Liouville integrability of the two-parameter deformed A_{2N} cluster maps for every N

For the two-parameter deformed A_{2N} cluster map, first integrals were known only for N ≤ 3, the case N ≥ 4 being open with only zero algebraic entropy available as evidence. We prove that the map is Liouville integrable for every N. First, the N = 4 (eight-dimensional) map: four explicit first integrals, proved invariant, functionally independent and in involution with respect to the invariant log-canonical Poisson bracket, all by exact polynomial identities over ℤ[a₁^{±1}, a₈^{±1}, x^{±1}]. Second, two integrals H₁, H₂ in closed form for every N, proved invariant and Poisson-commuting, together with the proof that the map preserves the bracket; the proofs rest on one observation: the map is exactly the row shift of a unimodular chain — a Coxeter-type frieze of width 2N with constant boundary entries a₁ and a_{2N} — whose quiddity the shift merely permutes. Third, and this settles the family, a generating function of all N integrals for every N: in the reduced coordinates y_i = x_{i+1}/x_{i−1}, in which the map becomes a continuant recursion, I_N(λ) = α(y₁)TE(y₂)TO(y₃)⋯TE(y_{2N})β is a transfer-matrix product with a four-dimensional state space. Its invariance is proved by an explicit discrete zero-curvature (gauge) representation, the involution of its coefficients by a finite closure certificate, and their independence by a degeneration with an explicit leading term; the λ¹-coefficient is (b/a)·H₁ and a² times the value at λ = 1 is H₂. All identities are exact. Prepared with the help of an AI assistant (Claude Fable 5.1, Anthropic), which assisted in structuring the arguments, drafting the text and carrying out the exact symbolic and computer verifications; the mathematical content and the responsibility for it rest with the author. Source (Markdown) and PDF are both included.