USSFT Paper III: Coarse-Graining Origin of the SRI. Why the Short-Range Inhibitor Takes the Form (Box Phi) ²: A Consistency Check

Within the Unified Space-Time and Scale-Dependent Field Theory (USSFT) framework, Paper I (DOI: 10. 5281/zenodo. 19622931) establishes the fundamental U-field dynamics on a discrete hypercubic mesh, with the Tier-1 hypercubic NLO correction operator derived from the lattice Taylor expansion. This paper provides an independent consistency check: starting from a continuum field with finite resolution imposed via Lorentz-invariant coarse-graining, we show that the leading higher-derivative operator consistent with (i) finite operational resolution, (ii) infrared Lorentz symmetry, (iii) low-energy stability, and (iv) analyticity takes the (BoxPhi) ² form, the isotropic Tier-2 limit of the fundamental hypercubic operator. The Tier-2 EFT Lagrangian adopts the sign LSRI = + (eta LSRI² / 2) (BoxPhi) ² (with eta = 1/12 for the hypercubic stencil), as determined by the lattice Taylor expansion in Paper I. We compare common UV-regularization alternatives within the local-EFT setting. The explicit coarse-graining construction shows how the LSRI² scaling emerges from the resolution kernel. This construction fixes the leading local operator form and scaling; the physical sign and eta value are fixed by the discrete lattice Taylor expansion in Paper I. This analysis supports the physical interpretation of LSRI as a resolution scale and confirms that the isotropic limit of the discrete theory matches continuum EFT expectations. Status: (B) derived consistency check under stated assumptions. This is Paper III in the 18-paper USSFT technical series; for a conceptual overview, see Paper 0 (DOI: 10. 5281/zenodo. 17852167, published in Int. J. Quantum Found. 12 (2), 667-718, 2026).