Universal Spectrum and Edge-Crowding Transitions for Discrete Difference Operators

This preprint studies the discrete Euler map E a = a + V((x−ζ)a) built from theLagrange difference operator (Va)_i = Σj≠i (a_i − a_j)/(x_i − x_j) on m distinctreal nodes. The spectrum of the induced operator on the quotient by constants isuniversal — always {1, …, m−1}, independent of the node geometry — while the maximalvariance amplification A_m is sharply geometry-dependent. Main results:(1) A three-regime phase diagram for symmetric midpoint power grids with edgeexponent β: A_m ≍ m² for β<1, m² log² m at β=1, and m2β for β>1, with the sharpcritical limit A_m/(m² log² m) → 1/4.(2) For β>1, operator-norm convergence of the normalized matrices to a direct sum ofcompact quasinilpotent Cauchy-difference edge operators H_β.(3) Edge universality for smoothly regularly varying grids (unequal exponents,slowly varying corrections), a dominant-edge principle, and an explicit optimal pivot.(4) For Chebyshev roots, an exact triangular representation, a weighted Volterralimit, and the closed-form constant lim A_m/m⁴ = 1/(4 j²−1/4,1), where j−1/4,1is the first positive zero of the Bessel function J−1/4.(5) Consequently, the exact norm identity ‖H₂‖² = π⁴/(16 j²−1/4,1) for thequadratic Cauchy-difference edge operator on odd-square nodes. These results isolate a mechanism by which fixed, node-independent eigenvaluescoexist with arbitrarily large nonnormal singular amplification determined entirelyby the local node-spacing law. Reproducibility: this deposit includes the full manuscript (PDF), the scriptreproduce.py, and the CSV tables regenerating every numerical entry and figure inIEEE double precision (NumPy, SciPy, Matplotlib). No asymptotic constant is fittedto matrix data. Status: Version 1 — preprint, not peer-reviewed. An updated version will bedeposited under the same concept DOI. Comments are welcome.