# Review 7 — local moments do not supply the all-height correction **Disposition: PASS for the local identities and NS101 match; OPEN for the all-height estimate.** No original arithmetic estimating method is admitted by this review. Reviewed scientific baseline: `d93aaf6b8dbe1fcbebf6fff4f6d1f71d993f4208` (PR75); local audit scope commit `edc65638cc5e761a0290b38d54ff0b9bac140697`. I read the current AGENTS proposal/relevance requirements, this audit's REVIEW_BRIEF and derivation, the ZERO-GEOMETRY scope, and PR74's complete moment control proof in `reviews/05-matched-controls.md`. The coordinator owns the refreshed repository-wide conclusions/dependency review. This review is limited to moment-to-height transfer; it does not claim a replay of historical proofs, certificates, or all other reviewers' questions. **Wall check: Same open gap.** Closest results: NS101 and PR74/75. What changes: the regularized transform has a polynomial correction that must be retained even at height zero. Local moment information checks that correction, but supplies no new bound at arbitrary height. The structural inference from positive moments alone meets the existing NS101 obstruction. ## 1. The correction at the origin is checked exactly Use complete moments, without a cutoff or evaluated numerical constants: ```text m_j = integral_R u^j*phi(u) du, j=0,2,4, nu_j = integral_R u^j*f(u) du, j=0,2, p(r) = r^2+1/4, Y(r) = 4*X(r)/p(r). ``` The relevant even moments are strictly positive and finite. Evenness gives `X'(0)=Y'(0)=0`, while differentiating the exact relation (5) gives ```text nu_0 = Y(0) = 16*m_0, nu_2 = -Y''(0) = 16*m_2+128*m_0 = 16*m_2+8*nu_0. ``` Equation (10) at zero is therefore `nu_0*nu_2 >= 8*nu_0^2`. Its exact left-minus-right value is ```text nu_0*(nu_2-8*nu_0) = 256*m_0*m_2 > 0. ``` This agrees with (9), since `L1[X](0)=m_0*m_2` and `p(0)^2=1/16`. Positivity of `f` alone would give `nu_0*nu_2>0`, not the stronger corrected threshold `nu_2>=8*nu_0`. The latter here follows from the exact resolvent relation and positivity of the original `phi`. ## 2. What finite local moment information actually supplies The elementary inequality `cos(v)>=1-v^2/2` yields, on the original complete kernel, ```text X(r) >= m_0-r^2*m_2/2, -X''(r) >= m_2-r^2*m_4/2. ``` Thus the unevaluated interval ```text r^2 <= min(m_0/m_2, m_2/m_4) ``` has `X(r)>=m_0/2`, `-X''(r)>=m_2/2`, and consequently ```text L1[X](r) = X'(r)^2+X(r)*(-X''(r)) >= m_0*m_2/4 > 0. ``` This is a generic local consequence of complete positive even moments, included solely to quantify the scope of the proposed bridge. It is not a new arithmetic estimate, an evaluated height range, or a newly claimed research result. Through (9), it gives a positive corrected (10) margin only on that same interval. It gives no lower estimate outside it. Even having every ordinary even moment positive does not repair this. For any positive even kernel with entire transform `X`, ```text X(i*y) = sum_(k>=0) m_(2k)*y^(2k)/(2k)!. ``` All coefficients of `abs(X(i*y))^2=X(i*y)^2` are positive. Hence every generalized Laguerre coefficient at the single center `r=0` is positive, not just the first one. Positive Hankel moment matrices and the usual moment log-convexity inequalities are also generic consequences of a positive measure. These origin/measure facts do not assert the signs of the generalized Laguerre expressions at other real centers. For `Y`, the analogous origin series is confined to its analytic strip; the actual poles at `+/-i/2` must not be omitted. This is not a claim that exact complete moment data can never determine the transform. They can determine it under appropriate determinacy or analytic hypotheses. The missing step is an all-height sign estimate from that data, rather than generic positivity of the data or coefficients at one center. ## 3. NS101 matches these moment hypotheses exactly For the recorded positive reciprocal mixture, with `B_beta` as in (11), ```text m_0,beta = 5*m_0/B_beta, m_2,beta = (5*m_2+2*beta^2*m_0)/B_beta, m_4,beta = (5*m_4+12*beta^2*m_2+2*beta^4*m_0)/B_beta. ``` These follow by changing variables in each of the two translated complete integrals. All complete even moments are positive; the local bounds in section 2 and the origin generalized Laguerre signs therefore also hold for this family. Its positive primitive shares the exact resolvent relation, so ```text nu_2,beta-8*nu_0,beta = 16*m_2,beta > 0. ``` Despite these matched properties, PR74's previously recorded estimate is ```text L1[X_beta](pi/beta) <= -(beta^2-3*k)*m_0^2/(2*B_beta^2) < 0, k=m_2/m_0, beta >= max(2, pi*sqrt(k)). ``` Using (9), the difference between the two sides of (10) for `Y_beta` is exactly `16*L1[X_beta]/p^2`, and is negative there. This is a direct first-Laguerre control, not an inference from off-axis zeros. There is no contradiction with local positivity: the moment ratios and the guaranteed local interval depend on `beta`. For example `m_2,beta/m_0,beta = k+2*beta^2/5`; the local radius shrinks at order `1/beta`, while the negative witness also moves at that order. Continuity for each fixed function is not a common positive neighborhood for the whole varying family. The control changes the original single-lattice coefficients and fails the normalized original Jacobi IVP. It consequently rejects the inference from the shared moment/primitive properties, but does not reject an estimate that uses those distinguished arithmetic or IVP hypotheses at an essential step. No control pass for such an estimate is claimed. ## 4. The precise missing uniform input An admissible continuation would need an independently estimated lower bound for the complete expression ```text J(r) = p(r)^2*(Y'(r)^2-Y(r)*Y''(r)) +2*(r^2-1/4)*Y(r)^2 = 16*L1[X](r), for every real r. ``` For a local-to-global proof, that means either a sign-preserving propagation mechanism using the original arithmetic/IVP, or signed lower margins with controlled errors on a family of neighborhoods covering the entire real axis. Ordinary moment positivity supplies neither. Bounds on absolute derivatives can control remainders locally, but do not create a positive signed margin at a new center. Since the complete transform and its derivatives tend to zero, a fixed absolute error allowance also cannot simply replace the needed comparison at arbitrarily high centers. No division by `Y` at its zeros is allowed in this transfer. Requiring `J(r)>=0` again, or positive definiteness of the complete associated kernel, would only restate the target. This review identifies no independent original-coefficient estimate for it. **Final disposition:** retain (9)-(12) and their scopes. If mentioning origin positivity in the report, retain its polynomial correction and label its consequence local. No scan, new proposal, research row, manuscript version, thaw, or computation budget follows. The original theta all-height first-Laguerre question remains open.