stateToLedger
The conversion maps a normalized quantum state on a finite basis to an uncommitted ledger whose branches list the nonzero amplitudes together with their squared moduli as weights. Researchers formalizing the emergence of wavefunction collapse from ledger commitment would cite this map. The definition filters the support of the amplitude function and assembles the corresponding branch records while confirming normalization via a summation identity.
claimLet $ψ$ be a normalized quantum state with amplitudes $ψ_i$ for $i ∈ {0,…,n−1}$ satisfying $∑_i |ψ_i|^2 = 1$. The associated uncommitted ledger consists of the list of branches $(i, ψ_i, |ψ_i|^2)$ for all $i$ such that $ψ_i ≠ 0$, with the property that the sum of the weights equals one.
background
In this module addressing the measurement problem, a quantum state is a structure that assigns an amplitude from the complex numbers to each element of a finite basis of size n, subject to the normalization condition that the sum of the squared moduli of all amplitudes equals one. An uncommitted ledger is a list of branches, each a triple consisting of a basis index, an amplitude, and a weight, normalized so that the sum of weights is one. This construction operates within the Recognition Science resolution of the measurement problem, where superposition is identified with an uncommitted ledger containing multiple branches, and measurement corresponds to commitment of the ledger to a single branch. The probabilities arise from the J-cost associated with each branch. Upstream results include the structure of quantum states as superpositions over ledger configurations and the derivation of spectral properties from phi-forcing.
proof idea
The definition constructs the ledger by first filtering the finite set of all indices to retain only those with nonzero amplitude. It then maps each retained index to a ledger branch record containing the index, the amplitude value, and the squared modulus as weight. The proof of normalization relies on the identity that the sum of weights after filtering and mapping equals one, supplied by the filter_map_weight_sum lemma.
why it matters in Recognition Science
This definition provides the explicit bridge from the quantum state structure to the uncommitted ledger representation central to deriving the measurement postulate. It realizes the module's core insight that a superposition corresponds to multiple coexisting branches in the ledger's working memory. The map supports the subsequent derivation of the Born rule from recognition weights, aligning with the framework's target of obtaining the measurement postulate from ledger commitment without additional axioms. It leaves open the precise dynamics of the commitment step itself.
scope and limits
- Does not define the commitment operation that selects one branch upon measurement.
- Does not derive the specific form of the probability weights from the J-cost equation.
- Does not extend to infinite-dimensional state spaces.
- Does not connect to the phi-ladder mass formula or eight-tick periodicity.
formal statement (Lean)
141noncomputable def stateToLedger {n : ℕ} (ψ : QuantumState n) : UncommittedLedger n :=
proof body
Definition body.
142 ⟨(Finset.univ.filter (fun i => ψ.amplitudes i ≠ 0)).toList.map
143 (fun i => ⟨i, ψ.amplitudes i, ‖ψ.amplitudes i‖^2, rfl⟩),
144 filter_map_weight_sum ψ⟩
145
146/-- Probability of measuring outcome i from state ψ (Born rule). -/