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\vspace*{2em}
{\Large\bfseries Online Appendix to\\[0.5em] ``The Cost Frontier of Cryptographically Relevant Quantum Computing: \\ Capital Intensity, Technological Expectations, and Security Investment''}

\vspace{4em}

\textbf{Stand-alone supplementary material}

\vspace{2em}

\emph{Prepared for Research Policy}

\vspace{6em}

\begin{minipage}{0.85\textwidth}
\noindent\textbf{Note on stand-alone reading.} This document is intended to be readable independently of the main manuscript. Every symbol, model expression, and definition used in any proof or computation is restated within this document. Numerical results are reproducible from the replication package accompanying the main manuscript.
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%=========================================================
\section{Preliminaries: the cost-frontier model}\label{app:prelim}
%=========================================================

This appendix restates the cost-frontier model and its standing assumptions. All subsequent appendices reference the equations and lemmas of Appendix~\ref{app:prelim}; the appendices are intended to be read in order.

\subsection{The cost-per-key expression}

The cost-per-successful-key expression is

\begin{equation}
C_{\text{key}}
\;=\;
\left[\,\frac{(K - S/(1+r)^{n})\,\mathrm{CRF}(r,n) + F}{8760\,u} \;+\; v_{h} \;+\; OC_{h}\,\right]\,E[T_{\text{success}}] \;+\; C_{\text{setup}} \;+\; C_{\text{target}},
\label{eq:Ckey}
\end{equation}

with expected successful-run time

\begin{equation}
E[T_{\text{success}}] \;=\; T_{s} \;+\; \frac{1-p}{p}\,T_{f},
\qquad p \in (0,1],\; T_{f} \geq 0,\; T_{s} > 0,
\label{eq:Esuccess}
\end{equation}

and capital-recovery factor

\begin{equation}
\mathrm{CRF}(r,n) \;=\; \frac{r\,(1+r)^{n}}{(1+r)^{n} - 1},
\qquad r \in (0,1),\; n \in \mathbb{N}_{\geq 1}.
\label{eq:CRF}
\end{equation}

\subsection{Definitions of every symbol}

\begin{itemize}
\item $K > 0$: capital cost of the operational CRQC system, in USD.
\item $r \in (0, 1)$: annual discount rate, dimensionless.
\item $n \in \mathbb{N}_{\geq 1}$: economic life in years.
\item $S \in [0, K]$: salvage value, in USD.
\item $F \in [0, \infty)$: annual fixed O\&M cost, in USD per year.
\item $u \in (0, 1]$: economically usable utilisation, dimensionless.
\item $v_{h} \in [0, \infty)$: variable hourly cost, in USD per hour.
\item $OC_{h} \in [0, \infty)$: opportunity-cost premium per attack hour, in USD per hour.
\item $T_{s} \in (0, \infty)$: successful-run wall-clock time, in hours.
\item $p \in (0, 1]$: per-attack success probability, dimensionless.
\item $T_{f} \in [0, \infty)$: failed-attack wall-clock time, in hours.
\item $C_{\text{setup}} \in [0, \infty)$: per-attack setup cost, in USD.
\item $C_{\text{target}} \in [0, \infty)$: per-attack target acquisition cost, in USD.
\end{itemize}

\subsection{Auxiliary quantities}

For compactness in proofs:

\begin{align}
A &\equiv (K - S/(1+r)^{n})\,\mathrm{CRF}(r,n) + F, \\
H(u) &\equiv \frac{A}{8760\,u}, \\
G(p, T_{f}, T_{s}) &\equiv T_{s} + \frac{1-p}{p}\,T_{f}, \\
B(u, v_{h}, OC_{h}) &\equiv H(u) + v_{h} + OC_{h},
\end{align}

so that (\ref{eq:Ckey}) becomes

\begin{equation}
C_{\text{key}} \;=\; B(u, v_{h}, OC_{h})\,G(p, T_{f}, T_{s}) \;+\; C_{\text{setup}} \;+\; C_{\text{target}}.
\label{eq:Ckey-compact}
\end{equation}

\subsection{Lemmas used throughout}

\paragraph{Lemma 0.1 (positivity of $A$).} If $K > 0$, $r \in (0, 1)$, $n \in \mathbb{N}_{\geq 1}$, $0 \leq S \leq K$, and $F \geq 0$, then $A > 0$.

\textit{Proof.} (i) $\mathrm{CRF}(r, n) > 0$: numerator $r(1+r)^n > 0$, denominator $(1+r)^n - 1 > 0$ since $(1+r)^n > 1$ for $r > 0$, $n \geq 1$ by induction. (ii) $K - S/(1+r)^n \geq 0$ since $(1+r)^n \geq 1$ so $S/(1+r)^n \leq S \leq K$. (iii) Strict positivity: if $S < K$, then $K - S/(1+r)^n > 0$ strictly; if $S = K$, then $K - K/(1+r)^n = K(1 - 1/(1+r)^n) > 0$ since $(1+r)^n > 1$. In either case the first term of $A$ is strictly positive, hence $A > 0$. $\square$

\paragraph{Lemma 0.2 (positivity of $G$).} If $T_s > 0$, $p \in (0, 1]$, and $T_f \geq 0$, then $G(p, T_f, T_s) > 0$.

\textit{Proof.} $G = T_s + ((1-p)/p)T_f$; first term $T_s > 0$; second term non-negative since $(1-p) \geq 0$, $p > 0$, $T_f \geq 0$. Hence $G \geq T_s > 0$. $\square$

\subsection{Axioms used in decision-theoretic propositions (4 and 5)}

\paragraph{Axiom 1 (Expected-cost minimisation).} An agent choosing between actions $a_1, a_2, \ldots, a_k$ with finite expected costs $\mathbb{E}[c(a_1)], \ldots, \mathbb{E}[c(a_k)]$ selects $a_i$ such that $\mathbb{E}[c(a_i)] \leq \mathbb{E}[c(a_j)]$ for all $j \neq i$. Ties are broken in favour of the action of lower index.

This is the standard expected-utility rule (von Neumann and Morgenstern 1944) specialised to costs. It is adopted as an axiom, not derived.

\subsection{Migration-threshold decision problem (used in C.4)}

A defender at time $t$ chooses between $\{\text{migrate},\,\text{not migrate}\}$. Quantities at time $t$:

\begin{itemize}
\item $\lambda(t) \in [0, 1]$: probability that a CRQC-based attack capable of compromising the defender's keys is executed in the period beginning at $t$, conditional on no prior migration.
\item $L(t) \geq 0$: cost incurred by the defender if attacked and compromised at $t$.
\item $M(t) \geq 0$: cost of migration at $t$.
\end{itemize}

Migration removes future exposure (assumption of perfect substitution).

\textbf{Cost of ``migrate at $t$'':} certain immediate amount $M(t)$. Therefore $\mathbb{E}[c(\text{migrate}\mid t)] = M(t)$.

\textbf{Cost of ``not migrate at $t$'' (one period):} probability $\lambda(t)$ of compromise with cost $L(t)$; probability $1 - \lambda(t)$ of no compromise with zero cost. Therefore $\mathbb{E}[c(\text{not migrate}\mid t)] = \lambda(t)\,L(t)$.

\subsection{Social-welfare decision problem (used in C.5)}

Distinguish private loss $L_p(t) \geq 0$ from social loss $L_s(t) \geq L_p(t)$, the latter capturing systemic, interoperability, and third-party harms not internalised by the private decision. Define

\begin{equation}
\Lambda(t) \;\equiv\; \lambda(t)\,L_p(t), \qquad \Sigma(t) \;\equiv\; \lambda(t)\,L_s(t),
\end{equation}

with $\Sigma(t) \geq \Lambda(t) \geq 0$, equality iff $L_s = L_p$. Let $\Phi(t) \geq 0$ denote the social cost of intervention.

\newpage

%=========================================================
\section{Notation reference}\label{app:notation}
%=========================================================

\input{appendix_B}

\newpage

%=========================================================
\section{Full proofs of Propositions 1--5}\label{app:proofs}
%=========================================================

\input{appendix_C}

\newpage

%=========================================================
\section{Source reliability rubric and per-source classification}\label{app:sources}
%=========================================================

\input{appendix_D}

\newpage

%=========================================================
\section{Cost-component inclusion audit}\label{app:components}
%=========================================================

\input{appendix_H}

\newpage

%=========================================================
\section{Sensitivity tables and robustness sweeps}\label{app:sensitivity}
%=========================================================

\input{appendix_I}

\newpage

%=========================================================
\section{Monte Carlo robustness study}\label{app:montecarlo}
%=========================================================

\input{appendix_J}

\newpage

%=========================================================
\section{Sociology-of-expectations theoretical frame}\label{app:expectations}
%=========================================================

\input{appendix_K}

\newpage

%=========================================================
\section{Risk management, insurance, and disclosure}\label{app:risk}
%=========================================================

\input{appendix_L}

\newpage

%=========================================================
\section*{References}
%=========================================================

\noindent von Neumann, J. and Morgenstern, O. (1944). \emph{Theory of Games and Economic Behavior}. Princeton University Press.

\bigskip
\noindent References to the cryptanalytic, technological-expectations, and innovation-policy literature cited in this appendix are listed in the main manuscript reference list. Where a specific work is cited in this appendix, the citation matches the form used in the main manuscript.

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