"The τ₀ Theorem: Unifying Newtonian Mechanics, Relativity, and Information Theory via Response Grammar"
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This paper proposes and validates the τ₀ theorem—a universal response function (\hat{S} = \frac{\hat{\tau}}{\hat{\tau}_0 + \hat{\tau}}) constrained by thermodynamic limits, informational delays, and control-theoretic stability. By demonstrating the irreducible minimal response time τ₀, the theorem unifies Newtonian mechanics (instantaneous action at \tau_0 \to 0), Einsteinian relativity (speed-of-light limit with \tau_0 = d/c), and Shannon information theory (bandwidth delay \tau_0 = 1/B) as special cases of a single framework. Empirical evidence shows that systems ranging from enzyme kinetics (Michaelis-Menten equation), neural adaptation (divisive normalization), to behavioral discounting all obey τ₀ constraints. This discovery not only reformulates cross-disciplinary response models mathematically but also proves that energy, information, and control jointly dictate the universe’s response grammar.
