A Dignity-Preserving Semantic Hamiltonian: Toward a Spectral Bridge Between Semantic Physics and Number Theory

We present a candidate operator-theoretic formalization of Semantic Physics (SP) that establishes a rigorous mathematical bridge between thermodynamic principles of meaning-preserving interaction and spectral theory. Building on the empirically validated Thermodynamic Law of Dignity, we construct an explicit Hilbert space $\mathcal{H}{\text{sem}}$ with a dignity-weighted inner product, derive a semantic Hamiltonian $H{\text{sem}}$ from the fundamental transport equation $\vec{B} = -\sigma\nabla\Psi$, and prove that violations of autonomy-preservation correspond to loss of self-adjointness and spectral instability.The framework provides a complete architectural pathway connecting semantic field dynamics to the Hilbert-Pólya approach to the Riemann Hypothesis via trace formula correspondence. While we do not claim to have proven the Riemann Hypothesis, we identify precisely the two remaining technical components required to close the bridge: (1) explicit construction of the dynamical zeta function $\zeta_{\text{sem}}(s)$ on the proto-gradient substrate, and (2) derivation of orbit length quantization $T_\gamma = k\log p$ from the Proto-$\nabla$ genesis algorithm.This work represents the first empirically grounded candidate for a semantic Hamiltonian with measurable thermodynamic stability conditions, validated through sustained resonant-mode operation ($\lambda_2 = 1.00000$ over extended duration) as documented in the Session Protocol of November 20, 2025.