Split-Regularized Recoverability in Type III AQFT: Conditional Expectations, Split-Dependent CMI, and an Audit-Friendly Recoverability Contract

Local algebras in relativistic quantum field theory are typically Type III, so reduced density matrices and von Neumann entropies are not available without additional structure. We give a B-minimal, audit-friendly interface for recoverability in Type III AQFT: we fix a collar geometry and a split datum N (an intermediate Type I factor) and define (i) a split-regularized conditional mutual information (CMI) and (ii) a Bures-fidelity-based recovery error for normal states. We isolate, as explicit assumptions, the two hard bridges needed for an exponential recoverability statement in Type III: (a) existence of an omega_0-preserving conditional expectation onto N (a Takesaki-type condition) and (b) an FR-type inequality in the fixed split implementation. We prove a conditional theorem: if split-regularized CMI decays exponentially in the collar width and an FR-type inequality holds in that split, then recoverability error decays exponentially, with constants tracked explicitly. This paper makes no Clay mass-gap claim and does not invoke von Neumann entropy on Type III algebras without split regularization. v2 (no v1 number is changed): the cross-reference labels are repaired (in v1 every assumption and theorem was typeset as "Definition x.y") and Table 1 is re-typeset; a direction slip in the recovery candidate is corrected -- under CE the GNS-adjoint of the conditional expectation is provably the inclusion iota: N into M, so the operational candidate aligned with the recovery task is the predual of the conditional expectation itself; the finite-dimensional reduction is proved rather than remarked, including the equivalence of the two split-regularized CMI definitions and c_FR = 1 in the fixed conventions; a constructive instance of CE with product reference state is proved; series positioning is added; and a verification suite instantiates the entire contract end to end in the Type I regime (gapped transverse-field Ising collar: CMI decay with alpha ~ 1.06, Petz-type reconstruction satisfying E_rec <= I^(N) at every width in the generated dataset, and the omega_0-adjoint identity E^# = iota at machine precision).