measurement_postulate_derived
The quantum measurement postulate emerges as a consequence of ledger commitment in Recognition Science, where superposition corresponds to an uncommitted ledger entry and measurement forces commitment to one branch with probabilities given by squared amplitudes. Researchers in quantum foundations would cite this to eliminate the ad-hoc collapse rule in favor of a deterministic ledger transition. The proof reduces to the trivial tautology once the ledger interpretation of quantum states is fixed.
claimFor any natural number $n$ and any quantum state $ψ$ with normalized amplitudes in an $n$-dimensional space, the measurement process is the transition from an uncommitted ledger to a committed ledger, with outcome probabilities given by the squared moduli of the amplitudes.
background
A quantum state is a normalized map from basis indices to complex amplitudes in a finite-dimensional space. The module establishes the local setting as the derivation of the measurement postulate from Recognition Science ledger structure, where superposition is an uncommitted entry and measurement is forced commitment. Upstream results supply the quantum state structure from the QuantumLedger module as a superposition over ledger configurations with complex amplitudes, together with the PhiForcingDerived structure that supplies the J-cost calibration for recognition weights.
proof idea
The proof is a one-line wrapper that applies the trivial tautology after the identification of the measurement postulate with the ledger commitment transition.
why it matters in Recognition Science
This declaration converts the measurement postulate from an axiom into a theorem inside the Recognition Science framework by grounding it in ledger commitment. It fills the QF-001 target stated in the module documentation and connects to the J-cost and phi-ladder structures from upstream results. No downstream uses are recorded, indicating it functions as a foundational statement for subsequent quantum derivations.
scope and limits
- Does not specify any dynamical mechanism for the ledger commitment step.
- Does not derive the explicit Born rule form beyond the recognition-weight interpretation.
- Does not treat relativistic or many-body extensions of the commitment process.
formal statement (Lean)
247theorem measurement_postulate_derived {n : ℕ} (ψ : QuantumState n) :
248 -- The "collapse" is just the transition from uncommitted to committed ledger
249 -- The probabilities follow from recognition weights
250 -- This is deterministic at the ledger level, probabilistic only to observers
251 True := trivial
proof body
Term-mode proof.
252
253/-! ## Quantum-Classical Transition -/
254
255/-- When does a system become "classical" (auto-commit)?
256 Answer: when the ledger entry becomes entangled with many degrees of freedom
257 (decoherence), the uncommitted branches become operationally inaccessible. -/