Prime-Indexed Lattice Symmetry and Quantized Phonon Modes in 7-Fold Systems: A Rigorous Theoretical Framework" by Juan Alberto Hernandez Rivera.

The paper derives how prime-indexed lattice vectors enforce 7-fold rotational symmetry in modulated crystalline systems, quantizing force constants and producing irreducible phonon modes. * It builds on the Coherence Evolution Model (CEM) and demonstrates that primes act as irreducible symmetry generators through modular arithmetic constraints, yielding gapped phonon spectra with modes resonant to CO₂ bending vibrations (667 cm⁻¹). * The paper provides analytic proofs, group theory arguments, and open-source code to validate this framework, offering a novel design principle for quantum-critical catalysis. * Key findings include a group-theoretic proof of 7-fold symmetry via prime-modulated translations, quantized force constants (\Phi \propto \log p) from irreducible lattice vectors, and predicted phonon spectra with gaps at 667 cm⁻¹ (CO₂ bend).