PrimerField Mathematical Foundations
The papers collected here present the formal mathematical framework underlying PrimerField (PF) Theory. PF Math Modules 0–9 progressively define the geometry, scaling relations, boundary conditions, and deterministic constraints implied by the theory, moving from foundational field definitions to explicit, experimentally testable limits. The accompanying Transverse Sensitivity paper provides the empirical basis for the transverse interaction scaling employed throughout the modules, fixing numerical constants through measurement rather than assumption. Together, these works translate PF Theory from a physical model into a rigorously constrained mathematical system, explicitly stating its required conditions, derived predictions, and points of potential falsification to enable independent analysis and experimental evaluation.
Transverse Sensitivity Scale of Photons
When light passes near the edge of an object, the edge affects where photons are detected—even at surprising distances. This paper answers a simple question: how far sideways from an edge can light still be affected? The answer is about 2 millimeters for visible light, which represents thousands of wavelengths. This work establishes precise, quantitative constraints that any theory of light must be able to reproduce.
In addition, this paper provides the mathematical framework used within PrimerField (PF) Theory to determine the physical transverse extent of a photon’s electromagnetic field. These quantitative constraints serve as the empirical anchor for estimating the size of the PrimerField surrounding an individual photon and are used consistently throughout the PF literature
PF Math Module 2 — Summary
PF Math Module 2 extends the geometric framework established in Modules 0–1 by introducing a minimal, deterministic matter–field coupling and formal confinement logic within PrimerField geometry. Matter is modeled as a state evolving under a Lyapunov-style interaction energy with the PF field, producing overdamped, field-aligned motion without invoking fluid dynamics or probabilistic behavior. The module defines the flip ring (FR) as an analytically well-posed event surface, imposing a discrete polarity inversion through a canonical flip operator when matter crosses the FR transversely. Confinement is formalized using a barrier-function approach within the dome-bounded volume, allowing internal drive to accumulate deterministically until thresholded escape occurs. Two and only two escape paths—an upper axial exit and a lower axial exit—are rigorously defined as exit sets in state space, yielding a precise “two-exit” confinement theorem under stated assumptions. Throughout, Module 2 deliberately separates geometry, kinematics, and thresholds from detailed microphysics, providing a mathematically explicit but still testable framework for confinement, inversion, and escape that later modules can refine or specialize.
PF Math Module 0 — Formal Definitions of PF Fields and Confinement Geometry
This paper establishes the mathematical foundation of PrimerField (PF) Theory by defining the core geometric and analytic structures on which all later PF results depend. The PF magnetic field is treated as an axisymmetric vector field on ℝ³, from which the canonical PF features—Flip Point (FP), Choke Ring (CR), Flip Ring (FR), and Confinement Dome (CD)—are defined using measurable scalar quantities.
All PF structures are specified analytically through zero-crossings, separatrices, extrema, and smooth level sets, ensuring they are uniquely identifiable, testable against field data, and free from qualitative ambiguity. The confinement region is defined purely geometrically, without invoking dynamics or matter behavior.
This module converts PF concepts into precise mathematical objects and serves as the required foundational reference for all subsequent PF math modules.
PF Math Module 3 — Summary
PF Math Module 3 formalizes PrimerField theory’s definition of wavelength by modeling a light ray as a one-dimensional chain of coupled photon field systems whose equilibrium axial spacing constitutes the wavelength. Rather than invoking oscillatory electromagnetic waves, the module defines wavelength as the stable separation that arises from an explicit photon–photon interaction potential exhibiting short-range repulsion, long-range attraction, and a unique convex equilibrium. Using an overdamped gradient-flow formulation, the module shows that uniform photon spacing is a dynamically stable equilibrium and introduces an effective “magnetic spring constant” governing elastic response to perturbations. Medium interaction is treated as a modification of the interaction potential, producing reduced equilibrium spacing inside matter and automatic recovery upon exit, thereby encoding refractive slowing without introducing new dynamics. The framework also establishes the mathematical origin of dispersion by allowing the interaction potential to depend on photon field size, yielding different equilibrium spacings for different colors. Throughout, Module 3 deliberately limits itself to axial spacing and energy structure, leaving transverse diffraction, refraction geometry, and detailed microphysical potentials to later modules.
PF Math Module 1 — Summary
PF Math Module 1 formalizes the topological structure of PrimerField configurations using the analytic anchors introduced in Module 0. It defines how space is partitioned by the axial field component BzB_zBz, identifying separatrix surfaces (Bz=0B_z = 0Bz=0) as true topological boundaries that divide the domain into distinct sign regions. The module rigorously defines the roles of the flip point (FP), choke ring (CR), flip ring (FR), and confinement dome (CD), establishing each as a geometric or topological object rather than a dynamical or material feature. Using these elements, the module introduces dome-bounded subdomains, axial channel neighborhoods, and a canonical separatrix graph that uniquely characterizes PF field architecture independent of scale or polarity. It also extends the framework to paired (“two-bowl”) systems, defining inter-bowl confinement regions purely as bounded connected components of the combined field topology. Throughout, Module 1 deliberately excludes any claims about particle motion, flow, confinement dynamics, or transport, providing a strictly geometric and topological foundation for later modules to build upon.