Foundation Born Rule Forcing Fourth Power Sum Pos Of Normalized

A small positivity fact

In quantum mechanics, the Born rule says that the probability of finding a system in a particular state is the square of the state's amplitude. The declaration fourth_power_sum_pos_of_normalized proves a small but necessary fact about that rule: for any normalized eight-component signal, the sum of the fourth powers of the component magnitudes is strictly positive.

Here, a normalized signal means its total squared magnitude equals one, a standard condition that makes the signal a valid probability distribution. The lemma states that if the sum of squares of eight nonnegative magnitudes is 1, then the sum of their fourth powers is greater than 0. This is not a deep result on its own, but it is a required step in the framework's proof that the Born rule, not some other weighting, is forced.

The framework, called Recognition Science, models measurement as a discrete ledger of recognition events. Its machine-checked library of formal theorems proves that a unique probability measure on eight-mode sectors satisfies four conditions: normalization, phase invariance, additivity, and consistency with a two-branch calibration. That unique measure is the standard Born rule, the sum of squared amplitudes. The fourth-power positivity lemma is one of the technical bridges that makes that uniqueness proof go through.

It does not claim that the Born rule is the only possible rule in all of physics, nor does it say anything about the physical meaning of the fourth power itself. It is a formal, internal step within a larger derivation.

private theorem fourth_power_sum_pos_of_normalized (ψ : Signal8)
    (hψ : IsNormalized ψ) : 0 < ∑ j : Fin 8, ‖ψ j‖ ^ 4 := by
  have hexists : ∃ k : Fin 8, ψ k ≠ 0 := by
    by_contra hnone
    have hall : ∀ k : Fin 8, ψ k = 0 := by
      intro k
      by_contra hk
      exact hnone ⟨k, hk⟩
    have hsum : ∑ k : Fin 8, ‖ψ k‖ ^ 2 = 0 := by
      simp [hall]
    rw [hψ] at hsum
    norm_num at hsum
  rcases hexists with ⟨k, hk⟩
  exact Finset.sum_pos'
    (fun j _ => pow_nonneg (norm_nonneg (ψ j)) 4)
    ⟨k, Finset.mem_univ k, pow_pos (norm_pos_iff.mpr hk) 4⟩

/-- A signal ψ is normalised when the sum of squared norms is 1. -/
def IsNormalized (ψ : Signal8) : Prop :=
  ∑ k : Fin 8, ‖ψ k‖ ^ 2 = 1

What this page does not claim

This lemma does not prove the Born rule itself. It does not assign any physical meaning to the fourth power of amplitudes. It does not claim the Born rule is the only possible rule in all physical contexts.

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

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