Disposition: OPEN target; STOP this proposed estimating mechanism. No independent original-arithmetic estimate survives this bounded review. The boundary repair in the draft does not supply a positivity-preserving evolution for its corrected expression (10).
Reviewed remote baseline: d93aaf6b8dbe1fcbebf6fff4f6d1f71d993f4208
(PR75); local audit scope commit edc6563. I read the review brief,
draft derivation, AGENTS proposal/relevance rules, current ZERO-GEOMETRY
record, evidence ledger, PR72’s two Jacobi source/transfer notes, PR73’s
pair report, and PR74’s local nonlinear certificate review. I also checked
the stated source scope and Jacobi-system definitions in
Planat–Sole, arXiv:2608.19160v1, Sections 1 and 5.
The coordinator’s all-node review is inherited, not represented here as my
independent review of every historical proof. No source certificates were
replayed. Both original-evidence recovery groups remain OPEN.
Wall check: Same open gap. Closest: NS100/101, PR72–75. The precise question is whether the distinguished real-clock evolution can transport positive definiteness from coincident factors to the complete shifted pair. This is a paper applicability check, not a proposed new sign lemma or an attempted proof of one.
Keep the complete original kernel and its normalization:
C(t) = integral_R s^2*phi(s+t)*phi(s-t) ds,
L1[X](r) = 4*integral_R C(t)*cos(2*r*t) dt.
The draft’s corrected requirement (10) is exactly L1[X](r)>=0 at every
real height. Equivalently, the complete kernel C must be positive definite:
sum_(i,j) c_i*conj(c_j)*C(t_i-t_j) >= 0
for every m>=1, every (t_1,...,t_m) in R^m,
and every (c_1,...,c_m) in C^m.
The possible transport mechanism is: start with coincident factors and separate their two Jacobi clocks while preserving this matrix cone. A concrete way to expose the proposed clock is
C_lambda(t) = integral_R s^2*phi(s+lambda*t)*phi(s-lambda*t) ds
= C(lambda*t), 0<=lambda<=1.
At lambda=0 the kernel is the positive constant C(0), so every matrix is positive semidefinite. The proposal would need a legitimate evolution from that positive initial kernel to lambda=1, whose positivity follows from independently estimated properties of the actual Jacobi orbit. The equality above is only an examination of this transport interpretation; it is not a new research candidate or an estimate.
PR72’s checked clock is L=2u; write P for the logarithmic derivative
in that clock and V for its three-state vector field. The kernel slope
is phi'(u)/phi(u)=2*P(2u). At the separated clocks, differentiation uses
the multiplier
2*t*[P(2*(s+lambda*t))-P(2*(s-lambda*t))].
Thus an arithmetic step beyond NS101 would have to use the distinguished trajectory and its fixed initial data to prove that the integrated, weighted multiplier generates an inward-pointing evolution for every finite matrix and every coefficient vector. A scalar sign of P or V(P) does not do this: the required property is a sign of arbitrary mixed quadratic forms, not of individual entries. The source’s real-kernel curvature cone is a different cone and a different target.
Two concrete problems prevent this from becoming an estimating method:
C_lambda is already
equivalent to positive definiteness of C: arbitrary real nodes remain
arbitrary after division by lambda. There is no weaker intermediate
all-matrix claim to propagate through positive lambda values.partial_lambda C_lambda=(t/lambda)*partial_t C_lambda for lambda>0.
It is singular at lambda=0. The constant limiting kernel does not supply
a well-posed, positivity-preserving initial-value problem there. All
smooth complete associated kernels have this constant limit, including
the NS101 family that fails the actual first-Laguerre target in PR74’s
stated parameter range.This does not rule out a different nonsingular arithmetic evolution. It
shows why “start on the positive diagonal and run the original ODE” is not
one as stated. In particular the modular initial state of the original
three-state ODE is not initial data for all complete matrix observables:
even at lambda=0 the integral ranges over the full clock 2*s.
Such a proof would need all of the following, none supplied by this audit:
c^*M'c whenever Mc=0, with
the regularity needed to infer invariance. Naming this condition is not
establishing it.The unresolved item is the third, with no independent estimating mechanism for it from the fourth. The current regularization only resolves transform domains. It neither provides a closed generator nor supplies an initial positive kernel at a nonsingular parameter. Accordingly this list is a precise stop report, not a conditional candidate offered for a new scan.
PR72 already found that integrating the two-clock Jacobi dynamics creates an unestimated signed moment and then a hierarchy. Its exclusion concerns finite constant-matrix closures containing a specified slice; it does not exclude all nonlinear or infinite evolutions. It nevertheless prevents silently treating the original three-state closure as closure after integration.
PR74 review 02 assessed an actual local alternative: remove a total
derivative using the original logarithmic slope and require the residual
kernel to be positive semidefinite. Its explicit drift-cancelling choice
fails already on the original large-coordinate diagonal. The positive
primitive does not repair that original-phi certificate. Applying a new
certificate to the primitive would also have to retain the polynomial
correction in draft (9)–(10); proving only L1[Y]>=0 would miss the
positive right-hand side in abs(r)<1/2.
NS101/PR74 supplies a matched control for the generic transport inference: its complete positive even kernels have the same constant diagonal limit, the same dilation identity, and the draft’s positive primitive, envelope, unit tail and corrected algebra. Its first-Laguerre target fails for the already recorded beta range. Therefore those shared ingredients cannot justify transport from lambda=0. It changes the original coefficients and does not solve the distinguished Jacobi IVP, so it does not refute an actual IVP-specific generator inequality. That missing hypothesis is a concrete mismatch, not a passed control.
NS100 excludes per-slice positivity; this review retains the integrated pair and does not infer a complete sign from slice signs. Davenport– Heilbronn has different coefficients, conductor and completion and is not a matched original-IVP model. NS74/83 concern NB coefficient/rate domains and are not controls of the present matrix evolution. No new control was run and no sampled pass is asserted.
No independent method is admitted. Stop the direct “positive initial diagonal plus Jacobi clock” transport proposal. The repaired primitive is useful algebra and boundary accounting, but it does not change this stop. An original-arithmetic invariant-cone estimate remains possible in principle and unprovided here. There is no basis for funding a numerical orbit scan, a larger matrix table, or source-certificate replay from this review alone.
Success of a separately supplied complete generator certificate would prove only the stated first-Laguerre target through this dependency, not RH or the all-order criterion. The present failure changes route selection only: it rejects this transport justification and admits no replacement. Budget: one paper review, now complete; no computation budget follows. Only this assigned internal note was written. No row, thaw, manuscript version, commit, push, outreach, or extra agent was started.
Final wall check: Same open gap – ZERO-GEOMETRY, NS100/101 and PR72–75.