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 Annealing Bridges Collatz Conjecture and Zermelo Navigation — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 25, 2026• 0 citations
Publisher

Abstract

FINDING: Collatz Conjecture remains the canonical "simple-to-state, impossible-to-prove" problem; Zermelo's navigation problem shows quantum annealing can yield efficient classical solutions via Landau-Zener transitions. | MATH: Collatz map: T(n) = n/2 if n even, 3n+1 if n odd; no closed-form invariant known. Zermelo: optimal control Hamiltonian H(x,p,t) = p·(v_flow + u_control) + λ·(constraints); Landau-Zener transition probability P = exp(-πΔ²/(2ℏ|dE/dt|)). | CONNECTION: Collatz dynamics exhibit a logarithmic spiral-like growth/decay ratio near 3/2 vs 1/2 — the ratio of odd-step multiplier (3) to even-step divisor (2) is 1.5, not a golden ratio, but the orbit's average contraction rate (3/4)^k for k odd steps suggests a hidden scaling near 0.75, not 0.618. Zermelo's optimal paths in uniform flow are cycloids — related to brachistochrone, whose curvature ratio involves π, not φ. No direct 0.382/0.618/0.786/1.618/2.618 or base-60 or crystallographic symmetry appears in either. | DEPTH: Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.22090611

Related papers

Quantum Annealing Solves Zermelo Navigation; Collatz Conjecture Remains Open — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Collatz Conjecture remains unsolved; Zermelo navigation problem yields efficient classical solution via quantum annealing and Landau-Zener approximation. | MATH: Collatz: f(n) = n/2 if n even, 3n+1 if n odd; Zermelo: optimal trajectory in flow field, Landau-Zener transition probability ~ exp(-π Δ²/(2ℏ|dE/dt|)). | CONNECTION: No direct geometric ratios or crystallographic symmetries found in these results. Collatz involves modular arithmetic and 2-adic dynamics; Zermelo solution uses adiabatic quantum optimization, not harmonic ratios. | DEPTH: 3 — Collatz is a famous unsolved problem but yields no new constants or geometric structure; Zermelo solution is algorithmic, not a fundamental mathematical discovery. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Aug 6, 2026•0 citations

Collatz Conjecture and Quantum Annealing: A Unified View — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The Collatz Conjecture (3n+1 problem) remains the canonical "simple-to-state, impossible-to-prove" problem; Zermelo's navigation problem shows a quantum-annealing path to classical efficiency via Landau-Zener transitions. | MATH: Collatz map: \( T(n) = \begin{cases} n/2 & \text{if } n \equiv 0 \pmod{2} \\ 3n+1 & \text{if } n \equiv 1 \pmod{2} \end{cases} \). No closed-form solution; known to be equivalent to a dynamical system on the 2-adic integers. Zermelo: optimal control Hamiltonian \( H(x,p,t) = \min_u \{ \langle p, f(x,u) \rangle + L(x,u) \} \); Landau-Zener formula for transition probability \( P = \exp(-\pi \Delta^2 / (2\hbar |\dot{\epsilon}|)) \). | CONNECTION: Collatz dynamics exhibit a logarithmic growth ratio ~ \( \log_2(3) \approx 1.585 \) (not a golden ratio, but a base-2/3 irrational exponent). The 2-adic structure is a lattice of dyadic integers — a crystallographic-like symmetry under \( \mathbb{Z}_2 \). Zermelo's solution via quantum annealing exploits avoide Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Sep 2, 2026•0 citations

Collatz Unsolved; Quantum Annealing Solves Zermelo Navigation Problem — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Collatz Conjecture remains unsolved; Zermelo navigation problem gains quantum annealing solution. | MATH: Collatz: f(n) = n/2 if n even, 3n+1 if n odd; Zermelo: optimal trajectory in flow field, Landau-Zener approximation yields classical solution. | CONNECTION: No direct geometric harmony ratios or base-60 links in these findings. Collatz involves simple integer iteration; Zermelo solution is algorithmic, not harmonic. | DEPTH: 2 — No new equations, constants, or geometric patterns extracted. The Zermelo paper offers a computational method, not a mathematical constant or symmetry. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Aug 12, 2026•0 citations

Quantum Annealing Tackles Zermelo Navigation, Not Collatz — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The search results are a mix of popular expositions on famous unsolved problems (Collatz, etc.) and one specific arXiv paper on Zermelo's navigation problem solved via quantum annealing; no new "impossible" problem was actually solved in the provided evidence. | MATH: Collatz map: \( T(n) = n/2 \) if \( n \) even, \( T(n) = 3n+1 \) if \( n \) odd; no closed-form solution or proof of termination for all \( n \). Zermelo navigation: minimize \( \int_0^T \sqrt{\dot{x}^2 + \dot{y}^2} \, dt \) subject to \( \dot{\mathbf{x}} = \mathbf{v}_{\text{flow}} + \mathbf{u}_{\text{boat}} \), \( |\mathbf{u}| \leq 1 \); Landau–Zener transition probability \( P = \exp(-\pi \Delta^2 / (2\hbar \dot{\epsilon})) \) used to map to quantum annealing. | CONNECTION: No direct geometric harmony ratios (0.382, 0.618, 0.786, 1.618, 2.618) appear in the evidence. However, the Collatz map's structure is a 2-adic dynamical system — its orbits are tied to the 2-adic integers, which are a lattice-like structure Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Oct 11, 2026•0 citations

Quantum Annealing Solves Zermelo Navigation via Landau-Zener Approximation — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum annealing reveals efficient classical solution to Zermelo's navigation problem via Landau-Zener approximation | MATH: Landau-Zener transition probability \( P = 1 - \exp(-\pi \Delta^2 / (2 \hbar v)) \), where \(\Delta\) is minimum gap, \(v\) is sweep rate; optimal trajectory reduces to geodesic in flow field | CONNECTION: No explicit geometric ratios or crystallographic symmetries; problem involves continuous optimization in vector fields, not discrete harmonic ratios | DEPTH: 6 — Bridges quantum optimization and classical control, but lacks direct link to fundamental constants or sacred geometry. FINDING: Mathematicians' ethical engagement framework (four levels) | MATH: No equations or constants; qualitative taxonomy | CONNECTION: None | DEPTH: 1 — Philosophical, not mathematical. FINDING: LMMs trained to overcome false premises via "See, Say, and Segment" | MATH: No equations; empirical fine-tuning | CONNECTION: None | DEPTH: 1 — Applied AI, no mathematical essenc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Aug 19, 2026•0 citations

Optimal CHSH Violation at 45° Tied to Crystallographic Rotation Restrictions — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: CHSH inequality optimal violation occurs at 45° measurement angle, linked to crystallographic rotation symmetry restrictions. MATH: CHSH inequality: \( S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2 \) (classical bound). Quantum optimal violation: \( S = 2\sqrt{2} \approx 2.828 \) at angle \( \theta = 45^\circ \) (π/4 rad). Crystallographic restriction theorem: allowed rotation orders \( n = 1,2,3,4,6 \) (angles 360°/n). 45° corresponds to 8-fold symmetry (forbidden in periodic crystals). CONNECTION: 45° = π/4 = 0.7854 rad ≈ 0.786 (geometric harmony constant). Ratio 0.786 appears in golden ratio-based systems (e.g., 1/φ ≈ 0.618, φ² ≈ 2.618, 0.786 = √(φ)/2?). 45° is half of 90°, relates to square lattice symmetry (4-fold) but not to 8-fold. The CHSH optimal angle aligns with a non-crystallographic rotation, hinting at quantum geometry beyond classical lattice constraints. DEPTH: 8 — Direct link between quantum nonlocality (2√2) and forbidden crystallographic symmetry (8 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Aug 1, 2026•0 citations
FINDING: Collatz Conjecture remains the canonical "simple-to-state, impossible-to-prove" problem; Zermelo's navigation problem shows quantum annealing can yield efficient classical solutions via Landau-Zener transitions. | MATH: Collatz map: T(n) = n/2 if n even, 3n+1 if n odd; no closed-form invariant known. Zermelo: optimal control Hamiltonian H(x,p,t) = p·(v_flow + u_control) + λ·(constraints); Landau-Zener transition probability P = exp(-πΔ²/(2ℏ|dE/dt|)). | CONNECTION: Collatz dynamics exhibit a logarithmic spiral-like growth/decay ratio near 3/2 vs 1/2 — the ratio of odd-step multiplier (3) to even-step divisor (2) is 1.5, not a golden ratio, but the orbit's average contraction rate (3/4)^k for k odd steps suggests a hidden scaling near 0.75, not 0.618. Zermelo's optimal paths in uniform flow are cycloids — related to brachistochrone, whose curvature ratio involves π, not φ. No direct 0.382/0.618/0.786/1.618/2.618 or base-60 or crystallographic symmetry appears in either. | DEPTH: Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Collatz Conjecture remains unsolved; Zermelo navigation problem yields efficient classical solution via quantum annealing and Landau-Zener approximation. | MATH: Collatz: f(n) = n/2 if n even, 3n+1 if n odd; Zermelo: optimal trajectory in flow field, Landau-Zener transition probability ~ exp(-π Δ²/(2ℏ|dE/dt|)). | CONNECTION: No direct geometric ratios or crystallographic symmetries found in these results. Collatz involves modular arithmetic and 2-adic dynamics; Zermelo solution uses adiabatic quantum optimization, not harmonic ratios. | DEPTH: 3 — Collatz is a famous unsolved problem but yields no new constants or geometric structure; Zermelo solution is algorithmic, not a fundamental mathematical discovery. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: The Collatz Conjecture (3n+1 problem) remains the canonical "simple-to-state, impossible-to-prove" problem; Zermelo's navigation problem shows a quantum-annealing path to classical efficiency via Landau-Zener transitions. | MATH: Collatz map: T(n)={n/2if n≡0(mod2)3n+1if n≡1(mod2) T(n) = \begin{cases} n/2 & \text{if } n \equiv 0 \pmod{2} \\ 3n+1 & \text{if } n \equiv 1 \pmod{2} \end{cases} T(n)={n/23n+1​if n≡0(mod2)if n≡1(mod2)​. No closed-form solution; known to be equivalent to a dynamical system on the 2-adic integers. Zermelo: optimal control Hamiltonian H(x,p,t)=min⁡u{⟨p,f(x,u)⟩+L(x,u)} H(x,p,t) = \min_u \{ \langle p, f(x,u) \rangle + L(x,u) \} H(x,p,t)=minu​{⟨p,f(x,u)⟩+L(x,u)}; Landau-Zener formula for transition probability P=exp⁡(−πΔ2/(2ℏ∣ϵ˙∣)) P = \exp(-\pi \Delta^2 / (2\hbar |\dot{\epsilon}|)) P=exp(−πΔ2/(2ℏ∣ϵ˙∣)). | CONNECTION: Collatz dynamics exhibit a logarithmic growth ratio ~ log⁡2(3)≈1.585 \log_2(3) \approx 1.585 log2​(3)≈1.585 (not a golden ratio, but a base-2/3 irrational exponent). The 2-adic structure is a lattice of dyadic integers — a crystallographic-like symmetry under Z2 \mathbb{Z}_2 Z2​. Zermelo's solution via quantum annealing exploits avoide Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Collatz Conjecture remains unsolved; Zermelo navigation problem gains quantum annealing solution. | MATH: Collatz: f(n) = n/2 if n even, 3n+1 if n odd; Zermelo: optimal trajectory in flow field, Landau-Zener approximation yields classical solution. | CONNECTION: No direct geometric harmony ratios or base-60 links in these findings. Collatz involves simple integer iteration; Zermelo solution is algorithmic, not harmonic. | DEPTH: 2 — No new equations, constants, or geometric patterns extracted. The Zermelo paper offers a computational method, not a mathematical constant or symmetry. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: The search results are a mix of popular expositions on famous unsolved problems (Collatz, etc.) and one specific arXiv paper on Zermelo's navigation problem solved via quantum annealing; no new "impossible" problem was actually solved in the provided evidence. | MATH: Collatz map: T(n)=n/2 T(n) = n/2 T(n)=n/2 if n n n even, T(n)=3n+1 T(n) = 3n+1 T(n)=3n+1 if n n n odd; no closed-form solution or proof of termination for all n n n. Zermelo navigation: minimize ∫0Tx˙2+y˙2 dt \int_0^T \sqrt{\dot{x}^2 + \dot{y}^2} \, dt ∫0T​x˙2+y˙​2​dt subject to x˙=vflow+uboat \dot{\mathbf{x}} = \mathbf{v}_{\text{flow}} + \mathbf{u}_{\text{boat}} x˙=vflow​+uboat​, ∣u∣≤1 |\mathbf{u}| \leq 1 ∣u∣≤1; Landau–Zener transition probability P=exp⁡(−πΔ2/(2ℏϵ˙)) P = \exp(-\pi \Delta^2 / (2\hbar \dot{\epsilon})) P=exp(−πΔ2/(2ℏϵ˙)) used to map to quantum annealing. | CONNECTION: No direct geometric harmony ratios (0.382, 0.618, 0.786, 1.618, 2.618) appear in the evidence. However, the Collatz map's structure is a 2-adic dynamical system — its orbits are tied to the 2-adic integers, which are a lattice-like structure Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum annealing reveals efficient classical solution to Zermelo's navigation problem via Landau-Zener approximation | MATH: Landau-Zener transition probability P=1−exp⁡(−πΔ2/(2ℏv)) P = 1 - \exp(-\pi \Delta^2 / (2 \hbar v)) P=1−exp(−πΔ2/(2ℏv)), where Δ\DeltaΔ is minimum gap, vvv is sweep rate; optimal trajectory reduces to geodesic in flow field | CONNECTION: No explicit geometric ratios or crystallographic symmetries; problem involves continuous optimization in vector fields, not discrete harmonic ratios | DEPTH: 6 — Bridges quantum optimization and classical control, but lacks direct link to fundamental constants or sacred geometry. FINDING: Mathematicians' ethical engagement framework (four levels) | MATH: No equations or constants; qualitative taxonomy | CONNECTION: None | DEPTH: 1 — Philosophical, not mathematical. FINDING: LMMs trained to overcome false premises via "See, Say, and Segment" | MATH: No equations; empirical fine-tuning | CONNECTION: None | DEPTH: 1 — Applied AI, no mathematical essenc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: CHSH inequality optimal violation occurs at 45° measurement angle, linked to crystallographic rotation symmetry restrictions. MATH: CHSH inequality: S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2 S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2 S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2 (classical bound). Quantum optimal violation: S=22≈2.828 S = 2\sqrt{2} \approx 2.828 S=22​≈2.828 at angle θ=45∘ \theta = 45^\circ θ=45∘ (π/4 rad). Crystallographic restriction theorem: allowed rotation orders n=1,2,3,4,6 n = 1,2,3,4,6 n=1,2,3,4,6 (angles 360°/n). 45° corresponds to 8-fold symmetry (forbidden in periodic crystals). CONNECTION: 45° = π/4 = 0.7854 rad ≈ 0.786 (geometric harmony constant). Ratio 0.786 appears in golden ratio-based systems (e.g., 1/φ ≈ 0.618, φ² ≈ 2.618, 0.786 = √(φ)/2?). 45° is half of 90°, relates to square lattice symmetry (4-fold) but not to 8-fold. The CHSH optimal angle aligns with a non-crystallographic rotation, hinting at quantum geometry beyond classical lattice constraints. DEPTH: 8 — Direct link between quantum nonlocality (2√2) and forbidden crystallographic symmetry (8 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com