Decision: stop the three proposed sign estimators; retain a potentially admissible bounded diagnostic for the quotient, subject to its own completed proposal and controls. No computation is authorized or performed by this review. A conjecture does not need a proof before admission. Here the problem is that heat and contour formulas supply no estimating technique, while the specified duplication cone is false. The quotient pole implication does, however, supply a complete rejection decision that an experiment could test.
Wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR76–78.
No original all-height signed estimate is obtained. The conditional pole
obstruction is useful route assessment, not a positivity estimate. A future
fixed-domain pole diagnostic could be classified Distinct test against
PR78: it tests general positive definiteness of k=C/M, rather than the
already rejected positive sech-power mixture. It would not test RH.
Reviewed baseline 221ceec9da825bdc787111447badad961860760a, scope commit
b5147e82803cccdb641bcb611b3c90c3c76cf51e; AGENTS, BRIEF/review record,
reviews 01–04, PR78 conclusions and the proposal template. The coordinator
owns the full-register refresh. This is an admission/dependency audit, not
historical proof replay. Both original-evidence recovery groups remain OPEN.
Heat: stop this formulation. The complete identity
Q_0 = G_A*Q_A - 2*integral_0^A G_a*S_a da,
Q_a=L1[H_a], S_a=L1[H_a'], A=1/8
has the correct adverse source and literature scaling. A coefficient-level
upper estimate for the accumulated source could qualify as an unproved
candidate. None is supplied; its comparison with G_A*Q_A merely restates
Q_0>=0. The shortcut S_a<=0 fails at zero height, and a bounded
S_a/Q_a comparison imports restrictions near possible multiple zeros.
This auxiliary Gaussian clock is not the original nonlinear Jacobi clock.
No claim that the latter route has been excluded follows.
Duplication: reject the unrestricted sector cone; defer an unspecified
nonlinear alternative. The exact original sectors satisfy
v=A(r)*X(r). Their full Hermitian Laguerre matrix is congruent to a fixed
matrix with inertia (2 positive, 1 negative) whenever X!=0. Its
all-sector positivity is false for the original data. Subtracting the known
phase contribution leaves L1[X]*A*A^*, which is exactly the scalar target.
Nonlinear or infinite refinement would need an actual invariant signed
inequality and a complete derivative-controlling limit. No such proposal
is present. Negative sector inertia is not negative original L1[X].
Contour positivity: stop this formulation. Moving the complete quotient integral to an interior horizontal line exposes every residue and a signed boundary density. A pole-free strip, an absolute tail bound, or a contour identity alone estimates no lower sign. Requiring the shifted integral to have the desired sign in a pole-free strip is equivalent to requiring the original Fourier sign. There is no independent arithmetic estimator here.
Retain the original completed theta kernel and complete real-s integrals:
M(z)=integral phi(s+z)*phi(s-z) ds,
C(z)=integral s^2*phi(s+z)*phi(s-z) ds,
k=C/M, K(xi)=integral_R k(t)*exp(i*xi*t) dt.
Review 03’s strip argument gives holomorphic M/C on abs(Im z)<pi/4,
positive M on both coordinate axes, finitely many zeros on closed
substrips, and integrable horizontal quotient lines avoiding poles.
The complex saddle estimates and the complete tails are material premises;
a finite theta prefix cannot replace them.
An uncancelled denominator zero in that strip gives an upper-half-plane pole by symmetry. The lowest upper pole height has finitely many poles; none lies on the imaginary axis. Shifting above that height gives the leading Fourier contribution
exp(-b*xi)*xi^(m-1)*T(xi),
where T is a nonzero real trigonometric polynomial with no constant term. Its positive and negative values recur away from zero; the smaller polynomial and exponential terms do not remove both signs. Thus
original M zero with C nonzero, abs(Im z)<pi/4
=> K changes sign => k is not positive definite.
This is not a claim that such a zero exists. Nor does it contradict the
original target: K>=0 is sufficient, not necessary, in
L1[X](r)=(1/pi)*integral X(r-xi/2)^2*K(xi) dxi.
A signed K can still produce nonnegative convolution. Success in finding an uncancelled pole would close the general quotient-PD sufficient construction, a stronger rejection than PR78’s mixture failure. Failure to find one in a bounded region would leave that construction undecided.
No actual candidate rectangle is supplied. Therefore immediate original zero certification is not ready. But AGENTS does not require somebody to supply a zero before an experiment: that would confuse discovery with verification. A separately pre-registered bounded locator can satisfy the gate because the rejection implication above is complete and useful. An externally supplied rectangle is one option, not a repository rule.
A concrete possible diagnostic scope, for the coordinator to accept or revise before computation, is the single rational rectangle
1/8 <= Re z <= 2, 1/8 <= Im z <= 5/8.
It stays away from the known zero-free coordinate axes and the strip edge, where the displayed theta series ceases to converge. This is a tractability choice, not a prediction that a root lies there or a coverage theorem. Pre-register at most a 9-by-9 evaluation grid, refinement of at most four cells selected by a stated winding/residual rule, and at most 12 local refinement steps per cell. No enlarged region or adjacent second scan is implied by a null result. Initial 128-bit calculations and 256-bit replay can be diagnostic only; retain original theta coefficients, complete integral/tail error treatment and cutoff replay. Floating winding estimates or apparent roots are not certificates.
A subsequent certificate must enclose a specific compact rectangle wholly
inside the strip, prove M nonzero on its entire boundary, and rigorously
enclose integral_boundary M'/M /(2*pi*i) around a positive integer.
Prove C nonzero throughout that rectangle, or otherwise rigorously
separate the particular M zero from C zeros. Enclose both lattice and
real-integration tails uniformly on the complex domain. Use Arb interval
arithmetic, two working precisions, no midpoint solves, and explicit
quadrature/subdivision errors. No simplicity assumption is necessary.
An inconclusive C enclosure is not evidence of cancellation.
Before any original locator, validate the actual pole/cancellation pipeline on two exact synthetic controls:
M_0(z)=z^4+1, C_0(z)=1,
K_0(xi)=pi/sqrt(2)*exp(-abs(xi)/sqrt(2))
*[cos(abs(xi)/sqrt(2))+sin(abs(xi)/sqrt(2))].
The toy box 7/10 <= Re z, Im z <= 18/25 encloses its upper-right
pole and lies inside the strip. This quotient has a changing Fourier sign.
It shares contour calculus and axis nonvanishing, not theta arithmetic.
For the removable-pole control take C_1=M_0*exp(-z^2). Then
k_1=exp(-z^2) has positive Fourier density. A denominator-zero-only
rejection must fail this control. Neither toy supplies original evidence.
NS101 shares quotient/convolution algebra and strip regularity, but changes
the original lattice/IVP. Its known negative Laguerre value forces its own
quotient to fail PD; it does not prove that poles cause that failure.
Do not require a pole locator to find one there without a separate witness.
NS100’s per-slice obstruction is not this complete quotient assertion.
Davenport–Heilbronn changes coefficients, gamma factor and completion;
its own complex strip/tails need checking before transplanting this
algorithm. The unchanged controls/example_screen.py tests a different
derivative expression and cannot screen this proposal. Record these
mismatches explicitly. NS74/83 constrain NB loads/rates and are inapplicable
to this theta quotient; no NB gain claim is being made.
Current budget: one paper review, zero scans/certificates. A possible later
locator budget is 90 minutes including control implementation, with the
fixed evaluation caps above and artifacts under evidence/diag_*. A
promising box would require a separately scoped certificate budget, capped
at two hours initially. Exhaustion or no witness means inconclusive and
stop, not PD, not a positive target estimate, and not an automatic wider
search. No row, thaw or manuscript version follows from this review.
Final disposition: stop heat, reject the stated sector cone, stop the unestimated contour positivity claim; retain a potentially admissible bounded quotient-falsification diagnostic for explicit preregistration. The original first-Laguerre inequality, RH, G2 and cofinal arithmetic bounds remain open. Final wall check: Same open gap.