The Pythagorean Angle Lattice: Algebraic, Logarithmic, and Transcendental Constants through Metallic Renormalization and Character Spectra

Three families of constants—algebraic quarter-angles, logarithmic partners, and the universal four-radian boundary—emerge from a single continuous governing parameter of the Pythagorean Angle Lattice through metallic renormalization and character spectra: The Pythagorean Angle Lattice (PAL) was introduced as an arithmetic angle framework generated by primitive Pythagorean triples and Gaussian-integer geometry. In the present work we develop a continuous right-triangle extension of its governing parameter and show that three apparently different layers—algebraic quarter-angle values, rational-argument logarithmic values, and boundary/special-value data—arise from one half-angle coordinate. For a right triangle with smaller acute angle θ, shorter leg a, longer leg b, hypotenuse c, and n = 2a/(c − b), we prove the universal identity n = 2 cot(θ/2), θ = 2 arctan(2/n). For primitive Pythagorean triples the quarter-angle quantity X = cot(θ/4) = [a + √(a² + (c − b)²)]/(c − b) is algebraic of degree at most two and satisfies (c − b)X² − 2aX − (c − b) = 0. The classical metallic means form the distinguished integer-governing-parameter sector, with the primitive sectors c − b ∈ {1, 2, 8}. We give the exact rationality criterion for the exceptional degree-one cases. Writing t = log(n/2) produces the circular–hyperbolic pair θ(t) = 2 arctan(e^{−t}), H(t) = 2 artanh(e^{−t}), with θ = gd(H) and nθ < 4 < nH, nθ ↗ 4, nH ↘ 4. Thus the PAL boundary constant and its degree measure are Θ_PAL = 4 radians, Θ_PAL^(deg) = 720/π. We then show that exact angle halving induces the metallic renormalization n ↦ R(n) = n + √(n² + 4) = 2M(n), which is explicitly linearized by Schröder and Abel coordinates. On analytic germs at the attracting boundary, the associated composition operator has point spectrum 2^{−r} and eigenfunctions (arctan x)^r. The four-radian defect decomposes into the even part of this dyadic spectrum. Finally, we introduce character fields F_χ(t) = 2 ∑_{m≥1} χ(m)/m e^{−mt}, whose Mellin transforms are 2Γ(s)L(s + 1, χ). The angular and logarithmic PAL channels correspond respectively to the primitive character modulo 4 and the principal character modulo 2. Every character field has the same normalized boundary value 4, while root-of-unity projections yield a cyclotomic hierarchy with arbitrarily high algebraic convergence order: Q_q(n) = 4 + 2^{q+2}/(q + 1) n^{−q} + O(n^{−2q}). The resulting framework separates the genuinely new PAL geometry and renormalization statements from the classical special-function, Dirichlet-L, theta, and functional-equation machinery used to analyze them.