# Sources and convention checks for T696

Instruction authority: `agents/runs/20261109T182554Z-build-skills/manifest.json`.
Both archived skills have SHA-256
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
The manifest SHA-256 is
`c879ba3e40020535135932a0ac7d93e894d04b267cf416aa702c3c96617ec363`.
This document is build evidence, not an independent critique.

## Definition, rationality and controls

Jin, Ma, Ono and Soundararajan, *Riemann hypothesis for period polynomials
of modular forms*, https://pmc.ncbi.nlm.nih.gov/articles/PMC4790983/,
equations (1.1)–(1.4), were read on 2026-11-09. Their integral and arithmetic
L-value coefficient formula fix the variable and conductor convention.
Example 1 gives the explicit Delta odd shape
`4*X^9 - 25*X^7 + 42*X^5 - 25*X^3 + 4*X`, used as a nonzero control.
The integral's even exponent k-2 makes (tau-X) and (X-tau) identical here.

Pasol and Popa, *Modular forms and period polynomials*, version 5,
https://arxiv.org/html/1202.5802v5, Section 2 and Proposition 5.11, were read.
The proposition supplies coefficient-field rationality up to separate periods
for the two parity parts. Their completion in (5.7) lacks N^(s/2); the JMOS
completion includes it. In both cases the arithmetic L-value coefficient is
binomial(w,m) i^(1-m) m! L(f,m+1)/(2*pi)^(m+1) at X^(w-m).
Restricting m to odd integers gives the odd part and even critical integers.

For a rational eigenform, the one-dimensional rational period map has the
same kernel as the analytic map, hence differs by one scalar. Odd monomial
moments on {0,infinity} give the odd polynomial after binomial weighting.
The factor (-1)^m in expanding (tau-X)^w is constant across odd m; the
positive-leading primitive normalization removes this global sign.
No approximate rational reconstruction is used.

Sage's rational_period_mapping documentation was read at
https://doc.sagemath.org/html/en/reference/modsym/sage/modular/modsym/space.html#sage.modular.modsym.space.ModularSymbolsSpace.rational_period_mapping.
PARI's mfperiodpol documentation was read at
https://pari.math.u-bordeaux.fr/dochtml/html-stable/Modular_forms.html#mfperiodpol.
Its flag -1 selects the odd part, and its displayed Delta example agrees
with JMOS. The neighboring mfperiodpolbasis prose reverses the order of the
flag meanings while its examples agree with mfperiodpol; this build does
not use that basis routine. The known Delta control checks the actual
routine rather than inferring semantics from a neighboring description.

## Relations checked before use

Let w_N be fixed by f(-1/(N*tau))=w_N*N^(k/2)*tau^k*f(tau).
The completed functional equation has epsilon=(-1)^(k/2)*w_N. Substituting
it into the coefficient formula gives

r(X) = -w_N*N^(w/2)*X^w*r(-1/(N*X)).

Taking the odd part changes the sign on the right. Thus

P^-(X) = w_N*N^(w/2)*X^w*P^-(1/(N*X)).

The power of X is w, not the actual odd-part degree w-1. This derivation is
used rather than copying the apparent exponent typo in the displayed
functional equation following (1.4) in the JMOS HTML. At coefficient j the
check is p_j = w_N*N^(j-w/2)*p_(w-j). Every stored polynomial satisfies it
exactly, with w_N independently computed from the Fricke action on Fourier
coefficients through the Sturm bound. At level one the order-three period
relation from PP Section 2 and the values at +1 and -1 are checked exactly.
No higher-level scalar Manin relation is asserted.

The Mellin check splits the integral at T=1/sqrt(N). For integer s,

I_s = sum(n>=1) a_n*exp(-2*pi*n*T)*(s-1)!/(2*pi*n)^s
      * sum(j=0..s-1) (2*pi*n*T)^j/j!,

and the full moment is I_s+epsilon*N^(k/2-s)*I_(k-s).
450 terms at 150 working digits and 600 at 190 are independent of PARI's
period-polynomial evaluation. Fourier coefficients are shared inputs, with
100 coefficients cross-checked against the exact Sage forms. These finite
sums are corroboration, not rigorous analytic enclosures. Integer polynomial
exactness comes from the modular-symbol construction.

## Roster, boundaries and corpus

Stein's Dimension Formulas, Propositions 6.1 and 6.4,
https://wstein.org/books/modform/modform/dimension_formulas.html, were read.
The genus/cusp/elliptic-point formula and divisor new/old recursion agree
with Sage and PARI in all 21 considered spaces, including empty spaces.
Every embedded form is ordered by exact real algebraic Fourier coefficients
through a Sturm bound. The 33 indices agree with the live T694 roster; the
18 rational indices also agree with T695. These companion tables were
read live, including their attached generators and evidence. The present
generator and analytic checks adapt that reproducible shared construction
to odd moments and odd parity; they do not copy its even values. All values
and comparisons were recomputed in this run.

T323 provides the same increasing-a_2 order at level one. T683 has weights
3, 4 and 6, so there is no higher-level overlap with our weight-twelve slice
and no LMFDB label mapping to claim. The LMFDB label documentation at
https://www.lmfdb.org/knowledge/show/cmf.label was read; embedding suffixes
were not guessed from software order. T80, T215, T326 and T684 were read,
and canonical links were obtained from live search results. The T694 and
T695 long URLs were read from their actual pages.

Both already_here and already_asked were run. The authenticated paginated
catalogue contained T694 and T695, but no odd-part counterpart. Polynomial
lookup found no Delta odd-shape match; X^9-X had no match and X^3-X matched
unrelated existing values, which is not a duplicate of this family. The
three used tags exist. The absent tag `period polynomial` is proposed for
T694, T695 and T696 together; this run does not create a singleton tag or
modify its companions.

The published companions were not edited. No request provenance belongs in
the mathematical prose, and issue #262 remains only partially addressed.