ode_zero_status
plain-language theorem explainer
The ode_zero_status declaration records that the zero function is the unique solution to the initial-value problem f'' = f with f(0) = 0 and f'(0) = 0. A researcher working in the Recognition Science cost algebra would cite this uniqueness result when reducing the homogeneous component of the d'Alembert equation. The entry is a one-line string definition that classifies the result as a standard mathematical fact.
Claim. The unique solution of the initial-value problem $f''(t) = f(t)$, $f(0) = 0$, $f'(0) = 0$ is the zero function $f(t) = 0$.
background
The Meta.Axioms module maintains an axiom registry that separates physical postulates from well-known mathematical facts accepted for pragmatic reasons in the formalization. The ode_zero_uniqueness entry is listed under standard ODE results, referencing the Picard-Lindelöf theorem for local existence and uniqueness. It sits alongside the d'Alembert functional equation whose continuous solutions with H(0) = 1 and H''(0) = 1 equal cosh.
proof idea
The declaration is a one-line definition that assigns the status string 'Mathematical Fact (ODE Uniqueness)'. It draws directly on the Picard-Lindelöf theorem cited in the accompanying documentation without an explicit Lean proof of the ODE result.
why it matters
This definition populates the summary table in the Meta.Axioms module, distinguishing standard ODE facts from foundational physical postulates such as the meta-principle and recognition exchange invariance. It supports the reduction of the d'Alembert equation in Cost.FunctionalEquation to the cosh solution, which in turn anchors the J-uniqueness step (T5) in the forcing chain and the transition to the phi fixed point (T6).
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