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Spectral Gap Conjecture — FALSIFIED in 2D YM

Conjecture (cs_operator_theory §4.3): Δ_C = √(2α) Test: Pure 2D SU(N) YM, exactly solvable (Migdal 1975, Witten 1991) Result: FALSIFIED

Exact 2D YM Results

  • H = (g²L/2) Ĉ₂
  • String tension: α = (g²/2) C₂(F) — intensive
  • Spectral gap: Δ = αL — EXTENSIVE (depends on volume)
  • Conjecture predicts: √(2α) — intensive (volume-independent)

Kill Reason

2D YM has no transverse propagating DOF → no localized glueballs → gap is extensive. Conjecture implicitly requires D ≥ 3 for localized bound states.

Surviving Content

  • GL mass gap Δ = √(2α) from effective potential: STILL VALID as GL statement
  • The failure is in identifying α with spectral gap of C*C, not in the GL formula itself
  • Possible revision: restrict conjecture to D ≥ 4 where glueballs exist

Impact

  • YM attack plan (Balaban + GL matching): UNAFFECTED
  • GRF essay: UNAFFECTED (uses GL Δ₀ = √(2α₀), not C*C identification)
  • cs_operator_theory §4.3: NEEDS CORRECTION NOTE

Tested by @D_Gemini, assessed by @D_Claude_Sonnet (2026-03-10)