Equations of Motion for an Economy: Capital Deepening, Technology, and Firm Survival

We derive equations of motion for capital deepening in a competitive economy directly from accounting identities, without assuming a production function. A profit imperative ${\eta }^{\ast }\equiv (w\mathrm{/}\kappa +1\mathrm{/}\tau )\mathrm{/}(1-f_p)$ sets the minimum viable capital productivity, where $\eta =Y\mathrm{/}K$ [yr${}^{-1}$] is capital productivity, $\kappa =K\mathrm{/}L$ is capital per worker, $w$ is the wage rate, $\tau$ is the capital lifetime, and $f_p$ is the production tax share. Four coupled relaxation equations govern $\kappa$, $\eta$, the frontier productivity ${\eta }_{\mathrm{n}\mathrm{e}\mathrm{w}}$ of new investment, and the labor share $q\equiv w\mathrm{/}y$, with the sandwich constraint ${\eta }^{\ast }\le {\eta }_{\mathrm{n}\mathrm{e}\mathrm{w}}\le \eta$ maintained as an exact invariant. The frontier equation separates two physically distinct channels: a structural cheapening channel ($\mu$, always active, drives ${\eta }_{\mathrm{n}\mathrm{e}\mathrm{w}}$ downward) and a productivity channel ($\phi$, historically zero). Calibration against BEA 2-digit NAICS sector data (1998–2023) confirms $\phi =0$ for all identifiable sectors over 25 years; the 75-year postwar record extends this finding across four capital lifetimes. A step $\phi =0.01$,yr${}^{-1}$ – a 1%/yr improvement in new-capital productivity, modest but historically unprecedented – nearly doubles the aggregate growth rate within one capital lifetime, a falsifiable prediction with a precise observable signature: upward-curving $\eta (t)$ in BEA sector data. Firms near the zero-profit threshold have a cash martingale, predicting establishment exit rate $\sim t^{-1\mathrm{/}2}$; convolved with the Zipf firm-size distribution~\cite{WP}, this yields firm exit rate $\sim t^{-1\mathrm{/}2}\text{\hspace{0.17em}⁣}\log t$ with apparent exponent $b=0.295\pm 0.03$, confirmed against BDS data with no free parameters.