measurementProbability
The definition assigns to each basis outcome i the squared modulus of its amplitude inside a normalized finite-dimensional quantum state ψ. Researchers deriving the measurement postulate from ledger commitment in Recognition Science cite it as the explicit Born-rule probability map. It is realized by a direct one-line projection onto the amplitudes field of the QuantumState structure.
claimLet ψ be a normalized quantum state in an n-dimensional Hilbert space with complex amplitudes ψ_i. The probability of obtaining measurement outcome i is |ψ_i|^2.
background
QuantumState is the structure recording a finite list of complex amplitudes together with the normalization condition that the sum of squared moduli equals one. The module frames superposition as an uncommitted ledger entry whose branches coexist until measurement forces commitment to a single definite value. Probabilities are then extracted from the relative recognition costs of those branches, yielding the Born rule as P(i) proportional to |amplitude_i|^2.
proof idea
One-line definition that returns the squared norm of the amplitude component at index i inside the QuantumState record.
why it matters in Recognition Science
It supplies the concrete probability function required by the downstream non-negativity, normalization, and cost-probability theorems. Within the Recognition Science framework it realizes the module claim that measurement probabilities arise directly from J-cost weights on ledger branches, closing the derivation of the measurement postulate from the eight-tick ledger structure.
scope and limits
- Does not prove non-negativity or normalization of the resulting probabilities.
- Does not extend to continuous spectra or infinite-dimensional Hilbert spaces.
- Does not derive the dynamical mechanism of wavefunction collapse itself.
formal statement (Lean)
147noncomputable def measurementProbability {n : ℕ} (ψ : QuantumState n) (i : Fin n) : ℝ :=
proof body
Definition body.
148 ‖ψ.amplitudes i‖^2
149
150/-- **THEOREM (Born Rule)**: Probabilities are non-negative. -/