Pythagorean Angle Lattice: From Gudermannian Duality and Silver–Pell Extremal Dynamics to Periodic Words and Quadratic-Form Similitudes

Gudermannian duality and silver–Pell extremal dynamics on primitive Pythagorean triples yield a complete classification of periodic words and quadratic-form similitudes in the Berggren–Romik system, with sharp bounds (π/2)(1+√2) < nθ < 4. Let (a, b, c) be an ordered primitive Pythagorean triple, 0 < a < b < c, with smaller acute angle θ, and define the governing parameter n = 2a/(c − b). Then tan(θ/2) = 2/n, θ = 2 arctan(2/n), so the normalized angular observable is A = nθ = 2n arctan(2/n). We prove the sharp range (π/2)(1 + √2) < nθ < 4. Using Euclid’s parameters and the hyperbolic coordinate t = log(u/v), we obtain a two-branch Gudermannian law, a Gaussian involution, and the silver fixed point L = log(1 + √2). We show that the rational governing parameter is a complete coordinate on ordered primitive triples and obtain an explicit inverse together with the generalized 1–2–8 law c − b = λ(P)Q² for reduced n = P/Q. The lower endpoint is approached by the consecutive-leg negative-Pell orbit (2a_k + 1) + c_k √2 = (1 + √2)^{2k+1}, which is proved to give the global record minimizers of nθ under a hypotenuse-height constraint. Its governing parameters satisfy n_{k+1} = (6n_k + 4)/(n_k + 2), with contraction (1 + √2)^{-2}. The same structure extends to fixed-gap hyperbolas (a + b)² − 2c² = −(b − a)², modular labels r² ≡ 2 (mod d), and a Dirichlet series R₂(s) = ζ(s)L(s, χ₈)/((1 + 2^{-s})ζ(2s)). We then embed the Silver dynamics into the complete three-branch Berggren/Romik system in the half-angle coordinate. For a word W in {A, S, C} we prove the exact content formula cont(M_W) = 2^{⌊(#S + #C)/2⌋}, so the primitive projective determinant belongs to {1, −1, 2, −2}. To every non-scalar matrix M we attach the quadratic form Q_M(X, Y) = m₂₁X² + (m₂₂ − m₁₁)XY − m₁₂Y² and prove Q_M(Mv) = det(M)Q_M(v), disc(Q_M) = (tr M)² − 4 det M. Conversely, proper integral similitudes of a primitive indefinite form Q = [A, B, C] are parametrized by x² − Δy² = 4μ. The identity Q_M Q_{(x,y)} = yQ separates matrix discriminant into field discriminant, order conductor, and form content. The Silver branch is therefore the distinguished μ = 2 prototype of the four-sector periodic-word theory. We explicitly separate this arithmetic classification of associated form similitudes from the further dynamical selection problem of determining least positive symbolic returns.