From Coherence Alone: No Historical Order from Naked Samples, No New Cut, and the Complete Canonical Geometry of the Space of Cuts (Fubini–Study, Flat Three-Torus, Berger Sphere) — with an Impossibility Corollary for Precedence under Four Conditions

Technical Notes I-D and I-E0 of the RCD program, Act I (continues I-C, DOI 10.5281/zenodo.24023903; I-B1, 10.5281/zenodo.23990720; I-B0, 10.5281/zenodo.23965760; Act I, 10.5281/zenodo.23961107). The principal fixed the primitive of the program as coherence alone, (𝔄₀, ω₀, G₀) — an algebra, an invariant state and its symmetry group — postulated and not derived, with any extension (such as a differentiation gesture 𝒟_δ) to be declared explicitly and never counted as a derivation from coherence alone. Under that rule, three results in the finite bank (M_d, ρ = Σλ_aP_a, G₀ = ∏U(n_a)/U(1); 17 control states in d = 2, 3, 4; theorems with numerical control; no hardware). (I-D) Theorem of historical non-identifiability from naked samples: for two samples the two-point homogeneity of PU(r) on ℂP^{r−1} forces every covariant binary relation to be symmetric, so no non-trivial precedence exists; for a multiset of i.i.d. samples the law is exchangeable and the generation order is not identifiable (Pr(Π = π | M) = 1/n!); intrinsic shape orders can be manufactured but are not historical precedence; the only invariant odd under transposition is the Bargmann phase, which is cyclic (a tournament with 3-cycles in 100 % of configurations), not an order. I-D closes “naked sample”, not “differentiation”. (I-E0A) No-New-Cut Theorem: the G₀-invariant projections of M_d are the spectral projections already encoded in W*(ρ); inside a homogeneous block no new cut is selected; the only G₀-commuting automorphisms act by phases between blocks; M_d admits no proper unital endomorphism and no Cuntz pair. Every new concrete division is a sample or is put in by hand. (I-E0B) Theorem of the Canonical Geometry of the Space of Cuts: for every finite ρ, (M_d, ρ) determines the orbit 𝒪_ρ, its homogeneous measure and the Riemannian metric induced by Fubini–Study, ds² = Σ_a p_a‖du_a‖² − |Σ_a p_a⟨u_a,du_a⟩|² with p_a = n_aλ_a, with no free coefficient. In the tracial sector it is Fubini–Study on ℂP^{r−1} (unique up to scale by symmetry; distance arccos√Tr(QR); curvature 4 on holomorphic and 1 on totally real planes; volume equal to the law μ_ρ of I-A; Bargmann phase = 2∫ω_FS, the Pancharatnam holonomy of geodesic triangles). Outside it the space of cuts can be three-dimensional with its geometry completely fixed by the spectrum: a flat three-torus with metric matrix diag(p) − ppᵀ for a non-degenerate state in d = 4, and a Berger sphere S³ with horizontal scale p = 2λ₁ and Hopf-fibre scale pq = 2λ₁λ₂ for spectral multiplicities (2,1) in d = 3; verified to 10⁻¹⁴. Recorded as the geometry of the space of possible distinctions, not as physical space. Impossibility corollary: no mechanism starting only from (𝔄₀, ω₀, G₀) produces a non-trivial precedence, not encoded in the input or in a sample choice, with “before/after” and no clock. The endomorphic route (proper infiniteness as a candidate algebraic realization of the program's hypothesis H-1, “every part contains the whole”) remains a pre-brief without a run. Scripts with SHA-256 manifest; all run incidents and all charges of the adversarial rounds documented. English and Spanish versions; the Spanish text prevails.