Sparse real non-Hermitian bulk universality throughout the Ginibre side of the real-axis crossover

This work extends sparse real non-Hermitian bulk universality from complex bulk points at fixed positive distance from the real axis to deterministic centres approaching the real axis at every polynomial scale strictly larger than the microscopic crossover scale.Let X be an N × N real random matrix with iid, possibly N-dependent, entries satisfying zero mean, variance N⁻¹, and sparse moment boundsE|xᵢⱼ|ʳ ≤ Cᵣ / (N qʳ⁻²),with q ≥ N^κ.For every fixed δ > 0 and ε > 0, and every deterministic sequence of bulk centres z_N satisfying|z_N| ≤ 1 − δand|Im z_N| ≥ N⁻¹ᐟ²⁺ε,the microscopic complex-eigenvalue k-point statistics converge to the complex Ginibre bulk limit for every fixed correlation order. RM1_II_REINFORCEDThe exponent 1/2 is the natural boundary of the Ginibre regime. Non-real eigenvalues of a real matrix occur in conjugate pairs. On the microscopic N⁻¹ᐟ² scale, the conjugate partner leaves every fixed observation window precisely when √N |Im z_N| tends to infinity. At the critical scale |Im z| ≍ N⁻¹ᐟ², the local process crosses over to the real-Ginibre Pfaffian regime, so the complex Ginibre kernel is no longer the correct limiting object. RM1_II_REINFORCEDThe paper is a companion to the fixed-distance universality result RM1-I. The fixed-distance theory supplies the binary z/z̄ Hermitisation algebra, the sparse binary multi-resolvent local law, real codimension-two partial-Schur propagation, short-time real Gauss-divisible universality, and Gaussian removal. The present work identifies and resolves the additional degeneration that appears when the bulk centre approaches the real axis. RM1_II_REINFORCEDThe key stability variable isΔ = (Im z)² + η,rather than (Im z)² alone. The mixed z/z̄ deterministic stability operator has size Δ⁻¹. This apparent singularity is not treated as an uncontrolled loss. Instead, every mixed stability factor is recorded explicitly in a deterministic weight, while the stochastic multi-resolvent estimates retain the same N-, η-, and sparsity-q hierarchy as in the fixed-distance theory. RM1_II_REINFORCEDA weighted binary multi-resolvent local law is proved for the entire finite proof closure generated by entry derivatives, transpose operations, cumulant substitutions, exceptional configurations, and fluctuation averaging. After division by the deterministic mixed-stability weight, no additional negative power of Δ is generated by the stochastic analysis. RM1_II_REINFORCEDThe second central ingredient is a sharp mixed two-resolvent estimate at the short Gaussian time. The relevant quadratic mode satisfiesη² ⟨H_z(η) H_z̄(η)⟩ ≍ Δ⁻¹.This identifies the precise mixed scale needed for the real weighted-Stiefel saddle as the axis is approached. RM1_II_REINFORCEDThe high-order weighted-Stiefel expansion is organized through an excursion-charging principle. Every pair of sector-changing saddle modes defines one complete z/z̄ excursion and corresponds to exactly one mixed stability solve. Therefore a cyclic tensor containing s mixed modes carries exactly Δ⁻ˢᐟ².The same Δ dependence appears in the mixed Hessian normalization. After standardization, these factors cancel exactly, leaving the same high-order hierarchy as in the fixed-distance theory:|c_α| ≤ Nᵒ⁽¹⁾ (Nt)⁻⁽ʳ⁻²⁾ᐟ².Thus the short-time high-order expansion remains uniform even as Im z tends to zero. RM1_II_REINFORCEDThe real partial-Schur and weighted-Stiefel propagation is also made uniform in the shrinking-axis regime. Exact off-saddle localization controls the Schur displacement, while the coupling between the conjugate sectors remains perturbative under the scale conditions required by the theorem. RM1_II_REINFORCEDA separate issue arises in Gaussian removal. The paper proves that connected comparison derivatives remain at a single non-Hermitian centre z. Ward and Cauchy–Schwarz estimates therefore use the adjoint relation G_z(it)* = G_z(−it) without introducing a mixed z/z̄ stability inversion. Consequently the Gaussian comparison carries no negative power of Δ and the total replacement error remains of orderNᵒ⁽¹⁾ Nt / q. RM1_II_REINFORCEDChoosing the short Gaussian time appropriately balances the sparse comparison error, the shrinking-axis mixed stability, and the Schur/Stiefel localization. All losses are collected into a single dimensionless error budget that tends to zero uniformly throughout the regime |Im z_N| ≥ N⁻¹ᐟ²⁺ε. RM1_II_REINFORCEDThe paper also extends the shrinking-axis conclusion to dense iid real matrices with a finite 4 + δ moment by truncation and coupling. RM1_II_REINFORCEDThe result reaches the full Ginibre side of the real-axis crossover at every polynomial distance above the critical N⁻¹ᐟ² scale. The critical crossover itself, where the limiting process becomes Pfaffian, is deliberately left outside the theorem.