EVGENY THEOREM — HISTORICAL FACT · CONFIRMED 2026

EVGENY THEOREM

A Gauge-Invariant Fourth Spectral
Moment on the Sierpiński Gasket

A closed-form identity for the fourth moment of a noncommutative SU(2) connection on the Sierpiński gasket — proven by computation, verified to 1e−13, reproducible by anyone.

Δ_m(H⁴, θ) = −16 · (3^m−1 + 1) · sin²(θ/2)

Read the Theorem ↓ Verify the Numbers

VERIFICATION LAYERS

SCROLL

CONVERGENCE

Seven levels to −8/9

lim_m→∞ I_m(π/2) = −8/9

I_m(π/2) → −8/9GEOMETRIC RATE 1/3

◈ AITKEN Δ²-EXTRAPOLATION

−8/9 = −0.8888889… exact limit

residual3.3×10⁻⁶

convergence rate1/3 — vertex growth

NOT the spectral factor1/5 ✗

The limit is governed by the SG vertex-growth factor 3, not by spectral decimation's 1/5 — a structural fingerprint of the non-Abelian bundle, not of the bare graph.

THE STRUCTURE

The full matrix, exposed

The Sierpiński gasket matrix, the SU(2)-bundle torus knot, the orbiting spectral paths — and the invariant core, separated into its own sphere at the center.

SG(m) · SU(2) BUNDLE ◈ CORE INVARIANT — I = −8/9 Δ_m(H⁴,θ) = −16(3^(m−1)+1)·sin²(θ/2)

RESEARCH PIPELINE

From claim to reproducible evidence

Every RFT-SIRM result travels the same six-stage pipeline. Click any stage to trace how the Evgeny Theorem moved from a mathematical claim to a machine-checkable identity.

THE OBJECT

SG(m) with a non-Abelian heartbeat

The standard Sierpiński-gasket approximation graph at refinement level m, carrying an SU(2)-valued connection with genuinely non-commuting holonomy.

GRAPH SG(m)

Built by recursive triangle subdivision. Every count is closed-form and checked in tests/test_levels.py. Flux is injected through exactly one designated edge per elementary triangle; every edge (a,b) carries a unitary U(a,b) ∈ SU(2) with U(b,a) = U(a,b)†.

n(m) = (3^m+1+3)/2vertices

3^m+1edges

3^mfaces

Hilbert space: Cⁿ ⊗ C² · dim(H) = 3^m+1 + 3

C Non-commuting

Rotation axis on the flux edge cycles x/y/z by triangle index mod 3. Verified to produce genuinely non-commuting holonomy between triangles sharing a vertex — ‖[U_i, U_j]‖ = 1.0, the maximal value for unitary 2×2 matrices.

axis(t) ∈ {x, y, z}, t mod 3

C′ Commuting control

Identical, but the axis is fixed to z for every triangle. All edge matrices commute — exactly equivalent to two decoupled U(1) magnetic SG Laplacians at flux ±θ/2.

axis(t) = z ∀t · C′ ≡ U(1) ⊕ U(1)

θ = 0 Reduction check

At θ = 0 both configurations reduce exactly to two decoupled copies of the plain SG Laplacian — pinned by a dedicated test in the suite.

C(0) = C′(0) = SG ⊕ SG

THE STATEMENT

Exact. Closed-form. Gauge-invariant.

THEOREM — EVGENY THEOREM

Let Δ_m(H⁴, θ) = Tr(H_C⁴) − Tr(H_C′⁴) be the raw trace defect of the fourth power between the non-commuting configuration C and the commuting control C′ on SG(m), at local rotation angle θ. Then, for every tested (m, θ) pair:

Δ_m(H⁴, θ) = −16 · (3^m−1 + 1) · sin²(θ/2)

INTENSIVE INVARIANT — normalizing by dim(H) = 3^m+1 + 3: I_m(θ) = Δ_m(H⁴,θ) / dim(H) = −8(3^m−1+1)·sin²(θ/2) / (3^m+1+3)

EXACT LIMIT — at θ = π/2, geometric convergence with rate 1/3: lim_m→∞ I_m(π/2) = −8/9 · rate = 1/3 (vertex growth, not spectral 1/5)

◈ LIVE CLOSED FORM — O(1) EVALUATION

EVALUATING

Rotation angle θ θ = 1.5708

Refinement level m m = 4

Δ_m(H⁴, θ) = −
dim(H) = −
I_m(θ) = −
I_m(π/2) → −8/9 = −0.888888…

// any (m, θ) in O(1) arithmetic — no 2n(m)-dimensional operator is ever constructed

EVIDENCE

Verification at a glance

Three independent lines of evidence: the refinement sequence at θ = π/2, a held-out parameter cross-check, and a random SU(2) gauge transformation.

m	dim(H)	I_m(π/2)	I_m / I_(m−1)
1	12	−1.333333	—
2	30	−1.066667	0.8000
3	84	−0.952381	0.8929
4	246	−0.910569	0.9561
5	732	−0.896175	0.9842
6	2190	−0.891324	0.9946
7	6564	−0.889701	0.9982

→ Aitken Δ²-extrapolation of m = 5,6,7: −0.8888855 vs. exact −8/9 = −0.8888889 — difference 3.3×10⁻⁶.

m	θ	Direct computation	Closed form	\|Δ\|
2	0.7	−7.5250500069	−7.5250500069	2.2×10⁻¹²
3	1.9	−105.8631653491	−105.8631653491	8.2×10⁻¹³
5	2.5	−1181.5502117988	−1181.5502117988	1.9×10⁻¹¹
1	3.0	−31.8398799456	−31.8398799456	8.2×10⁻¹⁴
4	0.3	−10.0046264358	−10.0046264359	2.5×10⁻¹¹

→ Parameters never used to derive the formula. Agreement to 1e−11 … 1e−14 — the identity holds, not the fit.

Check	Result
max \| spec(H) − spec(H_gauge) \|	7.99×10⁻¹⁵
M₄ difference under gauge transformation	exactly 0

→ Random SU(2) gauge transformation at level 3. The fourth moment is a spectral, gauge-invariant quantity.

ACCEPTANCE PROTOCOL

Six criteria. Not fewer.

I_m was accepted as a genuine structural invariant only after passing all six of the following checks.

CRITERION i ✓

Stable limit as m → ∞

Geometric convergence to −8/9, confirmed to refinement level m = 7 (dim 6564).

CRITERION ii ✓

Distinct from A and B

Differs from plain SG and U(1)-magnetic SG — the quantity does not exist in A or B by construction.

CRITERION iii ✓

Survives normalization

Computed as a per-state quantity throughout — the intensive invariant I_m, not the raw trace.

CRITERION iv ✓

Gauge-invariant

Verified to 8×10⁻¹⁵ under a random SU(2) gauge transformation; M₄ difference exactly 0.

CRITERION v ✓

Not a trivial counting function

C and C′ are identical in dim / edges / faces / flux-density — only the axis choice differs.

CRITERION vi ✓

Vanishes in the commuting limit

I_m = M₄(C) − M₄(C′) is zero exactly at moments p = 1,2,3 — first nonzero at p = 4.

HONEST SCOPE

What the closed form does —
and what it does not claim

✓ In scope

O(1) evaluation of Δ_m(H⁴, θ) for any (m, θ) — no operator construction required.
Applies strictly to the single defined quantity: the fourth spectral moment.
Accepted as a structural invariant only after passing all six criteria of the acceptance protocol — stability, distinctness from simpler models, gauge invariance, and more.
Full reproducibility: formula, code, tests, data — all independently checkable.

◇ Explicitly out of scope

Not a complexity-theory result. The full spectrum and spectral gap still require direct construction and diagonalization; the closed form covers only the fourth moment.

REPRODUCTION

Run it yourself

Every claim above is a pytest check. Clone, install, run — expected: all green, residual ~1e−13 … 1e−12 at level 7.

zsh — Evgeny-Theorem

$ git clone https://github.com/RFT-SIRM/Evgeny-Theorem.git $ cd Evgeny-Theorem $ python3 -m venv .venv && source .venv/bin/activate $ pip install -r reproducibility/requirements.txt $ pytest tests/ -m "not slow" -v ~60 checks — seconds $ pytest tests/ -m slow -v level-7 convergence — ~1–4 minutes ================== ALL GREEN ================== residual ~1e-13 … 1e-12 at level 7

Evgeny-Theorem/ ├── THEOREM.md exact statement + 6-criterion protocol ├── VERIFICATION.md full numerical tables, held-out set, gauge check ├── src/ │ ├── graph/ SG graph construction (3-connected, tree-of-triangles) │ ├── su2/ SU(2) rotation utilities │ ├── operators/ A (plain SG), B (U(1)-magnetic), C (SU(2)-bundle) │ └── moments/ moments, heat trace, counting function, I_m(θ) ├── tests/ pytest suite — every claim above is a test ├── data/ raw + computed reference spectra ├── figures/ convergence plot └── reproducibility/ environment, requirements, regeneration scripts

EXPECTED → all green · residual ~1e−13–1e−12 at level 7 · closed-form match to |Δ| ≤ 1e−11 on the held-out set

A Gauge-Invariant Fourth SpectralMoment on the Sierpiński Gasket

Created. Verified. Confirmed.