Again, I am genuinely curious to see your corrections to my graph, Econ.

How do you get "crushing columns" other than by "alignment of the columns and a sequence of buckling failures"?

What I was trying to say is that the reference to columns in the paper just follows from Schneider's use of Bazant's language as a point of departure. He ends up just working with mass density and resistance force, irrespective of the structure and mechanism of the towers.

The assumption of "homogeneous" is false and it is the fundamental error we are disagreeing with.

I don't understand how you can describe the intact WTC towers as anything other than homogeneous structures. The load is more or less equally distributed across 110 floors and capacity increases as we get lower on the building in a regular way. The buildings are also symmetrical.

Modelling them in terms of smooth functions of mass density, resistance force, and height is perfectly reasonable.

Neither Schneider's nor your smoothed graphs reflect the actual discontinuities of the real event.

I think we mean different things by discontinuity here. My assumption (which I think has to be true) is that there were no additional inputs of destructive energy on the way down. (This is also Schneider's "for the sake of argument" assumption.) What that means is that you must be able to predict the collapse curves (roofline and front) from initial assumptions you make about the mass density and resistance force functions.

The real event involved four stages - of which "ROOSD" is a subset of the "progression" stage and the ROOSD explanation pre-dated the clear distinction of the two sub-stages of progression.

The stages have no explanatory, only descriptive, power. Since no new causes are introduced, the explanation for the transition from one stage to the next is simply the state of the structure at the (essentially arbitrary) end of one stage and begining of the next.

If you think describing the event in stages constitutes explaining it then we are not at all on the same page.

Neither Schneider's nor your smoothed graphs reflect the actual discontinuities of the real event.

And yet a blue and red line

**must be possible** even in a ROOSD regime. Just draw them and specify the function that produces them from a set of initial conditions you prefer.

Since the whole buildings ultimately ended up on the ground, the lines must be "smooth" (i.e., there's no place where the crushing front "skips" ahead fifty meters, though it might steepen to freefall along the way, I guess.)

I would argue that someone who can't draw those curves at all, simply doesn't understand the collapses. And, yes, the deficiencies of my curves reveal deficiencies in my understanding.

Again, I'd love to see your corrections. Just draw your curves right on top of mine.