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6173 bytes, as of the version from 2026-10-29 08:32 (current). Recorded here, not run.
# Computation and independent checks for T604
Instruction provenance: `agents/runs/20261029T075206Z-build-skills/manifest.json`;
local and live SHA-256 both
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
## Fixed object
The nonnegative scalar operator is -div grad on the round unit hemisphere
H^d = {x in R^(d+1): |x|=1, x_(d+1)>=0}. Neumann means zero outward normal
derivative on the equator, or at both endpoints when d=1. There is exactly
one constant zero mode, excluded from the determinant. Positive eigenvalues
are k(k+d-1), k>=1, with m_N(k)=binomial(k+d-1,d-1). The multiplicities grow
as k^(d-1), so the defining zeta sum converges for Re(s)>d/2. Its continuation
is regular at zero, and L_N=-Z_N'(0), D_N=exp(L_N). Real logarithms are used.
Dowker equation (2) gives this by taking a=alpha=(d-1)/2 and removing m=0.
For Dirichlet a=(d+1)/2; translating the index produces
m_D(k)=binomial(k+d-2,d-1). Equation (3) is the sum of the two spectra.
## Finite formula used by the generator
Cunha and Freitas Theorems 2.7 and 2.8 give L_S for the unit sphere; Theorem
3.2 gives L_D for the hemisphere. `finite_formulas.py` computes these with
signed first-kind Stirling integers, exact rational harmonic sums and central
factorial coefficients, then Arb zeta derivatives and elementary functions.
Its arithmetic was adapted from the downloaded T595 and T603 generators,
with the exact source bytes recorded in `source-evidence.json` and the formulas
checked against the paper, including the scalings in Appendix A.5 and A.6.
The Neumann result is L_N=L_S-L_D. No rounded stored number is subtracted.
At d=1 the two hemisphere spectra are zeta(2s), giving D_N=2*pi.
The splitting is a direct sum of spectra, not an assertion about determinants
of a product of operators. The identity
m_N(k)+m_D(k)=binomial(k+d,d)-binomial(k+d-2,d)
is checked with exact arithmetic at d+1 distinct k for each d<=20; since the
polynomials have degree d-1, this proves the identity for every k. Determinant
and logarithm splitting are checked with the independent Neumann computation.
## Independent spectral continuation
`independent.py` imports no code or coefficients from the finite-formula
implementation. It was adapted from T603's direct-continuation checker, whose
Neumann branch already expands the defining Neumann multiplicities. This is
algorithmic independence from the generator, not independent actor review.
Set a=(d-1)/2, x=k+a and expand exactly
m_N(k) = product_(j=1)^(d-1) (x+j-a)/(d-1)! = sum_j c_j x^j.
The polynomial is independently tested at d+1 positive k. Separate k<K,
where K=40, and set q=K+a. Expand the tail as
Z_tail(s) = sum_j c_j sum_(r>=0) (s)_r/r! a^(2r)
* zeta(2s+2r-j,q).
The r=0 derivative contributes 2*zeta'(-j,q). For r>=1 with 2r-j!=1,
the contribution is a^(2r)/r*zeta(2r-j,q). At a pole, use
(s)_r/r! = (s/r)*(1+H_(r-1)*s+O(s^2)),
zeta(1+2s,q) = 1/(2s)-psi(q)+O(s).
Thus that term contributes a^(2r)/r*(H_(r-1)/2-psi(q)). Handling the pole
analytically matters: substituting s=0 into the unexpanded factors is invalid.
The finite part of Z'(0) is -sum_(1<=k<K) m_N(k)*log(k(k+d-1)).
The negative of the combined derivative is L_N.
For N>=floor(d/2), t=(a/q)^2<1, the omitted derivative is bounded in absolute
value by the exact rational
E_N = t^(N+1)/((N+1)*(1-t))
* sum_j |c_j| q^j * (1+q/(2N+1-j)).
For p>1, the decreasing-integrand estimate gives
zeta(p,q)<=q^(-p)*(1+q/(p-1)). Apply it to p=2r-j for r>=N+1, bound 1/r and
the denominator using r=N+1, then sum the geometric series. This also gives
local uniform convergence of the differentiated expansion near s=0, after
the finitely many singular factors are resolved. The integer N is increased
until E_N<10^(-170), with every decision in exact rational arithmetic. Arb
at 240 working decimal digits evaluates the finite sum and Hurwitz zeta
coefficients; E_N is added to the resulting ball as an error radius. For d=1,
a=0 makes every positive-r contribution vanish, and the same method still
passes the interval control.
## Checks actually run
`checks.py` first verifies the nonzero interval control L_N(1)=log(2*pi).
It also checks zeta'(-2n) using the functional equation for n=1,...,9.
All 40 entries at 150 working digits overlap the 200-digit recomputations and
the direct spectral enclosures. Both operands must be finite and have radius
less than 10^(-130) before comparison. The minimum relative accuracy is
504 bits for the production values and 560 bits for the independent values.
Changing K from 40 to 50 in dimensions 2 and 20 gives consistent enclosures.
A deliberate relative corruption of 10^(-90) is rejected for each entry.
The shared exactness, measurement and ring-import checks all pass.
Two low-dimensional formulas tested independently of the coefficient code are
L_N(2) = -zeta'(0)-2*zeta'(-1)+1/4,
L_N(3) = (1/2)*log(pi/2)-zeta'(-1)-zeta'(-2)+1/2.
`postfill.py` will additionally test each written decimal interval against
the independent enclosure, verify every database entry, and check the product
and sum identities using entries read from T604, T603 and T595. These stored
checks must complete before the build handoff is written.
## The source discrepancy
Dowker's 1993 preprint, printed page 7 (PDF page index 6), lists
Z_N'(0)-Z_D'(0) = 2*zeta'(-1) + 1/2 + log(2), d=3.
For its stated spectra the constant must be -1, not +1/2. Indeed the direct
Neumann continuation, the finite sphere-minus-Dirichlet evaluation, and the
independent low-dimensional expression agree with
Z_N'(0)-Z_D'(0) = 2*zeta'(-1) - 1 + log(2).
Substituting d=3 in the Barnes formula (8) gives
Z_D'(0)=zeta'(-2)-zeta'(-1)+zeta'(0)+1/2. Combining this with the sphere
formula (16) yields the corrected difference. The wrong printed difference
misses by exactly 3/2; it is explicitly rejected by the numerical check.
This is a discrepancy in a derived expression, not ambiguity in the family,
metric, boundary condition or zero-mode convention. We did not use the printed
difference as an oracle. The separate critic should inspect this reconciliation.