\section{Computational methods}
\label{app:algorithms}

All code is in \texttt{src/agi\_ge/} of the replication package (Python 3.14, NumPy, SciPy, numba); the
one-command reproduction is described in Appendix~\ref{app:replication}.

\subsection{Static block}

Given the predetermined stocks $(K^N_t,K^Q_t,K^E_t,K^H_t)$ and the period's $(\alpha_{i,t},\varepsilon_t)$, the
competitive allocation of labor and compute across tasks is the solution of the concave program
$\max_{L,Q\ge0}Y$ subject to $\sum_{i\in\Omega(s)}L_i=H_s$ and $\sum_iQ_i=Q$, with variables scaled to $[0,1]$
($l_i=L_i/H_{s(i)}$, $q_i=Q_i/Q$). It is solved by sequential least-squares programming with analytic gradients
and then \emph{polished}: the active set identified by the first solver (which tasks use labor, which use
compute) is fixed and the Karush--Kuhn--Tucker equalities $F_i=w_{s(i)}$ for tasks with labor, $F_i/\alpha_i=A$
for tasks with compute, and the two adding-up constraints are solved by Newton's method to machine precision.
The polished solution is accepted only if no inactive variable has a positive shadow value. Factor prices are the
multipliers (envelope theorem): $w_s=\max_{i\in\Omega(s)}F_i$, $A=\max_iF_i/\alpha_i$, $R^N=ZG_VV_K$,
$R^T=ZG_T$, $P^E=ZG_E$. Energy supply is predetermined by $K^E_t$ and allocated residually to final production;
if the marginal value of compute falls below its energy cost, utilization $u_t<1$ is found by bisection in
$\log u$ on the condition $A_t=\varepsilon_tP^E_t$. The housing rent follows from aggregate goods consumption.
A static solve takes 1--3 milliseconds; the Karush--Kuhn--Tucker violation is below $10^{-6}$ in every test and
the income identity \eqref{eq:identity} holds to $10^{-8}$.

\subsection{Perfect-foresight transition (two-class economy)}

Unknowns are, for $t=0,\ldots,T-1$, the logarithms of the four stocks $K^k_{t+1}$, the logarithm of capitalists'
expenditure $x^K_t$, and $r_{t+1}$: six per period, $6T$ in total. Residuals are the four capital Euler equations
\eqref{eq:capital_euler}, the goods-market clearing condition and the capitalists' Euler equation
\eqref{eq:euler}. Writing expenditure as an unknown rather than as the goods-market residual keeps the system
smooth where investment booms would otherwise drive the residual expenditure through zero. Beyond $T$ the
investment rates, the interest rate and the growth rate of expenditure are held at their period-$(T-1)$ values
and land prices take their Gordon values. Each residual depends on unknowns at $t-1$, $t$, $t+1$ only, so the
Jacobian is banded; it is computed by colored finite differences (30 evaluations of the full path) and factored
sparsely. Newton steps use a backtracking line search. The solution is reached by homotopy in the size of the
technology shock ($\lambda\in\{0,0.25,0.5,0.75,1\}$ scaling $g_{pQ}$, $g_\varepsilon$ and $g_\alpha$), with the base
point re-calibrated at each $\lambda$ so that 2026 stays consistent with the anticipated price path, on the full
horizon of 275 years (2026--2300). Convergence is declared when the maximum residual is below $10^{-9}$.
Truncation is checked by re-solving with 50 additional years and comparing the first 225 years of the stock and
interest-rate paths (Table~\ref{tab:diagnostics}).

\subsection{Heterogeneous households}

Given paths of returns, wages, transfers and the price index, the household problem is solved backward from the
stationary policy at the terminal prices by the endogenous grid method on a 300-point asset grid (curvature 4,
upper bound 400 times the 2026 average labor income, detrended by the efficiency factor), for the two skills,
three discount types and five productivity states. The distribution is simulated forward with the lottery method
of non-stochastic simulation. The general-equilibrium path of returns is found by relaxation: the production
side is re-solved with the return path given (the same banded Newton method with five unknowns per period), the
household block is re-solved, and $r_{t+1}$ is reduced in proportion to the log excess demand for claims at $t$,
until the maximum log excess demand is below $10^{-4}$.

\subsection{Tests}

The test suite (\texttt{pytest}, Appendix~\ref{app:replication}) checks: Propositions~1--6 of the source in the
benchmark; the Karush--Kuhn--Tucker conditions of the static block; the income identity, and that it fails on a
deliberately inconsistent accounting; the exactness of the frozen-technology balanced growth path; the
accuracy of the banded Jacobian against a dense one; the Euler-equation residuals, goods-market and asset-market
clearing of every saved scenario; and the horizon check.