# Argument selection for T691

This build follows family proposal numberdb-data#307, read with `queue.py show 307`, and the coordinated skill archived in `agents/runs/20261109T133048Z-build-skills/manifest.json`. Both archived versions and the snapshot canonical file have SHA-256 `886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`. The skill manifest and the source snapshot are evidence for this build; an independent critic must make its own selection assessment.

## Quantity and natural domain

The quantity is the uncompleted primitive arithmetic degree-four L-value of two distinct-weight, normalized, level-one cuspidal eigenforms. Fourier expansions use q=exp(2*pi*i*z) and a_1=1. For k>ell the finite Euler roots are all four products of the roots of X^2-a_p(f)X+p^(k-1) and X^2-a_p(g)X+p^(ell-1). Every prime is included. Its completion has gamma factors Gamma_C(s) Gamma_C(s-ell+1), Gamma_C(u)=2(2*pi)^(-u)Gamma(u), conductor 1 and root number +1. Reflection is s -> k+ell-1-s. We store L, not the completion.

The eigenform indices use lexicographic real Fourier-coefficient order (a_2,a_3,...); a_2 is distinct throughout this finite roster. This agrees with T690 and with the increasing-a_2 ordering of T215, T323, T325 and T326. The natural domain is not restricted to weight 26 or to Delta; those are the present selection.

## Every free parameter

* k (discrete_index): all nonempty even-weight cusp spaces above weight 12 through weight 26, namely 16,18,20,22,24,26. Weight is an intrinsic discrete index, but the cutoff is a noncanonical computational and reference choice. Including the first quadratic coefficient field at weight 24 and the following one-dimensional space gives coefficient-field breadth before increasing weight depth. There is no weight-14 cusp space.
* i (discrete_index): every embedded normalized eigenform in each selected weight. There is one at 16,18,20,22,26 and two at 24; both real embeddings are included rather than retaining rational coefficients only. Sturm-bound identities and exact Hecke polynomials check the roster.
* ell (discrete_index): fixed at 12, the minimal nonempty level-one cusp weight. This anchored tensor slice is explicitly our reference scope, not a claim that every pair is singled out in an application. It limits the quadratic growth of all-pairs coverage while allowing comparisons with the same small cusp motive and holomorphic Eisenstein-product endpoint identities.
* j (discrete_index): fixed at 1, the sole embedded eigenform at weight 12, Ramanujan's Delta. Thus no embedding has been discarded in the anchor space.
* s (intrinsic): every integer ell <= s <= k-1. These are exactly the integers where both gamma factors and their reflected partners are regular. With ell=12 this gives s=12,...,k-1. Criticality is a gamma-pole condition, not an arbitrary integer mesh. No value is selected by magnitude, sign or numerical recognition.

There are 4+6+8+10+2*12+14=66 values across seven embedded tensor products. Enumeration is increasing k, then i, then ell,j (fixed), then increasing s. All variables name the forms and the argument of one named quantity. No continuous exploratory samples or mixed-role groups are included. The anchor and weight bound are notation-independent reference choices within discrete families.

## Breadth and exclusions

The slice includes all intervening nonempty weights and both embeddings of the first quadratic coefficient field. Raising the weight limit, varying both factors, changing level, or adding characters would introduce substantially larger coefficient sets or new ramified-factor conventions. Those extensions are deferred rather than absorbed into this proposal. Equal-weight pairs have an empty critical set for this gamma-pole definition. Symmetric squares are a separate quantity in T690. Ordinary L-values and norms remain T323/T326 and T683/T684. The campaign's statistical quantile exclusion is inapplicable to these critical integers of motivic L-functions.

## Checked supporting sources

* Lei, Loeffler and Zerbes, *Euler systems for Rankin–Selberg convolutions of modular forms*, Annals of Mathematics 180 (2014), sections 4.1 and 4.2, especially Proposition 4.1.2, Definition 4.1.4 and Proposition 4.1.5. Their Gamma_C omits the factor 2, so their completion is one quarter of ours. This constant does not change L or criticality. https://annals.math.princeton.edu/wp-content/uploads/annals-v180-n2-p06-p.pdf
* T215, exact level-one Hecke polynomials and the level-one dimension formula, plus the exact roster in T690. https://numberdb.org/T215 and https://numberdb.org/T690
* T323, T325 and T326, read live for arithmetic Fourier normalization, indexing and Petersson convention. Their stored numerical values are not 100-digit oracles.
* PARI L-function manual, data structure and generic-accuracy caveat. https://pari.math.u-bordeaux.fr/dochtml/html/_L_minusfunctions.html

The bound and anchor are justified here as reference choices; these sources are not claimed to document an application of every individual selected pair.