PASS for the draft’s source-scope conclusion; OPEN for equation (10).
Reviewed scientific baseline: d93aaf6b8dbe1fcbebf6fff4f6d1f71d993f4208
(PR75), with scope scaffold edc65638cc5e761a0290b38d54ff0b9bac140697.
Read AGENTS section 0 and proposal gate, REVIEW_BRIEF, derivation equations
(1)-(12), the applicable ZERO-GEOMETRY register scope, PR72’s source-chain
note, PR74’s arXiv-method review, and the primary arXiv passages cited below.
The parent performed the full 104-node review. I did not replay historical
proofs or external interval certificates. No scan or new candidate proof.
Wall check: Same open gap. Closest: NS100/101 and PR74/75. The primitive repairs the transform domain but does not estimate the signed expression.
Csordas 1309.0055v2: Definition 1.2(v) requires super-Gaussian decay of every derivative. Theorem 3.5 gives admissibility of associated kernels, not their positive definiteness; Theorem 3.7 needs every associated order for its real-zero criterion. Remark 2.1 and Theorem 4.2 concern moment/Turan consequences. Theorem 4.5 applies to the original theta kernel. Open Problem 4.7 asks first-Laguerre positivity at every real height. Open Problem 4.14 concerns a different, real-kernel second concavity. These are distinct quantified statements.
Planat–Sole 2608.19160v1, Theorem 1.1
and Section 7, state second-level concavity for
s(t)=Phi(sqrt(t)), t>0, with a double-Turan consequence for normalized
coefficients. Its nonlinear Jacobi phase space is in the real kernel
coordinate. This is not a claimed all-frequency first-Laguerre inequality
for the Fourier transform. I checked statement/scope here, not its finite
certificate implementation.
The following calculations use the draft’s definitions. They prevent an otherwise tempting import of a theorem with the wrong object or clock.
The primitive is outside the stated entire-transform class. Equation
(1) gives f(u) ~ exp(-abs(u)/2). Consequently no epsilon>0 makes
f(u)=O(exp(-abs(u)^(2+epsilon))). This already fails the decay premise;
deciding the separate monotonicity premise is unnecessary. Equation (5)
gives actual poles of Y=4X/(r^2+1/4) at r=+/-i/2, because
X(+/-i/2)=1/4. So the mismatch is substantive, not notation.
Equation (8), however, follows by ordinary integrals and remains valid.
An associated-kernel identity is not its sign estimate. With the
draft’s A_2, equation (8) gives
L1[Y](r)=4*integral A_2(t)*cos(2*r*t) dt. Positivity of A_2(t)
does not assign a sign to this cosine integral. Even establishing the
stronger L1[Y]>=0 would prove (10) only on abs(r)>=1/2; inside that
interval the correction is positive and must be estimated. A theorem
for the original phi does not automatically become one for its
nonlocal resolvent f=4*G_(1/2)*phi.
Correct theta normalization. Comparing the explicit theta series
gives phi(u)=Phi(u/2). If H(x)=integral_0^infinity Phi(v)*cos(x*v) dv,
then direct substitution and evenness give
X(r)=4*H(2*r),
L1[X](r)=64*L1[H](2*r),
C(t)=8*K_1(t/2),
K_1(v)=integral_R w^2*Phi(w+v)*Phi(w-v) dw.
Thus the whole-height question matches after positive scaling. It
does not become the real-coordinate curvature question merely because
both use the notation L1.
Coefficient/origin versus arbitrary frequency. Let
F_c(z)=H(i*sqrt(z)). Since H is even, this is unambiguous through its
entire power series. For x!=0, direct chain rule gives
H(x)=F_c(-x^2),
L1[H](x)=4*x^2*L1[F_c](-x^2)
+2*F_c(-x^2)*F_c'(-x^2).
Inequalities involving derivatives of F_c at zero do not bound this
expression at every -x^2<=0. Even an all-real-argument lower bound
for L1[F_c] alone would leave the displayed product term. The
primitive’s equation (10) has the analogous explicit correction.
No missing pointwise or signed-transform comparison is supplied by
restating coefficient inequalities.
In the retrieved HTML, the source definitions and coefficients are:
(3.11): F(x)=integral_R exp(i*t*x)*varphi(t) dt.
(3.7): K_n(t)=integral_R varphi(s+t)*varphi(s-t)*s^(2n) ds.
(3.14): L_n(x;F)=[2*2^(2n)/(2n)!]*integral_R K_n(t)*cos(2*x*t) dt.
(4.1): H(x)=integral_0^infinity Phi(t)*cos(x*t) dt.
(4.6): K_n(t;Phi)=integral_R Phi(s+t)*Phi(s-t)*s^(2n) ds.
(4.7): L_n(x;H)=[2*2^(2n)/(2n)!]*integral_R K_n(t)*cos(2*x*t) dt.
The HTML’s last coefficient repeats the full-line convention despite the displayed half-line H. I record this as a source-presentation normalization caution, not a finding that the underlying mathematical theorem is defective. I have not checked PDF/source versions or an erratum. The draft does not depend on the ambiguous displayed coefficient.
An independent check suffices. Put F=2H, the full-line transform. Then
L1[F]=4*L1[H], while a direct pair change of variables gives
L1[F](x)=4*integral_R K_1(t)*cos(2*x*t) dt. Therefore the half-line
version is L1[H](x)=integral_R K_1(t)*cos(2*x*t) dt. Combined with the
scaling above, it gives exactly the draft’s
L1[X](r)=4*integral_R C(t)*cos(2*r*t) dt. No draft factor correction
is required. Any later source quotation should name the transform
convention and rederive its prefactor.
The source exclusion in Section 3 is correctly narrow. It does not prohibit using elementary Fourier or resolvent identities for f, and does not exclude a future inequality derived from the distinguished original theta trajectory. The draft retains the actual correction, all heights and zeros. The existing NS101 family matches the positive-primitive/envelope/tail properties but changes the original coefficients and IVP; no source theorem checked here replaces that lost hypothesis with an all-height estimate.
No arXiv theorem reviewed here establishes (10) from the proposed premises. The missing step is an independent bound for its complete corrected expression, using original theta arithmetic or the distinguished IVP. This is a bounded applicability check, not an exhaustive search or an assertion about the current global literature status. No new research row, thaw, manuscript version, RH/G2 assertion, certificate or scan follows.