PCH - Toy Hamiltonian: Prime-Modulated Lattice Model

In this revision, I flesh out the Prime-Coherent Hypothesis into a testable, semi-rigorous physical model. I start by constructing a toy Hamiltonian for a prime-labeled lattice, where hopping terms are modulated by the logarithmic relationship between primes. This yields eigenstates that are naturally localized at primes—suggesting primes themselves may act as a kind of “coherence basis” in certain quantum systems. I then show, through a variational principle, that a probability distribution over primes minimizes entanglement entropy when weighted as . That distribution isn’t arbitrary—it aligns with the intuition that primes are minimally redundant. Next, I connect this to the Riemann zeta function, suggesting that the spectral statistics of a quantum system tied to exhibit level repulsion consistent with quantum chaos. I speculate that observable consequences might appear in momentum-space structures (e.g., ARPES) that exhibit prime-linked features, like gap nodes at . I argue that composite-number states, being tensor products of prime states, introduce redundancy and rapidly decohere—suggesting a possible information-theoretic reason for physical systems to “prefer” primes. ---