Abstract
Stick two lumps of clay together: counted, they make one; weighed, they make two lumps’ worth. If the two are brought together gently from rest and the heat given off escapes, the whole, weighed exactly, is lighter than two lumps’ worth by that heat. The mass of the whole at rest minus the sum of the masses of the parts, each taken apart and at rest, times c², is the energy left inside the whole. This paper lays out the direction and the size of this difference for atoms, nuclei, the proton, light and gravity along a single axis.
In nuclei and atoms the whole is lighter than the sum of its parts; every one of the 2548 nuclei measured in AME2020 is. With the CODATA 2022 values the deuteron is 0.118464 percent and the alpha particle 0.753409 percent lighter, and the hydrogen atom is lighter by 1.4485×10⁻⁸ of the sum of its parts. The binding energy per nucleon is largest for nickel-62 at 8794.5555 keV, followed by iron-58 and iron-56, and nickel-62 is about 0.937 percent lighter than its parts. Conversely, the proton of Paper 286 is 104.37 times as heavy as the conventional sum of its quark rest masses, 8.99 MeV, and the two photons of Paper 240 and Paper 241 become as heavy as 2E/c² (E being the energy of one photon) from parts of mass 0, depending on the opening angle. What sets the direction is the sign of the energy left inside.
The gravitational potential energy of a ball of density r⁻ⁿ as in Paper 01 and Paper 02 is −α(n)·GM²/R, with coefficient α(n) = (3−n)/(5−2n). Starting from the 3/5 of a uniform ball (Paper 30 and Paper 165), it equals exactly 1 only at n = 2, where the energy is −GM²/R. If all the energy released in coming together goes out, the whole is lighter than the sum of its parts by α(n)·GM²/(Rc²). In a ball held up by the pressure of its own gas, the virial balance keeps half of the released energy inside as heat, and the whole is lighter only by the other half, the one that goes out; the 3/10 of Paper 497 is that outgoing half for a uniform ball. The mass is finite for n < 3 (Paper 02), while the potential energy is finite only for n < 5/2. The outside share of the field energy is (5−2n)/(6−2n), going from the 5/6 of a uniform ball (Paper 1106) to exactly one half at n = 2. Charges of one sign held in the same shape make the whole heavier by the same coefficient.
Quantities with similar names and numbers that lie close are told apart. The mass excess (M−A·u) of the atomic mass table is 0 for carbon-12, positive for helium-4 and negative for iron-56, but the mass defect is positive for all of them; they are compared with different references. When a deuteron forms, the 2.22456637 MeV of rest mass that is lost splits into 2.223248712 MeV of outgoing light and 1.31766 keV of recoil (Paper 911). The 0.007 of Paper 497 is the 0.711852 percent from four hydrogen atoms, which differs in starting material and denominator from the 0.753409 percent from separate protons and neutrons.
The separator is whether the difference is binding energy that went out or field and motion energy confined inside. In terms of the six roots of Paper 16, "the mass of the whole equals the sum of the masses of the parts" is not an identity but the equality of an empirical law, and moreover holds only approximately. It misses by the order of 10⁻⁸ for the hydrogen atom and by 10⁻³ to 10⁻² for nuclei, and for the proton it misses in the opposite direction by more than a factor of a hundred.
Keywords: mass defect, binding energy, invariant mass, nuclear mass, self-gravity, density exponent, mass excess, separator.
Scope (what is not claimed)
This paper claims no new theorem or law. That the mass of a system at rest includes its internal energy, binding energy and mass defect, the virial theorem, the atomic mass evaluation AME2020 (Wang et al., 2021), the recommended values of the physical constants CODATA 2022 (NIST, 2024), the gravitational and electric self-energies of a uniform ball, writing a self-energy as field energy, and the nominal values of IAU 2015 Resolution B3 (Prša et al., 2016) are all standard. What is added is the arrangement of the difference between whole and parts along one axis by the sign of the energy left inside; the count of the coefficient α(n) and of the outside share of the field energy for a ball of density r⁻ⁿ, showing that they are 1 and one half at n = 2; the placing of the range n < 5/2 where the potential energy is finite beside the range n < 3 where the mass is finite; the separation of mass excess and mass defect by their reference; and the check of the 0.007 and the 3/10 of Paper 497 and the recoil of Paper 911 with the numbers of nuclei and of the Sun. Paper 240 sorted "mass" into seven, showed with two photons that invariant mass is not addition, and counted that words such as mass defect never appear in the main text of the corpus. Paper 241 showed that the same two parts can give different wholes, Paper 228 that the "+" sign wears at least eight operations, and Paper 16 that the equals sign has six roots. This paper sits beside them and counts in which direction, and by how much, the "+" of mass misses. The numbers for two photons are taken from Paper 240 and Paper 241 and the quark values for the proton and the neutron from Paper 286, and they were recomputed in the check. The binding fractions of celestial bodies were counted by Paper 165. The only celestial body used here is the Sun, which was recomputed with the IAU nominal values and agrees with the value of Paper 165 to five significant digits. Strong gravity in general relativity (such as the correct values for neutron stars), models of the nuclear force, quantum chromodynamics inside the proton, and the numbers of the mass difference in chemical bonds are not treated. The defect from the gravitational potential energy, |U|/c² when no heat stays inside, and the difference U/(2c²) between whole and parts for a ball held up by its own gas are both weak-gravity estimates (U being the Newtonian gravitational potential energy) and apply only when α(n)·GM/(Rc²) is much smaller than 1. Where n is close to 5/2, or where gravity is strong as in a neutron star, that estimate itself does not apply.
1. Adding the Parts Does Not Give the Whole
Stick two lumps of clay together and the number of lumps becomes 1 while the amount becomes 2. The "+" of counting applies only to things that stay separate, and the "+" of amount applies only when nothing goes in or out. If the two are brought together gently from rest and the heat given off escapes, the whole is lighter by that heat. Ordinarily it is too small to measure, but in nuclei it reaches the order of one part in a thousand to one part in a hundred.
Let $M$ be the mass of the whole at rest and $\sum m_i$ the sum of the masses of the parts, each taken apart and at rest. The difference
$$\Delta=M-\sum_i m_i$$times $c^2$ is the energy left inside the whole (the sum of the kinetic energy of the parts, the potential energy between them and the field energy). If $\Delta<0$ the whole is lighter; the sum of the parts minus the whole, $B=-\Delta c^2$, is called the binding energy and $-\Delta$ the mass defect. If $\Delta>0$ the whole is heavier than its parts. Parts that cannot be stopped, like light, are counted as mass 0, and parts that cannot be taken out alone, like quarks, are given their conventional values (the values used by Paper 286).
Paper 16 showed that the equals sign has six roots (definition, consequence of substitution, identity, conditional equation, empirical law, dimensionally invalid). $M=\sum m_i$ is neither a definition nor an identity but the equality of an empirical law, and moreover holds only approximately. Paper 228 showed that the "+" sign wears at least eight operations, and Paper 241 asked whether the value of a whole is a function of the values of its parts at all. Paper 240 sorted "mass" into seven and showed with two photons that invariant mass is not addition. Beside them, this paper counts in which direction and by how much the "+" of mass misses, for atoms, nuclei, the proton, light and gravity.
2. Nuclei and Atoms: The Whole Is Lighter
With the CODATA 2022 recommended values (NIST, 2024), compare the masses of four light nuclei with the sums of the masses of their protons and neutrons.
| nucleus | sum of parts (MeV) | whole (MeV) | $B$ (MeV) | $B/A$ (MeV) | fraction of the sum of parts |
|---|---|---|---|---|---|
| deuteron ${}^{2}\mathrm{H}$ | 1877.83751137 | 1875.61294500 | 2.22456637 | 1.112283 | 0.118464 % |
| triton ${}^{3}\mathrm{H}$ | 2817.40293331 | 2808.92113668 | 8.48179663 | 2.827266 | 0.301050 % |
| helion ${}^{3}\mathrm{He}$ | 2816.10960080 | 2808.39161112 | 7.71798968 | 2.572663 | 0.274066 % |
| alpha particle ${}^{4}\mathrm{He}$ | 3755.67502274 | 3727.3794118 | 28.29561094 | 7.073903 | 0.753409 % |
In all four the whole is lighter. The fraction of the sum of parts grows from 0.118464 percent for the deuteron with two nucleons to 0.753409 percent for the alpha particle with four, and the binding energy per nucleon grows from 1.1 MeV to 7.1 MeV.
The hydrogen atom, a proton and an electron bound together, is also lighter, by its ionization energy. Bohr’s formula with the reduced mass gives an ionization energy of $R_\infty hc \mu/m_e=13.5982873$ eV ($\mu$ is the reduced mass; relativistic and quantum-electrodynamic corrections are not included). As a fraction of the sum of parts, 938.7830883807 MeV, it is $1.4485\times10^{-8}$, far below the order of $10^{-3}$ for nuclei. For a $1/r$ force the virial balance $2K+U=0$ gives the proton and the electron a potential energy of $-27.1965745$ eV and a kinetic energy of relative motion of $13.5982873$ eV. Of the potential energy lost in binding, half goes out as light and half stays inside as motion (mostly the motion of the electron).
Arranging the 2548 nuclei of the atomic mass evaluation AME2020 (Wang et al., 2021) that have values which are not estimates, every one has a positive $B$ (even the smallest, ${}^{6}\mathrm{H}$, has about 960 keV per nucleon, with an AME uncertainty of 42 keV), and the binding energy per nucleon is largest for nickel-62 at 8794.5555 keV, followed by iron-58 at 8792.2534 keV and iron-56 at 8790.3563 keV. The $B$ in the table below is the $B/A$ of the AME table times $A$, and the fraction is that $B$ divided by the sum of the masses of the protons and neutrons.
| nucleus | $B/A$ (keV) | $B$ (MeV) | fraction of the sum of parts |
|---|---|---|---|
| ${}^{2}\mathrm{H}$ | 1112.2831 | 2.2245662 | 0.11846 % |
| ${}^{4}\mathrm{He}$ | 7073.9156 | 28.2956624 | 0.75341 % |
| ${}^{6}\mathrm{Li}$ | 5332.3312 | 31.9939872 | 0.56792 % |
| ${}^{12}\mathrm{C}$ | 7680.1446 | 92.1617352 | 0.81798 % |
| ${}^{16}\mathrm{O}$ | 7976.2072 | 127.619315 | 0.84951 % |
| ${}^{56}\mathrm{Fe}$ | 8790.3563 | 492.259953 | 0.93618 % |
| ${}^{62}\mathrm{Ni}$ | 8794.5555 | 545.262441 | 0.93661 % |
| ${}^{208}\mathrm{Pb}$ | 7867.4530 | 1636.43022 | 0.83780 % |
| ${}^{235}\mathrm{U}$ | 7590.9151 | 1783.86505 | 0.80835 % |
| ${}^{238}\mathrm{U}$ | 7570.1262 | 1801.69004 | 0.80613 % |
The fraction of the sum of parts is largest around iron and nickel, about 0.94 percent, smaller for light nuclei and smaller again for heavy ones. Explaining the shape of this peak with models of the nuclear force and electric repulsion is standard, and this paper does not enter into the models.
Comparing $B$ for the same four nuclei between AME and CODATA, the deuteron and the triton (one proton) agree within 0.00017 keV and 0.00043 keV, while the helion and the alpha particle (two protons) are both off by 0.0516 keV and 0.0515 keV. AME counts with atomic masses (the atom subtracted from $Z$ hydrogen atoms and $N$ neutrons), so the difference between the electron binding of the helium atom and that of two hydrogen atoms remains. The electron binding of the helium atom obtained from the numbers of Section 7 is 79.0 eV; subtracting the share of two hydrogen atoms gives 51.8 eV. It differs from the 51.46 eV offset of the alpha particle by 0.34 eV, and where that 0.34 eV comes from is not settled in this paper. Whether one counts with atoms or with bare nuclei is a separator too.
3. Mass Excess Is Not the Reverse of Mass Defect
One column of the AME table is the mass excess, the atomic mass $M$ minus $A$ atomic mass units $u$ for mass number $A$, that is $M-A u$. Since $u$ is defined as one twelfth of the mass of a carbon-12 atom, the mass excess of carbon-12 is 0 by definition.
| nucleus | mass excess $M-A u$ (keV) | mass defect times $c^2$, $B$ (keV) |
|---|---|---|
| ${}^{2}\mathrm{H}$ | $+13135.722895$ | 2224.5662 |
| ${}^{4}\mathrm{He}$ | $+2424.91587$ | 28295.6624 |
| ${}^{12}\mathrm{C}$ | 0 (definition) | 92161.7352 |
| ${}^{56}\mathrm{Fe}$ | $-60607.163$ | 492259.9528 |
| ${}^{238}\mathrm{U}$ | $+47307.732$ | 1801690.036 |
The mass excess can be positive, zero or negative, but the mass defect is positive for all of them. The two are compared with different references: the mass excess with $A$ units of $u$, the mass defect with the separate parts. For carbon-12 the atomic mass of 12 u is 11177.9292446 MeV, adding six protons, six neutrons and six electrons gives 11270.0910619 MeV, and the difference of 92.1618173 MeV is the mass defect compared with separate protons, neutrons and electrons. The 92161.7352 keV of the table (AME) is the value with six hydrogen atoms and six neutrons as parts and is 0.08208 keV smaller; the difference is close to the 0.08159 keV binding of six hydrogen atoms. "Excess" and "defect" look like opposite words, but the sign of one does not decide the other. The definition that Paper 300 named as its strongest criterion states that numerical agreement alone does not make two things share a root; agreement of names does not either.
4. The Axis of Sign
| system | whole $-$ sum of parts | what stays inside | source |
|---|---|---|---|
| two photons (energy $E$ each, opening angle 180 degrees) | $+2E/c^2$ (both parts are 0) | motion of the light | Paper 240 and Paper 241 |
| proton | $+929.28$ MeV (the whole is 104.37 times the conventional 8.99 MeV) | confined field and motion | Paper 286 |
| ball of like charges (including what holds them) | $+\alpha(n) kQ^2/(Rc^2)$ | repelling field | Section 5 |
| hydrogen atom | $-1.4485\times10^{-8}$ of the sum of parts | attracting potential energy and motion | Section 2 |
| deuteron | $-0.118464$ % of the sum of parts | the same | Section 2 |
| alpha particle | $-0.753409$ % of the sum of parts | the same | Section 2 |
| ${}^{62}\mathrm{Ni}$ | $-0.93661$ % of the sum of parts | the same | Section 2 |
| the Sun (taken as a uniform ball of gas) | $-6.3675\times10^{-7}$ of the sum of parts | gravitational potential energy and heat inside | Paper 165 and Paper 497, Section 5 |
On the positive side are systems whose energy left inside is positive. Two photons of energy $E$, though their parts have mass 0, give a whole of mass $2E/c^2$ when they fly in opposite directions (Paper 240 and Paper 241). For the proton the conventional sum of the rest masses of its quarks (the value used by Paper 286, which moves with the convention) is less than 1 percent of the whole, and the rest is confined field and motion (Paper 286). On the negative side are systems whose parts attract each other and in which, of the potential energy lost in coming together, the part that did not stay inside as motion went out as light or heat.
What sets the direction is the sign of the energy inside. The same deuteron is lighter than the sum of its parts when a proton and a neutron are taken as its parts, and 91.54 times as heavy as its parts when quarks are taken as its parts, with the conventional values of Paper 286 (20.49 MeV as the quark sum of a proton, 8.99 MeV, and a neutron, 11.50 MeV). What is taken as the parts changes the energy counted inside, and with it the direction. Compared as fractions of the difference, the sizes span about ten orders of magnitude (9.853, the ratio of the 103.37 of the proton to the $1.4485\times10^{-8}$ of the hydrogen atom): the hydrogen atom is lighter at the order of $10^{-8}$, the Sun taken as a uniform ball of gas at $10^{-7}$, nuclei at $10^{-3}$ to $10^{-2}$, and the proton is heavier by more than a factor of a hundred.
5. The Gravitational Energy of a Ball of Density $r^{-n}$
Paper 02 wrote the mass of a ball of radius 1 with density $\rho(r)=r^{-n}$ in three dimensions as $m(n)=4\pi/(3-n)$ ($n<3$) and showed that at $n=2$ it is the full solid angle $4\pi$. The mass within radius $r$ is $M(r)=4\pi r^{3-n}/(3-n)$, obtained by integrating the $dm/dr=4\pi r^{2-n}$ of Paper 01. If this ball is built by bringing matter scattered at infinity in one shell at a time, the gravitational potential energy (the work done by the hand that brings the shells) is
$$U=-\int_0^R \frac{G\,M(r)}{r}\,dM(r)=-\alpha(n)\,\frac{GM^2}{R},\qquad \alpha(n)=\frac{3-n}{5-2n}$$(with $M=M(R)$). The work done by gravity is $-U=\alpha(n) GM^2/R$, the energy released in coming together.
If all the released energy goes out and no kinetic energy stays inside, then in the weak-gravity estimate the whole is lighter than the sum of its parts by $|U|/c^2=\alpha(n) GM^2/(Rc^2)$. This paper calls it the defect from the gravitational potential energy. In a ball held up by the pressure of its own gas (a monatomic ideal gas with zero pressure at the surface), the virial balance $2K+U=0$ keeps half of $|U|$ inside as heat, so the difference between whole and parts is $U/(2c^2)$. The $(3/10) GM^2/R$ that Paper 497 set as the reservoir of gravitational contraction is the other half, the one that goes out, for a uniform ball. Here $M$ is the sum of the masses of the parts.
| $n$ | $\alpha(n)=(3-n)/(5-2n)$ | shell-by-shell work | field energy | outside share (formula) |
|---|---|---|---|---|
| 0 | 0.6 | 0.6 | 0.6 | 0.833333333333 |
| 1 | 0.666666666667 | 0.666666666667 | 0.666666666667 | 0.75 |
| 1.5 | 0.75 | 0.75 | 0.75 | 0.666666666667 |
| 2 | 1 | 1.0 | 1.0 | 0.5 |
| 2.25 | 1.5 | 1.5 | 1.5 | 0.333333333333 |
| 2.4 | 3 | 3.0 | 3.0 | 0.166666666667 |
| 2.45 | 5.5 | 5.49999999796 | 5.49999999814 | 0.0909090909091 |
Numerical integration in two ways, as the work of bringing shells and as the field energy $-(1/8\pi G)\int g^2 dV$, agreed with the formula within a relative $1.38\times10^{-19}$ for $n\le2.4$. At $n=2.45$ the integrand is sharp at the centre, and the agreement stayed within $3.72\times10^{-10}$. $\alpha(0)=3/5$ is the value for a uniform ball, used by Paper 30 and Paper 165. $\alpha(n)=1$ requires $3-n=5-2n$, that is $n=2$ only. At $n=2$ and $R=1$, $M=4\pi$ and
$$U=-\frac{GM^2}{R}=-16\pi^2G=-157.913670417\,G .$$This $16\pi^2=(4\pi)^2$ is the same number as the factor Paper 1106 saw when putting $m=4\pi$ into a uniform ball, but it comes from a different place. There it is the ratio from setting the mass of a grain to $4\pi$; here it is the square of the mass $4\pi$ of the unit $1/r^2$ ball. The same number alone does not make them share a root.
Splitting the field energy into the inside and the outside of the ball, the ratio of inside to outside is $1/(5-2n)$ and the outside share is $(5-2n)/(6-2n)$. Paper 1106 saw that the outside share for a uniform ball is five sixths. At $n=2$ it is exactly half and half, and as $n$ approaches 5/2 the field energy gathers inside.
The mass is finite for $n<3$ (the $m(n)=4\pi/(3-n)$ of Paper 02), while the gravitational potential energy is finite only for $n<5/2$. At $n=2.4999$, $\alpha=2500.5$ while the mass is $4\pi/(3-n)=25.12771569$, staying at a finite value close to $8\pi$. At $n=2.6$ the mass is finite, but $|U|$ with balls of radius $10^{-2}$, $10^{-4}$ and $10^{-6}$ hollowed out at the centre keeps growing: 1435.870963, 6345.672515, 18885.057. In the window $5/2\le n<3$ the weak-gravity estimate of the defect grows without bound, so the estimate itself does not apply there. Paper 1106 showed that where an infinity sits is decided by how fast things thin out against the dimension. Here too the edges are set by $3-n$ and $5-2n$, and $4\pi$ has nothing to do with them.
Placing charges of one sign in the same shape, the hand that brings the shells does work against the repulsion, and $U=+\alpha(n) kQ^2/R$ ($k$ is Coulomb’s constant). The integral is the same and only the sign changes. Repelling charges do not stay at rest unless something holds them. If the energy inside what holds them does not change, the whole, including what holds them, counted by energy, is heavier than the sum of its parts by $\alpha(n) kQ^2/(Rc^2)$. Paper 908 counted that for the field alone of a charged spherical shell, counting by momentum gives $4/3$ times this, so that the two do not combine into one mass, and wrote that non-electromagnetic stresses are known to make up the difference. Whether the parts attract or repel decides whether the same coefficient makes a defect or a surplus.
Putting the nominal values of IAU 2015 Resolution B3 (Prša et al., 2016; solar radius $6.957\times10^{8}$ m, $GM=1.3271244\times10^{20}$ m$^3$ s$^{-2}$) for the Sun gives $GM/(Rc^2)=2.1225\times10^{-6}$. For a uniform ball the fraction of the defect from the potential energy is 3/5 of that, $1.2735\times10^{-6}$, agreeing with the value of Paper 165 to five significant digits. Taken as a uniform ball of gas, half stays inside as heat by the virial balance, and the difference between whole and parts is $-6.3675\times10^{-7}$ of the sum of parts. For density $r^{-2}$ the coefficient is 1 and the share from the potential energy is $GM/(Rc^2)$ itself, 5/3 of the uniform value. In the $r^{-n}$ family, the larger $n$ and the more concentrated toward the centre, the larger the coefficient. The coefficient of the real Sun is not given in this paper. The nominal values are conversion values, not the true values of the Sun themselves (Prša et al., 2016).
6. What Goes Out When Things Bind
Let a deuteron form from a neutron and a proton at rest, giving off one photon. Conserving energy and momentum, the energy of the light is $E_\gamma=(M_i^2-m_d^2)c^2/(2M_i)$ with $M_i=m_p+m_n$, and the deuteron recoils.
Of the rest mass lost, $B=$ 2.22456637 MeV, the light carries 2.223248712 MeV and the recoil of the deuteron 1.31766 keV. Together they make exactly $B$. Paper 911 showed that the rest mass of the emitter falls by more than the energy of the light, by the kinetic energy of the recoil. For the deuteron, the rest mass lost is 1.0005927 times the energy of the light. The mass defect is not the same as the light that went out; it is the light plus the recoil. What happens when a crystal, rather than a single atom, takes the recoil was counted by Paper 664 with the gamma ray of iron-57.
7. Two 0.7 Percents
Paper 497 set the nuclear reservoir of the Sun as hydrogen burning at an efficiency of 0.7 percent. Checked with nuclear numbers, there are two 0.7 percents. Making an alpha particle from two separate protons and two separate neutrons makes it 0.753409 percent lighter than the sum of its parts. Making a helium atom from four hydrogen atoms (with the atomic masses of AME), the energy that goes out is $Q=$ 26.7309684 MeV, 0.711852 percent of the four hydrogen atoms.
The difference of 0.04156 percentage points comes from
$$Q=B-2(m_n-m_p)c^2+2m_ec^2$$($2(m_n-m_p)c^2=2.58666502$ MeV, $2m_ec^2=1.0219979$ MeV; the remaining $2.46\times10^{-5}$ MeV is the electron binding of the helium atom minus that of four hydrogen atoms). Making it from hydrogen requires the mass to turn two protons into neutrons, and counting with atoms gives back the mass of the two leftover electrons. Rounded to one significant digit, 0.711852 percent (0.00711852) becomes 0.007 and 0.753409 percent (0.00753409) becomes 0.008. The 0.007 of Paper 497 is the number for making it from hydrogen. The numbers are close, but the starting material and the denominator differ.
8. Same Root or Different Roots
Paper 300 gathered the criteria the corpus has used to judge whether two things share a root. The definition it named as the strongest states that numerical agreement alone does not make them share one. The quantities of this paper are sorted by what is subtracted from what.
(1) The differences of wholes bound in nuclei, atoms and by gravity, the surplus of the proton and the mass of two photons are all "the mass of the whole at rest minus the sum of the masses of the parts," $\Delta$ (photons cannot be stopped, so their parts have mass 0; quarks cannot be taken out alone, so they have the conventional values of Paper 286), and all are the energy left inside divided by $c^2$. For gravity what stays inside is the sum of potential energy and heat, $U/2$ for a ball held up by its own gas. The mass defect and the binding energy are this $\Delta$ with its sign reversed. They share a root and differ in sign and size.
(2) The mass excess $M-A u$ is a bookkeeping quantity compared with $A$ units of $u$, a different reference from (1). Though the names are alike, the roots differ.
(3) The two 0.7 percents start from different material (separate nucleons or hydrogen atoms). Close numbers do not make them share a root.
9. The Separator
The separator is whether the difference is binding energy that went out or field and motion energy confined inside. In the first case the whole is lighter than the sum of its parts (nuclei, atoms, gravity); in the second it is heavier (the proton, two photons, repelling charges). Depending on what is taken as the parts, the same deuteron stands on both sides. The gravitational potential energy goes through a coefficient set by the density exponent, $\alpha(n)=(3-n)/(5-2n)$, and is exactly $-GM^2/R$ at $n=2$.
Closing
The mass of a whole is not the sum of the masses of its parts. The difference times c² is the energy left inside: if attracting potential energy goes out, the whole is lighter; if confined field and motion remain, it is heavier. The hydrogen atom is lighter at the order of 10⁻⁸, nuclei from the 0.118 percent of the deuteron to the 0.937 percent of nickel-62, and the proton is 104.37 times as heavy as its parts. The gravitational potential energy of a ball of density r⁻ⁿ is −α(n)·GM²/R, and α(n) = (3−n)/(5−2n) goes from the 3/5 of a uniform ball to exactly 1 at n = 2 and grows without bound as n approaches 5/2. In a ball held up by the pressure of its own gas, half of the energy released in coming together stays inside as heat, and the whole is lighter only by the other half. In terms of the six roots of Paper 16, "the mass of the whole equals the sum of the masses of the parts" is the equality of an empirical law that holds only approximately, and what sets the direction and size of its miss is what goes out and what stays inside when things stick together.
References
- National Institute of Standards and Technology (2024). CODATA Recommended Values of the Fundamental Physical Constants: 2022. NIST.
- Prša, A., Harmanec, P., Torres, G., Mamajek, E., Asplund, M., Capitaine, N., et al. (2016). Nominal values for selected solar and planetary quantities: IAU 2015 Resolution B3. The Astronomical Journal.
- Wang, M., Huang, W. J., Kondev, F. G., Audi, G. & Naimi, S. (2021). The Ame2020 atomic mass evaluation (II). Chinese Physics C.
- Yamagishi, Y. (2026). The Uniqueness Theorem of $n = 2$ for Spherically Symmetric Density Distributions [Paper 01]. YOI: https://www.yamagishiyuuki.com/yoi/1; DOI: 10.5281/zenodo.21403186
- Yamagishi, Y. (2026). The $4\pi$ Mass Cancellation Theorem [Paper 02]. YOI: https://www.yamagishiyuuki.com/yoi/2; DOI: 10.5281/zenodo.21403931
- Yamagishi, Y. (2026). The Equals Sign Has Distinct Roots Too [Paper 16]. YOI: https://www.yamagishiyuuki.com/yoi/16; DOI: 10.5281/zenodo.21600471
- Yamagishi, Y. (2026). The Yamagishi Double-Torus Stellar Fortress Equation [Paper 30]. YOI: https://www.yamagishiyuuki.com/yoi/30; DOI: 10.5281/zenodo.21601732
- Yamagishi, Y. (2026). There Are Three Equivalence Principles [Paper 165]. YOI: https://www.yamagishiyuuki.com/yoi/165; DOI: 10.5281/zenodo.22092957
- Yamagishi, Y. (2026). There Are Eight Plus Signs, and Seven of Them Become Real Addition in the Right Coordinate [Paper 228]. YOI: https://www.yamagishiyuuki.com/yoi/228; DOI: 10.5281/zenodo.22126254
- 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
- Yamagishi, Y. (2026). Before Asking Whether It Can Be Made into Addition, Ask Whether There Is a Binary Operation at All [Paper 241]. YOI: https://www.yamagishiyuuki.com/yoi/241; DOI: 10.5281/zenodo.22130189
- Yamagishi, Y. (2026). $99.04\%$ of the Proton’s Mass Is Not the Higgs [Paper 286]. YOI: https://www.yamagishiyuuki.com/yoi/286; DOI: 10.5281/zenodo.22171199
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. YOI: https://www.yamagishiyuuki.com/yoi/300; DOI: 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). Kelvin’s Calculation Was Never Refuted; What Was Replaced Was the Reservoir [Paper 497]. YOI: https://www.yamagishiyuuki.com/yoi/497; DOI: 10.5281/zenodo.22760194
- Yamagishi, Y. (2026). The Recoil Was Not Removed; the Crystal Took It Instead [Paper 664]. YOI: https://www.yamagishiyuuki.com/yoi/664; DOI: 10.5281/zenodo.23014706
- Yamagishi, Y. (2026). The 4/3 of the Electromagnetic Mass Comes Not from a Ball’s Volume but from Averaging over Field Directions [Paper 908]. YOI: https://www.yamagishiyuuki.com/yoi/908; DOI: 10.5281/zenodo.22804819
- Yamagishi, Y. (2026). Light Crossing a Box Carries a Weight $E/c^2$, and the Emitter’s Rest Mass Falls by More, by the Recoil [Paper 911]. YOI: https://www.yamagishiyuuki.com/yoi/911; DOI: 10.5281/zenodo.22804957
- Yamagishi, Y. (2026). A Physical Infinity Is Set by How Fast Things Thin Out against the Dimension, Not by the Mass [Paper 1106]. YOI: https://www.yamagishiyuuki.com/yoi/1106
Reference
Yamagishi, Y. (2026). A Mass Defect Follows the Sign of the Energy Left Inside, and the Gravitational Energy of a 1/r² Ball Is Exactly −GM²/R [Paper 1113]. YOI: https://www.yamagishiyuuki.com/yoi/1113
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 1113. This paper is a preprint and has not been peer reviewed.