Foundation Ledger Comparison To Composition Comparison Cost Self Zero

The self-comparison cost

The declaration comparison_cost_self_zero is a machine-checked theorem in the Recognition Science framework's library of formal theorems. It states that when a recognition cost, a number assigned to the act of comparing two observable states, is normalized, the cost of comparing a state to itself is exactly zero. The proof is short: self-comparison yields the unit ratio, and a normalized cost assigns zero to that unit ratio. The theorem is not an assumption; it is derived from the definition of the comparison object and the normalization condition.

The comparison object itself is a positive ratio. In the framework, a closed observable framework carries a strictly positive observable r for each state. The comparison between two states s₁ and s₂ is the ratio r s₁ / r s₂. This ratio is strictly positive, so the domain of the cost is the positive ray. Swapping the two states inverts the ratio, which is the geometric origin of reciprocal symmetry. Self-comparison, where both states are the same, gives the ratio 1, the unit ratio. The theorem comparison_cost_self_zero uses exactly this fact: since self-comparison is the unit ratio, a normalized cost, one that assigns zero to the unit ratio, must assign zero to self-comparison.

The theorem is one piece of a larger closure. The framework's library proves that a comparison cost that is reciprocal, normalized, calibrated, and continuous, whose symmetric combination is realized through a primitive ledger-posting combiner, is forced to equal J(x) = (x + 1/x)/2 - 1. The self-zero theorem supplies the normalization condition in that chain. It does not, by itself, force the form of J. It only establishes that a normalized cost vanishes on self-comparison. The full forcing requires the other conditions and the composition law.

What the theorem does not claim is equally precise. It does not claim that self-comparison is free in any operational sense, only that the cost function assigns zero to it. It does not claim that the comparison ratio is always 1, only that it is 1 when the two states are the same. It does not claim that the cost J is the only possible cost, only that a normalized cost vanishes on the unit ratio. The theorem is a lemma in a larger proof, not a standalone statement about the world.

/-- **Normalization realized on the ledger.**  A normalized cost vanishes on
self-comparison, since self-comparison is the unit ratio. -/
theorem comparison_cost_self_zero (F : ClosedObservableFramework)
    (J : ℝ → ℝ) (hJ : IsNormalized J) (s : F.S) :
    J (compRatio F s s) = 0 := by
  rw [compRatio_self]
  exact hJ

/-- **Self-comparison is the unit ratio.** -/
theorem compRatio_self (F : ClosedObservableFramework) (s : F.S) :
    compRatio F s s = 1 :=
  div_self (ne_of_gt (F.r_pos s))

/-- **The comparison object is a positive ratio.**  Read off the framework's
strictly positive observable `r`: the domain of the recognition cost is the
positive ray, derived rather than assumed. -/
theorem compRatio_pos (F : ClosedObservableFramework) (s₁ s₂ : F.S) :
    0 < compRatio F s₁ s₂ :=
  div_pos (F.r_pos s₁) (F.r_pos s₂)

/-- **State swap inverts the comparison ratio.**  Swapping the two states sends the
ratio to its reciprocal, the geometric origin of reciprocal symmetry. -/
theorem compRatio_swap (F : ClosedObservableFramework) (s₁ s₂ : F.S) :
    compRatio F s₂ s₁ = (compRatio F s₁ s₂)⁻¹ := by
  unfold compRatio
  rw [inv_div]

What this page does not claim

The theorem does not claim that self-comparison is free in an operational sense, only that the cost function assigns zero to it. The theorem does not claim that the comparison ratio is always 1, only that it is 1 when the two states are the same. The theorem does not claim that J is the only possible cost, only that a normalized cost vanishes on the unit ratio.

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/LedgerComparisonToComposition.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: