IndisputableMonolith.Relativity.Calculus.FunctionalEquationDeriv
This module derives the second-order differential equation satisfied by solutions to the d'Alembert functional equation. Researchers connecting functional forms to wave equations in relativity or Recognition Science models cite the result. The argument differentiates the given relation twice with respect to the auxiliary variable and specializes the shift to zero.
claimIf $f(x+y) + f(x-y) = 2f(x)f(y)$, then $f''(x) = f''(0) f(x)$.
background
The module resides in the Relativity.Calculus section and supplies a differentiation lemma for the d'Alembert equation. This functional relation encodes the addition formula for cosine and hyperbolic cosine, which arise from the Recognition Composition Law when arguments are specialized. The setting assumes twice differentiability of $f$ and imports Mathlib for the underlying real analysis.
proof idea
The module differentiates the functional equation twice with respect to $y$, then evaluates at $y=0$. The cross terms cancel and the remaining factor yields the stated multiple of $f(x)$.
why it matters in Recognition Science
The lemma supports the passage from the Recognition Composition Law to differential equations along the forcing chain, feeding the sibling dalembert_deriv_ode and aiding T5 J-uniqueness. It supplies the local differential consequence required for subsequent wave-propagation arguments in the Relativity submodule.
scope and limits
- Does not determine the numerical value of $f''(0)$.
- Does not construct explicit solutions such as $\cos$ or $\cosh$.
- Does not address nondifferentiable or distributional solutions.
- Does not extend the relation to higher spatial dimensions.