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# Quadratic CM determinants of Hessian Mahler measures (T651)
The only free parameter is the real number `t`, of type R. It is a real
quadratic CM parameter for the Hessian curve C_t, with the positive square
root and orientation fixed by Samart–Tao, arXiv:2409.14714v3, Table 1,
rows 13–27. Its role is **arithmetic_structure**. Neither the source row
number nor the matrix coefficients a,b are free parameters. The radical
identities match the ordinary and modified companion tables T649 and T650.
The named quantity is the signed determinant
D(t) = det([[a nu(t), b nu(t^sigma)], [-nu(t^sigma), nu(t)]])
= a nu(t)^2 + b nu(t^sigma)^2.
The normalization is the source's equation (5.5), with no absolute value,
primitive rescaling, or replacement of an interior period by its ordinary
Mahler measure. Each a,b is fixed data attached to t. The coefficient of xy
in the Hessian polynomial is the real cube root of t; the degree bound is
on t itself. For -1<t<27, nu(t)=I(t)-2J(t); elsewhere nu(t)=N(t).
## Purpose and extent
This includes all fifteen oriented real quadratic cases in the source's
normalization catalogue. The source's Theorem 5.2 links these particular
matrices to special values of the CM curve's L-function; the arguments and
coefficient pairings are not independently sampled from a mesh. Exact
Hilbert class polynomial tests independently check complex multiplication
for both embeddings of each argument. There is no reference_sample group,
no decimal grid, and no arbitrary coupling hidden beside a discrete index.
The square roots are positive; changing their signs exchanges the two
embeddings but does not preserve the selected matrix normalization.
The degree-two boundary is a noncanonical computational extent. It covers
all quadratic orbits in the checked catalogue before the more involved
quartic matrices. Ordinary rational CM values do not add two-dimensional
conjugate determinants. Non-CM, complex-quadratic and quartic determinants
would need their own source-specific definitions and are not silently
included. This breadth across all fifteen quadratic orbits is preferable
to extending only one field or retaining only the best-known row 16.
The finite bounds are structural and independent of numeral spelling.
The full enumeration and prescribed coefficients are in catalogue.py.
It contains 15 oriented arguments; the scalar computations use 30 distinct
embeddings, of which 9 lie in (-1,27). The final completeness note gives
this same catalogue and extent. `generate.py` enumerates exactly this list.
The signed determinant is one quantity; N and the modified period are
inputs, not separate value columns. No repeats declaration applies: this
is a determinant of scalar periods, not a duplicate scalar computation.
## Checked sources and companions
* Samart–Tao v3, https://arxiv.org/html/2409.14714v3, Sections 1–3,
Theorem 5.2, equation (5.5), Table 1 rows 13–27 and Table 4 row 16:
exact arguments, fixed normalization, CM model and determinant identities.
* Samart corrected v3, https://arxiv.org/pdf/2301.05390v3,
Theorem 1 and Lemma 3: the signed modified period and root integrals.
* Tao–Guo v2, https://arxiv.org/html/2310.12510v2, Theorem 1.4 and its
proof: the independent row-16 identity uses m_3=3N; its determinant is
nine times D(t), and its L-value factor becomes 2187/pi^4 here.
* T649 and T650 were fetched with authenticated reads. Their exact entry
keys and definitions determine the companion links. Their canonical URLs
are https://numberdb.org/T649 and https://numberdb.org/T650; the searches
do not list these drafts. T129 supplies Hilbert class polynomials.
The live tag list has number theory, special values, elliptic curves,
integral and L-function, which are reused. A Mahler measure tag is proposed
for human consideration: T282, T283, T649, T650 and T651 would use it.
No new tag or published-table edits are made by this run.
## Archived instructions and independent review boundary
Run: 20261101T124411Z-build.
Manifest: agents/runs/20261101T124411Z-build-skills/manifest.json in the
source snapshot; an unchanged copy is attached as skill-manifest.json.
Both archived local and live skills have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The copies matched byte for byte; skill-evidence.json records their paths
and hashes. The current skill governs this selection, not the historical
policy digest in campaign-gate.md.
This is a builder's justification, not independent argument-review approval.
The separate critique must inspect the live draft and fingerprint at least
`generate.py`, `catalogue.py`, `analytic.py` and `argument-selection.md`, as
well as any other dependency it identifies as controlling the selection.
The builder neither creates nor accepts the critique's -arguments.json file.