norm_div_norm_eq_one
plain-language theorem explainer
The theorem states that for any nonzero complex number z the modulus of z divided by its own modulus equals one. Researchers formalizing wavefunction collapse as ledger commitment in Recognition Science would cite it when normalizing amplitudes prior to branch selection. The proof is a short tactic sequence that rewrites the quotient norm, notes that the modulus of a real scalar is itself, and applies the division rule for nonzero denominators.
Claim. For every nonzero complex number $z$, $||z/||z||| = 1$.
background
Recognition Science treats quantum states as ledger entries whose uncommitted branches represent superpositions. Amplitudes carry recognition weights whose squares determine collapse probabilities via the J-cost. The present auxiliary result ensures that division by the modulus produces a vector whose norm is exactly one, so that the committed ledger preserves total weight one after measurement.
proof idea
The tactic proof introduces the nonzero complex z, rewrites the norm of the quotient with norm_div, records that the norm of the real scalar ||z|| equals ||z|| itself via Complex.norm_real, and concludes by exact application of div_self to the nonzero denominator.
why it matters
The result is invoked inside the commit definition that models wavefunction collapse as selection of one ledger branch. It supplies the unit-norm condition required for the Born-rule interpretation derived from J-cost in the QF-001 program. The declaration therefore closes the normalization step that lets the committed ledger inherit the total recognition weight of the original superposition.
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