Estimate research notes - just first pass quick research - nothing double checked, since estimate came out in somewhat reasonable range:

See http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/spht.html for specific heat math.

Specific heat of paper is 1.17 to 1.34 kJ/(kg * K). Cotton, flax, and hemp are in the high end of that same range. Used the high end in the estimate, but I don't expect cardboard paper to really be at the high end. Source: Handbook of physical testing of paper, Volume 2; By Jens Borch, Richard E. Mark, M. Bruce Lyne; page 393; Table 1; as found on Google Books

Burning spot diameter with best lens right now is estimated as 0.5mm, so spot area is 0.19mm^2

Mass of paper under spot for first face of corrugated cardboard. inverse "Planar density" of C flute corrugated cardboard, from memory of measurements long ago = 8sqft/1lb -> 743 224.32 square millimeters/1lb -> 743 224.32 square millimeters/0.45359237 kilograms. So "planar density" = 6.1 x 10-7 kg/square millimeter.

But what fraction of that square millimeter's mass is from the top face? Faces are 0.12mm thick, flute paper is 0.07mm thick. first face mass fraction is 0.12/(0.12+0.12+0.07)=0.38, so 0.38 * 6.1x10-7 kg/sqmm = 2.36x10-7 kg/sqmm is first face "planar density".

Mass to heat under burning spot is then 2.36x10-7 kg/sqmm * 0.19sqmm = 4.5x10-8 kg.

Temperature of ignition for paper: Fahrenheit 451 = 232.78 deg C (used for delta Kelvins, so Celcius is fine). Ambient temp - say 75 deg F = 23.89 deg C (which will be on the low end for mid day, sunny day temps around here)

Aperture area in meters: 1.25 inch diameter = 0.03175 meters, so area= 7.9173x10-4 square meters.

Typical incident radiation around orange county, around the middle of the day, on sunny days: well, generic insolation is cited as approximately 1,000 watts per square meter for a surface perpendicular to the Sun's rays at sea level on a clear day (from http://en.wikipedia.org/wiki/Insolation). I downloaded a government database of historical, hourly insolation measurments at various locations, but haven't looked at that yet.

Burning lens spot size guess: about 0.5mm

Re: attenuation due to two glass mirrors and the lens glass

"Uncoated glass, for example, reflects about 4% of the light impacting it; in the case of an uncoated lens 4% of the light is lost at entrance to and at exit from the lens, for a total light loss of about 8%." from http://www.meade.com/catalog/uhtc/index.html

So two plain glass mirrors are cutting the light power by at least 16%? (probably more because of the 45 deg angles, and the scattering and less than 100% reflectivity of reflecting surface) 16% because the light must pass the surface of each mirror twice.

Wait - that's all wrong. First surface reflection on the mirrors is not a loss in this coelostat arrangement. The relavent mirror losses will be due to dispersion and absorption. Or, what? Go back and figure that out. Until then, keep the 16% loss figure to be conservative.

How about the coated lens? A plot on Thor Labs catalog page 699 (pdf: http://www.thorlabs.com/catalogPages/711.pdf) suggests that lens coatings can get each surface reflection down to about 0.25%. Lets say, then, that my coated optics (not from Thor) are reflecting a total of 1% from the two surfaces.

Optics attenuation is then rough estimated to be 17%, so "attenuation factor" is 0.83

Some additional unaccounted for factors:

negative factors

attenuation factor due to glass not passing all available wavelengths
what percent of available radiation is in wavelengths that pass through glass? simple scalar factor

water
Normal water in paper will slow heating. (Easy experiment to measure cardboard's normal water content.) Specific heat would then be weighted average of water and paper, weighted by percent of total mass

heat dispersion
Heat that disperses through the paper beyond the burning spot lowers the burning spot watt estimate. By how much? How fast does it dissipate? Classic problem - should be easy to estimate.

albedo
highest at first instant, falls as paper browns and blackens

smoke
diffusion of light

positive factors

Ignition starts exothermic reaction that may speed adjacent heating, though we want to avoid or suppress open, self-sustaning combustion