Cost Uniqueness Jcost Satisfies Composition Law

The composition law

The cost function J(x) = (x + 1/x)/2 - 1 is the unique function that satisfies five plain conditions: reciprocal symmetry, zero cost at unity, a forced composition law, calibration, and continuity. The composition law is the heart of the matter. It says that when you recognize a compound event, the cost is not simply the sum of the costs of its parts; instead, the cost of the whole is determined by the costs of the parts through a specific algebraic identity. Think of a ledger that records every recognition event, and the cost of a compound entry is forced by the costs of its components, not chosen freely.

The theorem Jcost_satisfies_composition_law proves that the candidate cost function J does obey this composition law. In the framework's library, a machine-checked collection of formal theorems, this is a proved statement, not an assumption. The proof rests on a chain of already-established results: Jcost satisfies the composition law exactly when it satisfies a related identity involving hyperbolic cosines, and that identity is itself a proved theorem. The declaration is one of several that together show Jcost meets all the requirements for the uniqueness theorem, which then concludes that any cost function satisfying the same five conditions must equal Jcost on the positive real numbers.

What the declaration does not claim is that the composition law alone forces the cost function. The law is a necessary condition, but not sufficient by itself. Without the other conditions, especially calibration and continuity, there exist other functions that satisfy the composition law but are not Jcost. The theorem is also not a claim about the physical world. It is a mathematical statement about a specific function and a specific algebraic identity. The framework's further claims, such as the appearance of the golden ratio or the number of spatial dimensions, depend on additional theorems and interpretations that go beyond this single declaration.

The practical upshot is that the composition law is a strong constraint. It rules out a vast family of possible cost functions, leaving only those that are structurally similar to Jcost. This is what makes the uniqueness theorem powerful: it shows that the cost of recognition is not a free parameter but is forced by a small set of natural requirements. The declaration is a key step in that proof, and it is one of the reasons the framework can make specific predictions rather than merely fitting data.

/-- `Jcost` satisfies the Recognition Composition Law. -/
theorem Jcost_satisfies_composition_law : FunctionalEquation.SatisfiesCompositionLaw Jcost :=
  (FunctionalEquation.composition_law_equiv_coshAdd Jcost).2 FunctionalEquation.Jcost_cosh_add_identity

/-- `Jcost` satisfies the Recognition Composition Law. -/
theorem Jcost_satisfies_composition_law : FunctionalEquation.SatisfiesCompositionLaw Jcost :=
  (FunctionalEquation.composition_law_equiv_coshAdd Jcost).2 FunctionalEquation.Jcost_cosh_add_identity

What this page does not claim

The composition law alone forces the cost function to be Jcost. The declaration makes any claim about the physical world or the framework's further derivations. The declaration proves that Jcost is the only function satisfying all five conditions; that is the role of the separate uniqueness theorem.

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/CostUniqueness.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: