Gauge-Couplings and Fine Structure Constant from a 7D 2 time PLI Grand Unification Model

We derive a discrete gauge–coupling dictionary from a rectangular two–torus compacti-
fication T 2 selected by the Principle of Least Information (PLI). The dictionary ties each
Standard-Model (SM) coupling αi to a compact geodesic length Li and a small integer mul-
tiplicity mi through a single compactification scale K, αi = K mi/L2
i . Using only integers
≤ 7, we find an isotropic baseline solution with R2/R1 = 1, axis integer k = 7, asymmetric
pair (p, q) = (5, 2), multiplicities (mx, my , mpq ) = (2, 1, 4), and K = 0.854580, which yields
α1(MZ ) = 0.017092, α2(MZ ) = 0.034183, α3(MZ ) = 0.117873,
i.e. an overall RMS error of 0.80% vs. the canonical targets. The electroweak ratio α2/α1 = 2
follows exactly from the multiplicity pattern on an isotropic T 2. For comparison, an
anisotropic accuracy leader with R2/R1 = p4/3 , k = 7, (3, 2), (3, 2, 3), and K = 0.564011
achieves RMS 0.40%. Both solutions use only integers and one common scale; percent-level
threshold corrections (KK/brane matching) can absorb the residuals without introducing
floats. We present the MDL (description-length) rationale, numerics at MZ . High Energy
fine structure constant calculated accurately, 1/127.

In the second Metrology Paper we apply discrete PLI based parameters in to the calculation of couplings to ascertain if the "No Float" belief that integers are the most PLI appropriate values to minimise Information required to describe the universe.