Foundation Dalembert Ultimate Consistency Defines Composition

The consistency requirement

The declaration consistency_defines_composition proves a specific fact about a particular mathematical function. The function is Cost.Jcost, defined for positive numbers x as J(x) = (x + 1/x)/2 - 1. This function measures a kind of cost or deviation from the value 1, where J(1) = 0. The theorem states that this function satisfies a property called multiplicative consistency: for any positive x and y, the sum J(xy) + J(x/y) can be expressed as a function P of J(x) and J(y) alone. The proof explicitly provides that function: P(u, v) = 2uv + 2u + 2v.

This result is one component within a larger framework. The framework's library contains a broader theorem, ultimate_inevitability, which states that any smooth, symmetric, normalized cost function satisfying this same consistency property must be exactly J. That theorem also proves the combiner P is uniquely the RCL formula. The declaration consistency_defines_composition is the narrower claim: it verifies that J itself meets the consistency condition, using the explicit P. It does not, by itself, establish uniqueness of J or of P.

The distinction matters for what the declaration can be cited for. The theorem proves an existence statement: a suitable combiner P exists for J. The broader inevitability theorem proves a uniqueness statement: no other combiner is possible. The consistency declaration is a lemma in that larger proof, not the proof itself. A reader who cites only this declaration can claim that J has multiplicative consistency; they cannot claim that J is the only function with that property.

/-- Consistency IS what defines compositional structure.
    If you don't have it, you don't have a compositional cost theory. -/
theorem consistency_defines_composition :
    HasMultiplicativeConsistency Cost.Jcost := by
  use fun u v => 2*u*v + 2*u + 2*v
  intro x y hx hy
  exact J_computes_P x y hx hy

/-- Consistency IS what defines compositional structure.
    If you don't have it, you don't have a compositional cost theory. -/
theorem consistency_defines_composition :
    HasMultiplicativeConsistency Cost.Jcost := by
  use fun u v => 2*u*v + 2*u + 2*v
  intro x y hx hy
  exact J_computes_P x y hx hy

What this page does not claim

This declaration does not prove that J is the unique cost function satisfying the three primitive requirements. This declaration does not establish that the combiner P is unique. This declaration does not state that multiplicative consistency is a definition rather than an 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/Foundation/DAlembert/Ultimate.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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