riemann

Review 03 — matched controls and scope

Result: PASS for the control argument; OPEN for the original sign target.

Reviewed scientific baseline: d93aaf6b8dbe1fcbebf6fff4f6d1f71d993f4208 (PR75). I read the review brief and current derivation, AGENTS section 0 and the proposal gate, the relevant research-map scopes and freeze, the original missing-evidence ledger, PR74 review 05 and PR72’s published Jacobi mismatch argument. The coordinator performed the global register/dependency review. This review independently checks the paper control calculation below; it does not replay historical certificates, evaluate moments, or run a scan. Both original-evidence recovery groups remain OPEN.

Wall check: Same open gap. Closest results: NS101 and PR74/75. The repaired primitive has legitimate transforms, but the exact original arithmetic sign in derivation (10) is still missing. NS101 excludes only an inference from the retained generic primitive properties to that sign.

1. Exact match for the primitive, including its normalization

Take beta>0 and B=3+2*cosh(beta/2). Let T_beta g(u)=[3g(u)+g(u-beta)+g(u+beta)]/B. Then

T_beta[2*cosh(u/2)] = 2*cosh(u/2),
T_beta[phi] = phi_beta,
T_beta[f] = f_beta = 2*cosh(u/2)-T_beta[h].

Thus derivation (11) is the same normalized homogeneous subtraction for the actual NS101 family, not an independently chosen positive comparison function. Translation commutes with 1/4-partial_u^2 and its decaying Green inverse. Hence (1/4-partial_u^2)f_beta=4phi_beta, and its transform is Y_beta=q_beta*Y/B=4X_beta/p with exactly the same p=r^2+1/4.

Smoothness, evenness, positivity and the derivative decay needed for (7)-(9) are inherited by finite translation and positive summation. For u>=0,

e^(u/2)e^(-|u-beta|/2) <= e^(beta/2),
e^(u/2)e^(-|u+beta|/2) = e^(-beta/2).

Together with the strict original envelope these give 0<f_beta(u)<e^(-u/2). Evenness supplies all negative u. For fixed beta and u>beta, the three positive tail coefficients are 3, e^(beta/2), e^(-beta/2); their sum is exactly B. Therefore f_beta(u)/e^(-|u|/2)->1 at both ends. The envelope and unit leading tail are truly shared. Constants in derivative bounds may depend on beta; this does not affect the finite-beta control or require a uniform-beta bound.

2. Strict failure of the corrected inequality

The product identity is

B^2 L1[X_beta](r)=q_beta(r)^2 L1[X](r)
                 +2beta^2(2+3cos(beta*r))X(r)^2.

At r_beta=pi/beta, this becomes B^2 L1[X_beta]=L1[X]-2beta^2 X^2. Define the complete unevaluated positive moments m0=int phi, m2=int u^2 phi and k=m2/m0>0. If beta>=pi*sqrt(k), then r_beta^2 k<=1 and elementary moment inequalities give

X(r_beta)>=m0/2,
|X'(r_beta)|<=r_beta*m2, |X''(r_beta)|<=m2,
L1[X](r_beta)/X(r_beta)^2 <= 4r_beta^2 k^2+2k <=6k.

Consequently

L1[X_beta](r_beta)
 <= -(beta^2-3k)m0^2/(2B^2)<0.

This confirms derivation (12), including its negative-margin direction. The stronger stated range beta>=max(2,pi*sqrt(k)) also retains the NS101 critical-strip hypothesis. The estimate makes no separate assertion about the historical beta=2 member, since k has not been evaluated here.

For clarity, define the corrected residual in (10) as

R_beta(r)=L1[Y_beta](r)
          -2(1/4-r^2)/(r^2+1/4)^2 * Y_beta(r)^2.

Equation (9) says exactly R_beta(r)=16L1[X_beta](r)/(r^2+1/4)^2. The denominator is strictly positive for every real r. At the displayed parameter and height this gives the explicit bound

R_beta(r_beta)
 <= -8(beta^2-3k)m0^2/[B^2(r_beta^2+1/4)^2] <0.

No division by X or Y at their zeros is involved in the identity. The moment proof divides by X only at a point where X>=m0/2>0 was established. The failure is of the complete corrected expression, including the low-height term, rather than an inferred consequence of off-axis zeros.

3. What the control loses

The normalized reciprocal theta mixture has its slowest exponential at kappa=pi*exp(-2beta), with positive coefficient c=2exp(-beta/2)/B. For Theta_beta(v)=1+c exp(-kappa v)+o(exp(-kappa v)), with its differentiated expansion through order three, the original normalized Jacobi expression has leading term

Q[Theta_beta](v)
 =c^2 kappa^4(kappa^2-pi^2)exp(-2kappa v)
  +o(exp(-2kappa v)).

This is strictly negative eventually for every beta>0. One can check the coefficient directly: the leading T^2 contributes c^2 kappa^6, the -pi^2 Theta_beta^10 S^2 term contributes -pi^2 c^2 kappa^4, and 32S^3 is smaller. Thus the mixture cannot satisfy the original normalized Jacobi equation, and a fortiori does not match its distinguished IVP. It also changes the original single-lattice coefficients. Neither a general positivity conclusion nor failure of the original target follows from this mismatch. An estimate actually using that original IVP remains outside the control’s scope.

4. Other recorded controls

NS100: correctly scoped. The actual negative original slice excludes positivity of each separate slice. It does not determine the sign after the full weighted integration used by C, and the draft does not assert such slice positivity.
Davenport-Heilbronn: correctly not claimed as a full match. Generic transform/product calculus transfers when the analytic hypotheses hold; the original theta coefficients, conductor/completion and distinguished IVP do not. A positive phi and the resulting positive Green primitive have not been established for that analogue here. It is therefore not presented as a positive-primitive counterexample. No unchanged screen script has been treated as a screen of this different target.
NS74: not applicable to this formula. Its inner-factor control preserves Gram geometry of a particular NB approximation family while altering target approximation. No such Gram/target inference occurs in the primitive calculation. Its general warning does not make its theorem a direct theta-transform obstruction.
NS83: not applicable to this formula. Its fixed-smoothing NB error and dyadic relative-gain restrictions differ from the real frequency variable and first-Laguerre target here. This audit neither proposes a per-doubling rate nor bypasses that restriction.

Disposition

No correction to derivation (11)-(12) or the control classifications is required. Explicitly declaring beta>0 at the family definition would make the domain easier to read; the sufficient range already imposes it. The result is an application of PR74’s established control to additional shared primitive/envelope/tail properties, not a new original sign bound. No candidate, research row, thaw, manuscript version or further scan is licensed. Final wall check: Same open gap.

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