First Distinction and Canonical Minimal Orbit: No-New-Distinction Theorem and the Orbit of Elementary Distinctions Determined by a Symmetric State, with Uniqueness of the Covariant Measure under Ergodicity or for ρ = P/r — RCD Program, Act I, Technic

Technical Note I-A0/I-A1 of the RCD program, Act I (continues MEM-0, DOI 10.5281/zenodo.23926847). The Act I target sheet (v0.3, frozen, included) asks how a perfectly symmetric coherence can produce an inheritable distinction with no clock, no geometry and no external cut, and names three walls: I-A (Schur, fixed-point algebra), I-B (Petz, recoverability), I-C (Wiesbrock–Borchers, inheritance). This note closes the first two diligences of wall I-A in the finite control species: the algebra M_d(C), a state ω_ρ(A) = Tr(ρA) and the unchosen symmetry group G₀ = Aut(M_d, ω_ρ) ≅ ∏ U(n_a)/U(1). Results (theorems with numerical control in d = 2, 3, 4; no hardware): (I-A0) No-New-Distinction Theorem — the fixed-point algebra equals the double commutant {ρ}″ = W*(ρ) = ⊕ C P_a; no covariant distinction is born de novo: the maximally mixed state has none (and trivial modular flow), a non-tracial state has exactly the 2^{m−1} − 1 sums of spectral projections, a pure state exactly one; covariant channels preserve W*(ρ); general lemma: an invariant input cannot select, by an equivariant deterministic rule, a non-invariant member of an orbit. (I-A1, Theorem I) Canonical Minimal Orbit — the minimal projections (rank 1) fall into G₀-orbits classified by p_a(Q) = Tr(P_a Q); the Haar barycenter of an orbit is Σ (p_a/n_a) P_a; exactly one orbit has barycenter ρ, namely p_a = n_a λ_a, of dimension Σ_{λ_a>0}(2n_a − 1) − 1; symmetry selects none of its members but (M_d, ρ) determines the whole homogeneous space of elementary distinctions, with ρ as its barycenter (pure → a point; tracial → CP^{d−1}; ρ = P/r → CP^{r−1}). (I-A1, Theorem II) Measure — G₀-invariant measures on minimal projections with barycenter ρ correspond to measures ν on the simplex with mean p*; the measure is unique with no hypothesis iff ρ = P/r (p* a vertex), otherwise a convex set of covariant decompositions exists (covariant Schrödinger–HJW, explicit two-orbit mixtures exhibited); under ergodicity the measure is the homogeneous μ_ρ, unique for every ρ. First-moment identity ∫ Tr(QE) dμ_ρ = Tr(ρE) recorded as a linear detector, not as Born. Stated limits: finite species only; ergodicity is a declared hypothesis, not derived; nothing here is time, space, register, probability law or physical selection; I-B and I-C are not opened. Scripts with SHA-256 manifest. English and Spanish versions; the Spanish text prevails.