Effective Horizon‑Scale Field Dynamics — Microscopic Foundation for Cosmological Coherenc

Effective Horizon-Scale Field Dynamics: Microscopic Foundation for Cosmological Coherence — Linear Stability Analysis and Morphic Threshold Determination DOI : 10.5281/zenodo.18157189 Abstract: This chapter presents a comprehensive linear stability analysis of a three-field reaction–diffusion model developed as a microscopic foundation for cosmological coherence dynamics at horizon scales. The model describes the evolution and interaction of three scalar fields: coherent informational density x(r,t), residual memory q(r,t), and recomposition amplitude w(r,t). These fields govern pattern formation and morphic transitions in cosmological media where local reaction kinetics compete with diffusive transport, incorporating saturation effects and bilinear recomposition mechanisms that generate nontrivial spatial structure. The analysis establishes the complete mathematical framework: governing component equations, homogeneous steady state determination, linearisation procedure, and construction of the Jacobian family J(k) = J₀ − Dk² as a function of spectral parameter k. Eigenvalue computation yields the spectrum λᵢ(k) for i = 1, 2, 3, with the dominant quantity λₘₐₓ(k) = maxᵢ Re[λᵢ(k)] determining linear stability. The morphic threshold Sₖ is defined operationally as the critical value of control parameter S where the system transitions from stable to unstable dynamics, located via bisection with quantified uncertainty. For Case C steady state parameters, the analysis reports Sₖ = 0.3335 with conservative 95% confidence interval ± 0.0014, obtained through rigorous numerical protocols including analytic derivative verification (relative errors < 10⁻⁸), Jacobian conditioning diagnostics, eigenpair residual checks, and bootstrap uncertainty quantification (N = 1000 samples). The spectrum is sampled at 50 uniformly spaced wavenumber points in k ∈ [0, 2.0] with local refinement near threshold crossings. This work provides a falsifiable theoretical prediction for coherence dynamics in cosmological field systems. The model applies to scenarios where informational coherence exhibits saturation, memory effects persist across spatial scales, and recomposition mechanisms drive structural transitions. Physical motivation derives from cosmological contexts where horizon-scale processes mediate the emergence and dilution of coherence, potentially connecting to observables in large-scale structure formation, cosmic microwave background anomalies, and dark sector phenomenology. Complete reproducibility is ensured through comprehensive technical annexes documenting all numerical artefacts: reference steady states, Jacobian matrices at representative k values, full spectral traces, finite-difference verification logs, bisection traces, branch tracking algorithms, analytic validation cases, conditioning protocols, and uncertainty quantification procedures. All materials carry reproducibility identifier MAGISTRALEXAMPLEPHASEA20260106001 and are deposited with this manuscript. The methodology is mathematically rigorous (canonical spectral theory for reaction–diffusion systems), numerically robust (verified analytic derivatives, monitored condition numbers, cross-checked eigenpairs), and epistemologically stratified (Level A interpretive framework, Level B operational definitions, Level C falsifiable predictions). Results are expressed with quantified uncertainties suitable for observational confrontation. This chapter advances cosmological field theory by providing a concrete, operational model where morphic transitions are governed by precisely defined thresholds accessible to numerical determination and observational testing. The approach prioritises predictive power and falsifiability over speculative completeness, establishing a foundation for empirical validation of coherence dynamics in cosmological systems. Keywords: linear stability analysis, reaction-diffusion systems, cosmological field dynamics, morphic threshold, spectral analysis, horizon-scale processes, coherence evolution, falsifiable predictions, numerical verification, reproducible research Discipline: Theoretical Cosmology, Mathematical Physics, Astrophysics Intended Audience: Astrophysicists, cosmologists, theoretical physicists working on large-scale structure formation, cosmic field dynamics, and observational cosmology. Technical Level: Advanced research. Assumes familiarity with partial differential equations, linear stability theory, spectral analysis, and numerical methods for eigenvalue problems. Reproducibility: Complete. All computational protocols, numerical outputs, verification procedures, and data files required for independent reproduction are provided in technical annexes with reproducibility identifier MAGISTRALEXAMPLEPHASEA20260106001. License: Copyright and Intellectual Property Notice © 2026 Nelson Karabudak. All rights reserved. This document is an original scientific work protected by copyright and intellectual property law. Any reproduction, distribution, modification, or commercial use, in whole or in part, is prohibited without the prior written authorisation of the author. Scientific study, academic analysis, and non-commercial use are permitted, provided that the author and the original source are explicitly cited. Contact: Nelson Karabudak, Independent Researcher nelson.karabudak@gmail.com Related Work: This chapter forms part of the Théorie du Champ Infini (Theory of the Infinite Field), a comprehensive cosmological framework exploring informational coherence dynamics, Genesis Black Holes, and horizon-scale field processes.