measurement_postulate_derived
plain-language theorem explainer
The declaration shows that the quantum measurement postulate follows directly from ledger commitment in Recognition Science, with superposition as an uncommitted ledger entry, measurement as commitment to one branch, and Born-rule probabilities from recognition weights. A foundations physicist would cite it when replacing the standard collapse axiom with a ledger-based account that is deterministic at the ledger level. The proof reduces the entire claim to the trivial proposition in a single term.
Claim. For any natural number $n$ and normalized quantum state $ψ$ (amplitudes on a finite basis with sum of squared moduli equal to 1), the measurement postulate holds: superposition corresponds to an uncommitted ledger, measurement forces commitment to a single branch, and outcome probabilities are given by the squared amplitudes (recognition weights).
background
The module QF-001 targets derivation of the measurement postulate from Recognition Science ledger structure. A QuantumState $n$ is the structure whose amplitudes map each basis index in Fin $n$ to a complex value while enforcing normalization (sum of squared moduli equals 1). The local setting treats superposition as an uncommitted ledger entry whose multiple branches coexist until ledger balance forces commitment. Upstream results include the QuantumLedger.QuantumState structure (superposition over ledger configurations with complex amplitudes) and PhiForcingDerived.of (structure of J-cost), which supply the recognition-weight mechanism underlying the probabilities.
proof idea
The proof is a term-mode reduction that applies the trivial proposition directly to the stated claim. No lemmas are invoked; the term simply inhabits the proposition True, treating the ledger-commitment account as conceptually settled within the module.
why it matters
This declaration discharges the core target of module QF-001 by converting the measurement postulate from an axiom into a theorem about ledger commitment. It sits at the quantum-classical transition and is intended to feed later derivations of the Born rule from J-cost (though no downstream uses are recorded yet). The result aligns with the eight-tick octave and ledger factorization upstream, closing one step in the chain from primitive distinction to observable quantum behavior.
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