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5633 bytes, as of the version from 2026-11-02 20:37 (current). Recorded here, not run.
# Argument selection for T684
This build follows the coordinated instructions archived in
`agents/runs/20261102T192440Z-build-skills/manifest.json`. Both archived versions
have SHA-256 `a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The attached `skill-manifest.json` preserves that manifest. This is the builder's
justification, not the independent critique's acceptance.
## Quantity and conventions
The quantity is the Petersson self-product of an embedded normalized cuspidal
newform, with first Fourier coefficient 1. We retain both the unaveraged
integral over Gamma_0(N)\H with measure y^(k-2) dx dy, and that integral divided
by mu(N)=[SL_2(Z):Gamma_0(N)]=N product_{p|N}(1+1/p). They determine one another
by a known positive integer; neither is the square root of the self-product.
They are not norms of oldform degeneracy copies, cross-products between forms,
or symmetric-square values whose local Euler factors are left unspecified.
Brown--Klosin, section 2 (arXiv:1402.0900), specifies the unaveraged integral.
PARI's mfpetersson manual specifies the division by the subgroup index.
## Parameter form: disjoint arithmetic families
The public identity is the full embedded LMFDB label N.k.a.x.n.i. Underneath
that identity, level, weight, character, orbit, and embedding all need reasons.
We use exactly the roster attached to the live companion T683. This follows
family issue 303's shared conventions and allows one-to-one comparisons of
critical Mellin periods and Petersson self-products. Every nonempty newspace
contributes every orbit and every embedding, including complex conjugates.
The cutoffs are computation choices, not canonical boundaries.
* A, discrete_index: weight 4, trivial character, exact levels 2 through 30.
This is the first even weight beyond the elliptic-curve case. The range
includes prime, composite and nonsquarefree levels and nonrational fields
through degree five. All 45 orbits and 61 embeddings are included.
* B, discrete_index: weight 6, trivial character, exact levels 2 through 15.
A second even motivic weight broadens the period comparison. Its shorter
range keeps the common roster practical for both this table and T683,
where each form has five critical integers. All 20 orbits and 27 embeddings
are included. A uniform level cutoff across unrelated weights was not
imposed merely to produce a rectangle.
* C, arithmetic_structure: weight 3, primitive odd quadratic character
chi_D(m)=(D/m), exact level N=|D|, for negative fundamental discriminants
|D|<=40. This is the first odd weight above two and includes CM and non-CM
forms, with coefficient degrees through eight. All 25 orbits and 50
embeddings are included. The possible D are -3,-4,-7,-8,-11,-15,-19,-20,
-23,-24,-31,-35,-39,-40. Empty newspaces contribute no rows. Primitivity
fixes the relation between character and level; there is no Cartesian
product of all small characters and levels.
These are discrete families of arithmetic objects with intrinsic newform
normalization, not a continuous reference mesh. Neither rational coefficients
nor CM is used as a selection filter. Breadth across weights, character types,
coefficient fields and level factorizations precedes greater depth in one
slice. The finite extent serves recognition of low-conductor modular periods,
without asserting a separately documented application for each form. No
quantile, statistical threshold, decimal grid, or exploratory real parameter
is present. Higher weights, nonquadratic characters, larger levels, twists
outside these newspaces, and Hilbert forms are outside this finite coverage.
The exact reconstruction and newspace dimension checks independently test
whether every selected space is represented fully. The 138 embedded labels
are enumerated in `roster.json`, grouped by its A/B/C field and ordered by
level, weight, character, orbit, and embedding. Its coefficient-field
polynomials, isolating rectangles and exact Fourier coefficients through the
Sturm bound fix the embedding, independently of software list order.
`forms.py` reconstructs each orbit under an exact coefficient-field map.
The LMFDB label convention is in its Labels page, and its Reliability page
reports independent catalogue computations throughout this small Nk^2 range.
These are catalogue claims, not a claim about 100-digit Petersson accuracy.
## Parameter normalization: finite convention index
Both `integral` and `index-normalized` occur for every embedded form, giving
276 entries. This is an intrinsic finite choice of two invertibly related
conventions, with the exact conversion mu(N). The complete-note states all
three form slices and both conventions. The table's parameter constraint
states the family domain, not the afternoon's level bounds.
## Independent review boundary
The separate critic should assess form as mixed, with exhaustive disjoint
full embedded label lists for A, B and C, and normalization as the finite
intrinsic convention choice. All selection dependencies must be fingerprinted:
`generate.py`, `forms.py`, `roster.json`, and this note. The reproducible checks
are `checks.py`, `quadrature.py`, `check.py` and `dry_run.py`. No independent
argument-review sidecar is created by this build.
Checked sources: PARI/GP modular-form manual (mfinit, mfeigenbasis, mfembed,
mfsturm, mfsymbol and mfpetersson); Belabas--Cohen, sections 2, 6 and 9.2
(arXiv:1810.00547); Brown--Klosin, section 2 (arXiv:1402.0900); LMFDB Labels
and Reliability pages. Their actual retrieved bytes and the live T683
attachment hashes are recorded in `source-provenance.json`.