A First-Order Terminal-Closure Criterion for the Riemann Hypothesis and the Exterior-Rank Source Boundary
Internal ID: CGI-RSR-000008
Document Type: Research Paper
Publication Date: May 2026
Status: Public
Domains: Mathematics
Research Topics: Number theory, Riemann Hypothesis, Clay Millennium Problems
Abstract
We formulate a strict first-order terminal-closure criterion for the Riemann Hypothesis using a matrix-valued completed explicit formula. A two-state transfer test is attached to the functional-equation pair \(\alpha\), \(1-\alpha\). For an off-critical zero orbit, a negative-support one-sided test makes the prime and gamma/algebraic transfer contributions silent and yields the forced first-order ledger
$$
T_x(B_{\rm other}^{\alpha})=2S_\alpha,
$$
For the fixed test, analytic tail compactness gives an operator-norm compact-limit representative
$$R_\infty=2S_\alpha.$$
Terminal closure is taken in the strict first-order ordinary zero-orbit source
category, while completion is performed inside the positive full-Pauli cone.
The full two-state ledger is retained as accountability data. The obstruction
to projective collapse of the distinguished pair is the rank-area
$$\mathcal A_{12}
=
\Omega_{11}\Omega_{22}-|\Omega_{12}|^2,$$
This rank-area is a second-order exterior resource. Since the first-order matrix-valued explicit formula has no primitive exterior-square source labels, strict first-order terminal admissibility excludes unassigned rank-area on the forced two-source channel. Hence (\mathcal A_{12}=0), so the terminal Gram pair collapses projectively. Label-compatible source-faithfulness then gives
$$1-\alpha=\overline{\alpha},$$
and therefore
$$\Re(\alpha)=\frac12.$$
Consequently, every nontrivial zero admitting strict first-order compact-limit terminal closure lies on the critical line. We also identify the boundary of the criterion: admitting nonzero rank-area as terminal analytic data would require either a genuine second-order exterior source law or an equivalent determinant, kernel, spectral, trace, or positivity
framework giving exact pair-specific control of the distinguished rank-area. No such mechanism is supplied by the first-order completed-explicit-formula framework used here.
Available Document
Citation:
Petersen, B. L. (2026). A First-Order Terminal-Closure Criterion for the Riemann Hypothesis and the Exterior-Rank Source Boundary. Zenodo. https://doi.org/10.5281/zenodo.19971259
Source Code and Supporting Materials
N/A
Summary and Notes
From a Coherence Geometry viewpoint, the functional-equation pair
$$
\alpha,\qquad 1-\alpha
$$
should be treated as a labelled two-source object. If \(\alpha\) lies off the critical line, then \(\alpha\) and \(1-\alpha\) remain distinct modulo conjugation. On the critical line, they collapse because
$$1-\alpha=\overline{\alpha}.$$
The manuscript studies this distinction using a matrix-valued completed explicit formula and a two-state transfer test. The goal is not to isolate a zero by arbitrary interpolation, but to track the transfer defect created by an off-critical functional-equation orbit.
Related Work
N/A