Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Tunneling and Coherence: No New Mathematical Constants Emerge — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 23, 2026• 0 citations
Publisher

Abstract

FINDING: Quantum tunneling and macroscopic quantum coherence are experimentally accessible via atom interferometry and entangled mechanical oscillators, but no new mathematical constant or ratio emerges from these sources. | MATH: Standard Schrödinger equation \( i\hbar \partial_t \psi = \hat{H}\psi \); 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

DOI
10.5281/zenodo.22065153

Related papers

Quantum Tunneling and Entanglement: Standard Formalism, No New Physics — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum tunneling and macroscopic entanglement are experimentally validated, but no new mathematical formalism or universal constant emerges from these sources; the core math remains standard quantum mechanics (Schrödinger equation, tunneling probability). | MATH: Tunneling probability \( T \approx e^{-2\kappa L} \), where \( \kappa = \sqrt{2m(V_0-E)}/\hbar \); macroscopic entanglement tomography uses Wigner functions and Gaussian state covariance matrices; transfer learning in hybrid networks uses parameter-shift rules and gradient descent on \( \mathrm{Tr}(\rho_{\text{out}} O) \). | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 appear. However, the exponential decay \( e^{-2\kappa L} \) is structurally analogous to the golden ratio's logarithmic spiral (both are exponential forms), and the Wigner function's symplectic structure relates to the Heisenberg group, which has a lattice (crystallographic) representation in phase space — but Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Sep 15, 2026•0 citations

Quantum Tunneling and Coherence Validated via Atom Interferometry — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum tunneling and macroscopic quantum coherence are experimentally validated, with atom interferometry providing the key mathematical framework for wave-function evolution at scale. | MATH: Schrödinger equation \( i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi \); 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

•Aug 26, 2026•0 citations

Quantum Tunneling and Superposition: Experimental Frontiers, No New Math — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum tunneling and macroscopic quantum superposition are being actively pursued via levitated nanoparticles and atom interferometry, but no new mathematical constant or equation emerges from these search results — they are experimental/educational, not theoretical breakthroughs. | MATH: No new equations or constants extracted. Relevant known formalism: tunneling probability \( T \approx e^{-2\kappa L} \), \(\kappa = \sqrt{2m(V_0-E)}/\hbar\); macroscopic superposition requires maintaining coherence time \(\tau \propto \hbar^2/(m k_B T x^2)\) (decoherence rate scaling). Atom interferometry phase: \(\Delta\phi = (m/\hbar) \oint \mathbf{g}\cdot d\mathbf{r}\, dt\). | CONNECTION: None directly stated. However, note that levitated nanoparticle superposition experiments often use optical lattices — which are periodic structures with crystallographic symmetry (e.g., 1D lattice spacing \(d = \lambda/2\) — ratio 0.5, not golden). No 0.382/0.618/0.786/1.618/2.618 or base-60 appears in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Oct 19, 2026•0 citations

Quantum Tunneling and Coherence: No New Constants Emerge — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum tunneling and macroscopic quantum coherence are being actively explored via atom interferometry and entangled mechanical oscillators, but no new mathematical constant or closed-form equation emerges from these search results. | MATH: Standard framework — Schrödinger equation \( i\hbar \partial_t \Psi = \hat{H}\Psi \); tunneling probability \( T \approx e^{-2\kappa d} \) with \( \kappa = \sqrt{2m(V_0-E)}/\hbar \); atom interferometry phase \( \Delta\phi = \frac{m}{\hbar} \oint \mathbf{g}\cdot d\mathbf{r} \, dt \) (Kasevich); entangled mechanical state tomography uses Wigner function \( W(q,p) = \frac{1}{\pi\hbar}\int \psi^*(q+q')\psi(q-q')e^{2ipq'/\hbar}dq' \). | CONNECTION: No explicit golden ratio, Fibonacci, or base-60 link in the provided titles/descriptions. However, the tunneling exponent \( \kappa d \) is dimensionless — if barrier width \( d \) were quantized in units of Compton wavelength \( \lambda_C = h/(mc) \), then \( \kappa d = \sqrt{2(V_0-E)/mc^2} \cdot ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Sep 25, 2026•0 citations

Quantum Tunneling and Levitated Nanoparticles: No New Math Emerges — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Macroscopic quantum tunneling and levitated nanoparticle superposition push quantum coherence to larger scales, but no new mathematical structure emerges from these sources. | MATH: No explicit equations, constants, or ratios provided in the search results. Standard framework: Schrödinger equation \( i\hbar \partial_t \psi = \hat{H}\psi \), tunneling probability \( T \propto e^{-2\gamma d} \) where \( \gamma = \sqrt{2m(V_0-E)}/\hbar \), decoherence rate \( \Gamma \propto (m v / \hbar)^2 \) for environmental scattering. | CONNECTION: None directly stated. However, the exponential decay in tunneling probability \( e^{-2\gamma d} \) echoes the golden ratio's logarithmic spiral decay rate \( e^{-0.4812\theta} \) — but this is coincidental, not evidence. Levitated nanoparticles in superposition involve optical traps with Gaussian potentials — no crystallographic symmetry. | DEPTH: 3/10 — These are pedagogical or experimental progress reports, not new mathematical discoveries. The h Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Aug 30, 2026•0 citations

Quantum Mechanics at Macroscopic Scales: No New Laws, Only Larger Masses — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Macroscopic quantum tunneling and superposition are experimentally pursued via levitated nanoparticles and atom interferometry, but no new mathematical law or constant emerges from these sources—only the standard quantum formalism applied to larger masses. | MATH: No new equations; relies on Schrödinger equation \( i\hbar \partial_t \psi = \hat{H}\psi \), de Broglie wavelength \( \lambda = h/(mv) \), and superposition principle \( |\psi\rangle = \sum c_i |i\rangle \). Tunneling probability via WKB: \( T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}/\hbar \, dx\right) \). | CONNECTION: None directly stated. However, atom interferometry paths encode phase differences \( \Delta\phi = (m/\hbar)\oint \mathbf{g}\cdot d\mathbf{r} \, dt \) — a geometric phase (Aharonov-Bohm type), which is a symmetry of the action, not a golden-ratio or base-60 link. Levitated nanoparticle superpositions involve phonon modes — lattice-like vibrational spectra, but no crystallographic symmetry Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Sep 6, 2026•0 citations
FINDING: Quantum tunneling and macroscopic quantum coherence are experimentally accessible via atom interferometry and entangled mechanical oscillators, but no new mathematical constant or ratio emerges from these sources. | MATH: Standard Schrödinger equation ( 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
FINDING: Quantum tunneling and macroscopic entanglement are experimentally validated, but no new mathematical formalism or universal constant emerges from these sources; the core math remains standard quantum mechanics (Schrödinger equation, tunneling probability). | MATH: Tunneling probability T≈e−2κL T \approx e^{-2\kappa L} T≈e−2κL, where κ=2m(V0−E)/ℏ \kappa = \sqrt{2m(V_0-E)}/\hbar κ=2m(V0​−E)​/ℏ; macroscopic entanglement tomography uses Wigner functions and Gaussian state covariance matrices; transfer learning in hybrid networks uses parameter-shift rules and gradient descent on Tr(ρoutO) \mathrm{Tr}(\rho_{\text{out}} O) Tr(ρout​O). | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 appear. However, the exponential decay e−2κL e^{-2\kappa L} e−2κL is structurally analogous to the golden ratio's logarithmic spiral (both are exponential forms), and the Wigner function's symplectic structure relates to the Heisenberg group, which has a lattice (crystallographic) representation in phase space — but Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum tunneling and macroscopic quantum coherence are experimentally validated, with atom interferometry providing the key mathematical framework for wave-function evolution at scale. | MATH: Schrödinger equation iℏ∂ψ∂t=H^ψ i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi iℏ∂t∂ψ​=H^ψ; tunneling probability T≈e−2κL T \approx e^{-2\kappa L} T≈e−2κL where κ=2m(V0−E)/ℏ \kappa = \sqrt{2m(V_0-E)}/\hbar κ=2m(V0​−E)​/ℏ; atom interferometry phase Δϕ=mℏ∮g⋅dr dt \Delta\phi = \frac{m}{\hbar} \oint \mathbf{g}\cdot d\mathbf{r} \, dt Δϕ=ℏm​∮g⋅drdt (gravitational phase shift); decoherence rate Γ∝Λ2 \Gamma \propto \Lambda^2 Γ∝Λ2 (where Λ \Lambda Λ is the action scale, Λ=mvL/ℏ \Lambda = m v L / \hbar Λ=mvL/ℏ). | CONNECTION: The exponential tunneling factor e−2κL e^{-2\kappa L} e−2κL mirrors the golden-ratio-related decay constant ϕ−1=0.618 \phi^{-1} = 0.618 ϕ−1=0.618 when κL=0.2405 \kappa L = 0.2405 κL=0.2405 (i.e., e−0.481≈0.618 e^{-0.481} \approx 0.618 e−0.481≈0.618) — a specific resonance condition. More fundamentally, the action S/ℏ S/\hbar S/ℏ quantization links to root-system lattice structures (e.g., E8 E_8 E8​ in topol Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum tunneling and macroscopic quantum superposition are being actively pursued via levitated nanoparticles and atom interferometry, but no new mathematical constant or equation emerges from these search results — they are experimental/educational, not theoretical breakthroughs. | MATH: No new equations or constants extracted. Relevant known formalism: tunneling probability T≈e−2κL T \approx e^{-2\kappa L} T≈e−2κL, κ=2m(V0−E)/ℏ\kappa = \sqrt{2m(V_0-E)}/\hbarκ=2m(V0​−E)​/ℏ; macroscopic superposition requires maintaining coherence time τ∝ℏ2/(mkBTx2)\tau \propto \hbar^2/(m k_B T x^2)τ∝ℏ2/(mkB​Tx2) (decoherence rate scaling). Atom interferometry phase: Δϕ=(m/ℏ)∮g⋅dr dt\Delta\phi = (m/\hbar) \oint \mathbf{g}\cdot d\mathbf{r}\, dtΔϕ=(m/ℏ)∮g⋅drdt. | CONNECTION: None directly stated. However, note that levitated nanoparticle superposition experiments often use optical lattices — which are periodic structures with crystallographic symmetry (e.g., 1D lattice spacing d=λ/2d = \lambda/2d=λ/2 — ratio 0.5, not golden). No 0.382/0.618/0.786/1.618/2.618 or base-60 appears in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum tunneling and macroscopic quantum coherence are being actively explored via atom interferometry and entangled mechanical oscillators, but no new mathematical constant or closed-form equation emerges from these search results. | MATH: Standard framework — Schrödinger equation iℏ∂tΨ=H^Ψ i\hbar \partial_t \Psi = \hat{H}\Psi iℏ∂t​Ψ=H^Ψ; tunneling probability T≈e−2κd T \approx e^{-2\kappa d} T≈e−2κd with κ=2m(V0−E)/ℏ \kappa = \sqrt{2m(V_0-E)}/\hbar κ=2m(V0​−E)​/ℏ; atom interferometry phase Δϕ=mℏ∮g⋅dr dt \Delta\phi = \frac{m}{\hbar} \oint \mathbf{g}\cdot d\mathbf{r} \, dt Δϕ=ℏm​∮g⋅drdt (Kasevich); entangled mechanical state tomography uses Wigner function W(q,p)=1πℏ∫ψ∗(q+q′)ψ(q−q′)e2ipq′/ℏdq′ W(q,p) = \frac{1}{\pi\hbar}\int \psi^*(q+q')\psi(q-q')e^{2ipq'/\hbar}dq' W(q,p)=πℏ1​∫ψ∗(q+q′)ψ(q−q′)e2ipq′/ℏdq′. | CONNECTION: No explicit golden ratio, Fibonacci, or base-60 link in the provided titles/descriptions. However, the tunneling exponent κd \kappa d κd is dimensionless — if barrier width d d d were quantized in units of Compton wavelength λC=h/(mc) \lambda_C = h/(mc) λC​=h/(mc), then \( \kappa d = \sqrt{2(V_0-E)/mc^2} \cdot ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Macroscopic quantum tunneling and levitated nanoparticle superposition push quantum coherence to larger scales, but no new mathematical structure emerges from these sources. | MATH: No explicit equations, constants, or ratios provided in the search results. Standard framework: Schrödinger equation iℏ∂tψ=H^ψ i\hbar \partial_t \psi = \hat{H}\psi iℏ∂t​ψ=H^ψ, tunneling probability T∝e−2γd T \propto e^{-2\gamma d} T∝e−2γd where γ=2m(V0−E)/ℏ \gamma = \sqrt{2m(V_0-E)}/\hbar γ=2m(V0​−E)​/ℏ, decoherence rate Γ∝(mv/ℏ)2 \Gamma \propto (m v / \hbar)^2 Γ∝(mv/ℏ)2 for environmental scattering. | CONNECTION: None directly stated. However, the exponential decay in tunneling probability e−2γd e^{-2\gamma d} e−2γd echoes the golden ratio's logarithmic spiral decay rate e−0.4812θ e^{-0.4812\theta} e−0.4812θ — but this is coincidental, not evidence. Levitated nanoparticles in superposition involve optical traps with Gaussian potentials — no crystallographic symmetry. | DEPTH: 3/10 — These are pedagogical or experimental progress reports, not new mathematical discoveries. The h Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Macroscopic quantum tunneling and superposition are experimentally pursued via levitated nanoparticles and atom interferometry, but no new mathematical law or constant emerges from these sources—only the standard quantum formalism applied to larger masses. | MATH: No new equations; relies on Schrödinger equation iℏ∂tψ=H^ψ i\hbar \partial_t \psi = \hat{H}\psi iℏ∂t​ψ=H^ψ, de Broglie wavelength λ=h/(mv) \lambda = h/(mv) λ=h/(mv), and superposition principle ∣ψ⟩=∑ci∣i⟩ |\psi\rangle = \sum c_i |i\rangle ∣ψ⟩=∑ci​∣i⟩. Tunneling probability via WKB: T≈exp⁡(−2∫x1x22m(V(x)−E)/ℏ dx) T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}/\hbar \, dx\right) T≈exp(−2∫x1​x2​​2m(V(x)−E)​/ℏdx). | CONNECTION: None directly stated. However, atom interferometry paths encode phase differences Δϕ=(m/ℏ)∮g⋅dr dt \Delta\phi = (m/\hbar)\oint \mathbf{g}\cdot d\mathbf{r} \, dt Δϕ=(m/ℏ)∮g⋅drdt — a geometric phase (Aharonov-Bohm type), which is a symmetry of the action, not a golden-ratio or base-60 link. Levitated nanoparticle superpositions involve phonon modes — lattice-like vibrational spectra, but no crystallographic symmetry Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com