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.
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
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 complete associated kernel is
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
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.
The archived Jacobi data are specific, not merely reflection symmetry:
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,
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.
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.
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.
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:
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.