Finite-core Volterra reductions for a Weyl-positive Riemann phase kernel

Abstract: We record a Weyl-positive reduction and certificate framework for the Riemann phase kernel associated with the even Riemann kernel $Φ$. The manuscript does not present a complete proof of the Riemann hypothesis. Its immediate analytic target is a concrete positivity theorem for a Weyl kernel whose quantum characteristic function satisfies the Kastler--Loupias--Miracle-Sole condition in all numerical tests performed so far. Several natural factorizations are ruled out. In particular, the positive anti-Wick density route is obstructed by a local heat-deconvolution test, and several natural finite-core reductions are excluded by explicit counterexamples. The surviving structure is a finite-core Volterra program upgraded to a closed-trace quotient certificate for the full kernel. We derive exact same-sign finite-core formulae, the second-order theta-mode identity $$φ_n(t)=(\partial_t<sup>2-1/4)(e<sup>{t/2}e<sup>{-πn<sup>2e<sup>{2t}})$$ for $n\ge1$, a Volterra boundary-plus-tail representation, and a quotient Schur factorization for the normalized full-$Φ$ source/Volterra model. The latest certificate closes the active trace-range condition, the full-continuum source-inactive domination, and the Douglas/Moore--Penrose Schur hypotheses in the normalized model. What remains outside that certificate is explicitly separated: the quotient-to-original Weyl lift, uniform $ω$-coverage for $|ω|&lt;1/2$, and the final bridge from Weyl/KLM positivity to the intended de Branges or RH-side formulation.

• The paper constructs a closed normalized quotient-Schur certificate connecting Volterra positivity to Weyl/KLM positivity for the full Riemann kernel, while explicitly stopping short of proving the Riemann hypothesis.
• The analysis resolves an endpoint Hardy obstruction by splitting off a finite-dimensional jet space, certifying active-range rank with interval arithmetic, and obtaining a Douglas constant that falls from 1.36×10⁻⁶ to 6.90×10⁻⁸ under refinement.
• The paper rules out several local and finite-mode positivity shortcuts and identifies the uniform ω-cover, unconditional quotient-to-original lift, and KLM-to-de Branges intertwiner as the principal unresolved steps.

Scope and architecture

This manuscript records a reduction-and-certificate program for proving Weyl positivity of a Riemann phase kernel, explicitly disclaiming a proof of the Riemann hypothesis. Starting from the even Riemann kernel $\Phi$, the author constructs a phase-space Weyl symbol $\sigma_\omega(x,\xi)$ and its quantum characteristic function $Q_\omega(s,t)$, and targets the Kastler–Loupias–Miracle-Sole (KLM) positive-type condition, equivalent to positivity of the Weyl quantization, for $|\omega|<1/2$. The intended logical chain is

Volterra Schur positivity $\Rightarrow$ Weyl/KLM positivity $\Rightarrow$ de Branges kernel positivity $\Rightarrow$ shifted-$\Xi$ zero exclusion,

and the paper's central discipline is the separation of proved identities, certified theorems (with reproducible finite/interval inputs and operator-theoretic closure statements), numerical evidence, and external bridges. The internal certificate — a closed-trace quotient Schur factorization for the normalized full-$\Phi$ source/Volterra model — is claimed to be closed; the quotient-to-original Weyl lift, uniform $\omega$-coverage, and the final KLM-to-de Branges bridge are explicitly quarantined as open.

The kernel and the parity contraction target

The Weyl kernel is built from the positive-side theta-mode expansion $\sigma_\omega(x,\xi)$0, with the symbol $\sigma_\omega(x,\xi)$1 formed by integrating $\sigma_\omega(x,\xi)$2, where $\sigma_\omega(x,\xi)$3 is a cosine transform of $\sigma_\omega(x,\xi)$4. On the half-line, parity decomposition reduces Weyl positivity to a sharp contraction problem: with $\sigma_\omega(x,\xi)$5 the same-sign kernel and $\sigma_\omega(x,\xi)$6 the reflected cross-kernel, the target is $\sigma_\omega(x,\xi)$7 and $\sigma_\omega(x,\xi)$8, equivalently $\sigma_\omega(x,\xi)$9. This formulation is forced by the data: $Q_\omega(s,t)$0 itself is indefinite (minimum eigenvalues of order $Q_\omega(s,t)$1 to $Q_\omega(s,t)$2), while after projection off $Q_\omega(s,t)$3 the contraction spectrum is contained in $Q_\omega(s,t)$4 to roundoff, with numerical rank of $Q_\omega(s,t)$5 equal to 7 at $Q_\omega(s,t)$6 samples. The paper is candid that these are roundoff-level observations, not proofs.

Obstructions to natural factorizations

A substantial part of the paper's value is its systematic elimination of plausible shortcuts, each with explicit counterexamples:

• Positive anti-Wick density is obstructed by a local heat-deconvolution test: at $Q_\omega(s,t)$7 near $Q_\omega(s,t)$8, $Q_\omega(s,t)$9, one has $|\omega|<1/2$0 but $|\omega|<1/2$1, so even $|\omega|<1/2$2 is negative. The sign of the first deconvolution correction is too large for this to be a resolution artifact.
• Finite-core Hermite–Biehler fails: for the three-mode core at $|\omega|<1/2$3, the inequality has the wrong sign near $|\omega|<1/2$4, consistent with Pólya's warning about one-sided theta truncations.
• Layerwise local source positivity is false ($|\omega|<1/2$5 for the three-mode Green-source kernel).
• First-order score integration by parts after splitting $|\omega|<1/2$6 yields an indefinite kernel $|\omega|<1/2$7 (minimum near $|\omega|<1/2$8).
• Exact finite negative index for the anti-Loewner boundary is false: high-precision tests show a confluent tail of tiny negative eigenvalues rather than exactly two negative squares.

The conclusion drawn is structural: positivity must be proved at operator level, keeping the Volterra integral and full $|\omega|<1/2$9 weight intact, not layer-by-layer or branch-by-branch.

Exact identities and the zero-slope core

Three exact results anchor the finite-core analysis. First, the same-sign Weyl kernel of any exponential core admits an exact expansion in incomplete-gamma functions $\Rightarrow$0 and its $\Rightarrow$1-derivative; this formula exposed that earlier Simpson quadrature produced false negative alarms by missing endpoint layers with diagonal entries near $\Rightarrow$2. Second, the second-order theta identity $\Rightarrow$3 reduces the first-order score problem to a second-order Volterra problem for the base atom $\Rightarrow$4. Third, the exact Volterra boundary-plus-tail identity expresses $\Rightarrow$5 as a boundary term plus an integral of $\Rightarrow$6 along Volterra rays — proved via a single differentiation along the ray, with both boundary and fixed-$\Rightarrow$7 layers individually indefinite (minima near $\Rightarrow$8 and $\Rightarrow$9), so only the integrated object is positive.

The zero-slope corrected core $\Rightarrow$0, with $\Rightarrow$1, is not cosmetic: a nonzero $\Rightarrow$2 produces a delta-mass jump in the reflected mixed kernel at the boundary-crossing point, and $\Rightarrow$3 removes it exactly. On $\Rightarrow$4 the tail satisfies $\Rightarrow$5, and on a witness vector at $\Rightarrow$6 the third mode contributes $\Rightarrow$7 against a two-mode defect of $\Rightarrow$8, leaving $\Rightarrow$9; the $\Rightarrow$0 tail contributes only $\Rightarrow$1.

The endpoint Hardy problem and its resolution

The hardest analytic layer is a one-mode truncated Mellin monotonicity theorem at the endpoint $\Rightarrow$2, equivalent to positivity of the self-adjoint anti-commutator form $\Rightarrow$3 on $\Rightarrow$4. The paper shows this cannot be closed by all-order sign-regularity: a Wronskian computation gives $\Rightarrow$5, so the scalar Laguerre kernel fails reverse sign-regularity inside the endpoint rectangle. It also cannot be closed by point-kernel positivity: closed-form high-precision evaluation of the Green kernel $\Rightarrow$6 finds a genuine confluent negative mode of order $\Rightarrow$7 at $\Rightarrow$8, and a Taylor-jet computation shows the corrected $\Rightarrow$9-branch endpoint kernel has $\Xi$0 at $\Xi$1. Crucially, the full three-mode reduced Volterra kernel is positive at the same jets ($\Xi$2 at $\Xi$3), so the endpoint defect is a distributional jet defect removed by the finite-core structure, not a robust negative eigenspace.

The resolution proceeds by splitting the test space into an endpoint-jet space $\Xi$4 — the closed span of Taylor functionals $\Xi$5 built from the lowest eigendirection of the ninth-order jet, active for $\Xi$6 — and its smooth complement. On the complement, positivity of the Volterra moment form $\Xi$7 follows from a dilation-monotonicity proposition: if $\Xi$8 is Loewner monotone in $\Xi$9, then $\Phi$0, and finite-difference tests recover the moment matrix. On the endpoint space, a Moore–Penrose/Douglas quotient factorization $\Phi$1 is constructed, with the finite Douglas constant $\Phi$2 decreasing under refinement ($\Phi$3 at basis 10 down to $\Phi$4 at basis 16) and zero normalized range residual throughout.

Active range inclusion and the source-side noncollapse theorem

The remaining bottleneck — whether the active source rows factor through the closed trace range — is reduced to a rank statement: $\Phi$5 on the two-dimensional active source eigenspace, with $\Phi$6 at $\Phi$7. The paper closes this in the certified finite model through a chain of increasingly rigorous steps: a Riesz-projection/Davis–Kahan perturbation argument with a certified spectral gap $\Phi$8; a composite-trapezoid source-quadrature bound with a Bernstein-ellipse tail estimate, made machine-rigorous via complex interval arithmetic ($\Phi$9, against a tolerance of $\omega$0); and a Krawczyk/interval-collocation enclosure of the endpoint Plücker chart showing $\omega$1, which certifies full row rank of the endpoint map $\omega$2 and hence solvability of the adjoint Green boundary-value problem. The full-theta tail is then absorbed: the $\omega$3 derivative envelopes are below $\omega$4, interval propagation through the normalized source-row map gives $\omega$5, and the continuum-inflated inactive estimate $\omega$6 fits inside the finite Schur budget $\omega$7 with slack $\omega$8 (about 23% of the budget consumed).

Quotient-Schur assembly and the trace-fiber contraction

With active range inclusion and the source-inactive tail in hand, the closed-trace quotient theorem applies: on $\omega$9 the active source component vanishes and the inactive component is dominated, giving $\sigma_\omega(x,\xi)$00 on $\sigma_\omega(x,\xi)$01; polarization and the same bound give the bounded $\sigma_\omega(x,\xi)$02-factorization of the cross form. The output is the factorization $\sigma_\omega(x,\xi)$03, so $\sigma_\omega(x,\xi)$04 implies $\sigma_\omega(x,\xi)$05 — the normalized full-$\sigma_\omega(x,\xi)$06 quotient Schur certificate.

The repair-free reduction to the original kernel is then handled via the trace-fiber Schur kernel $\sigma_\omega(x,\xi)$07, which admits an exact but signed Volterra–Green feature representation $\sigma_\omega(x,\xi)$08. The paper identifies the closed-form multiplier $\sigma_\omega(x,\xi)$09, $\sigma_\omega(x,\xi)$10, with $\sigma_\omega(x,\xi)$11, and closes the contraction $\sigma_\omega(x,\xi)$12 by a density/closure argument in the completed Volterra graph norm, using the Euler–Lagrange identity $\sigma_\omega(x,\xi)$13 for $\sigma_\omega(x,\xi)$14. Finite tests show the top generalized eigenvalue $\sigma_\omega(x,\xi)$15 climbing toward 1 ($\sigma_\omega(x,\xi)$16 at the $\sigma_\omega(x,\xi)$17 stress point), indicating the constant is sharp. Since $\sigma_\omega(x,\xi)$18, the primitive endpoint compatibility closes conditionally: $\sigma_\omega(x,\xi)$19 and $\sigma_\omega(x,\xi)$20 together give the quotient-to-original lift.

Limitations and open questions

The paper is unusually explicit about what remains. Three external links are open. First, although the algebraic lift and endpoint compatibility are marked closed conditionally, the boundary-repair comparison $\sigma_\omega(x,\xi)$21 rests on the completed-domain closure argument, and the finite diagnostics for $\sigma_\omega(x,\xi)$22 are Galerkin-level with constants approaching sharpness. Second, all generated full-$\sigma_\omega(x,\xi)$23 certificates are at the stress value $\sigma_\omega(x,\xi)$24; the uniform-in-$\sigma_\omega(x,\xi)$25 argument via the $\sigma_\omega(x,\xi)$26-independent multiplier $\sigma_\omega(x,\xi)$27 and super-exponential theta decay is plausible but is presented as a packaging layer, not a certified parameter cover. Third, and most substantially, the KLM-to-de Branges bridge is unresolved: the Hermite–Biehler inequality $\sigma_\omega(x,\xi)$28 is itself RH-facing, and every tested explicit intertwiner — coherent-packet pullbacks (best relative residual $\sigma_\omega(x,\xi)$29), trace-lifted primitives, plus-branch and coupled-branch normal equations (residuals $\sigma_\omega(x,\xi)$30), coarea and resolvent/heat transmutation dictionaries, and diagonal Mellin atom matching (residual $\sigma_\omega(x,\xi)$31) — fails. The exact Mellin convolution identity shows the obstruction is structural: the incomplete-gamma boundary prefix is essentially the full atom at the relevant base point and cannot be routed through the existing $\sigma_\omega(x,\xi)$32 trace family (nullspace residual $\sigma_\omega(x,\xi)$33). An augmented trace $\sigma_\omega(x,\xi)$34 with the Mellin primitive functional kills the prefix and admits a positive finite Schur repair, but the final evaluation pullback $\sigma_\omega(x,\xi)$35 with $\sigma_\omega(x,\xi)$36, or its closed-cone limit, remains unconstructed; the Hardy-side truncation limit is closed (entry tail $\sigma_\omega(x,\xi)$37 at $\sigma_\omega(x,\xi)$38) but the KLM-side branch Grams are not matched by any tested candidate.

Conclusion

The manuscript delivers what it claims: a set of exact identities (the second-order theta identity, the incomplete-gamma same-sign formula, the Volterra boundary-plus-tail decomposition), a disciplined set of disproved shortcuts, and a closed normalized quotient-Schur certificate for the full-$\sigma_\omega(x,\xi)$39 Weyl/Volterra model, with the endpoint Hardy obstruction resolved by a jet-space Schur split and a machine-rigorous Plücker-chart rank certificate. Its honesty about the boundary between certified content and external bridges is a methodological strength. The remaining work is precisely delimited: the uniform $\sigma_\omega(x,\xi)$40-cover, the unconditional closure of the quotient-to-original lift, and above all the non-circular construction of the KLM/de Branges intertwiner, for which the paper's own diagnostics indicate that no scalar, diagonal, or generic-kernel ansatz will suffice and that a Mellin-convolution-derived transmutation with an augmented trace repair is the required object.