proposition. A uniform finite-horizon compute envelope [ftip-00O2]
✍️source

    Suppose for every \(z\in \mathcal Z\), almost surely, its nonnegative compute service rate \(u_z(t)\) has monetary price \(p_z(t)\geq p_*>0\), and

    \[\int _0^T p_z(t)u_z(t)\,dt\leq F_T,\qquad u_z(t)\leq q(t),\qquad u_z(t)\leq e(t)P(t).\]

    The deterministic functions are measurable and nonnegative; \(0\leq F_T<\infty \), \(q\) and \(eP\) are integrable. Here \(q\) bounds usable hardware throughput, \(P\) available power, and \(e\) compute per unit energy. These bounds hold over all admitted investments, prices, implementations and efficiency changes. Then

    \[\overline B(T;\mathcal Z)\leq \min \left \{F_T/p_*,\int _0^T q(t)\,dt, \int _0^T e(t)P(t)\,dt\right \}.\]

    Proof. 

      For each execution integrate the two rate inequalities. The expenditure inequality gives \(p_*\int _0^T u_z(t)\,dt\leq F_T\) almost surely. Taking the essential supremum for each execution and then the supremum over executions preserves all three bounds.

      This is a necessary envelope. It does not establish that its minimum can be spent on an arbitrary schedule or that its inputs describe a particular economy. An excluded financing source or a permitted efficiency improvement that violates the displayed premises invalidates that application of the bound.