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# Argument selection for T650
The sole parameter is the real Hessian parameter t, with domain -1 < t < 27.
The coefficient of xy is its real cube root k. Neither k nor a CM discriminant
is an independently varied parameter. This table holds the signed modified
period I(t)-2J(t), one named scalar, in the normalization of issue 285 and T649.
The entire selected group has role `arithmetic_structure`. Its arguments are
exactly the interior members of the three rational CM parameters and both
embeddings of all fifteen quadratic orbits in Samart–Tao v3, Table 1, rows
13–27. The rational candidates are -216,24,54. The quadratic catalogue is
transcribed in catalogue.py, and selected() tests -1<t<27 in Sage's exact real
algebraic field AA. This yields 24 and the interior conjugate from rows
14,17,19,22,23,24,25,26,27, ten entries in total. All ten happen to be positive;
positivity was not an additional selection filter. Their exact identities are:
- 24
- 18-6*sqrt(3)
- -27/2+27/2*sqrt(5)
- -4320+1944*sqrt(5)
- -864+216*sqrt(17)
- -163296+35640*sqrt(21)
- 4+4*sqrt(33)
- -16740+2916*sqrt(33)
- -55296+8640*sqrt(41)
- -13500000+1431000*sqrt(89)
The arithmetic reason is complex multiplication of
C_t: Y^2=X^3-27t^2X^2+216t^3(t-27)X-432t^4(t-27)^2.
These modified periods occur in the rational and quadratic CM L-value
identities of the cited papers. The argument is not that simple radicals are
intrinsically useful. cm_checks.py verifies the model's j-invariant formula
symbolically and H_D(j(C_t))=0 exactly for every candidate, using independently
computed Hilbert class polynomials. Source row and discriminant are provenance,
not extra table parameters.
The degree-two extent is an explicitly noncanonical computational boundary.
It covers the rational case and every real quadratic orbit in the chosen
catalogue before extending to the more elaborate quartic catalogue. It is not
an assertion that all CM parameters of arbitrary degree are covered, and the
completeness flag is no. The domain restriction removes conjugates where the
modified period is not defined; the ordinary companion retains all embeddings.
The sole rational value is shown first, followed by the source-catalogue order,
using exactly the machine identities in T649.
Breadth was reviewed before depth: the selection spans nine quadratic orbits
in seven real quadratic fields plus the rational CM case, rather than
extending one radical family. Intrinsic t=0, non-CM integer coefficients, and
quartic CM arguments would be legitimate candidates for a later extension of
this same scalar table. They are not needed to complete this proposal's
arithmetic slice. The torus boundaries -1 and 27 belong to the ordinary table;
they are outside this table's open domain. No reference mesh, probability
input, arbitrary coupling or Cartesian product is present. Ten is the result
of the exact domain test, not a target count or minimum.
Sources checked: Samart, corrected arXiv:2301.05390v3, introduction, Theorem 1
and Lemma 3, for the branch, sign and hypergeometric formula; Samart–Tao,
arXiv:2409.14714v3, introduction, Section 2, equation (1.7), Section 3,
Theorem 5.2 and Table 1 rows 13–27, for the parameter normalization, CM
catalogue and arithmetic use. The builder read the live T649 document and
its exact keys, plus T129, T9 and the surrounding corpus.
Instruction provenance: agents/runs/20261101T114635Z-build-skills/manifest.json.
Both archived skill-local.md and skill-live.md have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
This note is the builder's justification. It is not the required independent
argument-selection critique, which belongs to the controller's next session.