EXAFUSE
Programme note

A Published Formalism for a Gravitational-Analogue Force

A spin-2 effective field theory of the unreduced electromagnetic near field has been accepted at Classical and Quantum Gravity. It is not a theory of gravity, and it does not claim a nuclear effect. What it provides is the thing a thirty-year-old proposal in condensed matter nuclear science has been missing: a formalism.

ExaFuse · Programme note for the Condensed Matter Nuclear Science community · October 2026
How to read this note. The effective field theory described here is now peer-reviewed published physics, and its internal results stand or fall on their own stated assumptions, which the paper itself tabulates. Takaaki Matsumoto's observational claims are a contested part of the CMNS literature and are reported here as his claims, not as established results. The proposal that the mode described in the paper is the mediator Matsumoto's mechanism requires is an ExaFuse hypothesis. Acceptance of the formalism is not evidence for the application.

In 1989 Takaaki Matsumoto, a nuclear engineer at Hokkaido University, published a three-page note in Fusion Technology proposing that small clusters of deuterons inside a loaded metal could undergo many-body reactions.2 Over the following decade the proposal grew into something far stranger. By 1995 he was describing a process he called electro-nuclear collapse, in which a hydrogen cluster is squeezed by an attractive interaction that behaves like gravity but is enormously stronger, and by 1998 he had presented that mechanism as the core of his model at ICCF-7.34

The reception was predictable. Matsumoto reported micro-craters, ring marks on nuclear emulsions and traces he interpreted as tiny black holes and white holes.5 Those interpretations have not been independently confirmed, except for S.V Adamenko's group who used relativistic vacuum diodes to generate transmuted products with masses in excess of 1000 amu. The detailed experimental account is found in Controlled Nucleosynthesis. This note takes no position on them except to point out that both driven, condensed matter systems require a large deviation from standard quantum field theory. That deviation from standard QFT is the subject here.

Matsumoto needed an attractive, short-ranged interaction between composite objects that behaves like gravity in its tensor structure but is not gravity in its strength. He never had a formalism for it. For thirty years nobody did. The purpose of this note is to report that one now exists in the peer-reviewed literature, and to be precise about what that does and does not settle.

1. What was accepted

The paper is an effective field theory for a collective excitation of the electromagnetic vacuum.1 Its starting point is a technical observation about how we normally describe light. The standard treatment of electromagnetism keeps the part of the field that escapes to infinity and discards the part that stays bound to its source. That discarding is legitimate for scattering calculations and is exact for the quantities those calculations predict. It is not, however, a statement that the discarded structure does not exist.

The paper retains it. What survives is a medium: a structured vacuum with its own rest frame, its own defects, and its own collective oscillations. The paper then asks what excitations that medium supports, and derives the answer channel by channel. Most channels carry nothing that propagates. One does. It carries two polarisations and transforms as a spin-2 object, which is to say it has the same tensor character as a gravitational wave.

The central object

The paper calls this excitation the dyality plasmon. It is a collective mode of a structured vacuum, in the same sense that a sound wave is a collective mode of air and a plasmon is a collective mode of an electron gas. It is explicitly not a graviton, it defines no spacetime geometry, and it does not modify general relativity.

2. Why a journal of gravitation published it

The natural question is why a gravity journal took a paper about electromagnetism. The answer is that spin-2 is a structural property, not a proprietary one. Any field that carries two transverse-traceless polarisations has gravitational tensor structure, whether or not it has anything to do with spacetime. Condensed matter physics has understood this for decades: Grigory Volovik's work on superfluid helium-3 shows an entire emergent relativistic structure, with analogue gravitational and gauge fields, arising inside a non-relativistic liquid.8 The gauge structure the paper uses is older still, reducing to the two-potential formulation of electromagnetism that Daniel Zwanziger wrote down in 1971.6

What makes the present case worth a gravity journal's attention is that the medium in question is the vacuum rather than a liquid, and the analysis required confronting the hard questions that come with that: whether the mode respects causality, whether it can be evolved consistently in time, and how it relates to gravity without being gravity. Those occupy most of the paper, and they are the parts the referees pressed hardest on.

3. The force, and who feels it

The mode is massive, which means it has a finite range. Its dispersion relation is

\[ \omega^2 = c_h^2\left(\mathbf{k}^2 + m_h^2\right) \]

Here \(\omega\) is the frequency of the mode, \(\mathbf k\) its wavevector, \(m_h\) its gap, and \(c_h\) the velocity characterising the medium's response. The gap is what gives the force a finite reach: the interaction it mediates falls away beyond a coherence length \(1/m_h\), which under the conditions estimated in the paper is of order a centimetre in free vacuum.

The more consequential property is the selection rule. The mode couples to the quadrupole moment of whatever it is talking to, in the same way gravity couples to mass quadrupoles. For a composite object with spin, that coupling takes the schematic form

\[ \mathcal{L}_{\rm int} \supset g_Q\, h^{\rm TT}_{ij}\, Q_{ij}, \qquad Q_{ij} \sim S_i S_j - \tfrac13 \delta_{ij} s(s+1) \]

with \(h^{\rm TT}_{ij}\) the tensor mode, \(g_Q\) a coupling constant, \(S_i\) the spin operators of the composite object, \(s\) its total spin, and \(Q_{ij}\) its spin quadrupole. The quadrupole vanishes identically when \(s=\tfrac12\). A single electron therefore does not couple to this mode at all, and neither does a spin-singlet pair. A spin-triplet pair, with \(s=1\), does.

This is a strong statement and it cuts both ways. It explains why such a force could have gone unnoticed in ordinary matter, since the overwhelming majority of electronic states are dark to it. It also means the force cannot be invoked freely: it acts on a specific and identifiable class of objects, and any proposal that calls on it must show those objects are present.

4. The gap Matsumoto left open

Return to electro-nuclear collapse. Matsumoto's requirement was an attractive interaction between hydrogen nuclei and their surrounding electrons, strong enough and short-ranged enough to compress a cluster past the point where Coulomb repulsion would normally stop it, with a tensor character he described as gravitational.3

Stated that way, the proposal has a clear shape and an obvious problem. Ordinary gravity is some thirty-six orders of magnitude too weak between two protons to matter at nuclear scales. Invoking gravity directly is therefore not an option, and invoking something gravity-like without saying what it is leaves no way to calculate anything, predict anything, or be wrong about anything. That is the state the proposal has remained in.

What the published formalism supplies is not a demonstration that this mechanism occurs. It supplies the missing middle term. There is now a specific, peer-reviewed candidate for a spin-2 mediator that is not the graviton, with a stated dispersion relation, a stated range, a stated selection rule for what it couples to, and a stated experimental signature. A proposal of Matsumoto's type can now be written down in equations rather than in words.

Whether it then survives contact with those equations is a separate question, and an open one. This is the ordinary situation in physics: having a formalism is what lets a speculative mechanism become a calculation that can fail.

5. The honest ledger

The paper is unusually explicit about its own epistemic structure. It carries a table classifying every principal statement as a definition, a derived result, or a conditional claim resting on a named hypothesis. That table is the most useful page in it for anyone building on the work, and it is worth stating plainly what sits in each column.

Derived: the channel structure, meaning that only the tensor branch propagates and the others are empty or belong to separate branches. The causal analysis, which establishes that nothing in the construction transmits a signal faster than light, even though the medium's internal response velocity exceeds c in the same way a refractive index can. That analysis had to accommodate a known subtlety: a transverse-traceless variable is spatially nonlocal and is not by itself an observable, even in ordinary general relativity.7 The existence of the collective pole, derived within a specified response model whose assumptions are stated.

Conditional: the numbers. The velocity, the gap, the coherence length and the coupling all depend on a constitutive assumption about how the medium's stiffness and inertia are apportioned, and on an ultraviolet completion the paper describes as schematic rather than constructed. These are benchmarks, not predictions, and the paper says so in the abstract.

The referees pushed hard on exactly this boundary across two rounds, and the final revision moved several statements from the first column to the second.

6. The dark medium

This brings us to what we think is now the central problem, and the reason this note exists.

The mode does not propagate through empty space. It propagates through a structured medium whose constituents are topological defects: points where the retained near-field structure winds around itself and cannot be smoothed away. The paper establishes that these defects carry a quantised charge, that they organise into a connected web, and that their charge survives even when the dynamics around them becomes chaotic. The picture of that web is an old one in nonlinear dynamics, going back to Michael Berry's lectures on regular and irregular motion.9 The last of these took a substantial revision to state correctly, and the published version derives it from the fact that a topological winding cannot be removed by a dynamical event.

We call this the dark medium, and the term needs immediate qualification.

A necessary disambiguation

We do not identify the dark medium with dark matter. It is not a proposal about galactic rotation curves or cosmological mass budgets. It is called dark because its constituents are source-bound and non-radiating: they are invisible to scattering experiments by construction, since the standard radiative description discards precisely the structure they are made of.

Almost nothing is known about this medium quantitatively. The paper fixes its kinematics, which is to say what kind of object a defect is, what charge it carries, how the charge is conserved, and what collective mode the ensemble supports. It does not fix the properties that would matter for any application: the defect density in ordinary matter, how that density responds to strain, temperature, electromagnetic drive or chemical environment, how the web reorganises near a lattice defect or a crack tip, or how strongly it couples to real condensed-matter systems.

Those are the characteristics that determine whether a tensor-mediated mechanism is irrelevant at nuclear scales or decisive there, and they are presently unmeasured.

7. What follows from having a framework

The practical change is that these questions are now well-posed. Before, one could only ask whether a gravity-like force exists, which is not a question any experiment can answer. Now one can ask for a number, and know what would count as an answer.

Three programmes follow, and they are separable. Each can be pursued without committing to the others, and each produces a result whether the CMNS application is correct or not.

The first is direct detection of the mode in vacuum. The paper specifies this and the design is unusual, because the medium has a rest frame and the Earth moves through it. The predicted signature is therefore a daily modulation of a resonance as the apparatus rotates, with a fractional depth fixed by the ratio of the laboratory velocity to the mode velocity, independent of the coupling strength. A null result at that depth constrains the rest frame directly, in the manner of the classical ether-drift experiments. This is a difficult measurement in a difficult frequency band and is a proposal rather than a built apparatus.

The second is characterisation of the defect web in condensed matter. The paper gives the kinematic framework; what is missing is the material physics. This is where the connection to our earlier guide on coherent charge clusters lies, since the objects that couple to the mode are spin-triplet pairs, and the question of whether such pairs exist in a hydrogen-loaded metal defect is independently interesting and independently testable.11 It also intersects Edmund Storms' nuclear-active-environment framework, which locates the action at nanoscale defects on quite different grounds.10

The third is the calculation Matsumoto could not do. With a mediator specified, one can compute the actual tensor-mediated interaction between hydrogen nuclei and surrounding paired electrons, at a stated defect density, and determine whether it is anywhere near sufficient. We regard this as the decisive test of the application, and we expect it to be demanding, because an interaction strong enough to do what Matsumoto needs is also strong enough to have left traces elsewhere.

8. A practical map

Now establishedA spin-2 collective mode of the unreduced electromagnetic near-field sector admits a consistent effective field theory, with a derived channel structure and a causal analysis, published in a peer-reviewed gravitation journal.
Now establishedThe mode couples to spin quadrupoles. Single electrons and singlet pairs are dark to it; triplet pairs are not.
Conditional in the paperThe velocity, gap, coherence length and coupling, which rest on a stated constitutive assumption and a schematic ultraviolet completion.
Contested CMNS claimMatsumoto's electro-nuclear collapse, and the observational reports accompanying it. Not independently confirmed; reported here as his claims.
ExaFuse hypothesisThat the mode described in the paper is a candidate for the gravitational-analogue interaction Matsumoto's mechanism requires.
Open and unmeasuredEvery quantitative property of the defect web in real matter: density, response to strain and drive, behaviour near crack tips, coupling strength to condensed-matter systems.

9. What acceptance means

It means the formalism survived sustained adversarial reading by referees who were not sympathetic to begin with and who required two rounds of revision. It means the internal consistency of the construction, its causal structure and its canonical structure have been checked by people whose job was to find the holes. It means the conditional claims are labelled as conditional in the published text.

It does not mean the mechanism occurs in nature. It does not mean anomalous heat in loaded metals has an explanation. It does not confirm a single observational report in the CMNS literature, Matsumoto's included. Peer review establishes that an argument is sound given its assumptions, which is a smaller thing than being right and a necessary precondition for finding out.

What has changed is the kind of conversation that is now possible. A proposal that existed for thirty years as a description in words can be written as a Lagrangian with a stated range, a stated selection rule, and a falsifiable signature. If it is wrong, it can now be shown to be wrong, which it could not be before.

The next step is not another paper about the formalism. It is measurement of the medium the formalism describes.

References and starting points

  1. C. Scott, Spin-2 Sector of the Dyality-Ordered Electromagnetic Vacuum: An Effective Field Theory, accepted for publication in Classical and Quantum Gravity (2026). doi:10.1088/1361-6382/aeb594 . Paper and supplementary calculations available on request.
  2. T. Matsumoto, “‘Nattoh’ Model for Cold Fusion,” Fusion Technology 16, 532–534 (1989), doi:10.13182/FST89-A29119.
  3. T. Matsumoto, “Mechanisms of Cold Fusion: Comprehensive Explanations by the Nattoh Model,” Memoirs of the Faculty of Engineering, Hokkaido University 19(2), 201–224 (1995), Hokkaido University repository.
  4. T. Matsumoto, “Mechanisms of Electro-Nuclear Collapse: Comprehensive Explanation by the Nattoh Model,” distributed at ICCF-7, Vancouver (1998).
  5. T. Matsumoto, “Experiments of One-Point Cold Fusion,” Fusion Technology 24(3) (1993), OSTI record. The interpretations offered in this and related papers have not been independently confirmed.
  6. D. Zwanziger, “Local Lagrangian quantum field theory of electric and magnetic charges,” Physical Review D 3, 880–891 (1971), doi:10.1103/PhysRevD.3.880. The two-potential formalism the paper's gauge structure reduces to.
  7. Y.-Z. Chu and Y.-W. Liu, “The transverse-traceless spin-2 gravitational wave cannot be a standalone observable because it is acausal,” Classical and Quantum Gravity 37, 055001 (2020), doi:10.1088/1361-6382/ab60b6. The result that shaped the paper's treatment of causality.
  8. G. E. Volovik, The Universe in a Helium Droplet, Oxford University Press (2003). The standard reference for emergent relativistic and gravitational structure inside a condensed-matter system.
  9. M. V. Berry, “Regular and irregular motion,” in Topics in Nonlinear Dynamics, AIP Conference Proceedings 46, 16–120 (1978), doi:10.1063/1.31417. The source of the phase-space picture used in the paper's appendix on the defect web.
  10. E. Storms, “A New and Improved Understanding of Cold Fusion Based on the Observed Behavior,” Journal of Condensed Matter Nuclear Science 41, 1–18 (2026), doi:10.70923/001c.162973. An alternative CMNS framework for the nuclear-active environment.
  11. ExaFuse, What Is a Coherent Charge Cluster?, concept guide (August 2026). Treats the spin-triplet pairing question and the selection rule discussed in Section 3 above.