proposition. A conditional asymptotic portfolio advantage [ftip-00OI]
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    Under Definition [ftip-00OH], suppose actual assisted schedules exist and, for integers \(n\geq 1\), constants \(C_s,C_c,c_0>0\), exponents \(a,b,d\geq 0\) and \(r>0\),

    \[S(n)\leq C_s n^a,\qquad c(n)\leq C_c n^b,\qquad R(n)=\lceil n^r\rceil ,\qquad L_{R(n)}(n)\geq R(n)c_0 n^d.\]

    If \(\max \{a-r,b\}<d\), then

    \[\frac {U_{R(n)}(n)}{L_{R(n)}(n)}\longrightarrow 0.\]

    Proof. 

      Since \(R(n)\geq n^r\),

      \[0\leq \frac {U_{R(n)}(n)}{L_{R(n)}(n)} \leq \frac {C_s}{c_0}n^{a-r-d} +\frac {C_c}{c_0}n^{b-d}\longrightarrow 0.\]

      For instance, \(a=2,b=0,r=2,d=1\) obeys the exponent condition. The displayed lower bound on autonomous portfolios is still a hypothesis; the calculation does not establish it for conceptual discovery. The result identifies a possible macroeconomic route: sustained expertise serves many tasks while each recipient acquires a comparatively inexpensive contribution. If autonomous preparation is equally reusable, the proposed \(L_R\) may fail.