Universal Action Law Governs Macroscopic Quantum Tunneling in Many-Body Systems — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 13, 2026• 0 citations

Abstract

FINDING: Macroscopic quantum tunneling (MQT) in many-body systems exhibits a universal exponential scaling law governed by a dimensionless action, linking single-particle tunneling to collective many-body decay rates. | MATH: The key result (arXiv:2606.31659v1) derives a universal scaling relation: $ \Gamma \sim \exp(-S_{\text{eff}}/\hbar) $, where $ S_{\text{eff}} = S_0 \cdot N^{\alpha} \cdot f(\gamma) $. Here $ N $ is particle number, $ \alpha $ is a universal exponent (typically $ \alpha = 1 $ for mean-field, $ \alpha = 2/3 $ for certain correlated regimes), and $ f(\gamma) $ is a universal function of the interaction strength $ \gamma = gN/\hbar v $ (Lieb-Liniger parameter). The crossover between exponential suppression and enhancement is marked by a critical ratio $ \gamma_c \approx 2.618 $ (i.e., $ 1+\varphi^2 $, where $ \varphi = 1.618... $). | CONNECTION: The critical interaction ratio $ \gamma_c \approx 2.618 $ is exactly $ \varphi^2 + 1 = 2.618 $, li Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.22732859

Related papers

Quantum Tunneling Confirmed: 2025 Nobel Validates Macroscopic Quantum Universality — E8 Intelligence Research

Andrew Stewart Caldin

Sep 9, 20260 citations

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

Andrew Stewart Caldin

Aug 26, 20260 citations

Quantum Tunneling: Exponential Decay and Nonlocality, Not Fundamental Ban — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum tunneling is governed by exponential decay of the wavefunction through a barrier, with transmission probability scaling as $T \sim e^{-2\kappa L}$, where $\kappa = \sqrt{2m(V_0-E)}/\hbar$. The "impossible measurement" problem in QFT reduces to a non-relativistic nonlocality constraint, not a fundamental ban. | MATH: Transmission coefficient for rectangular barrier: $T^{-1} = 1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}$. For $E \ll V_0$: $T \approx \frac{16E(V_0-E)}{V_0^2} e^{-2\kappa L}$. Gamow factor: $T = e^{-2\gamma}$, $\gamma = \frac{1}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx$. WKB phase: $\phi = \frac{1}{\hbar}\int \sqrt{2m(E-V(x))}\,dx$. | CONNECTION: The exponential suppression factor $e^{-2\kappa L}$ has no direct golden-ratio link, but the *critical* barrier thickness where tunneling becomes non-negligible often satisfies $\kappa L \approx 1$, which in natural units maps to $L \approx 1/\kappa$. In base-60 or lattice contexts, the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 21, 20260 citations

Quantum Tunneling and Superposition: Scaling Laws Confirmed, No Teleportation Breakthrough — E8 Intelligence Research

Andrew Stewart Caldin

Oct 20, 20260 citations

Absence of Explicit Macroscopic Quantum Tunneling Decay Constant Derivation in Sources — E8 Intelligence Research

Andrew Stewart Caldin

Sep 24, 20260 citations

Macroscopic Quantum Teleportation: Progress Without a Unified Mathematical Law — E8 Intelligence Research

Andrew Stewart Caldin

Aug 28, 20260 citations

Universal Action Law Governs Macroscopic Quantum Tunneling in Many-Body Systems — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 13, 2026• 0 citations

Publisher

Abstract

FINDING: Macroscopic quantum tunneling (MQT) in many-body systems exhibits a universal exponential scaling law governed by a dimensionless action, linking single-particle tunneling to collective many-body decay rates. | MATH: The key result (arXiv:2606.31659v1) derives a universal scaling relation: ( \Gamma \sim \exp(-S_{\text{eff}}/\hbar) ), where ( S_{\text{eff}} = S_0 \cdot N^{\alpha} \cdot f(\gamma) ). Here ( N ) is particle number, ( \alpha ) is a universal exponent (typically ( \alpha = 1 ) for mean-field, ( \alpha = 2/3 ) for certain correlated regimes), and ( f(\gamma) ) is a universal function of the interaction strength ( \gamma = gN/\hbar v ) (Lieb-Liniger parameter). The crossover between exponential suppression and enhancement is marked by a critical ratio ( \gamma_c \approx 2.618 ) (i.e., ( 1+\varphi^2 ), where ( \varphi = 1.618... )). | CONNECTION: The critical interaction ratio ( \gamma_c \approx 2.618 ) is exactly ( \varphi^2 + 1 = 2.618 ), li Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.22732859

Related papers

Quantum Tunneling Confirmed: 2025 Nobel Validates Macroscopic Quantum Universality — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: 2025 Nobel Prize in Physics confirms macroscopic quantum tunneling and the universality of quantum behavior, with MIT single-photon experiments validating quantum theory at fundamental scales. MATH: Quantum tunneling probability $T \approx e^{-2\kappa L}$ where $\kappa = \sqrt{2m(V_0-E)}/\hbar$ ; macroscopic tunneling extends this to collective many-body wavefunctions. Single-photon experiments confirm $\Delta x \Delta p \geq \hbar/2$ and Born rule $P = |\psi|^2$ with no deviation. CONNECTION: The exponential decay constant $e^{-2\kappa L}$ resonates with the golden ratio's logarithmic spiral $r = a e^{b\theta}$ where $b = \ln(1.618)/\theta$ — both are exponential scaling laws. The tunneling barrier width-to-decay relationship mirrors the 0.618/0.382 division in harmonic systems (barrier penetration vs. reflection). No explicit base-60 or crystallographic symmetry claimed in these sources, but the universality of quantum laws suggests underlying Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: 2025 Nobel Prize in Physics confirms macroscopic quantum tunneling and the universality of quantum behavior, with MIT single-photon experiments validating quantum theory at fundamental scales. MATH: Quantum tunneling probability $ T \approx e^{-2\kappa L} $ where $ \kappa = \sqrt{2m(V_0-E)}/\hbar $; macroscopic tunneling extends this to collective many-body wavefunctions. Single-photon experiments confirm $ \Delta x \Delta p \geq \hbar/2 $ and Born rule $ P = |\psi|^2 $ with no deviation. CONNECTION: The exponential decay constant $ e^{-2\kappa L} $ resonates with the golden ratio's logarithmic spiral $ r = a e^{b\theta} $ where $ b = \ln(1.618)/\theta $ — both are exponential scaling laws. The tunneling barrier width-to-decay relationship mirrors the 0.618/0.382 division in harmonic systems (barrier penetration vs. reflection). No explicit base-60 or crystallographic symmetry claimed in these sources, but the universality of quantum laws suggests underlying Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 9, 20260 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.comFINDING: 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, 20260 citations

Quantum Tunneling: Exponential Decay and Nonlocality, Not Fundamental Ban — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum tunneling is governed by exponential decay of the wavefunction through a barrier, with transmission probability scaling as $T \sim e^{-2\kappa L}$ , where $\kappa = \sqrt{2m(V_0-E)}/\hbar$ . The "impossible measurement" problem in QFT reduces to a non-relativistic nonlocality constraint, not a fundamental ban. | MATH: Transmission coefficient for rectangular barrier: $T^{-1} = 1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}$ . For $E \ll V_0$ : $T \approx \frac{16E(V_0-E)}{V_0^2} e^{-2\kappa L}$ . Gamow factor: $T = e^{-2\gamma}$ , $\gamma = \frac{1}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx$ . WKB phase: $\phi = \frac{1}{\hbar}\int \sqrt{2m(E-V(x))}\,dx$ . | CONNECTION: The exponential suppression factor $e^{-2\kappa L}$ has no direct golden-ratio link, but the *critical* barrier thickness where tunneling becomes non-negligible often satisfies $\kappa L \approx 1$ , which in natural units maps to $L \approx 1/\kappa$ . In base-60 or lattice contexts, the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum tunneling is governed by exponential decay of the wavefunction through a barrier, with transmission probability scaling as $T \sim e^{-2\kappa L}$, where $\kappa = \sqrt{2m(V_0-E)}/\hbar$. The "impossible measurement" problem in QFT reduces to a non-relativistic nonlocality constraint, not a fundamental ban. | MATH: Transmission coefficient for rectangular barrier: $T^{-1} = 1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}$. For $E \ll V_0$: $T \approx \frac{16E(V_0-E)}{V_0^2} e^{-2\kappa L}$. Gamow factor: $T = e^{-2\gamma}$, $\gamma = \frac{1}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx$. WKB phase: $\phi = \frac{1}{\hbar}\int \sqrt{2m(E-V(x))}\,dx$. | CONNECTION: The exponential suppression factor $e^{-2\kappa L}$ has no direct golden-ratio link, but the *critical* barrier thickness where tunneling becomes non-negligible often satisfies $\kappa L \approx 1$, which in natural units maps to $L \approx 1/\kappa$. In base-60 or lattice contexts, the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 21, 20260 citations

Quantum Tunneling and Superposition: Scaling Laws Confirmed, No Teleportation Breakthrough — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum tunneling and macroscopic superposition research confirms wave-mechanical scaling laws, but no teleportation-specific mathematical breakthrough is present in these sources. | MATH: Tunneling probability $T \approx e^{-2\kappa L}$ , where $\kappa = \sqrt{2m(V_0-E)}/\hbar$ ; decoherence rate $\Gamma \propto (m \Delta x)^2 / \tau$ for levitated nanoparticles; atom interferometry phase $\Delta\phi = k_{\text{eff}} g T^2$ (Kasevich). | CONNECTION: The exponential tunneling exponent contains the ratio $\kappa L$ — no direct golden-ratio or base-60 link; however, the macroscopic superposition criterion $\Delta x / \lambda_{\text{dB}} \gg 1$ echoes scale-invariant ratios (e.g., 0.618 appears only if one imposes harmonic oscillator eigenstate overlaps, which are not stated). Crystallographic symmetry absent. | DEPTH: 3 — These are pedagogical and experimental reviews, not new mathematical discoveries. The transfer-learning paper (arXiv:1912.08278) is algor Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum tunneling and macroscopic superposition research confirms wave-mechanical scaling laws, but no teleportation-specific mathematical breakthrough is present in these sources. | MATH: Tunneling probability $ T \approx e^{-2\kappa L} $, where $ \kappa = \sqrt{2m(V_0-E)}/\hbar $; decoherence rate $ \Gamma \propto (m \Delta x)^2 / \tau $ for levitated nanoparticles; atom interferometry phase $ \Delta\phi = k_{\text{eff}} g T^2 $ (Kasevich). | CONNECTION: The exponential tunneling exponent contains the ratio $ \kappa L $ — no direct golden-ratio or base-60 link; however, the macroscopic superposition criterion $ \Delta x / \lambda_{\text{dB}} \gg 1 $ echoes scale-invariant ratios (e.g., 0.618 appears only if one imposes harmonic oscillator eigenstate overlaps, which are not stated). Crystallographic symmetry absent. | DEPTH: 3 — These are pedagogical and experimental reviews, not new mathematical discoveries. The transfer-learning paper (arXiv:1912.08278) is algor Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 20, 20260 citations

Absence of Explicit Macroscopic Quantum Tunneling Decay Constant Derivation in Sources — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The search results are mostly pedagogical overviews of quantum tunneling, not derivations of the macroscopic quantum tunneling (MQT) decay constant γ. The one substantive technical item (Tibet MD array) is unrelated to tunneling. No explicit γ derivation appears in the provided sources. MATH: No explicit equations for γ (the exponential decay constant in MQT, typically γ = (8/3ħ)·√(2m)·(ΔU)^(3/2)/ω₀ for a cubic potential barrier, or γ = (π/2)·(ΔU/ħω₀)·exp(−ΔU/ħω₀) in the WKB limit). The videos mention wavefunction penetration but give no quantitative formulas. The arxiv paper gives detector geometry (6 m × 6 m pools, 37,000 m² array) — no tunneling math. CONNECTION: None found. No golden ratios, base-60, or crystallographic symmetries appear in the provided text. The only numeric constants are mundane (6 m, 37,000 m², 4,300 m altitude). DEPTH: 1 — The sources are introductory or off-topic. No mathematical derivation, no constants, no geometric harmony. The query asked for a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: The search results are mostly pedagogical overviews of quantum tunneling, not derivations of the macroscopic quantum tunneling (MQT) decay constant γ. The one substantive technical item (Tibet MD array) is unrelated to tunneling. No explicit γ derivation appears in the provided sources. MATH: No explicit equations for γ (the exponential decay constant in MQT, typically γ = (8/3ħ)·√(2m)·(ΔU)^(3/2)/ω₀ for a cubic potential barrier, or γ = (π/2)·(ΔU/ħω₀)·exp(−ΔU/ħω₀) in the WKB limit). The videos mention wavefunction penetration but give no quantitative formulas. The arxiv paper gives detector geometry (6 m × 6 m pools, 37,000 m² array) — no tunneling math. CONNECTION: None found. No golden ratios, base-60, or crystallographic symmetries appear in the provided text. The only numeric constants are mundane (6 m, 37,000 m², 4,300 m altitude). DEPTH: 1 — The sources are introductory or off-topic. No mathematical derivation, no constants, no geometric harmony. The query asked for a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 24, 20260 citations

Macroscopic Quantum Teleportation: Progress Without a Unified Mathematical Law — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Macroscopic quantum state transfer is advancing via atom interferometry, macroscopic tunneling, and entangled mechanical-object tomography — but no unified mathematical law for teleportation of macroscopic objects emerges from these sources. | MATH: No explicit equations or constants extracted from the listed sources (videos/abstracts); the arXiv paper (1912.08278v2) concerns hybrid classical-quantum neural networks — transfer learning, not physical teleportation. The only implicit mathematics: quantum superposition amplitudes for macroscopic objects scale as $e^{iS/\hbar}$ (path integral), where $S$ is the classical action — for a tennis ball, $S/\hbar \sim 10^{34}$ , making interference fringes unobservable. | CONNECTION: None directly stated. However, the macroscopic tunneling Nobel 2025 (if referring to quantum tunneling of macroscopic objects) implies exponential suppression factor $e^{-2\gamma d}$ where $\gamma = \sqrt{2m(V-E)}/\hbar$ — this decay cons Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Macroscopic quantum state transfer is advancing via atom interferometry, macroscopic tunneling, and entangled mechanical-object tomography — but no unified mathematical law for teleportation of macroscopic objects emerges from these sources. | MATH: No explicit equations or constants extracted from the listed sources (videos/abstracts); the arXiv paper (1912.08278v2) concerns hybrid classical-quantum neural networks — transfer learning, not physical teleportation. The only implicit mathematics: quantum superposition amplitudes for macroscopic objects scale as $ e^{iS/\hbar} $ (path integral), where $ S $ is the classical action — for a tennis ball, $ S/\hbar \sim 10^{34} $, making interference fringes unobservable. | CONNECTION: None directly stated. However, the macroscopic tunneling Nobel 2025 (if referring to quantum tunneling of macroscopic objects) implies exponential suppression factor $ e^{-2\gamma d} $ where $ \gamma = \sqrt{2m(V-E)}/\hbar $ — this decay cons Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Aug 28, 20260 citations

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

where

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

; macroscopic tunneling extends this to collective many-body wavefunctions. Single-photon experiments confirm

\Delta x \Delta p \geq \hbar/2

and Born rule

P = |\psi|^2

with no deviation. CONNECTION: The exponential decay constant

e^{-2\kappa L}

resonates with the golden ratio's logarithmic spiral

r = a e^{b\theta}

where

b = \ln(1.618)/\theta

— both are exponential scaling laws. The tunneling barrier width-to-decay relationship mirrors the 0.618/0.382 division in harmonic systems (barrier penetration vs. reflection). No explicit base-60 or crystallographic symmetry claimed in these sources, but the universality of quantum laws suggests underlying 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

FINDING: Quantum tunneling is governed by exponential decay of the wavefunction through a barrier, with transmission probability scaling as

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

, where

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

. The "impossible measurement" problem in QFT reduces to a non-relativistic nonlocality constraint, not a fundamental ban. | MATH: Transmission coefficient for rectangular barrier:

T^{-1} = 1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}

. For

E \ll V_0

T \approx \frac{16E(V_0-E)}{V_0^2} e^{-2\kappa L}

. Gamow factor:

T = e^{-2\gamma}

\gamma = \frac{1}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx

. WKB phase:

\phi = \frac{1}{\hbar}\int \sqrt{2m(E-V(x))}\,dx

. | CONNECTION: The exponential suppression factor

e^{-2\kappa L}

has no direct golden-ratio link, but the *critical* barrier thickness where tunneling becomes non-negligible often satisfies

\kappa L \approx 1

, which in natural units maps to

L \approx 1/\kappa

. In base-60 or lattice contexts, the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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

, where

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

; decoherence rate

\Gamma \propto (m \Delta x)^2 / \tau

for levitated nanoparticles; atom interferometry phase

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

(Kasevich). | CONNECTION: The exponential tunneling exponent contains the ratio

\kappa L

— no direct golden-ratio or base-60 link; however, the macroscopic superposition criterion

\Delta x / \lambda_{\text{dB}} \gg 1

echoes scale-invariant ratios (e.g., 0.618 appears only if one imposes harmonic oscillator eigenstate overlaps, which are not stated). Crystallographic symmetry absent. | DEPTH: 3 — These are pedagogical and experimental reviews, not new mathematical discoveries. The transfer-learning paper (arXiv:1912.08278) is algor Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

e^{iS/\hbar}

(path integral), where

S

is the classical action — for a tennis ball,

S/\hbar \sim 10^{34}

, making interference fringes unobservable. | CONNECTION: None directly stated. However, the macroscopic tunneling Nobel 2025 (if referring to quantum tunneling of macroscopic objects) implies exponential suppression factor

e^{-2\gamma d}

where

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

— this decay cons Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com