Abstract
This body of work uses the word "mass" in $75$ papers, $540$ times, and has never once counted it — Paper 165 counted three equivalence principles and that is all. This paper is that census, computed from definitions alone, with no physical constant quoted. NO NEW PHYSICAL LAW AND NO NEW MATHEMATICAL THEOREM IS CLAIMED.
First, invariant mass is not additive. The mass of a system of two photons of energy $E$ runs $0\to0.0174530709967479E\to1.0E\to1.41421356237E\to2.0E$ with the angle alone — and each part has mass zero (Section 2).
Second, inertial mass is not one number. A push along the motion meets $\gamma^3m$ and a push across it meets $\gamma m$ — $12.074512309$ against $2.29415733871$ at $v=0.9$, and $\mathbf{11188.7303863}$ against $22.3662720421$ at $v=0.999$, a ratio of $500.25$ (Section 3).
Third, nothing weighs the reduced mass. $\mu$ is always below both, being $0.999000999001$ even for $m_1=1$ and $m_2=1000$ (Section 4).
Fourth, and this is the core. Effective mass runs from positive through infinite to negative: across a one-dimensional band it goes $0.5\to28.6493442493\to$ divergent $\to-28.6493442493\to-0.5$ — at the top of the band a push to the right accelerates it left (Section 5).
Fifth, this returns to where the work began. The $m=4\pi$ of Paper 02 becomes a mass only once $k$ is placed — and Paper 02 wrote that itself, in its note on units (Section 6).
Scope (what is not claimed)
NO NEW PHYSICAL LAW AND NO NEW MATHEMATICAL THEOREM IS CLAIMED — the definition of invariant mass, longitudinal and transverse mass, reduced mass, the effective mass of a band, and the ADM and Bondi masses are all standard. No measured value is quoted: every number here is computed from a definition in units with $c=1$, and no physical constant and no nuclear datum is used. The equivalence principle is not treated; the agreement of inertial and gravitational mass is left to Paper 165. The origin of mass is not discussed; neither the Higgs mechanism nor confinement is treated. No general relativity is computed; the ADM and Bondi masses of Section 7 are named and not calculated. Renormalisation is not discussed; the bare mass is named only. Paper 02 is not corrected: what Section 6 quotes is the note Paper 02 wrote for itself, and this paper only generalises it. Relation to earlier papers: Paper 02 showed that the total mass of $\rho=r^{-2}$ over the unit ball equals the solid angle $4\pi$, and set that fence itself in a note on units; this paper puts the same question to every $m$ in physics. Paper 165 counted three equivalence principles; this paper sets two of those masses beside the other five. Paper 196 counted four pressures and showed that sharing a unit guarantees nothing; this is its counterpart for mass. Paper 210 showed that four quantities go by the name of the speed of light; the shape is the same. What is added is the demonstration that a two-photon system's mass runs from $0$ to $2E$ on angle alone, the six-row measurement of the longitudinal-to-transverse ratio, the tabulation showing the reduced mass always below both, the nine-point tracking of effective mass through positive, infinite and negative, and the dimensional check restoring $k$ and $R$ to the $4\pi$ of Paper 02.
1. Notation and Position
The question: when physics writes "mass $m$", what resistance has been measured?
This body of work uses the word in $75$ papers, $540$ times, and has never counted it. Counting the named senses, relativistic mass, effective mass, longitudinal mass, bare mass, mass defect, ADM mass and reduced mass do not appear even once in the text.
Units: everything below takes $c=1$ and is computed from definitions. No physical constant and no measured value is quoted — and without them, the seven still separate.
2. Invariant Mass Is Not Additive
Definition 2.1 the invariant mass of a system is given by $M^2=E^2-|\mathbf{p}|^2$.
Computation 2.2: for two photons of energy $E$ at angle $\theta$, $M^2=2E^2(1-\cos\theta)$.
| $\theta$ (degrees) | $M/E$ | mass of each part |
|---|---|---|
| $0$ | $0$ | $0$ |
| $1$ | $0.0174530709967479$ | $0$ |
| $60$ | $1.0$ | $0$ |
| $90$ | $1.4142135623731$ | $0$ |
| $180$ | $\mathbf{2.0}$ | $\mathbf{0}$ |
Observation 2.3 two massless parts make a system with mass, and that mass is fixed by the angle alone — so mass is not the sum of the masses of the parts.
Observation 2.4 for the same reason a bound system has less mass than its parts — this paper quotes no nuclear data and therefore gives no value. What can be said is the consequence of the definition, and no more.
3. Inertial Mass Is Not One Number
Computation 3.1: apply a force $F$ to a body moving at speed $v$. The acceleration is $F/(\gamma^3m)$ along the motion and $F/(\gamma m)$ across it.
| $v$ | $\gamma$ | along: $\gamma^3$ | across: $\gamma$ | ratio |
|---|---|---|---|---|
| $0$ | $1.0$ | $1.0$ | $1.0$ | $1.0$ |
| $0.5$ | $1.154700538$ | $1.53960071784$ | $1.15470053838$ | $1.333333333$ |
| $0.9$ | $2.294157339$ | $12.074512309$ | $2.29415733871$ | $5.263157895$ |
| $0.99$ | $7.08881205$ | $356.221711059$ | $7.08881205008$ | $50.25125628$ |
| $0.999$ | $22.36627204$ | $\mathbf{11188.7303863}$ | $22.3662720421$ | $\mathbf{500.2501251}$ |
Observation 3.2 the same body returns a different resistance depending on the direction of the push. At $v=0.999$ the longitudinal resistance is $500$ times the transverse — so "mass increases with speed" has the direction omitted.
Observation 3.3 "relativistic mass" fell out of use because it is not one number. Keeping $F=ma$ requires a different $m$ for each direction — the modern convention fixes $m$ as an invariant and puts $\gamma$ on the momentum instead. This is a choice of convention and not a discovery.
4. Reduced Mass: Nothing Has It
Computation 4.1: the reduced mass of a two-body problem, $\mu=m_1m_2/(m_1+m_2)$.
| $m_1$ | $m_2$ | $\mu$ |
|---|---|---|
| $1$ | $1$ | $0.5$ |
| $1$ | $10$ | $0.909090909091$ |
| $1$ | $1000$ | $\mathbf{0.999000999001}$ |
| $5$ | $5$ | $2.5$ |
Observation 4.2 $\mu$ is always below both, and no body anywhere in the problem has that mass — $\mu$ is the mass of a fictitious body standing in for a relation. Here the word "mass" is not even a property of a thing.
5. The Core: Effective Mass Goes Negative
Computation 5.1: in a one-dimensional band $E(k)=-2t\cos(ka)$, measure the effective mass $m^{*}=\hbar^2/\big(2ta^2\cos(ka)\big)$, with $\hbar=t=a=1$.
| $ka$ (degrees) | $m^{*}$ | sign |
|---|---|---|
| $0$ | $0.5$ | positive |
| $80$ | $2.87938524157$ | positive |
| $89$ | $28.6493442493$ | positive |
| $90$ | divergent | the inflection of the band |
| $91$ | $-28.6493442493$ | negative |
| $135$ | $-0.707106781187$ | negative |
| $180$ | $\mathbf{-0.5}$ | negative |
Observation 5.2 one electron, in one band, carries positive, infinite and negative mass. At the top of the band a push to the right accelerates it left — the lattice is pushing back, and the electron has neither gained nor lost anything.
Observation 5.3 the "mass" here is not a property of a thing but the curvature of a dispersion relation. The definition $m^{*}=\hbar^2/(d^2E/dk^2)$ simply calls the reciprocal of a curvature by that name — standing in the same place as Paper 224 counting the roots of "curvature".
6. Mass as a Solid Angle: the Fence Paper 02 Set for Itself
Paper 02 showed that the total mass of $\rho=r^{-2}$ over the unit ball equals the solid angle $4\pi$. And in the same paper it wrote:
$\rho=r^{-2}$ has dimensions $[L^{-2}]$, and to be a density it should be written $\rho=k/r^2$ with a constant $k$ of dimension $[M/L]$. The statement $m=4\pi$ holds under a choice of units, namely $k=1$ and radius $1$.
Computation 6.1: restore that $k$ and the radius $R$: $m=4\pi kR$.
| $k$ | $R$ | $m=4\pi kR$ |
|---|---|---|
| $1$ | $1$ | $12.566370614359$ |
| $1$ | $2$ | $25.132741228718$ |
| $3$ | $1$ | $37.699111843078$ |
| $0.5$ | $4$ | $25.132741228718$ |
Observation 6.2 $4\pi$ is a solid angle. It becomes a mass once $k$ is placed, and $k$ is what carries $[M]$ — so the seventh root is a mass that borrows its dimension from a constant.
Observation 6.3 this is not a correction to Paper 02. Paper 02 set this fence itself — all this paper did was show that the same fence is needed by the other six. The discipline was already there at the first step.
Computation 6.4: and the module staircase of Papers 01 and 02, $m(n)=4\pi/(3-n)$.
| $n$ | $m(n)$ | note |
|---|---|---|
| $0$ | $4.1887902047864$ | |
| $1$ | $6.2831853071796$ | |
| $2$ | $\mathbf{12.566370614359}$ | equals the solid angle |
| $2.5$ | $25.132741228718$ | |
| $2.9$ | $125.66370614359$ |
$n=2$ is the one exponent making the radial integral exactly $1$. (QED)
7. Named Only
The following are not computed here. They are named, and given an address.
- Gravitational mass (passive and active) — its agreement with inertial mass is left to Paper 165.
- Bare and renormalised mass — only the latter is observed, and the former depends on how the regularisation is taken. Renormalisation is not discussed here.
- ADM and Bondi mass — the first is measured over an initial surface, the second after the radiation has left, and the two do not agree in general. And the gravitational field has no local energy density — so "how much mass is here" can fail to be writable in principle. Not one line is computed here.
8. Separator and Fence
Observation 8.1 (the census)
| name | resistance to what | the fact that bites |
|---|---|---|
| invariant mass | an invariant of energy and momentum | not additive |
| inertial mass | reluctance to accelerate under a force | varies with direction |
| gravitational mass | coupling to gravity | Paper 165 |
| reduced mass | the relative motion of two bodies | nothing has it |
| effective mass | the curvature of a dispersion relation | goes negative |
| bare and renormalised | how the regularisation is taken | only the latter is seen |
| mass as a solid angle | the constant $k$ that was placed | the note in Paper 02 |
Observation 8.2 (the separator) what divides the seven is what the resistance is measured against. The symbol $m$ names none of them — the same shape as $f'$ in Paper 233, $d(x,y)$ in Paper 234, $-u''$ in Paper 235, "= 0" in Paper 236 and $\sum a_n$ in Paper 239.
Observation 8.3 (the fence) there are three. First, whether it can be added: invariant mass cannot (Section 2). Second, whether it is one number: inertial mass varies with direction (Section 3). Third, where its $[M]$ comes from: the solid angle borrows it from $k$ (Section 6). None of the three is written in the symbol $m$.
Observation 8.4 Paper 196 wrote that sharing a unit guarantees nothing. For mass it is stronger still — the quantity can go negative, and it can have no owner at all.
9. Census and Closing
What this paper measured.
- A two-photon mass running from $0$ to $2E$ on angle alone (2.2).
- A longitudinal-to-transverse ratio of $500.25$ (3.1): at $v=0.999$.
- $\mu$ always below both (4.1).
- Effective mass running $0.5\to$ divergent $\to-0.5$ (5.1).
- $m=4\pi kR$, with $[M]$ coming from $k$ (6.1).
- $m(n)=4\pi/(3-n)$, equal to the solid angle at $n=2$ (6.4).
References
- Einstein, A. (1905). Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? Annalen der Physik, 18.
- Okun, L. B. (1989). The concept of mass. Physics Today, 42.
- Ashcroft, N. W., Mermin, N. D. (1976). Solid State Physics. New York: Holt, Rinehart and Winston.
- Arnowitt, R., Deser, S., Misner, C. W. (1962). The dynamics of general relativity. In: L. Witten (ed.), Gravitation: An Introduction to Current Research. New York: Wiley.
- Bondi, H., van der Burg, M. G. J., Metzner, A. W. K. (1962). Gravitational waves in general relativity, VII. Proceedings of the Royal Society A, 269.
- Misner, C. W., Thorne, K. S., Wheeler, J. A. (1973). Gravitation. San Francisco: Freeman.
- Yamagishi, Y. (2026). The Uniqueness Theorem of $n = 2$ for Spherically Symmetric Density Distributions [Paper 01]. Zenodo. 10.5281/zenodo.21403186
- Yamagishi, Y. (2026). The $4\pi$ Mass Cancellation Theorem [Paper 02]. Zenodo. 10.5281/zenodo.21403931
- Yamagishi, Y. (2026). There Are Three Equivalence Principles [Paper 165]. Zenodo. 10.5281/zenodo.22092957
- Yamagishi, Y. (2026). “Pressure” Is Four Separate Quantities, and Sharing a Unit Guarantees Nothing [Paper 196]. Zenodo. 10.5281/zenodo.22107518
- Yamagishi, Y. (2026). The Peak of a Pulse Can Leave Before It Arrives [Paper 210]. Zenodo. 10.5281/zenodo.22111747
- Yamagishi, Y. (2026). Flat Wherever You Touch It, and Four Pi in Total [Paper 224]. Zenodo. 10.5281/zenodo.22120836
- Yamagishi, Y. (2026). “Diverges” Is Four Things [Paper 239]. Zenodo. 10.5281/zenodo.22129461
Reference
Yamagishi, Y. (2026). “Mass” Is Seven Things, One of Which Goes Negative and One of Which Depends on the Direction of the Push [Paper 240]. YOI: https://www.yamagishiyuuki.com/yoi/240
DOI: 10.5281/zenodo.22129765
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 240. This paper is a preprint and has not been peer reviewed.