This paper substituted m₀=4π into the relativistic energy–momentum relation to write E²=(pc)²+(4πc²)², and the first edition called the rearranged form E²−(pc)²−(4πc²)²=0 a "zero-tension condition", claiming an isomorphism with the light cone ds²=0. The second edition withdrew that isomorphism. The 3rd edition names why it could not be called a condition at all.
First, zero-tension does not act on an identity. As Paper 89 identified, zero-tension — a word the author brought from the world of angling — denotes, within this series, a stance towards inverse problems: hold a null hypothesis that can be violated (the state in which nothing is happening) and read the deviations from it as the cause, exactly as one takes the slack of the line as the baseline and reads whatever departs from it as the movement of the fish. But E²−(pc)²−(m₀c²)²=0 is the energy–momentum relation itself, rearranged, and holds for every particle always. The deviation is therefore guaranteed to be zero. With nothing to read by definition, the stance cannot be applied to this expression — it is not that the line does not go slack; there is no line. In the census of the 3rd edition of Paper 16, the first edition read root (c), an identity, as root (d), a conditional. The form "=0" looked like a condition. Form does not tell you the root.
Second, the standing of the same expression depends on which theory one stands inside. Standing inside special relativity, the mass shell is a rearranged identity and cannot be violated. Standing outside it, the same expression becomes a testable null hypothesis: some candidate theories of quantum gravity predict modified dispersion relations E²=p²c²+m²c⁴+η p³c³/E_QG, which shift the group velocity of high-energy photons slightly and therefore appear as a difference in arrival time for distant gamma-ray bursts (Amelino-Camelia et al. 1998). The estimate here assumes a first-order effect δv/c ≃ E/E_QG; with E=10 GeV and E_QG=E_Pl=1.22×10¹⁹ GeV this gives δv/c=8.2×10⁻¹⁹, so that over a redshift z~1 (a distance ~3.9×10²⁵ m) the arrival-time difference is Δt ≃ (D/c)(δv/c) ≈ 0.1 s. A delay of that size is within observational reach, and the Fermi-LAT observation of GRB 090510 constrained E_QG for the first-order effect to above the Planck energy (Abdo et al. 2009). The mass shell therefore becomes a violable null hypothesis only when viewed from outside relativity, and only there can a deviation be defined. This is where zero-tension acts. Whether something is an identity or a hypothesis is not settled by how the equation looks, but by where one stands. This paper, however, performs no such test: it proposes no modified dispersion relation and claims nothing about E_QG. What it shows is confined to a question of position — from where can a deviation be read.
Third, the relation between the four-momentum and the four-displacement is neither an "isomorphism" nor "unrelated". For a massive particle p^μ = m₀u^μ with u^μ = dx^μ/dτ and u^μ u_μ = −c², so that p^μ p_μ = m₀²(u^μ u_μ) = −m₀²c² follows identically; meanwhile dx^μ dx_μ = −c² dτ², so ds²=0 ⟺ dτ=0. The two are therefore not "different vectors" but norms, under the same metric, of the tangent to the same worldline, related by the factor m₀/dτ. And as m₀→0 the parametrisation by τ ceases to exist at all — a null worldline has no proper time. The light cone is not a matter of a different vector: it is where the correspondence p^μ = m₀ dx^μ/dτ itself ceases to be definable. The first edition's "isomorphism" went too far; the second edition's "unrelated" did not go far enough. Precisely, they are two parametrisations of the same object, and the light cone is the boundary at which one of them disappears. This is of the same type as the module boundary of Paper 113, though as with the ν→0 of Paper 03 it is not a smooth limit. Numerical check (m₀=4π, c=1): for v/c=0, 0.5, 0.9, 0.999 the value of E−pc is 12.566, 7.255, 2.883, 0.281 — approaching 0 but never reaching it.
Fourth, the type of the false link to Paper 16 is named. The first edition substituted p=4πv from Paper 16 to obtain E²=(4πc)²(v²+c²), but p=4πv is a non-relativistic momentum whereas the relativistic relation requires p=γm₀v. Re-verified here (m₀=4π, c=1), the ratio of the correct E=γm₀c² to the E obtained from p=m₀v is 1.00 at v/c=0.01, 1.00 at 0.30, 1.03 at 0.50, 1.71 at 0.90 and 5.04 at 0.99. As the 3rd edition of Paper 16 noted, p=4πv is root (b), a consequence of substitution, and its range of validity cannot exceed that of the formula substituted into (Newtonian mechanics); the first edition carried it into a relativistic equation without writing that range. And √(v²+c²) is not a Lorentz scalar — what appears in relativistic kinematics is √(1−v²/c²), and a quantity adding a particle speed to the invariant speed is not an invariant.
Fifth, reading 4πc² as "the surface area of a sphere of radius c" requires budget. [4πc²]=(L/T)²=L²T⁻², and to read this as an area L² one must have [c]=L, i.e. fix the unit of time. But as the 3rd edition of Paper 39 established, the three slots of the budget against M, L, T are already spent on m=4π, G=4π, ħ=1/(4π), so c is no longer choosable. Reading c as a radius is itself an operation Paper 63 took as its subject and is not forbidden, but it cannot be used without declaration — the same type of oversight the 3rd edition of Paper 17 identified for ħ=1. The 3rd edition therefore records "4πc² takes the form of a surface area" as a resemblance of form only, and does not say that it is an area.
Finally, where this paper belongs within the series. Paper 108 took a census showing that "the geometry of relativity" is not one thing: (i) Minkowski's flat invariant interval (Paper 108), (ii) the hyperbolic geometry of velocity space (Paper 60), (iii) proper time (Paper 82) — three distinct roots. This paper's E²=(pc)²+(m₀c²)² belongs to (i); the √(v²+c²) the first edition tried to import resembles a quantity of the domain of (ii) but is not one of hyperbolic geometry (the composition of velocities is not Euclidean addition); and the light cone is where (iii) disappears. The first edition's confusion lay in treating all three as one "geometry of relativity". What the 3rd edition adds is where a name works: zero-tension can be applied only to what can be violated, and what can be violated is settled not by the equation but by where one stands. All numbers were computed here (no external tables consulted). This paper does not modify relativity and claims no isomorphism with the light cone.
(scope note) Scope of this paper: This paper offers a structural interpretation of existing physical law. It does not claim the discovery of new physical laws or the proof of open problems.
Note on units: Substitutions such as m₀=4π are a choice of natural units. G and ħ are dimensionful; their numerical values depend entirely on the unit system. Setting G=4π is legitimate in exactly the same sense as setting G=1 (geometric units) or G=6.674×10⁻¹¹ (SI); physics does not single out one of these as "correct." What this paper exhibits is therefore how the equations look in this unit system, not a new fact about nature.
A note on the aim of this work. This is a structural reinterpretation of well-established physics, not a proposal for a new physical law. Every relation shown is a substitution into, or a geometric re-reading of, equations that are already known and that remain unchanged. The aim is a modest one: to make a few recurring geometric structures — in particular the solid angle 4π and the exponent n = 2 — easier to see within a single picture. Where a particular unit convention is adopted (for example G = 4π), this is a choice of notation, not a claim about nature. Nothing here is intended to contradict existing results.
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). Zero-Tension Does Not Act on an Identity [Paper 21]. Zenodo.
DOI: 10.5281/zenodo.21601143
ORCID: 0009-0006-8587-3450