Foundation Primitive Recognition Calculus Delta Probability Expectation Const

The constant expectation

In ordinary probability, if a random variable takes the same value at every possible outcome, its expectation, or average, is simply that value. The declaration expectation_const establishes exactly this for the framework's primitive finite layer: it proves that on a space with N plus one possible outcomes, the expectation of a function that returns the same rational number c at every outcome equals c. This is not a new physical law but a basic, necessary property of averages, stated and verified in the framework's machine-checked library of formal theorems.

The setting is deliberately minimal. An event is a finite distinction event, a yes or no question about a finite set of alternatives. Probability is defined as a rational counting ratio: the number of outcomes where the event holds divided by the total number of outcomes. The framework proves the standard properties of such finite probabilities, that the impossible event has probability zero, the certain event has probability one, and every event has probability between zero and one. The expectation of a rational-valued observable is then the sum of its values across all outcomes divided by the number of outcomes.

Within this finite setting, expectation_const is a theorem, not an assumption. It follows directly from the definitions: summing a constant N plus one times and dividing by N plus one leaves the constant unchanged. The proof is short and the statement is exact. The declaration does not extend to infinite probability spaces, where a constant function still has that constant as its expectation under the usual measure-theoretic definition, but the framework's proof here covers only the finite case it defines.

The consequence for a reader is simple: the framework's primitive probability layer is internally consistent at its most basic point. Averages behave as averages must, and the machine-checked library records that fact. This is a foundation stone, not a surprising discovery. It means that when the framework later builds more elaborate structures, the expectation of a constant observable will not be a hidden assumption but a proved lemma available for use.

theorem expectation_const {N : ℕ} (c : ℚ) :
    expectation (N := N) (fun _ => c) = c := by
  have h : ((N + 1 : ℚ) ≠ 0) := by positivity
  simp [expectation, Finset.sum_const]
  field_simp [h]

/-- Uniform finite probability, as a rational counting ratio. -/
noncomputable def prob {N : ℕ} (E : Event N) : ℚ :=
  (count E : ℚ) / (N + 1 : ℚ)

/-- **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⟩

What this page does not claim

This declaration does not define expectation for infinite or continuous probability spaces. It does not introduce a new physical law; it verifies a basic property of averages in a finite setting. It does not claim that all observables in the framework are constant functions.

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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