# Review 06 — admission decision and a scoped quotient falsification test **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. ## 1. Decisions on the three mechanisms **Heat: stop this formulation.** The complete identity ```text 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. ## 2. The pole implication has a complete, narrower decision edge Retain the original completed theta kernel and complete real-s integrals: ```text 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) 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 ```text 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. ## 3. Certificate readiness differs from diagnostic admissibility 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 ```text 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. ## 4. Matched controls and budget Before any original locator, validate the actual pole/cancellation pipeline on two exact synthetic controls: ```text 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.**