Primary Organizational Field Theory

This upload replaces the earlier versions of the article associated with DOI 10.5281/zenodo.19736041.

Version v.3 presents the revised canonical formulation of the Primary Organizational Field Theory, in which physical reality is modelled as a dynamic spectral-phase field $\Phi (\mathbf{x},\omega ,t)$ and persistent objects are treated as emergent organizations rather than primitive substances. The revision preserves the key methodological principle that topology belongs to the space of solutions, not to the defining Lagrangian. It introduces a rigorous hierarchy linking phase flow, circulation, vortical non-integrability, toroidal closure, intrinsic angular momentum, and spin-related observables, while clearly separating these distinct levels of organization. The theory of memory is substantially expanded: memory is defined not merely as a delayed local trace, but as persistent recoverable content, learned relational access, structural closure or redundancy, and dynamostatic return. The document also develops informational impedance, adaptive transport geometry, the spectral coordinate, organizational observables, and explicit validation criteria. 

ANNEX
This numerical annex extends The Canonical Theory of the Primary Organizational Field, Version 5.2 by developing a constitutive account of rheology, memory, impedance, and transport in the complex spectral-phase field (\Phi). It consolidates a thirty-stage full-field research programme showing that a persistent toroidal organizer exhibits anisotropic, chiral, nonlinear, and history-dependent response, requiring distinct LOCAL, LONG, and RETURN constitutive regimes.

The study further demonstrates that recovery of field amplitude does not necessarily restore the original force–response law, and that apparent reversals of the odd response channel can arise from applying a local decoder outside its domain of identification. Direct reduced and full-tensor probes instead support retention of the physical fixed-axis pseudovector throughout the tested recovery trajectories.

The annex also identifies a CORE–SHELL transport topology in which shell conductance, boundary throughput, and backflow regulate the absolute level of mobility and informational impedance. The results support a hierarchy of universality: topology, winding, sector structure, and chiral orientation are more transferable than exact constitutive coefficients. All quantitative claims remain restricted to the declared dimensionless numerical model and are not presented as SI-calibrated viscosity, particle mass, gravitation, or a replacement for established physical theories.

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Old version v.2 description:
The revision updates the formal foundation of the work. The previous reduced architecture based on the integrated object ((U,\chi,A_\mu)) is retained only as an effective sector chart, while the primary object is now defined as the full spectral-phase field (\Phi(x,\omega,t)). The replacement clarifies the role of phase-spectral currents, closure defects, informational impedance, low-impedance resonant channels, memory return, and boundary admissibility certificates.

The purpose of the replacement is not to claim experimental validation or a completed derivation of the Standard Model, but to make the theoretical ontology and numerical artefacts consistent with the current DIFT formulation. The new version also includes updated publication figures and tables generated by the revised full-field demonstrator, including four toroidal ((5,1))-like attractors, low-impedance resonant paths, and anti-phase neutron-response structures.

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Old version v.1 description:
What is the smallest field structure capable of sustaining topological identity, amplitude–phase flow, gauge closure, shell organization, and composite binding without collapsing into a single scalar? This work answers that question by constructing an integrated three-sector field Ψ=(U,χ,Aμ), where U∈SU(2) carries topological identity, χ=ρeiθ governs shell currents, and Aμ provides gauge-mediated closure.

The paper delivers a complete formal framework: action, Euler–Lagrange equations, topological baryon current, explicit DIFT functionals (coherence, informational impedance, dynamostasis), dimensionless reduction, and linear stability analysis. But the centerpiece is a first-of-its-kind three-dimensional validation on a validated DIFT carrier:

• H1 (Real-time persistence): A shell-bound χ sector remains dynamically dominant for >90% of the evolution, with final ηshell≈0.9946, ηedge≈0.0018, and exact mass conservation.

• H2 (Reduced-mode stability): Projected Hessian analysis reveals no negative eigenvalue in a physically interpretable 9‑dimensional fluctuation basis.

• H3 (Computational trajectory): A continuous homotopy from weak to strong coupling yields explicit terminal observables: E∗=−62.63, R∗=4.51, rshell,∗=5.07, and ω∗=3.13.

This is not a claim of particle physics or full quantum theory. It is a rigorous, reviewer‑safe demonstration that a shell‑bearing regime can be dynamically persistent, modally stable (in a reduced sector), and connected to well‑defined observables — turning a conceptual DIFT extension into a computationally testable substrate for further research.

Keywords: DIFT, topological field theory, Skyrme model, gauge fields, shell persistence, reduced‑mode stability, computational trajectory
DOI: 10.5281/zenodo.19736041