Metadata
bibkey: joshi2025sturmian
authors: Gandhar Joshi
year: 2025
title: Automatic sequences from algebraic, dynamical, and computational perspectives
doi: 10.21954/ou.ro.00106631
url: https://oro.open.ac.uk/106631/
claim: Section 6.1 asks whether d ≤ qₙ implies A(d) ≤ A(qₙ) for every positive n and every Sturmian slope, without the hypothesis g(d) ≤ α of Proposition 4.4.4.
strata_touched:
- D5/S1/Words/SturmianMapCumulativeRefutation
license: citation-only
triage: anchor
Verified locator
DOI: https://doi.org/10.21954/ou.ro.00106631
Source: https://oro.open.ac.uk/106631/
The Open University PhD thesis, §6.1, p. 118; Definitions 2.2.3 (p. 12) and 2.6.1–2.6.2 (pp. 27–28), and Proposition 4.4.4. The source transcription is recorded in https://github.com/the-omega-institute/trureturing/issues/17837. The preregistration’s PDF reading uses an Internet Archive capture because the publisher page presents a bot challenge.
Source statement
Section 6.1, p. 118, “Stronger cumulative bound on A(d).”
“The data generated by the A(d) formulae in Corollary 4.3.13 heuristically directs to a conjecture that for every n ∈ N, d ≤ qₙ ⟹ A(d) ≤ A(qₙ) (Proposition 4.4.4) could be true for all g(d). As done for the specific case in [32, Corollary 4.17], this possibility for all Sturmians will be assessed in the author’s future work.”
Proposition 4.4.4: “Let n ∈ N. For all d ≤ qₙ, if g(d) ≤ α, then A(d) ≤ A(qₙ).” Chapter 4 takes irrational 0 < α < 1/2. The denominators satisfy q₋₁ = 0, q₀ = 1 and qₙ = aₙqₙ₋₁ + qₙ₋₂. N denotes positive integers. Definition 4.3.2 sets g(d) := min{⟨dα⟩, 1 − ⟨dα⟩}.
Objects
Definition 2.2.3, p. 12: “Let α ∈ [0, 1) be an irrational number. Let the interval [0, 1) be partitioned into I_a := [0, 1 − α) and I_b := [1 − α, 1). The n-th term of a Sturmian sequence s = (sₙ)ₙ≥0 is defined by sₙ = a if ⟨(n + 1)α⟩ ∈ I_a, b if ⟨(n + 1)α⟩ ∈ I_b.”
Definition 2.6.1, p. 27: “When a symbol a in a sequence x repeats ℓ times with a difference d, we call it a monochromatic arithmetic progression of difference d and length ℓ. That is, for some i ≥ 0, x_i = x_{i+d} = ⋯ = x_{i+(ℓ−1)d} = a.”
Definition 2.6.2, pp. 27–28: “A_x(d) = max{ℓ | there is a MAP of difference d and length ℓ in x}” denotes the longest such length. “Here, we have A_x(d) = ∞ if no finite maximum exists.” The formal maximum is the supremum in ℕ∞ of all positive finite lengths, over all nonnegative starts and both letters. The existing lower mechanical word has slope and intercept α, with true representing b.
Result and boundary
The repository refutes the unrestricted comparison using α = 2 − √3, n = 1, q₁ = 3 and d = 2. The proof establishes A(2) ≥ 5 and A(3) ≤ 4 without assuming density or the thesis’s maximum-length formulas. It does not prove the optional equalities A(2) = 8 and A(3) = 4. The retained-hypothesis proposition and the Fibonacci special case are outside this refutation.
The literature screen in issue #17837 and the probe inspects the thesis, Joshi–Rust arXiv:2501.05830v2 and the author’s later arXiv papers 2506.13337v1, 2507.16956v2 and 2604.02404v2. No prior settlement was located there. Completeness is ASSUMED-UNVERIFIED: rate limits and unavailable search results leave citation-graph and non-arXiv coverage incomplete.