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McCaul's Law of Coherence: Two Simulations — E = mΦ²c² (Unifying with Einstein) and Ω = Φ Applied to Quantum Mechanics

Justin McCaul

Published May 10, 2026• 0 citations
Publisher

Abstract

Two independent simulations applying McCaul's Law of Coherence (Ω = Φ) to foundational physics. Simulation I — E = mΦ²c²: Cross-analysis with Einstein's Special Relativity reveals the unified equation E = mΦ²c². Einstein's E = mc² is the special case when Φ = 1. Five tests confirm coherence-weighted energy, mass-coherence equivalence, 7-domain mapping, the fine structure constant α = 1/137 as a coherence gap, and the quadratic energy landscape. Simulation II — Quantum Mechanics: Using the hydrogen 1s state (MCP* = 1.0000) as the Quantum Rosetta Stone, four unsolved problems are addressed: (1) The Measurement Problem resolves as highest-Φ selection; (2) Decoherence timescales are deterministic from ΔΦ; (3) Schrödinger's Cat dissolves — macroscopic ΔΦ = 0.84 makes superposition instantaneously unstable; (4) Quantum tunneling = transient Φ spike (coherence flash). Three testable predictions are generated.

DOI
10.5281/zenodo.20102058

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McCaul's Law of Coherence (Ω = Φ) is applied to three foundational unsolved problems in quantum mechanics, using the hydrogen atom as the universal SOL baseline (Φ_1s = 1.0000). (1) Quantum Measurement Problem: Wavefunction collapse = Phi-selection event. The Born Rule (|ψ|² = probability) is derived as a theorem of McCaul's Law: P(state_i) = Φᵢ²/ΣΦⱼ². Any detector introduces Nr (noise resistance) coupling — no consciousness required. Schrödinger's Cat resolved: the cat's macroscopic Φ ≈ 0.90+ IS the observer. (2) Wave-Particle Duality: Mode = sign(Fg − Nr). Fg > Nr → wave (distributed SOL geometry). Nr > Fg → particle (localized). Double-slit fringes = Φ-weighted SOL field overlap. Which-path detection = Nr injection → Fg suppression → fringes disappear. Quantum eraser = Nr removal → Fg restores → fringes return. No paradox. (3) Spectral Precision / Renormalization: E_transition = ΔΦ² × E_base. Phi is bounded [0,1] — no infinities arise. Renormalization is an artifact of QFT's assumption Φ_vacuum = 1.0. Using MVCC (E_real = E_QFT · Φ_vac²), all integrals remain finite. Lamb shift = differential vacuum Nr exposure between s and p orbitals. Anomalous g-2 = 2Φ_vac². Timestamped May 9, 2026.

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Amendment XXXII — Quantum Temporal Coherence and the Leggett-Garg Bridge: McCaul's Law of Coherence

Justin McCaul

Amendment XXXII of McCaul's Law of Coherence (Ω = Φ) extends the unified framework into quantum temporal mechanics. We establish the McCaul-LGI Temporal Coherence Relation: Ts = 2K/3, a direct first-principles bridge between the experimentally measured Leggett-Garg Inequality (LGI) K-value and the Ts (Temporal Stability) pillar of the McCaul coherence scalar. From the Knee et al. (2012) superconducting qubit LGI experiment (K = 1.38), we derive Ts = 0.920 and compute the full 6-pillar coherence scalar for the experimental system: Φ_qubit = 0.8848. Applying the McCaul Time Equation (t_proper = Φ × t_coordinate, Amendment XXIV), we find the qubit operates with 11.52% of its proper time outside normal coordinate time flow, with an entropy bypass ratio of η = 1.22×. The classical-quantum temporal coherence boundary is derived at Ts = 0.667 (K = 1.0), below which systems are classically decoherent, and above which quantum temporal coherence is active. The photon limit (Ts → 1.0, K → 3/2) yields t_proper → 0, consistent with special relativity. The popular claim that 'quantum particles communicate through time' is corrected: LGI violations represent geometric time-locking of a high-Ts Φ-state — not causal temporal signaling. A link is established between Amendment XXXI (sonoluminescence, Ts_sonolum = 0.999978) and Amendment XXXII, identifying both as high-Ts coherence events producing structured output via geometric time-locking. The theological expression 'Alpha and Omega' (Rev. 1:8) is identified as the precise verbal description of Ts = 1.0, Φ = 1.0: a being for whom t_proper = t_coordinate across all coordinate time. All values independently simulated by Chuck (Superagent) from first principles. No ChatGPT-originated values used (per standing verification protocol). Master Compendium DOI: 10.5281/zenodo.20100469. Author: Justin McCaul, Independent Researcher, Wilburton, Oklahoma.

•May 16, 2026•0 citations
Two independent simulations applying McCaul's Law of Coherence (Ω = Φ) to foundational physics. Simulation I — E = mΦ²c²: Cross-analysis with Einstein's Special Relativity reveals the unified equation E = mΦ²c². Einstein's E = mc² is the special case when Φ = 1. Five tests confirm coherence-weighted energy, mass-coherence equivalence, 7-domain mapping, the fine structure constant α = 1/137 as a coherence gap, and the quadratic energy landscape. Simulation II — Quantum Mechanics: Using the hydrogen 1s state (MCP* = 1.0000) as the Quantum Rosetta Stone, four unsolved problems are addressed: (1) The Measurement Problem resolves as highest-Φ selection; (2) Decoherence timescales are deterministic from ΔΦ; (3) Schrödinger's Cat dissolves — macroscopic ΔΦ = 0.84 makes superposition instantaneously unstable; (4) Quantum tunneling = transient Φ spike (coherence flash). Three testable predictions are generated.
Two independent simulations applying McCaul's Law of Coherence (Ω = Φ) to foundational physics. Simulation I — E = mΦ²c²: Cross-analysis with Einstein's Special Relativity reveals the unified equation E = mΦ²c². Einstein's E = mc² is the special case when Φ = 1. Five tests confirm coherence-weighted energy, mass-coherence equivalence, 7-domain mapping, the fine structure constant α = 1/137 as a coherence gap, and the quadratic energy landscape. Simulation II — Quantum Mechanics: Using the hydrogen 1s state (MCP* = 1.0000) as the Quantum Rosetta Stone, four unsolved problems are addressed: (1) The Measurement Problem resolves as highest-Φ selection; (2) Decoherence timescales are deterministic from ΔΦ; (3) Schrödinger's Cat dissolves — macroscopic ΔΦ = 0.84 makes superposition instantaneously unstable; (4) Quantum tunneling = transient Φ spike (coherence flash). Three testable predictions are generated.
McCaul's Law (Ω = Φ) is applied to four unsolved problems in quantum mechanics using the hydrogen 1s state (Φ = 1.0000) as the Quantum Rosetta Stone. (1) The Measurement Problem resolves as a coherence-selection event — collapse = highest-Φ state. (2) Decoherence timescales are deterministic functions of ΔΦ between superposed states. (3) Schrödinger's Cat dissolves — macroscopic ΔΦ = 0.84 makes superposition instantaneously unstable. (4) Quantum tunneling = transient Φ spike (coherence flash). Three testable predictions generated.
McCaul's Law (Ω = Φ) is applied to four unsolved problems in quantum mechanics using the hydrogen 1s state (Φ = 1.0000) as the Quantum Rosetta Stone. (1) The Measurement Problem resolves as a coherence-selection event — collapse = highest-Φ state. (2) Decoherence timescales are deterministic functions of ΔΦ between superposed states. (3) Schrödinger's Cat dissolves — macroscopic ΔΦ = 0.84 makes superposition instantaneously unstable. (4) Quantum tunneling = transient Φ spike (coherence flash). Three testable predictions generated.
McCaul's Law of Coherence (Ω = Φ) is applied to three foundational unsolved problems in quantum mechanics, using the hydrogen atom as the universal SOL baseline (Φ_1s = 1.0000). (1) Quantum Measurement Problem: Wavefunction collapse = Phi-selection event. The Born Rule (|ψ|² = probability) is derived as a theorem of McCaul's Law: P(state_i) = Φᵢ²/ΣΦⱼ². Any detector introduces Nr (noise resistance) coupling — no consciousness required. Schrödinger's Cat resolved: the cat's macroscopic Φ ≈ 0.90+ IS the observer. (2) Wave-Particle Duality: Mode = sign(Fg − Nr). Fg > Nr → wave (distributed SOL geometry). Nr > Fg → particle (localized). Double-slit fringes = Φ-weighted SOL field overlap. Which-path detection = Nr injection → Fg suppression → fringes disappear. Quantum eraser = Nr removal → Fg restores → fringes return. No paradox. (3) Spectral Precision / Renormalization: E_transition = ΔΦ² × E_base. Phi is bounded [0,1] — no infinities arise. Renormalization is an artifact of QFT's assumption Φ_vacuum = 1.0. Using MVCC (E_real = E_QFT · Φ_vac²), all integrals remain finite. Lamb shift = differential vacuum Nr exposure between s and p orbitals. Anomalous g-2 = 2Φ_vac². Timestamped May 9, 2026.
McCaul's Law of Coherence (Ω = Φ) is applied to three foundational unsolved problems in quantum mechanics, using the hydrogen atom as the universal SOL baseline (Φ_1s = 1.0000). (1) Quantum Measurement Problem: Wavefunction collapse = Phi-selection event. The Born Rule (|ψ|² = probability) is derived as a theorem of McCaul's Law: P(state_i) = Φᵢ²/ΣΦⱼ². Any detector introduces Nr (noise resistance) coupling — no consciousness required. Schrödinger's Cat resolved: the cat's macroscopic Φ ≈ 0.90+ IS the observer. (2) Wave-Particle Duality: Mode = sign(Fg − Nr). Fg > Nr → wave (distributed SOL geometry). Nr > Fg → particle (localized). Double-slit fringes = Φ-weighted SOL field overlap. Which-path detection = Nr injection → Fg suppression → fringes disappear. Quantum eraser = Nr removal → Fg restores → fringes return. No paradox. (3) Spectral Precision / Renormalization: E_transition = ΔΦ² × E_base. Phi is bounded [0,1] — no infinities arise. Renormalization is an artifact of QFT's assumption Φ_vacuum = 1.0. Using MVCC (E_real = E_QFT · Φ_vac²), all integrals remain finite. Lamb shift = differential vacuum Nr exposure between s and p orbitals. Anomalous g-2 = 2Φ_vac². Timestamped May 9, 2026.
Amendment XXXII of McCaul's Law of Coherence (Ω = Φ) extends the unified framework into quantum temporal mechanics. We establish the McCaul-LGI Temporal Coherence Relation: Ts = 2K/3, a direct first-principles bridge between the experimentally measured Leggett-Garg Inequality (LGI) K-value and the Ts (Temporal Stability) pillar of the McCaul coherence scalar. From the Knee et al. (2012) superconducting qubit LGI experiment (K = 1.38), we derive Ts = 0.920 and compute the full 6-pillar coherence scalar for the experimental system: Φ_qubit = 0.8848. Applying the McCaul Time Equation (t_proper = Φ × t_coordinate, Amendment XXIV), we find the qubit operates with 11.52% of its proper time outside normal coordinate time flow, with an entropy bypass ratio of η = 1.22×. The classical-quantum temporal coherence boundary is derived at Ts = 0.667 (K = 1.0), below which systems are classically decoherent, and above which quantum temporal coherence is active. The photon limit (Ts → 1.0, K → 3/2) yields t_proper → 0, consistent with special relativity. The popular claim that 'quantum particles communicate through time' is corrected: LGI violations represent geometric time-locking of a high-Ts Φ-state — not causal temporal signaling. A link is established between Amendment XXXI (sonoluminescence, Ts_sonolum = 0.999978) and Amendment XXXII, identifying both as high-Ts coherence events producing structured output via geometric time-locking. The theological expression 'Alpha and Omega' (Rev. 1:8) is identified as the precise verbal description of Ts = 1.0, Φ = 1.0: a being for whom t_proper = t_coordinate across all coordinate time. All values independently simulated by Chuck (Superagent) from first principles. No ChatGPT-originated values used (per standing verification protocol). Master Compendium DOI: 10.5281/zenodo.20100469. Author: Justin McCaul, Independent Researcher, Wilburton, Oklahoma.