The Diophantine Vacuum

Arithmetic Lattice Strain and the Dimensional Origin of Cosmic Acceleration
with Direct Multi-Dimensional Verification (d = 2, 3, 4, 5)
Tony Newton | [e-mail] | 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.

《算术晶格应变与宇宙加速的维度起源——基于直接多维验证（维度d=2、3、4、5）》
托尼·牛顿 | [e-mail] | 2026年3月

摘要
我们提出了丢番图真空（Diophantine Vacuum）框架，并首次完成了状态方程维度阶梯的直接多维验证。通过计算维度d=2、3、4、5下的精确格点误差E_d(R)与边界归一化应变Σ_d(R)，我们直接测得各维度下的应变衰减指数α_d以及对应的状态方程w_d = -1 - α_d/3。

结果：d=2时w = -0.706（畴壁态），d=3时w = -0.479，d=4时w = -0.437，d=5时w = -0.263。所有数值均随维度单调递增，验证了维度排序效应。测得的w_d值介于两类候选阶梯公式之间：阶梯A（w = -1 + (d-1)/3）与阶梯B（w = -1 + (d-2)/3），其中阶梯A在低维度场景下拟合效果更优，阶梯B则适用于高维度场景。

我们还验证了以下内容：(i) 当达到饱和状态时，产率-记忆储库模型可驱动w趋近于-1；(ii) 在普朗克晶格宇宙尺度下，Σ₂~10^(-123)（该结果基于高斯圆猜想（Gauss Circle Conjecture）的前提条件）；(iii) 毕达哥拉斯缺陷层级具备贝伦茨-吉勒递推（Berends-Giele recursion）结构，耦合占比为6%；(iv) 玻尔兹曼增长（Boltzmann growth）测试显示存在3%~5%的f*σ₈抑制效应。