Natural-Law Acceleration of VPO

This paper proves a natural-law acceleration result for the spread of a Viable Predictive Organization (VPO) under auditable, model-agnostic floor conditions. We study two audited floor processes—minimal connectivity Dmin⁡(t)D_{\min}(t)Dmin(t) and minimal local net improvement Lnet+(t)L_{\mathrm{net}+}(t)Lnet+(t)—and analyze the speed lower bound vLB(t)=2Dmin⁡(t) Lnet+(t)v_{\mathrm{LB}}(t)=2\sqrt{D_{\min}(t)\,L_{\mathrm{net}+}(t)}vLB(t)=2Dmin(t)Lnet+(t). The core technical contributions are: (i) a line-integral representation for ΔvLB(t)\Delta v_{\mathrm{LB}}(t)ΔvLB(t); (ii) an almost-sure ratio band for Dmin⁡/Lnet+D_{\min}/L_{\mathrm{net}+}Dmin/Lnet+ with deterministic constants; (iii) a signed-coefficient inequality that correctly handles positive and negative increments; and (iv) a vanishing-average negative-variation condition (M1′) that can be enforced and audited via floor design. Assuming multiplicative tempering (R) and bounded one-step updates (BD), we give two audit routes: Route A (predictable drifts for both floors) and Route B (predictable drift for Lnet+L_{\mathrm{net}+}Lnet+ plus a connectivity-improvement lower density A5++ for Dmin⁡D_{\min}Dmin). In both routes we derive explicit constants mD,mL,MD,MLm_D,m_L,M_D,M_LmD,mL,MD,ML and a strictly positive acceleration constant c⋆c_\starc⋆, yielding linear growth of E[vLB(T)]\mathbb{E}[v_{\mathrm{LB}}(T)]E[vLB(T)] and a quadratic lower bound on cumulative coverage. All constants are log-extractable (falsifiable), and optional conditional mgf assumptions are used only to sharpen concentration. The result is natural-law—it follows from auditable regularities and floors rather than a crafted global objective.

Keywords
Artificial Intelligence; natural-law acceleration; Viable Predictive Organization; auditable floors; signed-coefficient inequality; Cesàro acceleration; martingale SLLN; concentration inequalities; predictable drift; improvement density.