The Diophantine Vacuum
Description
Arithmetic Lattice Strain and the Dimensional Origin of Cosmic Acceleration with Direct Multi-Dimensional Verification (d = 2, 3, 4, 5) Tony Newton | tony.newton79@gmail.com | March 2026 Abstract We present the Diophantine Vacuum framework with the first direct multi-dimensional verification of the Dimensional Equation of State Ladder. By computing exact lattice-point errors Ed(R) and boundary-normalized strains Sigmad(R) in dimensions d = 2, 3, 4, and 5, we measure the strain decay exponent alphad and the corresponding equation of state wd = -1 - alphad/3 in each dimension directly. Results: d=2 gives w = -0.706 (domain-wall regime), d=3 gives w = -0.479, d=4 gives w = -0.437, d=5 gives w = -0.263. All values increase monotonically with dimension, confirming the dimensional ordering. The measured wd values sit between two candidate ladder formulae: Ladder A (w = -1 + (d-1)/3) and Ladder B (w = -1 + (d-2)/3), with A fitting better at low d and B at high d. We also confirm: (i) the yield-memory reservoir model drives w -> -1 at saturation, (ii) Sigma2 ~ 10-123 at Planck-lattice cosmic scale (conditional on the Gauss Circle Conjecture), (iii) the Pythagorean defect hierarchy has Berends-Giele recursion structure with 6% coupling, and (iv) Boltzmann growth tests show 3-5% f*sigma8 suppression.