PRIMES AT THE LEGENDRE SCALE: UNIVERSAL SIDON RIGIDITY, THE 2D MAIER MATRIX, AND ALGEBRAIC PIVOTS

Abstract

We present a rigorous structural theory of primes at the critical short-interval exponent $\theta=1/2$ (the Legendre scale), strictly delineating unconditional geometric rigidity from conditional topological transfers. First, we embed the classical $X^{1/2}$ intervals into a two-parameter polygonal gnomon array---the non-linear Maier Matrix. We prove unconditionally that these intervals possess a universal combinatorial skeleton forcing exact Multiplicative Sidon rigidity, and that the valid domain of the array asymptotically exceeds the exponent threshold of the classical Iwaniec--Laborde theorem. Second, we mathematically isolate the fundamental analytic obstruction at this scale. By developing the exact corrected phase $\Phi^*$ and analyzing the 2D Hessian, we establish an absolute $L^2$ precision-floor of $\asymp X^{1/2}\log^{3/2}X$. We rigorously frame this as a ``No-Go'' theorem, explicitly demonstrating that it strictly blocks product-grouped Cauchy-Schwarz dispersion while preserving other $L^2$ architectures. We confirm these dynamics numerically via discrete summation over 5,000 empirical Riemann zeroes. To map the exact geometric coordinates of this obstruction without resorting to unprovable heuristics over the integer line, we introduce two rigorous control geometries. Over $\mathbb{F}_q[t]$, the structural mechanisms operate unconditionally as a theorem via Deligne's Weil bounds and Katz-Sarnak monodromy. Unconditionally over $\mathbb{Z}[\omega]$, utilizing angular Hecke Gr\"ossencharacters, we establish that almost all narrow sectors of area $X^{7/12+\varepsilon}$ contain the expected prime density. Finally, we rigidly quarantine the $\mathbb{Z}$ transfer. Rather than claiming an unconditional analytic resolution, we formalize the Weil-\'Etale Monodromy Hypothesis for $\mathrm{Spec}(\mathbb{Z})$. We demonstrate that under this absolute geometric axiom, the Legendre, Oppermann, and Andrica Conjectures cease to be independent combinatorial anomalies, organically degenerating into a single, topologically classified universality class governed strictly by maximal geometric monodromy.