I may have misunderstood, but you seem to assume that the "true" value of an F-coin is the energy that was used to make the bitcoinium the F-coin is made of; so the overvaluation of the F-coin over bitcoin is inherently "fraudulent".  However, the F-coins also have a value that derived from their utility as currency, and that may be arbitrarily higher than that of bitcoin, if the F-coins have some technical advantages (e.g. faster block time).

Ever since paper money was invented, it usually was first backed by precious metals, and had value equal to its backing.  But then more of it was printed without the required backing, and soon the purchasing power of the paper was below that of the original. 

However, the opposite could conceivably happen.  Just after it was first isolated, aluminum was extremely expensive.  Suppose that, at that time, the Republic of Paflagonia issued aluminum 1 pataca coins.  Af first, people would assign to them a value equal to the metal's value.  Suppose that, ten years later, aluminum was only 1/10 as costly, but there had been little further issuing (or counterfeiting) of patacas.  Then the coin would still be accepted for almost the same value it had originally (just as paper dollars are accepted today.)

This may be an alternative view of your F-coin scenario, with the difference that the overvaluing of the pataca would not have been fraudulent, but a decision of the market.   The same applies to my GNC scenario, or to sidechain coins (as far as I understand them): an increase in value of the secondary coin relative to the primary coin need not be fraudulent, just a plain free market decision.

In fact, the same applies to BTC itself.  Today the market has decided that 1 BTC is worth 312 USD, even though most BTCs in existence cost less than 10$ to make, and even today the marginal cost (minus miner profits) may be 250 $ or less.  And there is nothing hard to prevent the price of a bitcoin from falling well below that cost.