From Right Triangles to Fibonacci-Type Spirals via the Hypotenuse-Axis Intercept (HAI): Seed Ratios, Möbius Maps, and the Golden 2:1 Triangle

The Hypotenuse-Axis Intercept (HAI) rotates a right triangle's hypotenuse about an acute vertex until its endpoint meets the adjacent leg axis extension. Four canonical branches yield a positive seed (S1, S2). Adjoining squares generates a rectangle cascade with spans satisfying S(n+1) = S(n) + S(n-1); quarter-circle arcs of radius R(n) = S(n) (n ≥ 2) form a Fibonacci-type spiral. The triangle sets the initial ratio ρ2 = S2/S1; thereafter, ρ(n+1) = 1 + 1/ρ(n) is a universal Möbius update in PGL2(Z) ⊂ PGL2(R) driving ρ(n) → φ at a sharp rate Θ(φ^(-2n)). A complex lift reveals centre-step vectors satisfying z(k+1) = λ(k) u0 z(k) with u0 = ±i and λ(k) → φ; for golden seeds, the similitude z(k+1) = q z(k) with q = φ exp(±iπ/2) is exact. We prove ρ2 = φ iff the leg ratio is 2:1, i.e., (a, b, c) = (r, 2r, r√5). For parameter t = cot(β/2), the four seed ratios form a dihedral D4 orbit in PGL2(R) (restricting to PGL2(Q) for rational t); Fibonacci-Lucas and Pell-type identities emerge as geometric invariants.