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Projection-Covariant Dynamics in Coherence Geometry: Presymplectic Reduction, Mapping-Space Evolution, and Observable Descent

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Internal ID: CGI-RSR-000040
Author(s): Barry L. Petersen
Document Type: Research Paper
Publication Date: September 29, 2026
Original Creation Date: September 22, 2026
Revised Document Date: September 29, 2026
Status: Public
Domains: Mathematics, Mathematical Physics, General
Sub-Domain: Projection Geometry / Complete Source Dynamics / Observable Descent
Research Topics: Projection-Covariant Dynamics, Complete Source Dynamics, Presymplectic Reduction, Mapping-Space Geometry, Observable Descent, Active Quotients, Generator Descent, Hidden Source Directions, Projection-Induced Metrics, Projection-Covariant Stiffness

Abstract

This paper develops a projection-covariant mathematical framework for separating the dynamics of a complete Coherence Geometry Source from the dynamics of any particular observable projection derived from that Source.

A complete Source may contain structural and dynamical directions that remain invisible to a selected observable. Consequently, a dynamical law established for one observable realization does not automatically determine the evolution of the exhaustive Source.

For differential Source readouts, the paper develops a presymplectic formulation in which the selected Source curvature determines an observable-active quotient while leaving characteristic directions available for hidden Source evolution. In this setting, material self-transport constrains the observable-active sector without necessarily fixing every degree of freedom of the complete Source.

The framework also develops the metric counterpart of this reduction. Nonlinear projection generally prevents raw Source stiffness from commuting directly with observable projection. A projection-induced horizontal metric, connection, and projection-covariant stiffness construction are therefore introduced so that spatial refinement may descend consistently to the observable geometry.

The resulting architecture separates representation, projection, hidden fiber motion, generator descent, and closed observable evolution. It provides a common mathematical layer for analyzing fluid, reactive, electromagnetic, informational, thermodynamic, and other coherence-geometric projections.

Available Document

DOI: 10.5281/zenodo.23040342

Citation:
Petersen, B. L. (2026). Projection-Covariant Dynamics in Coherence Geometry: Presymplectic Reduction, Mapping-Space Evolution, and Observable Descent (Version 1.0). Coherence Geometry Institute. Zenodo. https://doi.org/10.5281/zenodo.23040342

Source Code and Supporting Materials

No separate computational package is included with this archival release.

The principal results are analytic and geometric and do not require a separate numerical dataset, computational notebook, or software package.

Primary file:
Projection_Covariant_Dynamics_in_Coherence_Geometry.pdf

Version 1.0 archival research paper.

A README file accompanying the Zenodo deposit records document provenance, scope, citation information, file integrity, and archival metadata.

Summary and Notes

Scope:

This paper develops the mathematical distinction between complete Source dynamics and the dynamics visible through a selected projection.

Within Coherence Geometry, an observable is generally a reduced readout of a richer Source configuration. The existence of a projection therefore does not imply that a dynamical law written for the observable determines every degree of freedom of the underlying Source.

For a complete Source field (Y) and a selected projection (\pi), the observable state has the form

$$
Z=\pi\circ Y.
$$

A complete Source generator descends to a closed observable generator only when the projected Source evolution depends solely on the observable state rather than on hidden position within the projection fiber.

The paper develops this problem through several related constructions:

  • differential Source readouts and selected Source curvature;
  • presymplectic geometry on Source mapping spaces;
  • observable-active quotients of complete Source dynamics;
  • local canonical coordinates for finite Source-current sectors;
  • hidden or characteristic Source directions;
  • generator descent and dynamical closure;
  • projection-induced horizontal metrics;
  • projection-induced connections;
  • projection-covariant spatial stiffness and refinement; and
  • compatibility conditions between complete Source evolution and observable dynamics.

For conservative transport, the selected curvature form may be degenerate. In that case, transport constrains the observable-active quotient while leaving characteristic Source directions undetermined. Ordinary material advection of the entire Source is recovered only when the selected curvature is nondegenerate on the sector under consideration.

This produces the general hierarchy

$$
\text{complete Source}
\longrightarrow
\text{projection}
\longrightarrow
\text{observable},
$$

while emphasizing that

$$
\text{projection}
\neq
\text{generator descent}
\neq
\text{closed observable dynamics}.
$$

The distinction is important whenever several physical or informational observables are derived from a common Source. A finite realization sufficient for one observable need not exhaust the Source, and dynamical laws associated with one realization should not be promoted automatically to complete-Source equations.

The framework is formulated generally rather than for one application. It therefore provides a common projection-geometric layer for fluid, reactive, electromagnetic, informational, thermodynamic, and other coherence-geometric systems.

Related Work

Foundational framework:

Petersen, B. L. (2026). Coherence Geometry Foundations, Part I: Orientation, Closure, and Algebraic Foundations(Version 0.1). Zenodo.
https://doi.org/10.5281/zenodo.20156531

Petersen, B. L. (2026). Coherence Geometry Foundations, Part II: Physical Projections. Zenodo.
https://doi.org/10.5281/zenodo.20156996

Related Source and projection work:

Petersen, B. L. (2026). Differential Global Coherence: Intrinsic Jet Persistence, Transport Covariance, and Projection Noncreation (Version 1.0). Coherence Geometry Institute. Zenodo.
https://doi.org/10.5281/zenodo.23040056

Petersen, B. L. (2026). Coherence Geometry as a Structured Generalization of Fluid Dynamics (Version 1.0). Coherence Geometry Institute. Zenodo.
https://doi.org/10.5281/zenodo.23043221

Petersen, B. L. (2026). Coherence Geometry Foundations, Volume II: Source Curvature, Book I: Source Geometry, Fluid Projection, and Regularity (Version 0.1). Coherence Geometry Institute. Zenodo.
https://doi.org/10.5281/zenodo.23056935

Petersen, B. L. (2026). Coherence Geometry Foundations, Volume II: Source Curvature, Book II: Projection Archetypes and Structural Realizations (Version 0.1). Coherence Geometry Institute. Zenodo.
https://doi.org/10.5281/zenodo.23057738

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