Bravyi-Kitaev encoding for fermion-to-qubit mappings.

 The BK transform uses a Fenwick tree structure to achieve O(log n) Pauli
 weight per ladder operator, compared to O(n) for the Jordan-Wigner transform.

 This module delegates to MajoranaEncoding with Fenwick-tree index sets.

 Reference:
   Seeley, Richard, Love — "The Bravyi-Kitaev transformation for quantum
   computation of electronic structure" (arXiv:1208.5986)

 A purely functional Fenwick tree (binary indexed tree) parameterized by
 an associative combining operation.

 The tree stores partial aggregates over an array of n values, supporting:
   - point update   : O(log n)
   - prefix query   : O(log n)
   - range query    : via prefix difference when the combine has an inverse

 The index sets used by the Bravyi-Kitaev encoding (update, parity,
 occupation, remainder) all fall out naturally from the Fenwick tree
 structure, making this a good foundation for the BK transform.

 Generic fermion-to-qubit encoding via Majorana decomposition.

 Every encoding in this family maps a single fermionic ladder operator
 to a pair of Majorana operators (c, d) built from three index sets:

   U(j, n)  — update set:     qubits that flip when occupation of mode j changes
   P(j)     — parity set:     qubits encoding the parity n₀ ⊕ … ⊕ n_{j−1}
   Occ(j)   — occupation set: qubits encoding whether mode j is occupied

 The Majorana operators are:
   c_j = X_{U(j)∪{j}} · Z_{P(j)}
   d_j = Y_j · X_{U(j)} · Z_{(P(j)⊕Occ(j))∖{j}}

 And the ladder operators follow:
   a†_j = ½(c_j − i·d_j)     a_j = ½(c_j + i·d_j)

 Different choices of index-set functions yield different encodings:
   Jordan-Wigner :  U = ∅,              P = {0…j−1},     Occ = {j}
   Parity        :  U = {j+1…n−1},      P = {j−1}?,      Occ = {j−1,j}?
   Bravyi-Kitaev :  U, P, Occ from Fenwick tree structure

 General tree-based fermion-to-qubit encodings.

 This module implements two approaches:

 (1) Index-set approach (Havlíček et al. arXiv:1701.07072):
     The generic tree → index-set construction below is CAR-valid ONLY for
     rooted STAR trees (one central node adjacent to all others). An
     independent census over all rooted labelled trees finds exactly n of the
     n^(n−1) trees satisfy CAR for n=3..6 — precisely the rooted stars.
     Fenwick, chain, and balanced binary/ternary trees FAIL in general, and
     monotonicity of ancestor indices is neither necessary nor sufficient
     (a star rooted at node 0 passes with ancestor < child; a Fenwick tree
     fails with ancestor > child). This construction is a demonstration of the
     EncodingScheme abstraction, NOT the library's production tree encoder.

 (2) Path-based ternary-tree approach (Bonsai: arXiv:2212.09731,
     Jiang et al.: arXiv:1910.10746):
     Works for ANY ternary tree. Constructs Majorana strings directly
     from root-to-leg paths. Each node has 3 descending links labeled
     X, Y, Z.  Each leg yields a Pauli string by collecting the labels
     along its root-to-leg path. Paired legs give the two Majoranas
     for each fermionic mode. This is what the exported tree encoders
     (ternaryTreeTerms, balancedBinaryTreeTerms, vlasovTreeTerms) use.

 Canonical encodings are provided directly and do NOT rely on the generic
 index-set construction above:
   - Jordan-Wigner:  JordanWigner.fs (and jordanWignerScheme)
   - Bravyi-Kitaev:  BravyiKitaev.fs via a persistent Fenwick tree
   - Balanced ternary tree: O(log₃ n) Pauli weight via the path-based method