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2026
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| Online Access: | https://doi.org/10.5281/zenodo.19134172 |
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| _version_ | 1866901571070066688 |
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| author | Smith, David B |
| author_facet | Smith, David B |
| contents | <p>We prove the substrate H-theorem for a single Fourier mode of the substrate one-form bμ under stochastic Langevin dynamics derived from the substrate action of Smith (2026). The dissipation and noise are not postulated: they arise from the coupling chain substrate → electron (via the guidance hypothesis pμ = bμ) → electromagnetic vacuum. The spectral density of this chain is J(ω) ∝ αω⁵, super-Ohmic with n = 5. The substrate temperature is T = Λfₐ², fixed without new free parameters by the MICROSCOPE-constrained WEP parameter fₐ ∼ 3×10⁻¹⁵. The fluctuation-dissipation relation is derived from the Kubo-Martin-Schwinger condition in Lorentzian signature; no Wick rotation is used. The Fokker-Planck equation has a unique fixed point, the von Mises distribution f_eq ∝ exp(Λ/T cosθ) with sharpness Λ/T = 1/fₐ² ~ 10²⁸. The relative entropy H[f] decreases monotonically. Crucially, J(0) = 0 exactly for any super-Ohmic spectral density, making the von Mises fixed point exact for the full non-Markovian dynamics. Via the Madelung transformation and the identification Φ0 = ℏ, substrate equilibrium forces ρ = R²: the Born rule. Quantum probability is substrate thermodynamics. Both open problems are now closed. Gap 1 (mode coupling, Section 9) is resolved by mean field theory: the global shift symmetry of the substrate potential forces the self-consistent mean field to zero, reducing the interacting multi-mode problem to the original single-mode problem exactly. Gap 3 (multi-particle extension, Section 10) is resolved by consistent histories on the static eternal substrate: N-particle wavefunctions, decoherence, the Born rule, and entanglement all emerge from the substrate path integral without postulate, and the preferred foliation problem that has blocked relativistic pilot-wave theory for seventy years is dissolved by the static-eternal ontology. The substrate framework is complete: a single geometric object bμ generates quantum mechanics for any number of particles, the Born rule, decoherence, entanglement, general relativity, and Planck’s constant, with no quantum postulates assumed and no configuration space fundamental.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19134172 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Substrate H-Theorem Emergence of the Born Rule from Substrate Equilibrium Smith, David B <p>We prove the substrate H-theorem for a single Fourier mode of the substrate one-form bμ under stochastic Langevin dynamics derived from the substrate action of Smith (2026). The dissipation and noise are not postulated: they arise from the coupling chain substrate → electron (via the guidance hypothesis pμ = bμ) → electromagnetic vacuum. The spectral density of this chain is J(ω) ∝ αω⁵, super-Ohmic with n = 5. The substrate temperature is T = Λfₐ², fixed without new free parameters by the MICROSCOPE-constrained WEP parameter fₐ ∼ 3×10⁻¹⁵. The fluctuation-dissipation relation is derived from the Kubo-Martin-Schwinger condition in Lorentzian signature; no Wick rotation is used. The Fokker-Planck equation has a unique fixed point, the von Mises distribution f_eq ∝ exp(Λ/T cosθ) with sharpness Λ/T = 1/fₐ² ~ 10²⁸. The relative entropy H[f] decreases monotonically. Crucially, J(0) = 0 exactly for any super-Ohmic spectral density, making the von Mises fixed point exact for the full non-Markovian dynamics. Via the Madelung transformation and the identification Φ0 = ℏ, substrate equilibrium forces ρ = R²: the Born rule. Quantum probability is substrate thermodynamics. Both open problems are now closed. Gap 1 (mode coupling, Section 9) is resolved by mean field theory: the global shift symmetry of the substrate potential forces the self-consistent mean field to zero, reducing the interacting multi-mode problem to the original single-mode problem exactly. Gap 3 (multi-particle extension, Section 10) is resolved by consistent histories on the static eternal substrate: N-particle wavefunctions, decoherence, the Born rule, and entanglement all emerge from the substrate path integral without postulate, and the preferred foliation problem that has blocked relativistic pilot-wave theory for seventy years is dissolved by the static-eternal ontology. The substrate framework is complete: a single geometric object bμ generates quantum mechanics for any number of particles, the Born rule, decoherence, entanglement, general relativity, and Planck’s constant, with no quantum postulates assumed and no configuration space fundamental.</p> |
| title | The Substrate H-Theorem Emergence of the Born Rule from Substrate Equilibrium |
| url | https://doi.org/10.5281/zenodo.19134172 |