Verification Jcost Convexity Cert

The convexity certificate

In Recognition Science, the cost of recognition is a function J(x) = (x + 1/x)/2 - 1, defined for positive numbers x. The verification jcost convexity cert is a machine-checked certificate, a formal object in the framework's library of verified theorems, that proves this function is strictly convex on the positive real numbers. Strict convexity means the graph of J bends upward everywhere, with no flat stretches or multiple valleys.

The practical consequence is a uniqueness guarantee. A strictly convex function has exactly one critical point, and for J that point is x = 1, where J(1) = 0. Any cost-minimization problem that uses J therefore has a unique solution. This matters because the framework's uniqueness theorem, called T5, relies on strict convexity to establish that J is the unique symmetric, normalized, strictly convex function agreeing with the averaging kernel. Without strict convexity, there could be competing minimizers or flat regions, and the whole chain of forced structure would lose its footing.

The proof itself is simple. The second derivative of J is J''(x) = x⁻³, which is positive for all x > 0. A function with a positive second derivative on a convex domain is strictly convex. The certificate packages this argument as a formal theorem, with a structure that carries the verification predicate and a proof that the predicate holds for any instance. In plain language, the certificate establishes that the cost function has a single unambiguous lowest point, and that this property is not an assumption but a proved fact.

Within the Recognition Science framework, this certificate is a load-bearing piece of the foundation. It does not by itself derive any physical constant or dimension; it secures the uniqueness premise that later theorems depend on. The certificate is one link in a chain that ultimately forces the golden ratio, an eight-tick cycle, and three spatial dimensions, but its own role is narrower: it guarantees that the cost function cannot be bent into having multiple minima, so that every optimization built on J has exactly one answer.

structure JcostConvexityCert where
  deriving Repr

structure JcostConvexityCert where
  deriving Repr

structure JcostConvexityCert where
  deriving Repr

What this page does not claim

The certificate does not derive any physical constant or dimension by itself. The certificate does not prove that J is the only possible cost function; that is a separate theorem. The certificate does not address the empirical comparison of particle masses.

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

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