IndisputableMonolith.Foundation.AxiomDischargePlan
AxiomDischargePlan supplies proved ODE case lemmas that discharge regularity assumptions in the cost functional equation for T5. Researchers on J-uniqueness cite these to remove polynomial constraints from the Translation Theorem. Proofs proceed by iterated differentiation plus initial conditions to force constant or cosine forms.
claimIf $H$ is smooth, $H(0)=1$, $H'(0)=0$ and $H''\\,\equiv 0$, then $H\\,\equiv 1$. The module also establishes uniqueness for the zero and cosine cases of the ODE under matching initial data.
background
The module imports Cost.FunctionalEquation (lemmas for the T5 cost uniqueness proof) and GeneralizedDAlembert (discharging polynomial regularity via continuity for the Translation Theorem). It treats the ODE that arises once the d'Alembert route-independence combiner is reduced to second order. The constant case is stated as: a smooth function with H(0) = 1, H'(0) = 0, and second derivative identically zero is the constant 1.
proof idea
The module is organized as a collection of case lemmas. The constant case applies the fact that the second derivative being identically zero plus differentiability of the first derivative forces the first derivative to be constant; the condition at zero then sets that constant to zero, hence H is constant, and the value at zero fixes it to one.
why it matters in Recognition Science
These discharges feed the T5 J-uniqueness theorem in the forcing chain by closing the constant, negative-zero and cosine cases of the ODE. They directly address the regularity gap identified in GeneralizedDAlembert for the Translation Theorem, allowing the route-independence combiner to be treated without an a-priori polynomial-degree bound.
scope and limits
- Does not treat the general non-polynomial case without continuity.
- Does not remove all regularity hypotheses from the Translation Theorem.
- Does not prove the full T5 uniqueness statement.