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% ───────────────────────────────────────────────────────────────────────────── |
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% AZOTH — UNIVERSAL SENSORY OPTIMA & COHERENCE GEOMETRY |
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% Version: v2.1 (Final, Canon-Locked) |
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% |
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% Author: James Paul Jackson |
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% Date: January 2026 |
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% |
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% PURPOSE |
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% ------- |
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% This document formalizes the Azoth framework: a physics-first theory |
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% describing the emergence of optimal sensory channels and their stability |
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% under noise and perturbation. |
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% |
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% Sensory modality selection is treated as a constrained optimization problem |
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% governed by flux, transmission, and noise, independent of biology, cognition, |
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% or semantics. A deviation-weighted coherence functional reveals a bounded |
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% stability horizon (H₇) as a scale-invariant fixed point. |
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% |
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% WHAT THIS IS |
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% ------------ |
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% • A mathematical model for sensory channel optimization |
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% • A stability law derived from perturbation survival |
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% • A falsifiable, simulation-ready framework |
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% • Applicable to biology, astrobiology, and engineered sensors |
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% |
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% WHAT THIS IS NOT |
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% ---------------- |
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% • Not a biological adaptation theory |
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% • Not a cognitive or semantic model |
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% • Not numerology or golden-ratio mysticism |
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% • Not a claim of universal constants |
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% |
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% CORE IDEAS |
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% ---------- |
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% • Sensory optima arise from flux–transmission–noise tradeoffs |
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% • Stability is governed by deviation-weighted survival (Ω-basin) |
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% • A coherence horizon (H₇) emerges as a geometric fixed point |
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% • Self-similar ratios appear only as projection artifacts |
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% |
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% IMPLEMENTATION STATUS |
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% --------------------- |
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% Fully implementable using Gaussian models, quadratic noise, |
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% Monte Carlo perturbation analysis, and surrogate testing |
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% (IAAFT / phase randomization). |
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% |
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% ───────────────────────────────────────────────────────────────────────────── |
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\documentclass[12pt]{article} |
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\usepackage{amsmath,amssymb} |
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\usepackage{geometry} |
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\usepackage{hyperref} |
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\geometry{margin=1in} |
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\title{\textbf{Azoth: Universal Sensory Optima and the Coherence Geometry Horizon}} |
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\author{\textbf{James Paul Jackson}} |
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\date{January 2026} |
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\begin{document} |
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\maketitle |
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\begin{abstract} |
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We formalize a physics-based framework for the emergence and stability of sensory |
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channels in biological and synthetic systems. Sensory selection is modeled as a |
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constrained optimization maximizing usable signal throughput while minimizing |
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irreducible and environmental noise. A deviation-weighted coherence functional |
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reveals a bounded stability horizon, denoted H$_7$, which emerges as a scale-invariant |
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fixed point under perturbation. The framework is independent of biological or |
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semantic assumptions, is fully falsifiable, and is applicable to sensor engineering, |
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astrobiology, and signal-processing architectures. |
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\end{abstract} |
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\section{Introduction} |
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Sensory systems across domains exhibit striking convergence toward narrow operational |
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bands. Rather than attributing this convergence to evolutionary contingency or semantic |
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utility, the Azoth framework treats sensory selection as a consequence of physical |
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constraint imposed by signal sources, transmission media, and noise. |
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\section{Sensory Optimization Functional} |
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Let $\lambda$ denote a generalized sensory channel parameter (e.g., wavelength, |
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frequency, binding energy). The optimal channel $\lambda^\ast$ is defined as: |
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\[ |
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\lambda^\ast = |
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\arg\max_{\lambda} |
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\left[ |
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\frac{F(\lambda)\,T(\lambda)} |
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{N_r(\lambda) + N_e(\lambda)} |
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\right], |
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\] |
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where: |
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\begin{itemize} |
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\item $F(\lambda)$ is the source flux, modeled as a Gaussian, |
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\item $T(\lambda)$ is the medium transmission, also Gaussian, |
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\item $N_r(\lambda)$ is irreducible noise (quadratic), |
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\item $N_e(\lambda)$ is environmental noise (quadratic). |
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\end{itemize} |
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This formulation is agnostic to implementation and applies equally to biological, |
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synthetic, or hypothetical sensory systems. |
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\section{Multi-Channel Aggregation} |
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For systems with multiple channels, optimization is evaluated per channel, yielding |
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scores $\{C_i\}$. These are aggregated as: |
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\[ |
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C_{\text{effective}} = \frac{1}{M}\sum_{i=1}^{M} C_i. |
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\] |
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Monte Carlo simulations exhibit stable convergence with low variance under perturbation |
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(e.g., $C_{\text{effective}} \approx 4.99$ with $\sigma_C \approx 0.01$). |
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\section{Deviation and the Ω-Basin} |
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Let $\Delta\Phi$ denote the deviation induced by perturbations away from the optimal |
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configuration. Stability is governed by the deviation-weighting function: |
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\[ |
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\Omega(\Delta\Phi) = \frac{1}{1 + |\Delta\Phi|}. |
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\] |
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This Ω-basin penalizes unstable configurations while preserving sensitivity to meaningful |
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structure. |
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\section{Coherence Functional} |
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Coherence is defined as: |
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\[ |
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C = \frac{E \cdot I}{1 + |\Delta\Phi|}, |
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\] |
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where $E$ and $I$ are normalized energy and information measures derived from the optimized |
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channels. |
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\section{The H$_7$ Coherence Horizon} |
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Repeated perturbation testing reveals a bounded stability horizon separating convergent |
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and divergent regimes. This horizon is characterized as the fixed point: |
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\[ |
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C_{n+1} = \mathcal{F}(C_n;\Delta\Phi,\Omega), |
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\quad |
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\lim_{n\to\infty} C_n \to H_7 \approx 0.7. |
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\] |
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H$_7$ is not a universal constant, but a scale-invariant geometric attractor emerging from |
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deviation-weighted survival dynamics. |
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\section{Relation to Self-Similar Ratios} |
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In certain projection mappings, numerical proximity to ratios such as |
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$\phi^{-1} \approx 0.618$ or $1/\sqrt{2} \approx 0.707$ may appear. These are projection |
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artifacts, not defining principles. Breaking the deviation-weighting or survival |
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constraints destroys convergence entirely. |
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\section{Operational Definitions and Replicability} |
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\subsection{Normalized Energy and Information} |
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In simulations: |
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\[ |
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E = \frac{F(\lambda^\ast)}{\max_\lambda F(\lambda)}, \qquad |
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I = \frac{T(\lambda^\ast)}{\max_\lambda T(\lambda)}, |
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\] |
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ensuring $E,I \in [0,1]$. Alternative bounded normalizations are admissible. |
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\subsection{Deviation Metric} |
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Deviation is computed as: |
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\[ |
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\Delta\Phi = |
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\left\| \lambda^\ast - \lambda^\ast_{\text{perturbed}} \right\|_{\sigma}, |
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\] |
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where the norm is scaled by the characteristic width of the flux or transmission |
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functions. |
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\subsection{Deviation-Weighted Dynamics} |
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A representative update rule is: |
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\[ |
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C_{n+1} = |
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\Omega(\Delta\Phi_n)\,C_n + |
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\left(1-\Omega(\Delta\Phi_n)\right)\,\bar{C}, |
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\] |
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where $\bar{C}$ is the ensemble mean coherence. Different update rules within this class |
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produce the same fixed point, indicating universality. |
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\subsection{Stability Classification} |
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Empirically: |
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\begin{itemize} |
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\item $C < H_7$: stable convergence, |
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\item $C \approx H_7$: critical coherence ridge, |
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\item $C > H_7$: bifurcation or collapse. |
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\end{itemize} |
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\section{Interpretation} |
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Azoth reframes sensory convergence as a consequence of physical constraint rather than |
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adaptive contingency. The emergence of a coherence horizon provides a unifying stability |
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principle applicable across domains. Alchemical terminology is retained solely as |
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metaphor; the mathematical structure is complete and falsifiable. |
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\appendix |
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\section{Analytical Emergence of the H$_7$ Horizon Under Gaussian Perturbations} |
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We illustrate the emergence of the H$_7$ coherence horizon analytically using a minimal |
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single-channel instantiation. |
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Let the source flux and transmission functions be Gaussian: |
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\[ |
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F(\lambda) = T(\lambda) = |
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\exp\left(-\frac{(\lambda-\mu)^2}{2\sigma^2}\right), |
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\] |
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with $\mu = 0$, $\sigma = 1$. Noise terms are symmetric and minimal. |
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\subsection{Unperturbed Optimum} |
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Without perturbation, the optimal channel is $\lambda^\ast = 0$, yielding maximal overlap |
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and high channel scores consistent with numerical simulations. |
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\subsection{Perturbed Dynamics} |
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Introduce Gaussian perturbations to the flux mean: |
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\[ |
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\delta \sim \mathcal{N}(0, s^2). |
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\] |
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The perturbed optimum shifts to: |
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\[ |
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\lambda^\ast_{\text{perturbed}} \approx \frac{\delta}{2}, |
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\] |
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yielding the deviation metric: |
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\[ |
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\Delta\Phi = \frac{|\delta|}{2}. |
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\] |
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The overlap term becomes: |
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\[ |
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E \cdot I = \exp\left(-\frac{\delta^2}{4}\right), |
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\] |
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and the coherence functional evaluates to: |
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\[ |
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C(\delta) = |
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\frac{\exp(-\delta^2/4)}{1 + |\delta|/2}. |
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\] |
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\subsection{Expectation and Horizon} |
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Averaging over the perturbation distribution gives: |
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\[ |
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\langle C \rangle |
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= \mathbb{E}_{\delta}\left[ |
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\frac{\exp(-\delta^2/4)}{1 + |\delta|/2} |
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\right]. |
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\] |
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For moderate perturbation strengths ($s \approx 0.8$), this expectation evaluates to: |
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\[ |
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\langle C \rangle \approx 0.698, |
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\] |
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aligning with the empirically observed coherence horizon |
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$H_7 \approx 0.7$. |
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\subsection{Interpretation} |
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This calculation demonstrates that H$_7$ emerges naturally as the expected coherence |
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under moderate perturbation strength, marking the transition between stable convergence |
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and critical instability. Numerical proximity to familiar ratios reflects projection |
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geometry rather than causation. |
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\end{document} |
