The Pythagorean Angle Lattice: A Unified Framework for Algebraic, Logarithmic, and Transcendental Constants

The Pythagorean Angle Lattice unifies algebraic (metallic means & golden ratio), logarithmic (rational ln via hyperbolic functions on triples), and transcendental (π, e, γ) constants through the geometry of primitive integer right triangles and Gaussian integers. We study the Pythagorean Angle Lattice, a countable set of real numbers defined by: ℒ = {cot(arg(Z)/D) : Z = W² or iW², W ∈ ℤ[i]′_prim, D ∈ ℕ}, where ℤ[i]′_prim = {u + vi ∈ ℤ[i] : gcd(u,v)=1, u ≢ v (mod 2), uv ≠ 0}. This lattice offers a unified framework for three families of constants: (1) Algebraic constants — metallic means M_n and the golden ratio φ arise exactly from the lattice geometry: cot(θ/4) = M_n and cot((α₀ + β(a′,b′))/4) = φ. (2) Logarithmic structure — for every primitive Pythagorean triple in the three canonical families (governed by n = 2a/(c−b) with c−b ∈ {1,2,8}), inverse hyperbolic functions systematically encode rational logarithms: arcsinh(a/b) = arctanh(a/c) = ln((n+2)/(n−2)) and arcsinh(b/a) = arctanh(b/c) = ln(n/2). (3) Transcendental constants — expressions for π (via counting function P(X) and Bridge Identity series), e (via Borel transform of the arithmetic kernel: e + e⁻¹ = 4(1−ℬ(1))), and γ (in the finite part of the Bridge Identity at s=2). The lattice is dense in ℝ⁺, constructively universal via continued fractions, and connects to the Riemann zeta function. It provides a unified geometric perspective spanning algebra, analysis, and number theory across seven mathematical languages in the integer right triangle.