Parameterized Reciprocal Quadrance Structures on the Null Conic in Universal Hyperbolic Geometry

This research paper presents a dynamic, parameterized formulation of projective metrical invariants on the boundary conic of Universal Hyperbolic Geometry (UHG). Building directly upon the author's previous studies on static reciprocal parallax symmetries (Coady, 2026; https://doi.org/10.5281/zenodo.20277798) and anharmonic metrical orbits (Coady, 2026; https://doi.org/10.5281/zenodo.20302803), this paper introduces a rational parameterization 't' on the absolute null conic:

b1(t) = [t² - 1 : 2t : t² + 1]

By constructing a pencil of cevian lines through the moving null point b1(t) to the vertices of a static, fully-right diagonal triangle efg, the paper determines the dynamic intersection coordinates b2(t), b3(t), and b4(t) along the side boundaries.

The principal mathematical result of this paper is that the induced adjacent quadrances evolve as exact reciprocals:

q(g, b2(t)) * q(g, b3(t)) = 1

forcing the quadrea of the associated right triangles to remain identically conserved at value A = 1 under the rational deformation.

Key Theoretical Contributions:

1. Geometric Polarity & Involution: Rather than presenting reciprocity as a mere coordinate-dependent accident, this work establishes that the polar correspondence of the diagonal triangle under Coxeter's projective involution theorem naturally induces a projective correspondence, giving a geometric explanation for the metrical reciprocity.

2. Projective Conjugation via the Cayley Transform: The paper proves that the Cayley transform conjugates the non-linear metrical reciprocity q -> 1/q into a simple linear sign-reflection w -> -w. Crucially, we show that the Cayley transform operates as a purely rational projective transformation, maintaining strict finitist field generality over any field F (where characteristic of F is not equal to 2) without requiring complex numbers.

3. Formal Algebraic Invariance (Noether Analogue): Using the formal derivative D = d/dt of Algebraic Calculus, we demonstrate that the derivative of the quadrea vanishes identically (D(A) = 0) without requiring continuous manifolds or analytical limits, establishing an algebraic analogue of symmetry-induced conservation.

4. Field-Dependent Degeneracy Analysis: We explicitly identify the projective degeneracies of the configuration at t = 1 +/- sqrt(2), showing that over the finite field F_p these degeneracies split if and only if 2 is a quadratic residue modulo p, which occurs precisely when p = +/-1 mod 8.

An exact symbolic verification of all coordinates, quadrances, and derivative cancellation identities is provided in Python using the 'sympy' library. This work contributes a complete, limit-free framework for algebraic spacetime kinematics over exact coordinate fields, offering an illustrative, coordinate-independent metrical model for relativistic structures.