A proof of a conjecture of R. Ford: semi-meanders with two mountain ranges and Euler's totient function

Abstract

We prove a conjecture of R. Ford stated in 2017 in the OEIS entry A000010 (Euler's totient function): for n > 1, the number of semi-meander solutions for n whose top arches contain exactly 2 mountain ranges and exactly 2 arches of length 1 equals phi(n). In Ford's setting, the points 1, ..., 2n lie on a line, the bottom arches form the rainbow of n nested arches, and a semi-meander solution is a system of n noncrossing top arches that, together with the bottom rainbow, forms a single closed curve (these are counted by the semi-meander numbers, OEIS A000682). A mountain range is an exterior top arch together with the arches below it (Ford's "top arch groupings" in A259689), and an arch of length 1 joins two consecutive points. The proof has two steps. First, a top arch system with exactly two mountain ranges and exactly two arches of length 1 is the concatenation of two rainbows, of sizes a and b = n - a, because every mountain range contains an arch of length 1 and a mountain range with only one such arch is a rainbow. Second, two rainbows of sizes a and b over the bottom rainbow form exactly gcd(a, b) closed curves: following a curve along one bottom arch and then one top arch is the rotation i -> i + 2a of Z/2nZ, and each curve consists of exactly two orbits of this rotation. This count is a known fact about rainbow meanders (Fiedler and Castañeda, 2012; Karnauhova and Liebscher, 2017); the note gives a self-contained proof. Hence the solutions correspond to the a in {1, ..., n-1} coprime to n, and there are phi(n) of them. We also observe that, for n >= 2, every semi-meander solution with exactly two arches of length 1 has exactly two mountain ranges, so the statement does not depend on the reading of "mountain range". The proof is elementary. Its interest lies in the exact count, which singles out a family of semi-meanders counted by a classical arithmetic function, while no closed formula is known for the semi-meander numbers. As of October 5, 2026 the statement was still marked as conjectural in the OEIS entry A000010, and we have not found a proof of it in the literature. The note also explains the methodology of the author's project on open statements in the OEIS (translation of the statements into formulas for the reasoning engine SyntheticMind, with time budgets by type of computation; a log of obstacles that decides which methods to implement next; recursive splitting into subgoals; independent verification; a review by a second AI system) and the steps that led to this proof. The use of AI is described in the paper. Files: Blanco_Gomez_2026_Ford_semimeanders_EN.pdf is the paper; Blanco_Gomez_2026_Ford_semimeanders_ES.pdf is the Spanish version (same content); verify_ford.py is the independent verification script (Python, no external libraries; it enumerates arch systems and traces the closed curves directly, reproduces the semi-meander numbers A000682 and Ford's triangle A259689 for n <= 12, and checks the structural lemma, the gcd count for n <= 200 and the theorem for n <= 300; it runs in about fifteen seconds and prints ALL CHECKS PASSED).