The Universal Boundary Mechanism

A separate French PDF is provided. We establish λ = 2 as a unique algebraic boundary in the family of order-2 linearr ecurrences xn = xn−1 + c · xn−2. This boundary is characterised by the condition that thed ominant characteristic root equals the coecient (r1 = c = 2), a property that fails for all other positive integers. The framework identies canonical identities D = r1 − r2 = 3 (discriminant) and S = r 2/1 + r 2/2 = 5 (sum of squares) that predict cosmological ratios: ΩDM/Ωb = D3/S = 27/5 = 5.40. The theory's central insight is that the Universal Boundary represents an irreducible gap approximately 0.11%between the integer-rational structures of discrete growth and the irrational requirements of geometric closure. This gap, arising from the impossibility of equating 3/2 with 2 7/12, is not a aw but the fundamental engine of structural stability across physics, biology, and acoustics. Crucially, the October 2024 paper by Banerjee, Brzeminski, and Hook (arXiv:2410.22412) independently derives that ΩDM/Ωb emerges as a ratio of beta function coecients in composite axion models, with the formula 2N/(27 − 3N) for SU(N) gauge groups where the number 27 appears as 3 × β3, the QCD beta function coecient. This independent convergence of two frameworks on the same integer (27) governing dark matter abundance is noteworthy, though whether it reects deep connection or numerical coincidence remains an open question. Additionally, Ming's 1989 theorem on triangular Fibonacci numbers {1, 3, 21, 55} validates three of our ve bridge values, and Alekseyev's 2010 framework confirms such sequence intersections are finite.