Riemann project · October 5, 2026 · Audit v1
Theta addition: the variables and boundaries matter
A valid two-dimensional lattice identity keeps the original arithmetic intact. It still leaves the signed lower bound open. The calculation also shows exactly which boundary cancellations a proposed transform argument must retain.
1. The shifts change the modulus
The familiar theta addition formulas change an elliptic argument while holding the modulus fixed. Our factors have elliptic argument zero and these two moduli:
tau_plus = i*exp(2s+2t) tau_minus = i*exp(2s-2t) tau_plus/tau_minus = exp(4t)
Substituting s+t and s-t into the other argument would change the function. Kharchev–Zabrodin I makes this distinction explicit. Its multidimensional companion, Proposition 3.1, also gives formulas for integer multiples of a common modulus. Those can apply when exp(4t) is rational, with all residue classes retained. They do not supply the required sign estimate or its limiting control.
This is a limitation of the proposed substitution, not an exclusion of every addition identity or unequal-modulus method.
2. The lattice formula that does apply
Let Theta(x)=sum exp(-pi*n^2*x) over all integers, and x=exp(2s). Reindex the complete product by k=n+m, l=n-m:
Theta(x*exp(2t))*Theta(x*exp(-2t))
= sum_(k=l mod 2) exp[-pi*x/2 *
{cosh(2t)*(k^2+l^2)+2*sinh(2t)*k*l}].
The parity constraint and coupling term are essential. At t=0 the coupling vanishes and the familiar duplication identity returns. For general t, discarding it changes even the contribution corresponding to the original lattice point (1,0).
The quadratic matrix has positive eigenvalues exp(2t) and exp(-2t), and determinant one. That guarantees convergence of this lattice series. It does not establish positive definiteness of our associated kernel as a function of real translation differences. These are different positivity questions.
3. Keep the complete differential kernel
Define h(u)=exp(u/2)*Theta(exp(2u)) and phi=(h''-h/4)/4. The original pair is a differential expression in H(s,t)=h(s+t)*h(s-t):
phi(s+t)*phi(s-t) = P H(s,t)/256
P = partial_s^4 -2*partial_s^2*partial_t^2
+partial_t^4 -2*partial_s^2 -2*partial_t^2 +1.
Integrating s^2*H directly would diverge. A smooth exact subtraction retains the boundaries:
R(s,t) = H(s,t)-2*cosh(s)
M_j(t) = integral_R s^j*R(s,t) ds, j=0,2
C(t) = [(partial_t^2-1)^2*M_2(t)
-4*(partial_t^2+1)*M_0(t)]/256.
The operator annihilates the subtracted term. The remaining integrals converge, and integration by parts in s has vanishing boundary terms. The full derivation checks the factor 256, the parity reindexing, and both ends of the complete theta tail.
4. The moments cannot be transformed separately without further work
The subtraction makes the s integrals finite, but their values still grow with t. In fact, reciprocity and positivity of the complete theta sum give the exact lower bound
M_0(t) >= 2*(t-1)*exp(t), t >= 0.
The full differential bracket above equals 256*C and decays rapidly. Transforming its separate growing moments as ordinary integrable functions would therefore be invalid. Their cancellation must stay inside the calculation; a distributional treatment would need its own justification.
This is an exact boundary-accounting result. It is not a lower sign bound for C or its transform.
Decision
The original target remains X'^2-X*X'' >= 0 at every real height, with X=xi(1/2+ir)/2. The new representation preserves that obligation and does not estimate it. The positive lattice matrix and individual Gaussian terms cannot replace a signed comparison after differentiation and transformation.
NS100 still rules out positivity of each separate original slice. PR74’s reciprocal control retains general symmetry and transform calculus while failing the actual first-Laguerre condition in its stated parameter range; it does not retain the original Jacobi equation. Davenport–Heilbronn likewise has different arithmetic data. These scope distinctions leave the original evolution open, but do not establish a control pass for a missing method.
No candidate passes admission. The next useful input would be a concrete arithmetic estimate for the complete differential combination above, or for the original pair directly. Another dimension, residue-class expansion or height table would not supply that estimate. No scan follows from this audit.
Reviewed commit: 1f09dd8a8587506ea83c0c48573d225b7a323bc1. Full register review; paper algebra self-review only. No historical proof replay, new research row, node, thaw, certificate or manuscript version. Both original-evidence recovery groups remain OPEN. RH, G2 and the cofinal signed-arithmetic lower bound remain open.