USSFT Paper VI: Emergent Constants. The Planck Scale, Gravitational Coupling, and Quantum Action Scale from U-Field Parameters

This paper establishes the mapping between the fundamental U-field parameters Phi₀, LSRI, xi, lambdaU (Paper V, DOI: 10. 5281/zenodo. 19626955) and the laboratory constants hbar, G, and c. In the Unified Space-Time and Scale-Dependent Field Theory (USSFT), these constants are not fundamental; they are emergent coefficients governing how the U-field resolves constraints at different scales. Working with the effective combination LSRIₑff = sqrt (eta) LSRI (where eta = 1/12 is derived from lattice geometry), we express all results through an effective U-field action scale hbarₑff (defined in the EFT normalization and connected to laboratory hbar by a one-time calibration). Within the EFT framework established in Papers I-V, we show that: (1) the effective U-field action scale hbarₑff scales as Phi₀² LSRIₑff², with the numerical coefficient fixed by calibration; (2) Newton's constant G scales as 1/ (xi Phi₀²), following from the xi R Phi² coupling in the Lagrangian; and (3) the EFT description employs a universal limiting signal velocity c as part of its Lorentz-covariant structure (Paper IV addresses its dynamical preservation). The central result is the Planck-scale relation: combining these relations yields ellP = LSRIₑff sqrt (kappaₕbar kappaG / (8 pi xi) ). Interpreting this as ellP comparable to LSRIₑff is conditional on xi, kappaₕbar, kappaG being O (1) within the minimal EFT setup. This same relation is independently derived in Paper XV (DOI: 10. 5281/zenodo. 19691421) from black hole entropy matching, providing a non-trivial consistency check. We explicitly distinguish scaling relations (which follow from dimensional analysis and Lagrangian structure) from numerical coefficients (which require calibration or future calculation). Status: (B) derived emergent-constants framework under stated EFT assumptions, with explicit O (1) calibration dependencies. This is Paper VI 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).