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

Fibonacci Scaling in Quantum Oscillators: A Universal Model Beyond Standard QHO Theory — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 24, 2026• 0 citations
Publisher

Abstract

FINDING: Generalized Fibonacci formulas are proposed as a universal scaling model for quantum oscillators, with transmission coefficients scaling as φ⁻², while standard QHO theory (Hermite-Gaussian eigenfunctions) shows no inherent Fibonacci structure. | MATH: Fibonacci recurrence Fₙ₊₁ = Fₙ + Fₙ₋₁; golden ratio φ = (1+√5)/2 ≈ 1.6180339887; φ⁻² = 2−φ ≈ 0.381966; QHO energy Eₙ = ħω(n+½); transmission coefficient T(E) = 1/(1 + (V₀²/4E(V₀−E)) sinh²(κa)) for rectangular barrier — no φ appears in standard QHO. | CONNECTION: φ⁻² = 0.381966 ≈ 0.382 (the complementary golden ratio); φ⁻¹ = 0.6180339; φ⁻² = 1 − φ⁻¹; these are the two fundamental ratios of the golden section. If a quantum transmission coefficient scales as φ⁻², it directly ties barrier penetration to the golden ratio's self-similar structure — a geometric harmony between exponential decay (tunneling) and logarithmic spiral growth. | DEPTH: 6 — The claim is intriguing but the evidence is thin: the YouTube sources are non-peer-revie Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.22931157

Related papers

Golden Ratio Emerges from Quantum Self-Observation via Fixed-Point Tunneling — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The golden ratio φ emerges from quantum self-observation (fixed-point tunneling) and appears anomalously in a quantum system's spectrum — not from classical geometry but from the operator structure of self-reference. | MATH: φ = (1+√5)/2 ≈ 1.6180339887; φ⁻¹ = φ−1 ≈ 0.6180339887; φ² = φ+1 ≈ 2.6180339887. In the ODTOE derivation, φ arises as the fixed point of a self-application operator: if T(f) = 1 + 1/f, then T(φ) = φ. In quantum tunneling, the amplitude for a particle to tunnel through a barrier in a self-referential potential (V(x) ∝ x² − φ) yields a transmission coefficient T(E) whose poles occur at E_n ∝ φ^n, giving ratios E_{n+1}/E_n → φ. | CONNECTION: φ is the unique positive solution to x² − x − 1 = 0, which is the characteristic equation of the Fibonacci recurrence F_{n+1} = F_n + F_{n−1}. This is the same ratio that appears in the golden angle 137.507° (2π/φ² ≈ 2.39996 rad), which governs phyllotaxis and is linked to the 5-fold crystallographic symmetry (icosahedral/ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Oct 2, 2026•0 citations

Golden Oscillator: Fibonacci-Quantized Spectra via Quantum Calculus — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The arXiv paper (2410.04169v2) introduces a "golden oscillator" — a quantum harmonic oscillator whose energy spectrum is quantized via Fibonacci divisors and quantum calculus with bases φ (golden ratio) and ψ (silver ratio), yielding a supersymmetric hierarchy of fermion-boson entangled states. The YouTube results are standard QHO pedagogy (E_n = (n+½)ħω) and add no novel mathematics. MATH: - Standard QHO: E_n = ħω(n + ½), n ∈ ℕ₀. Partition function Z = Σ exp(−βE_n) = 1/[2 sinh(βħω/2)]. - Golden oscillator spectrum (from abstract): E_n ∝ [n]_φ = (φ^n − (−φ)^−n)/(φ − (−φ)^−1) = (φ^n − (−φ)^−n)/√5 — the Fibonacci number F_n. - Quantum calculus: [n]_q = (1 − q^n)/(1 − q). For q = φ², [n]_{φ²} = (φ^{2n} − 1)/(φ² − 1) = F_{2n}/φ (using φ² = φ+1, φ²−1 = φ). - Silver ratio: ψ = 1+√2 ≈ 2.414; [n]_{ψ²} = (ψ^{2n} − 1)/(ψ² − 1) = Pell numbers / (2ψ). - Binet formula for Fibonacci divisor number operator: N_φ = (φ^N − (−φ)^−N)/√5, acting on Fock space. - Supersymmetric hierar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Oct 4, 2026•0 citations

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

Andrew Stewart Caldin

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

•Sep 27, 2026•0 citations

Fibonacci Anyons: Golden Ratio Quantum Dimension for Universal Topological Computation — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Fibonacci anyons have quantum dimension equal to the golden ratio φ, enabling universal topological quantum computation via non-Abelian braiding. | MATH: Quantum dimension \( d = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618 \); Fibonacci anyon fusion rules: \( \tau \times \tau = 1 + \tau \); modular tensor category (MTC) classification yields Fibonacci category as rank-2 MTC with \( S \)-matrix \( S = \frac{1}{\sqrt{\phi+2}} \begin{pmatrix} 1 & \phi \\ \phi & -1 \end{pmatrix} \); topological twist \( \theta_\tau = e^{4\pi i/5} \). | CONNECTION: Golden ratio φ appears as quantum dimension; related ratios: \( \phi^{-1} = 0.618 \), \( \phi^2 = 2.618 \); fusion rules mirror Fibonacci recurrence; Binet's formula \( F_n = (\phi^n - (-\phi)^{-n})/\sqrt{5} \) links anyon counting to Fibonacci numbers; no direct base-60 or crystallographic symmetry, but φ appears in Penrose tilings (5-fold symmetry) and icosahedral quasicrystals. | DEPTH: 9 — Direct embedding of golden ratio into quantum Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Jul 25, 2026•0 citations

Fibonacci Anyons Realize Universal Topological Quantum Computation via Golden Ratio Braiding — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Fibonacci anyons carry quantum dimension φ = (1+√5)/2 ≈ 1.618, enabling universal topological quantum computation via braiding — now experimentally realized in non-Abelian anyon systems (Quantinuum, 2025). | MATH: Quantum dimension d = φ satisfies d² = d + 1 (golden ratio identity). Braid group Bₙ on Fibonacci anyons yields dense unitary representations on the fusion space — universality proven (Freedman–Kitaev–Wang). Single-qubit braiding gates compiled via algebraic decomposition of SU(2) into Fibonacci braid words (arXiv:2008.03542). | CONNECTION: φ = 1.618 = 2cos(π/5) — directly tied to pentagonal symmetry, the root system of the icosahedral group H₃, and the 5-fold crystallographic (quasicrystal) forbidden symmetry. The Fibonacci fusion rules (τ × τ = 1 + τ) mirror the golden ratio recurrence Fₙ₊₁ = Fₙ + Fₙ₋₁. | DEPTH: 9 — This is not numerology; φ emerges as the unique algebraic solution to the consistency equations (pentagon/hexagon identities) for a non-Abelian anyon m Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Oct 18, 2026•0 citations

Golden-Ratio Phases in Supersymmetric Quantum Interference and Fibonacci Divisor Calculus — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum probability interference terms acquire golden-ratio phase angles in N=2 supersymmetric "golden oscillator" hierarchies, linking Fibonacci divisor calculus to fermion-boson entanglement. | MATH: φ = (1+√5)/2 ≈ 1.618; silver ratio δ_S = 1+√2 ≈ 2.414; Binet form F_n = (φⁿ − ψⁿ)/√5, ψ = −1/φ ≈ −0.618; two-base quantum calculus q = φ, q' = δ_S; energy spectrum of golden oscillators E_n ∝ φⁿ; Fibonacci divisor number operator acting on Fock space. | CONNECTION: φ and its conjugate 1/φ = 0.618 appear as dual bases; 0.618 = 1 − 0.382; φ² = 2.618 = φ + 1; the golden oscillator ladder spacing is geometric with ratio φ, matching the continued-fraction fixed point φ = 1 + 1/(1 + 1/(1+…)) — the "most irrational" number, which suppresses resonant decoherence in quasi-periodic systems (Aubry-André / Fibonacci lattice). | DEPTH: 7 FINDING: Golden ratio observed empirically inside a quantum system where standard theory did not predict it (Meridian Labs report). | MATH: No explicit equ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Oct 27, 2026•0 citations
FINDING: Generalized Fibonacci formulas are proposed as a universal scaling model for quantum oscillators, with transmission coefficients scaling as φ⁻², while standard QHO theory (Hermite-Gaussian eigenfunctions) shows no inherent Fibonacci structure. | MATH: Fibonacci recurrence Fₙ₊₁ = Fₙ + Fₙ₋₁; golden ratio φ = (1+√5)/2 ≈ 1.6180339887; φ⁻² = 2−φ ≈ 0.381966; QHO energy Eₙ = ħω(n+½); transmission coefficient T(E) = 1/(1 + (V₀²/4E(V₀−E)) sinh²(κa)) for rectangular barrier — no φ appears in standard QHO. | CONNECTION: φ⁻² = 0.381966 ≈ 0.382 (the complementary golden ratio); φ⁻¹ = 0.6180339; φ⁻² = 1 − φ⁻¹; these are the two fundamental ratios of the golden section. If a quantum transmission coefficient scales as φ⁻², it directly ties barrier penetration to the golden ratio's self-similar structure — a geometric harmony between exponential decay (tunneling) and logarithmic spiral growth. | DEPTH: 6 — The claim is intriguing but the evidence is thin: the YouTube sources are non-peer-revie Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: The golden ratio φ emerges from quantum self-observation (fixed-point tunneling) and appears anomalously in a quantum system's spectrum — not from classical geometry but from the operator structure of self-reference. | MATH: φ = (1+√5)/2 ≈ 1.6180339887; φ⁻¹ = φ−1 ≈ 0.6180339887; φ² = φ+1 ≈ 2.6180339887. In the ODTOE derivation, φ arises as the fixed point of a self-application operator: if T(f) = 1 + 1/f, then T(φ) = φ. In quantum tunneling, the amplitude for a particle to tunnel through a barrier in a self-referential potential (V(x) ∝ x² − φ) yields a transmission coefficient T(E) whose poles occur at E_n ∝ φ^n, giving ratios E_{n+1}/E_n → φ. | CONNECTION: φ is the unique positive solution to x² − x − 1 = 0, which is the characteristic equation of the Fibonacci recurrence F_{n+1} = F_n + F_{n−1}. This is the same ratio that appears in the golden angle 137.507° (2π/φ² ≈ 2.39996 rad), which governs phyllotaxis and is linked to the 5-fold crystallographic symmetry (icosahedral/ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: The arXiv paper (2410.04169v2) introduces a "golden oscillator" — a quantum harmonic oscillator whose energy spectrum is quantized via Fibonacci divisors and quantum calculus with bases φ (golden ratio) and ψ (silver ratio), yielding a supersymmetric hierarchy of fermion-boson entangled states. The YouTube results are standard QHO pedagogy (E_n = (n+½)ħω) and add no novel mathematics. MATH: - Standard QHO: E_n = ħω(n + ½), n ∈ ℕ₀. Partition function Z = Σ exp(−βE_n) = 1/[2 sinh(βħω/2)]. - Golden oscillator spectrum (from abstract): E_n ∝ [n]_φ = (φ^n − (−φ)^−n)/(φ − (−φ)^−1) = (φ^n − (−φ)^−n)/√5 — the Fibonacci number F_n. - Quantum calculus: [n]_q = (1 − q^n)/(1 − q). For q = φ², [n]_{φ²} = (φ^{2n} − 1)/(φ² − 1) = F_{2n}/φ (using φ² = φ+1, φ²−1 = φ). - Silver ratio: ψ = 1+√2 ≈ 2.414; [n]_{ψ²} = (ψ^{2n} − 1)/(ψ² − 1) = Pell numbers / (2ψ). - Binet formula for Fibonacci divisor number operator: N_φ = (φ^N − (−φ)^−N)/√5, acting on Fock space. - Supersymmetric hierar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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≈e−2κLT \approx e^{-2\kappa L}T≈e−2κL, κ=2m(V0−E)/ℏ\kappa = \sqrt{2m(V_0-E)}/\hbarκ=2m(V0​−E)​/ℏ. Golden ratio: ϕ=(1+5)/2≈1.618\phi = (1+\sqrt{5})/2 \approx 1.618ϕ=(1+5​)/2≈1.618, ϕ−2=(3−5)/2≈0.382\phi^{-2} = (3-\sqrt{5})/2 \approx 0.382ϕ−2=(3−5​)/2≈0.382. No equation in the sources equates TTT to ϕ−2\phi^{-2}ϕ−2. | CONNECTION: The arXiv paper (2510.08934) uses Φ\PhiΦ as a model of stable local recurrence — operationally, Φ−1=ϕ−1≈0.618\Phi^{-1} = \phi-1 \approx 0.618Φ−1=ϕ−1≈0.618, and Φ−2≈0.382\Phi^{-2} \approx 0.382Φ−2≈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
FINDING: Fibonacci anyons have quantum dimension equal to the golden ratio φ, enabling universal topological quantum computation via non-Abelian braiding. | MATH: Quantum dimension d=ϕ=1+52≈1.618 d = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618 d=ϕ=21+5​​≈1.618; Fibonacci anyon fusion rules: τ×τ=1+τ \tau \times \tau = 1 + \tau τ×τ=1+τ; modular tensor category (MTC) classification yields Fibonacci category as rank-2 MTC with S S S-matrix S=1ϕ+2(1ϕϕ−1) S = \frac{1}{\sqrt{\phi+2}} \begin{pmatrix} 1 & \phi \\ \phi & -1 \end{pmatrix} S=ϕ+2​1​(1ϕ​ϕ−1​); topological twist θτ=e4πi/5 \theta_\tau = e^{4\pi i/5} θτ​=e4πi/5. | CONNECTION: Golden ratio φ appears as quantum dimension; related ratios: ϕ−1=0.618 \phi^{-1} = 0.618 ϕ−1=0.618, ϕ2=2.618 \phi^2 = 2.618 ϕ2=2.618; fusion rules mirror Fibonacci recurrence; Binet's formula Fn=(ϕn−(−ϕ)−n)/5 F_n = (\phi^n - (-\phi)^{-n})/\sqrt{5} Fn​=(ϕn−(−ϕ)−n)/5​ links anyon counting to Fibonacci numbers; no direct base-60 or crystallographic symmetry, but φ appears in Penrose tilings (5-fold symmetry) and icosahedral quasicrystals. | DEPTH: 9 — Direct embedding of golden ratio into quantum Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Fibonacci anyons carry quantum dimension φ = (1+√5)/2 ≈ 1.618, enabling universal topological quantum computation via braiding — now experimentally realized in non-Abelian anyon systems (Quantinuum, 2025). | MATH: Quantum dimension d = φ satisfies d² = d + 1 (golden ratio identity). Braid group Bₙ on Fibonacci anyons yields dense unitary representations on the fusion space — universality proven (Freedman–Kitaev–Wang). Single-qubit braiding gates compiled via algebraic decomposition of SU(2) into Fibonacci braid words (arXiv:2008.03542). | CONNECTION: φ = 1.618 = 2cos(π/5) — directly tied to pentagonal symmetry, the root system of the icosahedral group H₃, and the 5-fold crystallographic (quasicrystal) forbidden symmetry. The Fibonacci fusion rules (τ × τ = 1 + τ) mirror the golden ratio recurrence Fₙ₊₁ = Fₙ + Fₙ₋₁. | DEPTH: 9 — This is not numerology; φ emerges as the unique algebraic solution to the consistency equations (pentagon/hexagon identities) for a non-Abelian anyon m Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum probability interference terms acquire golden-ratio phase angles in N=2 supersymmetric "golden oscillator" hierarchies, linking Fibonacci divisor calculus to fermion-boson entanglement. | MATH: φ = (1+√5)/2 ≈ 1.618; silver ratio δ_S = 1+√2 ≈ 2.414; Binet form F_n = (φⁿ − ψⁿ)/√5, ψ = −1/φ ≈ −0.618; two-base quantum calculus q = φ, q' = δ_S; energy spectrum of golden oscillators E_n ∝ φⁿ; Fibonacci divisor number operator acting on Fock space. | CONNECTION: φ and its conjugate 1/φ = 0.618 appear as dual bases; 0.618 = 1 − 0.382; φ² = 2.618 = φ + 1; the golden oscillator ladder spacing is geometric with ratio φ, matching the continued-fraction fixed point φ = 1 + 1/(1 + 1/(1+…)) — the "most irrational" number, which suppresses resonant decoherence in quasi-periodic systems (Aubry-André / Fibonacci lattice). | DEPTH: 7 FINDING: Golden ratio observed empirically inside a quantum system where standard theory did not predict it (Meridian Labs report). | MATH: No explicit equ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com