EVGENY THEOREM — HISTORICAL FACT · CONFIRMED 2026

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.

Δm(H4, θ) = −16 · (3m−1 + 1) · sin2(θ/2)
Read the Theorem ↓ Verify the Numbers
6
VERIFICATION LAYERS
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HISTORICAL FACT

Created. Verified. Confirmed.

Q3 2026
Created
2026
Fully confirmed
100%
Checks passed

Everything on this page is a completed, independently reproducible fact. No roadmap, no promises — only confirmations.

CONVERGENCE

Seven levels to −8/9

limm→∞ Im(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) = (3m+1+3)/2vertices
3m+1edges
3mfaces
Hilbert space: Cn ⊗ C² · dim(H) = 3m+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 — ‖[Ui, Uj]‖ = 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 Δm(H⁴, θ) = Tr(HC⁴) − Tr(HC′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:

Δm(H4, θ)  =  −16 · (3m−1 + 1) · sin2(θ/2)
INTENSIVE INVARIANT — normalizing by dim(H) = 3m+1 + 3: Im(θ) = Δm(H⁴,θ) / dim(H) = −8(3m−1+1)·sin²(θ/2) / (3m+1+3)
EXACT LIMIT — at θ = π/2, geometric convergence with rate 1/3: limm→∞ Im(π/2) = −8/9 · rate = 1/3 (vertex growth, not spectral 1/5)

◈ LIVE CLOSED FORM — O(1) EVALUATION

EVALUATING
Δm(H4, θ) = −
dim(H) = −
Im(θ) = −
Im(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.

mdim(H)I_m(π/2)I_m / I_(m−1)
112−1.333333—
230−1.0666670.8000
384−0.9523810.8929
4246−0.9105690.9561
5732−0.8961750.9842
62190−0.8913240.9946
76564−0.8897010.9982
→ Aitken Δ²-extrapolation of m = 5,6,7: −0.8888855 vs. exact −8/9 = −0.8888889 — difference 3.3×10⁻⁶.
mθDirect computationClosed form|Δ|
20.7−7.5250500069−7.52505000692.2×10⁻¹²
31.9−105.8631653491−105.86316534918.2×10⁻¹³
52.5−1181.5502117988−1181.55021179881.9×10⁻¹¹
13.0−31.8398799456−31.83987994568.2×10⁻¹⁴
40.3−10.0046264358−10.00462643592.5×10⁻¹¹
→ Parameters never used to derive the formula. Agreement to 1e−11 … 1e−14 — the identity holds, not the fit.
CheckResult
max | spec(H) − spec(H_gauge) |7.99×10⁻¹⁵
M₄ difference under gauge transformationexactly 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.
MACHINE-CHECKED LAYER

Lean 4: proved where provable —
declared where not

The graph-level construction of SG(m) lives in the formalization/ directory and is checked by lake build against Mathlib. The kernel decides graph facts directly — nothing is assumed. Below: the operator layer exactly as stated in Lean, and precisely what is proved versus open.

lake build ✓
Machine-checked
vs Mathlib
0
sorry in the
proof chain
m ≥ 1
every level,
every θ
std
axioms only
(#print axioms)
formalization/EvgenyTheorem/ — key statements (proofs omitted)
/-- THE EVGENY THEOREM, operator level: PROVED for every m ≥ 1, every θ. -/ theorem traceDefect_closed (θ : ℝ) (m : ℕ) (hm : 1 ≤ m) : traceDefect m θ = -16 * ((3 : ℂ) ^ (m - 1) + 1) * ((Real.sin (θ / 2) : ℝ) : ℂ) ^ 2 /-- Coupled recursion over two orientations. -/ theorem defect_succ_num (m : ℕ) (hm : 1 ≤ m) (θ : ℝ) : defectO (m + 1) θ faceAxis (fun _ => 2) (oNum (m + 1)) = 2 * defectO m θ faceAxis (fun _ => 2) (oNum m) + defectO m θ faceAxis (fun _ => 2) (oTil m) theorem defect_succ_til (m : ℕ) (hm : 1 ≤ m) (θ : ℝ) : defectO (m + 1) θ faceAxis (fun _ => 2) (oTil (m + 1)) = defectO m θ faceAxis (fun _ => 2) (oNum m) + 2 * defectO m θ faceAxis (fun _ => 2) (oTil m) /-- Base case with the corner flux edge reversed. -/ theorem tilde_base (θ : ℝ) : defectO 1 θ faceAxis (fun _ => 2) (oTil 1) = 0
traceDefect_closedPROVED

For the concrete operators HC and HC′ on SG(m): Δm(H⁴, θ) = −16(3m−1+1) sin²(θ/2), for every m ≥ 1 and every θ. Kernel-checked, no sorry.

gluing + recursionPROVED

Three copies glued into the next level, corner data proved for every m, and the recursions Dm+1 = 2Dm + D̃m, D̃m+1 = Dm + 2D̃m close the induction.

scopeNOT CLAIMED

m = 0, and any moment other than the fourth: the H6/H8/H10 values stay numerical regression references. No overclaim.

ResultStatusBasis
Lean 4 graph infrastructure — SG(m), Vertex, Adj, degreeCONFIRMEDcomputable + decidable instances, lake build
H4 exact closed form · I∞(π/2) = −8/9DERIVEDexact formula, verified numerically to 1e−13
Gauge invariance · vanishes at p = 1, 2, 3CONFIRMED< 8e−15 under random SU(2) gauge · first nonzero at p = 4
H6 growth law α(θ)·3^m + β(θ) · recurrence · path-class zeroDERIVEDmachine-precision numerics, locked by tests
H6 / H8 / H10 closed formsOPENno claim made — reference values only
Lean 4 operator-level closed form (all m ≥ 1, all θ)CONFIRMEDmachine-checked, no sorry · traceDefect_closed
BEYOND H⁴ — OPEN PROGRAM

Higher moments: the frontier

H⁴ collapses to one triangular traversal per walk. H⁶ does not. What is established — and what stays open.

<10⁻⁸
fit from m=4,5 predicts m=3
<10⁻¹⁵
no 3^2m term in H6 growth
5280
C(π) = 2^5·3·5·11 — exact
486
walks in H6 class {0,1,2}
p = 4
first nonzero moment
H6·H8·H10
reference values locked by tests

✓ Established — numerically, to machine precision

  • Growth law Δ_m(H⁶,θ) = α(θ)·3^m + β(θ) — no 3^2m term (coefficient ratio < 10⁻¹⁵).
  • Recurrence Δ_m = 3·Δ_{m−1} + C(θ) for m ≥ 2 · C(π) = 5280 · α(π) = −4112/3, β(π) = −2640.
  • Path-class zero theorem — on SG(1), single- and two-triangle walk classes contribute exactly zero; only walks visiting all three triangles count.

◇ Open — explicitly no claim

  • Closed forms for H6, H8, H10. The 486 walks of class {0,1,2} carry different axis sequences whose SU(2) holonomies do not collapse to a finite trig polynomial.
  • Values below are regression anchors, locked by tests/test_higher_moments.py and tests/test_path_class_h6.py.
mΔ(H⁶)Δ(H⁸)Δ(H¹⁰)
1−3180.000000−246909.026268−14013358.982483
2−6956.588745−595053.284942−37686518.369790
3−18274.354980−1632052.331669−107772338.487566
4−52227.653685−4743049.471847−318029798.840894
→ θ = π/2, numerical reference values. Full analysis: docs/HIGHER_MOMENTS.md.
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 31 checks — seconds $ pytest tests/ -m slow -v level-7 convergence — ~1–4 minutes $ lake build Lean 4 — kernel decides graph facts vs Mathlib ================== 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 ├── docs/ │ ├── index.html this site │ └── HIGHER_MOMENTS.md H6/H8/H10 — growth law, recurrence, path classes ├── formalization/ Lean 4 — graph infra machine-checked (lake build) │ ├── EvgenyTheorem.lean delta, I, traceDefect_eq_delta, TraceDefectIdentity (abstract form) │ └── EvgenyTheorem/ Graph (SG(m), Faces) · SU2 · Operator · SanityChecks ├── 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/ every claim above is a pytest check │ ├── test_levels.py · test_theta.py · test_heldout.py │ ├── test_gauge_invariance.py · test_higher_moments.py │ └── test_path_class_h6.py ├── data/ raw + computed spectra (SG 4–6, plain SG 1–6) ├── 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 · Lean graph layer kernel-checked