In a competitive (free) market, Bitcoin mining consumes in cost to the miner what it creates in value to the miner, both in the issuance of new units and in the service of confirmation. This is the case whether a mined block reward reflects the miner's full return or otherwise.

The amount of computation performed in mining is probabilistically reflected in the block difficulty. This computation is referred to as work. A valid block header is probabilistic proof that this work was performed. This is the basis of the term "proof of work".

The amount of energy consumed in block production is not provable, either specifically or probabilistically. Energy efficiency varies. A block header does not reflect "proof of energy" consumed. Such claims are approximations.

A miner's return on block production is not fully reflected by the block. The mining of one's own transactions implies fees not necessarily reflected in the block, as do side fees generally. A miner may introduce transactions with arbitrarily high or low fees. The block reward does not represent a "proof of reward". Such claims are assumptions.

In a free market, the return on mining is the value of its reward, whether or not the amount is reflected in the block, and the fees earned are determined by demand to transact. This is a consequence of competition. So in this case it is correct to consider a valid block header to be "proof of cost", however the amount of the cost remains unknown. All that is knowable is that the miner earned a market rate of return on capital.

However, in the case of state monopoly, price is not controlled by competition. A monopoly may charge any price that the market will bear. The enforcement cost of monopoly is paid by the taxpayer. The price premium is another tax, paid by the consumer. The value of the tax is transferred to the monopoly.

In the case of state-sponsored Bitcoin censorship, both enforcement and price (fee) premium exist as taxes in the manner of monopoly. The fee level may exceed a market rate, and its enforcement is subsidized by taxes. Monopoly mining can produce seigniorage just as any monopoly money. The block header continues to provide a proof of work, but no longer provides a proof of market cost.

In the same manner, the existence of a valid unit of monopoly money provides sufficient proof of a real production cost, but provides no proof that the issuer did not earn a monopoly premium on this cost. There is a theory that Bitcoin’s production cost is "unforgeable", where seigniorage of state money represents "cost forgery". As has been shown, Bitcoin is also subject to seigniorage, invalidating the theory.

All goods have real production cost. Monopoly exists to raise price above cost. While Bitcoin is censorship resistant, the effectiveness of resistance is not guaranteed.