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Closed Difference-Loop Resonance as a Pre-Geometric Route to Gauge Structure, Quantum Phase, and Causal Emergence

Closed Difference-Loop Resonance as a Pre-Geometric Route to Gauge Structure, Quantum Phase, and Causal Emergence

Abstract

This paper introduces Closed Difference-Loop Resonance as an extension of Difference Skill-Tree Cosmogenesis. The framework treats particle-like sectors, gauge-like symmetries, quantum-like phase, causal order, and dark residual components as emergent structures of stable closed difference loops rather than as primitive assumptions. A particle-like excitation is modeled as a stable closed difference loop equipped with an allowed resonance mode. Gauge structure is formulated as the tensor automorphism group of stable vacuum loop families, with Standard-Model-like symmetry treated as the outcome of a vacuum stability competition rather than as a starting postulate. Quantum phase is introduced through stability-weighted complex histories, while the effective Planck scale is defined as the minimal nontrivial action period of stable closed-loop histories. Unitarity is imposed as a phase-norm preservation condition on closed quantum sectors, and measurement is interpreted as stability-recording decoherence by large observer networks. The paper does not claim a completed derivation of the Standard Model, Born statistics, or Lorentz invariance. Instead, it defines a formal derivation program and a minimal numerical route: rank-four tensor networks or four-dimensional hypergraph models should be tested for stable loop density, Lorentz-restoring residual flow, three-generation resonance windows, Standard-Model-like vacuum automorphisms, and phase-preserving quantum evolution. This work positions matter, quantum phase, gauge structure, and causal order as possible consequences of relational differences that close, resonate, fail, decohere, and stabilize.

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