Asymptotic Sufficiency Depth, Volume VI: Projective Holonomy Contact and Observation-Relative Information Loss

Version 2 correction. In Theorem 3.5, Eqs. (18) and (19) now distinguish equality of the leading asymptotic terms of the forward and reverse hidden Kullback-Leibler losses from equality of their exact values. The two directions share the coefficient (1/2) Jπ,K(u) and, when this Fisher contraction defect is positive, the hidden-loss order 2q; their higher-order terms may differ. The revision also makes the finite-relative-entropy and reachable-output conventions explicit, corrects the reference to Definition 2.1, and updates the citations and equation locators for Volume V v2. An exact directional-asymmetry regression test is added. The projective contact classification and the subcritical, homogeneous-boundary, actual-peripheral, and formal-versus-topological results are unchanged. This correction is independent of Volume V's Stokes seed phase correction. Version 1 remains available as the earlier version. This volume develops a pairwise projective holonomy contact order and connects it, under an explicitly declared statisticalization and common observation, to observation-relative hidden Kullback-Leibler loss. For projectively quasi-unipotent continuation, repeated holonomy orbits admit polynomial compactifications at inverse iteration number; the vanishing order of the wedge of two regularized projective orbit germs gives a conjugacy-, orientation-, repetition-, and resampling-invariant contact order. For nilpotent index three, the finite contact orders are exactly 0, 1, 2, and 3. The analytic and statistical layers are kept separate. A finite strictly-positive projective probability embedding is constructed explicitly. If two projective orbit germs have finite contact order q ≥ 1 and the common observation hides their leading statistical tangent, the hidden directional Kullback-Leibler loss has leading order proportional to z2q, with coefficient given by the Fisher-information contraction defect. Thus analytic contact does not determine statistical Asymptotic Sufficiency Depth without an explicit probability realization and observation. For the negative-cross polynomial family Fp,q = X2p + Y2p + Z2p − X2qY2qZ2q, the volume classifies where nonsemisimple continuation memory resides. Below resonance, the physical-origin Borel peripheral monodromy is nonsemisimple; at p = 3q the normalized scalar object is projectively trivial; above resonance, size-three Jordan memory survives in formal irregular monodromy while actual peripheral topological monodromy is finite-order semisimple. The paper does not identify Borel sheets with observation fibers, Stokes matrices with Markov kernels, formal monodromy with actual peripheral monodromy, or analytic contact order with statistical ASD intrinsically. The manuscript, documentation, metadata, and non-code data are licensed under CC BY 4.0. The original source code and validation scripts are licensed under the MIT License; see LICENSES.md in the archives.