Spectral Trace Defects of Noncommutative SU(2) Connections on Sierpiński-Gasket Graphs¶
Research dissertation¶
Author: Evgeny Zaprudnov
Research group: RFT-SIRM
Version: 1.0 — UltraCore edition
Date: 9 November 2026
Primary formal project: RFT-SIRM/Evgeny-Theorem
Source dissertation: upstream research manuscript
Abstract¶
This dissertation studies trace defects of recursively defined graph operators on finite Sierpiński-gasket approximations. Each vertex carries a two-component complex state, while selected graph edges carry SU(2)-valued connection matrices. Two connection configurations are compared on the same graph: a non-commuting configuration whose flux-edge axes cycle through \(x,y,z\), and a commuting control whose flux edges all use the fixed \(z\)-axis.
The principal result is an exact fourth-moment identity. For every refinement level \(m\ge1\) and every real angle \(\theta\),
The concrete statement is machine-checked in Lean 4 as traceDefect_closed in the primary project. Its proof is based on recursive gluing and two coupled orientation recurrences. The proof has no unproved placeholder; the formalization records its dependencies on the standard Lean axioms propext, Classical.choice, and Quot.sound.
After normalization by the matrix dimension, the fourth-moment defect has an exact limit
This is a theorem about one normalized trace observable. It is not a formula for the full spectrum, individual eigenvalues, the spectral gap, or computational complexity.
The dissertation also records distinct higher-moment results. H⁶ and H⁸ have Lean-proved all-angle recurrences for \(m\ge3\), combined with exact symbolic certificates for the exceptional transition \(m=2\to3\). Those finite bridges are exact but are not yet Lean-kernel theorems. H¹⁰ has an unconditional Lean-proved affine recurrence for \(m\ge4\) and affine form for \(m\ge5\), because the required radius-four transport is itself proved in Lean. The level-five correction remains an exact finite trace expression rather than an expanded trigonometric polynomial.
The central methodological point is that a Lean theorem, a finite exact symbolic certificate, and a floating-point regression are different kinds of evidence. Every claim in this dissertation is restricted to the graph, orientation convention, and operators defined by the primary project.
Keywords: Sierpiński-gasket graphs; trace defect; spectral moments; SU(2) connection; non-commuting holonomy; recursive gluing; exact symbolic computation; Lean 4.
1. Scope and status¶
The principal theorem is universal over the stated refinement levels and angles: it holds for every integer \(m\ge1\) and every \(\theta\in\mathbb R\). The universal statement is not inferred from the finite numerical tables; it is proved for the concrete matrices in the formalization.
The current proof hierarchy is:
| Moment | Established result | Evidence boundary |
|---|---|---|
| H⁴ | Exact all-level, all-angle closed form for \(m\ge1\). | Unconditional Lean-kernel theorem. |
| H⁶ | All-angle recurrence for \(m\ge3\); combined formula for \(m\ge2\). | Lean recurrence plus exact Gaussian-integer polynomial certificate for \(2\to3\). The finite bridge is not yet in the Lean kernel. |
| H⁸ | All-angle recurrence for \(m\ge3\); combined formula for \(m\ge2\). | Lean recurrence plus exact sparse-polynomial certificate for \(2\to3\). The finite bridge is not yet in the Lean kernel. |
| H¹⁰ | Unconditional affine recurrence for \(m\ge4\) and affine form for \(m\ge5\). | Lean-kernel theorems; radius-four transport is proved. The level-five correction remains an exact finite trace expression. |
The H⁶ and H⁸ transition certificates are exact and reproducible, but their finite \(2\to3\) bridges are not yet formalized inside the Lean kernel. This is different from H¹⁰: its transport hypothesis has itself been discharged in Lean. Simplifying the H¹⁰ correction coefficient would improve its presentation, but is not a missing hypothesis in the stated recurrence.
This work does not claim a formula for all eigenvalues, the spectral gap, a computational-complexity classification, or a physical law.
2. Graph and connection operators¶
2.1 Recursive graph and Hilbert space¶
Let \(SG(m)\) be the finite Sierpiński-gasket approximation graph at refinement level \(m\). It is formed recursively by assembling three copies of the preceding level and identifying their corner vertices. The combinatorial counts are
Each vertex carries two complex state components, so the Hilbert space and its dimension are
2.2 SU(2) edge data¶
Let \(\sigma_x,\sigma_y,\sigma_z\) denote the Pauli matrices. A flux edge with rotation axis \(a\in\{x,y,z\}\) carries the matrix
The reverse orientation carries the adjoint:
Exactly one designated edge of each elementary triangle carries this rotation; the other edges carry the identity.
Configuration \(C\) assigns the axes \(x,y,z\) cyclically by triangle index modulo three. Configuration \(C'\) keeps the graph and flux-edge locations fixed but assigns the \(z\)-axis to every triangle. Its edge rotations commute and split into two decoupled U(1)-magnetic Sierpiński-gasket operators with fluxes \(+\theta/2\) and \(-\theta/2\).
The operator is of graph-Laplacian form
where \(D\) is the degree matrix tensored with \(I_2\), and \(A\) is the connection adjacency matrix. At \(\theta=0\), both configurations reduce to two copies of the ordinary Sierpiński-gasket Laplacian.
2.3 Orientation is part of the definition¶
The theorem uses the vertex-index orientation convention of the primary project: Lean uses vIdx, and the Python implementation orders edge endpoints by vertex indices. This convention must be retained when reproducing the theorem.
The upstream orientation audit reports that a different, geometrically uniform direction convention gives, for the tested levels \(m=1,\ldots,4\),
That alternative is a finite-level comparison, not the all-level theorem stated here. Graph indexing, face order, flux-edge choice, and edge orientation must not be silently changed.
3. Trace-defect notation¶
For an even moment \(p\), define
For the higher-moment Lean statements, \(D_m^{(p)}(\theta)\) denotes the corresponding numbering-orientation quantity defined by the primary declarations defectO6, defectO8, and defectO10.
For the polynomial formulae below, use
4. Main theorem: fourth spectral moment¶
Theorem 4.1 — exact fourth-moment trace defect¶
For every integer \(m\ge1\) and every real \(\theta\),
The concrete Lean declaration is traceDefect_closed in formalization/EvgenyTheorem/TildeEntries.lean of the primary project.
The identity implies
and the defect vanishes exactly when \(\theta\in2\pi\mathbb Z\).
4.1 Normalization, exact finite-level formula, and limit¶
Normalize by the matrix dimension:
The scalar factor satisfies
Therefore the exact finite-level expression is
Consequently,
The exact finite-level remainder is
In particular,
For \(\theta\notin2\pi\mathbb Z\), the ratio of successive absolute remainders is
For \(\theta\in2\pi\mathbb Z\), both remainders vanish identically, and this ratio is undefined. This is a limit of the normalized fourth-moment trace defect; it is not a statement about eigenvalue convergence or the spectral gap.
5. Lean proof architecture¶
The general-level proof uses the recursive gluing structure and two orientation states. Let \(D_m\) be the numbering-orientation defect and \(\widetilde D_m\) the defect with the designated corner orientation flipped. Put \(s=\sin(\theta/2)\). The initial values are
The coupled recurrences are
Set
Then
Thus
Recovering \(D_m=(S_m+E_m)/2\) gives
which is the asserted theorem.
The formal proof uses recursive gluing, block-matrix trace identities, corner data, and orientation transport. The primary Lean development is organized across modules including GlueAlg, GlueBridge, GlueSG, GlueK, GlueCorner, FinalOrient, FinalTilde, FinalArith, and TildeEntries.
The machine-checked result should be cited through the declaration traceDefect_closed and its coupled-recursion proof route. The upstream formalization records the dependencies on the standard Lean axioms propext, Classical.choice, and Quot.sound. The earlier hand-oriented manuscript docs/PROOF.md is not the machine-checked proof.
6. Higher-moment results¶
6.1 Sixth moment¶
For the numbering-orientation quantity \(D_m^{(6)}(\theta)\), Lean proves
The exact finite certificate gives
and
Combining the Lean recurrence with the finite transition gives, for every \(m\ge2\),
Equivalently,
At \(\theta=\pi\), the recurrence coefficient is \(C_6(\pi)=5280\).
Proof boundary. The recurrence for \(m\ge3\) is proved in Lean. The exceptional \(m=2\to3\) transition is certified by exact Gaussian-integer polynomial arithmetic in computational/p6_level3_orientation_certificate.py and tested in tests/test_higher_moments.py. This finite bridge is exact symbolic evidence, but it is not yet a Lean-kernel theorem.
On \(SG(1)\), the exact length-six walk classification gives 66 walks in each one-triangle class, 168 in each two-triangle class, and 486 walks visiting all three elementary triangles. The first two groups contribute zero; the all-three-triangle group contributes nonzero. This is a finite-level path-class result, not an all-level path-class theorem.
6.2 Eighth moment¶
For \(D_m^{(8)}(\theta)\), Lean proves
The exact certificate gives
and
For every \(m\ge2\),
At \(\theta=\pi\), \(C_8(\pi)=277504\).
Proof boundary. The recurrence is Lean-formalized for \(m\ge3\). The exceptional \(m=2\to3\) identity is computed coefficient-by-coefficient over Gaussian integers in computational/p8_base_certificate.py. It is exact and reproducible, but not yet a Lean-kernel theorem.
6.3 Tenth moment¶
The tenth-moment development contains an ordered \(p=10\) trace expansion, exact word-stratum reductions, gluing assembly, orientation recurrences, and transport of the required corner rows through degree nine.
The proposition H10TransportAll is proved by H10TransportAll_proved for both the face-axis and fixed-\(z\) configurations at every \(m\ge4\). Lean therefore proves unconditionally
The corresponding affine form is
The exact correction \(C_{10}^{(5)}(\theta)\) is the finite trace expression named defectO10_num_residual_constant. It has not yet been expanded as a trigonometric polynomial. This is a presentation task, not a remaining hypothesis in the recurrence theorem.
7. Numerical and exact-computation records¶
At \(\theta=\pi/2\), direct fourth-moment matrix calculations give:
| \(m\) | \(\dim\mathcal H_m\) | \(I_m(\pi/2)\) |
|---|---|---|
| 1 | 12 | \(-1.333333\) |
| 2 | 30 | \(-1.066667\) |
| 3 | 84 | \(-0.952381\) |
| 4 | 246 | \(-0.910569\) |
| 5 | 732 | \(-0.896175\) |
| 6 | 2190 | \(-0.891324\) |
| 7 | 6564 | \(-0.889701\) |
The held-out H⁴ matrix-power comparisons recorded upstream have maximum listed absolute discrepancy \(2.54\times10^{-11}\). The higher-moment regression table at \(\theta=\pi/2\) covers \(m=1,\ldots,4\). These finite tables are numerical checks, not the universal proof.
The Python tests also check selected vertex-wise gauge transformations. At level three, the recorded maximum spectral discrepancy is \(7.99\times10^{-15}\); the fourth-moment difference is zero at displayed precision. These checks are not assumptions in the Lean theorem.
8. Reproducibility and proof artifacts¶
The source-of-truth formal and computational artifacts remain in RFT-SIRM/Evgeny-Theorem:
| Artifact | Role |
|---|---|
src/graph/sg_graph.py |
Recursive graph construction and combinatorial counts. |
src/operators/operators.py |
Python graph operators and SU(2) connection. |
formalization/EvgenyTheorem/Operator.lean |
Concrete Lean operators and trace-defect definition. |
formalization/EvgenyTheorem/TildeEntries.lean |
Theorem traceDefect_closed. |
formalization/EvgenyTheorem/FinalTilde.lean |
Coupled H⁴ recurrences and H⁶ results. |
formalization/EvgenyTheorem/FinalClosed10.lean |
Unconditional H¹⁰ transport and affine theorem. |
computational/p6_level3_orientation_certificate.py |
Exact H⁶ transition certificate. |
computational/p8_base_certificate.py |
Exact H⁸ transition certificate. |
docs/HIGHER_MOMENTS.md |
Higher-moment formulae and proof boundaries. |
docs/ORIENTATION_NOTE.md |
Orientation convention audit. |
VERIFICATION.md |
Numerical and held-out checks. |
From a clean checkout of the primary project, the documented checks are:
python3 -m venv .venv
source .venv/bin/activate
pip install -r reproducibility/requirements.txt
pytest tests/ -m "not slow" -v
pytest tests/ -m slow -v
lake build
The Python suite checks numerical regressions and exact finite certificates. The Lean build checks formal proofs. A successful run certifies only the assertions actually executed by that run. The primary project's lakefile.toml and lean-toolchain pin the Mathlib revision and Lean toolchain.
9. Limitations and non-claims¶
- No full-spectrum formula. The fourth-moment trace identity does not determine all eigenvalues or the complete spectral distribution.
- No spectral-gap theorem. No conclusion about the smallest positive eigenvalue follows from this identity alone.
- No complexity-theory result. The closed form evaluates the scalar trace defect without constructing the full matrix. This is not a complexity-class separation, lower bound, or bit-complexity theorem.
- No silent change of orientation. A different edge orientation defines a different operator family.
- No automatic extension to higher moments. Each higher moment requires its own recurrence and boundary argument.
- No conflation of evidence. Lean theorems, exact external certificates, and floating-point regressions are distinct.
- No cross-project identification. This work does not identify the SU(2) operators with signed \(Z_2\)-flux or spectral-decimation models.
- No physical-law claim. All statements concern the graph operators defined in the primary project.
The six-criterion protocol in the primary repository is a project-defined diagnostic for the normalized H⁴ invariant. It is not an external peer-review verdict and does not replace the theorem or its proof.
10. Conclusion¶
For the specified Sierpiński-gasket graph and vertex-index-oriented SU(2) connection, the fourth-moment trace defect has the exact all-level, all-angle formula
The result is machine-checked in Lean 4. Its normalized form has an exact limit and a fully explicit finite-level remainder. The H⁶ and H⁸ developments combine Lean recurrences with exact finite transition certificates; the H¹⁰ recurrence and affine form are unconditional Lean theorems.
This document consolidates the proved fourth-moment theorem and the current higher-moment advances. It does not claim that every related spectral question is closed. The full spectrum, spectral gap, Lean formalization of the H⁶/H⁸ finite bridges, and explicit trigonometric simplification of the H¹⁰ correction remain outside the principal H⁴ theorem.
References and primary research artifacts¶
- Evgeny-Theorem repository — source, formalization, certificates, tests, and research records.
- THEOREM.md — statement of the fourth-moment identity.
- Lean formalization guide — theorem declaration, proof architecture, axioms, and build instructions.
- Higher-moment results — detailed H⁶, H⁸, and H¹⁰ formulae.
- Orientation note — edge-orientation convention.
- Verification record — finite numerical and held-out checks.
This is a research dissertation/monograph about the mathematics maintained in RFT-SIRM/Evgeny-Theorem. It is not a claim of an academic degree and is separate from the broader “Theory of Everything” synthesis dissertation.