Modal Simplicity: the primitive and its mathematical surroundings

Route B characterized covariant perfect anticorrelation as the singlet cosine E(θ_A, θ_B) = −cos(θ_A − θ_B), whose optimized CHSH value is 2√2, attaining the Tsirelson bound, and reduced its foundational content to a single adopted axiom: Primitive Modal Simplicity (real-cyclic irreducibility — one active nontrivial harmonic sector). This paper maps the mathematical structure surrounding that primitive and reports the outcome of a dedicated attempt to derive it. Primitive Modal Simplicity is a local, selected-response condition; it is surrounded by two distinct global conditions — observable uniformity (End₀(V) irreducible), a genuine strengthening once the invariant mode is removed (w₀ = 0), and state homogeneity (G_cov transitive on pure states), a state-side condition the primitive does not imply, both holding for the physical qubit. We prove the observable side is sharply classified: for a nontrivial irreducible complex representation of a connected compact group with simple Lie algebra g, End₀(V) is irreducible if and only if (g, V) ≅ (su(d), d or d*); within this class observable uniformity implies state homogeneity, while a computed Sp(2) = USp(4) witness shows the converse false. Typing the primitive to the covariance group as PMS_G (real-cyclic irreducibility of the response's carrier), the same USp(4) carries, at the response level, an admissible perfectly anticorrelated dichotomic response whose cyclic carrier is reducible (5 ⊕ 10): state homogeneity does not force PMS_G for all admissible seeds, closing that implication in the negative while leaving a canonical-selector form open. State homogeneity is characterized as pure-state controllability by the covariance generators, and the residual Z_n ambiguity is resolved by automorphism transitivity (intrinsic non-selection). Finally, an external investigation asked whether a deeper informational principle grounds either condition. Its result is negative but precise: forgetting the designated covariance group provably loses the carrier (a non-identifiability theorem), and the one surviving canonical-enrichment escape — covariance completeness, identifying the covariance group with the connected reversible group — yields observable uniformity for a complex matrix factor, but only as a maximality choice with no independent informational ground. Primitive Modal Simplicity therefore remains adopted rather than derived, now with a precisely delimited conditional bridge and an explicit open problem. Code: the accompanying reproducibility suite (in modal_simplicity_note_package.zip) is licensed MIT; this manuscript is licensed CC BY 4.0.