Rendered report · Review record · Proposal record
Wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR76–78. No original all-height signed estimate is obtained. The three methods expose an unestimated heat source, reject an unrestricted sector-matrix condition, and leave a contour density unestimated. A separate useful rejection criterion emerges: any uncancelled interior pole of the complete original quotient C/M would rule out its positive definiteness. No such pole has been located.
Alex explicitly requested this six-agent continuation after PR78. Three
proposers assessed auxiliary heat, modular duplication and direct quotient
contours; three independently checked algebra, complex analysis, controls and
admission. Reviewed remote: 221ceec9da825bdc787111447badad961860760a.
The 104-node/14-input review concerns conclusions and dependencies, not a
replay of all historical proofs or certificates. Both recovery groups stay OPEN.
For the auxiliary Gaussian family H_a=Fourier(exp(a*u^2)*phi), let
Q_a=L1[H_a], S_a=L1[H_a'] and G_a be the ordinary heat kernel. Then
partial_a Q_a = -Q_a'' + 2*S_a,
Q_0 = G_A*Q_A - 2*integral_0^A G_a*S_a da, A=1/8.
The endpoint normalization is checked against the literature’s scaled kernel. It has nonnegative Q_A; no lower uniform margin is assumed. The source is subtractive in the required direction and has no independent arithmetic estimate. It is already positive at r=0. A locally bounded relative-source bound also imposes an unproved restriction near multiple zeros. This auxiliary heat clock is not the original nonlinear Jacobi evolution; neither route is generally excluded.
The completed theta2/theta3/theta4 sectors are exact translates of the original
completed kernel. Their full Hermitian Laguerre matrix has inertia (2+,1-)
whenever X is nonzero. This rejects unrestricted sector-matrix positivity.
Subtracting the known phase correction leaves precisely L1[X]*A*A^*.
All pair sectors and lower moments are retained; nonlinear and infinite
duplication methods remain unexcluded but have no supplied estimating cone.
For the contour analysis, define the complete original functions
M(z)=integral_R phi(s+z)*phi(s-z) ds,
C(z)=integral_R s^2*phi(s+z)*phi(s-z) ds,
k=C/M, K(xi)=Fourier(k)(xi).
M and C are holomorphic on abs(Im z)<pi/4; M is positive on both coordinate
axes. The complete complex saddle estimate makes M zero-free at distant
horizontal ends of every closed substrip. Its remaining zeros there are finite.
A nonremovable pole of k creates a leading oscillating residue contribution
to K. The lowest poles have no zero frequency because none is imaginary-axis;
their nonzero mean-zero trigonometric sum forces both signs. This includes
multiple poles and all complete tails.
M(z0)=0, C(z0)!=0, abs(Im z0)<pi/4
=> k has a nonremovable pole
=> K changes sign
=> k is not positive definite.
This rejects only a sufficient certificate. In the exact transfer
L1[X](r)=(1/pi)*integral_R X(r-xi/2)^2*K(xi) dxi,
a sign-changing K can still yield a nonnegative result. The original J, first Laguerre, RH, G2 and the cofinal arithmetic bounds remain open. Pole absence also does not imply positive definiteness; an explicit analytic control in review 05 demonstrates that distinction.
A bounded locator is useful in principle because it now has a complete
rejection decision. An already known or externally supplied root is not
required by AGENTS. The review’s possible discovery domain is
1/8<=Re z<=2, 1/8<=Im z<=5/8, with at most a 9-by-9 grid, four selected
cell refinements and twelve refinement steps per cell. Cap discovery, including
controls, at 90 minutes; do not expand the region after a null result. This
domain is a tractability choice, not evidence of a zero there.
Before running, a separate completed protocol must fix the evaluator,
complete-integral error handling, exact cell-selection/stopping rule and
matched control implementation. Use an uncancelled-pole control and a
removable-pole control, so detecting M=0 alone cannot reject k. Diagnostic
artifacts belong under evidence/diag_*; floating phase winding is not a
zero certificate. A promising box then needs complete Arb argument-principle
and C-nonvanishing enclosures, uniform tails, and two-precision replay.
Initial certification budget: at most two hours for one specified box.
No witness means inconclusive. A certified uncancelled pole closes the general quotient-PD sufficient construction, beyond PR78’s particular mixture; it does not disprove J. Failure to exclude zero in C is inconclusive, not cancellation. No locator or certificate was run in this paper round.
The synthesis resolves an admission wording difference: review 03’s suggested externally supplied box is an option, not a prerequisite for all diagnostic discovery. An instantiated box is required for an immediate certificate; a bounded locator requires its own prior protocol and control screen.
No research row, node, thaw, sign scan, interval certificate or manuscript version. Current budget: six paper reviews and one audit PR. No automatic unbounded search, neighboring mixture or global route pivot follows.