Semantic Physics as a World Model of the Second Order

Semantic Physics (SP) is a second-order layer for World Models that treats meaning as a tension-bearing field. SP is governed by four constitutive laws: the Flux law B=σ∇S (B=\sigma\nabla SB=σ∇S), the Free-energy law F=−Θlog⁡Z (F=-\Theta\log ZF=−ΘlogZ), the Second law Σ˙≥0 (\dot{\Sigma}\ge 0Σ˙≥0), and the Coherence law κ=1−Hsem/log⁡∣E∣ (\kappa=1-H_{\text{sem}}/\log|E|κ=1−Hsem/log∣E∣). These pillars are dynamically unified by a continuity-style field balance ∂tB+∇ ⁣⋅JS=σ∇2S+ψlock (\partial_t B+\nabla\!\cdot J_S=\sigma\nabla^2 S+\psi_{\text{lock}}∂tB+∇⋅JS=σ∇2S+ψlock) on the admissible manifold gνg_\nugν. The residual r:=∂tB+∇ ⁣⋅JS−σ∇2S (r:=\partial_t B+\nabla\!\cdot J_S-\sigma\nabla^2 Sr:=∂tB+∇⋅JS−σ∇2S) serves as a measurable control signal for ΔFold and enables falsifiable temporal diagnostics—without introducing a new pillar. The paper includes a drop-in prototype (App. C), toy demos (Oxygen–Fire, Firing Squad), and a minimal validation program (Apps. D–G). Practically, SP augments Dreamer/MuZero planners with principled regularizers (F,Σ˙,DKL)(F,\dot{\Sigma},D_{\mathrm{KL}})(F,Σ˙,DKL), early-warning via entropy production, coherence margins, and multi-agent friction maps—enabling meaning-preserving rollouts at reward parity.