Research audit · October 5, 2026 · Version 1
Six agents, one remaining sign estimate
The original nonlinear theta evolution remains unexcluded. We obtained a complete approximation-error bound, rejected a specific local certificate, and made the controls sharper. No method yet proves the original pair’s sign at every height.
Reviewed scientific baseline: 5434d96f0034f842b0229f1d69dbbf3fb7fea270. Six assisting agents worked in two rolling waves, followed by adversarial cross-review. The coordinator reviewed all 104 registered conclusions and 14 continuation inputs; this was a scope and dependency review, not a replay of all historical evidence.
The target we kept
The target retains the complete, smooth, even original theta kernel phi, both shifted factors and all tails. Approximations and controls are identified explicitly below. With the project’s normalization:
X(r) = integral_R phi(u) exp(i*r*u) du
= xi(1/2+i*r)/2
C(t) = integral_R s^2 phi(s+t) phi(s-t) ds
L1[X](r) = X'(r)^2 - X(r)*X''(r)
= 4 integral_R C(t) cos(2*r*t) dt
Target: L1[X](r) >= 0 for EVERY real r.
This is equivalent to positive definiteness of the complete associated kernel C. It is only the first Laguerre condition; proving it alone would not prove RH. The original Jacobi evolution retains the clock L=2u and its distinguished initial state.
What the six reviews established
- The previous pair algebra survives independent review. The factors and signs in PR73 check. Its tail cancellation includes prefix–tail cross terms: the tail’s own Laguerre expression is eventually negative. Smooth boundary cancellation by itself gives rapid decay, not a positive sign. Review 1
- A particular local nonlinear certificate fails. An integration-by-parts construction would suffice if its residual kernel were positive semidefinite for every finite set of real arguments. The drift-cancelling choice fails already on its diagonal for the original theta tail. This rejects that choice, not every local or nonlocal use of the nonlinear evolution. Review 2
- A smooth reciprocal completion has a complete error estimate. It avoids PR73’s cusp and admits explicit bounds on the transform and its first two derivatives at every real height. Its differential kernel has a negative artificial real-space tail; that fact alone does not determine its Laguerre sign. No lower sign margin for this approximation was obtained. Review 3
- A positive reference margin is available. Gasper’s Bessel square identity supplies an all-height positive Laguerre margin for a different smooth reference. The original-theta comparison needed to use it remains missing. The other three arXiv sources checked do not supply that comparison. Review 4
- The reciprocal-mixture family fails the actual first-Laguerre test. A sufficient parameter range follows analytically from complete theta moments. This avoids the invalid inference that off-axis zeros alone force first-Laguerre failure. The control does not satisfy the original Jacobi equation. Review 5
- The adversarial review admits no sign method. It checked the new error rates, local rejection, reference normalization and control threshold. The missing comparison remains an admission blocker. Review 6
Why the local certificate stops
Write p=phi'/phi, w(u,v)=(u+v)^2/4, and S=p(u)+p(v). For an admissible symmetric function h, integration by parts gives the residual:
R_h = w - (partial_u+partial_v)h - S*h
Choose h = w/S. On the diagonal:
R_h(u,u) = -u/p(u) + u^2*p'(u)/(2*p(u)^2)
= (u-u^2)/(2*pi*exp(2u))
+ O(u^2*exp(-4u)) < 0 eventually.
The complete original theta asymptotic supplies this rejection. A global smooth-quotient discussion is conditional on pointwise p'(t)<0; the diagonal rejection needs no such global premise. No numerical onset is asserted.
What the error bound does—and leaves open
The smooth completion blends a finite original theta sum with its reciprocal using w(u)=1/(1+exp(-(pi/2)*sinh(2u))). All derivatives of the weight and both omitted tails are retained. The detailed note defines every constant in:
|X^(j)(r)-X_M^(j)(r)| <= S_j(r) exp(-pi*|r|/8-beta*(M+1)), j=0,1,2 beta = pi/(2*sqrt(2)), M >= 1, r real. If M(r) >= ceil(|r|)+M0, M0 >= 1: |L1[X]-L1[X_M]| <= O((1+r^2) exp(-(pi/4+beta)*|r|)) + O((1+r^2)^2 exp(-(pi/4+2*beta)*|r|)).
Derivatives are always taken with M fixed, then the bound is evaluated at the chosen M(r). We never differentiate a discontinuous cutoff. The constants are given as convergent integrals in Review 3, rather than fitted from samples.
This bounds the absolute error. It does not prove that the approximation has a positive margin larger than that error. In particular, a zero lower margin cannot absorb a positive error allowance. The supporting estimate therefore does not establish the target.
A reference with a known positive margin
For B_a(r)=K_(i*r/2)(a), a>0, taking the quadratic coefficient in Gasper’s identity (2.6) gives:
L1[B_a](r)
= (1/4) integral_0^1 [-log(t)]/[t*(1-t)]
* K_(i*r/2)(a/sqrt(t))^2 dt > 0.
The reviewed note also derives a crude explicit lower bound. But B_a is not the smooth completion X_M. The error estimate above does not estimate X-B_a. Combining those two findings without a new comparison would be a false transfer.
For any real reference H, errors |(X-H)^(j)|<=epsilon_j would suffice if, at every real height:
2|H'|*epsilon_1 + |H|*epsilon_2 + |H''|*epsilon_0 + epsilon_0*epsilon_2 <= m_H <= L1[H].
That is a conditional comparison with the mixed terms retained, not an estimate we have proved. The original integer-square theta coefficients and nonlinear evolution would have to control these errors, or the signed cross terms directly.
The matched control, without a scan
For the NS101 reciprocal family, write X_beta=(3+2cos(beta*r))*X/B_beta, where B_beta is its positive normalization constant, unrelated to the Bessel reference. Set m_j=integral_R u^j*phi(u)du for j=0,2, and k=m_2/m_0>0.
beta >= max(2, pi*sqrt(k)) => L1[X_beta](pi/beta) < 0.
The proof uses complete moments and the exact product identity. It asserts no evaluated threshold and does not separately decide the archived beta=2 member. The family shares reciprocity, positive even kernels and order-one transform structure, while failing the original normalized Jacobi equation. A separate smooth, strictly log-concave control also shows that boundary cancellation and kernel shape alone cannot provide the desired sign. Neither control defeats a method that genuinely uses the original equation.
NS100 already excludes positivity of each separate original slice. Davenport–Heilbronn has different coefficients and completion, so it is not a counterexample satisfying the original theta initial-value premises. No unchanged example script was presented as a screen of this different target.
Decision and next admissible step
No original-theta sign candidate is admitted. The work separates three obligations: construct a smooth approximation, estimate its complete error, and prove a lower sign margin. The first two can now be made explicit for the displayed completion. The third remains unestimated.
The next useful proposal must supply a concrete method for a signed lower bound using the original two-shift Jacobi trajectory, or a complete original-theta-to-reference comparison. It must retain every real height and arbitrary finite translate coefficients. Another fitted reference or finite height table would not meet this requirement.
Budget decision: no scan is authorized by this audit. If an independent estimating mechanism is specified, allow one bounded applicability review before numerical work. Success would establish the first-Laguerre target only; failure would stop that mechanism under its stated hypotheses.
No new NS row, node, thaw, interval certificate or manuscript version. The manuscript remains v1.66; the required build has 358 pages and zero undefined or duplicate references. Both historical original-evidence recovery groups remain OPEN. RH, G2 and the cofinal signed-arithmetic lower bound remain open.
Archive index · Review and validation record · Classical Laguerre context