From Intuition to Judgment: Spectral Collapse, an Exact Crossover Law, the Symmetry–Positivity Dichotomy, and the Geometric Shape of the Wall Confronting the Riemann Hypothesis

This paper presents a transparent, fully reproducible study of the spectral structure underlying the Riemann zeros, organized around one question: which features of the problem are structurally available, and which constitute the genuine obstruction. It makes NO claim of proving the Riemann Hypothesis. Working in the random-matrix model S = H + gA (the elliptic GinOE ensemble), four results are established, each proved algebraically and verified numerically: (i) an exact crossover law for the variance of the real parts, which corrects a previously reported fractional exponent −0. 8 shown here to be a finite-window artifact; (ii) an exact dimensional collapse det Σ = 0 arising from the trace constraint; (iii) a mirror symmetry λ ↔ 1−λ* induced by the involution S = I − Sᵀ; and (iv) a symmetry–positivity dichotomy, proved with explicit matrices, showing that the functional-equation analogue (set-symmetry about Re = ½) does not force all eigenvalues onto the critical line. Across five independent avenues — logarithmic time, GUE statistics, the explicit formula, interacting-Majorana/SYK with Bott periodicity, and the metric/positivity structure — the general structure is shown to be uniformly available, while the precise determination of the zeros is uniformly protected by a single wall: positivity. This wall is then given a geometric shape: the analogue of the Riemann Hypothesis is a theorem exactly where a real curved space supplies positivity automatically (finite fields via the Hodge intersection form; Selberg surfaces via the Laplacian), and open for ζ precisely because the required space — a curvature on Spec (ℤ) whose closed orbits have lengths ln p — does not yet exist. The contribution is a rigorous, computable map of where the difficulty truly resides, with every positive claim verified under a fixed random seed and every failed conjecture recorded honestly. A companion verification script (masterᵥerification. py) re-checks all eighteen reported results. Reproducibility: all numerical values were computed with mpmath and NumPy under a fixed seed (12345) ; the verification script reproduces every result. Artificial intelligence (Claude, by Anthropic) was used as a tool for computation, simulation, refutation testing, figure generation, and typesetting; the intuitions, direction, and full responsibility for the content belong to the author.