Foundation Primitive Recognition Calculus Delta Probability

Finite distinction probability

Probability is a ratio of counts. Given a finite set of N+1 distinct alternatives, an event is a subset of those alternatives. Its probability is the number of alternatives in the subset divided by the total N+1. For example, if there are five alternatives and an event picks out two of them, its probability is 2/5. This is the classical definition of probability on a finite sample space, and it is the only notion of probability used here.

The basic properties any probability must have are established. The impossible event, which picks out nothing, has probability zero. The certain event, which picks out every alternative, has probability one. Every event has probability between zero and one inclusive. If one event is contained in another, its probability is no larger. And if two events share no alternatives, the probability of their union is the sum of their individual probabilities. These are the standard axioms of finite probability, proved for the specific setting of finite distinction spaces.

Recognition Science, the framework that builds physics from a discrete record of events, calls these finite alternatives distinction events: each event is a yes-or-no distinction among a fixed finite set of points. The headline theorem packages all five properties into one statement: at this native finite layer, probability is rational counting over finite distinction alternatives. The word "rational" matters: every probability here is a fraction of two whole numbers, not a real number from a continuum. The framework's deeper layers may introduce real-valued measures, but this foundation stays discrete.

The payoff is a clean base for everything later. Because the framework's cost function and forcing chain begin from finite recognition events, having a rigorous, machine-checked theory of probability on those events means the later continuous structures do not smuggle in assumptions about chance. This is a small, complete piece: it defines the objects, proves the standard laws, and stops. A reader who understands finite counting understands everything proved here.

/-- The impossible event has probability zero. -/
theorem prob_empty (N : ℕ) : prob (N := N) (fun _ => False) = 0 := by
  classical
  simp [prob, count_empty]

/-- The certain event has probability one. -/
theorem prob_univ (N : ℕ) : prob (N := N) (fun _ => True) = 1 := by
  classical
  have h : ((N + 1 : ℚ) ≠ 0) := by positivity
  rw [prob, count_univ]
  rw [show (((N + 1 : ℕ) : ℚ)) = (N + 1 : ℚ) by norm_num]
  exact div_self h

/-- **Delta-native probability headline.** Probability at the native finite layer
is rational counting over finite distinction alternatives: impossible event zero,
certain event one, and every event has probability in `[0,1]`. -/
theorem delta_probability_headline (N : ℕ) :
    prob (N := N) (fun _ => False) = 0
      ∧ prob (N := N) (fun _ => True) = 1
      ∧ (∀ E : Event N, 0 ≤ prob E ∧ prob E ≤ 1)
      ∧ (∀ E F : Event N, (∀ i, E i → F i) → prob E ≤ prob F)
      ∧ (∀ E F : Event N, (∀ i, ¬ (E i ∧ F i)) →
          prob (fun i => E i ∨ F i) = prob E + prob F) :=
  ⟨prob_empty N, prob_univ N, fun E => ⟨prob_nonneg E, prob_le_one E⟩,
    fun _ _ h => prob_mono h, fun _ _ h => prob_disjoint_or h⟩

/-- Event inclusion gives probability monotonicity. -/
theorem prob_mono {N : ℕ} {E F : Event N} (h : ∀ i, E i → F i) : prob E ≤ prob F := by
  unfold prob
  have hden : (0 : ℚ) < (N + 1 : ℚ) := by positivity
  have hcount : ((count E : ℚ) ≤ (count F : ℚ)) := by
    exact_mod_cast count_mono h
  exact div_le_div_of_nonneg_right hcount (le_of_lt hden)

/-- Disjoint finite events have additive probability. -/
theorem prob_disjoint_or {N : ℕ} {E F : Event N}
    (hdisj : ∀ i, ¬ (E i ∧ F i)) :
    prob (fun i => E i ∨ F i) = prob E + prob F := by
  unfold prob
  rw [count_disjoint_or hdisj]
  rw [Nat.cast_add]
  ring

What this page does not claim

This module does not derive the cost function J or any part of the forcing chain. This module does not define probability on infinite or continuous spaces. This module does not introduce any empirical or measured probability values.

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/PrimitiveRecognitionCalculus/DeltaProbability.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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