A Discrete Reflection Algebra and its Convergence to the Riemann xi-Function: The Matchstick Game, the Heat-Kernel Bridge, and the Remaining Step to RH

This paper introduces a finite combinatorial system, the Matchstick Game, built from two elementary operations on the half-line: expansion by integer multiples and reflection across positive ticks. The construction is purely discrete and uses no a priori notion of analytic continuation, functional equation, or critical line. Three results are established: 1. Reflection uniqueness (discrete). The reflection algebra generated by the two operations is shown to be uniquely determined by its anchor data, up to the explicit symmetry of the construction. This is a self-contained algebraic statement about the game. 2. Heat-kernel bridge (analytic). A spectral function psi(x) attached to the game is shown to coincide with the classical Jacobi theta function. Via Poisson summation and Mellin transform, the corresponding completed function Xi(s) is proved to equal Riemann’s xi(s) on the entire complex plane. In particular, the non-trivial zeros of Xi coincide with the non-trivial zeros of the Riemann zeta function. 3. Open residual step. The remaining gap to a full proof of the Riemann Hypothesis is reformulated geometrically as Conjecture (collapse): every off-equatorial four-orbit of the reflection algebra collapses onto the critical line Re(s) = 1/2. A numerical test (T up to 5000) is reported, together with an honest discussion of where a future argument would have to act (Hilbert-Polya-type spectral realization, positivity arguments, or an additional symmetry not yet identified in the discrete layer). The paper does not claim a proof of the Riemann Hypothesis. It claims a clean two-layer construction (discrete reflection algebra plus heat-kernel bridge) that reproduces the xi-function from finite combinatorial data, and it isolates the precise step that remains open. Falsification criteria are stated explicitly. The construction would be refuted by, among others, an off-equatorial zero of Xi or xi, or by a counterexample to the reflection-uniqueness theorem.