probability_equals_weight
plain-language theorem explainer
probability_equals_weight establishes that the probability of outcome i equals the squared amplitude weight for any finite-dimensional quantum state. Foundational quantum researchers seeking a ledger-based derivation of the Born rule would cite the result. The equality follows by reflexivity from the definition of measurementProbability.
Claim. For any natural number $n$, normalized quantum state $ψ$ with amplitudes $ψ : Fin n → ℂ$, and index $i$, the measurement probability of outcome $i$ satisfies $P(i) = |ψ_i|^2$.
background
QuantumState is the structure carrying amplitudes $Fin n → Amplitude$ together with the normalization condition that the sum of squared moduli over the basis equals one. In the module setting, superposition corresponds to an uncommitted ledger entry while measurement forces commitment to a single branch, with outcome probabilities arising as recognition weights. Upstream results supply the amplitude representation via the QuantumState definitions in QuantumLedger and NoCloning, plus the Measurement structure for the empirical interface.
proof idea
The proof is a direct term-mode reflexivity. It holds because measurementProbability is defined to return exactly the squared norm of the amplitude at index i, making the stated equality an instance of definitional equality.
why it matters
This supplies the explicit identification of Born-rule probabilities with ledger weights inside the derivation of the measurement postulate from Recognition Science ledger commitment. The module targets wavefunction collapse by equating probability to recognition cost rather than postulating it, supporting the irreversibility argument that follows. It touches the framework step where |ψ|² encodes relative J-cost on the phi-ladder.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.