Primerfield Math
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 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 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.
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 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 4 — Summary
PF Math Module 4 presents a concise, physical explanation of wavelength, refractive index, light-speed reduction in matter, time delay, dispersion, and refraction within PrimerField (PF) theory, without using electromagnetic wave models. Wavelength is treated as the equilibrium spacing between photons, and transparent materials are shown to compress this spacing in a deterministic way, directly producing reduced propagation speed and measurable delay.
The module explains refractive index and dispersion as consequences of how matter resists photon fields, with shorter-wavelength photons experiencing stronger resistance. Refraction is described as arising from asymmetric spacing compression at material boundaries, which produces lateral bending of photon paths. Empirical calibration tables for common optical materials demonstrate consistency with measured behavior. The scope is intentionally limited to axial photon spacing and propagation, leaving diffraction and deeper microphysical detail to later modules.
PF Math Module 5 — Summary
PF Math Module 5 extends the PrimerField framework from longitudinal photon spacing to transverse boundary interaction, establishing how photon fields couple laterally to sharp edges and apertures. Building directly on Module 4, it shifts focus from forward propagation and speed reduction to the spatial extent over which boundaries influence photon detection beyond simple geometric ray limits.
The module introduces a transverse sensitivity scale that defines how far, sideways from an edge or aperture, a boundary can measurably affect photon behavior as a function of wavelength and propagation distance. This scale is calibrated against classical diffraction experiments and serves as a numerical anchor for all later PF treatments of edges, apertures, diffraction, and interference. The analysis is intentionally minimal and does not attempt to derive full diffraction patterns, instead providing a physically grounded constraint that limits where boundary effects can occur.
Within PF theory, this transverse sensitivity is interpreted as a real photon-field influence envelope rather than wave interference or self-interaction. The module clearly defines what this scale represents, how it is measured experimentally, and what it does not represent, ensuring strict scope control. Module 5 therefore establishes the transverse geometric limits that govern photon–boundary coupling and prepares the foundation for subsequent interference and diffraction modules.
PF Math Module 6 — Summary
PF Math Module 6 completes the sequence begun in Modules 4 and 5 by defining the physical regimes where PrimerField (PF) edge-coupling behavior is valid, where it breaks down, and how it can be experimentally falsified. Rather than introducing new mechanisms, the module establishes clear validity limits, failure modes, and testable predictions tied directly to experimental conditions.
The module specifies the conditions required for reliable transverse boundary coupling, including sharp edges, isolation from competing boundaries, low absorption and scattering, narrow spectral bandwidth, and consistent measurement protocols. It then identifies breakdown scenarios—such as rough edges, nearby secondary boundaries, loss or scattering, and instrumental limits at very short or very long propagation distances—explaining how and why measured results deviate under those conditions.
Finally, the module provides explicit, falsifiable experimental predictions that distinguish intrinsic PF behavior from measurement artifacts, including saturation effects, roughness transitions, and loss-induced suppression. By clearly separating physical limits from experimental limitations, Module 6 establishes rigorous criteria for testing, validating, or refuting PF transverse edge-coupling claims and locks the canonical calibration used throughout later work.
PF Math Module 7 — Summary
PF Math Module 7 provides the formal foundation for treating photon wavelength as a stable equilibrium spacing produced by PrimerField (PF) magnetic field interactions. The module explains why photon spacing is fixed in free space, why it compresses inside matter, and why it restores immediately upon exit, all without invoking oscillatory waves or probabilistic mechanisms.
The module models adjacent photons as interacting through overlapping PF magnetic fields that produce long-range attraction and short-range repulsion. This balance creates a stable equilibrium spacing that defines wavelength. Near equilibrium, photon chains behave like elastic magnetic systems: compression or extension generates restoring forces, but no time-based oscillation is required. As a result, photon rays behave as magnetically coupled structures rather than wave trains.
When photons enter matter, transverse compression of PF fields shifts this equilibrium spacing in direct proportion to the material’s refractive properties. This naturally explains wavelength shortening in media and its instantaneous recovery on exit, without hysteresis or energy barriers. The module also clarifies how interference, diffraction, and other wave-like phenomena emerge from continuous field overlap and elastic spacing, not from traveling waves. Module 7 completes the PF photon force framework by grounding wavelength, stability, and medium response in deterministic magnetic equilibrium.
PF Math Module 8 — Summary
PF Math Module 8 extends the PrimerField (PF) equilibrium-spacing framework to provide a physical explanation of refractive index and dispersion based on wavelength-dependent field compression. Building on Modules 4–7, it defines refractive index as a direct measure of how strongly a material compresses photon fields and explains why different colors of light refract by different amounts.
Within the PF framework, photons propagate as discrete chains with a stable equilibrium spacing. When photons enter matter, their magnetic fields are compressed elastically by the medium, reducing photon spacing and propagation speed. The refractive index of a material is therefore interpreted as the inverse of this compression effect, rather than as a wave-based phase property. Because photon field size depends on wavelength, different colors experience different degrees of compression in the same material, producing dispersion.
The module shows that measured refractive-index curves directly encode how a material’s resistance to photon fields varies with wavelength. Normal and anomalous dispersion arise naturally from whether field compression weakens or strengthens with wavelength, without invoking oscillatory waves or phase mechanics. A worked reference example using standard optical glass illustrates how empirical data map cleanly onto PF field-compression behavior.
Overall, Module 8 unifies wavelength, refractive index, propagation speed, and dispersion into a single, physically consistent PF description based on magnetic field equilibrium and elastic compression, completing the optical scaling framework without reliance on electromagnetic wave models.
PF Math Module 9 — Summary
PF Math Module 9 formalizes the PrimerField (PF) explanation of the double-slit experiment by converting the qualitative claim “one photon influences both slits” into a precise, geometry-based, pass/fail condition. The module’s purpose is not to introduce new physics, but to define when the PF mechanism is geometrically possible and when it must fail.
The module establishes a strict requirement: for deterministic PF double-slit behavior to occur, the photon’s transverse interaction region must fully span both slits at the slit plane. This requirement is tied directly to the transverse sensitivity scale developed in earlier modules and depends on wavelength, effective source-to-slit distance, slit width, and slit separation. Partial overlap is shown to be insufficient, because PF redirection requires asymmetric field compression across the entire width of the second slit.
From this requirement, the module defines a critical slit separation. Below this separation, PF predicts deterministic double-slit behavior consistent with observed interference patterns. Above it, the PF mechanism is no longer geometrically admissible, and the double-slit effect must break down or change character. The module provides clear experimental guidance for testing this prediction using real slit geometries, controlled separation sweeps, and standard far-field or Fourier-plane measurements.
Module 9 closes by emphasizing that the result is falsifiable and audit-ready: the PF double-slit mechanism either satisfies the slit-spanning condition or it does not. The outcome depends only on measurable geometry and previously established transverse scaling, providing a clear experimental boundary between valid PF double-slit behavior and regimes where it cannot occur.