This paper develops an En-indexed analytic and representation-valued field framework with thedefining opposite-lift sector E_5 = E_6^- and constituent content boldtf{16} circleplus boldtf{9} circleplus boldtf{1} of total dimension 26. The main text covers E_5 through E15. The finite exceptional levels E_6, E_7 and E_8 are distinguished from the affine E_9, hyperbolic E_10, very-extended E_11, and proposed E_12—E_15 framework extensions. The eight-sector mGFT ansatz uses hypergeometric, exponential and monomial generator factors; its (0, 1, 1) sector recovers the weighted hull Hn,n−1(2ξ) when Q = ξ and Z = 2ξ. This supplies a candidate profile family, while internal symmetry is carried by explicitly chosen Lie-algebra representations. Integral ft-{ht,lsr}-mGFT is formulated as a generated-core Green—Duhamel correction with stated boundary data, quadrature and convergence conditions. The primary reduction chain is E_15->...->E_5; reverse operations are constructive lifts. Physical equations are discussed as theoretical contexts for the framework. The quantitative results are confined to fully specified mathematical and effective-action benchmarks: local profile and sum identities, an exact canonical scalar wall and its generated-coordinate double lift, constituent threshold arithmetic, prescribed quadratic masses, conditional two-body kinematics, a manufactured nonlinear boundary-value problem, and the E_10 Cartan spectrum. The supplied atlases distinguish three opposing labels, anti-, mirror and negative matter, relative to ordinary matter. Charge conjugation, copy labels, signed energy and gravitational sign are treated separately; signed constituent sums do not establish physical negative masses or photon rest masses. The separate supplement supplies the inputs, derivations and reproducible numerical checks.