Audit Gaussian chirp domain and exact block budgets

research/gaussian-chirp-domain-audit/CLAIM.md

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+# Sprint claim: Gaussian chirp domain and block budgets
+
+- Track: adversarial mathematical audit; OS-random assignment seed **64**
+  (`secrets.randbelow(100)`, drawn 2026-10-13 03:37 UTC).
+- Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+- Claimed at: 2026-10-13 03:43 UTC. Expected finish: 04:17 UTC.
+- Exact object: the hypotheses, integer windows and precision budgets of
+  `lem:chirped-gaussian`, as inserted by `scripts/make_fast_gaussian_patch.py`;
+  compare the domain caveat in PR #371 at
+  `988ef67986c313445936fc7103908b6163b9d47e`.
+- Method: follow generated source rather than the unmodified upstream header;
+  derive rational uniform bounds and use a separate standard-library checker
+  for bounded integer cases and deliberately invalid controls.
+- Deliverable: one source-pinned audit note, reproducible exact checker and
+  receipt. No new kappa or full analytic/tape theorem claim.
+- Time allocation: 40 minutes from 03:37 UTC; orientation ends before 03:45,
+  verification and handoff begin no later than 04:07.
+
+## Selection and overlap
+
+Three narrow candidates were considered, ordered as follows:
+
+0. Recheck the missing exact infeasibility evidence behind PR #620's P111
+   decoder floor. The review already identifies absent proof logs and
+   nonportable scripts; a portable solver audit would need more setup.
+1. Audit the Gaussian chirp domain and block budgets left conditional in
+   PR #371. This is an older unreviewed contribution with a small exact seam.
+2. Audit the normalized zero/power-of-two boundary in PR #573's eight-tape
+   handoff. The sources have substantial scoped Lean checks already.
+
+Seed `64 % 3 = 1` chooses candidate 1. It also determines the finite residue
+sampling order; the claim is useful because it checks a different obligation,
+not because it uses a different random seed.
+
+At selection, GitHub returned 491 open PRs. The central ledger marks #371
+unreviewed. Its final-grid counterexample and correction are predecessor work,
+not claimed here. PRs #479, #481 and #489 address different Gaussian mechanisms.
+The repository had no pinned issues and its sole open issue was #650
+(Discussions), so this claim is recorded locally. Pinned issues and #371's
+head/comments were refreshed immediately before recording the claim.
+
+## Result update
+
+Pending verification and handoff.

research/gaussian-chirp-domain-audit/INDEPENDENT-REVIEW.md

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+# Independent source and arithmetic review
+
+Reviewed on base `66849e1475c930fd7732b1a0067ffee7e1e95f60`, assignment
+seed 64. This review covers the generated Gaussian chirp domain, integer
+window containment and the two displayed working-precision budgets. It does
+not certify the analytic or tape-complexity theorem and claims no new kappa.
+
+## Source composition resolves the apparent domain mismatch
+
+The unmodified header in
+[`upstream/build/sections/07-resampling.tex`, lines 53–59](../../upstream/build/sections/07-resampling.tex#L53-L59)
+has the older condition `theta > p/alpha^4`. Reading that header together
+with the new chirp note would indeed leave the latter's upper bound on theta
+unexplained. They are not the composed manuscript.
+
+[`scripts/make_fast_gaussian_patch.py`, lines 23–33](../../scripts/make_fast_gaussian_patch.py#L23-L33)
+replaces that condition by `alpha^-2 <= theta <= 1/4`, while retaining integer
+`p > 100`, integer `2 <= alpha < sqrt(p)`, coprime integer `2 <= s < t < 2^p`,
+and `sigma = t/s = 1+theta`. The chirp note is inserted by the same function
+at line 160; dispatch occurs at lines 300–312. Calling `patched_files()` on
+the exact base produced these conditions at generated
+`build/sections/07-resampling.tex:59–65`, and the chirp lemma refers back to
+them at generated lines 392–400. Thus `sigma <= 5/4 < 2` is available.
+
+The generated assembly also supplies this domain: source
+[`scripts/make_fast_gaussian_patch.py`, lines 232–263](../../scripts/make_fast_gaussian_patch.py#L232-L263),
+generated `build/sections/08-assembly.tex:105–115,160–174`, gives
+`alpha=floor(sqrt(b/(8d)))`, `eta=1/(4d)`,
+`eta < theta_i < 1/(2d-1) <= 1/4` for `d>=3`, and
+`alpha^2 theta_i > b/(64d^2) >= 1` eventually. This verifies the particular
+domain transfer; it does not independently prove the prime-existence or
+asymptotic setup claims used by that assembly.
+
+PR [#371](https://github.com/CrocSwap/integer-mult-bounds/pull/371), immutable
+head `988ef67986c313445936fc7103908b6163b9d47e`, correctly keeps its rounding
+correction conditional on the other proof obligations. The fetched historical
+head's generator and upstream section 07 were compared with this audit's
+base and match byte for byte; the generator SHA-256 is
+`ce2013dc3a674b920ddacf25fd17329b262885af0a19279e084238318dff4760`.
+The note differs only by the proposed correction. Therefore the PR body's
+additional assertion that the referenced no-sort header is broader does
+**not** describe either its own composed construction or the construction at
+this audit's base. It reads the unmodified header without its replacement.
+
+## Universal arithmetic checks
+
+The following arguments use every integer `p>=101` and every integer
+`2<=alpha<sqrt(p)`, with `alpha^-2<=theta<=1/4`. They are written arithmetic
+arguments, not a formalization. The one-block error enclosure itself is an
+inherited premise, not re-proved here.
+
+- **Output and input windows.** In the S' block, put
+  `c=floor(k0/sigma)` and `j0=c-m`, where `m=ceil(sqrt(p))*alpha`.
+  For `0<=b<m`, every integer `j` with `|j-(k0+b)/sigma|<m` lies strictly
+  between `j0` and `j0+3m`; hence it is included in the `3m+3` input records.
+  In the E block, an output `ell=ell0+r`, `0<=r<m`, uses every required
+  `ell+h`, `|h|<=m`, inside `[ell0-m,ell0+2m]`, the `3m+1` input records.
+  For the selected outputs, monotonic nearest-integer rounding gives
+  `0<=q_(ell0+r)-q_ell0<=ceil(sigma*r)<=2r<=2m-2`;
+  this fits the `2m+2` correlation outputs. These are integer containments,
+  not a numerical Gaussian-tail test. See the
+  [note, lines 160–169 and 190–216](../../notes/fast-gaussian-resampling.tex#L160-L216).
+
+- **Exponential growth enclosures.** The elementary bounds `pi<22/7` and
+  `ln(2)>2*(1/3+1/81)=56/81` imply `pi/ln(2)<891/196`.
+  Consequently `(891/196)*(121/400)<7/5`,
+  `(891/196)*(25/4)<29`, and `(891/196)/4<57/50`.
+  These respectively justify the displayed S' `Gamma=ceil(1.4p)`,
+  E `Gamma=29p`, and E input scale `B=ceil(1.14 alpha^2)` from the
+  exponents in the [note, lines 170–205](../../notes/fast-gaussian-resampling.tex#L170-L205).
+  The S' step uses `sqrt(p)+1 < 1.1 sqrt(p)` for `p>100`; the E step uses
+  `alpha*(2m+2)<=sqrt(p)+4alpha<5sqrt(p)`. Both Gammas satisfy `Gamma<=2P`.
+
+- **S' precision.** `L_A=3 alpha ceil(sqrt(p))+3 < 4p` for `p>=101`.
+  For each `p=101,102,103,104`, `4p<=2^(floor(p/4)-1)`. Increasing p by
+  four doubles the right side and adds only 16 to the left, proving the
+  same inequality in all four residue classes. Thus
+  `ceil(log2 L_A)<=p/4`, and the block requirement is at most
+  `p+(7p/5+1)+p/4+11=(53/20)p+12<3p`.
+
+- **E precision.** `m=ceil(sqrt(p)/(2alpha))<=sqrt(p)/4+1` implies
+  `L_A=3m+1<=3sqrt(p)/4+4<=p`.
+  Hence `ceil(log2 L_A)<=p-1`. Because B is an integer,
+  `ceil(log2(F L_A))=B+ceil(log2 L_A)` for `F=2^B`.
+  Using `B<=1.14p+1` gives the source's bound
+  `p+29p+(1.14p+1)+(p-1)+11=32.14p+11<34p`.
+  The weaker printed bound `F<=2^(1.15p)` alone would instead give
+  `32.15p+10`; that also suffices, but is not the derivation of `32.14p+11`.
+  See the [note, lines 197–213](../../notes/fast-gaussian-resampling.tex#L197-L213).
+
+- **PR #371's corrected final extraction.** Conditional on its internal
+  enclosure `h/4`, the exact normalization by `2alpha>=4` leaves at most
+  `h/16`, and precision-`3p` rational division adds less than
+  `sqrt(2)*2^(-3p)`. Their sum is below `h/4` for `p>=101`.
+  Since `sqrt(2)<3/2`, the complete displayed budgets are strictly below
+  `(1+1/4+3/2)h<3h` for S' and `(3+1/4+3/2)h<5h<(p/3)h` for E.
+  This corroborates the propagation of the predecessor correction; its
+  original final-grid counterexample is not a new finding of this audit.
+
+## Independent calculation and limits
+
+A separate Python standard-library calculation, without importing the
+production numerical verifier, checked all 168,093 pairs
+`101<=p<=4096`, `2<=alpha<=floor(sqrt(p-1))`. Integer square roots located
+both windows; integer `bit_length` computed ceiling logarithms; rational
+arithmetic checked the three exponential-growth bounds above. It checked
+`L_A<4p` or `L_A<=p`, both precision requirements, `B<=1.15p`, and `m<2p`.
+All passed. Minimum integer precision slack was 40 bits for S' at
+`(p,alpha)=(101,8)`, and 277 bits for E at `(101,10)`. This finite replay is
+supporting evidence only; the preceding arguments establish the stated
+uniform arithmetic claims.
+
+The calculation is retained in
+[`independent_budget_check.py`](independent_budget_check.py), independently
+implemented from the main checker. From the repository root run:
+
+```sh
+python3 -B research/gaussian-chirp-domain-audit/independent_budget_check.py
+```
+
+Expected first line:
+
+```text
+PASS_INDEPENDENT_GAUSSIAN_BUDGETS pairs=168093 controls=5
+```
+
+The following JSON line records the fixed domain, stopping rule and minimum
+slacks above. Five negative controls reject `p=100`, `alpha^2=p`, the two
+insufficient sufficient-precision choices `P=2p` and `P=31p`, and an E window
+one integer too short at `(p,alpha)=(101,2)`. Runtime checks remain active
+under `python3 -O`; the independent script imports no repository verifier.
+
+This review has not independently re-established Gaussian tail estimates,
+transcendental approximation algorithms, exact packed multiplication,
+multitape movement, the full Neumann theorem, or the selected multiplication
+frontier. In particular, absence of a domain mismatch does not remove the
+genuine final-grid defect that PR #371 addresses.
+
+Credit: eumemic's fast-Gaussian construction; icekylinx's shifted-input error
+enclosure; Harvey and van der Hoeven's analytic interfaces; and Artus
+Krohn-Grimberghe / PR #371's disclosed GPT-6 Pro and Codex rounding work.
+Prepared with substantial OpenAI Codex assistance as a separate source and
+arithmetic review in Effy's research sprint. No independent human peer review
+or formal proof is claimed.

research/gaussian-chirp-domain-audit/README.md

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+# Audit: the generated Gaussian chirp domain closes PR #371's caveat
+
+**Research note and independent mathematical audit; no new kappa.** The
+generated Gaussian construction supplies `alpha^-2 <= theta <= 1/4` to
+`lem:chirped-gaussian`. Its integer windows and stated working precisions
+`P=3p` and `P=34p` satisfy the audited arithmetic obligations throughout
+that domain. The broader header left in `upstream/` is not the header used
+by this construction.
+
+This resolves the specific domain caveat in the body of
+[PR #371](https://github.com/CrocSwap/integer-mult-bounds/pull/371) at
+`988ef67986c313445936fc7103908b6163b9d47e`. That PR's separate final-grid
+rounding defect remains genuine and its correction is **not** incorporated
+by this audit. Passing this package does not certify the containing lemma,
+the selected construction or the full multiplication theorem.
+
+## Exact source and statement
+
+Base commit: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+Assignment seed: **64**, adversarial mathematical audit.
+The accepted certificate at this base is #626 with #543,
+`kappa=51110230636731/31250000000000000`; it is unchanged.
+
+The audited statement is the use of the no-sort domain by
+`lem:chirped-gaussian`, specifically its block windows, amplification
+ceilings and sufficient precision inequality
+
+```text
+P >= p + Gamma + ceil(log2(F L_A)) + 11.
+```
+
+The [source pins](SOURCES.json) identify the exact direct inputs, generated
+manuscript digests and PR #371's source hashes. The base commit pins the
+transitive generator dependencies. The source check executes the actual
+`patched_files()` composition in memory; it writes nothing to `upstream/`
+or existing certificates. It checks these links:
+
+1. [`make_fast_gaussian_patch.resampling`](../../scripts/make_fast_gaussian_patch.py#L23-L33)
+   replaces the original hypothesis `theta > p/alpha^4` by
+   `alpha^-2 <= theta <= 1/4`.
+2. The same function [inserts the note](../../scripts/make_fast_gaussian_patch.py#L157-L161)
+   verbatim. The chirp lemma inherits that new header, including integer
+   `p>=101`, integer `2<=alpha<sqrt(p)`, and `sigma=1+theta`.
+3. The [assembly replacement](../../scripts/make_fast_gaussian_patch.py#L251-L263)
+   establishes the upper and lower theta bounds eventually. This audit
+   does not re-prove its prime-existence or asymptotic setup arguments.
+
+The generator and unmodified upstream resampling file at PR #371's exact
+head are byte-identical to these files at the audit base. Thus the domain
+finding applies there too, rather than merely to a later repair. No pending
+PR is treated as an accepted stronger multiplication bound.
+
+## Concrete control: what the old header alone would permit
+
+Take `p=101`, `alpha=2`, `s=2`, `t=19`. These satisfy the old upstream
+header, including coprimality and `theta=17/2 > p/alpha^4=101/16`.
+For the **new chirp E algorithm**, however,
+
+```text
+m = ceil(sqrt(101)/(2*2)) = 3,
+L_B = 2m+2 = 8,
+q_0 = 0,
+q_1 = floor(19/2 + 1/2) = 10.
+```
+
+The first block contains outputs `ell=0,1`, but evaluates only local
+correlation coordinates `0,...,7`. It misses `q_1=10`. The actual generated
+header rejects this tuple because `theta>1/4`.
+
+This is a counterexample to applying the *new window recipe under only the
+old hypotheses*. It is not a counterexample to the original slower
+no-sort theorem, nor to the generated construction's restricted claim.
+It shows why source composition is mathematically significant.
+
+## Uniform arithmetic argument
+
+The [separate review](INDEPENDENT-REVIEW.md) gives the window-containment
+proofs and the uniform precision argument. A rational common enclosure is
+
+```text
+pi/ln(2) < (22/7)/(56/81) = 891/196.
+```
+
+For completeness, `ln(2)>56/81` follows from the first two positive terms
+of `ln(2)=2*sum_(k>=0) (1/3)^(2k+1)/(2k+1)`.
+Also `22/7-pi = integral_0^1 x^4(1-x)^4/(1+x^2) dx > 0`: polynomial
+division gives the integrand
+`x^6-4x^5+5x^4-4x^2+4-4/(1+x^2)`.
+These elementary analytic identities explain the enclosure; the finite
+checker evaluates only its exact rational consequences.
+
+With `theta<=1/4`, `alpha^2<p` and `p>100`, those consequences are
+
+| Obligation | Exact sufficient bound |
+|---|---|
+| Forward amplification | `(891/196)*(121/400) < 7/5`, so `Gamma=ceil(7p/5)` suffices |
+| Correction amplification | `(891/196)*(25/4) < 29`, so `Gamma=29p` suffices |
+| Correction input scale | `(891/196)/4 < 57/50`, so `B=ceil(57 alpha^2/50)` suffices |
+| Forward precision | `(53/20)p+12 < 3p` for every `p>=101` |
+| Correction precision | `(1607/50)p+11 < 34p` for every `p>=101` |
+
+The precision rows use `L_A<4p` for S', `L_A<=p` for E, and exact integer
+ceiling-log bounds. For E, use the sharper `B<=1.14p+1` when deriving the
+printed `32.14p+11` bound, rather than only `F<=2^(1.15p)`.
+
+## Reproduce
+
+From the repository root, using Python and its standard library (tested
+with Python 3.14.7):
+
+```sh
+python3 -B research/gaussian-chirp-domain-audit/verify.py
+```
+
+Expected output:
+
+```text
+PASS_SCOPED_GAUSSIAN_CHIRP_DOMAIN_AUDIT
+Exact parameter pairs: 21773
+Exact correction windows: 968049
+All 7 negative controls rejected; receipt matches
+```
+
+The deterministic [receipt](RECEIPT.json) records exact coverage, generated
+source digests, minimum precision slacks and control outcomes. The command
+recomputes its contents and rejects any mismatch. `--write` is available
+only for deliberate regeneration after reviewing a change.
+
+This is exhaustive finite validation within the following domains, not a
+search for a stronger construction:
+
+- Every `101<=p<=1088` and integer `2<=alpha<sqrt(p)`: 21,773 pairs.
+  The worst theta-dependent amplification envelope is checked at `theta=1/4`.
+- Every coprime `2<=s<=128`, `s<t<=5s/4`, and `1<=m<=9`; every starting
+  residue and every output offset. Correction residues include negative
+  periodic representatives; all forward residues are also checked. These
+  are generic integer-window cases, independent of the parameter-pair
+  loop; actual S' block sizes in that loop exceed 9. The written window
+  proof covers arbitrary positive integer `m`.
+- Seed 64 selected the task, sets the finite cutoffs, and rotates the
+  correction residue order. The stopping rule is completion of those
+  enumerations. The written arguments, not enumeration, address all sizes.
+
+Seven controls remove each source-domain bound, remove its assembly
+justification, underfund each stated precision budget, use the old-header
+counterexample above, and shorten an in-domain E output window. The two
+precision controls reject violations of the *sufficient certified budget*;
+they do not prove that those smaller precisions necessarily fail numerically.
+
+The separate review also preserves a second implementation with a different
+formula for the exact correction-window ceiling. Run
+`python3 -B research/gaussian-chirp-domain-audit/independent_budget_check.py`;
+it prints `PASS_INDEPENDENT_GAUSSIAN_BUDGETS pairs=168093 controls=5` and
+a JSON summary. Its wider arithmetic domain is `101<=p<=4096`; it does not
+repeat the source composition or residue-window enumeration.
+
+## Scope, attribution and handoff
+
+The new evidence is a source-bound resolution of the domain concern, a
+concrete window counterexample to the wrong source composition, written
+uniform arithmetic arguments, and independent exact finite implementations.
+This does not reopen a closed search or assert a new exponent.
+
+Inherited obligations include the one-block internal error enclosure,
+Gaussian tail bounds, transcendental approximation routines, packed
+integer multiplication, periodic tape movement, Neumann evaluation, and
+the selected suppliers' all-size and analytic interfaces. The tests
+neither implement the Gaussian maps nor establish those obligations.
+PR #371 remains unreviewed in the base ledger; this submission is scoped
+evidence for its review, not a request to mark the entire PR verified.
+
+Contributor: **Effy Elden (ineffyble)**, with substantial OpenAI Codex
+assistance, including a separate AI source/arithmetic reviewer. No human
+peer review or formal verification is claimed. Credit eumemic's PR #5
+construction, icekylinx's shifted error enclosure, Artus Krohn-Grimberghe
+and PR #371's disclosed GPT-6 Pro/Codex final-grid work, and David Harvey
+and Joris van der Hoeven's analytic algorithms. The new package is under
+the repository's Apache-2.0 license; predecessor notices remain intact.
+
+The [sprint claim](CLAIM.md) records candidate selection, overlap checks,
+timing and the final handoff. No existing source, selected certificate or
+review-ledger disposition is changed by this contribution.