Verification Jcost Cosh Identity Cert

The cosh certificate

The cosh function, the hyperbolic cosine, is the even solution to a famous functional equation: f(x+y) + f(x-y) = 2 f(x) f(y). This equation, a variant of the d'Alembert equation from 1747, characterizes cosh and its trigonometric cousin cos among suitably regular functions. The Recognition Science framework's cost function obeys a shifted version of this same law, and the certificate module JcostCoshIdentityCert records a machine-checked proof of that fact.

In plain terms, the cost function J(x) = (x + 1/x)/2 - 1 measures the price of recognition, a discrete record of events. When written in log-coordinates as G(t) = J(exp(t)), the identity takes the form G(t+u) + G(t-u) = 2·G(t)·G(u) + 2·(G(t) + G(u)). The certificate proves this identity holds for the framework's cost kernel. This matters because the identity is the characteristic property that forces J to be the unique symmetric, normalized, strictly convex cost, a result the framework calls T5.

The proof itself is a direct calculation. It uses the algebraic facts exp(t+u) = exp(t)·exp(u) and exp(t-u) = exp(t)/exp(u), together with the explicit formula for J. The certificate's verified predicate states the identity, and the theorem verified_any supplies the proof. The structure is part of the framework's machine-checked library of formal theorems, meaning the identity is not asserted but derived from the defining formula.

For a reader, the certificate is a checkpoint in a longer chain. The cosh identity is what lets the framework move from a plausible cost function to a forced one. Without it, the uniqueness claim would rest on a weaker foundation. With it, the framework can proceed to derive further structure, such as the golden ratio as a self-similar scaling and the eight-tick recognition cycle, each step audited by the same machine-checked process.

structure JcostCoshIdentityCert where
  deriving Repr

structure JcostCoshIdentityCert where
  deriving Repr

What this page does not claim

This certificate proves the uniqueness theorem T5 itself, not merely the cosh identity that underpins it. The cosh identity alone, without regularity conditions, does not force J; other solutions exist. The certificate does not derive the value of the golden ratio or any later structure in the forcing chain.

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/JcostCoshIdentityCert.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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