Proposal: Bounded diagnostic of an uncancelled complex zero of the ORIGINAL M(z)=int_R phi(s+z)phi(s-z)ds, C(z)=int_R s^2 phi(s+z)phi(s-z)ds, in 1/8<=Re z<=2, 1/8<=Im z<=5/8. A candidate is not a certificate.
Shared-input group: ZERO-GEOMETRY; validation of PR78/79’s sufficient quotient-PD construction, closest NS100/101. Status: frozen; no thaw or research-row claim. This is a falsification diagnostic of a specified sufficient mechanism, not a lower-bound proposal.
Arithmetic input beyond the functional equation: phi(u)=sum_(n>=1)(2pi^2n^4exp(9u/2)-3pin^2exp(5u/2))exp(-pin^2*exp(2u)), extended evenly. Keep the complete original coefficients and complete M,C integrals. They determine whether this actual denominator has an uncancelled zero. The generic contour lemma alone supplies no existence claim; altered arithmetic controls cannot supply this original witness.
Dependency edge and arithmetic use (before any new candidate computation): Complete original M(z0)=0, C(z0)!=0, abs(Im z0)<pi/4 => uncancelled pole of k=C/M => Fourier(k) changes sign by PR79’s complete-strip finite-pole argument => k is not PD. This rejects the general sufficient quotient-PD certificate. It does not refute original L1[X], RH or G2: L1X=(1/pi)int X(r-xi/2)^2 Fourier(k)(xi)dxi can be positive with a signed kernel. Diagnostic root/numerator samples do not establish these hypotheses. Certificate would additionally require full Arb integral/tail enclosures, a nonzero M boundary, rigorous positive argument count, and nonzero C throughout a box (or rigorously separated zeros), at two precisions. No simplicity assumption. A bounded null does not prove zero-freeness or PD.
Control screen (actual candidate, before proof/certification): Davenport-Heilbronn: not-applicable to this literal evaluator: different coefficients, gamma/completion and unverified complex-strip tail domain. The unchanged controls/example_screen.py computes a different derivative assertion and cannot screen a pole locator. NS100 log-concave order-2: not-applicable; the recorded obstruction is a per-slice cosine transform, while this tests the complete moment quotient. NS101 reciprocal dilation: shares generic quotient/convolution algebra and sufficient PD implication, changes the original lattice/Jacobi IVP. Known negative L1 implies its quotient is not PD but does not establish a pole; without a known pole it cannot validate locator sensitivity. Not run or called a pass. Additional control: SAME locator for M0=z^4+1,C0=1 on [7/10,18/25]^2 must find an approximate root with separated numerator samples. SAME M1=M0,C1=M0*exp(-z^2) must not label it an uncancelled pole. The first exact Fourier density changes sign; second quotient is exp(-z^2) with positive Fourier density. These share meromorphic/contour and axis-nonzero hypotheses, not theta arithmetic. Controls run before original evaluations; outputs archived. NS74/83: not-applicable: integer-dilation target sensitivity and NB per-doubling rate restriction; no NB approximation/rate claim. Command/artifact, precision, sample domain, values/residuals: evidence/diag_theta_pole_locator/locator.py controls first; original run only after control/reviewer gate. 128-bit discovery and same-cell256-bit replay. Full records and exact CLI to be archived with results. No original evaluation performed at preregistration. Full-scope obstruction, if any: PR78 excludes the positive sech-power mixture only, not general PD. None established for this original bounded search.
Wall check: Distinct test Closest result: PR78/79; NS100/101. What changes: search for an original uncancelled denominator zero, which would reject general quotient PD rather than only the already rejected mixture. No signed lower estimate supplied.
What success changes: a diagnostic candidate prepares one box for interval verification; only a verified uncancelled pole excludes general quotient PD. What failure changes: no witness/control or numerical failure => inconclusive; stop this bounded test, no automatic domain enlargement or adjacent scan; no manuscript version. Budget: 90 minutes discovery including controls, one PR; only if a promising candidate exists, at most one box and two hours for a separately scoped complete Arb certificate. Disposition before claim: admit bounded diagnostic subject to passing matched controls and independent protocol review; no research row claimed. Version bump expected: no for this diagnostic/audit.
9x9 equally spaced vertices. For each of64 cells use four-corner sampled principal-argument sum/(2*pi), not a certified winding. Rank by descending abs(round(sampled winding)), ascending min(abs(Mcorner))/max(abs(Mcorner)), then x/y indices. If the corner minimum is <=2^(-bits/2) times the maximum (or the scale is zero), sampled winding is marked unresolved and set to zero for ranking; residual ordering remains. First four cells only. Same-cell Newton from center, <=12 iterations, each proposed step halved at most16 times to stay inside the SAME cell; otherwise stop. Analytic derivative of normalized M. Preserve all failures. Replay the SAME selected cells at256bits; do not rescan or reselect.
Approximate-root label requires endpoint residual abs(M)/corner-scale<2^-40 at both precisions and endpoints agreeing within2^-32 times cell diagonal. Numerator sample separation requires abs(C)/corner-C-scale>2^-20 at both precisions and C agreement within2^-32 relative corner scale; otherwise numerator unresolved. These thresholds prove neither an M zero nor C exclusion on a neighborhood. All cell scales and raw values retained.
Evaluator: modular-even original phi with24 lattice terms initially,32 on replay. Gauss-Legendre48/96 nodes per segment, initial real-s breakpoints [0,1/8,1/4,1/2,1,2,4], replay adds9/2. Multiply M and C by the SAME entire nonzero B(z)=exp(2piexp(2z)-9z)/(4pi^4); include B’ in normalized M derivative. Normalization preserves zeros and exact boundary winding, but sampled phase can alias. Lattice and real-tail bounds explicit. Quadrature error is NOT certified. Precision/cutoff/node replay detects instability but is not a complete integral enclosure. This limitation is part of the protocol, not a later downgrade.
Draft PR80 created from preregistration commit728cc6c before original computation. Matched actual-pipeline controls passed: both toy denominator roots located, uncancelled numerator sample separated and removable numerator unresolved without pole flag. Independent reviewer03 admits the bounded diagnostic only. Initial unchanged manuscript build:358 pages, zero undefined/duplicate references. At that pre-run checkpoint, the original run remained pending until evaluator/CLI readiness.
Disposition after test: inconclusive; stop this bounded test. No row claimed. The actual matched controls passed. Original171evaluations (81grid +40initial refinements +10replayed corners +40replay refinements) produced no endpoint meeting2^-40. All four paths stopped at cell boundaries. No boundary extension, certificate attempt, or sign inference followed. An independent direct-term/tanh-sinh implementation reproduced the same four endpoints (256bits,n36,R5); largest observed relative discrepancy across M,C,normalized M’ is1.404e-26, not an interval bound. Complete tails are analytic formulas evaluated in ordinary arithmetic; finite quadrature and roundoff remain unbounded.
Six agents completed reviews01–06; no unresolved major finding. Main outcome and exact commands are in audits/theta-pole-locator-2026-10-05-v1/README.md and evidence/diag_theta_pole_locator/results.md. Working Python3.14.6/mpmath1.3.0; pyright:zero errors/warnings on three touched scripts. Manuscript unchanged, successful build358pages, zero undefined/duplicate references. No new version, so new-version manifest check not applicable.104 registered nodes and14 input demands unchanged; ZERO-GEOMETRY annotation/source only. General quotient PD and the original Laguerre estimate remain open.
What had to change: nothing in the frozen domain or candidate thresholds; no apparent root was found, so the conditional Arb stage was not entered. The blocker is absence of a witness under this locator, not a proved mathematical obstruction. No manuscript version for diagnostic/audit work.