# Argument selection for T649

This table has one free parameter, the real number t. Its value is the ordinary logarithmic Mahler measure N(t) of x^3+y^3+1-cuberoot(t)*x*y. The cube root is real, including for negative t. Natural logarithms and normalized Haar measure are used. The integration variables x and y are not parameters. The degree bound concerns t, not its cube root.

The selection is mixed, with two disjoint groups:

1. Intrinsic controls: t = -1, 0, 27. The real torus-zero region has boundary -1 and 27. At 0 the polynomial is the Fermat specialization, whose Mahler measure equals m(1+x+y). At 27 the cubic splits into three linear factors, giving three times that measure. These are intrinsic structural controls, not CM catalogue members.
2. Arithmetic structure: the rational CM parameters -216, 24, 54 and both real conjugates of every quadratic parameter in Samart–Tao, arXiv:2409.14714v3, Table 1, rows 13–27. The catalogue is transcribed exactly in generate.py. Every square root is positive before the displayed sign is applied. Representatives appear in source row order, followed immediately by their conjugates. The three controls appear first, then the three rational CM arguments.

There are 36 entries: 3 intrinsic, 3 rational CM and 30 quadratic CM. Exact algebraic comparisons classify ten CM arguments in (-1,27). All embeddings are retained, including those requiring ordinary integration inside that interval. This avoids silently replacing the ordinary measure by the modified period used in some arithmetic identities.

The finite extent is the full rational selection and the full fifteen-orbit quadratic catalogue in the chosen source, not a claim to cover every possible real argument or every presentation of a CM parameter. Degree two is a noncanonical computational boundary. It gives breadth across the published quadratic arithmetic families before expanding into quartic fields and harder embedding choices. No denominator mesh, decimal grid, Cartesian product, or exploratory reference sample has been added. The rows were not chosen from their output values or from a count target. All entries concern one named scalar N(t); modified periods and determinants belong to companion tables.

Supporting sources checked:

- Samart, arXiv:2301.05390v3, introduction (the Hesse and quadratic-polynomial Mahler measures agree), Lemma 3 and equation (2.1) (Jensen integral), Proposition 4 (torus crossings). The ordinary measure is used throughout; Theorem 1's modified period is not substituted for it.
- Samart–Tao, arXiv:2409.14714v3, introduction (the three rational CM parameters), equation (1.7) (the shared elliptic curve), Lemma 2.4 (j = t(t+216)^3/(t-27)^3), and Table 1 rows 13–27 (the full selected quadratic catalogue).
- Live T129, Hilbert class polynomials, provides an external exact check: every selected CM argument's j-invariant is annihilated by the appropriate stored Hilbert class polynomial, also recomputed with Sage. The exact checks and the discriminants are retained in cm-checks.json.
- Live T283 supplies the three-step random-walk Mahler measure used in the intrinsic controls. Live T145 supplies L(2,chi_-3) in its independent evaluation.

Skill evidence: agents/runs/20261101T105238Z-build-skills/manifest.json. Both archived local and live skills were read and hash-verified; each has SHA-256 a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278. The current balanced argument policy governs this selection. This build rationale is not an independent critique or an acceptance sidecar.