RIEMANN PROJECT · OCTOBER 5, 2026 · AUDIT V1
Repairing the theta-pair boundary
The complete theta pair can be rewritten so that every term has a legitimate ordinary Fourier transform. Recovering the original function preserves an explicit sign correction. The arithmetic inequality remains open.
Wall check: Same open gap. Closest: NS100/101 and PR #74–75. This resolves a boundary-accounting question. It supplies no all-height signed estimate.
1. A positive primitive that decays
Keep the complete original theta sum. With h(u)=exp(u/2)*Theta(exp(2u)), define
f(u) = 2*cosh(u/2)-h(u) (1/4-partial_u^2)*f = 4*phi f(u) = 4*integral_R exp(-abs(u-v)/2)*phi(v) dv.
The last identity uses the positive Green kernel with decay at both ends. It proves f>0. The complete theta series also gives f(u)<exp(-abs(u)/2), with ratio tending to one in both tails. No finite prefix or artificial cusp is introduced.
2. The pair calculation now has finite terms
Define the two moments of f(s+t)*f(s-t), weighted by 1 and s², as A_0(t) and A_2(t). They and the required derivatives decay exponentially. The complete original associated kernel is
C(t) = [(partial_t^2-1)^2*A_2(t)
-4*(partial_t^2+1)*A_0(t)]/256.
Every term can now be transformed separately. This addresses PR #75’s growing-moment difficulty while preserving the original kernel and both infinite boundaries.
3. The exact correction must remain
Let Y be the Fourier transform of f, and write p=r²+1/4. Recovering the original X=xi(1/2+ir)/2 gives X=p*Y/4. Two independent algebraic calculations give
L1[Y] = Y'^2-Y*Y'' 16*L1[X] = p^2*L1[Y] + 2*(r^2-1/4)*Y^2. Required, at EVERY real r: L1[Y] >= 2*(1/4-r^2)/(r^2+1/4)^2 * Y^2.
The correction changes sign at abs(r)=1/2. The formula stays valid at zeros, with no division by Y. Positivity of the real primitive and of its pair moments does not establish this inequality. In fact, Y(r)=-Lambda(1/2+ir), where Lambda(s)=pi^(-s/2)*Gamma(s/2)*zeta(s): this is the completed zeta factor with its quadratic prefactor removed.
4. The existing control retains the repair
The NS101 reciprocal mixture has the same smooth, positive primitive, the same exponential envelope and the same unit leading tail. Yet PR #74’s complete-moment argument makes its first Laguerre expression negative in the stated parameter range. The corrected expression above therefore fails for that control too.
This stops an inference from those shared structural properties. The control changes the original lattice coefficients and loses the distinguished Jacobi equation. It does not refute an estimate that actually uses that missing arithmetic input.
There is also a source-domain qualification: Y has poles at r=+/-i/2. Its kernel has exponential tails, so it falls outside the faster-decay, entire-transform class in Csordas’s Definition 1.2. Those theorems cannot be imported merely from positivity and smoothness.
5. Twelve focused agent reviews
The reviews check the new derivation and its nearby methods. They do not replay the project’s historical proofs or certificates. The primitive, pair constants and control argument passed the assigned checks. Clarifications retain the complete old-to-new boundary cancellation, distinguish integral convergence from analytic continuation, and make parameter-dependent constants explicit.
| Review | Finding |
|---|---|
| 1. Primitive and normalization | Factors, positivity and tails checked. |
| 2. Real boundaries | No omitted endpoint or artificial atom. |
| 3. Matched controls | NS101 retains the repair and fails the sign. |
| 4. Pair algebra | Jacobian and sign correction checked. |
| 5. Complex domain | Real-axis decay does not make Y entire. |
| 6. Nonlinear transport | No closed positivity-preserving evolution supplied. |
| 7. Moment estimates | A local interval does not give all heights. |
| 8. Euler-product input | No justified transfer to the critical line. |
| 9. Positivity distinctions | A positive integrand does not give the needed transform sign. |
| 10. Adversarial check | No formula defect found; scope clarified. |
| 11. Source applicability | Coefficient conclusions do not give the all-height inequality. |
| 12. Admission decision | No new estimating method admitted. |
Three estimating possibilities remain insufficient. Complete moments prove positivity near zero, with a radius determined by those moments; they do not control all heights. The proposed separation-clock transport is singular at its constant initial kernel and needs an independent positivity-preserving evolution. The logarithmic Euler derivative has an absolutely convergent prime series to the right of 1, with no justified sign transfer here to the critical line. None meets the arithmetic admission condition.
Decision
No original sign method is admitted. The boundary repair is valid; the resulting corrected inequality is exactly the existing first-Laguerre obligation. Further research needs a concrete estimate for the whole expression, using the original arithmetic or nonlinear trajectory at a step absent from the control. A new expansion or finite-height table does not supply it.
Reviewed commit: d93aaf6b8dbe1fcbebf6fff4f6d1f71d993f4208. No numerical scan, new research row, node, thaw, certificate or manuscript version. Both original-evidence recovery groups remain OPEN. RH, G2 and the cofinal arithmetic lower bound remain open.