Volume 9: The Continuum Limit

Volume 9 of 14. Establishes the bridge from the arithmetic lattice ℤ≥0 × P1 to continuous spacetime ℝ⁴: the Bloch Band Merger and the complete continuum limit. Proves Theorem 36 (Bloch Band Merger): the lattice propagator G_a(p) = a²/(2−2·cos(a·p)+(a·m_gap)²) converges exactly to the continuum propagator 1/(p²+m_gap²) as a→0, with error O(a²). The mass term m_gap² survives the limit — it is not a lattice artifact but the arithmetic mass gap expressed in physical units. Proves Theorem 37 (Strong Resolvent Convergence): H_a → H_W in the strong resolvent sense via the Trotter-Kato theorem. Proves Theorem 38 (Symmetry Restoration): discrete lattice symmetries extend to the full Euclidean group E(4) = SO(4) ⋉ ℝ⁴ in the continuum limit, then to ISO(3,1) via Wick rotation. Proves Theorem 39 (Schwinger Function Convergence): S_n⁽ᵃ⁾ → S_n as tempered distributions; continuum two-point function C·K₁(m·r)/r. Proves Theorem 40 (Vacuum Convergence): |Ω₀⟩ → Ω_W as a→0. Proves Theorem 41 (Action Convergence): a⁴·C(N) → S_YM[A] = (1/4)∫F²d⁴x as a→0. Establishes that the continuum limit does not destroy the arithmetic — it reveals it. The mass m_gap carries Δ=13 in its numerator. The structure constants carry the PSGW arithmetic. The continuum Yang-Mills theory on ℝ⁴ is the arithmetic of M = 30 expressed in the language of continuous spacetime.