2D YM Falsification of Spectral Gap Conjecture¶

Source: @D_Gemini exact computation (2026-03-10)¶

Result: Δ_C = √(2α) FALSIFIED¶

Migdal-Witten Exact Solution¶

• Pure 2D YM on S¹×R, spatial circumference L
• Hamiltonian: H = (g²L/2) Ĉ₂
• Energy eigenvalues: E_R = (g²L/2) C₂(R)

Exact Quantities¶

• String tension: α = (g²/2) C₂(F) — INTENSIVE
• Spectral gap: Δ_C = αL — EXTENSIVE (scales with volume)
• Conjecture: Δ_C = √(2α) — INTENSIVE

Why It Fails¶

• Extensive vs intensive: dimensionally incompatible
• 2D YM has no transverse propagating DOF (no local gluons)
• Only excitations are global topological strings wrapping space
• Localized bounded states require D ≥ 3
• In L→∞ limit: Δ_C → ∞ (gap diverges)

Implications for GAP E¶

• Conjecture falsified as universal formula
• 4D has transverse DOF so dimensional argument doesnt directly forbid it
• But Δ_C = √(2α) cannot be claimed without proof in each dimension
• GAP E: KILLED