{"id":2606,"date":"2026-05-03T15:32:12","date_gmt":"2026-05-03T15:32:12","guid":{"rendered":"https:\/\/coherencegeometry.com\/?p=2606"},"modified":"2026-05-09T14:05:48","modified_gmt":"2026-05-09T14:05:48","slug":"a-first-order-terminal-closure-criterion-for-the-riemann-hypothesis-and-the-exterior-rank-source-boundary","status":"publish","type":"post","link":"https:\/\/coherencegeometry.com\/index.php\/2026\/05\/03\/a-first-order-terminal-closure-criterion-for-the-riemann-hypothesis-and-the-exterior-rank-source-boundary\/","title":{"rendered":"A First-Order Terminal-Closure Criterion for the Riemann Hypothesis and the Exterior-Rank Source Boundary"},"content":{"rendered":"\n

Internal ID:<\/strong> CGI-RSR-000008
Document Type:<\/strong> Research Paper
Publication Date:<\/strong> May 2026
Status:<\/strong> Public
Domains:<\/strong> Mathematics
Research Topics:<\/strong> Number theory, Riemann Hypothesis, Clay Millennium Problems<\/p>\n\n\n\n

Abstract<\/h3>\n\n\n\n


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
$$R_\\infty=2S_\\alpha.$$<\/p>\n\n\n\n

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<\/p>\n\n\n\n

$$\\mathcal A_{12}\u2028=\u2028\\Omega_{11}\\Omega_{22}-|\\Omega_{12}|^2,$$

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.<\/p>\n\n\n\n

Available Document<\/h3>\n\n\n\n

DOI:<\/strong> 10.5281\/zenodo.19971259<\/a><\/code><\/p>\n\n\n\n

Citation:<\/strong>
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<\/p>\n\n\n\n

Source Code and Supporting Materials<\/h3>\n\n\n\n

N\/A<\/p>\n\n\n\n

Summary and Notes<\/h3>\n\n\n\n

From a Coherence Geometry viewpoint, the functional-equation pair<\/p>\n\n\n\n

$$
\\alpha,\\qquad 1-\\alpha
$$<\/p>\n\n\n\n

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<\/p>\n\n\n\n

$$1-\\alpha=\\overline{\\alpha}.$$<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

Related Work<\/h3>\n\n\n\n

N\/A<\/p>\n\n\n\n","protected":false},"excerpt":{"rendered":"

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