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 the State Is Pure — RCD Program, Act I, Technical Note I-B0 (v0.1)

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.