c as the Ball of Velocity Space — second edition: this was a result about c, not about four pi.
In special relativity the set of admissible velocities is a ball of radius c, and under the metric induced by the velocity-addition law it is hyperbolic space of curvature minus one over c squared (Varicak 1910). Here the reading of c as a radius holds literally, not as a patch. And that curvature is measured, as Thomas precession, the holonomy of velocity space. On this known geometry the first edition rewrote the rest energy of the mass-equals-four-pi representation (Paper 12) as the intrinsic quantity four pi divided by the magnitude of the curvature. NO NEW PHYSICAL LAW AND NO NEW MATHEMATICAL THEOREM IS CLAIMED.
First, the four pi standing in the energy formula cannot be a solid angle. Energy has dimension mass times length squared over time squared, while c squared has length squared over time squared, so the factor in front must carry a mass. A solid angle is dimensionless, of mass dimension zero, and cannot stand there. Under the patch that sets the mass to four pi, this four pi is the mass itself (Paper 116, the theorem that the power of the patch constant is the mass dimension) — that is, a convention and not a geometry (Paper 56, second edition).
Second, the intrinsic form requires the same reading. The reciprocal of the curvature magnitude has the dimension of c squared, a squared velocity and not an area. Read as a solid angle, four pi over the curvature magnitude would be a squared velocity and not an energy. The first edition’s gloss, that the four pi is the solid angle and a fixed geometric constant, is not available; and its reading of the expression as a solid angle times the squared intrinsic radius of curvature of velocity space contains two misplacements at once, making the left factor dimensionless and calling the right factor an area.
Third, the first edition’s fifth section therefore needs three entries and not two. It rightly separated the solid-angle four pi from the topological four pi that equals two pi times the magnitude of the Euler characteristic. The separation is right, but the census is short: what stands in the energy formula is neither, but a third thing — the mass patch, a convention. Listed together they are: the solid angle (the geometry of the sphere, a fact, mass dimension zero), the topological factor (the Euler characteristic, a fact, mass dimension zero), and the mass patch (a choice of units, a convention, mass dimension one). What stands in the area formula of the third section is the first of these. Within one paper, the first and the third were written with the same symbol, and that is the source of the first edition’s confusion. It is the shape Paper 51 exhibited for the shell theorem, where the four pi surviving in the answer was the convention and not the geometry.
Fourth, and what remains is stronger than the first edition supposed. The core is the geometric identity that c squared equals the reciprocal of the curvature magnitude, and it needs no patch whatever. Hence the rest energy equals the mass over the curvature magnitude for any mass at all. The four pi contributes nothing geometric; it merely names the unit of mass. This paper’s result was about c, not about four pi — and it is the more robust for it, holding also for a reader who does not adopt the convention.
The second edition also repairs two small inconsistencies. The first edition said that the rapidity equals the hyperbolic distance from rest; but rapidity is dimensionless, and the distance from the centre in the metric the first edition itself uses is c times the rapidity, which is exactly why its area formula reads as it does. The text disagreed with its own area formula by one factor of c. And the first edition wrote the Thomas rotation angle as the product of the curvature and the enclosed area; since the curvature is negative, what equals the angle defect is its magnitude times the area. The substance is unchanged.
What is unchanged: the hyperbolic geometry of velocity space; that the area of a geodesic sphere reduces to four pi r squared in the small-radius limit (verified here by series expansion); the genus-two Gauss-Bonnet arithmetic (verified here); and the paper’s strongest point, that the curvature is genuinely measured as Thomas precession, manifested in the Thomas factor of one half in atomic spin-orbit coupling. That section is not altered in the second edition.
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts (Layer 2). That velocity space is hyperbolic (Varicak 1910), Thomas precession (Thomas 1926), the area of a hyperbolic geodesic sphere, and Gauss-Bonnet are all standard. What the second edition adds is only a judgement about which four pi appears in the formula. What is not claimed: any new physical law (there is none) / any new mathematical theorem (there is none) / that the four pi of the energy formula is a solid angle (the mass dimensions do not match; it is a convention) / that the reciprocal curvature is an area (it is a squared velocity) / that two entries suffice for the census of four pi (three are needed). What remains: taking the patch away does not always weaken a result; sometimes it strengthens it.
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). $c$ as the Ball of Velocity Space [Paper 60]. Zenodo.
DOI: 10.5281/zenodo.21598012
ORCID: 0009-0006-8587-3450