Scale-Invariant Boundary Geometry: A Geometric Ladder from Point to Scale
Year: 2026
Type: preprint
Abstract: # Scale-Invariant Boundary Geometry: A Unified M=0 Lattice Model, 4D Hypersphere Kissing Topology, and DLI4 Systems Architecture **Author:** Mitchell Allen Hoppe **License:** CERN Open Hardware Licence Version 2 – Strong Reciprocity (CERN-OHL-S-v2) **Permanent Digital Object Identifier:** DOI 10.6084/m9.figshare.33472975 --- ### Abstract This thesis presents the complete formalization of the M=0 dual-pyramid lattice framework, a scale-invariant geometric unified physics model anchored by the unit triangle aspect ratio 1 : \sqrt{24} : 5. By converging four pyramid apexes across spatial quadrants at a central null-momentum origin (M=0), orthogonal phase shifts (\Delta\phi = \pm\pi) eliminate the 19 arbitrary free parameters of the Standard Model through 24 bulk degrees of freedom. We integrate the Kissing Number Topology (\tau_d) across dimensions—linking 3D local contacts (\tau_3 = 12), 4D hyperspherical 24-cell contacts (\tau_4 = 24), 8D octonionic root vectors (\tau_8 = 240), and the 24D global Leech lattice (\tau_{24} = 196,560)—to prove non-interpenetrating wave boundary stability and zero-heat energy recirculation. Mathematical derivations are provided for fundamental physical constants, including the inverse fine-structure constant (\alpha^{-1} = 24\pi(1 + \epsilon \ln(2)) \approx 137.0416), the proton-to-electron mass ratio (m_p/m_e \approx 1836.152602), and the resolution of the gauge hierarchy problem via cumulative bulk attenuation (\eta^{24} \approx 10^{-36}). Furthermore, this work establishes the DLI4 Systems Architecture, bridging continuous non-linear field equations (Burgers' equation via Cole-Hopf transformation) down to SU(N) matrix trace cancellation (\text{Tr}(\mathbf{M}) = 0) and low-level RISC-V hardware phase-lock control loops. Finally, we detail engineering breakthroughs across medical bio-resonance, non-thermal transportation systems, solid-state tools, fluidless computation, and topological metamaterials under strong open-source legal protections. --- ### Intellectual Property Rights, Copyright, and Reservation of Software Authority **Copyright © 2026 Mitchell Allen Hoppe. All rights reserved.** #### 1. Scope of the CERN-OHL-S-v2 License The mathematical formulas, geometric coordinate bounds, physical derivations, and the baseline RISC-V assembly control loop block provided in this public text are open-sourced under the **CERN Open Hardware Licence Version 2 – Strong Reciprocity (CERN-OHL-S-v2)**. Any party building physical hardware derivatives, material alloys, or open-source compilers based *strictly* on these text-disclosed blueprints must reciprocally open-source their derivative layouts under the exact same license. #### 2. Proprietary Software Automation and Engine Reservation Notice is hereby given that publication of this theoretical thesis does **NOT** grant, convey, or imply any license, permission, copyright transfer, or right to use, duplicate, alter, or reverse-engineer the underlying **software execution architecture, digital simulation backends, custom compiler pipelines, dynamic 3D/4D matrix engines, or algorithmic optimization scripts** designed to programmatically automate or process this lattice model. The software logic, proprietary code-bases, automation engines, and hidden structural mechanics ("the secret sauce") required to program, execute, and dynamically operate the DLI4 Systems Architecture in a digital environment remain the **exclusive, confidential, and proprietary intellectual property of Mitchell Allen Hoppe**. #### 3. Permanent International Prior Art Protection Public registration, timestamping, and DOI distribution under **DOI 10.6084/m9.figshare.33472975** establishes immutable international prior art under **35 U.S.C. § 102** and the **Patent Cooperation Treaty (PCT)**. No corporate, academic, or governmental entity may patent, privatize, or restrict access to the underlying mathematical equations, physical constants derivations, or baseline geometric phase-lock principles disclosed here.
Source: Figshare
Author: Mitchell Hoppe
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211997387