No, I didn't fail to understand the question, you failed to read my posts, or were incapable of comprehending basic sentence structure. Both of my posts mentioned--and accounted for--sidewall load. You claimed they did not account for the fact that there was another factor besides tire pressure. You're right, we can't have a rational discussion if you can't read and comprehend posts.
The fact is, we can safely ignore sidewall load--unless we're running run flat tires--as the load our sidewalls carry is negligible as a percentage of the total. Run flat tires and "E" rated tires have sufficient sidewall strength to invalidate our basic contact patch equation enough to matter.
There's no "magic" happening in sand that increases the contact patch. Try really hard to read and understand this: The tire is flexible. It has a given tire pressure. If the tire pressure exceeds the load one square inch of sand can hold, the tire will flex out until the tire runs into resistance stretching out. In other words it becomes round and is held in place by the sidewall. At this point, it is no longer carrying a load. The portion of the tire carrying the load is the portion of the tire that is flat against sand. That portion will adjust until the pressure from below is equal and opposite to the pressure in the tire. If the two forces are not equal and opposite, the tire will accelerate in the direction of applied force. (for instance when your car is bouncing.) Since your car isn't floating above the sand, you know with absolute certainty that the pressure applied by the sand is exactly equal and opposite to the pressure in the tire. Hence, contact patch = weight/(air pressure - sidewall load)--assuming the tire pressure isn't so low that you're running on the rims.
I'm not the one being irrational. You're just making up crap and upset that physics says you're wrong.