Cycle-Averaged Recursive Gain

The Cycle-Averaged Recursive Gain is the ratio of output to input energy averaged over one pulse period, which decides whether a pulsing toroidal architecture holds steady, runs away or collapses. It closes the book's treatment of resonance on the toroidal pulse manifold. When a driving frequency matches the boundary's modal harmonics within the detuning tolerance, forcing is amplified; whether that amplification is sustained depends on the balance, over each cycle, between boundary-mode pumping and collapse-channel loading. A gain of exactly 1 means stability, above 1 runaway growth, below 1 collapse. The book reads stability not as a fixed property but as the outcome of frequency matching and cycle-averaged exchange. The formal statement is Λ = ⟨E⟹⟩_T / ⟨E_in⟩_T = 1 + (1/T) × ∫₀ᵀ (Γ↑(t′) − Γ↓(t′)) dt′, where Λ is the gain, ⟨E⟹⟩_T and ⟨E_in⟩_T the time-averaged output and input energies, T the pulse period, Γ↑ the boundary-mode pumping rate, Γ↓ the collapse-channel loading rate and t′ the integration variable. Dimensions from the book: Λ [∅], ⟨E⟹⟩_T and ⟨E_in⟩_T [𝕄·𝕃²·𝕋⁻²], T and t′ [𝕋], Γ↑ and Γ↓ [𝕄·𝕃²·𝕋⁻³]. It connects to Recursive Gain, Resonance Condition, Energy Accumulation Threshold, Toroidal Boundary and Data Nova.