The 2026 World Cup in North America has now delivered the counterargument, and it arrives with a mathematical formula attached. When the video assistant referee disallowed Shoja Khalilzadeh’s late winner for Iran against Egypt, the on-field officials had ruled the goal onside. VAR’s three-dimensional reconstruction placed the tip of a foot beyond an invisible line. Under the tournament’s protocol, the machine’s numerical reading replaced the human’s binary judgment, and a nation’s qualification probability, which The Athletic had estimated at 92 percent after the draw with Egypt, collapsed to zero as none of the three subsequent results went Iran’s way. A referee’s authority, once final, now yields to a measurement — and the measurement can be wrong in ways nobody has bothered to quantify.
A Formula for When the Machine Should Stay Silent
The question the 2026 controversies raise is not whether VAR is accurate but whether accuracy alone justifies overruling a person. A paper posted to arXiv by Amir Ali Ahmadi of Princeton University and two high school students, Parsa and Radeen Alavi of Algonquin Regional High School, answers it with an explicit maximum-likelihood rule. [1] The referee perceives it through Gaussian noise with standard deviation sigma_H; the AI measures it through independent Gaussian noise with standard deviation sigma_AI. Only the sign of the referee’s internal perception is observed — the binary call — while the AI’s numerical value is observed in full. Maximizing the joint likelihood over the unknown margin yields a reversal threshold of the square root of two over pi, multiplied by sigma_AI squared and divided by sigma_H, when both error means are zero. An onside call is overturned precisely when VAR’s measured margin exceeds this threshold, not merely because VAR places the attacker beyond the line. The distinction matters: a measurement that crosses the geometric boundary by a hair does not, under the rule, carry the evidentiary weight to displace a human decision.
The structure of the formula encodes an asymmetry that is easy to miss. The AI’s error enters squared, while the human’s enters to the first power. Differentiating the Gaussian log-density for the numerical measurement introduces a factor of one over sigma_AI squared; differentiating the log tail probability for the binary call introduces a factor of one over sigma_H. A noisier machine raises the bar for overturning a person, but the bar rises steeply, because the machine’s uncertainty is penalized twice over. A noisier referee lowers the bar, because a less reliable human call deserves less deference. The threshold comes out small in absolute terms — roughly 0.32 centimeters in the paper’s zero-mean example — yet the point is not its size but its existence: there is a zone of measured offside beyond the line within which the correct decision is to let the human call stand.
The Bias Correction Nobody Has Implemented
Systematic bias shifts the picture further, and here the paper’s findings cut against the assumption that machines are neutral arbiters. If the referee tends toward offside calls, the crossover point at which an onside and an offside call become equally likely moves away from the true zero margin. The same logic applies to a biased machine: a VAR system that systematically overestimates attackers’ positions must clear a higher measured margin before its reading should count. None of this requires computing the maximum-likelihood estimate numerically. The machinery is heavy; the output is a line on a chart.
What the rule produces, in effect, is a formal account of when a human judgment survives contact with a superior instrument. At a true margin of zero, all three err with probability one half. Away from zero, the maximum-likelihood rule dominates. The human and the AI, when their error standard deviations are equal, err at identical rates — a symmetry that dissolves once the parameters diverge. With a human standard deviation of 10 centimeters, an AI standard deviation of 2 centimeters, and a human bias of 5 centimeters toward offside, the thresholds for overturning become approximately 0.2037 centimeters after an offside call and 0.4564 centimeters after an onside call. A value of z + c_ML equal to negative 0.1808 centimeters falls below both, and the human call stands. The arithmetic is unglamorous, which is precisely the point: the decision to override a person is treated as an estimation problem with a stated answer, not as a contest of institutional authority.
The Frame Rate Problem
Video recorded at 30 frames per second, the authors note, leaves an athlete moving at 5 meters per second traveling roughly 17 centimeters between consecutive frames. [1] Even small uncertainty in the relevant contact time — when exactly the ball left a teammate’s foot, when exactly a hair brushed a sensor — translates into consequential uncertainty in the measured position. The Portugal-Croatia incident illustrates the point. In the 90+13th minute, Josko Gvardiol equalized to force extra time, but the goal was disallowed after Mario Pasalic was deemed offside during the build-up. The alleged touch from Igor Matanovic was nearly imperceptible to television viewers; the sensor inside the Trionda ball registered a signal that VAR interpreted as contact with Matanovic’s hair, and the Croatian federation filed a complaint with FIFA challenging the overturn. The sensor trace, displayed during the review, is a number. The question the paper forces is whether that number’s precision was ever matched by its accuracy, and whether anyone had estimated the latter before using the former to erase a goal.
The same structural problem surfaced in tennis, where electronic line calling has its own version of the conflict. At the 2025 Madrid Open, Alexander Zverev disputed a call in the second set against Alejandro Davidovich Fokina, pointing to a ball mark on the clay that he argued showed the technology had erred by four to five centimeters. Umpire Mohamed Lahyani declined to inspect the mark because, under the electronic line-calling rules, the electronic call took precedence. Zverev retrieved his phone and photographed the mark, later posting it on social media. The episode is a clean illustration of what the paper calls the information structure shared across domains: a human binary judgment and an AI numerical measurement concerning the same underlying quantity, with a protocol that decides in advance which one wins. The protocol in tennis grants the machine unconditional precedence. The protocol in soccer grants it precedence above an unstated threshold.
Three Domains Where the Same Arithmetic Applies
In medical diagnosis, a physician classifies a lesion as benign or malignant while an AI system assigns a numerical malignancy score. In aircraft inspection, a maintenance engineer classifies a component as safe or requiring maintenance while an AI system produces a defect-severity score. In finance, a hedge fund manager is bullish or bearish on a stock while an AI model estimates expected return. Each case shares the structure the paper formalizes: a human binary call, an AI numerical measurement, independent Gaussian errors, and a decision about which source prevails when they disagree. The finance case has the useful property that the true margin becomes observable after the fact — the realized excess return’s sign determines the correct binary decision, and its magnitude measures the extent of outperformance or underperformance — which makes calibration data abundant. The medical and aviation cases lack that convenience, and the paper acknowledges that reference margins may require controlled plays, reconstructed incidents, or a higher-fidelity and more costly reference technology that need not be deployed in routine operation.
Calibration itself is treated as a maximum-likelihood problem. For the AI, with errors assumed independent and identically distributed Gaussian, the estimates are the sample mean and the sample standard deviation of the observed errors. For the human, the internal measurements are unobserved, so the likelihood is a product of Gaussian cumulative distribution functions evaluated at the reported call. A reparametrization — alpha equal to the bias over the standard deviation, beta equal to the reciprocal of the standard deviation — turns the problem into a probit regression, and a sufficient condition for a maximizer to exist is that the two types of calls overlap in reference margin while offside calls occur at larger margins on average than onside calls. The requirement of distinct reference margins is essential: if every calibration case sits at the same true margin, the two human error parameters cannot be identified separately, because the likelihood depends on them only through a single combined quantity.
What the Threshold Does to the Person
The deepest consequence of the paper is not technical but institutional, and it concerns what happens to the human role once a threshold exists. Under the current protocol, the referee’s call is provisional and the measurement is decisive above whatever margin the reconstruction displays. Under the paper’s rule, the referee’s call is a data point with a stated reliability, and the measurement is a second data point with its own stated reliability, and the decision between them follows from the ratio. The referee is no longer the authority whose judgment the machine corrects; the referee is a sensor whose output enters a likelihood. That reframing is what makes the human superfluous in the specific sense the 2026 controversies have made vivid: not because the machine is better, but because the protocol has stopped asking what the human contributes and started asking only whether the machine’s number crosses a line.
The authors thank their grandmother, Maryam Dara, for her support, disclose the use of ChatGPT for generating figures, additional proofreading, and sanity checks, and note that they followed their sqrt(2/pi) rule to ensure that AI did not overturn their human judgment. [1] The joke is precise: the rule that governs referees governs authors, and it works by specifying in advance how much evidence the machine needs before it gets to overrule the person. The alternative — the one the 2026 World Cup has been running — is a protocol in which the machine’s reading wins whenever it disagrees, and the human’s role is reduced to supplying the initial call that the machine then corrects or confirms. The paper argues that the question of when a human should be overruled has an answer, and that the answer depends on quantities — the two error standard deviations and the two biases — that, according to the paper, no major competition has yet estimated.
The Calibration That Has Not Happened
The next step follows directly from the paper’s calibration section rather than from any recommendation the authors make. The rule requires four parameters: the human error mean and standard deviation, and the AI error mean and standard deviation. For soccer, those parameters could be estimated from controlled plays or reconstructed incidents in which player positions and contact times are known, with the human calibration sample recording the referee’s binary call at each known margin and the AI sample recording the corresponding measurement. The two samples need not share observations or size. What the 2026 controversies reveal is that the parameters have not been estimated, and that the protocols in place substitute a default — the machine wins — for the calibration the rule demands. The paper’s authors frame their work as educational, aimed at undergraduates and advanced high school students who know basic probability and calculus. The framing is accurate and also a quiet indictment: the mathematics needed to decide when a machine should overrule a referee is within reach of a motivated teenager, and the institutions running the tournaments have not done it.