Abstract
Optics, electromagnetism, structural mechanics and thermodynamics each have a reciprocity theorem. What this paper shows is that the first three require only that an operator be symmetric, while the fourth requires reversing the sign of the magnetic field as well. No new theorem or law is claimed.
First, with a symmetric operator reciprocity is exact——for a symmetric stiffness matrix the difference between $f_{1}\cdot u_{2}$ and $f_{2}\cdot u_{1}$ is $\mathbf{0.00\times10^{0}}$ (Section 2).
Second, making it asymmetric breaks it——a difference of $\mathbf{0.029851}$, relatively $\mathbf{0.6667}$: the far side of the boundary (Section 2).
Third, and this is the core. With a magnetic field the symmetry is destroyed and reciprocity survives——$|L_{12}-L_{21}|=\mathbf{1.600000}$, and yet $L(B)$ matches the transpose of $L(-B)$ with difference $\mathbf{0.00\times10^{0}}$ (Section 3).
Fourth, switching off the field restores ordinary symmetry——at $B=0$, $|L_{12}-L_{21}|$ falls to the $10^{-15}$ level (Section 3).
Fifth, the conditions have different shapes——the first three can be stated about one matrix; the fourth needs two states, $B$ and $-B$, set side by side (Section 4).
Sixth, the separator is how many states are needed——one, or two (Section 4).
Keywords: reciprocity theorem, Onsager reciprocal relations, Helmholtz reciprocity, Betti’s theorem, time-reversal symmetry.
Scope (what is not claimed)
No new theorem or law is claimed——the reciprocity relations of Helmholtz, Lorentz, Betti and Onsager are all standard. The Onsager relations are not proved——the derivation from microscopic reversibility is not entered; only the shape of the relation is examined numerically. No real material is treated——the matrices are illustrative and are not the coefficients of any substance. Nonlinear cases are not covered——only linear response is examined. No non-reciprocal device is designed——isolators and circulators are not the subject. It is not claimed that four exhausts the list——four were counted, and no more are asserted. Neither “the same theorem” nor “unrelated” is claimed——what is shared is a shape, and the requirements differ. Relation to earlier papers: Paper 143 showed that minimum entropy production is not a principle but a theorem with a fence——one part of that fence is the Onsager relation, and this paper shows that relation differs in shape from the other three. Paper 77 treated fine-grained conservation and coarse-grained growth——the time reversal the Onsager relation requires lives on the fine-grained side. Paper 125 showed that there are three Hopfs——this is likewise a census, of a word rather than a name, and it divides. Paper 300 showed that sharing a root is decidable——the four split three against one under the criterion of what a quantity carries. What is added is confirming numerically that symmetry gives exact reciprocity, measuring a relative break of $\mathbf{0.6667}$ when symmetry is removed, showing reciprocity surviving a destroyed symmetry under a magnetic field, restoring it by switching the field off, and placing the separator on the number of states required.
1. Notation and the Question
Setting: a reciprocity theorem says that exchanging cause and effect leaves the response unchanged. Helmholtz has one in optics, Lorentz in electromagnetism, Betti in structural mechanics and Onsager in irreversible processes.
Observation 1.1 all say “the same when exchanged”. Sharing a shape suggests sharing a requirement——but requirements cannot be read off a shape.
Observation 1.2 there is a way to tell them apart. Break a condition and see whether reciprocity fails——different failures mean different requirements.
Question: do the four require the same thing?
2. Three Need Only a Symmetric Operator
Computation 2.1: checking Betti’s reciprocity on a two-degree-of-freedom elastic system, comparing $f_{1}\cdot u_{2}$ with $f_{2}\cdot u_{1}$.
| stiffness matrix | $f_{1}\cdot u_{2}$ | $f_{2}\cdot u_{1}$ | difference |
|---|---|---|---|
| symmetric | $-0.049180327868852$ | $-0.049180327868852$ | $\mathbf{0.00\times10^{0}}$ |
| asymmetric | $-0.044776119402985$ | $-0.014925373134328$ | $\mathbf{0.029851}$ |
Observation 2.2 symmetry gives it exactly. Not even at the rounding level: exactly zero——an identity rather than an approximation.
Observation 2.3 asymmetry breaks it. Relatively $\mathbf{0.6667}$——the far side of the boundary, confirming that reciprocity does not simply always hold.
Observation 2.4 optics and electromagnetism share the shape. A reciprocal medium makes the operator symmetric and the same argument runs——what is required is one property of one operator.
3. The Fourth Needs Two States
Computation 3.1: building a linear-response matrix with a term odd in the magnetic field and measuring symmetry and reciprocity separately.
| quantity | value |
|---|---|
| $L_{12}$ of $L(B)$ | $1.3000$ |
| $L_{21}$ of $L(B)$ | $-0.3000$ |
| broken symmetry $|L_{12}-L_{21}|$ | $\mathbf{1.600000}$ |
| difference between $L(B)$ and the transpose of $L(-B)$ | $\mathbf{0.00\times10^{0}}$ |
Observation 3.2 this is the core. The symmetry is broken by $\mathbf{1.600000}$ and reciprocity still holds exactly——but in the form $L(B)=L(-B)^{\top}$, not $L=L^{\top}$.
Observation 3.3 switching off the field restores it. At $B=0$, $|L_{12}-L_{21}|$ falls to the $10^{-15}$ level——the far side of the boundary: the shape of the condition changes only with a field present.
Observation 3.4 what is required is time reversal. Reversing time in the microscopic equations also reverses the field——the fine-grained property treated in Paper 77 appears here as a condition.
Observation 3.5 so “it holds because it is symmetric” is wrong here. It can hold without symmetry——carrying the explanation of the first three across to the fourth produces an error.
4. The Separator
Observation 4.1 the separator is how many states are needed. The first three are statements about one operator——the fourth cannot be written without two states, $B$ and $-B$.
Observation 4.2 three against one under Paper 300. The first three carry only an operator——the fourth carries the sign of an external field as well.
Observation 4.3 the difference shows in practice. Building a non-reciprocal device needs something that breaks time reversal, such as a magnetic field——breaking symmetry alone is not enough; the fourth condition must be broken.
Observation 4.4 this is one plank of the fence in Paper 143. There, minimum entropy production was a theorem with a fence——the Onsager relation is part of it, and the relation itself changes shape under a field.
Observation 4.5 no claim that four exhausts the list. Four were counted——to call a list exhaustive one must first fix how to exhaust it.
Observation 4.6 what is not said. The Onsager relations are not proved——the derivation from microscopic reversibility is not entered, and only linear response is examined.
5. Roll Call and Closing
- symmetric gives $\mathbf{0.00\times10^{0}}$ (Computation 2.1): an identity.
- asymmetric gives $\mathbf{0.029851}$ (Computation 2.1): relatively $\mathbf{0.6667}$.
- optics and electromagnetism share the shape (Observation 2.4): one property.
- broken symmetry $\mathbf{1.600000}$ (Computation 3.1): with a field.
- reciprocity still exact (Observation 3.2): the core.
- in the form $L(B)=L(-B)^{\top}$ (Observation 3.2): not $L=L^{\top}$.
- restored at $B=0$ (Observation 3.3): the far side.
- the requirement is time reversal (Observation 3.4): a fine-grained property.
- the separator is the count of states (Observation 4.1): one or two.
- three against one (Observation 4.2): different belongings.
- four does not exhaust (Observation 4.5): four were counted.
Closing: Optics, electromagnetism, structural mechanics and thermodynamics each have a reciprocity theorem. Three need only a symmetric operator——with a symmetric stiffness matrix, $f_{1}\cdot u_{2}$ and $f_{2}\cdot u_{1}$ differ by $\mathbf{0.00\times10^{0}}$, exactly zero rather than at the rounding level, so this is an identity. Making the matrix asymmetric opens a difference of $\mathbf{0.029851}$, relatively $\mathbf{0.6667}$, so reciprocity does not simply always hold. Optics and electromagnetism share the shape, since a reciprocal medium makes the operator symmetric, and what is required is one property of one operator. The fourth holds without symmetry——in a linear-response matrix with a term odd in the field, $|L_{12}-L_{21}|=\mathbf{1.600000}$ destroys the symmetry, and yet $L(B)$ matches the transpose of $L(-B)$ with difference $\mathbf{0.00\times10^{0}}$. What holds is $L(B)=L(-B)^{\top}$, not $L=L^{\top}$. Switching the field off returns $|L_{12}-L_{21}|$ to the $10^{-15}$ level, so the shape of the condition changes only when a field is present. The requirement is time reversal, which also reverses the field, so the fine-grained property of Paper 77 appears here as a condition. The separator is how many states are needed——the first three are statements about one operator, while the fourth needs $B$ and $-B$ side by side. Under Paper 300, the first three carry only an operator and the fourth also carries the sign of an external field, so they split three against one. The difference shows in practice: a non-reciprocal device needs something that breaks time reversal, and breaking symmetry alone will not do. Paper 143 found minimum entropy production to be a theorem with a fence, and the Onsager relation is one plank of it, itself changing shape under a field. So carrying the explanation of the first three across to the fourth produces an error. To be honest——the Onsager relations were not proved; the derivation from microscopic reversibility was not entered and only the shape of the relation was examined. The matrices are illustrative and belong to no substance. Only linear response was treated, nonlinear cases were not, and no isolator or circulator was designed. It is not claimed that four exhausts the list, nor that the four are the same theorem, nor that they are unrelated. What can be said is that they share a shape while requiring different things, and no further.
References
- Betti, E. (1872). Teoria della elasticita. Il Nuovo Cimento.
- Helmholtz, H. von (1856). Handbuch der physiologischen Optik. Voss.
- Lorentz, H. A. (1895). Het theorema van Poynting over de energie in het electromagnetisch veld en een paar algemeene stellingen over de voortplanting van het licht. Verslagen der Afdeeling Natuurkunde.
- Onsager, L. (1931). Reciprocal Relations in Irreversible Processes I. Physical Review.
- Yamagishi, Y. (2026). Liouville’s Theorem and the Arrow of Time: Fine-Grained Conservation and Coarse-Grained Growth [Paper 77]. Zenodo. 10.5281/zenodo.21722473
- Yamagishi, Y. (2026). There Are Three Hopfs [Paper 125]. Zenodo. 10.5281/zenodo.22079384
- Yamagishi, Y. (2026). Minimum Entropy Production Is Not a Principle but a Theorem with a Fence [Paper 143]. Zenodo. 10.5281/zenodo.22082367
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
Reference
Yamagishi, Y. (2026). “Reciprocity” Is Four Theorems: Three Need Only Symmetry, One Needs Time Reversal [Paper 542]. YOI: https://www.yamagishiyuuki.com/yoi/542
DOI: 10.5281/zenodo.23014145
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 542. This paper is a preprint and has not been peer reviewed.