What light cannot measure is a density – one conformal rescaling separates the four clocks by weight.
Paper 111 wrote that light determines only the conformal structure, that is, nine of the ten components of a metric. Papers 118, 119, 121 and 122 separated four clocks – the coordinate dt, proper time as a functional of a curve, entropy as a functional of a measure, and the parameter of Ricci flow – as different logical types. This paper puts the two together under a single test. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Weyl (1921), Ehlers, Pirani and Schild (1972), Ricci flow (Hamilton 1982), and the coordinate-dependence of differential entropy are all standard. It is not claimed that the metric is real, nor that the conformal structure is the more fundamental thing.
First, the component light leaves behind is a density. A symmetric two-tensor in dimension d has d times d plus one, over two, components, and the representative normalised to unit determinant has one fewer; the remaining one is a density. In four dimensions ten splits into nine and one. For the Schwarzschild metric the determinant is minus r to the fourth times sine squared, the normalised representative has determinant minus one, and the density is r squared times the absolute value of the sine (computed here symbolically). Light determines the former and not the latter.
Second, a conformal rescaling leaves the light cone alone and moves proper time. A null vector stays null for every conformal factor, while the interval of a timelike vector is multiplied by the square of the factor (computed here). That the light cone is conformally invariant is exactly why light reports only the conformal structure.
Third, and this is the core. One conformal rescaling separates the four clocks by weight. The coordinate dt is a function on a chart and does not change, so its weight is zero. Proper time is a function on curves and is multiplied by the factor, so its weight is one. The parameter of Ricci flow is a function on the space of metrics and is multiplied by the square of the factor, so its weight is two. The last point was checked on the round sphere: the extinction time of the n-sphere of radius r nought is r nought squared divided by twice n minus one, and multiplying the radius by the factor multiplies the extinction time by its square (computed here). That weight of two agrees with what Paper 119 obtained by an entirely different route, namely that this clock has the dimension of a length squared. So the difference that Paper 118 recorded as a difference of logical type appears here as a single number. And what fixes the weight is what the clock is a function of, that is, the test object of Paper 148.
Fourth, the third clock splits in two under the same test. Comparing the differential entropy of a three-dimensional Gaussian before and after stretching the space by the conformal factor, the shift is exactly the dimension times the logarithm of the factor (6.907755 at a factor of ten, against a theoretical 6.907755; computed here). Yet the Shannon entropy of the same information read as a probability on a finite set is 1.207974 nats and does not move when the space is stretched. So entropy is two quantities. The entropy of a probability measure has weight zero, while the differential entropy has no weight at all and shifts additively. The reason is plain: differential entropy is not a function of the measure alone but of the ratio of the measure to a volume element, and that volume element is precisely the density of the first section. The separator is whether a standard has been carried in from outside.
Fifth, by Weyl’s theorem the factor is forced to be constant. Writing down how the connection changes under a conformal rescaling and comparing with the condition for sharing unparametrised geodesics, contraction of the indices forces the gradient of the conformal factor to vanish. Concretely, with a flat metric and a factor depending on the first coordinate alone, the component with upper index one and lower indices two and two is minus the gradient of the logarithm of the factor on one side and zero on the other, so the two agree only when the gradient vanishes, that is, only when the factor is constant (verified here). Light, the conformal structure, and free fall, the projective structure, therefore fix the metric together up to a constant, and neither does so alone. Paper 111’s statement that light reaches nine gains a second leg.
The census comes out as follows. The light cones are observed by light rays and fix the conformal class – nine of ten, and an object that pulls back. Unparametrised geodesics are observed by free fall and fix the projective structure – again pulling back. The last one is carried in by a clock or a ruler and fixes a density – one of ten, and an object that pushes forward. The component light cannot reach is therefore exactly what Paper 149 placed on the pushing-forward side, the side that carries the absolute determinant. Light reports only what pulls back; a standard of volume has to be brought in from outside, which is also why the differential entropy shifted in the fourth section. None of this touches whether a metric is real: it is only a count of which observation fixes which structure.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that the metric is real (not claimed) / that the conformal structure is fundamental (not claimed). What is added: putting the four clocks through one conformal rescaling and separating them by weight, showing that the third clock splits in two under that test, and writing that the component light leaves undetermined is a density, on the pushing-forward side of Paper 149’s sorting. What remains: light reports only what pulls back. The conformal structure pulls back and a density pushes forward, so no amount of measuring with light will ever hand over a standard of volume. The four clocks fell apart into zero, one and two under a single rescaling, and one of them split in two, because they differ in which of them carries in a standard from outside. A clock was the name of a standard brought in from outside.
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). What Light Cannot Measure Is a Density [Paper 150]. Zenodo.
DOI: 10.5281/zenodo.22083989
ORCID: 0009-0006-8587-3450