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The research aims to validate the scaling law for a 49-cell STNT lattice and its implications for quantum architecture.

April 18, 2026Open Access

Chapter 6 Figure₃: N=49, 7x7 Scaling law proof.

Key Points

The research aims to validate the scaling law for a 49-cell STNT lattice and its implications for quantum architecture.
Analysis of a 49-cell STNT lattice with visualization of Hilbert spaces and energy shells.
Derivation of spectral results linked to the Riemann zeros using numeric and geometric foundations.
Reproducible numerical experiments with Python code for exploring eigenvalue spectra.
The outer states of the lattice exhibit lower energy than the core, showcasing the spectral compression effect.
Demonstrated scale invariance of the STNT topology across different lattice sizes.
Validated 18 non-trivial eigenvalues correspond to the first nine Riemann zero attractors with high precision.

Abstract

V3. 0 Figure₃: N=49, 7x7 Scaling law proof. (Chapter 6) The 49-cell STNT lattice (N=49, L=7). * The 64-node Hilbert space exhibits a stable 9-state st core (red) identical to N=25, surrounded by two energy shells. This demonstrates the L² → (L+2) ² scale invariance of the STNT topology. The outer st states (10st-49st) are shown to have lower energy than the core, explaining the spectral compression effect where Deltaₙ decreases for higher n. K=112. What's New in v2. 0This version adds the geometric foundation of STNT. Chapter 4-5 provided the theory. Figures 1 & 2 provide the visual proof. Any researcher can now reconstruct the operator from the lattice diagrams alone, verifying that the spectral results are not fitted but derived. This record contains Chapter 4 and Chapter 5 of STNT Vol. 1. Chapter 5 includes the Nine-Zero Hilbert-Polya Bracketing results showing 0. 20% error versus the first 9 Riemann zeros. This preprint presents Chapters 4-5 of STNT Vol. 1: The 3×3 Causal Core and Quantum Architecture by Ehab Ramkh. Chapter 4 provides the complete spectral representation of the STNT framework, establishing the first geometric structure directly linked to the Riemann zeros. This chapter includes: 1. Formal definition of the STNT Adjacency Operator and proof of Hermiticity from topology2. The complete 36-eigenvalue spectrum, partitioned into three bands: Trivial, Fracture, and Non-trivial3. Numerical experiments demonstrating band formation and correspondence to distinct Riemann zeros4. Reproducible Python code for all experiments5. Scaling evidence via the 25-Cell Operator (N=25) and convergence analysis The numerical results in Section 5. 3 demonstrate that the 18 non-trivial eigenvalues correspond to the first 9 Riemann zero attractors with high numerical precision. This work is part of the larger STNT Vol. 1 framework. The full book will be published commercially on Amazon KDP. This DOI establishes priority for the spectral methods and numerical results contained herein. Related to: https: //doi. org/10. 5281/zenodo. 19587559

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