A limit exists because two forces carry the same exponent – a census of the Chandrasekhar mass.
A white dwarf has an upper limit of mass. This paper writes why it appears as a limit at all, and which three layers the number is made of. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Chandrasekhar (1931), the Lane-Emden equation and the ultra-relativistic degeneracy pressure are all standard.
First, and this is the core. A limit exists because two forces carry the same power of the radius. Writing the equation of state as pressure proportional to density raised to Gamma, expressing hydrostatic balance by scaling and solving for the mass, the mass depends on the radius as the minus third power at Gamma equal to five thirds, the minus first power at three halves, and exactly the zeroth power at four thirds (confirmed here symbolically). For an ultra-relativistic degenerate gas, in other words, gravity and pressure enter with the same power of the radius, so the contest is settled by the coefficient alone, that is, by a single mass. Put another way, at five thirds every mass finds a radius that balances it: compress and pressure wins, expand and gravity wins. At four thirds there is no such arbiter: below one value of the mass every radius is supported, and above it none is. Paper 117 wrote that one root of scale invariance is exponents lining up, and here the two terms’ exponents line up, the balance loses its ability to fix a radius, and a mass is fixed instead. Equal exponents are often the sign that something is not determined, and next to what is not determined, something else becomes determined.
Second, the number of order one comes from the Lane-Emden equation. Integrating the Lane-Emden equation of polytropic index three by fourth-order Runge-Kutta at a step of one millionth, with the singularity at the origin avoided by a series start, gives a first zero at 6.89685 and a mass constant of 2.01824. Neither number follows from dimensional analysis; they appear only on solving the differential equation, and what is usually called a number of order one is generally this (Paper 101).
Third, the limit is the Planck mass cubed over the nucleon mass squared. Assembling it from the ultra-relativistic degeneracy coefficient and the mass formula for a polytrope of index three gives, at two nucleons per electron, 2.854031 times ten to the thirtieth kilograms, that is 1.4350 solar masses. And taking the Planck mass to be 2.176434 times ten to the minus eighth kilograms, dividing the Chandrasekhar mass by the Planck mass cubed over the nucleon mass squared gives 3.097972. The limit is therefore the Planck mass cubed over the nucleon mass squared, times a number of order one. The exponents three and two are decided by dimension (Paper 116), and the nucleons per electron stands in the denominator. The ratio of the Planck mass to the nucleon mass is about 1.3 times ten to the nineteenth, and the limit is a macroscopic mass because that ratio is enormous; as Paper 10 said, only dimensionless quantities can be tuned, and the dimensionless quantity here is that ratio.
Fourth, the famous 1.4 solar masses contains a choice of composition. Varying the number of nucleons per electron gives 5.7399 for hydrogen, 1.4350 for helium, carbon or oxygen, and 1.2373 for iron-56. The number moves with the chemistry, so the statement that the Chandrasekhar mass is 1.4 solar masses is a statement about carbon-oxygen white dwarfs and not a constant of nature; for hydrogen it would be 5.74. The value two, incidentally, restates the fact that nuclei have equal numbers of protons and neutrons, so that two comes from nuclear physics and from neither gravity nor degeneracy. As Paper 53 said of the exponent two, the same number can arrive from a different place.
Fifth, there are three layers. The form, the three-halves power of the combination of Planck’s constant, the speed of light and the gravitational constant, divided by the square of the nucleons per electron times the nucleon mass, is forced by dimensional analysis; the number of order one, 3.098, is computed from the Lane-Emden solution; and the nucleons per electron is an input supplied by the star’s composition. Paper 140 wrote three layers of geometry (forced), dynamics (measured) and phase (convention), and here they are forced, computed and supplied; what is the same is that a dichotomy does not suffice. And finally, 1.435 is the leading-order value and not the observed bound: Coulomb corrections, general-relativistic effects and inverse beta decay all lower it. The figure 1.4 must not be handled as a constant of nature, being an approximate value, a compositional value, and a value before corrections.
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 Chandrasekhar’s result is in error (it is not) / that the figure 1.4 is wrong (it is not; the point is one of standing, that it is a leading-order value including a composition). What is added: writing why the bound falls on the mass rather than the radius as a coincidence of exponents, integrating the Lane-Emden constants here, writing the limit as the Planck mass cubed over the nucleon mass squared, and separating the three layers. What remains: where the radius stops being determined, a mass becomes determined. When the exponents of two terms line up, the balance can no longer choose a length and chooses a single mass instead. A limit is the other face of an indeterminacy.
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). A Limit Exists Because Two Forces Carry the Same Exponent [Paper 141]. Zenodo.
DOI: 10.5281/zenodo.22082127
ORCID: 0009-0006-8587-3450