Earlier substrate-theory papers [1-3] used the intrinsic decoherence formula γ = (m/MP)2 × νP, motivated by F1/F2 sampling at a per-mass rate. This formula gives reasonable BMV-scale predictions but over-predicts decoherence for composite quantum systems by orders of magnitude. The most severe tension is with C60 molecular interferometry [4]: the per-mass formula predicts decoherence at 108 Hz for a C60 molecule (m ≈ 720 amu), whereas observed coherence times in Talbot-Lau interferometers extend to milliseconds. The 5-order-of-magnitude tension is real and unresolvable within the per-mass framework. This paper formally introduces the energy-eigenstate decoherence formula: γeff = (ΔE / MP c2)2 × νP (1) where ΔE is the energy difference between superposed states. This formula: (i) Reduces to the per-mass formula in the special case of particle creation/annihilation (ΔE = mc2). (ii) Gives ΔE = 0 → γeff = 0 for translation-symmetric position superpositions (same kinetic energy in both branches). Resolves the C60 tension. (iii) Predicts γeff = (kBT / MPc2)2 × νP for thermal molecular beams, giving ~100 Hz at T = 300 K. Testable in current molecular interferometry. (iv) Predicts γeff ≈ 1.4 × 10−20 Hz for B0-B̄0 oscillations — essentially perfect coherence, consistent with all observations but not a discovery target. (v) Gives γeff ≈ 1037 Hz at inflationary energy scales — cosmologically relevant. The formula is derived from substrate's F1/F2 ontology [5]: F2 events sample between substrate-mode energy eigenstates with rate proportional to (ΔE/MPc2)2, representing the rate at which the F1 wavefunction's off-diagonal coherence between energy eigenstates is sampled. Position-basis decoherence is not intrinsic to substrate framework; it emerges from environmental decoherence as in standard QM. This paper audits the existing substrate corpus for predictions affected by the formula change: • B-meson anomalies paper [v3, ref. 6]: B-B̄ decoherence prediction (5.4 × 10−4 amplitude suppression) replaced by essentially zero. The decoherence prediction is withdrawn; substrate consistent with observations. • Heisenberg uncertainty paper [ref. 1]: BMV-scale predictions revised. Old: γ ~ 10−8 Hz at m = 10−14 kg from per-mass formula. New: γ depends on gravitational interaction energy ΔE = Gm2/r, giving γ ~ 10−24 Hz at BMV scales. Far below BMV sensitivity; prediction effectively withdrawn at lab scales. • F1/F2 paper [ref. 5]: Framework presentation updated to specify energy-eigenstate sampling. • Inflation paper [ref. 7], MOND paper [ref. 8], dark energy paper [ref. 9], cluster mass paper [ref. 10]: no decoherence content affected. The energy-eigenstate formula opens new testable predictions in experimental programs that the per-mass formula did not access cleanly: thermal molecular interferometry (γ ∝ T2), trapped-ion motional-eigenstate superpositions (γ ∝ ΔE2), and atomic clock internal-state superpositions. The substrate decoherence sector thus becomes richer and more falsifiable under the new formula, even as some previous predictions (B-B̄, BMV) are no longer accessible signatures. Section VII presents a specific falsification test: thermal molecular interferometry with varying beam temperature, measuring whether the coherence time scales as τ ∝ 1/T2 (substrate prediction) or τ ∝ 1/m2 (CSL/GRW prediction) or shows no intrinsic decoherence (standard QM with only environmental decoherence). The same apparatus can test all three competing intrinsic-decoherence proposals. Keywords: substrate theory; intrinsic decoherence; energy-eigenstate sampling; F1/F2 ontology; molecular interferometry; CSL; GRW; BMV protocol; corpus audit.
Earlier substrate-theory papers [1-3] used the intrinsic decoherence formula γ = (m/MP)2 × νP, motivated by F1/F2 sampling at a per-mass rate. This formula gives reasonable BMV-scale predictions but over-predicts decoherence for composite quantum systems by orders of magnitude. The most severe tension is with C60 molecular interferometry [4]: the per-mass formula predicts decoherence at 108 Hz for a C60 molecule (m ≈ 720 amu), whereas observed coherence times in Talbot-Lau interferometers extend to milliseconds. The 5-order-of-magnitude tension is real and unresolvable within the per-mass framework. This paper formally introduces the energy-eigenstate decoherence formula: γeff = (ΔE / MP c2)2 × νP (1) where ΔE is the energy difference between superposed states. This formula: (i) Reduces to the per-mass formula in the special case of particle creation/annihilation (ΔE = mc2). (ii) Gives ΔE = 0 → γeff = 0 for translation-symmetric position superpositions (same kinetic energy in both branches). Resolves the C60 tension. (iii) Predicts γeff = (kBT / MPc2)2 × νP for thermal molecular beams, giving ~100 Hz at T = 300 K. Testable in current molecular interferometry. (iv) Predicts γeff ≈ 1.4 × 10−20 Hz for B0-B̄0 oscillations — essentially perfect coherence, consistent with all observations but not a discovery target. (v) Gives γeff ≈ 1037 Hz at inflationary energy scales — cosmologically relevant. The formula is derived from substrate's F1/F2 ontology [5]: F2 events sample between substrate-mode energy eigenstates with rate proportional to (ΔE/MPc2)2, representing the rate at which the F1 wavefunction's off-diagonal coherence between energy eigenstates is sampled. Position-basis decoherence is not intrinsic to substrate framework; it emerges from environmental decoherence as in standard QM. This paper audits the existing substrate corpus for predictions affected by the formula change: • B-meson anomalies paper [v3, ref. 6]: B-B̄ decoherence prediction (5.4 × 10−4 amplitude suppression) replaced by essentially zero. The decoherence prediction is withdrawn; substrate consistent with observations. • Heisenberg uncertainty paper [ref. 1]: BMV-scale predictions revised. Old: γ ~ 10−8 Hz at m = 10−14 kg from per-mass formula. New: γ depends on gravitational interaction energy ΔE = Gm2/r, giving γ ~ 10−24 Hz at BMV scales. Far below BMV sensitivity; prediction effectively withdrawn at lab scales. • F1/F2 paper [ref. 5]: Framework presentation updated to specify energy-eigenstate sampling. • Inflation paper [ref. 7], MOND paper [ref. 8], dark energy paper [ref. 9], cluster mass paper [ref. 10]: no decoherence content affected. The energy-eigenstate formula opens new testable predictions in experimental programs that the per-mass formula did not access cleanly: thermal molecular interferometry (γ ∝ T2), trapped-ion motional-eigenstate superpositions (γ ∝ ΔE2), and atomic clock internal-state superpositions. The substrate decoherence sector thus becomes richer and more falsifiable under the new formula, even as some previous predictions (B-B̄, BMV) are no longer accessible signatures. Section VII presents a specific falsification test: thermal molecular interferometry with varying beam temperature, measuring whether the coherence time scales as τ ∝ 1/T2 (substrate prediction) or τ ∝ 1/m2 (CSL/GRW prediction) or shows no intrinsic decoherence (standard QM with only environmental decoherence). The same apparatus can test all three competing intrinsic-decoherence proposals. Keywords: substrate theory; intrinsic decoherence; energy-eigenstate sampling; F1/F2 ontology; molecular interferometry; CSL; GRW; BMV protocol; corpus audit.
Substrate framework's energy-eigenstate decoherence formula γeff = (ΔE/MPc2)2 × νP [1] depends on the energy gap between superposed states, not on cosmological distance directly. But typical ΔE values for thermal modes, particle physics processes, and inflaton field fluctuations are functions of cosmic energy density, which evolves dramatically over cosmic history. This paper develops the cosmological evolution of substrate intrinsic decoherence across the principal epochs: • Planck era (t ~ 10−44 s): Substrate framework breaks down; γ formula not applicable. • Inflation era (ΔE ~ 1016 GeV): γ ~ 1037 Hz. Catastrophically fast decoherence drives primordial perturbation classicalization — providing the microscopic mechanism for the well-known but mechanistically unexplained quantum-to-classical transition of cosmological perturbations. • EW epoch (ΔE ~ 100 GeV): γ ~ 109 Hz. Coherence time τ ~ ns; particle physics processes decohere rapidly. • QCD epoch (ΔE ~ 100 MeV): γ ~ 103 Hz. τ ~ ms. • Nucleosynthesis (ΔE ~ MeV): γ ~ 0.1 Hz. τ ~ 10 s. • Recombination (ΔE ~ 0.3 eV): γ ~ 10−15 Hz. τ ~ 30 Myr, longer than the age of the universe at that time — the threshold at which the universe becomes capable of sustaining cosmologically long quantum coherence. • Present era (ΔE ~ kBTCMB ~ 0.2 meV): γ ~ 10−21 Hz. τ ~ 30 Gyr, longer than the current age of the universe. Coherent quantum systems can persist for cosmological timescales. • de Sitter future: Floor of γ set by dark energy; in the limit, decoherence approaches a constant minimum tied to the cosmological constant. Three principal results: (i) The decoherence rate has dropped by ~58 orders of magnitude over cosmic history, from γ ~ 1037 Hz at inflation to γ ~ 10−21 Hz today. The universe has become exponentially more capable of sustaining coherent quantum behavior as it has cooled. (ii) Coherent quantum behavior is a late-universe phenomenon. Modern quantum-mechanical experiments (atom interferometry, molecular interferometry, atomic clocks, B-meson oscillations) require a low-energy-density universe to operate. They could not have been performed in the radiation-dominated era because thermal energies were too high. (iii) Substrate framework provides a microscopic mechanism for primordial perturbation classicalization. Inflaton field quantum fluctuations decohered at γ ~ 1037 Hz during inflation, becoming classical perturbations that seeded the large-scale structure observed today. This resolves a long-standing puzzle in cosmology, where the quantum-to-classical transition is traditionally treated as a black box. Additionally, the cosmological floor on momentum uncertainty Δpmin ≥ ℏH(t)/c evolved from Δpmin ~ MPc at Planck era to Δpmin ~ 10−61 kg·m/s today — a quantum bound that has weakened by ~60 orders of magnitude. In the early universe, quantum uncertainty operated at vastly different scales than today. These results imply that the substrate framework's intrinsic decoherence sector is fundamentally a cosmological history sector: the framework's quantum-mechanical behavior changes character with cosmic epoch. Quantum coherence is not a universal property but rather a feature of low-energy-density universes. The framework's testable predictions in the modern era (thermal molecular interferometry, atomic clocks) are made possible by the universe having sufficiently cooled. Section II clarifies what does and doesn't depend on cosmic time. Section III develops the cosmological floor evolution. Section IV develops the decoherence rate across cosmic history with a master table. Section V addresses the early universe's rapid classicalization. Section VI addresses the inflation era specifically and primordial perturbations. Section VII addresses the present era. Section VIII addresses the de Sitter future. Section IX synthesizes the cross-epoch comparison of quantum properties. Section X concludes. Keywords: substrate theory; cosmological decoherence; primordial perturbations; inflation; quantum-to-classical transition; cosmological floor; energy-eigenstate formula; cosmic evolution of quantum capacity.
This paper develops the substrate-theoretic mechanism of intrinsic decoherence in three connected directions: (i) the description of the double slit experiment for electrons vs photons under the energy-eigenstate formula [1, 2]; (ii) the microscopic mechanism by which substrate time-quantum F2 events interact with matter (radial-mode displacements per postulate P6); and (iii) the role of the 5D cascade structure (which makes matter stable [3]) in shaping the decoherence process. Three principal results: (i) Electrons and photons both show full interference in the standard symmetric double slit under the energy-eigenstate formula. ΔE = 0 between branches → γeff = 0 for both. But substrate framework distinguishes them at the field level: electrons are localized radial-mode displacements of the substrate (per P6, mass IS substrate displacement); photons are excitations of the substrate gauge sector with no radial displacement. Both yield identical observable interference patterns despite different substrate-level identities. (ii) The decoherence mechanism is F2 sampling between substrate-mode energy eigenstates, with rate γ = (ΔE/MPc2)2 × νP. F2 events occur uniformly throughout substrate at the Planck clock rate νP; they detect substrate field configurations including matter (radial-mode displacements). The ΔE2 scaling emerges because pairs of F2 events are needed to consistently distinguish two energy eigenstates — analogous to how Born-rule probabilities scale as amplitude squared. Mass-as-displacement gives matter substrate presence that F2 events sample, but mass alone does not drive decoherence — the energy gap between superposed states does. (iii) The 5D cascade structure prevents matter decay but does not directly suppress decoherence. Matter at the IR brane (cascade site 15) has wavefunction overlap with the bulk radial mode suppressed by q15 ≈ 10−15, giving decay-rate suppression q30 ≈ 10−30 — necessary for matter stability. But F2 sampling is a global substrate process occurring at νP throughout the substrate, not localized to specific 5D positions. The 5D structure determines what energy spectrum matter has access to (which sets the available ΔE values), but does not suppress the decoherence rate for a given ΔE. Decoherence and decay are therefore distinct processes: decay is energy transfer from matter (IR brane) to bulk radial mode (throughout 5D), requiring wavefunction overlap, suppressed by 5D localization; decoherence is information transfer between superposed energy eigenstates via local F2 sampling, not requiring bulk overlap, not suppressed by 5D localization. Implications for experimental tests: • Standard double slit (electrons or photons): substrate predicts full interference, ΔE = 0 in symmetric setups. Consistent with all observations. • Asymmetric interferometers (potential difference, frequency shift): substrate predicts γ ∝ (ΔE)2, but for lab-scale ΔE ~ eV, γ ~ 10−13 Hz, undetectable. • Thermal molecular interferometry: substrate predicts γ ∝ T2, mass-independent — uniquely substrate among intrinsic-decoherence frameworks [2]. Principal near-term test. • Cosmological-scale superpositions (inflaton fields, primordial perturbations): substrate predicts γ ~ 1037 Hz at inflation, providing microscopic mechanism for quantum-to-classical transition. Keywords: substrate theory; double slit; electrons; photons; F2 sampling; mass-as-displacement; 5D cascade; matter stability; decoherence mechanism; energy-eigenstate formula.
A complete walkthrough, accessible to undergraduates and the general reader, of how the substrate framework explains the double-slit experiment for an electron and photon. We follow the electron and photon from emission to detection, tracking what happens in Frame 1 (the continuous quantum reality) and Frame 2 (the discrete observed reality), and we calculate every step. By the end of this chapter, you should understand: why interference happens, why "which-slit measurement" destroys it, and how the substrate framework explains both without postulating mysterious wavefunction collapse. For all existing matter-wave interferometry experiments (atoms, molecules, electrons), the framework predicts substrate-induced decoherence rates FAR below environmental decoherence. The framework therefore predicts that no current matter-wave experiment can distinguish substrate decoherence from environmental decoherence. The BMV-scale regime (m ~ 10⁻¹⁴ kg, γ ~ 10² Hz) is where the framework first becomes directly testable in laboratory experiments. The (m/M_P)² scaling means coherence time τ_coh = 1/γ scales as 1/m². Doubling the mass quarters the coherence time. This is a sharp prediction that distinguishes the framework from collapse models.
We propose a trajectory density formulation of the double-slit interference pattern that reproduces observed fringe intensities without wave superposition. The intensity at screen position x is the weighted density of particle trajectories arriving there, suppressed exponentially by the deflection cost in energy space: I(x) = V_{Trit}(KE) \times \sum_n w_n \cdot \delta\!\left(x - \frac{n\lambda L}{d}\right) \cdot e^{-k|n|} where k = dθ_eff/λ_dB is the coherence parameter set by the effective illumination semiangle θ_eff, and V_Trit(KE) = 2γβ/(1+γ²β²) is a new energy-dependent visibility factor from the Trit framework [1–3]. The parameter k connects to the effective source temperature through the emission physics: higher source temperature → larger emission area → larger θ_eff → larger k → fewer visible fringes. We test the formula against five independent experiments across five orders of magnitude in particle energy and three particle types. For thermionic and Schottky electron sources the formula outperforms standard QM by a factor of 2–6× in R². For cold FEG sources and C60 molecules the formula correctly reduces to QM (k → 0). The C60 result leads to a clean new distinction: V_Trit(KE) applies to elementary particles, where the energy space geometry directly governs wave-particle balance; for composite particles V_Trit = 1 and interference is from the centre-of-mass de Broglie wave. This difference is falsifiable: varying electron energy should reveal a visibility peak at KE = 211.66 keV; varying C60 velocity should show no such peak.
Note on Theoretical Context and Terminology: The mathematical formulations and notations presented in this version reflect an earlier stage of development. In subsequent works and theoretical revisions, all concepts previously referred to under localized personal naming conventions (e.g., specific individual equations or theories) have been formally standardized and integrated under the unified framework of the Stochastic Decoherence Model. Readers are advised to interpret the underlying velocity-dependent decoherence mechanisms, stochastic differential equations, and cosmological implications within this updated conceptual framework. This preprint presents a hypothesis of material-dependent quantum decoherence within an emergent information-based framework. The work derives quantitative predictions for coherence-time ratios between crystalline and amorphous forms of several materials, including silicon, carbon and silica. The central prediction is that structural disorder enhances intrinsic decoherence through material-dependent coupling parameters linked to experimentally measurable dielectric response functions. Unlike standard quantum mechanics and conventional mass-based collapse models, the framework predicts measurable differences between materials of identical chemical composition but different structural order. The proposed effects are experimentally testable using optomechanical resonators, nanomechanical oscillators and matter-wave interferometry. The manuscript explicitly discusses limitations, open theoretical questions and falsification criteria, and is intended as a contribution to ongoing research on quantum foundations, decoherence and collapse models. The author welcomes constructive feedback, criticism and potential collaboration from researchers interested in quantum foundations, decoherence theory, condensed matter physics and precision quantum experiments.
Part III completes the three-layer architecture of State Locality Theory 2 (SLT-2) by providing a phenomenological framework that translates the structural ontology and effective dynamics of SLT-2 into concrete, experimentally testable predictions. No new free parameters are introduced beyond the localization constant α inherited from SLT-1. The document formulates an operational protocol for applying SLT-2 to real systems, constructs a regime map in mass–path-separation parameter space, derives quantitative visibility predictions for molecular interferometry, nanoparticle interferometry, and superconducting systems, and specifies falsifiability criteria. SLT-2 predicts gravitationally enhanced decoherence in the 10⁵–10⁶ amu range (ΔV ∼ 0.5–5%), suppressed sensitivity in superconducting phases, a pressure-independent visibility plateau in ultrahigh vacuum, and full coherence of neutrinos over astronomical baselines. The structural mass hierarchy is discussed honestly as a scale of node complexity, not a mass formula.