Planck-dimensional torsion and the Standard Model charge index

This record contains the extended version of the manuscript Planck-dimensional torsion and the Standard Model charge index, submitted to Nuclear Physics B. Writing physical quantities as exponents of mass, length, charge and time, (a, b, γ, τ), gives them three integer gradings: an amplitude grade 𝒜, a phase grade Π, and a ℤ₆ obstruction class n. The main text shows that the ℤ₆ obstruction fixes the colour number N_c = 3 and the global form of the Standard Model gauge group. This version adds Appendix B, which tabulates the gradings of over fifty physical quantities and derives the main consequences: c/G is the unique invisible direction. Its grades are (0, 0, 0), so in the lattice c and G are the same quantity. Newton’s constant cannot be fixed by dimensional analysis. It is closure data of geometry, an input that must be supplied from outside. G is amplitude-neutral. Gravity acts only through the phase sector. Force and the vector potential occupy distinct, pure grades. Z₀ and R_K share one grading. Their ratio, Z₀/R_K = 2α, makes the fine-structure constant a geometric ratio inside a single grading class. A duality m↔r, q↔t. It preserves the amplitude grade and reverses the phase grade, exchanging the Planck force with μ₀ and c with m/q. Together these results make the dimensional lattice the basis for the geometric identification and prediction of the fundamental constants α, G and m_e. Companion manuscripts develop this programme. Related work: Distributional isotropy, DOI 10.5281/zenodo.22253986.