"The reason is this: there aren't enough atoms of fixed carbon in the biosphere to replace fossil fixed carbon at the rate we are currently burning them. By a factor of several hundred, near as I can gather from articles occasionally linked from TOD."

I would really like to see this calculation. It would be an interesting exercise to see what kind of turnover is nessisary in biomass growth to sustain current consumption.

So would I - that's the about as baseline as it gets. We'll find out real fast whether we have a physics problem or an engineering problem.

I was thinking of a remark posted some time in the last six or eight months stating that we are blowing through 400 years' worth of fossil hydrocarbon every year. Perhaps that source was talking about the natural sequestration rate in peat bogs and such.

I may have overstated the ratio, as the Wikipedia says the biosphere has "about 1900 gigatonnes of carbon". So if we throw every last polar bear, housecat, and blade of grass into the TDP retort, we may juuust be able to make our 85mbpd of artificial fuel for awhile... :)

The problem will solve itself.
But not in a nice way.

Fixing carbon from CO2 in the atmosphere is certainly possible using solar energy. The problem is not carbon, the problem is converting sunlight to energy in an efficient manner. No one doubts that there is enough energy hitting the earth.
I doubt that the economy can survive waiting until 2030 or 2040 for a new fleet of electric cars, so we need to solve the biofuel problem somehow to power our soon-to-be-downsized-increasingly-vanpooled fleet.

The problem is not carbon, the problem is converting sunlight to energy in an efficient manner.

No, the problem is doing it (conversion to useful energy) in a cost effective manner. Efficiency per se would only otherwise be important if sunlight (and land to collect it on) were scarce. As it stands, efficiency is important only to the extent it affects the capital cost of the system per unit of capacity.