# Review 7 — complete moments, Jensen transfer and the absent estimate Reviewed remote baseline: `18d324703969b5bcc9aa94043cfd90a6e8211f34` (PR76), in the coordinator's isolated worktree. I read AGENTS/proposal requirements, current README/NEXT_STEPS, relevant ZERO-GEOMETRY map scope, the missing-original ledger, PR74's matched-control proof, PR76's derivation and moment review, and the September 29 Jensen audit. The coordinator owns the repository-wide review. This is a paper dependency assessment, not a historical proof/certificate replay. Both original-evidence groups stay OPEN. **Wall check: Same open gap.** Closest results: NS100's Jensen proposal, NS101, PR74 and PR76. What changes: specify the integration constants and critical-value inequalities omitted by a proposed backward-in-shift transfer. No new original-coefficient estimate is supplied. **Disposition: no candidate admitted; stop the moment/Jensen shortcut.** ## Exact original input and target Keep the complete original integer-square theta series, its positive even differential kernel `phi`, and ```text m_(2n) = integral_R u^(2n)*phi(u) du, a_n = n!*m_(2n)/(2n)!, n = 0,1,2,..., X(r) = sum_(n>=0) a_n*(-r^2)^n/n! = xi(1/2+ir)/2, p(r) = r^2+1/4, Y(r)=4*X(r)/p(r), J(r) = p(r)^2*(Y'(r)^2-Y(r)*Y''(r)) +2*(r^2-1/4)*Y(r)^2 = 16*(X'(r)^2-X(r)*X''(r)). ``` The target is `J(r)>=0` for **every real r**, including zeros. The factorial normalization in `a_n` is essential; raw moment log-convexity is a different inequality. PR76 already proves origin positivity and a bounded local interval from `m_0,m_2,m_4`. Neither covers all heights. Work with entire `X`; meromorphic `Y` has poles at `+/-i/2` and cannot silently replace it in an entire-function theorem. ## A complete transfer exists, but its arithmetic premises are unproved Define the normalized Jensen polynomials ```text P_(d,n)(z) = sum_(j=0)^d binom(d,j)*a_(n+j)*z^j. P'_(d,n)(z) = d*P_(d-1,n+1)(z). ``` For the simple-critical-point case, suppose `P_(d-1,n+1)` has distinct negative roots `c_1<...= 0, i=1,...,d-1. (*) ``` The bracket is precisely `P_(d,n)(c_i)`. Alternating critical values, together with the leading coefficient's signs at the two infinities, give the required roots by monotonicity and the intermediate value theorem, counting a zero extremum twice. Necessity follows from interlacing. Multiple critical roots require the corresponding multiplicity conditions; no strict-root premise may be inferred from weak inequalities during an induction. Strict versions of (*) are a sufficient induction premise but stronger than necessary. Already `d=2` requires `a_(n+1)^2>=a_n*a_(n+2)`: ordinary normalized Turan inequalities settle this degree only. Their validity cannot replace (*) for unbounded degree. The derivative recurrence discards `a_n`; differentiation universality does not bound the lost constant against the alternating integral areas. For a concrete algebraic warning, `z^2+1` has a real-rooted derivative but fails its single required critical-value sign. If an independent argument supplied negative real roots for `P_(d,0)` for every `d>=1`, set ```text Q_d(r) = P_(d,0)(-r^2/d). ``` These have real roots and satisfy `Q_d'^2-Q_d*Q_d''>=0` on the real line. Because `binom(d,j)/d^j -> 1/j!` and is at most `1/j!`, the absolutely convergent complete moment series gives `Q_d -> X` locally uniformly in the complex plane; Cauchy's formula supplies convergence of two derivatives. Passing to the limit proves `J(r)>=0` at each arbitrary real `r`. This transfer needs neither division by `X` nor an all-height fixed absolute margin. Its **missing premise** is unshifted all-degree hyperbolicity, an RH-strength obligation, not an easier moment bound. The first-Laguerre conclusion alone is not RH. ## Prior-art and matched-control scope [Holland, arXiv:2608.08682v1, Theorem 1.1](https://arxiv.org/html/2608.08682v1#S1) states hyperbolicity in `n^3*log(n+2)^2>=K*d^5`. Its coefficients are `gamma(n)=2*a_n` in our normalization. The positive scaling changes no roots. That region excludes `n=0,d>=1`; its proof supplies no estimates for the successive missing critical values (*). The September 29 audit already records this backward-in-shift gap. This source check imports the stated scope, not a fresh verification of its proof. [Csordas, arXiv:1309.0055v2](https://arxiv.org/html/1309.0055v2) separates the first Laguerre inequality from the complete real-zero criterion. No full positive-definite matrix hierarchy is being proposed here as an easier estimate. NS101 shares complete positive moments, their Hankel positivity, origin generalized-Laguerre positivity and the primitive relation. PR74 nevertheless proves, with `k=m_2/m_0`, ```text L1[X_beta](pi/beta) <= -(beta^2-3*k)*m_0^2/(2*B_beta^2) < 0, beta>=max(2,pi*sqrt(k)), B_beta=3+2*cosh(beta/2). ``` Thus those shared moment properties cannot supply the target. No claim is made that this control satisfies all normalized Turan inequalities, Holland's complete quantitative region, or (*). It changes the original lattice coefficients and Jacobi IVP. Any proposed proof of (*) must use those original data at an identifiable estimating step; their mere appearance in `a_n` supplies no such step. NS100 excludes per-slice positivity, which the conditional transfer above never assumes. ## Decision and bounded future budget The exact missing inequalities (*) are **gap identification**, not an admitted arithmetic candidate. No mechanism estimates their alternating areas using the original integer-square coefficients. Stop: no new scan, computation, row, thaw or manuscript version. Only if an explicit original-coefficient estimate for (*) is proposed, allow a **two-hour paper admission audit**: check its unbounded-degree quantifiers and multiplicities, identify the arithmetic step, then compare NS101 premises. Success would admit that particular estimate for further proof review, not prove it or RH; failure leaves the present stop intact. Do not spend this budget on more eventual-hyperbolicity tables. **Final wall check: Same open gap.** RH, G2, the cofinal signed-arithmetic lower bound, and the original all-height first-Laguerre sign remain open.