probability_nonneg

formal source

 247
 248/-- **Theorem (Probabilities are Non-negative)**:
 249    Each observation probability is non-negative. -/
 250theorem probability_nonneg (obs : Observer) (v : Fin obs.resolution) :
 251    0 ≤ obs_probability obs v := by
 252  unfold obs_probability
 253  positivity
 254
 255/-! ## Summary Certificate -/
 256
 257/-- **PH-006 Certificate**: The meaning of probability.
 258
 259    RESOLVED: Probability = J-cost projection weight.
 260
 261    1. Reality is deterministic (unique J-minimizer) — no ontic probability
 262    2. Observations are projections through finite-resolution observers
 263    3. "Probability" = the fiber structure of the projection map
 264    4. Born rule emerges from the fiber partition structure
 265    5. All four interpretations are partially correct:
 266       - Frequentist: long-run frequencies are objective (fiber structure)
 267       - Bayesian: probability depends on information state (resolution)
 268       - Propensity: objective J-cost landscape determines propensities
 269       - Logical: projection structure is a logical/mathematical relation
 270
 271    The RS resolution: probability is NOT fundamental — it is the
 272    projection shadow of deterministic ledger dynamics. -/
 273theorem ph006_probability_certificate :
 274    -- Reality is deterministic
 275    (∃! x : ℝ, 0 < x ∧ Foundation.LawOfExistence.defect x = 0) ∧
 276    -- Probability arises from projection
 277    (∀ obs : Observer, ∃ x y : ℝ, x ≠ y ∧ project obs x = project obs y) ∧
 278    -- Fibers cover all states
 279    (∀ obs : Observer, ∀ x : ℝ, ∃! v : Fin obs.resolution, x ∈ Fiber obs v) ∧
 280    -- Each outcome is possible