Abstract
That a force has finite range is the same statement as that its mediating particle has mass. What this paper shows is that the same relation had already been measured inside superconductors, with penetration depth and mass tied as reciprocals by one formula. No new theorem or law is claimed.
First, the penetration depth follows from the electron density——$\mathbf{53.14}$ nm at $1\times10^{28}$ per cubic metre, $\mathbf{30.68}$ at $3\times10^{28}$, and $\mathbf{16.80}$ at $1\times10^{29}$ (Section 2).
Second, these are of the measured order——lead at roughly $37$ nm and aluminium at roughly $16$ nm fall in this range (Section 2).
Third, and this is the core. The depth can be read as a mass——$m=\hbar/(\lambda c)=\mathbf{1.1465\times10^{-35}}$ kg, or $\mathbf{6.4316}$ eV, some $\mathbf{1.259\times10^{-5}}$ of an electron’s rest mass (Section 3).
Fourth, the inversion returns——the depth comes back from the mass to $12$ digits and the density from the depth to $10$ (Section 3).
Fifth, the same formula fits the weak force——the $80.379$ GeV of the $W$ boson gives a range of $\mathbf{2.4550\times10^{-3}}$ fm, $\mathbf{1.250\times10^{10}}$ times smaller than the penetration depth (Section 4).
Sixth, the separator is what is screening the field——$mLc/\hbar$ is $\mathbf{1.000000000000}$ in both, so the relation is one; what differs is the screen (Section 5).
Keywords: Meissner effect, London equations, photon mass, Anderson-Higgs mechanism, superconductivity.
Scope (what is not claimed)
No new theorem or law is claimed——the London equations, the Meissner effect and the Anderson--Higgs mechanism are standard. Quantum field theory is not entered——no derivation of how a gauge field acquires mass; only the reciprocal relation between length and mass is examined numerically. The mechanism of superconductivity is not discussed——how pairs form is not the subject; the calculation stays inside London’s phenomenology. Historical priority is not adjudicated——who noticed first is not at issue, only that the same relation occupies two places. Types of superconductor are not distinguished——the difference between type I and type II and the ratio to the coherence length are not treated. Effective mass is not used——the bare electron mass was used; in a real metal the effective mass differs. It is not asserted that “the photon acquires mass”——this is an effective description inside a medium and says nothing about the photon in vacuum. Relation to earlier papers: Paper 286 showed that $99.04$ per cent of the proton’s mass is not the Higgs——that paper divided the origins of mass; this one shows mass and length to be two readings of one quantity. Paper 175 showed that “symmetry breaking” is not one word——the breaking in a superconductor and the breaking in vacuum share a name and differ in what breaks. Paper 270 showed that the two in the flux quantum is a count of carriers——the same pairs appear there as a two and here as a penetration depth. Paper 300 showed that sharing a root is decidable——the two settings share a root under the criterion of a correspondence that preserves identity. What is added is building the depth from density and comparing with measured values, reading it as a mass of $\mathbf{6.4316}$ eV, confirming the inversion to $12$ and $10$ digits, applying the formula to the $W$ boson for a separation of $\mathbf{1.250\times10^{10}}$, and placing the separator on what screens the field.
1. Notation and the Question
Setting: a superconductor expels magnetic field, which decays exponentially over a distance $\lambda$ from the surface, the London penetration depth,
$$\lambda=\sqrt{\frac{m}{\mu_{0}\,n\,e^{2}}}$$The other formula: a force mediated by a particle of mass $m_{\gamma}$ has range $L=\hbar/(m_{\gamma}c)$.
Observation 1.1 both give a length at which a field stops reaching. One from a property of matter, the other from a particle’s mass——matching shapes suggest two readings of one quantity.
Observation 1.2 there is a way to check. Read the depth as a mass and put the mass back——if it does not return, the two are different things.
Question: is it one relation?
2. Building the Depth from Density
Computation 2.1: computing the depth for several densities from the electron mass and charge and the permeability of free space.
| electron density (per cubic metre) | penetration depth |
|---|---|
| $1\times10^{28}$ | $\mathbf{53.14}$ nm |
| $3\times10^{28}$ | $\mathbf{30.68}$ nm |
| $1\times10^{29}$ | $\mathbf{16.80}$ nm |
Observation 2.2 these are of the measured order. Lead at roughly $37$ nm, aluminium at roughly $16$ nm——both fall inside the table, checking formula and inputs together. Not a claim.
Observation 2.3 it goes as the square root of density. A factor of ten from $1\times10^{28}$ to $1\times10^{29}$ takes $\mathbf{53.14}$ to $\mathbf{16.80}$ nm, about a $3.16$th——matching $\sqrt{10}=3.162$.
Observation 2.4 the bare electron mass was used. Real metals have a different effective mass——so agreement of order is claimed, and agreement of value is not.
3. Reading the Depth as a Mass
Computation 3.1: putting $\lambda=30.68$ nm into $m=\hbar/(\lambda c)$.
| quantity | value |
|---|---|
| mass | $\mathbf{1.1465\times10^{-35}}$ kg |
| energy | $\mathbf{6.4316}$ eV |
| ratio to the electron’s rest mass | $\mathbf{1.259\times10^{-5}}$ |
Observation 3.2 this is the core. Inside a superconductor the electromagnetic field behaves like a field of mass $\mathbf{6.4316}$ eV——heavy for a quantum of light, of the same order as the energy of one visible photon.
Computation 3.3: inverting. The depth returns from the mass to $12$ digits, and the density from the depth to $10$.
Observation 3.4 returning is the check. If the reading were not reversible, the two would not be one quantity——it returned, so depth and mass are two readings of one thing.
Observation 3.5 it is not said that “the photon acquires mass”. This is an effective description inside a medium——nothing is claimed about the photon in vacuum, so as not to mix the two breakings that Paper 175 separated.
4. The Same Formula for the Weak Force
Computation 4.1: feeding the same formula the $W$ boson mass of $80.379$ GeV.
| setting | mass | length |
|---|---|---|
| superconductor | $\mathbf{1.1465\times10^{-35}}$ kg | $\mathbf{3.0681\times10^{-8}}$ m |
| $W$ boson | $\mathbf{1.4329\times10^{-25}}$ kg | $\mathbf{2.4550\times10^{-18}}$ m |
| ratio of lengths | — | $\mathbf{1.250\times10^{10}}$ |
Observation 4.2 ten orders apart. $30.68$ nm against $\mathbf{2.4550\times10^{-3}}$ fm——a laboratory length and a length smaller than a nucleus, entirely different worlds.
Computation 4.3: measuring the dimensionless $mLc/\hbar$ in both, obtaining $\mathbf{1.000000000000}$ each time.
Observation 4.4 it is one relation. Two settings ten orders apart return the same dimensionless value——by the second criterion of Paper 300 the correspondence can be written down, so this is not agreement of numbers.
Observation 4.5 the historical order follows. Measurement in superconductors preceded the understanding of the weak force——though no priority is adjudicated here; only that the relation occupies two places.
5. The Separator
Observation 5.1 the separator is what is screening the field. Electron pairs in a superconductor, another field in vacuum——one relation, different screens.
Observation 5.2 so one of them can be removed. Warm the superconductor and the depth diverges and the mass vanishes——the vacuum does not behave so. Removability is what separates them.
Observation 5.3 the pairs of Paper 270 are at work here. There the two in the flux quantum was a count of carriers——the same pairs appear there as a two and here as a length. One object surfacing in two quantities.
Observation 5.4 the orientation differs from Paper 286. That paper divided the origins of mass——this one unites mass and length as two readings. One divides, the other binds.
Observation 5.5 what is not said. Quantum field theory is not entered——no derivation of mass generation, no mechanism of superconductivity, no distinction between type I and type II.
6. Roll Call and Closing
- $\mathbf{30.68}$ nm at $3\times10^{28}$ (Computation 2.1): between lead and aluminium.
- square root of density (Observation 2.3): ten gives a $3.16$th.
- bare electron mass (Observation 2.4): order, not value.
- mass $\mathbf{1.1465\times10^{-35}}$ kg (Computation 3.1): the core.
- energy $\mathbf{6.4316}$ eV (Computation 3.1): the order of visible light.
- $\mathbf{1.259\times10^{-5}}$ of an electron (Computation 3.1): heavy for light.
- inversion returns to $12$ and $10$ digits (Computation 3.3): a check on reversibility.
- $W$ range $\mathbf{2.4550\times10^{-3}}$ fm (Computation 4.1): $\mathbf{1.250\times10^{10}}$ apart.
- $mLc/\hbar=\mathbf{1.000000000000}$ in both (Computation 4.3): one relation.
- the separator is the screen (Observation 5.1): not the relation.
- one vanishes on warming (Observation 5.2): removability.
Closing: That a force has finite range is the same statement as that its mediator has mass. The relation had already been measured inside superconductors——electron densities of $1\times10^{28}$, $3\times10^{28}$ and $1\times10^{29}$ per cubic metre give penetration depths of $\mathbf{53.14}$, $\mathbf{30.68}$ and $\mathbf{16.80}$ nm, and lead at roughly $37$ nm and aluminium at roughly $16$ nm fall inside. It goes as the square root of density, so a factor of ten gives about a $3.16$th, matching $\sqrt{10}$. The bare electron mass was used, so agreement of order is claimed and agreement of value is not. The depth can be read as a mass——$\lambda=30.68$ nm gives $m=\hbar/(\lambda c)=\mathbf{1.1465\times10^{-35}}$ kg, or $\mathbf{6.4316}$ eV, some $\mathbf{1.259\times10^{-5}}$ of an electron’s rest mass, which is heavy for a quantum of light and of the order of one visible photon. The depth returns from the mass to $12$ digits and the density from the depth to $10$, so the reading is reversible and the two are one quantity. It is still not said that the photon acquires mass: this is an effective description inside a medium, and the two breakings separated in Paper 175 must not be mixed. The same formula fits the weak force——the $80.379$ GeV of the $W$ boson gives a range of $\mathbf{2.4550\times10^{-18}}$ m, that is $\mathbf{2.4550\times10^{-3}}$ fm, some $\mathbf{1.250\times10^{10}}$ times smaller than $30.68$ nm. A laboratory length and a length below nuclear scale, and yet $mLc/\hbar$ returns $\mathbf{1.000000000000}$ in both. By the second criterion of Paper 300 the correspondence can be written, so this is not agreement of numbers. The separator is what screens the field——electron pairs in a superconductor, another field in vacuum: one relation with different screens. So one can be removed: warm the superconductor and the depth diverges and the mass vanishes, while the vacuum does not oblige. The pairs counted as a two in Paper 270 appear here as a length, one object surfacing in two quantities. Paper 286 divided the origins of mass; this paper binds mass and length as two readings. To be honest——quantum field theory was not entered, and no derivation of how a gauge field acquires mass was given; only the reciprocal relation between length and mass was examined. The mechanism of superconductivity was not discussed and the calculation stayed inside London’s phenomenology. Type I and type II were not distinguished, the coherence length was not treated, and the bare rather than the effective electron mass was used. No priority was adjudicated. What can be said is that one relation occupies two places ten orders apart, and that one of them vanishes on warming, and no further.
References
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- Higgs, P. W. (1964). Broken Symmetries and the Masses of Gauge Bosons. Physical Review Letters.
- London, F., & London, H. (1935). The Electromagnetic Equations of the Supraconductor. Proceedings of the Royal Society A.
- Meissner, W., & Ochsenfeld, R. (1933). Ein neuer Effekt bei Eintritt der Supraleitfahigkeit. Naturwissenschaften.
- Yamagishi, Y. (2026). “Symmetry Breaking” Is Not One Word [Paper 175]. Zenodo. 10.5281/zenodo.22097873
- Yamagishi, Y. (2026). The Two in the Flux Quantum Is Neither a Dimension nor an Exponent but a Count of Carriers [Paper 270]. Zenodo. 10.5281/zenodo.22169893
- Yamagishi, Y. (2026). $99.04\%$ of the Proton’s Mass Is Not the Higgs [Paper 286]. Zenodo. 10.5281/zenodo.22171199
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Reference
Yamagishi, Y. (2026). Light With Mass Had Already Been Found Inside a Superconductor [Paper 541]. YOI: https://www.yamagishiyuuki.com/yoi/541
DOI: 10.5281/zenodo.23014143
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 541. This paper is a preprint and has not been peer reviewed.