Definition. A finite-horizon learning account [ftip-00O8]
Definition. A finite-horizon learning account [ftip-00O8]
Let \(b(t)\) be an absolutely continuous liquid balance with \(b(0)=b_0\geq 0\). Let \(f(t)\geq 0\) be admitted inflows, \(o(t)\geq 0\) other outlays, \(u(t)\geq 0\) the counted compute rate, and \(p(t)>0\) its price. Assume these flows are measurable and integrable, and the balance equation holds almost everywhere. The account satisfies
\[\dot b(t)=f(t)-p(t)u(t)-o(t),\qquad b(T)\geq 0.\]All quantities are measured over the same institutional boundary and time interval. Inflows include operating receipts, equity, grants and debt proceeds whenever admitted; debt service and terminal obligations enter outlays or a stronger terminal-balance condition. Receipts recycled within this account are not counted again as external funding. A restriction to retained earnings is a special financing model, not a property of AI.
For a resource envelope, require a uniform bound \(\int _0^T f(t)\,dt\leq F\) over the admitted policies, with \(0\leq F<\infty \). Expected inflows alone do not impose this almost-sure bound. Unlimited refinancing violates this premise unless some separate constraint bounds its cumulative proceeds.