Information H Uniqueness Verified

The verified uniqueness statement

The declaration H_UniquenessVerified is a formal statement inside the Recognition Science framework's machine-checked library of formal theorems. It asserts that any function F from real numbers to real numbers, if it satisfies the framework's InformationCost property, must equal the specific function J(x) = (x + 1/x)/2 - 1 for every positive input x. In plain terms: among all cost functions that meet the framework's definition of an information cost, there is exactly one, and it has this precise algebraic form. The declaration is a definition of a proposition, not a proof of it; the proof lives in a separate module called JCostNecessity, which the aggregator docstring marks as established.

The statement is deliberately narrow. It quantifies over functions F that already satisfy InformationCost, so it does not claim that every conceivable cost function collapses to J. It only claims uniqueness within that pre-defined class. The declaration also says nothing about whether J is computable, whether the domain extends to zero or negative inputs, or whether the property holds for functions that fail the InformationCost axioms. Each of those is a separate question, and the declaration itself does not settle any of them.

In Recognition Science, this uniqueness statement is the hinge for a chain of derived results: the golden ratio as a self-similar scaling, an eight-tick recognition cycle, and the number 2^3. But the declaration itself does not contain those consequences. It is a single quantified equation, and the framework's own docstrings mark the full unification with the aggregator as a TODO. The declaration is a verified building block, not a complete theory of information.

/-- **HYPOTHESIS**: J-Cost Uniqueness.
    The J-cost is the unique symmetric minimal information cost.

    STATUS: SCAFFOLD — Proof established in `Information.JCostNecessity`.
    TODO: Fully unify the uniqueness theorem with the aggregator. -/
def H_UniquenessVerified : Prop :=
  ∀ (F : ℝ → ℝ), InformationCost F → (∀ x > 0, F x = Cost.Jcost x)

-- Legacy axiom eliminated. See CostUniqueness.T5_uniqueness_complete.

/-- **HYPOTHESIS**: J-Cost Uniqueness.
    The J-cost is the unique symmetric minimal information cost.

    STATUS: SCAFFOLD — Proof established in `Information.JCostNecessity`.
    TODO: Fully unify the uniqueness theorem with the aggregator. -/
def H_UniquenessVerified : Prop :=
  ∀ (F : ℝ → ℝ), InformationCost F → (∀ x > 0, F x = Cost.Jcost x)

-- Legacy axiom eliminated. See CostUniqueness.T5_uniqueness_complete.

What this page does not claim

H_UniquenessVerified does not prove that J is the unique information cost; the proof is attributed to a separate module, not contained in this declaration. The declaration does not claim uniqueness among all functions, only among those satisfying the InformationCost property. It does not assert anything about the golden ratio, the eight-tick cycle, or any derived consequence; those are separate results in the framework.

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/Information.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

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