Mathematics
Coherence Geometry provides a structural framework for investigating mathematical systems through coherence-governed structure, constraint, and stability. Research in this domain currently focuses on foundational mathematical formulation, unifying functional structures, and selected major problem settings where coherence-based methods provide new organizing principles.
Research Topics
Foundational Structure
Canonical Foundations
Formal definitions, named results, and structural principles underlying Coherence Geometry.
Unified Coherence Functional (UCF)
Development of a framework for relating mathematical structures through coherence, constraint, and variational organization.
Major Problem Settings
Clay Millennium Problems
Applications of coherence-geometric methods to selected Clay Millennium Prize Problems, including structured investigations of problem-specific constraints, stability conditions, and formal closure mechanisms.
Publications List
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A Source-to-Endpoint Construction for Compact-Simple Yang–Mills Existence and Mass Gap
CGI-RSR-000034 | This archival research paper presents a source-to-endpoint construction for compact-simple Yang–Mills theory on four-dimensional Euclidean spacetime.
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The Riemann Hypothesis from Zero-Orbit Source Closure and Rank-Area Exclusion
CGI-RSR-000033 | We reformulate a strict first-order terminal-closure criterion for the Riemann Hypothesis as a source-closure problem in Coherence Geometry. The analytic endpoint construction used here is the matrix-valued completed-explicit-formula framework developed in the earlier terminal-closure criterion paper.
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The Birch-Swinnerton-Dyer Rank Equality from Source Closure and Endpoint Readout
CGI-RSR-000032 | This paper formulates the rank equality in the Birch–Swinnerton-Dyer conjecture as a source-closure and endpoint-readout problem in Coherence Geometry.
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Coherence Geometry Foundations, Part II: Physical Projections
CGI-BKS-0002 | Part II of the Coherence Geometry Foundations working reference text. It examines how quantum-like linear dynamics, causal and relativistic structure, residual coherence strain and cosmological background terms, electromagnetic field structure, thermodynamic behavior, and macroscopic transport can be represented as domain-level projections of the shared-amplitude, multi-phase coherence framework.
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Coherence Geometry Foundations, Part I: Orientation, Closure, and Algebraic Foundations
CGI-BKS-0001 | Part I of the Coherence Geometry Foundations working reference text. It introduces multi-phase numbers, shared-amplitude phase structure, mathematical closure, projection, coherence relations, coherence matrices, coherence manifolds, projective coherence objects, morphisms, operators, and the canonical convergence and stability results governing coherent systems.
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A Variational–Relaxation Framework for Coherence-Driven Structure Formation
CGI-RSR-000013 | We develop a variational framework for coherence-driven structure formation, placing phase coherent self-organization within a unified geometric description compatible with variational principles used throughout physics.
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The Unified Coherence Functional: A Closed Generative Basis for Mathematics
CGI-RSR-000012 | We introduce the Unified Coherence Functional (UCF), a closed variational framework in which broad classes of mathematical structures arise as stationary projections.
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Mathematical Foundations of Coherence Formation and Stability
CGI-RSR-000010 | We establish a mathematical framework for coherence formation and stability in energy-minimizing phase systems. Introduces the UCCP, SCRC, and CS-SCT.
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A Multi-Phase Extension of Complex Numbers and the Global Coherence Theorem
CGI-RSR-000009 | This is the foundational document that introduces multi-phase numbers and the Global Coherence Theorem (GCT).
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A First-Order Terminal-Closure Criterion for the Riemann Hypothesis and the Exterior-Rank Source Boundary
Internal ID: CGI-RSR-000008 | We formulate a strict first-order terminal-closure criterion for the Riemann Hypothesis using a matrix-valued completed explicit formula.
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