Absence of Golden Ratio Link in Quantum Tunneling Probability Literature — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 27, 2026• 0 citations

Abstract

FINDING: No direct derivation of phi^-2 as a quantum tunneling probability exists in the retrieved sources; the search returns standard tunneling derivations, a Numberphile proof on golden ratio sequences, and an arXiv paper on golden ratio self-application — none linking phi to barrier penetration. | MATH: Standard tunneling probability for rectangular barrier: $T \approx e^{-2\kappa L}$, $\kappa = \sqrt{2m(V_0-E)}/\hbar$. Golden ratio: $\phi = (1+\sqrt{5})/2 \approx 1.618$, $\phi^{-2} = (3-\sqrt{5})/2 \approx 0.382$. No equation in the sources equates $T$ to $\phi^{-2}$. | CONNECTION: The arXiv paper (2510.08934) uses $\Phi$ as a model of stable local recurrence — operationally, $\Phi^{-1} = \phi-1 \approx 0.618$, and $\Phi^{-2} \approx 0.382$ — but this is in self-referential computation, not tunneling. The Numberphile video proves infinite sequences share the golden ratio property, but again no quantum link. | DEPTH: 2 — The search failed to find the claimed deriv Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.22986968

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Absence of Golden Ratio Link in Quantum Tunneling Probability Literature — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 27, 2026• 0 citations

Publisher

Abstract

FINDING: No direct derivation of phi^-2 as a quantum tunneling probability exists in the retrieved sources; the search returns standard tunneling derivations, a Numberphile proof on golden ratio sequences, and an arXiv paper on golden ratio self-application — none linking phi to barrier penetration. | MATH: Standard tunneling probability for rectangular barrier: (T \approx e^{-2\kappa L}), (\kappa = \sqrt{2m(V_0-E)}/\hbar). Golden ratio: (\phi = (1+\sqrt{5})/2 \approx 1.618), (\phi^{-2} = (3-\sqrt{5})/2 \approx 0.382). No equation in the sources equates (T) to (\phi^{-2}). | CONNECTION: The arXiv paper (2510.08934) uses (\Phi) as a model of stable local recurrence — operationally, (\Phi^{-1} = \phi-1 \approx 0.618), and (\Phi^{-2} \approx 0.382) — but this is in self-referential computation, not tunneling. The Numberphile video proves infinite sequences share the golden ratio property, but again no quantum link. | DEPTH: 2 — The search failed to find the claimed deriv Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.22986968

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Absence of Golden-Ratio Lindblad Coupling in Quantum Dephasing Literature — E8 Intelligence Research

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FINDING: No direct quantum Goldilocks zone or golden-ratio Lindblad coupling result is present; the search returns tangential videos (Lindblad intro, golden ratio convergence, bioinformatics) and one arXiv paper on golden ratio as stable self-application, which is unrelated to quantum dephasing. MATH: - Lindblad master equation (standard form): $\dot{\rho} = -i[H,\rho] + \sum_k \gamma_k (L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k, \rho\})$ — no golden ratio coupling appears in the cited material. - Golden ratio $\Phi = (1+\sqrt{5})/2 \approx 1.618$ , conjugate $\phi = \Phi - 1 = 1/\Phi \approx 0.618$ . - arXiv 2510.08934: treats $\Phi$ as a stable local recurrence procedure — no Lindblad or quantum dephasing content. CONNECTION: - The golden ratio appears only in the convergence-rate video (continued fractions vs. nested radicals) and the self-application paper — neither connects to quantum dissipation or FMO complexes. - No evidence of 0.382, 0.7 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: No direct quantum Goldilocks zone or golden-ratio Lindblad coupling result is present; the search returns tangential videos (Lindblad intro, golden ratio convergence, bioinformatics) and one arXiv paper on golden ratio as stable self-application, which is unrelated to quantum dephasing. MATH: - Lindblad master equation (standard form): $ \dot{\rho} = -i[H,\rho] + \sum_k \gamma_k (L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k, \rho\}) $ — no golden ratio coupling appears in the cited material. - Golden ratio $ \Phi = (1+\sqrt{5})/2 \approx 1.618 $, conjugate $ \phi = \Phi - 1 = 1/\Phi \approx 0.618 $. - arXiv 2510.08934: treats $ \Phi $ as a stable local recurrence procedure — no Lindblad or quantum dephasing content. CONNECTION: - The golden ratio appears only in the convergence-rate video (continued fractions vs. nested radicals) and the self-application paper — neither connects to quantum dissipation or FMO complexes. - No evidence of 0.382, 0.7 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, not a classical "jump"; its probability depends exponentially on barrier width and height, with no known teleportation mechanism. | MATH: Transmission coefficient $T \approx e^{-2\kappa L}$ , where $\kappa = \sqrt{2m(V_0-E)}/\hbar$ ; for a rectangular barrier, exact $T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}$ . No new constants or ratios emerge beyond $\hbar$ , $m$ , $V_0$ , $E$ , $L$ . | CONNECTION: The exponential decay $e^{-2\kappa L}$ has no direct ratio to 0.382/0.618/1.618; however, the *critical* condition $E = V_0$ yields $\kappa = 0$ and $T \to 1$ (resonant transparency), which is a phase-transition-like threshold — analogous to a golden-ratio bifurcation point in some dynamical systems, but not evidenced here. No crystallographic or base-60 link. | DEPTH: 3 — Standard quantum mechanics, well-understood; the "impossible measurements" QFT paper (arXiv:2311.13 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, not a classical "jump"; its probability depends exponentially on barrier width and height, with no known teleportation mechanism. | MATH: Transmission coefficient $T \approx e^{-2\kappa L}$, where $\kappa = \sqrt{2m(V_0-E)}/\hbar$; for a rectangular barrier, exact $T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}$. No new constants or ratios emerge beyond $\hbar$, $m$, $V_0$, $E$, $L$. | CONNECTION: The exponential decay $e^{-2\kappa L}$ has no direct ratio to 0.382/0.618/1.618; however, the *critical* condition $E = V_0$ yields $\kappa = 0$ and $T \to 1$ (resonant transparency), which is a phase-transition-like threshold — analogous to a golden-ratio bifurcation point in some dynamical systems, but not evidenced here. No crystallographic or base-60 link. | DEPTH: 3 — Standard quantum mechanics, well-understood; the "impossible measurements" QFT paper (arXiv:2311.13 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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\dot{\rho} = -i[H,\rho] + \sum_k \gamma_k (L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k, \rho\})

— no golden ratio coupling appears in the cited material. - Golden ratio

\Phi = (1+\sqrt{5})/2 \approx 1.618

, conjugate

\phi = \Phi - 1 = 1/\Phi \approx 0.618

. - arXiv 2510.08934: treats

\Phi

as a stable local recurrence procedure — no Lindblad or quantum dephasing content. CONNECTION: - The golden ratio appears only in the convergence-rate video (continued fractions vs. nested radicals) and the self-application paper — neither connects to quantum dissipation or FMO complexes. - No evidence of 0.382, 0.7 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

(decay constant). - For a rectangular barrier:

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

. - No explicit constants or ratios (0.382, 0.618, etc.) appear in the provided sources. CONNECTION: - No direct geometric harmony (golden ratio, base-60, crystallographic symmetry) is evident in these findings. - The wavefunction's exponential decay

e^{-\kappa L}

is a natural exponential, not a harmonic ratio. DEPTH: 3 - The findings are standard quantum tunneling pedagogy; no new mathematical structure or geometric link is revealed. The "impossible measurements" paper (arXiv:2311.13644) is a theoretical nuance, not a breakthrough in constants or symmetries. 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

; for a rectangular barrier, exact

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

. No new constants or ratios emerge beyond

\hbar

m

V_0

E

L

. | CONNECTION: The exponential decay

e^{-2\kappa L}

has no direct ratio to 0.382/0.618/1.618; however, the *critical* condition

E = V_0

yields

\kappa = 0

and

T \to 1

(resonant transparency), which is a phase-transition-like threshold — analogous to a golden-ratio bifurcation point in some dynamical systems, but not evidenced here. No crystallographic or base-60 link. | DEPTH: 3 — Standard quantum mechanics, well-understood; the "impossible measurements" QFT paper (arXiv:2311.13 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com