IndisputableMonolith.Foundation.PolynomialityFromLogic
The module defines closed under iteration for combining rules on subsets of the reals. Researchers tracing functional forms back to logical consistency in cost functionals would cite it. The structure imports d'Alembert inevitability for uniqueness of the equation and Aczel smoothness to secure continuity of solutions.
claimA combining rule Φ is closed under iteration on a set S ⊆ ℝ if Φ(x, y) ∈ S whenever x, y ∈ S and Φ is continuous in both arguments.
background
The module sits in the Foundation domain and imports the d'Alembert inevitability result, which shows that multiplicative consistency of any cost functional F : ℝ₊ → ℝ forces the equation H(t+u) + H(t-u) = 2 H(t) H(u). It also imports Aczel's theorem classifying continuous solutions with H(0) = 1 as either the constant 1 or cosh(λt), both C^∞. In this setting the module introduces the auxiliary notion of a combining rule Φ being closed under iteration on S.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the definition of closure under iteration that supports derivation of polynomiality from logic. It rests on the d'Alembert inevitability and Aczel smoothness results and prepares iterated-closure statements for later use in the foundation chain, though no direct downstream declarations are recorded here.
scope and limits
- Does not derive the explicit polynomial form of the combining rule.
- Does not address discontinuous solutions or non-real domains.
- Does not connect the closure property to the phi-ladder or physical constants.
- Does not prove that every logically consistent rule must be closed under iteration.