IndisputableMonolith.Foundation.AxiomDischargePlan
AxiomDischargePlan collects proved ODE case lemmas that close regularity gaps in the foundation. Researchers on T5 uniqueness or the Translation Theorem cite these to weaken the polynomial-degree assumption on the route-independence combiner. The arguments use repeated differentiation on smooth functions with fixed initial conditions H(0)=1 and H'(0)=0.
claimIf $H$ is smooth, $H(0)=1$, $H'(0)=0$ and $H''=0$, then $H=1$. Parallel uniqueness statements hold for the cosine case, negative-zero case, and related ODE reductions under the same initial conditions.
background
The module belongs to the Foundation domain and imports Cost.FunctionalEquation for T5 helpers plus GeneralizedDAlembert for regularity discharge. GeneralizedDAlembert states: 'Move 3: discharge polynomial regularity using continuity. The existing Translation Theorem requires that the route-independence combiner P be a polynomial of total degree at most two.' The supplied lemmas therefore target the ODEs that appear once the functional equation is reduced to second-order form with the given boundary data.
proof idea
The module proceeds by case analysis on the ODE. For the constant case the argument runs: deriv(deriv H) ≡ 0 plus differentiability of deriv H forces deriv H constant; the condition deriv H 0 = 0 then yields deriv H ≡ 0; differentiability of H plus this forces H constant; finally H 0 = 1 fixes the value. Parallel direct differentiation or rescaling steps handle the cosine and other listed cases.
why it matters in Recognition Science
These lemmas feed the parent axiom-discharge results that support the generalized d'Alembert theorem and ultimately T5 J-uniqueness. They address the open regularity question left by the quartic-log counterexample, allowing the Translation Theorem to operate with continuity rather than the stronger polynomial-degree bound.
scope and limits
- Does not treat non-smooth or distributional solutions.
- Does not prove the general functional equation without case splitting.
- Does not discharge all regularity assumptions in one step.