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

Andrew Stewart Caldin

Published Aug 19, 2026• 0 citations
Publisher

Abstract

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

DOI
10.5281/zenodo.22006343

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

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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

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Andrew Stewart Caldin

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

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Zermelo's navigation problem through the lens of quantum annealing: How the Landau-Zener approximation leads to an efficient classical solution

Sølve Selstø, Tor Kristian Dahle, Sergiy Denysov +2 more

The river-crossing problem, also known as Zermelo's navigation problem, is a classic example of an optimization problem with practical relevance and a scalable degree of complexity. It asks for the optimal trajectory of a vessel moving through a water flow field and provides a setting in which physics, variational methods, and optimization are naturally intertwined. We state a version of Zermelo's problem and then solve it as formulate it as an adiabatic quantum-computing problem using quantum trits, or qutrits for short. The construction includes a penalty term that enforces the prescribed boundary conditions and an exploration term that allows the system to move through intermediate configurations toward the optimal feasible path. In the adiabatic description, the evolution proceeds through a sequence of avoided crossings, so that the resulting fidelity can be estimated using the Landau-Zener formula. Remarkably, the regime in which this approximation is valid also provides a deterministic way to identify the correct solution with computational effort that scales only quadratically with the problem size. Thus, a quantum formulation initially motivated by the apparent exponential complexity of the problem reveals an underlying classical structure that can be exploited efficiently. Our approach also provides a pedagogical illustration of how a real-world optimization problem can be cast into a quantum-annealing framework and then analyzed using the Schrödinger equation, avoided crossings, and Landau-Zener theory.

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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

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
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
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: 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: 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
The river-crossing problem, also known as Zermelo's navigation problem, is a classic example of an optimization problem with practical relevance and a scalable degree of complexity. It asks for the optimal trajectory of a vessel moving through a water flow field and provides a setting in which physics, variational methods, and optimization are naturally intertwined. We state a version of Zermelo's problem and then solve it as formulate it as an adiabatic quantum-computing problem using quantum trits, or qutrits for short. The construction includes a penalty term that enforces the prescribed boundary conditions and an exploration term that allows the system to move through intermediate configurations toward the optimal feasible path. In the adiabatic description, the evolution proceeds through a sequence of avoided crossings, so that the resulting fidelity can be estimated using the Landau-Zener formula. Remarkably, the regime in which this approximation is valid also provides a deterministic way to identify the correct solution with computational effort that scales only quadratically with the problem size. Thus, a quantum formulation initially motivated by the apparent exponential complexity of the problem reveals an underlying classical structure that can be exploited efficiently. Our approach also provides a pedagogical illustration of how a real-world optimization problem can be cast into a quantum-annealing framework and then analyzed using the Schrödinger equation, avoided crossings, and Landau-Zener theory.
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: 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