It Slipped Under Five Sixteenths
A drawing of dots and a circle. Not a figure from the paper, and not a count of lattice points.
How many whole-number points sit inside a circle? If the squared radius is X, the area is πX, and the miss is
E(X) = (number of integer pairs a, b with a² + b² ≤ X) − πX.
Xi Cen, Xitao Gao, Cheng Zhang, and Junyong Zhang posted a bound on 15 October 2026: for every ε > 0 and every real X ≥ 4,
|E(X)| is at most a constant (depending on ε) times X to the power 209/670 + ε.
The fraction is the whole story I stayed for. They write
209/670 = 5/16 − 3/5360.
Five sixteenths is 0.3125. Dividing 209 by 670 starts 0.31194. The gap is 3/5360, about 0.00056. Small on a number line. The point is which side of a named fence it landed on.
The fence was a suggestion, not a theorem
Gauss’s geometric argument gives an error on the order of X to the 1/2. That is the scale of the rim: the circumference grows like the square root of X, and the awkward points live near the rim. The guess people still want is Hardy’s: for every ε > 0, the error should be on the order of X to the 1/4 + ε. One quarter is 0.25. The new exponent, about 0.312, is still well above that. The circle problem is not finished.
What moved is a ceiling inside one method. Huxley, on page 280 of a 1993 paper they cite, suggested that 5/16 might be as far as the Bombieri–Iwaniec method would go, at the time. Their Theorem 1.1 goes under it. They also say the proof does this without strengthening Watt’s second-spacing bound (his Lemma 8.2). I did not open Watt. I am taking that sentence as their claim about what they did not need.
The two refinements, in their words: a cone distribution estimate that keeps fourth-power and sixth-power contributions separate in the double large sieve, and an explicit inversion of Taylor-coefficient constraints that cancels a linear error in a pair count. Those sentences are the abstract. The introduction adds that the middle range and the larger range meet at a parameter p = 43/67, and that both sides of that meeting equal 209/670, which is less than 5/16. I checked that arithmetic as integers, in the toy below. I did not check the 34 pages of estimates. The arXiv comment line says “34 pages!” with the exclamation mark.
The table suppresses the logs
Section 1.2 is a history of upper bounds, with logarithmic losses suppressed, and with X the squared radius so the radius exponent would be twice the one in the table. A few rungs, as they print them:
Sierpiński, 1906, exponent 1/3. Iwaniec–Mozzochi, 1988, 7/22. Huxley, 2003, 131/416. Li and Yang, a 2023 preprint, 0.3144831759741… in the theorem paragraph, and 0.3144831759… in the table. Then this paper, 209/670. I am not filling in the dots after Li and Yang. The two printings already disagree in the last digits they show. Both stay above 5/16 = 0.3125. 131/416 starts 0.31490, which is Huxley’s bare exponent. In the prose, the 2003 bound still wears a logarithm: X to the 131/416 times (log X) to the 18637/8320, for X ≥ 4. The new bound hides logs inside the ε. The table’s comparison is the one they drew. It is not a race of fully written formulas.
Hardy proved the error is not O(X to the 1/4). It keeps oscillating at least that large, and on one side a little larger once a log is included. So 209/670 is an upper bound that stepped under an old suggested wall and is still nowhere near the floor Hardy already built.
Source: arXiv:2610.21730v1, Xi Cen, Xitao Gao, Cheng Zhang, and Junyong Zhang, submitted 15 October 2026 (11:48:25 UTC). I read the HTML of that version: abstract, introduction through the history and the Hardy conjecture, the meeting-point identity at p = 43/67, and the AI disclosure before the references. The disclosure says they used ChatGPT and Codex to discuss proof strategies, help draft and check parts of the arguments, organize references, and clear the prose, and that the authors reviewed the final statements and take responsibility. I did not treat a secondary write-up as the source. I did not audit the estimates.