This paper started with a simple question that wouldn't leave me alone: how come black holes are able to divide by zero? When you push general relativity to the center of a black hole, the math fails. Density goes infinite. Tyson says the limit is on our theory of the universe, not on the math itself. He's right, but I don't think that goes far enough. The math fails because the math was built for things that persist. We invented counting to track objects that don't change while we're counting them. Sheep, apples, coins. Then we generalized counting to everything. We made arithmetic the foundation of mathematics and along the way we quietly imported an assumption that wasn't in the original tool: that the things being counted stay. But most things in the universe don't stay. Atoms decay. Stars burn out. Cells die. Black holes evaporate. The universe runs intrinsic ending operations everywhere, all the time. We just refused to put one in the integers. I started wondering what an arithmetic with an intrinsic-ending operator would even look like. Two apples minus two apples requires another two apples to do the subtracting. But what if the apples just decompose? What if integers can commit arithmetic suicide, die from within, because of what they are, not because something subtracts them? This paper is what came out of that question. Eight computational tests. Four different ways to define a unary degradation operator on the integers. A closed-form result that ties the evaporation time to the Riemann zeta function. A differential equation that at one specific rate exponent matches Hawking's law for black hole mass loss. A geometric finding that the empty zone the operator creates around the integers has the same shape, and the same five structural properties, as the numeral zero, the topological unit circle, and the philosophical monad that Pythagoras, Brahmagupta, Leibniz, and the Mayans all drew when they drew nothing. Three independent paths in the framework point to the integer n=2 as the structural anchor. The Hourglass equation n+n = n×n forces it. The Möbius survival rate of squarefree integers gives 6/π² = 1/ζ(2). And under both multiplicative and additive arithmetic, n=2 occupies the same lower boundary of the Forbidden Zone. Four further consistency checks return the same integer once the framework is set up. The Hawking isomorphism is the part I keep coming back to. Reading (c) of the operator at rate exponent α=-2 gives dΩ/dt = -λ/Ω², which is Hawking's dM/dt = -k/M² under the substitution of multiplicative depth for mass. I'm not claiming integer arithmetic is black hole physics. I'm claiming the equation that governs evaporation in both cases is the same equation. Whatever causes things to shed at a rate inversely proportional to their internal complexity is a pattern that appears in two different regimes, with the same dynamical form. That is the kind of universality that ought to be noticed. Recent gravitational physics is converging on similar structural pictures from a completely different direction. Gralla (2026) finds that matter re-emerges as a stable end-state from evaporating charged black holes rather than collapsing into a singularity. Alford (2026) rigorously establishes that extremal black holes may not completely evaporate. The KM3-230213A neutrino event is still under investigation as a possible primordial-black-hole signature, with the interpretation disputed. All three results were developed independently of this framework. They describe the same structural pattern from the physics side that the framework describes from the arithmetic side: decay terminates at a stable irreducible end-state, not at vanishing. This paper is the answer to a question I left open at the end of The Language of the Universe: what is geometry, that a Man may know it, and a Man, that he may know geometry, and how can he understand it. The answer was always going to be by demonstration, not by deduction. Geometry was asked. Geometry has spoken. The math in this paper is transparent and reproducible. Test scripts, raw data, and the full reasoning chain are openly available. Anyone with Python can verify the closed form, the survival rates, the Markov MFPT, the Hawking isomorphism, the gap-width calculations, and the eight derivations in an afternoon. The math itself is not in question. What's in question is whether the structural pattern the framework identifies, namely operator-relative mortality, the Forbidden Zone as the geometric trace of discreteness, n=2 as the unique structural anchor, and ζ as the convergence governor of arithmetic mortality, gets taken seriously enough to test further. Companion papers: A Theory of Geometric Structure: The Classification Law (R₂ ∧ C ≠ C₀): https://zenodo.org/records/18756471 The Sieve Firewall - The True Reimann Hypothesis Solved: https://zenodo.org/records/18854321 AXONLang Labs Internal Research Paper: https://zenodo.org/records/19651828 The Classification Law as a Diagnostic: https://zenodo.org/records/19843822 The Language of the Universe - Geometry and the Rosetta stone that is Mathematics: https://zenodo.org/records/20298106
Daniel Santiago (Sat,) studied this question.