Somewhere between one and one hundred billion. That is the range of a number called the condition number in the calibration software behind REACH, a radio experiment built to detect the faint hydrogen signal from the universe's first stars. A condition number measures how much a small wobble in the input to a matrix calculation gets amplified in the output. Values around 10 are unremarkable. Values of 10 to the ninth through 10 to the eleventh mean the calculation is so delicately balanced that ordinary rounding differences between two computers can change the answer.
Saswata Dasgupta and colleagues at Cambridge report exactly that in a preprint posted to arXiv on 29 July 2026. Their finding is not about the sky. It is about the machinery used to read the sky, and about a failure mode that would be easy to miss if nobody went looking.
The science at stake is one of the harder measurements in cosmology. Neutral hydrogen gas emits and absorbs radio waves at a wavelength of 21 centimetres. As the universe expanded, that wavelength stretched, so hydrogen from the Cosmic Dawn (when the first stars lit up) and the Epoch of Reionization (when their light stripped electrons from the surrounding gas) now arrives as a broad, shallow dip in the radio spectrum. The problem is that our own galaxy blazes in the same radio band, thousands of times brighter than the signal. To find the dip, an experiment has to understand its own receiver to a precision far below the brightness of the foreground it is looking through.
The two knobs that look like one
REACH does that understanding with what radio engineers call the noise wave formalism: a model of the receiver's low-noise amplifier described by five parameters, each allowed to vary with frequency, and all fitted together in a Bayesian analysis against several reference sources of known temperature. Fit them well and you can subtract the instrument's own contribution from the data.
The trouble the team identified sits in the fitting itself. They used singular value decomposition, a standard way of asking which combinations of parameters the data actually constrains, and found two of them pulling in nearly the same direction. One tracks the temperature of an excess noise source used as a reference. The other tracks the temperature of a load, a component of known thermal properties also used as a reference. In the mathematics of the fit, the columns describing these two are nearly collinear, meaning the data can barely tell them apart. Trade a bit of one for a bit of the other and the model fits about as well. That near-degeneracy is what inflates the condition number, and it is what makes solutions non-reproducible across computing environments.
The authors' fix comes in two steps, both built on a Chebyshev basis, a way of writing frequency-dependent functions that behaves better numerically than raw powers of frequency. First, they pin the noise source temperature to a single number instead of letting it vary with frequency. That collapses the ambiguity, and the condition number drops to roughly 60, which is to say the calculation becomes ordinary. The cost is that a real noise source is not perfectly flat across frequency, and the simplified model gives that up.
So the second step puts the frequency dependence back, but from a different direction. Rather than fitting it, they measure it: the noise source temperature as a function of frequency is recovered directly from the hot-load calibration measurement the experiment already takes. Tested on mock data, this approach keeps the numerical steadiness of the simplified model while reaching what the authors describe as comparable calibration accuracy. They also found that masking out narrow slices of frequency, where signals reflecting back and forth along the cables (standing waves) create their own local degeneracies, cleans up further artefacts.
Why it matters
The honest headline here is about trust rather than discovery. A detection of the global 21-cm signal would be a genuinely important measurement, and claims in this field have been contested before. If a calibration pipeline can give materially different answers depending on which machine ran it, then reproducibility, the plainest test any result has to pass, is not available. The authors are not reporting that REACH's conclusions are wrong. They are reporting that a particular step was fragile enough to deserve fixing before it mattered.
The wider point is that the offending degeneracy is not a REACH quirk. Because the near-collinearity between those two parameters is built into the noise wave formalism itself, the authors argue their approach is relevant to other global 21-cm experiments using the same modelling framework. Several do.
Two caveats belong with this. The paper is a preprint, not yet peer reviewed, and the authors note it is to be submitted to MNRAS. And the demonstration runs on mock data, simulated observations rather than real sky measurements, which is the appropriate place to test a calibration method but not the same as showing it works on a season of observing. What the work offers is a calibration procedure that is stable, reproducible, and driven by measurements the experiment already makes. In a field where the quarry is buried thousands of times deeper than the noise above it, that is worth having.