Proof-of-work (PoW) blockchains rely on computational expenditure to secure a ledger supporting a native cryptocurrency. In existing systems such as Bitcoin, this expenditure is intentionally useless: the computation secures consensus but produces no external economic output. An emerging alternative -- proof of useful work (PoUW) -- enables the same computation to simultaneously secure the blockchain and generate economically valuable output. However, PoUW is often criticized on economic grounds: if the work is useful, attackers might be "paid to attack," potentially weakening security. We develop a competitive-equilibrium model of a PoUW blockchain in which compute can be allocated across pure mining, pure useful work -- instantiated as machine-learning inference -- or "duplex" work that produces both with computational overheads. We provide a complete closed-form characterization of equilibrium allocations and prices as a function of the duplex overheads and a single economic parameter -- the token-inference ratio -- measuring token adoption relative to the inference market. This characterization reveals three regimes: "Bitconia," in which the economy reduces to classical PoW; "Fortessia," in which duplex replaces mining, increasing security while useful output remains unchanged; and "Duplexia," in which token rewards subsidize inference, lowering prices and expanding inference supply. Contrary to the common strawman argument, PoUW does not make attacks economically cheap: once equilibrium prices are taken into account, the economic cost of a majority attack remains tied to the block reward. Moreover, in Duplexia, block rewards act as rebates on inference prices, generating additional socially useful computation that would not arise without the blockchain -- an expansion monotonically increasing in token adoption and technological efficiency.

翻译：工作量证明（PoW）区块链依靠计算开销来保护支持原生加密货币的账本。在比特币等现有系统中，这种开销是有意无用的：计算用于保障共识，但不产生外部经济产出。一种新兴的替代方案——有用工作量证明（PoUW）——使相同的计算既能保障区块链安全，又能产生具有经济价值的产出。然而，PoUW 常受到经济学层面的批评：若工作量有用，攻击者可能“拿钱攻击”，从而潜在地削弱安全性。我们构建了一个 PoUW 区块链的竞争均衡模型，其中计算资源可分配至纯挖矿、纯有用工作（以机器学习推理为例），或同时产生两者的“双效”工作（附带计算开销）。我们提供了均衡分配和价格的完整闭式表征，其作为双效开销和单一经济参数——代币-推理比率（衡量代币采用程度相对于推理市场规模的指标）的函数。该表征揭示了三种状态：“比特国”，经济体系退化为经典 PoW；“堡垒国”，双效工作取代挖矿，安全性提升而有用产出保持不变；“双效国”，代币奖励补贴推理，降低价格并扩大推理供应量。与常见的稻草人论证相反，PoUW 并不会使攻击在经济上变得廉价：一旦考虑均衡价格，多数攻击的经济成本仍与区块奖励挂钩。此外，在双效国中，区块奖励充当推理价格的回扣，产生了若无区块链便不会出现的额外社会有用计算——这种扩张随代币采用程度和技术效率单调递增。