The Algebraic Horizon: Why the p9 Breakdown Proves Computational Finitism

The Polynomial On Residue Classes (PORC) conjecture, proposed by Graham Higman in 1960, posits that the number of isomorphism classes of groups of order behaves as a polynomial function of the prime for sufficiently large. For decades, this held true for ≤ 8. However, in 2010, du Sautoy and Vaughan-Lee proved that the conjecture fails at exactly = 9, where the enumeration function becomes dependent on the number of rational points on elliptic curves over finite fields. We argue that this algebraic breakdown is not a mathematical anomaly, but a rigorous proof of Computational Finitism. Just as physical metrics (Lorentz factors, Schwarzschild curvature) saturate at a finite limit, algebraic counting functions saturate at a complexity threshold. The transition from polynomial regularity to modular arithmetic at = 9 mirrors the = 9 limit observed in discrete physics. Pure mathematics independently confirms that infinite, smooth scaling is unsustainable; finite substrates enforce structural completion.