Folded-Lambda Eichler Extensions: Framed Odd-Zeta Coordinates and the Cusp-Detection Defect

Let H_s be the Legendre differential module in the reflection coordinate s = 1 - 2 lambda. For every even k >= 4, we construct a linear folded-lambda realization from M_k(SL_2(Z); C) to regular-singular extensions of 1 by Symᵏ⁻² H_s. Its scalar source is an explicit rational function determined by the lambda pullback polynomial of the modular form. After fixing the continuation action and the weight-one period lift, its Riemann-Hilbert class is exactly the restriction to Gamma(2) of the regularized Eichler class. We prove that this realization is injective. For a finite graded family W = direct sum_k W_k, the resulting differential Galois group is [direct sum_k Symᵏ⁻²(C^2) ⊗ W_k^*] semidirect PSL_2(C). All positive even symmetric powers have the same projective Legendre Picard-Vessiot field; over this field, a basis of the selected scalar cyclic jets is algebraically independent. In contrast, restriction to any one cusp detects only the Eisenstein coefficient: on a fixed-weight subspace its rank is zero or one, and a common-pure realization has Jordan form Jₖ₋₁(1) plus J_1(1)oplus dim W_k in the cuspidal case and J_k(1) plus J_1(1)oplus (dim W_k - 1) otherwise. This gives an exact local-global defect: every nonzero cuspidal direction is globally non-split but invisible at every cusp. The full connected differential Galois group, global non-splitting, and the jet-independence count persist after arbitrary algebraic base extension. On every finite cover, cuspidal directions remain locally split at each lifted cusp while staying globally non-split. For the Ramanujan-Eisenstein line, the same normalization gives the explicit parabolic cocycle, maximal cusp monodromy, and a standard-framed CM-reflection coordinate for the classical odd-zeta fixed-point formula. The general unipotent-radical and relative-completion mechanisms are used as established theory; the contribution is their faithful all-weight scalar realization and the simultaneous local-global calculation in one explicit frame.