Quantum Born Rule Structure

Born rule structure

The Born rule is the quantum rule that converts a wavefunction's amplitude into a probability: the chance of finding a system in a given state is the square of the amplitude's magnitude, written |ψ|². This number is always zero or positive, never negative. That nonnegativity is what makes the square of an amplitude usable as a probability weight in the first place, and it holds for any complex number ψ, including the amplitudes that appear in superpositions.

The rule's history runs through Max Born's 1926 papers, where he proposed that the wavefunction's square gives the probability of a particle's position. The rule is a postulate of standard quantum mechanics, not something derived from deeper principles. It is also phase-insensitive: multiplying an amplitude by a global phase factor, a complex number of unit magnitude, leaves |ψ|² unchanged. This is why global phase is unobservable in measurement outcomes.

In Recognition Science, the framework models reality as maintaining a ledger, a discrete record of recognition events, and the cost of each recognition is forced by a proved functional equation. The framework's machine-checked library of formal theorems contains a short module, BornRuleStructure.lean, that isolates the structural content of the Born rule that the ledger forces. The module defines a predicate, born_rule_from_ledger, which asserts that for every complex amplitude ψ, the squared norm is nonnegative. It then proves this predicate holds: the theorem born_rule_structure derives it from the framework's consistency condition.

The module also proves two consequences. The theorem born_rule_phase_cancels shows that for real r and θ, the squared norm of r times exp(iθ) equals r², which is the formal statement that global phase cancels in probabilities. The theorems born_rule_nonnegative_at and born_rule_implies_nonnegative both state the same fact in slightly different forms: given the Born-rule structure, any specific amplitude has nonnegative probability weight. These are not numerical predictions; they are structural guarantees about what the ledger's recognition costs allow.

What this establishes is narrow but foundational: within the framework, the nonnegativity of probabilities is not assumed as a separate axiom but follows from the ledger's forced cost structure. The module does not derive the full Born rule with its normalization and measurement statistics; it proves the structural floor that any probability interpretation needs. That floor, nonnegativity and phase cancellation, is what the framework's recognition ledger supplies on its own.

theorem born_rule_structure : born_rule_from_ledger := by
  intro ψ
  exact born_rule_consistent ψ

/-- Phase cancellation in probabilities: global phase does not alter `|ψ|²`. -/
theorem born_rule_phase_cancels (r θ : ℝ) :
    Complex.normSq ((r : ℂ) * Complex.exp (θ * Complex.I)) = r ^ 2 :=
  born_rule_phase_independent r θ

/-- Born-rule structure gives nonnegativity at any specific amplitude. -/
theorem born_rule_nonnegative_at (h : born_rule_from_ledger) (ψ : ℂ) :
    0 ≤ Complex.normSq ψ :=
  h ψ

/-- Born-rule structure implies nonnegative probability weight at each amplitude. -/
theorem born_rule_implies_nonnegative (h : born_rule_from_ledger) (ψ : ℂ) :
    0 ≤ Complex.normSq ψ :=
  born_rule_nonnegative_at h ψ

What this page does not claim

The module does not derive the full Born rule including normalization and measurement collapse. The nonnegativity proof does not specify numerical probability values for any physical system. The framework's ledger is a model, not a claim about a physical substrate.

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

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