IndisputableMonolith.Foundation.AxiomDischargePlan
This module collects lemmas that discharge regularity assumptions for the Translation Theorem by reducing the functional equation to ODE cases and proving uniqueness in each. Researchers on the T5 J-uniqueness proof cite it to justify smoothness hypotheses. The argument proceeds by case analysis on solution forms, beginning with the constant case where zero second derivative plus initial conditions forces constancy.
claimIf $H$ is smooth, $H(0)=1$, $H'(0)=0$, and $H''(x)=0$ for all $x$, then $H(x)=1$ for all $x$.
background
The module sits in the foundation layer and imports Cost.FunctionalEquation, which supplies lemmas for the T5 cost uniqueness proof, and GeneralizedDAlembert, whose doc states it performs Move 3 by discharging polynomial regularity using continuity for the Translation Theorem. The local setting concerns the constant case of a smooth function obeying the second-order ODE that arises after reduction from the generalized d'Alembert equation.
proof idea
The module organizes its content as a sequence of case lemmas. The constant case is handled by first noting that the second derivative being identically zero implies the first derivative is constant, then using the initial condition at zero to force that constant to be zero, and finally integrating to obtain constancy of H itself.
why it matters in Recognition Science
The lemmas feed the T5 J-uniqueness argument in the forcing chain by closing the constant-solution branch of the ODE reduction. They support the overall axiom-discharge plan that removes the polynomial-degree hypothesis from the Translation Theorem while preserving the Recognition Composition Law.
scope and limits
- Does not treat non-smooth or distributional solutions.
- Does not establish uniqueness for the full functional equation without the remaining case lemmas.
- Does not provide numerical checks or counterexamples outside the smooth regime.