# Exact coefficient computation for T607

The definition and normalization are those of Zhao (2010), introduction and
(2.1), and Komori-Matsumoto-Tsumura IV (2009), (1.2)-(1.4) and (2.4).
The values are Z_g(s)=M^s sum over positive a,b of P(a,b)^(-s), where

| Type | P(a,b) | N | M | Weyl group order |
|---|---|---:|---:|---:|
| A2 | ab(a+b) | 3 | 2 | 6 |
| B2 | ab(a+b)(a+2b) | 4 | 6 | 8 |
| G2 | ab(a+b)(a+2b)(a+3b)(2a+3b) | 6 | 120 | 12 |

N is the number of positive roots. The coordinate variables are positive
integers shifted by the Weyl vector. At (1,1) the dimension P/M is 1.
Since s=2m is even, the summand is invariant under Weyl reflections. Regular
integral weights have one representative in each Weyl chamber. Thus the
positive-chamber sum is 1/|W| times the sum over all (a,b) in Z^2 with no zero
factor. This is the lattice-sum identity in Zhao (2.1).

Set w=Ns. For fixed nonzero b, P(a,b)^(-s)=b^(-w) f(a/b), with
f(x)=L^(-s) product_(r in R) (x-r)^(-s). Here R is {0,-1} for A2,
{0,-1,-2} for B2, and {0,-1,-2,-3,-3/2} for G2. L=1,1,2 respectively.
Exact Taylor expansion at each pole gives
f(x)=sum_(r,k=1..s) c_(r,k)/(x-r)^k.
The generator computes these coefficients by finite convolution over Q;
checks.py verifies each resulting rational-function identity exactly.

For even k>=2, the full integer sum of (a-rb)^(-k), omitting its pole when
integral, equals 2*zeta(k) when rb is integral and
2*(2^k-1)*zeta(k) when rb is half integral. For odd k it is zero under
symmetric summation. In particular the k=1 principal values vanish for
integer and half-integer shifts. Finite partial fractions preserve the inner
symmetric limit; the original rational function is absolutely summable.
Every outer series used after evaluation has exponent at least w-s>=4,
so no rearrangement of a divergent outer series is needed.

The excluded integers a=q*b for q != r must also be removed. Such a point
exists exactly when den(q) divides b. Its total contribution, after summing
nonzero b, is 2*c_(r,k)*(q-r)^(-k)*den(q)^(-w)*zeta(w).

Write e_j=zeta(j)/pi^j for even j, obtained exactly by Euler's Bernoulli
formula (DLMF 25.6.2). The full lattice coefficient of pi^w is

sum_(r,k even) 4*c_(r,k)*e_k*e_(w-k)*h_(r,k)
 - sum_(r,k) sum_(q != r) 2*c_(r,k)*(q-r)^(-k)*den(q)^(-w)*e_w,

where h=1 for integral r and
h=2^(-(w-k))+(2^k-1)*(1-2^(-(w-k))) for the sole half-integral pole.
Multiply this by M^s/|W| to obtain the representation-sum coefficient.
A1 is zeta(2m). All arithmetic up to multiplying by pi^w is exact rational
arithmetic. The final multiplication uses Arb balls at bits(100,losing=96).
All coefficients are positive rationals; positive powers of pi make these
values transcendental, so none of the entries should be written as a rational.

## Independent checks

* A1: Euler's four known rational coefficients and Arb's zeta evaluation.
* A2: Zhao (2010), equation (1.1), using an independent triangular Bernoulli
  algorithm for exact comparison and a separate 600-bit Arb zeta-product sum.
* B2: the four factored exact pi-power coefficients on page 8, section 4 of
  Zhao, *Alternating Euler sums and special values of Witten multiple zeta
  function attached to so(5)*, MPIM 2009-42 / arXiv:0903.0473. Multiply by 6^(2m).
* G2: the four exact rational coefficients of KMT IV, Example 2.2, page 5.
  Multiply by 120^(2m).

These source coefficients are checker data only. The generator does not
import them. Every exact coefficient must agree before decimal agreement
is considered. A perturbed value must fail the narrow-ball comparisons.
The checker's zeta(2)=pi^2/6 control is nonzero and is run first.

For additional checks against the original positive sums, all three rank-two
series are summed over 1<=a,b<=B=256 with 600-bit balls, at all four arguments.
Since P>=Ca*a^(N-1)*b and P>=Cb*a*b^(N-1), the omitted positive tail is at most

M^s*(Ca^(-s)+Cb^(-s))*zeta(s)*B^(1-(N-1)*s)/((N-1)*s-1).

(Ca,Cb) is (1,1), (1,2), (2,18) for A2,B2,G2. This follows by a union bound
on a>B or b>B and the decreasing-series integral bound. The finite sum is a
strict lower bound. Both limits must contain the exact-formula ball, and the
tail must be less than 10^(-5); adding 10^(-4) must fail. A one-dimensional
zeta(2) finite-sum control tests the same inequality direction first.
These modest-precision tail checks verify interpretation and normalization;
the exact independent formulas verify the full claimed precision.

The check is repeated on the decimal intervals read from the website.
Each 600-bit independent ball must lie strictly within the stored last-place
interval, and a ten-interval-width perturbation must fail. All values decrease
strictly with m and exceed 1, as follows from the positive representation sum.

Skill manifest: agents/runs/20261029T103821Z-build-skills/manifest.json.
Both archived skills have SHA-256
28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199.