Six Boundaries, Four Resonances: Relative Normal Forms for Exceptional Series Diagrams

We study graphs built from a Lie bracket and an invariant metric, subject to the local relations of the classical exceptional series with parameter a. Over K = ℚ(a), we prove that the Morrison–Snyder–Thurston candidate lists span all such diagrams with at most six external legs, regardless of the number of internal vertices. Closed diagrams form a commutative coefficient ring S. The four-, five- and six-boundary spaces are S-modules isomorphic to S⁵, S¹⁶ and S⁸⁰, respectively. The proof classifies the abstract algebra of boundary operations on a possible six-boundary remainder as ℚ × M₅(ℚ) × M₅(ℚ) × M₁₅(ℚ), where Mᵣ(ℚ) denotes the algebra of rational r × r matrices. Its central element ξ, corresponding to a(1 − a) in the diagram presentation, acts on these factors by −6/25, −2, −6, −20, respectively. We also identify the endomorphism algebra of an explicit three-strand summand as a product of seven copies of S, describe the possible remainders at E₆ and E₈, and construct Lie superalgebra evaluations detecting diagrams killed by the corresponding ordinary Lie-algebra evaluations at A₁, A₂ and D₄. The normal forms are relative to S; generic nonvanishing S ≠ 0 and the stronger identification S ≅ K remain open.