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# Full one-form determinant continuation for T633
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## Full operator and zero modes
Hodge decomposition is an orthogonal direct sum of the exact and coexact
one-form spectra. The nonzero exact spectrum is the nonzero scalar spectrum,
by d_u and its inverse on each positive eigenspace. Flat-line one-cohomology
vanishes on these quotients: pullback is a subspace of H^1(S^3,C)=0.
Thus Z1(s)=Z0(s)+Zperp(s), with no harmonic one-form term, and
log D1=log D0+log Dperp. Additivity follows first for convergent spectral sums
and then by continuation. This is a direct sum, not a multiplication of
operators or an assumption about multiplicative anomalies.
The code computes both spectra afresh, using no stored companion decimals.
The scalar and coexact algorithms are adapted from the attached generators
of T632 and T631, retrieved from the authoritative website in this run.
## 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.
## Coexact continuation
The eigenvalue m^2, m>=2, carries V_m tensor V_(m-2) plus
V_(m-2) tensor V_m. Both helicities are counted. In the weight notation
used for the scalar projector, let A_a count W_m0 and B_a count W_(m0-2)
in each residue of parity m0 modulo P=2p. At m=m0+P*k each count increases
by 2*k. Consequently the complete projected multiplicity is
sum_[e(a,b)=u] [(A_a+2k)(B_b+2k)+(B_a+2k)(A_b+2k)]
= c0+c1*k+c2*k^2,
c0=sum (A_a*B_b+B_a*A_b),
c1=sum 2*(A_a+B_b+B_a+A_b), c2=sum 8.
With a=m0/P and alpha=(c0-c1*a+c2*a*a,c1-2*c2*a,c2), its zeta
function is P^(-2s)*sum_j alpha_j*zeta(2s-j,a). Therefore
log Dperp=sum_(m0,j) 2*alpha_j*[log(P)*zeta(-j,a)-zeta'(-j,a)].
This finite formula, added to the scalar formula, is the production full
one-form result. The exact counts prove the quasipolynomials, rather than
fitting sample multiplicities. Arb computes rigorous Hurwitz zeta derivative
balls, and all logarithms and exponentials propagate those errors.
## Independent numerical checks
independent.py shares no production multiplicity or continuation functions.
It evaluates scalar SO(4) traces on residue classes modulo p and coexact
SU(2) traces on residue classes modulo 2p in 175-digit mpmath arithmetic.
Scalar traces are [cos(m*beta)-cos(m*alpha)]/[cos(beta)-cos(alpha)]
for distinct cosines, m*sin(m*alpha)/sin(alpha) for a noncentral equal-angle
element, m^2 at the identity and (-1)^(m-1)*m^2 at the antipode.
Here alpha=2*pi*j/p and beta=2*pi*q*j/p. Character projection uses
cos(2*pi*u*j/p)/p; opposite group elements have the same real trace.
Coexact traces are the sum of SU(2) character products of dimensions
m+1,m-1 in the two orders, at angles pi*(1+q)*j/p and pi*(1-q)*j/p.
At an integer multiple k*pi, a d-dimensional character is (-1)^((d-1)*k)*d.
Each residue trace is polynomial in m, so finite Hurwitz continuation gives
independent scalar and coexact logarithms, a and b.
Kitaoka's torsion gives t=log(2*pi^2/p^2) for u=0 and
log(4*sin(pi*u/p)*sin(pi*v/p)) for u!=0, v=u*q^(-1) mod p in 1,...,p-1.
Two independent full logarithms are 3*a-2*t and 3*b/2+t. These are compared
with each other and the production result for every selected pair.
Floating checks are corroboration; only production Arb balls justify the
`proven` rigour and recorded digits.
Nash–O'Connor equation (1.1) gives a=log(2)-3*zeta(3)/(2*pi^2) and
b=-2*log(2)-3*zeta(3)/pi^2 at (2,1,1). The nonzero full control is
-log(2)-9*zeta(3)/(2*pi^2). These controls run first. The covering-space
full logarithm is -log(4*pi)+3*zeta(3)/(2*pi^2), obtained directly from
sphere multiplicities m^2 and 2*(m^2-1). The scalar sum gives
log(pi)+zeta(3)/(2*pi^2), and the coexact sum gives
-4*zeta'(-2)+4*zeta'(0)=zeta(3)/pi^2-2*log(2*pi).
Every returned ball must be finite with at least 400 relative-accuracy bits;
150-digit recomputations must overlap and retain at least 550 bits.
The entire ball must fit inside the intended 100-digit decimal interval.
The independent answer is compared to that interval and a perturbation of ten
last places must fail. All exact multiplicity symmetry coefficients and
covering sums are checked over the unreduced labels. Scalar harmonic-monomial
counts provide another independent low-mode check.
The stored-check mode verifies every live entry and applies the independent
comparisons, Hodge and torsion identities, and all 46 covering identities to
the read-back decimals with their last-place uncertainties. All 1328 scalar
and coexact companion entries and 286 published torsions are checked too.
The evidence reports actual results and accuracy. The independent campaign
critique and its fingerprint-bound selection sidecar remain a separate stage.