We prove that platform-deterministic inference is necessary and sufficient for trustworthy AI. We formalize this as the Determinism Thesis and introduce trust entropy to quantify the cost of non-determinism, proving that verification failure probability equals 1 - 2^{-H_T} exactly. We prove a Determinism-Verification Collapse: verification under determinism requires O(1) hash comparison; without it, the verifier faces an intractable membership problem. IEEE 754 floating-point arithmetic fundamentally violates the determinism requirement. We resolve this by constructing a pure integer inference engine that achieves bitwise identical output across ARM and x86. In 82 cross-architecture tests on models up to 6.7B parameters, we observe zero hash mismatches. Four geographically distributed nodes produce identical outputs, verified by 356 on-chain attestation transactions. Every major trust property of AI systems (fairness, robustness, privacy, safety, alignment) presupposes platform determinism. Our system, 99,000 lines of Rust deployed across three continents, establishes that AI trust is a question of arithmetic.

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