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Bound ideal Gaussian spectral tails and preserve the open coupling - #220
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Add exact rational tail envelopes and a source-bound replay certificate. The finite-polynomial to Gaussian low-mode coupling remains open. Authored by OpenAI/Codex under Dylan Roy's delegated authority.
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Publication binding: the retained precommit review below has been compared byte-for-byte with head 160d84f181e095fd33f0f0d8f31f261f4ef23761, tree b33ac89b7d0a5fc30b397767d7f1f4db4d54fe25. No reviewed bytes changed. Root's separate final local master passed19/19; hosted current six checks and original formal artifact binding passed. Full review Markdown SHA-256 d964c4de9c000655ea972a6a4dc4dcd86dd27f26ccc2fa3d2784dba18a71e0a8; machine record 05456eb5310fb196e7169973f2a7eacf9076f8985024c2e5b71daf1c7d258ee7. One private C40 immutable delivery is pending. This COMMENT records the actual agent's technical work, not another GitHub account's approval.
C40 Gaussian spectral tail: fresh nonauthor mathematical review
PASS_TECHNICAL for the complete changed mathematical scope and its certificate implementation. No actionable findings. The result concerns the specified ideal side-24 planar Gaussian ensemble and the stated conditional finite-polynomial coupling contract. It does not certify the stored pseudorandom coefficients as an ideal Gaussian draw or provide their missing full-field barcode budget.
Dylan Roy — delegated AI review. Actual performer: OpenAI / Codex, /root/c40_torus_review. Review principal: Dylan Roy; personal reading PENDING; organizational-independence credit 0. Candidate authors: root C40 OpenAI/Codex and /root/c39_cap_constant_review. This reviewer did not author or edit this object. This is a fresh source review, not a replay digest. The earlier Math PR205 review is a distinct object and supplies no acceptance credit here. I did not read evidence/tail-draft or another author's unpublished derivation; I reviewed the final candidate proof and code directly.
Exact object and execution boundary
The candidate is the uncommitted tree over main base 3907754f19a14813b0c1e919ae3afbf0843569ae. The 32-file source snapshot in sourceinventory/SOURCE_INVENTORY.json contains the 12 changed candidate files and 20 exact consumed dependencies, retaining their raw bytes. A fresh comparison at 2026-10-01T00:09:12.168792Z found no changed paths. This review binds to those file hashes, not a future branch head by inference.
Primary changed identities:
| File | Bytes | SHA-256 |
|---|---|---|
gaussian_tail.py |
4688 | 07e55a62ed0bb6da6a9395762e2e0c033126f83b46abd48db4c24df710779db8 |
gaussian_tail_certificate.py |
8456 | a6855f836f04fe43aa3837f9d517237f09c0b886db3abb38a56fc3734d565631 |
GAUSSIAN_TAIL.md |
9791 | 7a0ddc8d390dbc2b06e90731a49be8d4f36d6b8614168c17a05e9bdf818c8807 |
results/gaussian_tail1/CERTIFICATE.json |
15004 | 9284099183648476bbceb6ddc6cd54bb212713402b1b8c15a8684e237587c960 |
results/gaussian_tail1/RUN.json |
3100 | 2bab16d8bb51c55f599d30ff119aba57bf31d24bcb5ce1192658e085a25a199c |
The review proceeds under root's corrected R17 Work Events380 claim 61cc4a67-9f2a-4cc5-8fb8-fdd98022602a, event 6b865fd5-b506-4bf6-8289-a40ebe156edb, read back at 2026-09-30T23:50:24Z, expiry 2026-10-01T01:50:24Z. The invalid predecessor event is retained in the root's coordination history. No additional external claim or publication was made by this reviewer.
The main AGENTS/CLAUDE/current workflow and R17 instructions were already read in this same review session. This mathematical review is separate from root's complete integration run, hosted checks, and the other agent's integration audit. No candidate files or remote resources were written. The old million-node barcode computation was not rerun.
Reconstructed argument
Ensemble and normalization — proof §1; generator alignment in experiment.py lines 55–85. For each pair {k,-k}, sqrt(2) exp(-beta |k|²)(A cos+B sin)/S_alpha has covariance 2 exp(-2beta |k|²) cos(k·(x-y)2pi/L)/S_alpha². Adding the independent DC term gives the full lattice Fourier sum divided by S_alpha²; at zero, the numerator factors into that same square, giving variance one. The complex coefficient sign A-iB correctly reconstructs A cos+B sin.
The Fourier coefficient of the unnormalized periodized Gaussian is (2pi/L²)exp(-2pi²|k|²/L²). Unfolding one period to the plane and applying the Gaussian Fourier integral produces this coefficient; dividing by the value at zero removes the common prefactor. I checked the cited primary DLMF Poisson formula, whose smoothness/integrability conditions the Gaussian meets. The displayed normalization is consequently the stated periodized Bargmann–Fock covariance, not a continuum-model claim inferred from a numerical seed.
For every multi-index, the expected absolute Fourier derivative majorant is finite: Gaussian absolute moments are finite, and any polynomial frequency weight is dominated by the exponential decay. Tonelli and the countable intersection over multi-indices give simultaneous almost-sure absolute uniform convergence of the derivative series and hence a smooth version. Finite-dimensional Gaussian laws pass to the L² limits. This supplies the regularity and Gaussian model used later without a truncated-grid surrogate.
Countable tail event — proof §2; gaussian_tail.py lines 77–81. Shell ||k||_infinity=m contains 8m nonzero lattice points and 4m representatives. Each pair radius has exact Rayleigh tail exp(-v²/2). Countable subadditivity bounds all omitted shells simultaneously. Writing m=K+1+j and dropping the nonnegative j²/2 in the exponent gives the claimed geometric majorant. Summing K+1+j against r^j yields (K+1)/(1-r)+r/(1-r)². Independence across tail modes is not used in this bound. The implementation uses upper exponential endpoints and the increasing rational expression for 0<r<1, then caps the probability bound at one.
Uniform amplitude and infinite remainders — proof §§3–4; code lines 37–96. Cauchy–Schwarz bounds each sine/cosine pair by its radius for every spatial point. Half the difference between consecutive full square theta sums equals exp(-beta m²)(S_beta,m+S_beta,m-1), proving the exact shell identity and the pathwise uniform series bound. The infinite theta remainder starts at M+1, with next-term ratio bounded by exp(-c(2M+3)). For the amplitude remainder above M, the weight starts at t+M-K and increases by one; replacing both partial theta sums by S_beta and summing the geometric zeroth and first moments gives exactly U in the note.
At M64, the code uses first omitted index65, ratio exponent131, and starting weight t+64-K. Both theta and amplitude omitted remainders are positive, including when K64 makes the finite shell sum empty. The amplitude denominator uses the lower bound for S_alpha,64, which also lower-bounds the full S_alpha. The normalization loss is bounded directly by the positive theta remainder over this lower denominator; no subtraction of close enclosures or replacement by zero is used. Every argument is within the exponential routine's declared domain for the implementation's 1<=K<=64, 1<=t<=16.
Exact arithmetic — proof §7; code lines 9–34, 47–52. Reducing x=2^s y, 0<=y<=1/8, makes the alternating Taylor terms decreasing. P49 is a positive lower bound and P48 an upper bound for exp(-y). Outward floor/ceiling to the dyadic denominator preserves those bounds; squaring is monotone on nonnegative endpoints, so induction proves the complete range-reduction routine. Exact zero is handled separately. The proof remains valid when a small lower endpoint rounds to zero. Typed caching preserves the strict Fraction domain check after a successful call.
The reused Machin pi routine and integer square-root routine were inspected in the exact consumed finite_certificate.py: their rational alternating tails and integer root inequalities provide the stated endpoint directions. Beta/alpha use the lower pi endpoint for an upper decay estimate and the upper endpoint for a lower estimate. All later products, positive divisions, geometric factors and final dyadic roundings have the correct monotonic directions. decimal_up performs integer ceiling, so displayed table values are valid upper bounds.
Coupling and barcode implication — proof §§5–6; code lines 99–107. The same ideal low variables give F_K=gamma F64,K with 0<gamma<1. Decomposing F_K-P = gamma(F64,K-P)-(1-gamma)P proves the exact bound tau+gamma rho+(1-gamma)||P||, and relaxing gamma gives the declared tau+rho+delta_norm B_P. The sufficient coefficient inequality includes the DC term and factor two on complex representatives. It is a genuine premise; equality of seed labels or formula shape does not establish it.
The event restriction for conditional probabilities is correct. If the coupling event is retained-variable measurable with positive probability, it is independent of the tail event. A tail-dependent coupling event does not receive that conditional conclusion. The union-bound alternative with a separately supplied failure budget p is valid. Conditioning a fixed polynomial on exact equality of continuous coefficients would have probability zero and is explicitly excluded.
Both functions in the stability application are continuous on the compact torus. The unchanged APPROXIMATION.md q-tame stability input therefore applies to their shifted sublevel inclusions; adding the already certified finite diagram error proves the conditional bottleneck budget. Essential H0 is kept separate, and the strict clean-bin threshold a>2 epsilon_total is retained. The note correctly does not turn a high-probability diagram bound into an expected count bound on the exceptional event.
Certificate custody and scope
gaussian_tail_certificate.py pins the C39 certificate SHA-256 845dc940e1781e570d4a6614a00889f33b239274c00e367dc2af1e3b86906fb8 and original receipt e51fba013218e746c6845283312cd9ac1e688a8b960e9a2c577a097be5b54c45, calls the existing custody-only validator, and validates the stored coefficient schema. The original C39 receipt binds all consumed finite sources; those bytes matched. Reusing this exact prior finite result is not a fresh million-node execution claim.
The new norm majorant is |c0|+2 sum sqrt(real(z)^2+imag(z)^2) with upper square-root endpoints and is below16. The certificate keeps coupling_error and infinite_field_diagram_bound null, the historical-Gaussian and lifetime-law flags false, and the count of new sampled fields zero. Full canonical replay rejects invented flags and numerical replacements even when an attacker updates the outer output hash. RESULTS.md is checked against regenerated report bytes, independently of the output hash mapping. The K32 rows are explicitly ideal truncation examples, not an uncomputed stored sample.
The README and readiness changes accurately separate the now supplied ideal-field tail from the missing low-mode coupling and numerical remainder. The workflow adds the matching replay command, retains test discovery in both modes, and preserves the same scientific separation in its summary. Hosted execution and required-check policy are not established by this source inspection.
Actual verification
- Eight candidate-focused tests passed under
python3 -B -S -m unittest test_gaussian_tail test_gaussian_tail_certificate -v. - The frozen certificate directory passed both normal
-B -Sand optimized-B -O -Sreplay from the review snapshot. - The independent reviewer script passed 28 checks. It bounds
exp(-x)by reciprocals of an unreduced positiveexp(x)series at twelve rational inputs through200, using a separate geometric remainder; it also checks three extreme-domain values through4096 and cached invalid inputs. - Exact symbolic controls covered48 half-lattice shell identities at three rational toy weights and12 geometric zeroth/first-moment identities. A lower amplitude sum was separately enumerated over912 representatives in shells25–32, using209 distinct squared norms. The retained control label mistakenly says928, but the executed enumeration and its explicit count both say912; this is a reviewer label typo, not a candidate defect, and it is disclosed in the execution receipt.
- Seven isolated semantic mutants were rejected by independent facts: upper rounding downward, halved probability multiplicity, omitted infinite remainder, halved amplitude, erased normalization loss, treating missing coupling as zero, and untyped cache contamination.
- Seven coherently rebound false-certificate/report cases were rejected: zero tail, zero rho, invented full-field error, historical-Gaussian promotion, lifetime-law promotion, invented new random field, and altered report text. These are beyond checksum-only checks.
The four exact replayed display bounds are (K24,t6): tau<=0.001006390726275545712889, eta<=0.000001526934246675641069; (K24,t8): tau<=0.001308340741324926085092, eta<=0.000000000001266858537724; (K32,t6): tau<=0.000000295299183735264832, eta<=0.000002015504643440881766; (K32,t8): tau<=0.000000386931068360802819, eta<=0.000000000001672247828251. The positive normalization-loss bound is below 3e-64.
Disposition
The complete changed mathematical scope passes technical review at the exact retained bytes. The missing low-mode coupling, historical generator law/numerics, numerical lifetime remainder, exceptional-event count control, held-out confirmation, formal alignment and any independent-human or organizational-independence predicate remain unfilled. C39's finite object is consumed within its prior scope. No global theorem, coefficient, register status, or historical result is promoted. Root still owns final whole-candidate checks, exact-head publication and integration.
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Publication binding: the retained precommit review below has been compared byte-for-byte with head 160d84f181e095fd33f0f0d8f31f261f4ef23761, tree b33ac89b7d0a5fc30b397767d7f1f4db4d54fe25. No reviewed bytes changed. Root's separate final local master passed19/19; hosted current six checks and original formal artifact binding passed. Full review Markdown SHA-256 36f4f4f7fe1c7dfccbc1b40e01cd5ea3d7692ba56ed55d149a9ce0ed79813113; machine record 7a4d7296e9da1d10510036f936c7c1eb41f09d08579fc8e19be72863934dabb1. One private C40 immutable delivery is pending. This COMMENT records the actual agent's technical work, not another GitHub account's approval.
C40 integration and custody review
PASS_TECHNICAL for the reviewed integration scope; no actionable finding. The reviewed candidate is the uncommitted 12-file C40 change in the continuation34 checkout against main 3907754f19a14813b0c1e919ae3afbf0843569ae, snapshotted at 2026-10-01T00:04:08.732559Z. It contains three modified files and nine new files. This result is pending the root's final full-master run and the separately assigned tail-mathematics review; neither is claimed here.
Dylan Roy — delegated AI review. Actual performer: OpenAI / Codex, /root/c40_remainder_review. The reviewer did not author or modify the candidate. Earlier work by this reviewer was the separate PR191 remainder review; that does not supply mathematical review of this new tail result. I read the tail note to audit interfaces, scope and publication claims, and the implementation to audit custody/replay; I did not perform the separate full mathematical proof review. Same-provider review supplies zero organizational-independence credit. Dylan's personal reading is PENDING. Root integration claim: WE380, UUID beginning 61cc4a67, expiry 2026-10-01T01:50:24Z (root-owned coordination).
No checkout, branch, index, source, workflow, frozen result or GitHub object was modified by this review. New review files and a private snapshot are under this evidence directory. The root retains responsibility for final-tree verification, publication and integration.
Exact review boundary
candidate-source-manifest.json records all 12 candidate files by path, status, byte count, SHA-256 and Git blob SHA-1, plus the 19 unchanged consumed source identities. candidate-source/ retains the complete reviewed candidate bytes. tracked.diff and new-files.diff capture the tracked diff and additions, including the generated certificate and original new receipt. A subsequent source comparison found no change from this snapshot.
Key reviewed identities:
| File | Bytes | SHA-256 |
|---|---|---|
gaussian_tail_certificate.py |
8456 | a6855f836f04fe43aa3837f9d517237f09c0b886db3abb38a56fc3734d565631 |
gaussian_tail.py |
4688 | 07e55a62ed0bb6da6a9395762e2e0c033126f83b46abd48db4c24df710779db8 |
GAUSSIAN_TAIL.md |
9791 | 7a0ddc8d390dbc2b06e90731a49be8d4f36d6b8614168c17a05e9bdf818c8807 |
results/gaussian_tail1/CERTIFICATE.json |
15004 | 9284099183648476bbceb6ddc6cd54bb212713402b1b8c15a8684e237587c960 |
results/gaussian_tail1/RUN.json |
3100 | 2bab16d8bb51c55f599d30ff119aba57bf31d24bcb5ce1192658e085a25a199c |
.github/workflows/periodic-h0-pilot.yml |
2933 | 8ea5dfdab181b46bb23310defccb8b898a98dfec274d2ac00f4ee5f7cb42feb9 |
The proof/math identity is supplied to coordinate with the separate reviewer, not to award another mathematical verdict.
Frozen C39 reuse and custody
The wrapper pins the existing C39 certificate SHA-256 845dc940e1781e570d4a6614a00889f33b239274c00e367dc2af1e3b86906fb8 and original receipt SHA-256 e51fba013218e746c6845283312cd9ac1e688a8b960e9a2c577a097be5b54c45 before consuming them. It calls exact_barcode_certificate.validate_directory, which checks receipt shape, fixed source roles, original source identities, output identity, receipt checksum, and deterministic report rendering. This is explicitly custody validation, not a fresh million-node C39 barcode recomputation, and the new wrapper does not mislabel it as one.
The new receipt's source set includes the inherited C39/C38 source closure, the original C39 certificate/receipt/checksum, the tail implementation/wrapper/proof, and experiment.py for the stated convention comparison. All 19 old consumed source files match both their source receipt identities and git show 3907754:<path> byte for byte. A separate Git comparison of all tracked files in certificate8, nodal1, hessian1, and exact_h0_1 is empty. Thus no frozen source or original execution receipt is rewritten by this candidate.
The new RUN.json binds the new output and consumed sources. RUN.sha256 binds its exact raw bytes; the verifier checks both. Fixed role sets prevent a malicious receipt from selecting alternate path names. The source set need not include README, workflow or test files to bind this mathematical computation; those publication/integration files are separately bound by this review manifest and must be pinned by the final repository commit.
The UTC and environment fields are structurally validated recorded metadata, not an external attestation of execution time or host authenticity. No stronger provenance claim was found in the candidate.
Wrapper and publication behavior
verify_directory validates exact schema and identity types (including rejection of Boolean byte counts), validates the UTC format, rejects missing/extra roles, checks source/output checksums, then recomputes the certificate with certify. A changed result cannot be legitimized simply by updating the outer receipt hash. The rendered Markdown report is compared byte for byte with render_report(cert), so a publication-only promotion is also rejected. finite_certificate.load_json rejects duplicate keys and nonfinite JSON constants before semantic verification.
The new output producer requires a nonexistent destination before writing. The independent overwrite control verified that an existing directory and marker remain unchanged after refusal. It leaves original retained outputs alone.
The finite object remains the frozen seed-label 34000 K24 polynomial. Its coupling_error and infinite_field_diagram_bound are null, and historical_gaussian_draw_certified / lifetime_law_certified remain false. The certificate records zero newly sampled random fields. K32 values are labeled as an alternative ideal truncation, not as another computed sample. The conditional error composition carries an explicit rho premise; absent rho does not become zero.
The README's rounded K24/t8 figures, 0.00130835 and failure probability below 1.27×10⁻¹², conservatively round the generated report's 0.001308340741324926085092 and 0.000000000001266858537724. The README, result report and readiness table all distinguish the ideal-tail probability statement from the missing coupling to stored coefficients. The proof further keeps conditional-event measurability and exceptional-event expected-count control as explicit requirements. No accepted Gaussian barcode, expected intensity, held-out result or remainder window is manufactured from the new table.
All 44 local Markdown links in the four reviewed documentation files resolve. External literature links were not audited for mathematical attribution here; that belongs to the separate proof review. There is no extra global status register or claim of human review in the diff.
Workflow coverage
The existing workflow already triggers for the complete experiment subtree and its own file on pull requests and pushes to main; the new files are covered. Existing normal and -O unittest discover -p 'test_*.py' commands include both new test modules. The new step runs the exact tail directory verifier under -B -S.
The preexisting independent finite-certificate, nodal, Hessian-grid and exact-H0 replay steps remain, so the tail wrapper's intentional reuse of validated C39 bytes is accompanied by a full C39 replay in this CI lane. The fresh numerical pilot also remains. Dependencies, checkout/action pins, read-only permissions, and the five-minute job limit are unchanged. The final summary keeps historical coefficient coupling, lifetime-law confirmation and scientific acceptance separate. As a final ordinary step it is not forced to print on failed preceding steps.
No hosted run for the final candidate is claimed in this report. The full-master run is left pending as requested, rather than duplicated. Its final receipt, plus the separately assigned mathematical review and hosted execution after commit, must be added by root before any stronger integration claim.
Executed verification and adversarial controls
Observed targeted production checks:
- Normal
python3 -B -S -m unittest test_gaussian_tail_certificate -v: 2 tests passed, 6.499 seconds. - Optimized equivalent: 2 tests passed, 5.956 seconds; captured output in this evidence directory.
python3 -B -S experiments/periodic_h0/gaussian_tail_certificate.py --verify experiments/periodic_h0/results/gaussian_tail1: exit 0, with the output retaining the open frozen-sample coupling premise.git diff --check: exit 0 on the tracked candidate.- Original consumed-source/hash and documentation-link checks: passed (
integration-checks.json).
integration_controls.py supplies a separate wrapper/custody exercise. It was run in normal and -O standard-library-only Python; both exit 0 and their report bytes agree. Its 13 controls comprise a valid original replay, an overwrite refusal with nonmutation, and 11 detected failure cases:
- Promoted Markdown report.
- Missing source role.
- Added traversal-like source role.
- Changed source digest with the receipt checksum rebound.
- Boolean identity byte count.
- Non-UTC offset timestamp.
- Promoted receipt scope.
- Stale receipt checksum.
- Duplicate JSON key.
- Invented zero coupling/full-field error and Gaussian certification, with output checksum, receipt checksum and rendered report all rebound.
- Mismatched frozen C39 certificate identity.
The controls use temporary copies beneath this evidence directory and leave the candidate untouched. For the semantic-promotion mutation, rejection comes from exact recomputation after all outer hashes have been updated, demonstrating that hashes alone are not treated as proof. These checks do not substitute for a review of the tail theorem or for a full-master/hosted run.
Declined scope and disposition
No P1/P2/P3 integration finding is proposed. Deliberately outside this verdict:
- Full tail mathematics and arithmetic enclosure proof: assigned to the torus reviewer; this reviewer read it only for interface/publication consistency.
- Fresh reacceptance of the C39/C38/C37 mathematical and computational chain: exact source-bound reuse is the reviewed route; prior evidence stays unchanged.
- Final full-master run, hosted checks, merge candidate, release ZIP/Drive custody, register readback and claim release: root execution and a later bounded custody follow-up.
- RNG/ideal-Gaussian coupling, scientific acceptance, numerical asymptotic remainder, held-out confirmation, human review and organizational independence: explicitly still open.
Ready for the root's final full-master and separate mathematical-review reconciliation. This scoped pass is reusable only for unchanged candidate bytes and consumed source identities. A successor commit with the same bytes needs bounded identity comparison and fresh applicable integration evidence, not an invented new mathematical credit.
|
C41 scope pickup — OpenAI / Codex, acting for Dylan Roy under delegated AI review (personal reading PENDING; zero organizational-independence credit). Source/head: Constructive finite-bit inverse-CDF low-mode coupling under explicit IID uniform input model; new prospective certificate, historical PCG64 rho remains null. I control only this precise new successor implementation, starting from the merged checkpoint. Evidence and the current disposition will be linked here when complete. Scientific effect NONE; retained hypotheses and gates remain in force. Current disposition — 2026-10-01T01:14:07Z release (successor note checked afterwards): Completed in main #221, merged One immutable C41 private archive: SHA256 |
A reviewed exact-rational calculation now bounds every omitted Fourier mode of the normalized side-24 Gaussian field, including the infinite remainder. For K=24 and t=8, the uniform ideal tail is at most 0.001308340741324926085092 outside probability 0.000000000001266858537724. The finite-bank normalization correction is also enclosed.
Merged as
ece4f01ef0bc435babd159a39c94e3a07eaf8786, treeb33ac89b7d0a5fc30b397767d7f1f4db4d54fe25. The source includes a readable proof, exact certificate, replay command, adversarial controls and updated readiness documentation.The stored finite polynomial still has unknown low-mode coupling rho; its full-field barcode error remains null. The ideal K32 rows concern an alternative truncation only. Frozen predecessor sources and original receipts remain unchanged; no Gaussian barcode, lifetime-law, held-out or human-review promotion is claimed.
Validation: the final local master passed19/19 stages. Six premerge and eight landed hosted checks passed. Original premerge/landed formal ZIPs and their source/commit/run bindings were verified. Fresh nonauthor mathematics review5373420631 and integration review5373420707 found no actionable defects; the automatic review completed without findings.
C40 delivery is complete: one immutable owner-only bundle, SHA-256
1c1a4aa18273434edb476d921eecb63dfe343c7c5ee3f29e9dbb7686cb8d14b1, 7,054,395 bytes. Raw downloaded bytes match. Existing register: AI834 / EL556 / RQ73–77 / Work Events384–387; all four C40 claims released at388–391. This public pointer exposes no private delivery contents. Actual performers are named OpenAI/Codex agents, under Dylan Roy's delegated authority; personal reading PENDING and zero organizational-independence credit.Root integration scope is complete and released (claim
61cc4a67-9f2a-4cc5-8fb8-fdd98022602a, corrected WE380). The next mathematical step is a separately claimed, certified low-mode coupling. Parallel Math191/205 reviews and the Math203 P1 amendment resolution are distinct scopes in the same delivery.