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Three Long-Standing Quantum Mysteries Resolved via McCaul's Law of Coherence (Ω = Φ): The Measurement Problem, Wave-Particle Duality, and Hydrogen Spectral Renormalization

Justin McCaul

Published May 10, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.20102680

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Three Long-Standing Quantum Mysteries Resolved via McCaul's Law of Coherence (Ω = Φ): The Measurement Problem, Wave-Particle Duality, and Hydrogen Spectral Renormalization

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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.
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 (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.
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.
We demonstrate that eight major quantum mechanical anomalies share a unified resolution: they are projection artifacts, not ontological mysteries. When classical descriptions are recognized as partial projections from fuller coherence manifolds, apparent contradictions dissolve. No new physics required, no quantum mechanics modified, no hidden variables introduced. The framework applies three mutually reinforcing perspectives to each anomaly: the Translation Principle (projection geometry), Coherence-Gated Formation (threshold conditions), and the Coherence Propagation Constant (persistence requirements), all connected through the RMAxt gradient invariant. The eight resolutions: - Entanglement: Shared manifold structure. Spatial separation is projection artifact. Bell violations indicate non-factorizable relational structure, not nonlocal causation. Correlation survives projection; control does not — explaining why no faster-than-light signaling is possible. - Wave-particle duality: Non-commuting projections from single manifold. Pi_wave discards localization, preserves phase relations. Pi_particle discards phase, preserves localization. The experiment chooses which information to discard, not which reality to create. - Measurement problem: Coupling-induced projection stabilization. Measurement couples the system to an external channel enforcing a specific projection operator. Increases D:C ratio, pushes Upsilon out of coherence band, forces reprojection into minimum-cost basis. No collapse postulate needed. - Double-slit paradox: Path versus phase projection incompatibility. Which-path detection increases correction load beyond coherence capacity. Coherence loss is cumulative, not binary — weak measurements produce partial interference loss, quantum eraser restores it. - Quantum tunneling: Coherence-induced barrier narrowing via CGF. Coherence does not lower barriers — it narrows them. Tunneling probability proportional to rho_c * exp(-barrier width). Explains why tunneling in noisy environments is suppressed even when energy permits it. - Decoherence: Upsilon band exit forcing reprojection. Environmental coupling increases delay paths, injects phase noise, raises dissipation relative to correction. When Upsilon exceeds Upsilon_max, system must reproject into lower-cost basis. Irreversibility is thermodynamic — not forbidden by law, but overwhelmingly improbable to reverse. - Preferred basis: Minimum correction cost under environmental coupling. Position survives because it is the least expensive basis to maintain under noise. Not fundamental — privileged by stability. Phase becomes real in systems that actively protect coherence (superconductors, lasers, BEC). - Pointer states: Formation attractors that keep Upsilon viable without intervention. Not selected by environment as special rule — they survive because everything else is too expensive to maintain. Environmental coupling continuously re-selects them. Four testable predictions: coherence precursors correlating with measurement outcomes, decoherence rate hierarchy matching D:C ratio predictions, tunneling rate dependence on coherence density beyond standard QM, and pointer state universality across physical domains. Resolution chain is complete with no circularity: quantum-classical transition (CPC) -> preferred basis (CPC + CGF) -> pointer states (CGF) -> wave-particle duality (Translation) -> entanglement (Translation) -> measurement (all three). No leftover mystery, no new axioms. Part of the Coherence Institute publication sequence (Track B10). This paper applies the Translation Principle ontology to resolve the specific paradoxes that have driven quantum foundations debates for a century.