# Next-step review 02: complete pair transport **Decision: STOP the currently specified transport mechanism. No independent estimating lemma is admitted.** The complete original Jacobi evolution remains unexcluded, but PR76 does not give it a positivity-preserving generator on the required integrated observables. **Wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR72–76.** What changes: nothing in the arithmetic input or required inequality. This is an admission assessment of whether the repaired primitive creates a usable transport mechanism, not another attempt to derive or compute its sign. The generic separation-clock justification is already defeated under its stated shared hypotheses. That scoped stop is not a proved wall for the original Jacobi IVP. ## Reviewed record and scope The coordinator refreshed and reviewed remote commit `18d324703969b5bcc9aa94043cfd90a6e8211f34`; the shared local interpretation checkout reports HEAD `32f599ced1abb2bf2376a715042e6e4101591639`. I read the current AGENTS section 0 and proposal/relevance gate, README checkpoints, NEXT_STEPS frozen inputs and October 5 entries, the ZERO-GEOMETRY and HEAT-TRANSPORT register scopes, evidence/MISSING.md, PR76's derivation and review 06, PR72's two source/transfer notes, PR73's current README, and PR74's review 02. The coordinator's full-register review is inherited; this is not an independent replay of every historical proof. Both original-evidence recovery groups remain OPEN. No new external theorem is imported here. The exact target and domain remain ```text p(r) = r^2 + 1/4, X(r) = p(r)*Y(r)/4, J(r) = p(r)^2*(Y'(r)^2-Y(r)*Y''(r)) + 2*(r^2-1/4)*Y(r)^2 >= 0, every r in R. ``` Here Y is the transform of the complete original smooth primitive f from PR76, not a prefix or substituted kernel. The assertion includes zeros of Y and r=+/-1/2 without division. Since J=16*L1[X], this is precisely the first Laguerre inequality. It is not RH, an all-order Laguerre criterion, G2, or the cofinal signed-Weil floor. ## The transport cone is already identified; its generator is missing The complete associated kernel is ```text 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. ``` With the complete decay and inversion hypotheses established in the existing record, the required sign is equivalent to positive definiteness of C: every finite matrix `[C(t_i-t_j)]` must be positive semidefinite for arbitrary real nodes and arbitrary complex coefficient vectors. Pointwise C>=0, pointwise pair positivity and positivity of the primitive are different properties. They do not establish this matrix condition. PR76 review 06 already checked the natural factor-separation family ```text C_lambda(t) = integral_R s^2*phi(s+lambda*t)*phi(s-lambda*t) ds = C(lambda*t), 0 <= lambda <= 1. ``` Its lambda=0 value is the positive constant C(0). For every lambda>0, however, arbitrary real nodes can be rescaled by 1/lambda, so positive definiteness of C_lambda is equivalent to the final target. Its reduced evolution `partial_lambda C_lambda=(t/lambda)*partial_t C_lambda` is singular at lambda=0. The constant endpoint does not define an independent positive initial state for a well-posed evolution to lambda=1. Changing the clock or replacing phi by the positive primitive does not repair this fact. In particular, taking small positive lambda is not a weaker all-matrix lemma: it already assumes the full all-height conclusion. Taking finitely many nodes instead changes the quantifier and requires an unprovided uniform limit. No smaller separation scan is recommended. ## What the original arithmetic actually supplies The archived Jacobi data are specific, not merely reflection symmetry: ```text da/dL=(U+a^2-1)/2, dU/dL=2*U*(a+chi), dchi/dL=a*chi-U, a(0)=chi(0)=0, U(0)=Gamma(1/4)^8/(64*pi^4), phi'(u)/phi(u)=2*P(2u). ``` P is the rational logarithmic-slope function of the distinguished state in PR72's support note; its clock is L=2u. This particular orbit and its original single-lattice theta coefficients exclude the exact NS101 reciprocal mixture. They could, in principle, supply an additional signed estimate. But closure of the two pointwise state vectors is not closure of their integrated pair observables. PR72 gives, with complete integrals, ```text A_s(r) = integral_0^infinity phi(s+t)*phi(s-t)*cos(2*r*t) dt, B_s(r) = integral_0^infinity P(2*(s+t))*P(2*(s-t)) *phi(s+t)*phi(s-t)*cos(2*r*t) dt, partial_s^2 A_s(r) + 4*r^2*A_s(r) = 16*B_s(r). ``` Differentiation of B introduces the next mixed moment containing the Jacobi vector field applied to P. That moment has no independently supplied lower estimate. It is signed both through the slope product across t=s and through the cosine. The modular state at one clock does not provide the initial values of all these integrated observables: even the coincident pair integral samples all clocks 2s. The PR72 closure obstruction excludes finite constant-matrix linear/affine systems containing its specified complete zero-frequency slice. It does not exclude variable-coefficient, nonlinear or infinite systems. Nothing in this assessment enlarges that obstruction. PR74's actual local alternative, the drift-cancelling residual `h=w/(p(u)+p(v))` in that note's slope notation, fails on the original large-coordinate diagonal. It therefore cannot serve as the missing invariant inequality either; arbitrary corrected or nonlocal certificates remain outside that specific failure. ## Why a heat or positivity label cannot fill the transfer Ordinary heat-flow preservation of nonnegative real-space functions is about a different cone from positive definiteness of the complete associated kernel. The signed cosine transform is not an order-preserving map for pointwise comparisons. The original Jacobi clock has not been identified with a heat generator on C or on the corrected target J. Such an identification, its direction, domain, initial data and invariant-cone statement would all need proof; no heat theorem is imported on the strength of terminology. Similarly, f>0 implies that its transform Y is positive definite in the frequency variable, not that Y is pointwise positive or that L1[Y]>=0. Even a separate proof of L1[Y]>=0 would still need `p^2*L1[Y] >= 2*(1/4-r^2)*Y^2` for |r|<1/2: the positive right-hand threshold offsets a negative polynomial term in J. Transport must recover the complete J, retaining both terms. PR76's meromorphic Y is also not silently placed in an entire-transform class. These are applicability checks, not additional sign obstructions. ## Matched control and exact stop - **NS101 / PR74–76:** the exact translated mixture has the same smooth positive primitive, envelope, unit leading exponential tail, complete pair identities, constant coincident-factor limit and dilation evolution. For beta>=max(2,pi*sqrt(m2/m0)), the already recorded complete-moment argument proves its corrected J is negative at r=pi/beta. This directly rejects the implication from those shared hypotheses to positivity transport. It does not rely only on off-axis zeros. The mixture changes the original theta coefficients and fails the distinguished Jacobi IVP; it is not a counterexample to an IVP-specific generator estimate. That mismatch is not a control pass. - **NS100:** its negative original cosine slice rules out a generator argument that silently assumes every A_s is nonnegative. It does not rule out the complete s^2-weighted integral or a valid signed invariant inequality. - **Davenport–Heilbronn:** the original conductor, coefficients and completion differ, so it is not an instance of the specified Jacobi IVP. No sampled control pass or first-Laguerre conclusion is claimed for it. - **HEAT-TRANSPORT / NS52 and NB controls:** these have different targets and domain/rate hypotheses. The exterior-cluster estimate and NB exclusions are not transferred to this pair problem. Thus the precise existing obstruction matches the available structural transport proposal, while the original arithmetic generator inequality is an open input. There is no surviving proposed lemma here whose success would be independently testable before that input is supplied. ## Admission decision and bounded budget **No new transport research is recommended from these inputs alone.** The following is a completeness test for a future supplied mechanism, not a candidate lemma or a renamed version of J>=0. Budget at most **two hours of paper review**, allocated as follows: 1. **30 minutes:** specify the actual nonsingular evolution and the complete observables on which it closes; identify its original theta/arithmetic coefficient dependence. A scalar Jacobi ODE or formal moment hierarchy alone fails this step. 2. **45 minutes:** exhibit an independently known positive initial object and a concrete generator estimate at every null matrix direction. The estimate must cover arbitrary real translates and complex coefficient vectors, with the regularity/uniqueness needed for cone invariance. The full signed estimate may not be assumed as a premise. 3. **30 minutes:** prove the final identification with complete C, or with J including its correction, retaining the two infinite tails, mixed terms and all real heights. A finite matrix or bounded-height statement needs an explicit uniform extension. 4. **15 minutes:** screen the full implication against NS101 and NS100 with exact hypothesis matching; reject any claim based only on a mismatch. Stop at the first missing dependency; do not use the rest of the budget for nearby clocks, larger matrices or coefficient scans. The present proposal already stops at step 1 and supplies no starting generator for a new session. These are all missing transfer steps, not implied deliverables of a two-hour proof attempt. **Success consequence:** a complete original-data cone certificate would prove the stated first-Laguerre target through the displayed identity; it would not prove RH. **Failure consequence:** reject that specified mechanism and retain the open original target. Current outcome: no independently established generator, invariant inequality or arithmetic lower bound; retain the ZERO-GEOMETRY freeze and do not fund a sign computation. Only this assigned internal note was written. No calculation run, scan, certificate, new research row/node, manuscript version, commit, push, outreach or agent spawn. Other work in the shared checkout was preserved. **Final wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR72–76.** RH, G2 and the actual cofinal signed-arithmetic lower bound remain open.