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.
Keep the complete original integer-square theta series, its positive even
differential kernel phi, and
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.
Define the normalized Jensen polynomials
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<...<c_(d-1). The exact additional inequalities for
P_(d,n) to have only real roots are
(-1)^(d-i) * [a_n + d*integral_0^(c_i) P_(d-1,n+1)(t) dt] >= 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
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.
Holland, arXiv:2608.08682v1, Theorem 1.1
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 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,
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.
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.