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# Exact continuation and independent check for T631
The build uses the coordinated skill manifest
`../20261030T204529Z-build-skills/manifest.json`, SHA-256 for both skill copies
`10c1d8bd62d6ddfa2c016375ad82879151f0ab7a9516c2f371432108a85be009`.
## Coexact spectrum
Put m=n+1 >= 2. Dowker--Chang (arXiv:0904.0744), equations (6)-(9), give
sphere eigenvalue m^2 and SO(4) representation
V_m tensor V_(m-2) direct sum V_(m-2) tensor V_m,
where V_j is the SU(2) representation with weights -j,-j+2,...,j and dimension
j+1. The total dimension is 2(m^2-1). For quotient exponents 1,q the angles
in the two SU(2) factors can be taken as pi(1+q)/p and pi(1-q)/p. The weight
pair (a,b) has character exponent
e(a,b) = (a(1+q)+b(1-q))/2 mod p.
The numerator is even, since a,b have the same parity. The chi_u projector
keeps precisely e(a,b)=u modulo p. This is a count of complex dimensions;
there is no further factor two. Both tensor products must be included.
## Proof of the quasipolynomial, without interpolation
Let P=2p. On one residue class write m=m0+Pk, 2 <= m0 <= P+1, k >= 0.
For each residue a modulo P of parity m0, let A_a count its occurrences in
-m0,-m0+2,...,m0 and let B_a count its occurrences in
2-m0,4-m0,...,m0-2. On increasing m by P, each string gains exactly two
occurrences of every allowed residue. Its counts therefore become A_a+2k
and B_a+2k. Character exponents depend only on these residues, so
d_u(m0+Pk) = sum_[e(a,b)=u]
[(A_a+2k)(B_b+2k)+(B_a+2k)(A_b+2k)]
= c0+c1*k+c2*k^2.
This proves the formula for every k, not just the modes tested. The generator
forms the three integer coefficients directly from this identity.
Let a=m0/P and set
alpha0=c0-c1*a+c2*a^2,
alpha1=c1-2*c2*a,
alpha2=c2.
Then, initially for Re(s)>3/2 and hence by continuation,
Z_perp(s)=P^(-2s) sum_m0 sum_(j=0)^2 alpha_j zeta(2s-j,a),
log D_perp=2 sum_m0 sum_(j=0)^2
alpha_j [log(P) zeta(-j,a)-zeta'(-j,a)].
The finite sum is regular at zero. All rational inputs and coefficients are
exact. Sage's Arb polynomial `_zeta_series` computes both the Hurwitz values
and derivatives with enclosures. Arithmetic, logarithms and exponentials
propagate those enclosures. `proven` refers to the resulting numerical bounds;
it does not make correctness of the mathematical model automatic.
## Independent scalar computation
The checks use a different representation of a different operator. For a
group element let alpha=2*pi*j/p and beta=2*pi*q*j/p. Its scalar trace at
eigenvalue m^2-1 is
[cos(m beta)-cos(m alpha)]/[cos beta-cos alpha]
when the denominator is nonzero. At equal cosines the noncentral limiting
trace is m*sin(m alpha)/sin(alpha). At the identity it is m^2, and at the
antipodal element it is (-1)^(m-1)*m^2. Averaging with cos(2*pi*u*j/p)
projects the character because conjugate group elements have the same real
trace. This produces, on each residue class modulo p,
d0(m)=A_r*m^2+B_r*m+C_r.
No production weight counts are used. Direct counts of invariant homogeneous
monomials in z1,z1bar,z2,z2bar, with the degree-(n-2) radial multiples
subtracted, independently check all selected characters at m=2,...,7.
For m>=2, factor m^2-1=(m-1)(m+1). In this setting regularizing the two
linear factors gives the same derivative at zero as regularizing the product.
Here is the necessary check rather than an assumption about determinants.
Let F(z)=sum_(m>=2) d0(m)*m^(-z). Its possible poles are at 1,2,3; the pole
at 2 has residue proportional to sum_r B_r. The B terms are sine sums from
noncentral equal-angle elements, so sum_r B_r=0. The checks also verify this
cancellation numerically. The identity and antipodal traces contribute no B.
Thus F is regular at every positive even integer. The convergent binomial
expansions, after meromorphic continuation, give
[sum d0(m)(m^2-1)^(-s)]'(0)
= 2 F'(0) + sum_(k>=1) F(2k)/k,
[sum d0(m)((m-1)^(-s)+(m+1)^(-s))]'(0)
= 2 F'(0) + sum_(k>=1) F(2k)/k.
The sums over k converge geometrically because m>=2. No even-pole term is
present, so there is no multiplicative anomaly in this factorization. On
m=r+pk, r=2,...,p+1, each shifted sum reduces to Hurwitz zeta derivatives
at (r-1)/p and (r+1)/p. These are evaluated with mpmath at 175 decimal digits,
independently of the Arb calculation and with a different residue period.
Finally use the independently known spectral torsion:
log D_perp = 2(log D0-log T),
T = 2*pi^2/p^2 (u=0),
T = 4*sin(pi*u/p)*sin(pi*v/p) (u!=0),
v = u*q^(-1) mod p, 1<=v<p.
Kitaoka (SIGMA 18 (2022), 091), (4.3) and Proposition 5.1, supplies these
normalizations; the nontrivial values also agree with published T599. The
scalar p=2 and sphere controls are checked against Nash--O'Connor (1.1) and
published T600/T595. The mpmath comparison is corroboration only, not the
basis for the rigorous stored error claim.
## Further controls
Before broad comparisons, nonzero closed-form scalar and coexact values at
(p,q,u)=(2,1,1) must pass. All exact polynomial coefficients are compared
under u -> -u, q -> -q and (q,u) -> (q^(-1),u*q^(-1)). Summing all character
multiplicities yields 2(m^2-1) coefficient by coefficient. Summing the
computed logarithms gives the sphere logarithm
zeta(3)/pi^2 - 2 log(2*pi).
All balls must be finite and have small measured radii. Production values
are recomputed at two precisions. After filling the draft, the checks read
its actual decimals and compare the independent scalar/torsion result with
the interval denoted by each decimal's last place. A ten-last-place
perturbation must fail, so a vacuous comparison cannot silently pass.