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Research Monographs

UltraCore-RFT links two complementary research dissertations maintained in their primary repositories. They address related graph-operator themes, but they do not make the same theorem claims.

1. Focused mathematical dissertation

Spectral Trace Defects of Noncommutative SU(2) Connections on Sierpiński-Gasket Graphs

Primary project: RFT-SIRM/Evgeny-Theorem
Date: 9 November 2026 (UltraCore edition)
Scope: the specified non-commuting SU(2)-connected operator on Sierpiński-gasket graphs and its commuting control.

The central result is the exact fourth-moment trace-defect identity

\[ \Delta_m(H^4,\theta) =\mathrm{Tr}(H_C^4)-\mathrm{Tr}(H_{C'}^4) =-16(3^{m-1}+1)\sin^2\left(\frac{\theta}{2}\right), \qquad m\ge1,\ \theta\in\mathbb R. \]

After normalization by the Hilbert-space dimension \(3^{m+1}+3\),

\[ I_\infty(\theta) =\lim_{m\to\infty} \frac{\Delta_m(H^4,\theta)}{3^{m+1}+3} =-\frac{16}{9}\sin^2\left(\frac{\theta}{2}\right). \]

The focused UltraCore dissertation is a theorem-centred edition prepared from the primary source records. The machine-checked Lean declarations, computational scripts, and detailed source history remain in Evgeny-Theorem. It also documents the distinct proof status of H⁶, H⁸, and H¹⁰. The H⁴ statement is the primary all-level theorem; higher moments must be cited at their own levels and with their own proof boundaries.

Primary verification sources: THEOREM.md · Lean formalization · VERIFICATION.md · higher-moment status.

2. Broader synthesis dissertation

Recursive Graph Operators, Spectral Trace Defects, and Higher-Moment Recurrence

Primary project: RFT-SIRM/Evgeny-s-Theory-of-Everything
Date in source document: 15 October 2026
Scope: a broader research synthesis covering recursive graph operators, trace-defect moments, finite Fourier-coefficient certificates, numerical recurrence checks, and a separate signed \(Z_2\)-flux spectral-decimation programme.

This document is a dated research snapshot. Its discussion of proof status predates some later formal results in the primary repositories. For current status, consult Evgeny-Theorem and Phi-Genesis, rather than treating the older synthesis text as a live status dashboard.

3. Keep the mathematical strands distinct

  • The SU(2) fourth-moment theorem is a statement about its specified connection operators and commuting control.
  • The higher-moment programme has separate H⁶/H⁸/H¹⁰ results and finite-certificate boundaries.
  • Phi-Genesis proves a separate least-eigenvalue/convergence theorem for signed and signless graph Laplacians at levels \(N\ge6\). Its Gate A exceptional-set problem and external \(\theta=2\pi\) identification remain open.
  • No equivalence between the SU(2) operators and the signed \(Z_2\)-flux operators is claimed.

The focused theorem dissertation is maintained in this repository as the UltraCore edition; its formal proof artifacts remain canonical in Evgeny-Theorem. The broader synthesis dissertation remains canonical in its own repository. This page is the navigation guide between the two distinct documents.

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