The exponent is the signature of what is conserved – a census of the Sedov-Taylor solution.
A shock driven by energy released into a uniform medium expands with its radius growing as the two-fifths power of time, the law used daily on supernova remnants. This paper writes where that exponent comes from and where it does not. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Taylor (1950), Sedov (1946), von Neumann, and Barenblatt’s distinction between the first and second kinds are all standard. The solution is treated here in the setting where it is used today, the expansion of supernova remnants.
First, the exponent moves with the dimension. Writing the radius as a product of powers of the energy, the time and the density and solving the dimensional equations in d dimensions, the radius is proportional to the energy times the square of the time divided by the density, all raised to the power one over d plus two, so that the radius grows as time to the power two over d plus two (computed here symbolically). The exponent is two thirds in one dimension, one half in two, two fifths in three and one third in four. The two fifths is therefore not a constant of nature but the solution of a linear system in the exponents, and it moves when the dimension changes. Paper 28 warned against mistaking an exponent for a dimension; here the exponent is a function of the dimension, and it is still not a dimension.
Second, and this is the core. The exponent is the signature of what is conserved. Taking the swept-up mass to be of order the density times the radius to the d, conserved energy reads as the density times the radius to the d times the square of the velocity being constant, and conserved momentum as the density times the radius to the d times the velocity being constant; these give the radius growing as time to the two over d plus two, and as time to the one over d plus one, respectively. In three dimensions that is two fifths against one quarter. Both integrate in closed form, so the local logarithmic slope was computed: at time ten it is 0.42553191 and 0.27027027, at a hundred 0.40241449 and 0.25188917, at ten thousand 0.40002400 and 0.25001875, and at a hundred million 0.40000000 and 0.25000000. The exponent alone distinguishes the two laws: knowing nothing of the details of the explosion, taking logarithms of radius and time and reading the slope tells which quantity is conserved. And real remnants change over, since once radiation begins carrying the heat away energy is no longer conserved and the shock leaves the two-fifths law for the momentum-conserving one. The exponent is a signature not of the explosion but of what is conserved now. Paper 135 said that whether an error is a drift or an oscillation is told by the scaling rather than the size, and the same holds here: the size says nothing and the exponent says everything.
Third, dimensional analysis cannot see the adiabatic index. The variables are the radius, the energy, the time and the density, four in all, and the independent dimensions are mass, length and time, three in all, so there is exactly one dimensionless group and hence exactly one undetermined constant. Adding the adiabatic index changes nothing about this, since it is already dimensionless: the count becomes five minus three, that is two, and the relation reads that the radius times the density over the energy times the square of the time, raised to the power one over d plus two, equals a function of the adiabatic index. Dimensional analysis therefore says nothing whatever about that index, a dimensionless quantity being invisible to it from the start. Here the other half of the famous story appears. Reading a radius and a time off a single photograph and recovering the energy by dimensional analysis alone is true, and it is half: the other half is a number depending on the medium, which comes only from solving the self-similar equations. The value of that constant is not computed in this paper, because numbers that have not been computed are not written (Paper 101).
Fourth, there are again three layers. The form is forced by dimensional analysis, the constant is computed from the self-similar solution, and the adiabatic index is an input supplied by the medium. This is the same three-layer shape as Paper 140 (symmetry, dynamics, phase) and Paper 141 (dimension, Lane-Emden, composition), and three papers in a row have now produced it: a forced form, a number that must be computed, and an input that must be supplied. The common lesson is that a dichotomy does not suffice.
Fifth, there is also self-similarity whose exponent does not come from dimensional analysis. The exponent of the Sedov-Taylor solution follows from dimensional analysis alone, which is called self-similarity of the first kind. But for problems such as a shock converging on a centre the exponent does not follow from dimensional analysis; it is fixed instead by the condition that the solution pass smoothly through a singular point, an eigenvalue problem, and this is called self-similarity of the second kind (Barenblatt). In the first kind the exponent is fixed by dimensional analysis and is rational; in the second it is fixed by an eigenvalue condition and is generally irrational. The single word self-similar thus covers two different origins of the exponent. Paper 117 listed four roots of scale invariance, of which root B is the dimensional matrix; the first kind belongs to root B and the second does not, and the separator is whether the exponent follows from dimensional analysis. A practical clue follows: solutions of a dimensional matrix are solutions of a linear system and are therefore always rational, so an observed exponent that is irrational did not come from dimensional analysis. The rationality of the exponent is itself a clue to its root.
IMPORTANT (scope): a census of established standard material (Layer 2). Every work cited is standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that the Sedov-Taylor solution is in error (it is not) / that dimensional analysis is powerless (it is not; it fixes the form completely). What is added: writing the exponent as the signature of a conservation law, deriving from the Buckingham count that the adiabatic index is invisible, confirming the convergence of the exponent by local logarithmic slopes, and setting the two kinds of self-similarity side by side as distinct roots. What remains: take logarithms of radius and time, read the slope, and what is conserved right now can be read off; and when an exponent stops being rational, that is the sign that it came from outside dimensional analysis.
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). The Exponent Is the Signature of What Is Conserved [Paper 144]. Zenodo.
DOI: 10.5281/zenodo.22082585
ORCID: 0009-0006-8587-3450