eventRatio
eventRatio supplies the likelihood ratio of two events inside a finite weighted Boolean reality by dividing their summed positive weights. Researchers embedding discrete Boolean structures into the positive reals for the LogicAsFunctionalEquation bridge cite it when moving from event weights to ratio comparisons. The definition is realized as a direct quotient of the two eventWeight sums.
claimFor a finite weighted Boolean reality $R$ over $n$ elements (with strictly positive component weights) and events $A,B : {0,1}^n$, the event ratio is $w(A)/w(B)$, where $w(E) = sum_i w_i$ if $E(i)$ holds and $0$ otherwise.
background
The module establishes a discrete-to-continuous bridge: finite Boolean events equipped with positive weights are compared via likelihood ratios that remain positive whenever both events have positive measure. FiniteBooleanReality is the structure consisting of a weight function $Fin n to R$ together with the axiom that every weight is strictly positive. eventWeight computes the measure of an event by summing the component weights exactly where the Boolean indicator is true.
proof idea
One-line definition that divides the eventWeight of A by the eventWeight of B.
why it matters in Recognition Science
This definition is the immediate prerequisite for eventRatio_pos and finite_boolean_logic_embeds_into_positive_ratios, which prove that nonempty events produce positive ratios and thereby embed finite Boolean logic into the positive-ratio domain. It supplies the concrete comparison map required by the module's stated purpose of bridging discrete Boolean realities to the continuous setting used in the Recognition Science functional equation.
scope and limits
- Does not guarantee that the ratio is positive.
- Does not extend beyond finite n.
- Does not incorporate J-cost or phi-ladder arithmetic.
- Does not assume any particular distribution on the weights beyond positivity.
formal statement (Lean)
52noncomputable def eventRatio {n : Nat} (R : FiniteBooleanReality n)
53 (A B : Fin n → Bool) : ℝ :=
proof body
Definition body.
54 R.eventWeight A / R.eventWeight B
55
56/-- Nonempty event comparisons land in positive ratios. -/