The Rational Substrate: Foundations of the Secant–Cosecant Barycentric Geometry

This document establishes the foundational geometric structure underlying the Rational Substrate program.We introduce the Secant-Cosecant Orthogonal Triangle, define its associated barycentric coordinate system, and prove its rationality properties when generated by Pythagorean data.All later arithmetic, spectral, and physical results depend exclusively on the constructions formalized here.Define a triangle T θ ⊂ R 2 with side lengths A := sec θ, B := csc θ, C := sec 2 θ + csc 2 θ.We refer to T θ as the Secant-Cosecant Orthogonal Triangle.This triangle is constructed purely from reciprocal trigonometric functions and does not presuppose any embedding in the unit circle beyond the definition of sec and csc. Metric PropertiesLemma 1.2 (Right-Angle Lemma).The triangle T θ is right-angled for all θ ∈ (0, π 2 ).Proof.By direct computation,By the Pythagorean theorem, the triangle with sides A, B, C is right-angled, with hypotenuse C. Rationality StructureDefinition 1.3 (Dream Number Angle).An angle θ ∈ (0, π 2 ) is called a Dream Number Angle if sin θ ∈ Q and cos θ ∈ Q. Definition 1.4 (Dream Number Triangle).A Secant-Cosecant triangle T θ is called a Dream Number Triangle if θ is a Dream Number Angle.