The Pythagorean Angle Lattice and the Omni-Metallic Framework

A single countable angle lattice built from squared Gaussian integers unifies metallic means, rational logarithms, and the constants π, e, γ inside one geometric–algebraic framework completed by the four-parameter Omni-Metallic family. 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), (u, v) ≠ (0, 0)}.We show that 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,cot((α₀ + β(a′, b′))/4) = φ. 2. Logarithmic structure: For every primitive Pythagorean triple in the three canonical families, the inverse hyperbolic functions arcsinh and arctanh systematically encode rational logarithms: arcsinh(a/b) = arctanh(a/c) = ln((n + 2)/(n − 2)),arcsinh(b/a) = arctanh(b/c) = ln(n/2), where n = 2a/(c − b) is the governing parameter. 3. Transcendental constants: The lattice provides expressions for π, e, and γ. The constant π arises via two approaches: the counting function P(X) and the Bridge Identity series. The constant e is derived via the Borel transform of the arithmetic kernel: e + e⁻¹ = 4(1 − B(1)). The Euler-Mascheroni constant γ appears in the finite part of the Bridge Identity at s = 2: FPₛ₌₂ Z(s) = γ/2 − 5π²/48. The framework is completed by the Omni-Metallic Family ℧m,r(n, N), a four-parameter algebraic generalization defined as the unique positive root of x^m − 2n xᵐ⁻¹ − N^r = 0, which rests on three master identities: 1. The Mirror Identity: ℧m,r(n, N) − N^r / ℧m,r(n, N)ᵐ⁻¹ = 2n. 2. The Crown Identity: ℧2,1(2a/(c − b), N) = 2a/(c − b) + √(8c/(c − b) + (N − 4)), generalized to all primitive triples. 3. The Bridge Identity: Z(s) = (1/2)ζ(s − 1) − (1/2)(1 + 2⁻ˢ)ζ(s).