Analytical microscopic matching for the interacting XXZ decoder

Establish an analytical microscopic matching for the outcome-resolved conformal decoder in the interacting critical spin- XXZ chain, deriving why the unchanged conformally weighted snapshot coordinate b4_L(m) determines the conditional Rb1nyi entropy through the compact-boson prediction with coupling g(94).

Background

The paper proves and benchmarks the outcome-resolved relation most explicitly at the free-fermion XX point, where exact conditional Gaussian formulas, replica factorization, and a two-cut Coulomb-gas representation connect the microscopic measurement record to the compact conformal coordinate b4_L(m).

For the interacting critical XXZ chain, the authors retain the same microscopic coordinate without interaction-dependent retuning and compare conditional entropies against the compact-boson function Fb1^g(h,b4_L), using the known Luttinger coupling g(94). Exact diagonalization and matrix-product-state calculations support this relation numerically, but the free-fermion microscopic derivation does not apply once determinantal structure is absent. The unresolved problem is therefore to derive analytically, directly from the interacting XXZ lattice model, the identification of b4_L(m) with the compact boundary zero mode and the resulting entropy decoder.