Foundation Ledger Comparison To Composition Comp Ratio Self

Self-comparison as the unit ratio

A comparison ratio is the number you get by dividing one observable state by another. In a closed framework, each state carries a strictly positive observable value, and comparing a state to itself means dividing that value by itself. The declaration compRatio_self establishes the plain fact that this self-comparison always equals 1. It is a theorem in the machine-checked library of formal theorems, proved from the definition of the ratio and the positivity of the observable.

This unit ratio is the origin of a normalization condition on recognition costs. A recognition cost, a number assigned to a comparison that measures how much work it takes to recognize one state from another, is said to be normalized when it assigns zero cost to comparing a state with itself. Because self-comparison is the ratio 1, any normalized cost vanishes on it. The theorem comparison_cost_self_zero makes this explicit: if a cost function is normalized, then its value on the self-comparison ratio is 0.

The statement does not claim that all costs vanish on self-comparison, only those that are normalized by definition. It does not say that the ratio 1 is the only input on which a cost can be zero; other ratios could also map to zero under a particular cost function. It also does not assert that self-comparison is the same as equality of states, only that the ratio of a state's observable to itself is the number 1.

What this fact changes is the footing of the normalization condition. Instead of being an extra assumption added to a cost function, the vanishing on self-comparison is read off from the ledger: the discrete record of observable states and their comparisons. The unit ratio is not chosen; it falls out of the definition of a ratio. This makes normalization a derived property of any comparison cost built on a ledger, not a free postulate.

/-- **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))

/-- **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

What this page does not claim

That all cost functions assign zero to self-comparison, only those that are normalized. That self-comparison is the only ratio on which a cost can be zero. That the ratio 1 implies the two states are identical, only that their observable values are equal.

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)

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