Canonical Register Interface and Petz Lock: The Square-Root Measurement on the Canonical Orbit of Elementary Distinctions, Its Petz Recovery as the Barycentric Preparation, and the Loss of Relative Sufficiency for Every Concrete Distinction Unless th

Technical Note I-B0 of the RCD program, Act I (continues I-A0/I-A1, DOI 10.5281/zenodo.23961108, and MEM-0, DOI 10.5281/zenodo.23926847). Act I left wall I-A resolved in the finite species: the pair (M_d, ρ) canonically determines the orbit 𝒪_ρ of elementary distinctions (rank-one projections) whose Haar barycenter is ρ, selecting none of them. I-B0 asks whether those same data generate, on their own, a register interface — a covariant quantum-to-classical channel whose outcome space is not added from outside — that loses the recoverability of every concrete distinction, with no clock, no preferred basis and no external cut. Results (theorems with numerical control in d = 2, 3, 4 on sixteen states; no hardware): (i) Canonical Register Interface Theorem — E_ρ(dQ) = ρ^{-1/2} Q ρ^{-1/2} dμ_ρ(Q) is a G₀-covariant POVM on 𝒪_ρ integrating to the support (the square-root / pretty-good measurement of the canonical ensemble); the channel 𝓜_ρ(σ)(dQ) = Tr(σ E_ρ(dQ)) satisfies 𝓜_ρ(ρ) = μ_ρ; and its Petz recovery map with respect to ρ is exactly the barycentric preparation ν ↦ ∫Q dν of I-A1, so that the I-A1 barycenter and the I-B0 Petz recovery are the same object seen from both sides; the chain (M_d, ρ) → 𝒪_ρ → μ_ρ → E_ρ → 𝓜_ρ requires no new hypothesis; canonicity is self-aligned uniqueness: among covariant rank-one-seed POVMs on 𝒪_ρ, E_ρ is the only one whose response to Q peaks at Q itself (the others are its central relabelings and, only for ρ = P/2, an anti-aligned measurement exhibited in the review round). (ii) Petz Lock — D(Q‖ρ) = S(ρ) on the whole orbit; the gap Δ_rec = D(Q‖ρ) − D(𝓜_ρ(Q)‖μ_ρ) = χ − I(E_ρ) is Q-independent and vanishes iff rank ρ = 1: for rank greater than one the channel returns ρ but is not sufficient for any pair (Q, ρ), i.e. it does not fully preserve any concrete distinction relative to the reference coherence (not absolute irrecoverability). (iii) Exact benchmark Δ_rec(P/r) = H_r − 1 (harmonic number), equal to the Holevo gap of the accessible information of the uniform ensemble (Jozsa–Robb–Wootters); closed form in d = 2, Δ_rec = S(ρ) − ln((1+s)/2) − 1 + s with s = |λ₁ − λ₂|; and a discontinuity of the gap at spectral degeneracy, 2 ln 2 − 1 ≈ 0.386 versus 1/2 at ρ = I/2, because the stabilizer and the canonical orbit change. Stated boundary: no concrete record is adjudicated — 𝓜_ρ(ρ) = μ_ρ, not δ_Q, and no deterministic equivariant rule can select δ_Q; stochastic selection remains open; the realization of an outcome is the question of I-B1, not opened here. Declared heritage: pretty-good measurement, Petz recovery, Barnum–Knill, Holevo's covariant measurements; what is RCD's own is the input, since the ensemble is derived from symmetry rather than postulated. Scripts with SHA-256 manifest. English and Spanish versions; the Spanish text prevails. Erratum (v0.2, 5 November 2026): v0.1 of this note and v0.2 of the I-B0 report said “sixteen states” and “thirteen of rank > 1”; there are 17 control states (3 pure and 14 of rank > 1), all contained in ib0.out and all passing the gates. Documentary error with no effect on any result or theorem. This version corrects the note and the I-B0 report; code and output are unchanged.