# Exact computation and independent checking of T687

The definitions and argument selection use the skill versions recorded in
`agents/runs/20261109T100832Z-build-skills/manifest.json` (also attached as
`skill-manifest.json`). Both SHA-256 values are
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.

## Exact algorithm

Write u_m^(N)(d) for the Gamma_0(N) trace with b fixed to any one root h of
-d modulo 4N, with h understood modulo 2N. Matsusaka (2017), Theorems 3.2
and 3.4, identifies the series

    sum_{n>=0,r in Z} u_m^(N)(4Nn-r^2) q^n z^r

as a weight-2 weak Jacobi form of index N for m=1,2 and these levels.
The prescribed singular coefficients are

    u_1(0)=2, u_1(-1)=-1;
    u_2(0)=6 (N odd), 10 (N even);
    u_2(-1)=-1, u_2(-4)=-2 (N odd), -4 (N even).

All other negative-discriminant coefficients vanish. No negative powers of
q occur: for each residue class of r, choose |r|<=N; the listed negative
discriminants have n>=0 whenever they are admissible. The trace theorem
supplies existence, including every imprimitive form.

The ring of even-weight weak Jacobi forms is generated over Q by E4,E6 and
A=phi_{-2,1}, B=phi_{0,1}; see Zagier, section 8, and Amir, Part II,
Proposition 13 and its subsequent construction. The generator explicitly
lists every monomial

    E4^u E6^v A^a B^(N-a), 0<=a<=N, 2u+3v=a+1.

It matches all singular coefficients with 0<=r<=N and 0<=n<=r^2/(4N).
Exact rational elimination asserts consistency and full column rank. Thus no
unexamined holomorphic Jacobi-form ambiguity is assumed away: uniqueness is
certified by that rank calculation on the complete basis, separately at each
level. The solution must equal the trace form given by the cited theorem.

The implementation uses the product

    A=(z-2+z^-1) product_{k>=1}
          (1-z q^k)^2(1-z^-1 q^k)^2/(1-q^k)^4

and B=-24 q*dA/dq+6 (z*d/dz)^2 A-5 E2*A. The latter formula follows from
the covariant heat operator: Richter, The action of the heat operator on
Jacobi forms, Proceedings AMS 137 (2009), 869–875, equation (3), DOI
10.1090/S0002-9939-08-09566-X. For weight -2 and index 1, multiplication of
the covariant derivative by -6 gives weight 0 and constant Laurent term
z+10+z^-1, fixing B. E2,E4,E6 are expanded by exact divisor sums.
Finite products and convolutions are truncated only beyond the required
q-degree; they have nonnegative q-degrees, so this does not affect any
retained coefficient. No numerical series evaluation enters the exact data.

Finally, Matsusaka, Lemma 4.2 and its remark for general f, gives

    t_m^(N*)(d)=u_m^(N)(d)/2^omega(gcd(N,d)).

The generator performs this division in QQ. Every repeated coefficient
with the same discriminant and |r|<=N is checked for agreement, including
the composite levels. All admissible d through 200 must be present.

## Independent CM calculation

`checks.py` uses a different algorithm: reduced integral positive definite
forms [a,b,c] of discriminant -d, without a primitivity filter, and every
projective first column in P^1(Z/NZ) for the level structure. A primitive
integer lift is completed to a determinant-one matrix. Transforming the
form by this matrix and requiring N to divide the first coefficient gives
all Gamma_0(N) classes after quotienting by the reduced form's automorphisms.
Their projective stabilizer orders are computed from these finite orbits.
The full modular stabilizers have orders 3,2,1 in the hexagonal, square,
and other cases, including integer multiples of the first two forms.

The sum over all Gamma_0(N) classes, divided by 2^omega(N), equals the
sum over the full Atkin–Lehner quotient by the orbit-stabilizer formula.
The attachment `orbit-evidence.json` deliberately records Gamma_0(N)
representatives and effective summation weights 1/(2^omega(N)*w_0), not
purported unique representatives of the larger quotient. It preserves all
base forms, level structures, transformed forms, and w_0.

CM arguments and Dedekind eta are evaluated at 640 bits with Arb.
The eta formulas are those of Matsusaka, section 1, including the 6A and
10A formulas. The Faber values are j_N^* and (j_N^*)^2-2c_N(1).
Every resulting complex sum is finite, has each radius below 10^-100,
contains the exact rational in its real part, and contains zero in its
imaginary part. No integer or rational is inferred from these digits.
The largest observed component radius is 5.3802565e-170.

All 810 entries pass. Also, partitioning the same Gamma_0(N) orbits by
b modulo 2N verifies all 1392 fixed-residue sums and the conversion factor
for every admissible root, independently of the Jacobi implementation.
Known-value controls are j(i)=1728, j_2^*(i)=544,
j_2^*((1+i)/2)=-104, j_2^*(i/sqrt(2))=152 and
j_3^*((-3+i*sqrt(3))/6)=-42. Deliberately wrong, nonfinite and wide balls
must fail before the comparisons are trusted. These controls give starred
traces -26, -23, 76 and -7 at (N,d)=(2,4),(2,7),(2,8),(3,3).
The ramified examples in Zagier's section 8 have different weights, so they
are used as individual CM-value controls, not copied as starred traces.

## Fourier checks

A separate exact Euler-product implementation computes the starred and
unstarred Hauptmodul expansions. Matsusaka's Theorem 1.1 is checked for
unstarred coefficients at n=1..50 at all seven levels (350 identities),
and for starred coefficients at n<=floor(200/(4N)) (74 identities).
The auxiliary nonpositive-discriminant trace values in that theorem are
used only inside these checks and are not table entries.

All 360 overlapping coefficients (both starred and unstarred, n=1..30,
levels 2,3,5,6,7,10) agree with the published T652 snapshot. T652 does not
cover level 13. Its absence is not filled by fabricated companion entries.

`checks.py --stored` repeats the CM and Fourier checks using the values
read from the website. `generate.py` separately runs verify(sample=None).
The checks are part of this build; the controller still requires a distinct
critique session and argument-selection sidecar before offering the draft.