Ill posed is not one word – what separates them is not uniqueness but the rate of decay.
The corpus has treated four inverse problems: GLM scattering (Paper 89), Calderon and EIT (Paper 99), Connes’s noncommutative geometry (Paper 102) and the Wiener filter (Paper 106). This paper adds the Radon transform as a fifth and identifies what separates them. NO NEW MATHEMATICAL THEOREM IS CLAIMED. The Radon transform and its inversion formula, filtered back-projection, the fact that Radon inversion is ill posed of order one half, and the exponential ill-posedness of the backward heat equation are all standard.
First, the Radon operator was built here and checked against known values. A forty-eight by forty-eight image, forty-eight directions and sixty-seven detector samples give a matrix of three thousand two hundred and sixteen by two thousand three hundred and four. For a disc of radius one half the chord at offset s is twice the square root of R squared minus s squared. At the centre the operator gives 0.99782 against 1.00000, at an offset of minus 0.1714 it gives 0.93594 against 0.93939, and at minus 0.3428 it gives 0.72358 against 0.72790. The worst difference is 0.0360 and occurs only at the rim of the disc, where the pixel boundary is not smooth.
Second, the filter is not cosmetic: it is the inversion. Projecting a phantom of three discs and inverting it two ways, back-projection alone gives a relative error of 0.6545, while passing the data through a ramp filter proportional to frequency before back-projecting gives 0.1167. Back-projection alone gives a blur. The filter must not be called pre-processing – it is the body of the formula. The remaining 0.1167 is a discretisation floor, from the grid and the finite number of directions, and is kept out of the discussion of noise.
Third, and this is the core. The separator is how the singular values decay. Normalised by the largest, the singular values of the Radon matrix are 0.436 at k of ten, 0.283 at fifty, 0.229 at one hundred, 0.161 at three hundred, 0.125 at six hundred and 0.106 at nine hundred. A straight line fitted in logarithms over k from five to one thousand three hundred and eighty-two gives a power law, k to the minus 0.372 (computed here). This is not claimed as a theorem: the continuum statement is that Radon inversion is ill posed of order one half, and the value here is measured on a discretised operator. All that is used is that the decay is a power law.
For contrast, the exponentially ill-posed class – the backward heat equation, where Calderon and EIT live – has singular values falling as the exponential of minus k squared times tau. At k of one the two read 0.9805 against 1.000, at five 0.6105 against 0.5222, at ten 0.1389 against 0.4364, at twenty 0.0003723 against 0.3579, and at thirty 1.926 times ten to the minus eight against 0.3250 – seven orders of magnitude apart. Both are called ill posed. But with a power law, giving up one step of fine structure buys back a fixed proportion of accuracy; with an exponential it buys back no orders at all. One phrase, two different situations.
Fourth, what that costs was measured. Adding relative noise to the sinogram, inverting by filtered back-projection and separating the discretisation floor of 0.1167 in quadrature, the noise part is 0.0017 at a noise level of 0.001 (a ratio of 1.67), 0.0076 at 0.003 (2.53), 0.0238 at 0.01 (2.38), 0.0670 at 0.03 (2.23) and 0.2228 at 0.1 (2.23). The amplification settles to a bounded constant near 2.2: ten times the noise gives roughly ten times the error and no worse. For an exponentially ill-posed problem no such constant exists, the amplification growing without bound as the resolution rises. That is the mathematical reason why, from one and the same stance – look inside a body from outside – CT became an everyday instrument and EIT did not.
This paper does not write that EIT is useless. Without a demand for resolution it serves perfectly well: the rate decides how finely one may look, not whether one may look at all.
The census. GLM scattering recovers a potential from scattering data; Calderon and EIT recover an interior conductivity from boundary voltages and currents; Connes’s noncommutative geometry recovers a geometry from a spectrum; Wiener recovers a signal buried in noise; and Radon recovers the function itself from its integrals along every line. The five share one stance – read a hidden cause from data at the edge – and their mathematics is five different things. What separates them is neither the stance nor uniqueness but the rate at which the singular values decay. Uniqueness cannot be the separator: it is proved for Calderon as well as for Radon. That being unique and being recoverable are different questions is the lesson of this whole family.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none) / the measured exponent as a theorem (it is not identified with the continuum order of one half). What is added: placing Radon as the fifth member of the corpus’s inverse family, building the operator independently and checking it against known chord lengths, putting the power law and the exponential in one table as a difference of seven orders, and measuring the noise amplification with the discretisation floor separated in quadrature. What remains: the solution is unique and the solution can be found are different sentences. All that stands between them is one sequence, how fast the singular values approach zero. And the gap between a power law and an exponential is not a difference of degree – it is the difference between seeing and not seeing.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). “Ill Posed” Is Not One Word [Paper 159]. Zenodo.
DOI: 10.5281/zenodo.22092229
ORCID: 0009-0006-8587-3450