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# Build evidence for T681
Run: `20261102T162707Z-build`. Table: Higher Mahler measure of cyclotomic polynomials. This is build evidence, not an independent campaign critique or acceptance. No offer, review, or publication is authorized or performed by this actor.
## Instructions and source provenance
The coordinated instruction archive is `agents/runs/20261102T162707Z-build-skills/manifest.json`. Both archived local and live skills were read. Both hash to SHA-256 `a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`; the snapshot's canonical file has the same hash. The manifest is also attached as `skill-manifest.json`; `skill-evidence.json` records the paths and the manifest's own hash. No policy was selected from a mismatching version.
The full family issue 302 was read before screening. Its conventions and coverage map were retained. The corpus was rechecked using `already_here`, `already_asked`, text searches, three numerical `api/lookup` requests, and live reads of T9, T14, T94, T95, T175, T282, T283, T499 and T522. No duplicate of the proposed family was found. The numerical overlap at pi^2/12 belongs to a different context (T459). No geometric-sum companion existed to link; none was invented. The complete 73-tag alphabetical list was read, and the five selected tags exist.
Primary sources were downloaded and read, not inherited from an older transcript:
- Lalín and Sinha, arXiv:1106.1304v1, Definition 1, Theorems 5 and 6, Lemma 10, Theorem 21, Proposition 22, and the proof of Theorem 6. PDF SHA-256 `b98d63d21acfc6a4a1ee09c45faf188af6742968aa7d5930497bae1142fa9826`.
- Kurokawa, Lalín and Ochiai, arXiv:0908.0171v1, Definition 1, Theorem 14 and section 4.2. PDF SHA-256 `1b6e625c06865eb0f06da804bd7ef8f4929c8a993027ec7b2e224155c90ccc10`.
`sources.json` records URLs and byte sizes. PyMuPDF 2.0, already installed in the shared dependency directory, extracted the PDF text. The HTML versions were also consulted to disambiguate displayed fractions. In particular, the Clausen series specify radians: C3(theta) = Re Li3(exp(i theta)), S2(theta) = Im Li2(exp(i theta)). The source's nearby prose prints an extra factor 2*pi in its description of the polylogarithm argument; its explicit Fourier series and Theorem 6 fix the convention, also confirmed by the live T175 definition and direct integration. Cyclotomic order n and polynomial degree d are kept distinct.
## Selection and method
The attached `argument-selection.md` independently reassesses both free parameters. There are 44 odd squarefree representatives n <= 105 and two moment orders k = 2,3, hence 88 entries in increasing n and then k. Both are genuine discrete indices. The cutoff is noncanonical, reaching the first three-prime structure without redundant prime-power or even-order rows. General parameter constraints remain n >= 1 and k >= 2.
Second moments use pi^2 phi(n) 2^omega(n)/(12n). Third moments use the finite sorted-root formula of Lalín–Sinha Theorem 6. Root angles and pair weights are exact Sage rationals. The generator folds angles by reflection and collects equal weights before evaluating Arb polylogarithms; the independent check does neither.
At 461 working bits (100 decimal digits plus 128 extra bits), every result is finite and retains at least 414 relative-accuracy bits, about 124 decimal digits. An explicit accuracy guard requires the requested precision plus 32 bits and increases working precision if needed. This is an error-bound computation, not a fixed-precision value placed into a ball. Every exported 100-digit decimal interval is checked to contain the entire computed ball by comparing exact rational endpoints. The longest stored numeric spelling is 103 characters.
## Completed pre-fill checks
- The repository's `dry_run.py`, `exactness`, `measure`, and `names_its_rings` passed. No Python float coefficients, nonfinite balls, suspicious trailing zeros, or problematic entry comments occurred. The 88 in-memory balls measure about 16.9 KB; the longest ball display is 156 characters. These are not the shorter exported decimal spellings.
- Every second-moment coefficient equals both the Möbius divisor sum (LS Proposition 22) and the exact Bernoulli root-pair sum (LS Lemma 10). `agrees_with` was used on the exact rational coefficients.
- Every third moment overlaps an independent 512-bit implementation that computes Clausen values from finite Hurwitz-zeta sums. That check uses uncompressed, unreflected root pairs. Both sides are checked for finiteness and at least 350 retained bits before an overlap is accepted. This tests the evaluation algorithms and assembly independently but still shares the published root formula; direct integration supplies a separate check of that formula.
- Known nonzero positive and negative controls are verified first: m2(Phi1) = pi^2/12 and m3(Phi1) = -3*zeta(3)/2. Fourier endpoint controls at zero and one-half also pass.
- Exact cyclotomic identities Phi_n(x) = Phi_rad(n)(x^(n/rad(n))) passed for all 1 <= n <= 420. Rotation identities passed for every selected odd representative, including the sign at n = 1. Every selected prime satisfies Phi_p = 1+x+...+x^(p-1) exactly.
- Mpmath tanh-sinh quadrature, at 50 decimal working places, independently integrates the Möbius log-sine expression for all 44 orders, obtaining the second and third moments together. It splits at all j/n and evaluates endpoint sine distances stably. The gamma-ratio derivatives are first checked independently, then the n = 1 integral is required to match the known controls within 1e-42. All 88 moments agree with the generator within the required absolute tolerance 1e-40; the maximum observed discrepancy is 2.1329065e-49. This is heuristic numerical evidence, not the proof of the stored error bounds. `quadrature-evidence.json` records every result and discrepancy.
`numerical-evidence.json` contains every exported value, its retained precision, and its independent 110-digit Hurwitz/divisor check. `checks.py` and `quadrature.py` are attached reproducible tests; `check.py` is the unmodified repository check helper required by `checks.py`.
## Final build checks
The fill process requires the draft to remain private and empty before writing, compares freshly computed entries to the pre-fill evidence, then writes the complete document and all 88 entries once. It attaches the generator and checks, and calls `verify(sample=None)`. A separate authenticated readback repeats all formula comparisons against the written intervals and rechecks the ordered selection. The exact results of those post-fill steps, the site audit, and the attachment fingerprint are in the run's final handoff and artifacts. Until those steps finish, this paragraph is a statement of the remaining procedure, not a claim they passed.
The argument-selection fingerprint is for a future independent critique. This builder does not create an acceptance sidecar. The controller alone may offer the draft after critique and repair.
## Tag recommendation
The live corpus has no `Mahler measure` tag. A subject tag could group T282, T283 and this table T681, along with other explicitly defined Mahler-measure families such as T649. The unrelated Mahler polynomials T182 are outside that scope. This is a recommendation only; no existing table or tag was changed.