# Build handoff: T664

Built **Mahler measures of the Bertin Q_k family**,
https://numberdb.org/T664, from numberdb-data family issue #293.
The table is a filled draft. This actor has not offered, reviewed, or published
it, and has not edited any published companion. The controller must run its
separate critique and repair stages before any offer. No independent critique
sidecar has been written by the builder.

## Instruction provenance

Read the family first with `python3 agents/queue.py show 293` and saved it as
`family-293.txt`. Read the coordinated archived skills governed by
`agents/runs/20261102T003143Z-build-skills/manifest.json` in this source snapshot.
Both valid archived versions, `skill-local.md` and `skill-live.md`, have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The bytes and manifest hashes were checked for equality. See the attached
`skill-evidence.json` and `numerical-evidence.json`; the source snapshot and
skills were not edited.

## Scope and conventions settled

The polynomial has the twelve nonconstant terms in Bertin's 2005 paper,
Bertin 2009 Theorem 1.1, and Bertin et al. 2012 Section 1. There is no
additional xz+(xz)^-1 pair. The family issue warns that Samart,
arXiv:1205.4803v2, Section 3, printed page 18, displays that extra pair.
This source discrepancy is preserved here; none of that section's identities
is used. The three directly read sources agree on the definition adopted.

The table holds one quantity: ordinary logarithmic Mahler measure, using
natural logarithms and normalized Haar measure, including the integrable torus
zeros. The real parameter k appears in the constant term -k. Its sign is
retained, and the number header is m(Q_k). The domain stays all real k.

The selected inputs are all integers -8 through 16 and -36, ordered by
absolute value, positive first at ties. The attached `argument-selection.md`
distinguishes the intrinsic group {-4,0,12}, arithmetic group {-3,-6,-36}, and
20 exploratory reference inputs. It supplies an explicit reference policy for
the latter, not an intrinsic-argument claim. The four-integer exterior extents
are a noncanonical practical boundary around the actual torus-zero interval;
-36 extends arithmetic breadth without filling a deeper grid. Completeness,
entry order, generator enumeration and the note agree.

## Numerical results and controls

There are 26 entries with 100 significant digits, at the honest common level
`heuristic (agreement-checked)`. Modular computations at 150 and 210 working
decimal digits agree to at least 149.3366 digits relative to max(1,|value|).
An explicit omitted-series bound is recorded for each nome, but does not
certify numerical inversion or all floating-point arithmetic.

Independent moment-ODE continuation checks 23 nonsingular arguments at 145
working digits. A separate real Gauss-hypergeometric endpoint integral checks
12 and, using the proved factor-four identity, 0. The minimum absolute agreement
is 139.3275 digits. At -4, the exact factorization plus an independent 650-bit
Arb evaluation of Smyth's zeta formula checks more than 190 digits. A separate
Arb Hurwitz-zeta calculation checks -3 to more than 185 digits.

Controls include a known-answer continuation from 32 to 64, 35 Domb coefficients
against exact binomial sums, both exact Laurent substitutions, every selected
exterior argument against the direct moment sum with explicit tails, Watson's
endpoint gamma product, and rejection of a deliberate 1e-90 perturbation.
The displayed standalone program was executed and checked on all five selected
arguments in its domain. Logs and source are attached.

An early unconstrained inverse-nome experiment produced a wrong sheet at -2 and
-1 despite tiny parameter residuals. Its cause is identified, and the final
quadrant/exterior branch restrictions follow continuation from a known CM
point. Independent moment continuation confirms all selected branches. The
artifact `lesson-proposal.md` records this general issue without editing the
canonical skill.

`dry_run.py` passed: 26 entries, no exactness/prose/disguised-rational findings,
longest internal enclosure 154 characters, 5.3 KiB before rounding to the
100-digit stored display. `verify(sample=None)` returned 26/26 matched, no
missing, differing or extra entries. Every stored decimal interval contains
both the higher-precision computation and its independent comparison; the
largest displacement is 0.483363 of a stored last-place unit. Stored-value
checks also pass the factor-four relation, the three-/four-step companion
relations, the relation to T663 at zero, and the T393 Watson value.

The final site audit is clean: `findings: []`, `clean: true`. Its only initial
finding was an install-command omission in the generator docstring, corrected
before the final fingerprint. Numerical code was unchanged.

## Corpus and independent-review handoff

The cheap duplicate checks used `already_here`, `already_asked`, live text
searches, numeric API lookups, and the live tag list. No same-family table was
found. T663 and all referenced existing companions were read. Actual slugs and
entry keys were recorded, rather than inferred from titles. The table links
to T663, T283, T145, T14, T522, T393 and T153 with specific relationships.

The existing tags are number theory, special values, integral, statistical
mechanics and random walks. **Proposed tag:** Mahler measure, absent from the
live list. It would connect at least T282, T283, T649, T650, T651, T658, T663 and
T664. No new tag or retagging of existing tables was performed.

`selection-fingerprint.json` records the live selection and attachment hashes,
without a verdict. `generate.py` has no local source dependencies; the
selection note and verification files are attached too. The separate critic
must inspect the live draft and regenerate the fingerprint, record its own
NUMBERDB_RUN_ID as reviewer_run, independently assess all three selection
groups, and write its own required sidecar. The builder has not impersonated
that independent assessment.

The controller handoff is
`agents/runs/20261102T003143Z-build.table.json`, containing only
`{"tid": "T664"}`. No proposal completion or publication receipt was changed.