Primitive Pythagorean Triples as a Bridge between Diophantine Geometry and Algebraic Number Theory: Quadratic Fields, Hyperbolic Angles, Class Numbers, and the Modular Surface

We develop an explicit correspondence connecting primitive Pythagorean triples to a broad range of quadratic algebraic structure, extending from elementary Diophantine geometry to the spectral theory of the modular surface. The central object is the governing parameter n = 2a/ (c−b) associated to each primitive Pythagorean triple (a, b, c), which serves as a canonical bridge between integer right-triangle geometry and algebraic number theory. The main results are organised into six themes. First, every quadratic equation x² − px − q = 0 with positive coefficients is canonically associated to a primitive Pythagorean triple via x² − (2αa/ (c−b) ) x − (α²+γ) = 0, where Q = α²+γ is a family invariant independent of the specific triple. Second, every real quadratic field Q (√D) corresponds to infinitely many primitive Pythagorean triples via an explicit surjective map. Third, the positive root satisfies x = √Q · exp (sinh⁻¹ (n/ (2√Q) ) ), and the PPT hyperbolic angles form a strictly increasing, asymptotically logarithmic, dense sequence in R⁺. Fourth, for algebraic number fields K = Q (ξ) of degrees d = 2, 3, 4, the norm form N₊/ₐ evaluated at PPT-derived rationals yields an explicit Diophantine equation of degree d (d−1), with a general framework established for all d. Fifth, for those positive integers n for which the PPT-connected constant is the fundamental unit of its associated quadratic field — a condition verified for n ≤ 13 in the metallic mean family and n ≤ 10 in the deca-metallic ratio family — the PPT hyperbolic angle equals the regulator, giving the PPT class number formula: h (n² + 4Q/α²) = √ (n²+4Q/α²) / (ln Q + 2 sinh⁻¹ (n/ (2√Q) ) ) · L (1, χ₍ℂ+₄ₐ/⏐ℂ), expressed in PPT-geometric quantities and the Dirichlet L-function value L (1, χD). Sixth, the PPT hyperbolic angle equals half the length of a primitive closed geodesic on the modular surface H/SL (2, Z), and the Selberg zeta function of the modular surface admits an explicit PPT parametrisation. As a structural consequence, proved via four independent naturality conditions, exactly two natural PPT-connected quadratic families exist: the classical metallic means (α=1, Q=1) and the deca-metallic ratios (α=2, Q=10), characterised by Q = |N (εK) |. The paper establishes primitive Pythagorean triples as a unifying arithmetic object connecting Diophantine geometry, polynomial algebra, algebraic number fields, hyperbolic analysis, class numbers, and the spectral geometry of the modular surface.