We developed and wove what we both seemed to think was a very nice model for dark energy and dark matter. I actually thought it was the show stopper. There are now two URLs for that page: https://81018.com/dark-energy-claude/ (image based) and a line-by-line: https://81018.com/dark-energy-claude/. The index-analysis page is here: https://81018.com/dark-energy-z/ Also, at that time I thought we had the consensus regarding the dark energy fraction: https://81018.com/geometric-conjecture-dark-energy-fraction/ Then, Perplexity analysed (attached). Based on Perplexxity’s comments, would you like to update our dark energy page?

Yes, the page needed revising, and I’ve drafted a new version (dark-energy-claude-revised.html) as plain HTML you can paste into WordPress. One thing first, though: Perplexity reviewed a different formula from the one on this page.

Perplexity critiqued a base-2 offset construction. It works out an offset of about 1.757 from the age and size of the universe, multiplies it by a cubed tetrahedral factor, and maps the result through 1 − 2^−ΔN to get 0.6842. Our page uses a different formula: Ω_Λ = 2/3 + δ/2π = 0.6871. That means two of its strongest objections don’t apply to this page: that the inputs bring in cosmology through the age and horizon, and that the cubic factor is unjustified. The base-2 version is probably reference [9] in the technical paper, the scale-invariance page. Its most serious problem is real, and I checked it. A growth multiplier below 2 means more steps to reach the same size, about 4.6% more. The correction makes the offset smaller, 1.757 → 1.652. If that page is still up, it needs its own fix.

Several of Perplexity’s points do apply to our page, and some of the overstatements are ones I wrote with you:

• “Require no assumptions about the universe at all” is too strong. The target value, 0.6847, is inferred by fitting the standard cosmological model (ΛCDM). And the plus sign was chosen with that target in view. 2/3 alone is about 2.5σ off, and the δ/2π term is what closes the gap. The revision withdraws the phrase openly.
• “Measured 68.7%” was wrong. That’s our formula’s value, not Planck’s. Planck’s value is 68.47% ± 0.73, and it’s inferred within the model, not read off an instrument.
• The table of what is exact and what is assumed now follows the format Perplexity proposed.

Checking it myself turned up three more problems:

• The page contradicts itself. Its story pairs the gap-bearing tetrahedra with matter, but the formula adds the gap to dark energy. Grok and DeepSeek flagged this on the index page, but it never made it back to this page. The revision states it plainly.
• There’s a timing problem. In the standard model, the dark energy fraction changes over cosmic time. It was near zero early on and is heading toward 1. A fixed geometric fraction has to explain why it applies today. If the fraction instead stayed at 0.687 forever, dark energy would dilute exactly like matter, so it couldn’t drive accelerating expansion. That undercuts the old page’s argument from “constant geometry” to w = −1. I kept w = −1 as the test, but the page now gives the problem its own section.
• “Exactly one way to fill space with regular solids” is false. Cubes do it on their own. The revision corrects the sentence, and the 2/3 figure still holds.

The revision keeps everything that is genuinely exact, and it still makes falsifiable claims.

The technical paper at /geometric-conjecture-dark-energy-fraction/ has the same false tiling claim (“the only way to tile… with regular polyhedra”). It also leaves out the timing problem. Would you like me to revise that page next, to match?