A Geometric Reformulation of the Riemann Hypothesis via a Triangulation of the Primes

This expository paper reformulates the Riemann Hypothesis in an elementary geometric language built from a triangulation of the prime numbers. Each prime is taken as the apex of a triangle spanning its two neighbours, and the chain of these triangles is measured against a fixed equilateral ideal whose side is the natural prime spacing log p. The accumulated deviation of the real chain from this ideal is shown, by a telescoping identity, to be the classical Chebyshev error term; the hypothesis that this deviation remains within a square-root band is then exactly the Riemann Hypothesis, via the classical equivalence of von Koch (1901), with the non-vanishing of the deviation a theorem of Littlewood (1914). The same picture is translated through Riemann's explicit formula, in which the deviation is a superposition of oscillatory modes indexed by the zeta zeros, and the hypothesis becomes the statement that every mode is a pure, non-growing oscillation; through Weil's criterion it becomes a global positivity, of which only a partial form (de la Vallée Poussin, 1896) is known. Elementary computations on the primes below a few million illustrate each feature. The paper contains no new theorem and no progress toward proving the Riemann Hypothesis; its sole contribution is a unifying geometric presentation in which the problem, its proven boundary, and its single missing ingredient appear together in one picture. Note: 11M26 (distribution of zeros of ζ, RH) and 11N05 (distribution of primes)