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# Argument selection for T664
This is the builder's justification, not an independent argument-review verdict.
The separate critique must assess the live enumeration and fingerprint its source
attachments. The controlling archived skill manifest is
`agents/runs/20261102T003143Z-build-skills/manifest.json` in the source snapshot.
Both `skill-local.md` and `skill-live.md` were read and have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The attachment `skill-evidence.json` records their full archive location.
The single free parameter is real k, the coefficient in the negative constant
term -k of the twelve-term Laurent polynomial in the definition. Values are
ordinary logarithmic Mahler measures, with natural logarithms and normalized
Haar measure. The parameter is not replaced by k+4, the square of a random-walk
distance, or a nome. Positive and negative k are different inputs.
The exact selected set is {-8,-7,...,16} union {-36}, 26 entries, ordered by
(|k|, k<0), positive before negative at equal magnitude. The three disjoint,
exhaustive groups are:
1. **Intrinsic:** -4, 0, 12. These are the finite singular parameters of the
K3 pencil (Bertin 2005, Section 3.1). The torus polynomial is the squared
length of a four-step planar walk minus (k+4); its zero interval is
[-4,12]. At -4 an exact factorization reduces the measure to twice Smyth's
four-step measure. The endpoint 12 also gives the face-centred cubic
spanning-tree entropy; this application is additional evidence for an
already intrinsically selected point (Guttmann and Rogers, Section 5.6).
2. **Arithmetic structure:** -3, -6, -36. Bertin 2009, Theorem 1.1, evaluates
-3 in terms of a quadratic Dirichlet L-value. Bertin 2005, Theorem 4.2,
studies the pair -6,-36 through an arithmetic L-series relation. Including
the whole displayed pair adds arithmetic breadth without extending every
intervening coefficient. No assertion that all integer inputs are CM is
made, and the relation discussed in that theorem is not presented as a
proved identity in the table.
3. **Exploratory reference sample:** -8,-7,-5,-2,-1,1,2,3,4,5,6,7,8,9,10,11,
13,14,15,16. These are the remaining integers in [-8,16]. Their role is
bounded reference coverage of simple integral Laurent polynomials in this
particular normalization. There is no claim that every member is a CM
point, appears in a particular application, or is individually famous.
The reference policy for group 3 is:
```json
{
"purpose": "Identify ordinary Mahler measures arising from small integral coefficients of this particular twelve-term K3 pencil.",
"bounds": "The integers -8 through 16 after removing the intrinsic and arithmetic members explicitly listed in groups 1 and 2.",
"coverage_rationale": "Complete the integral slice through the entire torus-zero interval [-4,12] and four coefficients into each exterior regime. Four exterior coefficients on each side are a practical noncanonical boundary, equal to one quarter of the spectral interval width, rather than a size target. The separate arithmetic pair adds breadth beyond this slice. No rational, radical, decimal, angular, or multidimensional mesh is added.",
"noncanonical_acknowledged": true,
"notation_independent": true,
"sources_checked": true
}
```
These bounds are a deliberate reference choice, independent of numeral base;
small integral coefficients alone do not prove a documented application. The
asymmetric interval in k reflects the actual spectral interval, not an assumed
sign symmetry. The arithmetic extension -36 is not a reason to fill a deeper
integer range. All important strata (both exteriors, interior, singular
parameters, and the cited arithmetic examples) are covered before depth is
extended. A quadratic-coefficient survey would require its own sources and
selection rationale and is deferred rather than manufactured from simple
radicals. The table has one named quantity and no quantity/convention selector.
## Sources actually checked
- Marie-José Bertin, *Mesure de Mahler d'hypersurfaces K3*,
https://arxiv.org/pdf/math/0501153, definitions and Theorem 1.1 on printed
page 2, Section 3.1, Theorems 4.1 and 4.2. The exact twelve-term definition,
eta parameter, singular parameters, and the -6,-36 pair were read.
- Bertin, Feaver, Fuselier, Lalín and Manes,
https://arxiv.org/html/1208.6240, Section 1, confirms both polynomial
definitions and Smyth's normalizations. Its six integer CM examples concern
the other P pencil; they were not imported as a selection for Q.
- Marie-José Bertin, https://arxiv.org/pdf/0901.0619,
Theorem 1.1 and Proposition 3.1, fixes the twelve-term Q_-3 and its Dirichlet
value and CM parameter.
- Mathew D. Rogers, https://arxiv.org/html/0704.2438,
equations (1.2), (2.11), (3.4), and the paragraph after (1.2), for the
face-centred cubic transformation, shifted parameter and moment expansion.
- Guttmann and Rogers, https://arxiv.org/html/1207.2815v2, Section 5.6,
gives the face-centred cubic entropy and endpoint hypergeometric formula.
- Borwein, Straub, Wan and Zudilin,
https://arminstraub.com/downloads/pub/walks-densities.pdf,
equations (35)-(37), for the Domb moments and their differential equation.
The family's warning about the fourteen-term Q printed in Samart's
arXiv:1205.4803v2 is preserved in the build handoff. No identities from that
section are used. Three directly read sources agree on the chosen polynomial.
The draft enumeration, completeness note and generator use this same set.
`generate.py` alone controls enumeration and values; it imports no local helper
or data file. `independent_checks.py` and `exact_checks.py` are verification
attachments. The critique should include this selection note in its attachment
fingerprint as well as all relevant source files.