Verification Dalembert Symmetry Cert

The symmetry certificate

The d'Alembert functional equation is a classical object: it is the equation whose solutions include the cosine and hyperbolic cosine functions. Written as H(t+u) + H(t-u) = 2·H(t)·H(u), it says that the average of a function at two points symmetric around t equals the product of the function's values. The classical fact, known since the 18th century work of Jean le Rond d'Alembert on vibrating strings, is that with the normalization H(0) = 1, the only continuous solutions are cosh(k·t) and cos(k·t).

The symmetry certificate in Recognition Science packages a sharper statement. It proves that the d'Alembert equation alone, with H(0) = 1, forces the function to be even: H(-t) = H(t) for every t. No continuity, no differentiability, no extra symmetry assumption is needed. This is a pure algebraic consequence of the equation itself. The certificate also proves a second fact: if an even function is differentiable at zero, then its derivative there is zero, H'(0) = 0. Together these two results give the correct initial conditions for the ordinary differential equation H'' = H, the equation that selects cosh and cos from the solution family.

In Recognition Science, this certificate plays a supporting role in the chain that forces the cost function J(x) = (x + 1/x)/2 - 1. The framework's ledger, a discrete record of recognition events, requires a cost function satisfying certain forced conditions. A key step in proving that J is uniquely determined passes through the d'Alembert equation. The certificate establishes that the even symmetry needed for that proof is not an additional assumption but a consequence of the equation itself.

The machine-checked library of formal theorems records this certificate as a structure with a verification predicate. The predicate states both results: d'Alembert plus H(0) = 1 implies evenness, and evenness plus differentiability at zero implies zero derivative. A single theorem, verified_any, proves that this predicate holds for any certificate instance. This means the two classical facts are not merely asserted but mechanically checked, with every logical step validated by the kernel.

What this changes is the epistemic status of the symmetry claim. In a typical textbook treatment, evenness of cosh and cos is either assumed or shown by inspecting their power series. Here it emerges as a forced consequence of the functional equation alone. The reader can now see that the symmetry is not a property of the specific functions but a structural constraint of the equation itself.

structure DAlembertSymmetryCert where
  deriving Repr

structure DAlembertSymmetryCert where
  deriving Repr

What this page does not claim

This certificate does not prove that the d'Alembert equation has only cosh and cos as solutions. This certificate does not establish the full cost function uniqueness theorem. This certificate does not show that H'(0) = 0 without the differentiability assumption.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Verification/DAlembertSymmetryCert.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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