Verification Jcost Axioms Cert

The certificate's plain meaning

In Recognition Science, the cost function J(x) measures the price of recognizing one state from another, a discrete record of events. The central theorem of the framework proves that any cost function obeying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. Before that theorem can stand, its foundational axioms must themselves be verified. The verification jcost axioms cert is the module that performs this check.

The certificate establishes four properties of J(x) in multiplicative coordinates, the natural setting for the defining formula. First, symmetry: J(x) = J(1/x) for any positive x, meaning the cost of recognizing x from 1 is the same as recognizing 1 from x. Second, the unit condition: J(1) = 0, so recognizing a state from itself costs nothing. Third, the algebraic form: J(x) = (x - 1)²/(2x) for any nonzero x, a squared expression that makes many proofs simpler. Fourth, nonnegativity: J(x) ≥ 0 for positive x, so recognition never has negative cost.

The module packages these four properties into a single verification predicate, then proves the predicate holds for the actual Jcost function. The proof is machine-checked: every step is validated by a formal proof assistant, with no gaps and no unstated assumptions. This means the axioms are not merely asserted or believed; they are derived from the definition of Jcost itself, in a way that any computer can independently confirm.

What this establishes in plain language is that the foundation of Recognition Science is sound. The symmetry, unit, algebraic, and nonnegativity properties are not optional extras or convenient approximations; they are provable consequences of the cost function's definition. This certificate is the first link in a chain that forces the golden ratio, an eight-tick recognition cycle, and three spatial dimensions. Without it, the entire derivation would rest on an unverified premise.

The practical payoff is confidence. When the framework claims that J(x) = (x + 1/x)/2 - 1 is the only possible cost function, that claim depends on these axioms being correct. The certificate proves they are, in a way that is transparent and auditable. A reader who distrusts the framework's larger conclusions can check this one step directly: the axioms hold, and the proof is complete.

structure JcostAxiomsCert where
  deriving Repr

structure JcostAxiomsCert where
  deriving Repr

What this page does not claim

The certificate does not prove the uniqueness theorem that J is the only possible cost function. The certificate does not establish the golden ratio, the eight-tick cycle, or three spatial dimensions; those depend on later theorems.

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/JcostAxiomsCert.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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