The Ideal Structure k+1+k: The Guaranteed Majorana Zero-Mode, the Parity-Split Variance Floor, and the Conditional Linear Response of the Strong-Coupling Real Elliptic Ginibre Ensemble

We study the real elliptic Ginibre ensemble S = H + gA in the strong-coupling limit, where H is a symmetric Gaussian matrix and A a real antisymmetric Gaussian matrix. We prove that the variance of the real parts of the eigenvalues converges to a deterministic, parity-split floor: (n−2) /2 (n−1) for even dimension and exactly 1/2 for odd dimension. This finite-n even/odd distinction — commonly assumed absent at leading order in the asymptotic literature — is shown to be forced by the Altland–Zirnbauer class-D structure of the operator B = iA, whose spectrum takes the exact form of an ideal structure n = 2k+1: k particle modes (+λ), k antiparticle modes (−λ) bound to them by exact particle-hole conjugation, and one neutral Majorana mode pinned at zero. The guaranteed neutral mode is the pivot of the entire structure. We establish that it exists by oddness, that it is its own particle-hole conjugate (a Majorana mode), that it is statistically independent of the side modes (kernel–image orthogonality), and that its real part has variance exactly one — the precise origin of the odd-dimensional excess. We then determine the conditional linear response EAΣ Re² = α·tr (H²) + β· (tr H) ² in closed form for both parities, finding that α carries the zero-mode fingerprint (αₑven = (n−1) /n (n+2), αₒdd = 1/ (n+2) ) while β = (n+1) /n (n+2) is parity-independent. Finally we localize the finite-coupling correction: the protected neutral mode carries only a 1/g² correction, whereas the 1/g term is carried by the near-real, persistently-defective modes. Every analytical result is derived from first principles (degenerate perturbation theory, Gaussian–Wick contraction, kernel–image orthogonality) and independently verified by high-statistics simulation at a fixed seed. The work positions these results against the current literature on real-eigenvalue statistics of the elliptic Ginibre ensemble (to mid-2026), and states explicitly the single quantity whose closed form remains open, together with the precise reason for the obstruction. The methodological discipline is strict throughout: no fitting presented as derivation, no manual adjustment, every limit of the method named honestly.