A physics engine, and a verification tool enforced on top of it. PhysWall inverts a closed, published, non-linear physical law at a single measurement point — and refuses when the inverse is not unique. Seven laws, one engine, and the same refusal in all of them.
A physical-logic engine that decomposes a measurement gap along four axes — the machine, the law, the object, and which definition — and runs every domain through one structure.
RF matching, PCB loss, orbital conversion, photonic scattering, stream gauges, ocean waves, basketball. Same engine, same refusal: when the measurement cannot carry a conclusion, it says so instead of answering.
A hot streak is almost always nothing.
Not a claim about effort or confidence — a claim about how many shots it takes before a shooting percentage means anything at all.
A streak that means nothing is one hypothesis among several, and noise is usually the one left standing.
Take a shooter whose true rate is 36%. Nothing about him varies. Here is how far his observed percentage can wander on noise alone:
attempts 95% range width
20 15% - 55% 40
yours ?
Over twenty attempts the same unchanged player can look like a 15% shooter or a 55% one. Both are ordinary. Neither is a streak.
⚠ This row read 18.9%–57.7% until an outside reviewer pointed out that neither is reachable: in twenty attempts the only possible percentages are multiples of five. Those figures are a Wilson interval, which answers “I observed 36% — what true rates are consistent?” The question here is the reverse: “the true rate is 36% — what can be observed?” That is the binomial, and it gives 3 to 11 makes: 15% to 55%. Same direction, less drama, and the right question.
That is one row. The width closes as attempts rise, and where it closes enough to carry a conclusion depends on the number you actually have.
A percentage and an attempt count. It says when the figure is usable and when it is mostly sampling noise.
Run those two extremes back through the geometry and they imply release scatter of 1.12° and 4.49° — a factor of 4.0 in hand steadiness. Which did not happen, because nothing about him changed.
That is the shape of the error: a difference in the noise gets read as a difference in the player.
Three-point rate settles somewhere around 750 attempts — Blackport, 2014. At 750 the range is still seven points wide.
Seven players in NBA history have ever taken 750 threes in a season.
So for nearly every player, in nearly every season, the honest answer to "did he improve" is that you cannot tell. Not "probably not" — cannot tell.
Everything else on this site derives a quantity nothing measures, and that is the point — but it also means nothing outside can contradict it. Basketball is the exception.
Two datasets exist that were produced without knowing this tool exists: 90,970 three-point attempts counted by the league's official statistician, and 49,562 free throws captured by optical tracking at 60 frames per second.
This tool treats release speed as the thing that varies, and holds release angle fixed — on the grounds that the angle sits at a stationary point where small errors barely matter.
That was a modelling choice, made from geometry. A physics group at UT Austin measured it independently from 21,964 tracked shots and wrote:
"launch angle enters through the cross term sinθ·cosθ.
Around 45°, the slope of this cross term is locally flat,
minimising the impact of small deviations in θ."
consistency in launch velocity r = 0.73
consistency in launch angle r = 0.35
— McGrath et al., arXiv 2512.08824
Same claim, arrived at separately. And speed matters roughly twice as much as angle, which is what holding the angle fixed assumes.
The release velocity needed to reach the centre of the hoop, computed here from geometry alone with nothing fitted, comes out at 14.94 mph. The tracking system measured 14.74.
The scatter figures do not line up, and the size of the gap is worth stating.
league free-throw average ~78%
pure-swish window from geometry ±1.20%
→ so the model needs σ = 0.49% (0.07 mph)
and tracking measured σ = 1.6-1.9% (0.24-0.28 mph)
a factor of 3.3 to 3.9
The reason is visible in the same paper: a shot that touches the rim frequently still drops. The real tolerance window is roughly three and a half times wider than a clean swish, and this model folds that forgiveness into σ rather than into the window.
So the σ here is not a measured release scatter. It is a swish-equivalent figure that absorbs how much the rim forgives. It ranks shooters correctly — and it is not the number a tracking system would hand you.
That distinction was not visible until somebody measured the real thing. It is written here rather than quietly corrected.
Most calculators return a number whatever you feed them. This one reports the range, and refuses outright when the sample cannot carry a conclusion. A confident answer from twenty attempts is worse than a straight no verdict, which at least tells you the sample is the problem.
Miller and Sanjurjo, whose 2018 correction this page rests on, show in their Appendix A.2 that the bias is negligible when an observer sees many sequences.
one shooter, 20 attempts the bias is here, and large a whole league, thousands negligible
⚠ So this page is about a short run by one player, and it does not carry over to a season-long or league-wide analysis. An analyst pooling thousands of sequences is not exposed to the effect this page describes, and saying otherwise would claim more than the paper supports.
Miller, J.B. & Sanjurjo, A. (2018), "Surprised by the Hot Hand Fallacy? A Truth in the Law of Small Numbers", Econometrica 86(6), 2019–2047. Freely available as arXiv:1902.01265. The earlier SSRN draft carries a different title and is not the published text.
PhysWall was developed and architected by Gadi Zion.
Built on PhysWall — the same engine reads antenna bandwidth, conductor loss, bit erasure and heat limits. It answers what the measurement implies, and refuses when the measurement cannot say.⚠ Check this instead of believing it. Every number here reproduces from a source that is named, and the claims that turned out wrong are still printed next to what replaced them. The same engine runs all of these — it asks how much a measurement allows you to conclude, and refuses the same way in every field. The same engine runs all of these — it asks how much a measurement allows you to conclude, and refuses the same way in every field. How to check each one →