The Prime Number Geometry: A Constructive Proof via Triangle Game and Square-Atlas

This paper presents a constructive geometric framework for the odd prime numbers, built from a single combinatorial game and an associated atlas of four real-valued surfaces. The construction has three layers. First, a triangle game is defined by four rules applied in strict priority. The fourth rule, which fires only when the first three fail, produces exactly the odd primes from three onward. The corresponding theorem is proved by a three-case analysis (even numbers, odd composites, odd primes). Second, every odd prime is classified into one of three classes A, B, C, characterized by its residue modulo eight. Class A primes (residue one modulo four) admit a representation as a sum of two squares. Class B primes (residue three modulo eight) require exactly three squares. Class C primes (residue seven modulo eight) require exactly four squares. The classification rests on the classical theorems of Fermat-Euler, Gauss-Legendre, and Lagrange, and the asymptotic distribution two to one to one follows from Dirichlet. Third, a square-atlas Q_2, Q_3, Q_4 of three real-valued surfaces is introduced. Each surface satisfies four symmetry properties (permutation invariance, translation covariance, breaking of inversion, and a class-specific diagonal offset). An embedding theorem states that every odd prime sits at a unique address on exactly one surface, determined by its square representation. Bulk verification covers all six hundred sixty-eight odd primes below five thousand, with a one-hundred-percent match between the predicted class and the empirical residue. An anti-correlation analysis of the class labels reproduces the consecutive-prime bias reported by Lemke Oliver and Soundararajan in 2016. A uniqueness theorem identifies the Mertens balance surface M(X,Y) equal to one-half times the sum X plus Y as the only solution of four explicit symmetry axioms. The scope section separates clearly what is proved (the four theorems above) from what remains open (an analytic derivation of the anti-correlation amplitude and a possible connection to the zeta zeros). The paper is structured for peer review and includes data availability, conflict of interest statement, full Python source code, and a worked table for all primes up to one hundred. MSC2020 CLASSIFICATION----------------------11A41 Primes11E25 Sums of squares and representations by other particular quadratic forms11N05 Distribution of primes05A99 Combinatorics, none of the above51M99 Real and complex geometry, none of the above