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# Scalar continuation and independent validation for T632
Policy evidence: `../20261030T212747Z-build-skills/manifest.json`. Both skill
copies have SHA-256
`10c1d8bd62d6ddfa2c016375ad82879151f0ab7a9516c2f371432108a85be009`.
## Exact scalar multiplicities
Write m=n+1>=2. On the unit three-sphere the scalar eigenvalue is m^2-1
and its eigenspace is V_(m-1) tensor V_(m-1), of complex dimension m^2.
Here V_j is the SU(2) representation with weights -j,-j+2,...,j.
Dowker–Chang, arXiv:0904.0744, equations (6)–(9), gives the spectrum and
character projection. For quotient exponents 1,q a weight pair (a,b) has
character exponent
e(a,b) = ((1+q)*a+(1-q)*b)/2 mod p.
The division is exact because a and b have the same parity. The chi_u
multiplicity counts weight pairs with e(a,b)=u. Complex dimensions are counted
once; the extra realification factor used in parts of that paper is absent.
At m=1 this count is 1 for u=0 and 0 otherwise; the computation starts at m=2
so it removes precisely the required scalar zero mode.
Let P=2p, m=m0+P*k, 2<=m0<=P+1. For a residue a modulo P of parity m0-1,
let A_a count its occurrences in 1-m0,3-m0,...,m0-1. Increasing m by P adds
two occurrences of every admissible residue, giving
d(m0+P*k) = sum_[e(a,b)=u] (A_a+2*k)*(A_b+2*k)
= c0+c1*k+c2*k^2,
c0=sum A_a*A_b, c1=sum 2*(A_a+A_b), c2=sum 4.
These exact integer formulas prove the entire quasipolynomial. They are not
interpolation of a finite list. The checks compare complete coefficient sets
under character conjugation, q negation and coordinate exchange, for all 1744
unreduced labels through p=20. Summing the coefficients over all characters
recovers m^2. Independently counting homogeneous monomials in
z1,z1bar,z2,z2bar and subtracting radial multiples verifies the scalar
multiplicities at m=1,...,10 for every selected character.
## Why factoring the eigenvalue does not create an anomaly
Define F(z)=sum_(m>=2) d(m)*m^(-z), continued by residue-class Hurwitz sums.
A quadratic quasipolynomial gives possible simple poles only at z=1,2,3.
The coefficient B_r of m in d(m)=A_r*m^2+B_r*m+C_r has zero mean, so the
possible pole at z=2 cancels. Here is a structural justification.
For a deck transformation, write alpha=2*pi*j/p and beta=2*pi*q*j/p.
The scalar trace, when cos(alpha) != cos(beta), is
[cos(m*beta)-cos(m*alpha)]/[cos(beta)-cos(alpha)].
This is periodic, with no m term. For equal cosines at a noncentral element
the trace is m*sin(m*alpha)/sin(alpha). Its coefficient of m has zero average
over a period. The identity and antipodal element contribute respectively
m^2 and (-1)^(m-1)*m^2, again with no m term. Taking the finite character
projection preserves the zero average. The generator also asserts the exact
identity sum_(m0) (P*c1-2*m0*c2)=0 for each computed character.
For |s| small the binomial expansions yield the meromorphic identities
Z0(s) = F(2s) + sum_(k>=1) (s)_k/k! * F(2s+2k),
G(s) = 2*F(s) + 2*sum_(k>=1) (s)_(2k)/(2k)! * F(s+2k),
G(s) = sum_(m>=2) d(m)*[(m-1)^(-s)+(m+1)^(-s)].
These may first be justified in their common half-plane of absolute
convergence and then continued. For k>=2 the F terms are absolutely
convergent and decay geometrically in k, since m>=2 and d(m)<=m^2. The k=1
term is regular near s=0 because F has no pole at 2. Hence differentiation
term by term near zero is legitimate, and both derivatives equal
2*F'(0) + sum_(k>=1) F(2k)/k.
Consequently log D0=-Z0'(0)=-G'(0). This is a justified special factorization,
not an assumption that zeta determinants are always multiplicative.
## The finite formula used in the generator
For each m0, choose shift t=-1 or +1, and put a=(m0+t)/P. Then
c0+c1*k+c2*k^2 = alpha0+alpha1*(k+a)+alpha2*(k+a)^2,
alpha=(c0-c1*a+c2*a^2, c1-2*c2*a, c2).
Since m+t=P*(k+a), the corresponding part of G(s) is
P^(-s)*sum_(j=0)^2 alpha_j*zeta(s-j,a). Therefore
log D0 = sum_(m0) sum_(t=-1,+1) sum_(j=0)^2
alpha_j*[log(P)*zeta(-j,a)-zeta'(-j,a)].
All a are positive rational numbers; no singular Hurwitz argument occurs.
The prime is differentiation in the first variable. Exact coefficients and
Sage's Arb polynomial `_zeta_series` enclosures produce certified intervals.
Ball logarithms, arithmetic and exponentiation propagate those bounds. The
calculation uses numberdb.bits(digits,losing=180), or 513 bits for 100 digits.
The complete range is rerun at 150 requested digits and its actual retained
accuracy is recorded in numerical-evidence.json. The finite formula requires
no truncated numerical infinite sum or claimed numerical tail bound.
## Independent coexact calculation
The checker never uses production multiplicities or the scalar factorization
for its independent values. The coexact eigenspace at eigenvalue m^2 is
V_m tensor V_(m-2) plus V_(m-2) tensor V_m. Its trace is
chi_(m+1)(theta_plus)*chi_(m-1)(theta_minus)
+ chi_(m-1)(theta_plus)*chi_(m+1)(theta_minus),
theta_plus=pi*(1+q)*j/p, theta_minus=pi*(1-q)*j/p,
chi_d(theta)=sin(d*theta)/sin(theta).
At theta=k*pi the exact limiting formula is (-1)^((d-1)*k)*d.
On each residue m=r modulo 2p these products are quadratic polynomials in m.
Projection with cos(2*pi*u*j/p)/p is valid because opposite group elements
have the same real trace. Each polynomial is continued directly at eigenvalue
m^2 using mpmath Hurwitz zeta derivatives at 175 decimal digits.
Kitaoka, SIGMA 18 (2022), 091, equation (4.3) and Proposition 5.1, gives
T=2*pi^2/p^2 for u=0,
T=4*sin(pi*u/p)*sin(pi*v/p), v=u/q mod p for u!=0.
In the second line v is the least positive residue of u*q^(-1). The trivial
formula uses g_std, not the metric 4*g_std used elsewhere in that paper.
The nontrivial acyclic torsion is unchanged under a constant metric scaling.
The independent scalar result is log T+(log D_perp)/2. This corroboration is
floating arithmetic and is not the basis of the proven enclosure claim.
## Controls and stored values
Before any broad comparison the coexact checker must reproduce
-3*zeta(3)/pi^2-2*log(2), and both scalar methods must reproduce
log(2)-3*zeta(3)/(2*pi^2), at (p,q,u)=(2,1,1). These are nonzero controls from
Nash–O'Connor, arXiv:hep-th/9212022, equation (1.1). The sum of all character
logarithms gives log(pi)+zeta(3)/(2*pi^2), the unit three-sphere scalar value
in T595 and Cunha–Freitas, arXiv:2404.12114v1, Corollary 2.9. The (2,1,0)
value agrees with T600, and every independent coexact value agrees with T631.
All 286 nontrivial torsions agree with T599, including its exact rationals.
Every returned ball must be finite and retain at least 400 accuracy bits;
low- and high-precision balls must overlap. Stored 100-digit decimal intervals
are checked against the independent answers, and a ten-last-place
perturbation must fail the same check. Covering identities are also checked
using the actual stored logarithms and their last-place uncertainty.
`Generator.verify(sample=None)` verifies all entries against the generator.
The controller, not this build, commissions the independent critique and
argument-selection review before any draft is offered.