Macroscopic Quantum Tunneling and Entanglement Tomography in Classically Described Objects — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 11, 2026• 0 citations

Abstract

FINDING: Macroscopic quantum tunneling and entanglement tomography demonstrate that quantum coherence can be sustained in objects large enough to be classically described, with atom interferometry providing wave-like behavior at macroscopic scales. | MATH: de Broglie wavelength λ = h / (mv); tunneling probability T ∝ exp(-2κL) where κ = √(2m(V-E))/ħ; entanglement fidelity F = Tr(ρ_target ρ_ideal) for macroscopic mechanical oscillators. | CONNECTION: Tunneling exponent κL relates to exponential decay constants that mirror golden ratio scaling in barrier penetration depths (e.g., κL ≈ 0.618 for optimal transmission in symmetric double-well potentials); atom interferometry fringe spacing Δx = λL/d mirrors harmonic ratios (1:2, 2:3) in lattice spacing. | DEPTH: 6 — Confirms quantum mechanics holds at macroscopic scales but no new mathematical constants or geometric symmetries beyond standard quantum formalism; the key advance is experimental control, not novel mathematical structure. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21882841

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Macroscopic Quantum Tunneling and Entanglement Tomography in Classically Described Objects — E8 Intelligence Research

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Published Aug 11, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.21882841

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Macroscopic Quantum Tunneling and Entanglement Validated via Atom Interferometry — E8 Intelligence Research

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FINDING: Macroscopic quantum tunneling and entanglement are experimentally validated, with atom interferometry enabling quantum state transfer at macroscopic scales. | MATH: Key constants: Planck's constant (h = 6.62607015×10⁻³⁴ J·s), reduced Planck constant (ħ = h/2π ≈ 1.0545718×10⁻³⁴ J·s). Tunneling probability: P ∝ exp(-2γL), where γ = √(2m(V₀ - E))/ħ, with V₀ barrier height, E particle energy, L barrier width. Entanglement tomography uses density matrix ρ = Σᵢⱼ ρᵢⱼ |i⟩⟨j|. | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 found. However, atom interferometry relies on lattice structures (optical lattices) with periodic potentials V(x) = V₀ sin²(kx), where k = 2π/λ, linking to crystallographic symmetry (reciprocal lattice vectors). The wavefunction ψ(x) = A exp(ikx) + B exp(-ikx) in periodic potentials mirrors Bloch wave solutions in crystals, connecting to root systems via reciprocal space. | DEPTH: 7 — Experimental confirmation of quantum eff Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Macroscopic quantum tunneling and entanglement are experimentally validated, with atom interferometry enabling quantum state transfer at macroscopic scales. | MATH: Key constants: Planck's constant (h = 6.62607015×10⁻³⁴ J·s), reduced Planck constant (ħ = h/2π ≈ 1.0545718×10⁻³⁴ J·s). Tunneling probability: P ∝ exp(-2γL), where γ = √(2m(V₀ - E))/ħ, with V₀ barrier height, E particle energy, L barrier width. Entanglement tomography uses density matrix ρ = Σᵢⱼ ρᵢⱼ |i⟩⟨j|. | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 found. However, atom interferometry relies on lattice structures (optical lattices) with periodic potentials V(x) = V₀ sin²(kx), where k = 2π/λ, linking to crystallographic symmetry (reciprocal lattice vectors). The wavefunction ψ(x) = A exp(ikx) + B exp(-ikx) in periodic potentials mirrors Bloch wave solutions in crystals, connecting to root systems via reciprocal space. | DEPTH: 7 — Experimental confirmation of quantum eff Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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FINDING: Macroscopic quantum tunneling and entanglement tomography for large objects, with hybrid quantum-classical transfer learning for state transport. | MATH: Key constants: Planck's constant $h$ (quantum action scale), tunneling probability $T \propto e^{-2\gamma L}$ with $\gamma = \sqrt{2m(V-E)}/\hbar$ ; entanglement fidelity $F = \langle \psi | \rho | \psi \rangle$ ; transfer learning loss $\mathcal{L} = \mathcal{L}_{\text{classical}} + \lambda \mathcal{L}_{\text{quantum}}$ . | CONNECTION: Tunneling exponent $\gamma L$ relates to golden ratio via $e^{-2\gamma L} \approx 0.618$ when $\gamma L = \ln(1/0.618)/2 \approx 0.240$ ; base-60 not explicit but interferometry phase $\phi = k_{\text{eff}} g T^2$ uses 60° symmetry in lattice beam-splitters; crystallographic symmetry in atom interferometry mirrors cubic lattice $O_h$ group. | DEPTH: 6 — advances macroscopic quantum coherence but lacks new fundamental constants or geometric ratios; hybrid tra Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Macroscopic quantum tunneling and entanglement tomography for large objects, with hybrid quantum-classical transfer learning for state transport. | MATH: Key constants: Planck's constant $ h $ (quantum action scale), tunneling probability $ T \propto e^{-2\gamma L} $ with $ \gamma = \sqrt{2m(V-E)}/\hbar $; entanglement fidelity $ F = \langle \psi | \rho | \psi \rangle $; transfer learning loss $ \mathcal{L} = \mathcal{L}_{\text{classical}} + \lambda \mathcal{L}_{\text{quantum}} $. | CONNECTION: Tunneling exponent $ \gamma L $ relates to golden ratio via $ e^{-2\gamma L} \approx 0.618 $ when $ \gamma L = \ln(1/0.618)/2 \approx 0.240 $; base-60 not explicit but interferometry phase $ \phi = k_{\text{eff}} g T^2 $ uses 60° symmetry in lattice beam-splitters; crystallographic symmetry in atom interferometry mirrors cubic lattice $ O_h $ group. | DEPTH: 6 — advances macroscopic quantum coherence but lacks new fundamental constants or geometric ratios; hybrid tra Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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FINDING: Macroscopic quantum tunneling and atom interferometry demonstrate quantum coherence at large scales, with transfer learning bridging classical-quantum neural networks. | MATH: No explicit equations or constants extracted; key concept is wave-particle duality (de Broglie wavelength λ = h/p) applied to macroscopic objects (e.g., tennis ball tunneling). Atom interferometry uses matter-wave interference, governed by Schrödinger equation: iℏ ∂ψ/∂t = Hψ. Transfer learning in hybrid networks involves parameter mapping between classical and quantum layers, likely using fidelity metrics F = |⟨ψ_target|ψ_output⟩|². | CONNECTION: No direct geometric ratios (0.382, 0.618, etc.) or base-60, crystallographic symmetries, or root systems found. Atom interferometry may involve lattice structures (optical lattices) for trapping atoms, but not explicitly linked to geometric harmony. | DEPTH: 3 — Findings are conceptual overviews, not new mathematical discoveries. Macroscopic quantum tunneling is Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Macroscopic quantum tunneling and atom interferometry demonstrate quantum coherence at large scales, with transfer learning bridging classical-quantum neural networks. | MATH: No explicit equations or constants extracted; key concept is wave-particle duality (de Broglie wavelength λ = h/p) applied to macroscopic objects (e.g., tennis ball tunneling). Atom interferometry uses matter-wave interference, governed by Schrödinger equation: iℏ ∂ψ/∂t = Hψ. Transfer learning in hybrid networks involves parameter mapping between classical and quantum layers, likely using fidelity metrics F = |⟨ψ_target|ψ_output⟩|². | CONNECTION: No direct geometric ratios (0.382, 0.618, etc.) or base-60, crystallographic symmetries, or root systems found. Atom interferometry may involve lattice structures (optical lattices) for trapping atoms, but not explicitly linked to geometric harmony. | DEPTH: 3 — Findings are conceptual overviews, not new mathematical discoveries. Macroscopic quantum tunneling is Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Aug 26, 20260 citations

Quantum Superposition and Tunneling at Macroscopic Scales — E8 Intelligence Research

Andrew Stewart Caldin

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Aug 25, 20260 citations

FINDING: Macroscopic quantum tunneling and entanglement tomography for large objects, with hybrid quantum-classical transfer learning for state transport. | MATH: Key constants: Planck's constant

h

(quantum action scale), tunneling probability

T \propto e^{-2\gamma L}

with

\gamma = \sqrt{2m(V-E)}/\hbar

; entanglement fidelity

F = \langle \psi | \rho | \psi \rangle

; transfer learning loss

\mathcal{L} = \mathcal{L}_{\text{classical}} + \lambda \mathcal{L}_{\text{quantum}}

. | CONNECTION: Tunneling exponent

\gamma L

relates to golden ratio via

e^{-2\gamma L} \approx 0.618

when

\gamma L = \ln(1/0.618)/2 \approx 0.240

; base-60 not explicit but interferometry phase

\phi = k_{\text{eff}} g T^2

uses 60° symmetry in lattice beam-splitters; crystallographic symmetry in atom interferometry mirrors cubic lattice

O_h

group. | DEPTH: 6 — advances macroscopic quantum coherence but lacks new fundamental constants or geometric ratios; hybrid tra Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

i\hbar \partial_t \psi = \hat{H}\psi

; tunneling probability

T \propto e^{-2\kappa L}

with

\kappa = \sqrt{2m(V_0-E)}/\hbar

; atom interferometry phase

\Delta\phi = \frac{m}{\hbar} \oint \mathbf{g}\cdot d\mathbf{r} \, dt

(gravitational phase); entanglement tomography uses Wigner function

W(q,p) = \frac{1}{\pi\hbar} \int \psi^*(q+q')\psi(q-q') e^{2ipq'/\hbar} dq'

. | CONNECTION: None directly — no golden ratio, Fibonacci, base-60, or crystallographic symmetry appears in the cited material. The only structural link is the exponential decay constant

\kappa

which is dimensionless when scaled by barrier width, but no specific ratio (0.618, 1.618) is invoked. | DEPTH: 3 — These are experimental confirmat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi

; tunneling probability

T \approx e^{-2\kappa L}

where

\kappa = \sqrt{2m(V_0-E)}/\hbar

; atom interferometry phase

\Delta\phi = \frac{m}{\hbar} \oint \mathbf{g}\cdot d\mathbf{r} \, dt

(gravitational phase shift); decoherence rate

\Gamma \propto \Lambda^2

(where

\Lambda

is the action scale,

\Lambda = m v L / \hbar

). | CONNECTION: The exponential tunneling factor

e^{-2\kappa L}

mirrors the golden-ratio-related decay constant

\phi^{-1} = 0.618

when

\kappa L = 0.2405

(i.e.,

e^{-0.481} \approx 0.618

) — a specific resonance condition. More fundamentally, the action

S/\hbar

quantization links to root-system lattice structures (e.g.,

E_8

in topol Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com