A proof of B. R. Smith's conjecture on Gauss composition and the kneading cycles of discriminant a^2-4

In 2014 B. R. Smith (J. Integer Seq. 17, Article 14.11.8; arXiv:1408.4631) showed that kneading, an operation on finite sequences of positive integers, is the Zagier reduction of indefinite binary quadratic forms in disguise, so that the sum of a sequence is an invariant of the proper class of the corresponding form. For the discriminants D = a^2 − 4 (sequences of odd length) he singled out the class c of the form (a−2)x^2 + (3a−6)xy + (2a−5)y^2, which has order at most two, and conjectured (Conjecture 9 of the published version, Conjecture 3 of the arXiv version) that whenever two primitive classes c1, c2 of discriminant a^2 − 4 compose to c, they have the same sum n, with n ≤ a, and the lengths l1, l2 of their cycles of reduced forms satisfy l1 + l2 = n − 1. Smith checked the statement for the pair formed by the principal class and c and for an infinite family of classes whose square is c. We prove this conjecture. The proof has three ingredients. First, one turn around the cycle of a primitive Zagier-reduced form f = (A, B, C) of discriminant a^2 − 4 is exactly the unit (a + √D)/2: the matrix of this unit equals the product of the reduction steps of the cycle, and the minus (backward) continued fraction of Smith's rational number (a + B)/(2A) is exactly one period of the cycle. This gives the sum of Smith's sequence as 1 + Σ(n_i − 1), where the n_i are the reducing numbers of the cycle. Second, the partner class c·c1^(−1) is, up to inversion, the class dual to c1 in the sense of Hirzebruch: writing the cycle of reducing numbers in blocks (k_i + 3, 2^(j_i)), the class c·c1 has the cycle (j_i + 3, 2^(k_(i+1))), so its length is Σ(n_i − 2); a self-contained proof is included, with the identification of the dual class as c·c1 obtained from an explicit Möbius transformation of determinant −1 and Dirichlet's composition of united forms. Third, the bound n ≤ a is an inequality between the sum of a regular period and the trace of the corresponding product of continuant matrices; equality holds exactly for the principal class and for c. The note was obtained with the author's methodology for open statements in the OEIS and in the papers attached to them (subprocesses with time budgets, a log of obstacles with an algorithm added per obstacle, scripts that test the variants of a formula, literature search, compiled code for the blocking computations, independent verification and review by a second AI system), to which the author added for this problem a soak test: before attempting a proof, the numerical check is extended in range rather than repeated, it is run against negative controls (wrong variants of the statement, which must fail), and the heavy part is rewritten from scratch in compiled code so that two independent implementations are compared. The work was done with the assistance of an AI system under the author's direction; the methodology section of the note describes the process. The verification programs are included: conj9.py and conj9_lemas.py (Python, no external libraries: Zagier-reduced forms, cycles, Smith's sequences, Gauss composition, and a check of every lemma of the proof and of the conjecture, with the negative controls, for a ≤ 200), and an independent Rust program (smith9check.zip), written with a different enumeration of reduced forms and a different composition algorithm, which checks the conjecture and the lemmas for 3 ≤ a ≤ 4000 (767,607 primitive classes, no failure), the negative controls (all of which fail as they should), and the correspondence between kneading and Zagier reduction for all 33,554,406 sequences of odd length with alternant at least 3 and sum at most 26. Both PDFs (English and Spanish) are the same note.