No Free Actualization: The Canonical Structure Derived from a Symmetric State Determines the Law of Elementary Registers but Not the Realized Sample — Deterministic No-Go, Instrument Non-Uniqueness, a Smoothing Detailed-Balanced Canonical Closure with Exact Spectrum, and Discontinuity at Spectral Coalescence — RCD Program, Act I, Technical Note I-B1 (v0.1)

Technical Note I-B1 of the RCD program, Act I (continues I-B0, DOI 10.5281/zenodo.23965761, and I-A0/I-A1, DOI 10.5281/zenodo.23961108). I-B0 derived, from the frozen input of Act I (an algebra M_d, a state ρ and its full symmetry group G₀), the space 𝒪_ρ of elementary distinctions, its canonical law μ_ρ, the register interface E_ρ and the channel 𝓜_ρ. I-B1 asks whether that structure can produce a concrete record — operationally a pure state δ_Q of the classical algebra C(𝒪_ρ) — without selection, clock, asymmetry or primitive randomness. Result (theorems with numerical control on 17 states in d = 2, 3, 4; no hardware): the No-Free-Actualization Theorem, with four distinct and complementary obstructions. (1) No deterministic G₀-equivariant rule takes μ_ρ to any δ_Q when rank ρ > 1; a concrete record reduces the state's stabilizer from G₀ to H_Q. (2) Every instrument compatible with E_ρ is measure-and-prepare (Haapasalo–Pellonpää), the covariant ones correspond to stabilizer-invariant states, and the segment tQ + (1−t)ρ is covariant with identical statistics: POVM plus symmetry do not fix the post-measurement update; the instrument preparing Q is selected by a declared consistency condition (H_PB, Petz–barycenter), legitimized by the unique affine weak-* continuous extension of the Petz recovery. (3) The canonical closure T_ρ = 𝓜_ρ∘𝓑_ρ is a smoothing kernel (T_ρν ≪ μ_ρ for every ν, never a delta), positive (0 ≤ T_ρ ≤ I) and reversible (detailed balance) with respect to μ_ρ; for ρ = P/r its spectrum is {1, 1/(r+1) with multiplicity r² − 1, 0} with exact contraction (1/(r+1))ⁿ⁻¹ after the first step and D(Q‖𝓑_ρ(T_ρδ_Q)) = ln((r+1)/2); in d = 2 the spectrum is {1, √(λ₁λ₂) double, 0}, jumping from 1/2 to 1/3 at degeneracy; and a coalescence theorem shows μ_ρ and its second moment discontinuous at every coalescence of positive eigenvalues (Var Tr(P₁Q) jumps from 0 to λ²n₁n₂/(n₁+n₂+1)); every possible distinction lies at the same distance from the barycenter, ‖Q − ρ‖² = 1 − Tr ρ². (4) μ_ρ is realized as Q = gQ₀g* with g Haar-distributed on G₀: the structure derives the law of the sample, not the sample. Stated boundary: it is not claimed that no extension of RCD can derive the event, nor that the event is a fundamental primitive; μ_ρ ⇝ δ_Q is recorded as an open debt; I-A/I-B derive new structure from the Act I input but do not derive that input from a structureless coherence (an upstream question). For I-C the event enters only as a conditional input, and the next wall is stated: can an event be registered without consuming the coherence needed for the next one? Includes an erratum for the two earlier deposits (17 control states, not 16; 14 of rank > 1, not 13). Scripts with SHA-256 manifest; the original run code with a corrected spectral-control error is kept for traceability. 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-A1 report stated 16 control states; there are 17 (3 in d = 2, 5 in d = 3, 9 in d = 4; 3 pure and 14 of rank > 1), all contained in ia1.out and all passing the gates. Documentary error with no effect on any result or theorem; the I-A0 count (13 cases) is correct. This version corrects the note and the I-A1 report; code and outputs are unchanged."