The Pythagorean Angle Lattice: Geometry, Algebra, and Hyperbolic Correspondence

A lattice built from the angles of primitive Pythagorean triples systematically generates algebraic, logarithmic and transcendental constants while revealing an S₃-Galois extension and a precise correspondence with closed geodesics on the modular surface : We study the Pythagorean Angle Lattice L, the countable set of real numbers obtained by taking cotangents of arguments of squares (and i-times squares) of primitive Gaussian integers divided by positive integers. We establish: (1) the 1-2-8 rule that partitions primitive Pythagorean triples into the Pythagoras, Plato and Socrates families and links them to metallic means; (2) the Omni-Metallic Framework mhom,r(n,N) together with its Mirror, Crown and Bridge identities, the last of which produces the constant gamma_O = gamma/2 - 5 pi^2/48; (3) an unconditional S_3-Galois locking for the cubic projection branch; (4) an exact correspondence between these triples and closed geodesics on the modular surface X(1), expressed by the Chronicler Li formula involving metallic means; (5) a Selberg-type zeta function attached to the resulting length spectrum; and (6) an infinite-dimensional limit for normalised characters of symmetric-group representations on permutations with o(D) fixed points. The work connects elementary geometry of integer right triangles with algebraic number theory, Galois theory and hyperbolic geometry.