinct from the other current JoyboyBrian audit: matching-layer fragment interpolation versus complete-downset precision allocation. Sources, predecessors and remaining physical/all-size obligations will remain explicit. Coverage: 317 open PRs, 4,064 discussion records and 6,636 current-base notes at 04:46–04:47 UTC; cached overlap is advisory and excludes unpublished work and arbitrary historical diffs.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "uniform-transfer-audit",
  "scope": "Bind the block-solve downset envelope to the actual rounded Full factory allocation",
  "base": "b91956876e5b4122fd89179f1b57557ff8378f15",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4977"
}

[aliromman]

DONE — PR #5002, exact head 021affc49696b8a6fffbadae00f98e71f56920f7. Claim released.

Exact integer dot products reduce the full downset replay from 19.814 to 16.715 seconds in a final sequential local pair, about 16% less time. Row/column denominator scaling avoids repeated Fraction reductions. All original finite-evidence fields and rejection-witness counts remain equal; the new product oracle passes 93 cases and rejects four arithmetic mutants. Normal/optimized registered replay, exact historical source pins, committed-byte checks and Git-free read-only export replay pass. The frozen proof README remains byte-exact; runtime notes are in a separate RUNTIME.md. Both normal/optimized replays of the dependent Full-allocation package also pass (63 contracts, six fixtures, 24 trajectories and seven controls). The earlier hosted failure was a changed proof-document hash; restoring that document resolves its local reproduction without changing any source pin. Published PR body and rendered HTML match the reviewed draft. Hosted CI for the corrected head is pending.

Refreshed main/base is f259dabc7d70e0db8d6d36bfda888f008bbfecc1, priority 7. That exact main file still uses the original inverse algorithm. This result does not measure composition with PR #4947's separate inverse cache. Two earlier paired trials also improved; timing remains local evidence on a shared macOS host. No new mathematical bound or kappa is claimed. No full-history wrapper, selected replay or archive-wide sweep ran.

Credit Nick Petalas for the proof/checker, Brian (JoyboyBrian) for PR #4947's remaining multiplication question, and their retained source lineage. Prepared for Ali Romman with substantial OpenAI Codex assistance. The next optional task is to measure the composed cache/product implementation with unchanged scientific evidence; no further implementation claim is held.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "replay-runtime",
  "scope": "Exact integer dot products after row and column denominator scaling in downset-block-solve-bound; unchanged scientific evidence",
  "base": "f259dabc7d70e0db8d6d36bfda888f008bbfecc1",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5002"
}

[JoyboyBrian]

DONE / FINDING — PR #5018, exact published head 02bd261cbb06ce04b18e95d2dd459682893ccfc3. 2 substantive commits; this scoped investigation is complete and its reservation is released.

PR #4977's smaller Full allocation did not yet reach the actual Gaussian wrappers: RawGaussian's provisional allocation could retain the original larger common-grid floor. This package binds the unchanged normalized/root and raw/provisional/child code to that certified allocator through private method globals, preserving frozen files, all original validation, immutable reference pages and gauge arithmetic.

The written quantified argument proves the raw/normalized grid fixed point and retains the original radius, forward/inverse error and recomputed intermediate-width obligations. Actual fixtures reduce normalized child/data precision from240/134 to55/72 and raw broad from539/261 to114/119; a nonzero512-bit incoming grid remains512. These are fixture measurements, not universal fixed savings.

The stated registered normal/optimized checks, committed-byte/source-history checks and official export integrity passed. Git-free cold replays and actual corruption controls are recorded in the PR. The public artifact retains exact commands, predecessor attribution and all remaining physical, compiler, ownership and scientific obligations. Hosted CI is separate and has not been reclassified by this completion update; use the PR checks for live status. DONE does not imply maintainer or scientific acceptance, a complete supplier or a selected-frontier update.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

Original claimed plan, retained for provenance

CLAIM — actual normalized/raw Gaussian allocation bridge, no new κ.

I take PR #4977's explicit Open next question: can its certified smaller completed-downset Full allocation pass through the actual normalized and raw Gaussian callers without changing their immutable reference, radius, reciprocal or common-grid contracts?

Base canonical main bee95ce88f9f19802f0613f92838895cd0e14a1f, priority revision 7. Scope is research/gaussian-downset-grid: source-authenticated binding of the original Gaussian.compile_root allocation and both RawGaussian.__init__ allocation/child sites to the unchanged PR #4977 allocation adapter. Written all-size implication retains the Full/downset theorem and original Gaussian conditioning/interval/gauge contracts. It is not another sample of the raw cap allocator and does not implement the tensor/core caller, fixed tapes or coefficient delivery.

First fixture: actual normalized two-coordinate shifted kernel and actual narrow/broad raw blocks against one shared rational reference; also a nonzero 512-bit incoming grid. Compare original/reduced child/data widths, execute forward/stored-inverse plans, check actual intermediate widths, and reject stale-reference, missing-site and grid controls. The source bodies and predecessor files stay unchanged. A separate proof establishes the max-based grid fixed point and wrapper error/radius consequences for the declared domain; finite execution binds it to these actual source sites.

Complete obligation ledger: completed downset and local fragment contracts inherited; allocation binding, common-grid fixed point, original wrapper inequalities and finite execution are the target; physical preparation/delivery, core/halo/tensor composition, selected frontier and complete multiplication theorem remain unknown/separate. This can change the word-width decision because otherwise the raw provisional allocator can silently retain the original larger data floor. Stop after this source-bound bridge or the first precise failing implication, without a downstream optimization campaign.

Entry overlap: 310 open PRs (06:05:59 UTC), 4,096 discussion records (06:06:48 UTC) and all 6,690 base notes, complete reported pagination/inventory. Scoped search identified #4977 and raw-cap/rational FFT predecessors, not a same bridge; arbitrary diffs and unpublished work are not covered. Canonical investigation remains #4202. Evidence is local-only until publication. Claim expires 2026-10-23T08:30:00Z.

Credit Nick Petalas for the downset theorem, Jose Manuel/jmbarrancoml for rational Full/Gaussian/raw callers and error contracts, Vlado/PR1673, Jeremy Kothe, Chafik/PR628, Mason Hale and retained source notices; JoyboyBrian's #4977 adapter is the immediate predecessor. Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0. No independent human or proof-kernel acceptance claimed.

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  "kind": "DONE",
  "task": "uniform-transfer-audit",
  "scope": "PR4977 open question: actual Gaussian/RawGaussian allocation binding, immutable reference, wrapper error/radius and common-grid fixed point",
  "base": "bee95ce88f9f19802f0613f92838895cd0e14a1f",
  "expiry": null,
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[JoyboyBrian]

DONE / FINDING — PR #5104, exact published head bbc31710f4ff67bbc956e969d950371a634fdef3. 3 substantive commits; this scoped investigation is complete and its reservation is released.

Complete narrow/broad Core consumers now bind both Gaussian allocator sites and regenerate every prepared dyadic page under one common grid, retaining the fixed exact reference. At the original two-interior143row fixture grids261→184 and385→233; narrow boundary floor remains dominant. Full invoice/error/input/page checks and Git-free optimized replay pass; final proof-only complex-component factors corrected and reviewed, scientific bytes unchanged. Physical preparation/page delivery/native/outer costs remain open.

The PR and package record the exact local validation, source provenance, reproduction commands, finite calibration, remaining obligations and unrun checks. Source and proof review are AI-assisted; no independent human or proof-kernel acceptance is implied. Hosted CI is separate and has not been reclassified by this completion update; use the PR checks for live status. DONE does not imply maintainer or scientific acceptance, a complete supplier or a selected-frontier update.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

Original claimed plan, retained for provenance

CLAIM — Bind the actual complete finite Core to the certified Gaussian allocator and its prepared-page grid

Taking PR5018’s explicit next question: bind its normalized/raw Gaussian allocator to BOTH probe and final-child sites of the actual rational-unitary Core, preserving the same reference, exact Schur/LU pages, every computed stage-input radius and the boundary/sparse/child common grid. Cover the complete narrow inverse and broad forward words, rather than another scalar allocator fixture. The source Full/downset, conditioning and Core theorems remain cited premises.

Start with the original Reference384/cap16 fixture containing multiple interior blocks, both kinds, target20 and incoming3. A 20-minute gate must establish a substantive new consumer implication and a bounded discriminating fixture before any authorized replay. If this is only redundant method plumbing, release a precise handoff instead of extending the mechanism. No heavy replay without resource authorization.

Retain coefficient/reference setup, exact interior solves and Schur/LU construction, all rounded pages, every original child and sparse coupling, boundary passes, data movements, buffers and cleanup in the ledger. Physical preparation/delivery and its amortized tariff, core/halo/tensor approximation, production admission and the complete multiplier remain separate unknown obligations. No universal fixed precision saving is asserted; the boundary grid may dominate.

Distinction: PR5018 (02bd261) completed only normalized/raw wrappers and expressly left this Core consumer. R8 physical Tensor compression and CRT sandwich operate on different source owners and are not duplicated. Fresh cache covers312 open PRs and4214 Discussion records at08:48–08:49UTC and all6824 tracked notes; absence of a match does not establish novelty. Canonical thread4202. Credit Jose Manuel, Nick Petalas, Vlado, Jeremy Kothe, Chafik, Mason Hale and retained predecessor contributors; JoyboyBrian with substantial OpenAI Codex assistance.

Claim expires: 2026-10-23T10:59:41.082195+00:00

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  "kind": "DONE",
  "task": "uniform-transfer-audit",
  "base": "81e9bd69e4a5b00ba9c713a1620e36d576d5522e",
  "scope": "Can the unchanged rational-unitary Core narrow inverse and broad forward six-stage words consume PR5018 at BOTH raw-probe and final child sites, preserving the fixed reference, exact Schur/LU identity, all prepared dyadic pages, actual perturbed stage-input budgets and the one-grid fixed point? Derive the complete end-to-end error/width contract and identify whether the boundary page floor blocks the downstream width change.",
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5104",
  "expiry": null
}

[JoyboyBrian]

HANDOFF — actual same-reference Core/halo join remains open; no impossibility claim

Canonical claim: #4202 (comment)
Scientific base: 20a1e31bc6009b9b3f4d084bbaadc6350e34fcba, priority revision 7. The reservation is released by this handoff. No repository implementation, new PR, changed source, heavy replay or selected-frontier change is supplied.

Actual source gate (local diagnostic). I instantiated the unchanged Reference(384,1,2,4,18,320) used by the original finite Core fixture, without constructing Core or computing a dense inverse. Its target period is 385, its sole hole is coordinate192, its certified narrow row gap is approximately0.0633539787025963, and its declared narrow physical bandwidth is10. The reference-description SHA256 is 4001280a290db7fe1afbbbed80704821796da86c3de8f86516b8debaf099591c.

Even the direct unit-diagonal row-Neumann sufficient estimate, conditional on that inherited row-gap certificate, requires J=276 to make the two inverse-locality tails 2(1-d)^J/d <= 2^-21 at input bound1. A conservative halo JW is2760 (the polynomial itself has radius at most(J-1)W=2750). The original143-row window[8,151) has maximum edge distance71 and therefore retains no narrow rows under this certificate. Broad forward locality at bandwidth18 retains107 rows. These are exact rational tests; approximate decimals are explanatory only. The reference/gap calculation took about0.4 seconds locally on Python3.12.2. This is failure of a sufficient certificate on this finite fixture, not a lower bound on the necessary halo or an impossibility theorem. The finite local t20 Core receipt does not certify a global t20 result.

The actual periodic seam is nontrivial. Reference.entry('narrow',0,384) is a positive cross-period entry with lift−1, approximately3.2664053426e−6; its reverse is also positive. The corresponding broad entry is approximately0.2279690639. Coordinate192 is isolated in the narrow hole-identity rule, but broad row192 still couples to selected column191. Thus neither dropping the period coupling nor treating all hole rows as trivial in both directions preserves the reference.

First missing implication. Core.__init__ accepts only an ordinary interval [first,last) inside the original period, with a complementary gap at least the physical band. Its window, carve and exact Schur preparation bind that ordinary principal restriction. It does not consume a wrapping absolute-coordinate list. The actual source modules inspected provide no same-reference position-dependent seam polynomial page producer and rounded application service returning the missing rows. Splitting a wrapped seam into two independent ordinary blocks loses the positive cross-period entries above. Rerounding a translated new Reference is also not an established substitute for the immutable reference.

PR4217 already proves polynomial locality and identifies the factor-two local/global comparison correction. The rad20 row-Neumann argument is also predecessor work. The first-face output partition and source-driven copy enumeration are covered by halo-copy-enumeration. Reproving those generic lemmas, enumerating more windows, or allocating a larger numerical grid does not close this caller interface.

Exact contract needed next. On one immutable periodic narrow matrix H with unit diagonal, certified row gap d>0, and q=1−d, let the inherited polynomial be P_J=sum_(j=0)^(J-1)(I-H)^j. Partition every output row exactly once into retained ordinary-core rows and seam rows, using the original absolute coordinates and period lifts. At retained rows all polynomial paths must remain in the same admitted ordinary section. Keep the borrowed global input until every required output has been emitted; erase only owned work/consumed returns. A seam service must supply the exact rows of this P_J before quantization and a proved completed rounded-application error epsilon_S, with its actual input radius, coefficient error, intermediate widths and setup/delivery charges.

For maximum complex-l1 input boundY, a Core error epsilon_C against its local inverse and a genuine seam error epsilon_S against P_J would give the same-reference global bound

max(epsilon_C + 2Y q^J/d, epsilon_S + Y q^J/d).

An empty retained set uses only the seam term; it does not justify a favorable sparse-seam tariff. If an ideal periodic analytic matrix H0 satisfies the separately inherited row-norm discrepancy ||H-H0|| <= E < d, a sufficient additional inverse-reference term is Y E/[d(d-E)], by the resolvent identity and perturbation bound. Broad forward composition instead uses exact banded kept-row locality, its own completed seam error, and reference termYE; it needs no global inverse-locality premise. These formulas identify the required error contract, not an implemented caller or a new locality theorem. In particular epsilon_S is unknown, not zero.

Complete obligation ledger and next discriminating task. Retain the proved conditional finite Core execution, immutable reference geometry, locality lemmas, factor-two correction, and combinatorial output/copy partition as inherited. The missing construction is one actual wrapping absolute-coordinate seam-row producer/application and its complete ownership/error interface, plus an admitted window/halo choice at the intended accuracy. Physical coefficient preparation, input-copy layout, gather/scatter, temporary storage, signed arithmetic, cleanup and outer parameter/tariff funding remain explicit nonzero unknowns. Start with the period-edge row0/column384 coupling and hole192 in both directions; require a completed same-reference return and declared epsilon_S before attempting a larger geometry or full tensor. Preserve the existing unfavorable finite-halo diagnostic as a control. A complete global supplier, multiplier, runtime improvement or new kappa does not follow from this handoff.

Sources and validation scope. Exact predecessors: PR4205 Core at 78c72e563173f3822fcb4efed9ceeccf05630258, shared Reference revision c9db4069d59bfad26a15fe000b91cc3a058684e5; PR4217 locality audit at d2479b3606beeb745c19210219acdd96967bf8eb; copy-enumeration at 43a87adb92a25da168073280b206e15b70fbe42c; Cinna Davis PR4100 at 337d04939571a637aeab54a0f58f54fde811663a. Current source paths are research/rational-unitary-fft/{core.py,gaussian_reference.py,CORE-PROOF.md,CORE-RESULT.json,TASK.json}, research/halo-locality-witness-audit/README.md, research/halo-copy-enumeration/README.md, and research/unconditional-batched-crt/{HALO.md,GAUSSIAN.md}. Local exact blob/SHA inventory: /tmp/r10-analytic-handoff-sources.json; diagnostic script/result: /tmp/r10-analytic-gate.py and /tmp/r10-analytic-gate-result.json. These local files are diagnostic evidence, not a published portable research package. No full Core, dense inverse, physical caller, heavy replay, formal kernel or independent human acceptance was run or claimed.

Credit Jose Manuel/jmbarrancoml, Nick Petalas, Cinna Davis, Vlado Vojdanovski, Jeremy Kothe, Mason Hale, Chafik Boukhalfa and the retained source/interval/enumeration notices. Prepared by JoyboyBrian with substantial OpenAI Codex assistance. The selected conditional kappa remains unchanged.

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  "kind": "HANDOFF",
  "task": "uniform-transfer-audit",
  "base": "20a1e31bc6009b9b3f4d084bbaadc6350e34fcba",
  "scope": "Can the actual rational-unitary Core narrow inverse / broad forward contract be joined to the original core/halo schedule on a single immutable periodic Reference, with retained ordinary-window admission, all core and seam output rows owned exactly once, and a complete analytic plus numerical error budget? The locality lemma is inherited, not a new result.",
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/discussions/4202#discussioncomment-19143431",
  "expiry": null
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[JoyboyBrian]

DONE / FINDING — PR #5159, exact published head ba435a39f69436f2f7ce5cc02f18879c8d8790f5. 2 substantive commits; this scoped investigation is complete and its reservation is released.

Complete actual385-row periodic Reference inverse service pays fullperiodsetup, J276/P33 roundedrecurrence and cache/scratch/outputcoexistence. All148225original entries, actualwrap/hole, source residual and mutationcontrols checked; normal32s andfinalGitfree optimized33.6s pass. This is a fixed-reference host-array service with inherited analytic gap/error, not physicaltape implementation, sparsehalo saving or newkappa.

The PR and package record the exact local validation, source provenance, reproduction commands, finite calibration, remaining obligations and unrun checks. Source and proof review are AI-assisted; no independent human or proof-kernel acceptance is implied. Hosted CI is separate and has not been reclassified by this completion update; use the PR checks for live status. DONE does not imply maintainer or scientific acceptance, a complete supplier or a selected-frontier update.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

Original claimed plan, retained for provenance

CLAIM — complete same-reference periodic rounded inverse service (uniform-transfer-audit)

I will address the actual wrapping-producer obligation left by my R10 handoff in this thread. The changed contract is an explicitly paid whole-period fallback: every output row is produced, without assuming that the original143-row ordinary Core window has a usable halo. I preserve that unfavorable sufficient-halo diagnostic.

The first source fixture is the unchanged Reference(384,1,2,4,18,320), period385 and hole192. I will bind its actual positive cross-period entries, materialize a once-quantized periodic coefficient table, and execute one complete rounded inverse action on signed complex data. The proof must retain coefficient error, every recurrence rounding, intermediate signed widths, the polynomial tail and an independent exact residual certificate against the same immutable reference. Any ideal analytic-reference error is a separate perturbation term under its explicit gap hypothesis.

The complete bill includes coefficient setup and every full-period pass, with explicit input lifetime, current/next buffer reuse and output disposal. This does not claim a sparse-seam tariff, Core speedup, physical tape realization, complete tensor/multiplier or new kappa. Generic Neumann/locality/rounding lemmas are inherited, not the contribution.

Sources: Reference c9db406; Core78c72e563173f3822fcb4efed9ceeccf05630258; locality audit d2479b3; R10 handoff comment19143431. Source base20a1e31bc6009b9b3f4d084bbaadc6350e34fcba. Credit Jose Manuel, Nick Petalas, Cinna Davis, Vlado Vojdanovski, Jeremy Kothe, Mason Hale, Chafik Boukhalfa and retained source contributors.

The20-minute gate is actual seam/hole binding, a funded quantization/tail budget and one complete sparse update against the original entries. I will stop at a missing source binding, unaffordable setup or unclosed numerical/ownership term rather than substitute a toy oracle. A useful precise handoff is an acceptable outcome. Full cached overlap at10:31–10:32UTC covers318 open PRs and4288 Discussion records; metadata search does not prove novelty. R7–R10 completed scopes and the current conditioned-tensor native claim remain separate.

Prepared for JoyboyBrian with substantial OpenAI Codex assistance.

Reservation checkpoint: canonical main a5df48c55ea806d7952b65acc9c76d878ff09b13; priority revision7 and selected result unchanged. Fresh coverage305openPRs (10:49UTC),4290Discussionrecords (10:50UTC),6918HEADnotes, complete reported pagination. Earlier scouting coverage remains historical. Prior source heads stay frozen; no novelty guarantee for unpublished work.

Execution: one initial20-minute feasibility gate within an approximately60-minute scoped session. Two to four substantive commits when warranted; no padded PR. Root coordinates the single heavy-replay slot. Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

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  "kind": "DONE",
  "task": "uniform-transfer-audit",
  "base": "a5df48c55ea806d7952b65acc9c76d878ff09b13",
  "scope": "Can the existing immutable periodic Gaussian Reference supply a complete quantized narrow inverse action on every output coordinate, including cross-period and hole rows, with an independently checked same-reference error and the actual whole-period bill?",
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5159",
  "expiry": null
}

[JoyboyBrian]

DONE / FINDING — PR #5183, exact published head 73660fc53b9910aa25523f37ec7d34e3fd9f083b. 3 substantive commits; this scoped investigation is complete and its reservation is released.

For fixed alpha, the rescaled full Gaussian inverse converges in operator norm to two half-line Volterra operators, with explicit remainder and exact norm prefactor. Growing-alpha uniformity and physical preparation/application remain open.

The PR and package record the exact local validation, source provenance, reproduction commands, finite calibration, remaining obligations and unrun checks. Source and proof review are AI-assisted; no independent human or proof-kernel acceptance is implied. Hosted CI is separate and has not been reclassified by this completion update; use the PR checks for live status. DONE does not imply maintainer or scientific acceptance, a complete supplier or a selected-frontier update.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

Original claimed plan, retained for provenance

DRAFT ONLY — not posted.

CLAIM: two-edge half-line operator limit for the broad Gaussian phase-cell inverse.

I will address the actual boundary-layer implication left open by Ryanson Jonathan #4435 and Nick Petalas #5150, using the exact q-LDU inverse and JoyboyBrian #5103 upper envelope. The target is operator-norm convergence of the rescaled full finite inverse to two decoupled half-line Volterra kernels, including the actual endpoint domains, middle/tail estimates and sharp limit. This is not another proof of theta^(-1/2) order or a finite norm table.

The first source fixture is alpha3,theta1/400,N400,beta0=-1/2. The first20minute gate is a full positive-sum asymptotic, justified half-line norm identity and operator-norm tail domination. I will stop with a precise handoff if only pointwise or numerical evidence is available. Fixed-alpha quantifiers will remain distinct from the production growing-alpha regime; exact-real conditioning does not supply coefficient preparation, rounded prefixes, physical storage/movement, native admission or outer recovery.

Base a5df48c, priority revision7. PR5150 source c16d2fc read in full. Fresh overlap329PR/4339Discussion11:33UTC checked; no same live operator-limit claim found in body search, which is not an exhaustive novelty proof. Credit Jose Manuel/#4193, Vlado Vojdanovski/#1673, Ryanson Jonathan/#4408/#4435, Nick Petalas/#5150 and retained classical q-LDU lineage. JoyboyBrian with substantial OpenAI Codex assistance. No new kappa claimed.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0. Original attribution retained. Claim expires 2026-10-23T13:39:09.347281+00:00.

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  "kind": "DONE",
  "task": "uniform-transfer-audit",
  "base": "a5df48c55ea806d7952b65acc9c76d878ff09b13",
  "scope": "For fixed alpha>0, theta tending to0 and N=floor(1/theta), identify the rescaled checkerboard inverse as two decoupled half-line Volterra operators in operator norm, with quantified tail and finite-boundary errors. Prove or correct the resulting sharp norm limit.",
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5183",
  "expiry": null
}

[JoyboyBrian]

DONE / FINDING — PR #5198, exact published head 6b985d37efd6211f2fcfcdf523ce0a8426c4fb93. 3 substantive commits; this scoped investigation is complete and its reservation is released.

Actual Axis and encoded old-kernel setup are reused across protected pages consumed by original Core, with pending-output guards, cache eviction, exact ownership and complete source-machine backing/cost bounds. Production grids and complete tensor/CRT composition remain unrun.

The PR and package record the exact local validation, source provenance, reproduction commands, finite calibration, remaining obligations and unrun checks. Source and proof review are AI-assisted; no independent human or proof-kernel acceptance is implied. Hosted CI is separate and has not been reclassified by this completion update; use the PR checks for live status. DONE does not imply maintainer or scientific acceptance, a complete supplier or a selected-frontier update.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

Hosted CI follow-up: CI on Ubuntu24.04 CPython3.12.15 timed out at the exact360-second normal replay after both guard checks passed. This metadata-only change raises normal and optimized caps to585 seconds each, preserving15-second guard caps and1200-second total. Scientific code, proof, receipts and pinned sources are unchanged. Prior local optimized241.91-second success is not Linux completion evidence. No scientific replay was rerun for this change; hosted completion remains unverified.

Original claimed plan, retained for provenance

CLAIM — actual reusable Gaussian old-kernel setup through protected page consumption

I will investigate a persistent physical preparation owner for PR #3851's actual Gaussian page/Core interface. Preparation.run_explicit rebuilds its Axis each call, and Window.prepare_schur repeatedly inverts identical old truncated-kernel blocks before erasing their setup. The intended change retains one genuine frozen Axis and reuses encoded old-kernel inverse setup keyed by actual interior length across a window stream, while emitting the unchanged protected page and calling actual Core.run.

The first 20-minute gate must establish the exact cache key, authenticated source activation, and a genuine second-window hit with unchanged page bytes. Start from the source axis (13,14,1,1), work precision24/target16, window (1,5,2,3), then shifted and period-crossing cases. Completion requires actual Core outputs and the whole paid bound: initial setup, hit/miss lookup and copies, encoded rational cost, page/LU regeneration, retained backing, borrowed heads, invalidation, protected output consumption, and final disposal. A host matrix dictionary, whole-page memoization, synthetic3x3 substitute, or cache without a demonstrated complete consumer/budget change is a stop condition, not a contribution.

This is distinct from #3872's host principal-inverse cache and #5119's comparison barrier; neither is already an encoded Schur-page supplier. It excludes the separate #5159 periodic sparse-MVM supplier and current CRT repair. I retain the production approximation premises and do not claim lower precision or a complete multiplication theorem. Credit Jose Manuel's physical Gaussian core, Dugongue's cache precedent, and their retained analytic/word/factorization sources. Fresh overlap coverage:329 open PRs/4339 discussion records; base a5df48c55ea806d7952b65acc9c76d878ff09b13. No production-size fixture will be run; any replay over30s requires the shared serialized slot.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0. Claim expires 2026-10-23T13:38:27.718209+00:00. Unknown remaining supplier costs stay explicit.

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  "kind": "DONE",
  "task": "uniform-transfer-audit",
  "base": "a5df48c55ea806d7952b65acc9c76d878ff09b13",
  "scope": "Can a persistent preparation owner retain one genuine Axis and reuse exact old truncated-kernel inverse setup across overlapping windows while emitting the unchanged structured page consumed by actual Core.run, with all cache ownership and lifecycle costs paid?",
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5198",
  "expiry": null
}

Hosted follow-up verified: exact head 6b985d37efd6211f2fcfcdf523ce0a8426c4fb93 passed the complete registered Ubuntu24.04/CPython3.12.15 replay and unchanged-input check, job120426206087, completed2026-10-23T13:08:08Z. This supersedes the earlier pending-status note; proof and production-scale limits are unchanged.

[JoyboyBrian]

DONE / FINDING — PR #5190, exact published head 13b76bd3c76b315a198136f3ff9d5688ec196f64. 2 substantive commits; this scoped investigation is complete and its reservation is released.

A complete fixed385-row periodic Gaussian update now pays coefficient layout and cyclic delivery, signed80bit accumulation, once-per-output truncation and all ownership cleanup.449445retained cells and exact structural invoice; original coefficient generation is separately authenticated,275update composition symbolic.

The PR and package record the exact local validation, source provenance, reproduction commands, finite calibration, remaining obligations and unrun checks. Source and proof review are AI-assisted; no independent human or proof-kernel acceptance is implied. Hosted CI is separate and has not been reclassified by this completion update; use the PR checks for live status. DONE does not imply maintainer or scientific acceptance, a complete supplier or a selected-frontier update.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

Original claimed plan, retained for provenance

CLAIM — Physical periodic Gaussian update with unchanged final rounding

Question: Can the actual385-row periodic update u-trunc(Ahat*y) use fixed resident tape roles, cyclic coefficient/input delivery and exact signed accumulation while preserving the original once-per-output truncation and paying all movement and cleanup?

Exact starting sources: PR5159 ba435a3; original Reference(384,1,2,4,18,320); retained scalar physical arithmetic.

First gate: one complete update at p33 over all385rows, eight offsets, actual wrap couplings and hole192. Offset-major preparation must be charged; no host gather or early per-offset rounding. Prove exact update equivalence and symbolic275-step composition only after that equivalence. Full275-update physical replay, host interval coefficient generation, general geometry/precision and all-size supplier remain separate.

This investigation starts with a20-minute actual-source feasibility gate, then completes one scoped implication or releases a precise handoff. Complete producer/consumer ownership and costs remain explicit; unknowns are not zero. Source proof, finite checks and physical/supplier claims will be distinguished. No frozen predecessor or selected-frontier change; no merge authority.

Fresh canonical base a5df48c55ea806d7952b65acc9c76d878ff09b13, priority7. Claim checkpoint covers329openPRs at11:32UTC,4339Discussion records at11:33UTC and6918HEADnotes; active chain-ring, exact-annihilator and persistent R437leaf claims were excluded. Coverage is advisory and omits unpublished work.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance. Original contributor attribution and Apache-2.0 notices retained. Claim expires 2026-10-23T13:36:25.599444+00:00.

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  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "uniform-transfer-audit",
  "base": "a5df48c55ea806d7952b65acc9c76d878ff09b13",
  "scope": "Can the actual385-row periodic update u-trunc(Ahat*y) use fixed resident tape roles, cyclic coefficient/input delivery and exact signed accumulation while preserving the original once-per-output truncation and paying all movement and cleanup?",
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5190",
  "expiry": null
}

[JoyboyBrian]

DONE / FINDING — PR #5216, exact published head 1ea9efdc1d1d36757a5491bf7fdaa93597dcea67. 2 substantive commits; this scoped investigation is complete and its reservation is released.

A source-compatible global weighted power bound includes every periodic crossing and rational-weight error, reducing the complete original385-row inverse from276terms/275updates to120terms/119updates at unchanged33bitgrid. Full gauge, numerical/analytic budget and host invoice retained; no production-family or physical supplier admission.

The PR and package record the exact local validation, source provenance, reproduction commands, finite calibration, remaining obligations and unrun checks. Source and proof review are AI-assisted; no independent human or proof-kernel acceptance is implied. Hosted CI is separate and has not been reclassified by this completion update; use the PR checks for live status. DONE does not imply maintainer or scientific acceptance, a complete supplier or a selected-frontier update.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

Original claimed plan, retained for provenance

CLAIM — source-bound weighted power contraction for the periodic narrow Gaussian inverse

I will investigate whether the actual periodic narrow matrix admits a global diagonal majorant that removes the target-precision multiplier from its sufficient Neumann iteration count. This changes the power estimate used by the consumer, rather than sharpening the constant in the existing scalar row-gap bound.

The starting source is Jose Manuel's unchanged gaussian_reference.py at c9db406, retained by PR5159's whole-period service at ba435a3. The first fixture remains Reference(384,1,2,4,18,320), including all385 rows, hole192 and the genuine wrap entries. I will test a global weight based on exp[-pi alpha²(1/4-beta_i²)/theta], with both rational weight approximation and downward coefficient quantization retained. The proposed power bound has an explicit exponential-in-1/theta prefactor and a contraction independent of theta; that prefactor must be paid rather than omitted.

The20-minute gate is a written majorant valid across every phase-cell and period crossing, followed by an exact sufficient iteration certificate on the original fixture. Success requires joining the new tail to the unchanged range, coefficient and every data-rounding allowance, and pricing the resulting complete consumer work. The gauge is for the proof only; applying a numerically huge gauge to data would introduce a different precision/physical contract. If the approximation or wrap terms defeat the claimed majorant or the complete error/work allowance does not improve, I will release a precise handoff rather than publish a fitted spectral estimate or another scalar-tail constant.

This does not reopen the failed local143-row halo, alter the physical update's truncation rule, extend the broad inverse half-line asymptotics, or claim a complete multiplier or new kappa. Any production-parameter asymptotic consequence requires separately verifying its actual theta/alpha/target family. Initial work is source-level and exact rational reasoning; heavy regeneration requires a separate resource slot.

Base ec2780e, priority7. Predecessor credit remains with Jose Manuel, JoyboyBrian, Vlado Vojdanovski, Jeremy Kothe, Cinna Davis, Mason Hale, James Reed and the retained Gaussian/precision lineage. Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

Claim checkpoint: canonical main ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c, priority 7; complete cached coverage of 314 open PRs at 12:20 UTC, 4388 Discussion records at 12:21 UTC and 6996 HEAD notes. Earlier scout counts are superseded by this checkpoint. Coverage is advisory and excludes unpublished work. This bounded investigation is assigned to JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0 and predecessor attribution retained. Claim expires 2026-10-23T14:31:05.571389+00:00.

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  "kind": "DONE",
  "task": "uniform-transfer-audit",
  "base": "ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c",
  "scope": "Prove or refute a source-compatible diagonal majorant for every actual narrow periodic entry, then replace the scalar row-gap Neumann tail with a theta-independent weighted contraction while retaining the exponential gauge prefactor and complete numerical/consumer bill.",
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5216",
  "expiry": null
}

[aliromman]

DONE — PR #5260, exact head 65cdd6b36e002ce9d13bd96dbc504da90f77f2c9. Claim released; priority 7.

The original 385-row periodic Gaussian Reference now has a complete certified direct consumer on the unchanged 33-bit input/output grid. A written all-input proof funds coefficient error, exact factor residual, both rounded solves, final grid truncation and uniform integer word size. The consumer uses 1,398 products versus 731,136 for the retained 119-update recurrence. Factor preparation adds 319 products, 349 divisions and 668 certificate products. Local median preparation+solve is 0.00823s versus 0.19839s; retained-factor solve is 0.00320s. Original source generation took 77.27s and remains paid. These are consumer improvements, not equal cold-start or multiplication speedups. Hosted CI is pending; no new kappa or physical/production-family admission is claimed.

Normal and optimized registered checks, all 148,225 original source entries, exact original-source residual, 105 independent rational right-hand-side checks, eleven control groups, four actual code mutations, exact source pins, base ancestry, committed bytes and Git-free read-only optimized export replay pass. Published body and rendered HTML match the reviewed draft. Full-history wrapper, human/proof-kernel review, physical tape and selected cold replay remain unrun.

Canonical main was refreshed to ff3cc58e370d0f678e529246f8c73e8094e18fcf before this submission; guidance, selection and consumed sources were unchanged. Final live overlap covered 323 open PRs, 4524 Discussion records and 134 of 7091 local notes. Its focused matches were this claim and task. Sparse omitted notes, arbitrary diffs and unpublished work remain limits.

Credit Jose Manuel for the Reference; JoyboyBrian for PR #5159 and PR #5216; Dugongue for the prior native band/Woodbury algorithm in PR #4554; Ryanson Jonathan for its audit; and their retained sources. Classical elimination and the existence of a periodic solver are not new claims. The actual large-row-norm reference consumer and its certificate are the added result. Prepared for Ali Romman with substantial OpenAI Codex assistance.

Open handoff: bind this factor owner to one admitted physical caller, retaining its 33-bit boundary, and charge preparation, ordered tape delivery, factor storage and restoration. A production-family proof must bound fill, certificate success and word widths across its actual parameters. The selected kappa remains 60403/20000000. No further implementation claim is held.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "uniform-transfer-audit",
  "scope": "Certified cyclic sparse LU consumer for the actual 385-row periodic Reference outside the predecessor small-norm contract",
  "base": "ff3cc58e370d0f678e529246f8c73e8094e18fcf",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5260"
}

[aliromman]

DONE — the sharp production-domain quadratic Neumann coefficient is 179.

This answers Open next question 1 of Nick Petalas's PR #5145, with the question and hypotheses pinned on current main. For every integer p >= 101,

32p (1-1/(16p))^(179p^2) < 2^(-16p).

The integer coefficient 178 fails at p=101, so 179 is the least nonnegative integer c that works in J=cp^2 on this whole domain. This is a scalar locality prescription. It improves 226p^2 on the specified production domain without contradicting #5145's sharp all-positive-precision result. It does not change a caller or kappa.

Proof of sufficiency. The convergent positive atanh series gives
693/1000 < log(2) < 13863/20000. Since log(1-x)<-x,

log[32p (1-1/(16p))^(179p^2) 2^(16p)]
  < log(32p) - (971/10000)p.

At p=101, log(3232)<9, because e>8/3 and (8/3)^9>3232, whereas 971*101/10000>9. The derivative of the last expression is 1/p-971/10000<0 for every real p>=101. This proves the strict bound on the entire stated domain.

Proof of necessity. The elementary integral bound -log(1-x)<=x/(1-x) gives, at p=101,

log[3232 (1-1/1616)^(178*101^2) 2^1616]
 >= log(3232) + 1616 log(2) - 178*101^2/1615
 > 7 + 1616*(693/1000) - 178*101^2/1615
 > 0.

Here e<3 and 3^7<3232. The last comparison is rational. The tail decreases with c, so every c<=178 also fails.

The following self-contained exact checker ran under CPython 3.12.13 in normal and optimized modes. It verifies the proof constants, then uses a separate rational logarithm enclosure to certify 25 positive instances and the coefficient/domain controls. Its finite checks corroborate the written universal proof. Copy the block to a file and run python3.12 -I -B FILE.py and python3.12 -I -O -B FILE.py.

from fractions import Fraction as F

def check(x, why):
    if not x: raise ValueError(why)

def log_unit(x, n=20):
    z=(x-1)/(x+1)
    check(0<=z<=F(1,3),'log reduction')
    lo=2*sum((z**(2*k+1)/(2*k+1) for k in range(n)), F(0))
    hi=lo+2*z**(2*n+1)/((2*n+1)*(1-z*z))
    return lo,hi

def log_bounds(x):
    k=0
    while x>=2: x/=2;k+=1
    lo,hi=log_unit(x);a,b=log_unit(F(2))
    return lo+k*a, hi+k*b

lo,hi=log_unit(F(2))
check(F(693,1000)<lo<hi<F(13863,20000),'log2 bounds')
a=F(179,16)-16*F(13863,20000)
check(a==F(971,10000),'decay coefficient')
check(F(8,3)**9>32*101,'base exponential bound')
check(101*a>9 and a>F(1,101),'base and derivative')
check(F(178*101*101,16*101-1)<16*101*F(693,1000)+7,'178 exclusion')
check(3**7<32*101,'lower logarithm')
checks=0
for p in list(range(101,121))+[128,256,1024,10000,1000000]:
    low,high=log_bounds(F(32*p)); l2,h2=log_unit(F(2)); lc,hc=log_unit(F(16*p,16*p-1))
    upper=high+16*p*h2-179*p*p*lc
    check(upper<0,'179 finite corroboration')
    checks+=1
p=101;low,high=log_bounds(F(32*p));l2,h2=log_unit(F(2));lc,hc=log_unit(F(16*p,16*p-1))
check(low+16*p*l2-178*p*p*hc>0,'independent 178 sign')
low,high=log_bounds(F(32));lc,hc=log_unit(F(16,15));check(low+16*l2-179*hc>0,'p=1 domain control')
print('PASS_PRODUCTION_NEUMANN coefficient=179 excluded=178 domain=p>=101 exact_log_cases='+str(checks)+' controls=2 no_new_kappa')

Expected marker: PASS_PRODUCTION_NEUMANN coefficient=179 excluded=178 domain=p>=101 exact_log_cases=25 controls=2 no_new_kappa.

Credit: Nick Petalas (@petalas) supplied the explicit production-domain question and the all-positive-precision optimum in PR #5145. Jose Manuel (@jmbarrancoml) supplied the contraction, prefactor and Gaussian caller in PR #3851; James Reed (@jwr404) supplied the earlier finite tail checks in PR #3864. The logarithm inequalities are classical. Their predecessor notices remain unchanged.

Limits and next step: the contraction, inverse norms, rounded evaluation, setup, movement, ownership and full error sum remain inherited obligations. The next useful step is to bind 179p^2 and its locality margin to the actual caller; that integration was not attempted here. No predecessor suite, selected cold replay, human proof review or proof assistant was run. Base ff3cc58, priority revision 7. Fresh main, live PR search, the current complete note inventory and the 14:14 UTC Discussion snapshot were checked; arbitrary diffs and unpublished work remain outside overlap coverage. Prepared by Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. This short scoped result is complete and unreserved.

[aliromman]

DONE — PR #5284, exact head 72269696a9c5ae5b1c0b469afdd3db08080635cf. Claim released; priority revision7.

A complete integer cyclic-band solver now has an all-size proof with variable row gap delta>0. Exact accumulation and one rounding per factor entry give logarithmic guards in N and inverse delta, at most4NW^2 factor products, at most8NW products per complex solve, and O(N(W+1)) numeric storage. The proof includes factor error, both solves and a uniform word range. The 4,097-row width-four, gap1/4096 fixture meets2^-20 error with60 fractional bits; proved signed width151, observed peak132.

Both registered modes,72 exact complex RHS comparisons, nine scaling cases,129 symbolic fill shapes, eight controls and three larger exact-residual cases pass. Four code mutations fail in both modes. Source pins, committed bytes, official export and both read-only Git-free replays pass with all nine files unchanged. Public links and published body/rendered-HTML readback pass. Hosted CI is pending; no independent human/proof-kernel review, full-history wrapper, selected cold replay, Lean or Docker claim.

Credit Dugongue #4554, Ryanson Jonathan's scoped audit, classical Wilkinson/Higham diagonal-dominance analysis, the noncyclic batched-window predecessor and #5260's fixed consumer. Matrix formation, source approximation, physical layout and the outer invoice remain open. No production-family admission, kappa or frontier change. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.

Original claim follows for provenance.

CLAIM — cyclic band solves with a variable diagonal-dominance gap.

Status: active, 2026-10-23 15:12 UTC; priority revision7. Base is current main ff3cc58. This takes the production-family limitation left by PR #5260, and complements Dugongue's PR #4554 constant-gap band/Woodbury interface.

Question: for every real dyadic cyclic band matrix with positive diagonal, row gap delta>0 and row norm at most M, can one prove a fixed-point direct solver with factor work O(N W^2), retained factors O(N W), and an explicit internal precision depending logarithmically on N and polynomially on inverse delta? I will use natural-order cyclic fill, exact integer accumulation with one rounding per factor entry, a uniform factor residual and a full complex-RHS solve/word-size bound. The initial proof target permits delta to shrink, including delta=1/(16p); it does not assume a constant near-identity norm or a small Woodbury correction.

Changed scope from uniform-transfer-audit: prove a reusable numerical interface instead of repeating the fixed385-row replay or asserting production coefficient generation. Success requires a quantified proof, a complete integer implementation, an independent rational solver on small cases, actual cyclic fill/cost checks and rejecting controls. Stop at that interface; no complete tape/native tariff, coefficient-generation bound, Gaussian-family admission, new kappa or selected-frontier change. The factor/solve arithmetic budget is the optimistic local budget; physical storage, coefficient setup and outer composition remain caller obligations. This is a bounded expansion supported by the first complete direct consumer, not another finite parameter census.

Current open PRs, Discussions, relevant main proofs and targeted closed-PR searches were checked. The existing ordinary-band constant-gap proof and its independent audit are predecessors, not missing results. Classical diagonal-dominance and Gaussian-elimination stability retain their credit; no global novelty claim. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. Claim expires 2026-10-23T17:12:00Z.

[aliromman]

DONE — PR #5288, head cd33c723408bb98875211ee3145582c507881f29. The structure-dependent forward/inverse/noise theorem and exact checker are public. Registered CPython 3.12.14 normal/optimized replay passes 24 circuits, 192 prefixes, 192 noisy steps, 16 large-budget circuits and five controls. Published body/rendered text, links and structure, and committed bytes were checked. Hosted CI is pending. No kappa or selected-frontier change. This scoped investigation is complete and its claim is released.

Original claim

CLAIM — coefficient errors that preserve elementary shear support.

Question: can the simultaneous prefix/inverse perturbation bound dispense with its small-total-error invertibility premise when every exact gate and rounded gate have the same elementary-shear support? The source is ywz65's simultaneous gate rounding note, which retains Vlado Vojdanovski's PR1673 circuit and prefix estimates as hypotheses.

I will prove or refute a finite ordered-product bound, including inverse prefixes and additive arithmetic noise, and test exact rational noncommuting shear words. The changed premise preserves the diagonal ones and structural zeros; arbitrary entry perturbations remain outside this extension. The expected artifact is a self-contained theorem and standalone exact checker. Stop at this contract; no new all-size compiler, physical arithmetic kernel, supplier, or kappa is claimed.

Base: canonical main ff3cc58; priority revision 7. This refines the precision interface of uniform-transfer-audit; it does not purport to close a #4100 caller. I checked the live handoff hub, 345 open PR bodies, 4,540 Discussion records, and 247 pinned precision-related README/TASK notes. General historical diffs and unpublished work remain outside coverage. The active FFT, rounded matching-layer, variable-gap and comparison-barrier scopes are distinct.

Prepared by Ali Romman with substantial OpenAI Codex assistance. Claim expires 2026-10-23T18:00:00Z. This is result 1 of a consecutive sequence; no later question is reserved.

[aliromman]

DONE — Complete proof and exact checker. Ordinary nearest coefficient rounding realizes quadratic closed-word amplification, including linear repeated-loop growth. A fixed-rational word works at every dyadic precision; a dyadic family gives a second construction. Both CPython 3.12.14 modes pass 288 cases and 6,876 literal prefixes. Public proof/code readback passes. Symbolic loop cancellation and data rounding are outside the lower-bound claim. No new kappa. Claim released.

Original claim

CLAIM — deterministic nearest rounding in the closed-word guard.

I take Open next question 2 of Nick Petalas's PR #5153, pinned question. The predecessor proves sharp powers for arbitrary bounded coefficient errors and explicitly leaves a specified deterministic rounding policy open.

Question: does componentwise nearest coefficient rounding still force quadratic prefix-norm amplification for exact closed words? I will test a dyadic three-shear cancellation loop, conjugated by bounded dyadic scale gates. The three coefficients avoid ties. Repeating the loop should test the gate-count factor without increasing the exact prefix bound. Expected artifact: an all-size construction/proof and exact rational rounded products. Stop there; no production caller, arithmetic-rounding or physical-cost claim.

Base ff3cc58, priority revision 7. Fresh main, open PR/Discussion snapshots and the actual predecessor question were checked. The active native and matching-layer work is separate. Claim expires 2026-10-23T18:25:00Z. Prepared by Ali Romman with substantial OpenAI Codex assistance. This is result 4 in a consecutive sequence, with results 1–3 complete.

[aliromman]

DONE — PR #5326, exact head 234142af90b5e7afe0b33c4b78c3c0bd622d5fc3. The source/consumer interface is complete at its conditional scope; claim released.

The unchanged narrow periodic Gaussian Reference now has an all-parameter direct consumer for every tuple that funds its explicit gap and analytic-error gates. The proof binds selected-coordinate cyclic support, all hole identities, coefficient rounding, variable-gap factor/solve precision, final output rounding and the analytic perturbation. It removes the fixed 16-bit gap floor. Source generation and caches, full target scans, gather/scatter, retained factors and cleanup remain paid. There is no production parameter funding, physical tape or complete kappa claim.

Both registered CPython 3.12.13 modes pass on eight actual sources, 103,717 original matrix entries, 24 complete signed complex inputs, six separate dense rational solutions, eleven rejection controls and seven small-gap checks. Four code mutations fail in both modes. All finite targets have an original-H internal residual certificate plus separately charged final rounding. The final-grid residual bound is too loose in two cases, and the receipt records that fact. The 258-row case uses 492 factor and 1,314 solve products; source setup remains separate.

Committed bytes, every declared pin's direct ancestry, archive integrity, both read-only Git-free exports and published body/rendered-HTML readback pass; all 22 exported files stay unchanged. Hosted run 39995095815 passed planning and the registered Linux replay for published head 234142af90b5e7afe0b33c4b78c3c0bd622d5fc3. Hosted execution is separate from independent scientific review. No full-history wrapper in this shallow checkout, independent human/proof-kernel review, analytic backend acceptance, production native/Tensor, selected cold replay, Lean, Docker or complete kappa calculation is claimed.

Credit Jose Manuel, JoyboyBrian (#5159/#5216), Dugongue (#4554), Ryanson Jonathan and the retained Gaussian/interval lineage. The fixed-source predecessor is #5260; the unchanged all-size solver is #5284. Exact source and proof links are in the PR. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.

Next handoff: pin a production parameter policy, fund its actual four error terms, and price source bit work together with ordered physical delivery. The selected frontier is unchanged. Final refreshed main remains ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority 7; the 17:25 UTC final overlap check found no competing same-scope result. Arbitrary diffs and unpublished work remain outside that coverage.

{"schema":"integer-mult-coordination/v1","kind":"DONE","task":"uniform-transfer-audit","scope":"All-parameter source-bound cyclic direct consumer for the narrow periodic Gaussian Reference","base":"ff3cc58e370d0f678e529246f8c73e8094e18fcf","expiry":null,"artifact":"https://github.com/CrocSwap/integer-mult-bounds/pull/5326"}