back to table · edit · history · where entries came from · files · download
3952 bytes, as of the version from 2026-11-02 01:23 (current). Recorded here, not run.
# Argument selection for T663
This note uses the coordinated local/live skill archived in
`agents/runs/20261101T235407Z-build-skills/manifest.json`.
Both files have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The only free parameter is the real coefficient k in the negative constant
term of P_k = x+x^(-1)+y+y^(-1)+z+z^(-1)-k. Natural logarithms and normalized
Haar measure are fixed. We use k >= 0 because changing all three torus
variables to their negatives proves m(P_-k)=m(P_k). Integer identities are
exact, even though the full parameter domain is continuous.
The enumeration is 0,1,2,3,4,5,6,7,8,9,10,11,12,18, in ascending order.
The mixed groups below are disjoint and exhaustive. Membership in one group
does not erase additional mathematical properties of that member.
* Intrinsic: 0,2,6. Zero is the symmetry center and has the Smyth evaluation.
The nonnegative critical values of 2 cos(u)+2 cos(v)+2 cos(w) are 2 and 6:
stationarity requires each angle to be 0 or pi. The value 6 is the torus
boundary and gives the simple-cubic spanning-tree entropy. All three also
occur in the published CM list.
* Arithmetic structure: 3,10,18. Together with 0,2,6 these cover all six
explicitly exhibited integer CM examples in Bertin, Feaver, Fuselier,
Lalin and Manes, arXiv:1208.6240, section 2.3. We do not assert that these
are the only CM parameters. Their quadratic tau values provide independent
modular computations, and the paper proves special L-value identities.
* Exploratory reference sample: 1,4,5,7,8,9,11,12. These complete the integer
coefficient slice 0 <= k <= 12 around the structural and arithmetic cases.
They are small integral Laurent polynomials in a fixed, unit-coefficient
normalization, suitable for identifying values encountered in comparisons
of this K3 pencil and its simple-cubic logarithmic potential. No particular
application or CM property is asserted for each of these inputs.
Reference policy:
- Purpose: bounded exploratory identification of the ordinary measures of
simple integral members of this particular six-term pencil.
- Bounds: the eight explicitly listed integers, completing 0 <= k <= 12.
No other fractions, radicals, or independent variables are sampled.
- Coverage rationale: the integer slice includes every integral interior
parameter, the transition at 6, and an exterior interval reaching twice the
edge of the torus spectrum. Its cutoff 12 is a practical, noncanonical
reference choice. The isolated value 18 adds a documented arithmetic
example rather than extending the entire slice. This prioritizes coverage
of the different regimes and CM examples before increasing exterior depth.
- noncanonical_acknowledged: true.
- notation_independent: true. The cutoff is described relative to the spectral
edge and the inputs are integral coefficients, independent of numeral base.
- sources_checked: true. Bertin arXiv:math/0501153, page 2 and Theorem 1.1;
Bertin et al. arXiv:1208.6240, sections 1 and 2.3; Rogers arXiv:0704.2438,
equation (1.1) and Theorem 2.5, with its stated domain retained. These
sources establish the function, conventions and arithmetic examples, not
a published use for every exploratory input.
Breadth review: both torus-critical values, the symmetry center, every
integer inside the torus-zero interval, a short exterior reference slice,
and all six cited integer CM examples are represented. Negative arguments
would duplicate values by a proved symmetry. Arbitrary rational and radical
meshes would require separate reasons. The Q_k pencil, derivatives, exponentiated
measures and modular L-values are different quantities and are not extra rows.
This is the builder's justification, not the independent argument-selection
review. The controller's separate critique must inspect the live enumeration,
this note, generate.py and checks.py and bind all relevant attachments.