# Numerical evidence for T663

Coordinated skill manifest: `agents/runs/20261101T235407Z-build-skills/manifest.json`.
Local and live SHA-256: `a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
Both archived copies were read and their bytes checked against the manifest.
The complete machine-readable pre-fill results are in `numerical-checks.json`.

Selection: k=0,...,12 and 18 (14 entries). See argument-selection.md.
Stored target: 100 significant digits, heuristic (agreement-checked).
Primary quadrature: 150 and 210 decimal working digits.
Minimum agreement normalized by max(1,abs(value)): 149.667429 digits.
Independent eta/q-series calculation: 140 working digits, minimum agreement
139.585134 digits. Every argument passes, including the torus-zero interval.
No modular evaluation was substituted for a quadrature result.

Controls were run before accepting comparisons. The density integrates to 1
and has second moment 4 (rounded residuals zero at 140 working digits).
The Hurwitz-zeta evaluation of Smyth's value agrees to 141.18 digits.
Adding 10^-90 to that control is rejected. All seven exterior arguments pass
the exact closed-walk sum with explicit omitted tails below 10^-153 and at
least 139.42 digits of agreement. Six times the right derivative at 6 agrees
with Watson's gamma product to 140.46 digits. The arithmetic relation
m(P_10)-3m(P_2)-2m(P_0) has residual 8.17e-202 in the 210-digit values.

The q-series tail bounds apply at the computed q. Root residuals are recorded,
but this does not certify parameter errors. Tanh-sinh convergence estimates
and repeated precision also are not proofs. The table's common rigour is
therefore heuristic (agreement-checked), even where a separate series has
an analytic tail bound.

The shared dry-run tool passed exactness, display, comment and import checks.
Its measurement was 14 entries, longest 155 characters including its comment,
and a 3.1 KB block before storage rounding. No exact-rational claim is made
from numerical digits; the k=0 value is a nonzero special value, not an exact integer mistaken for a decimal.

Software measured during checks: Python 3.12.3,
mpmath 1.4.1. Submission adds the installed NumberDB and
Sage/passagemath version record. All Sage computations used agents/sage.sh.

The definition, sign convention, Haar normalization, torus-zero treatment and
CM tau values were checked in Bertin (math/0501153), page 2, Theorem 1.1,
and Bertin et al. (1208.6240), sections 1 and 2.3. Guttmann and Rogers
(1207.2815v2), sections 3 and 5.3, give the lattice interpretation and walk
coefficients. DLMF 19.8.5 gives the AGM identity (after modulus-to-parameter
conversion). methods.md supplies the Jensen and density derivation.

Fresh searches and successful api/lookup queries found only the expected
normalization companions T283#3 and T153#spanning-tree,sc,entropy. No existing
table of this pencil was found. Invalid exploratory lookup request formats
were corrected, and their errors were not treated as empty search results.
The source snapshots, corpus documents, successful lookup responses and raw
logs are retained in this run's artifact directory.

The new tag Mahler measure is proposed for T663 and existing T282, T283,
T649, T650, T651 and T658. It did not exist in the live tag list, so only
existing tags were used. No existing table was edited. No unbuilt companion
was assigned an identifier. The family-reported fourteen/twelve-term Q_k
source discrepancy is outside this table, and no identity depending on that
discrepancy was imported.

This document is build evidence, not an independent critique or acceptance.

Installed NumberDB version: 0.1.10. The stored entries have at most 102 characters per numerical value; the JSON entries block is 2285 bytes including comments. The parameter identities are exactly the 14 enumerated arguments.

Read-back verification completed: all 14 stored decimal intervals contain the independent modular values. The stored arithmetic-relation residual is 3.0e-100, consistent with the aggregate last-place uncertainty. The full result is stored-checks.json. Generator verify(sample=None) returned 14/14 matched, 0 differing, 0 missing, 0 extra. Site audit: clean=true, findings=[]. Sage version: 10.9.