Continuum Conditions, Mode Leakage, and Curvature Handoff

SCQG paper 17 of 28. Paper 17 develops the conditions under which the microscopic system of Semantic Commitment / Quantum Gravity (SCQG) has a continuum description of spacetime curvature. A continuum description represents the system through fields on continuous spacetime. Its validity depends on physical support, causal propagation, stability, memory, fluctuations and controlled approximation errors. The paper separates the modes retained in this description from the unresolved modes. Eliminating unresolved modes leaves their effects in the equations through response, memory, forcing and initial correlations. The construction tracks these effects as the cutoff changes. Mode leakage means actual evolution out of the retained subspace; the paper gives a norm that measures it. The spatial spectrum and long-wavelength relaxation must each realize three spatial dimensions, expressed as dₛ = dₕ = 3. These conditions are tested together with causal propagation and convergence under refinement. The stochastic equations use the physical covariance of their sources, with mean-energy bounds and mean-square convergence for random source factors. If a continuum representation fails, the construction tests available alternative representations before returning to the microscopic dynamics. The paper also derives conditions for Ohmic response, in which the force spectral density grows linearly with frequency: J_C(Ω) ∝ Ω as Ω → 0. This result depends on the mode density, dispersion and the physical capacity coupling. Short-memory dynamics follow when the retained memory kernel admits a controlled local approximation. Finally, the paper couples the continuum equations to the equations governing how effective couplings change with scale. It gives sufficient conditions for a common local solution, uniqueness and continuation, together with bounds for validating a computed solution.