Sub-Planckian Discrete Spacetime Calibrated by Dirac's Large-Number Ratio: Effective Field Theory Bounds, Sector Invariants, and a Ten-Condition Falsifiability Roadmap

Abstract

We evaluate the continuum limit of a discrete spacetime substrate with fundamental spacing fixed by Dirac's large-number ratio Q ~ 4.17 x 10^42. At the sub-Planckian scale delta_p = L_P / Q ~ 3.88 x 10^-78 m, we examine electrodynamics recovery and the leading Lorentz-violating corrections within an effective-field-theory framework.Lattice spatial parity symmetry eliminates odd-order finite-difference terms, excluding dimension-5 operators and cubic energy corrections. The leading residual correction is quadratic in the grid spacing, O(delta_p^2), modifying photon dispersion at dimension 6. An angular prefactor g(theta, phi) encodes its directional structure, vanishing along the Cartesian axes and peaking at 2/3 along the space diagonals.Applying g(theta, phi) to the infrared condition (BC9) places the predicted dispersion correction roughly 55 orders of magnitude below current ultra-high-energy cosmic-ray bounds while predicting zero vacuum birefringence. Algebraic analysis isolates two scale invariants, zeta* ~ 10.91 and K* = Q^2, once unit-convention scale factors are removed. Matching against Newton's gravitational constant G_N reduces K* to a calibration constraint.The paper's primary contribution is organizational: a ten-condition falsifiability roadmap (BC1-BC10) converting these results into explicit, numerically checkable failure criteria. Among five evaluated conditions, BC1 fails under unconstrained bulk volume counting. BC2 satisfies spatial isotropy via cubic point-group symmetry. BC4's entropic construction underestimates G_N by Q^2. BC5 achieves topological quantization but underestimates the empirical fine-structure constant alpha by a factor of 80. Five conditions remain open.