Geometric Operator on the Eisenstein Lattice: GUE Classification, Bounded Spectrum, and the Merkabit as Irreducible Unit
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2026
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| author | Stenberg, Selina |
| author_facet | Stenberg, Selina |
| contents | <p>Paper 6 of the companion papers. We construct a zero-parameter Hermitian operator M on the Eisenstein lattice ℤ[ω] from geometric SU(2) Hopf-paired spinors encoding the E₆ McKay correspondence. The spectral program spans Analyses 14–37 and yields three categories of result.</p> <p><strong>GUE classification.</strong> Threading Peierls flux Φ = 1/6 per triangular plaquette drives spectral statistics from GOE to GUE: KS(GUE) = 0.052, p = 0.57 at L = 18. A GUE resonance island is confirmed at L = h·ξ to (h+1)·ξ = 36–39, architecturally determined by the Coxeter number h(E₆) = 12 and correlation length ξ = 3. Within the island, GUE symmetry strength and Riemann spectral correspondence are perfectly anti-correlated (Spearman ρ = -1.000): the symmetry-breaking mechanism and the arithmetic mechanism are separable properties competing for the same degrees of freedom.</p> <p><strong>Route C closed.</strong> α⁻¹ = 137.036 follows from rank(E₆) = 6, h(E₆) = 12, n_gates = 5 via gate modulation angles and Floquet return fidelity F = 0.696778. Every step is determined by the architecture. Zero free parameters.</p> <p><strong>The bounded spectrum.</strong> Across twelve lattice sizes from L = 18 to L = 72, the spectral radius converges to R∞ = 1.4793 ± 0.004, independent of lattice size. A rigorous seven-analysis investigation (Analyses 33–37) definitively established that M is not a Berry–Keating operator: spectral zeta zero spacings are indistinguishable from generic GUE (Fisher combined p = 0.954), and zeros do not converge to Re(s) = 1/2. The bounded spectrum is architectural — a consequence of exponential decay, bounded Hopf pairing, and fixed coordination — and makes M structurally incapable of encoding unbounded Riemann zeros.</p> <p>The bounded spectrum is not a failure. It is the mathematical expression of the merkabit as an irreducible computational unit. The spectral radius R∞ = 1.4793 sits within the operational window [4/3, 3/2], above the cooperative threshold of Paper 1 and below the standing-wave collapse region. The idealized window width 3/2 − 4/3 = 1/6 approximates the Peierls flux Φ = 1/6 — a near-exact self-consistency whose precise status in the thermodynamic limit remains open.</p> <p>The central question this paper opens: what is the spectral density of the infinite Eisenstein lattice with Hopf-paired spinors, and does its Mellin transform have zeros only on the critical line? The bounded spectrum makes this tractable — the limiting spectral measure has compact support on [−1.4793, +1.4793], a fundamentally different analytic object from the unbounded spectra traditionally sought in the Berry–Keating program.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19075162 |
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| publishDate | 2026 |
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| spellingShingle | Geometric Operator on the Eisenstein Lattice: GUE Classification, Bounded Spectrum, and the Merkabit as Irreducible Unit Stenberg, Selina Eisenstein lattice Merkabit architecture Hermitian operator Hopf pairing Dual pentachoron E₆ root system McKay correspondence Coxeter number SU(2) spinors GUE universality GOE to GUE transition Random matrix theory Bounded spectrum Spectral radius Berry-Keating conjecture Spectral zeta function Peierls flux Time-reversal symmetry breaking Fine structure constant Cooperative threshold Floquet operator Riemann zeros Montgomery pair correlation Critical line Arithmetic quantum chaos <p>Paper 6 of the companion papers. We construct a zero-parameter Hermitian operator M on the Eisenstein lattice ℤ[ω] from geometric SU(2) Hopf-paired spinors encoding the E₆ McKay correspondence. The spectral program spans Analyses 14–37 and yields three categories of result.</p> <p><strong>GUE classification.</strong> Threading Peierls flux Φ = 1/6 per triangular plaquette drives spectral statistics from GOE to GUE: KS(GUE) = 0.052, p = 0.57 at L = 18. A GUE resonance island is confirmed at L = h·ξ to (h+1)·ξ = 36–39, architecturally determined by the Coxeter number h(E₆) = 12 and correlation length ξ = 3. Within the island, GUE symmetry strength and Riemann spectral correspondence are perfectly anti-correlated (Spearman ρ = -1.000): the symmetry-breaking mechanism and the arithmetic mechanism are separable properties competing for the same degrees of freedom.</p> <p><strong>Route C closed.</strong> α⁻¹ = 137.036 follows from rank(E₆) = 6, h(E₆) = 12, n_gates = 5 via gate modulation angles and Floquet return fidelity F = 0.696778. Every step is determined by the architecture. Zero free parameters.</p> <p><strong>The bounded spectrum.</strong> Across twelve lattice sizes from L = 18 to L = 72, the spectral radius converges to R∞ = 1.4793 ± 0.004, independent of lattice size. A rigorous seven-analysis investigation (Analyses 33–37) definitively established that M is not a Berry–Keating operator: spectral zeta zero spacings are indistinguishable from generic GUE (Fisher combined p = 0.954), and zeros do not converge to Re(s) = 1/2. The bounded spectrum is architectural — a consequence of exponential decay, bounded Hopf pairing, and fixed coordination — and makes M structurally incapable of encoding unbounded Riemann zeros.</p> <p>The bounded spectrum is not a failure. It is the mathematical expression of the merkabit as an irreducible computational unit. The spectral radius R∞ = 1.4793 sits within the operational window [4/3, 3/2], above the cooperative threshold of Paper 1 and below the standing-wave collapse region. The idealized window width 3/2 − 4/3 = 1/6 approximates the Peierls flux Φ = 1/6 — a near-exact self-consistency whose precise status in the thermodynamic limit remains open.</p> <p>The central question this paper opens: what is the spectral density of the infinite Eisenstein lattice with Hopf-paired spinors, and does its Mellin transform have zeros only on the critical line? The bounded spectrum makes this tractable — the limiting spectral measure has compact support on [−1.4793, +1.4793], a fundamentally different analytic object from the unbounded spectra traditionally sought in the Berry–Keating program.</p> |
| title | Geometric Operator on the Eisenstein Lattice: GUE Classification, Bounded Spectrum, and the Merkabit as Irreducible Unit |
| topic | Eisenstein lattice Merkabit architecture Hermitian operator Hopf pairing Dual pentachoron E₆ root system McKay correspondence Coxeter number SU(2) spinors GUE universality GOE to GUE transition Random matrix theory Bounded spectrum Spectral radius Berry-Keating conjecture Spectral zeta function Peierls flux Time-reversal symmetry breaking Fine structure constant Cooperative threshold Floquet operator Riemann zeros Montgomery pair correlation Critical line Arithmetic quantum chaos |
| url | https://doi.org/10.5281/zenodo.19075162 |