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5552 bytes, as of the version from 2026-11-09 18:56 (current). Recorded here, not run.
# Argument selection for T695
The instruction authority is the coordinated archive at
`agents/runs/20261109T174608Z-build-skills/manifest.json`.
Both local and live skill copies have SHA-256
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
`skill-evidence.json` records the manifest digest and both files.
This build rationale is not an independent argument-review acceptance.
## Quantity and conventions
One value is the primitive integral polynomial with positive leading
coefficient proportional to the even-in-X part of
r_f(X) = integral from 0 to i infinity of f(tau)(tau-X)^(k-2) d tau.
The form is normalized by a_1=1, has trivial character and exact level N,
and has rational Hecke field. No change of variable, reciprocal operation,
forced-factor removal, or deletion of the constant term is made. Odd parts
and full-polynomial roots are distinct quantities. The parity shapes alone
do not supply the relative period scale needed to recover the full roots.
## Every free parameter
All three parameters have role `discrete_index`; none is a sampled real input.
Their joint domain is a roster of arithmetic objects, not a Cartesian grid.
* N is the exact level. Group A fixes N=1 to cover the first full-level cusp
forms. Group D fixes weight twelve and includes every exact level 2 through
10. This supplies conductor breadth, including prime, composite and
nonsquarefree levels, before extending the depth of the full-level slice.
The cutoff ten is a noncanonical reference and computation choice.
* k is the even weight. In A every nonempty cusp space at even k<=26 is
considered, then the rational-field restriction is applied. This covers the
first form Delta, all intervening spaces, and the next rational space after
the first nonrational space at weight 24. The resulting polynomial degrees
are at most 24. In D the fixed weight twelve is the first full-level cusp
weight and makes a direct comparison across levels with richer parity
structure than the small forced binomials in the lowest weights. The bound
26 is noncanonical; it is not a claim that later weights lack interest.
* i is the one-based lexicographic order of all embedded normalized newforms
in a space by real (a_2,a_3,...). All real embeddings of nonrational Hecke
fields participate in the ordering. Exact algebraic comparisons through
the Sturm bound distinguish them; only afterward are nonrational forms
excluded from the stored values. Every rational form in each selected
space is included, with original indices preserved. No index cutoff is
imposed independently of its space. This is the same index as T694.
The coefficient-domain restriction permits exact searchable Z[X] values.
It does not claim that nonrational forms are less interesting. Taking their
field norms would change the quantity; using a field generator as another
free polynomial variable would misrepresent it. T694 retains their full
complex roots. No arbitrary evaluations of these polynomials are included.
## Breadth and actual extent
The exact decomposition has 33 embedded forms in the considered nonempty
spaces: eight in A and 25 in D. Six in A and twelve in D have rational Hecke
field. All eighteen even parts have degree k-2 and are nonzero; there are no
vanishing-part omissions. Their identities are:
A: (1,12,1), (1,16,1), (1,18,1), (1,20,1), (1,22,1), (1,26,1).
D: (3,12,1), (4,12,1), (5,12,2), (6,12,1), (6,12,2), (6,12,3),
(8,12,2), (9,12,1), (9,12,3), (10,12,1), (10,12,2), (10,12,3).
Level two has no weight-twelve newforms. Level seven has five embedded
weight-twelve newforms, all nonrational. Weight 24 at level one likewise has
no rational form. These are exact domain-filter outcomes, not omissions
chosen to make the table smaller. Retaining the entire rational roster gives
more than famous examples while keeping the two meaningful slices. The
measured longest polynomial and block size are recorded in numerical-evidence.json.
The default enumeration and complete-note describe exactly this selection.
## Checked sources and relationship to companions
* Jin, Ma, Ono and Soundararajan, equations (1.1)-(1.4) and Example 1,
https://pmc.ncbi.nlm.nih.gov/articles/PMC4790983/ : integral and completion
conventions, and the explicit Delta parity shape.
* Pasol and Popa, sections 2 and 5.3, Proposition 5.11,
https://arxiv.org/html/1202.5802 : parity and coefficient-field rationality.
* Stein, Dimension Formulas, Propositions 6.1 and 6.4,
https://wstein.org/books/modform/modform/dimension_formulas.html : full/new
dimension formulas. These and an independent PARI decomposition verify
completeness, including empty spaces.
* Sage modular-symbol documentation, rational_period_mapping and
modular_symbol: exact arithmetic construction and monomial-path convention.
* T694 was read from the live website, including its attached exact roster.
Exact embedded Fourier comparisons check all 33 common form indices.
T323 uses the same ordering on A. T683 currently covers weights 3, 4 and 6,
so its roster has no overlap with this table's weight-twelve higher-level
slice; no LMFDB labels are guessed or assigned from software list order.
No `reference_sample` or mixed-role attestation is appropriate here: these
are discrete object indices. Finite arithmetic coverage can be noncanonical
without becoming a grid of continuous arguments. A separate critique must
independently assess these reasons and bind its sidecar to the live selection
and all generator source dependencies.