SSF Primers: Solid State Fusion & Numerical Methods

Introduction: The Empirical Center

An electrochemical cell sits in a temperature-controlled bath. Current goes in, the cell warms, and a calibration curve converts that warming into a number for heat output. Run the books: electrical power in, heat out, ordinary chemistry accounted for. In most cells the two sides balance. In a stubborn handful, the heat side comes out a little high. A few percent, sometimes a few hundred milliwatts, intermittent, almost never on command.

That small surplus is the empirical center of solid-state fusion (SSF), sometimes called cold fusion. Strip away three decades of argument and what sits at the bottom is a single quantity: an excess power that is either real or an artifact of the measurement. The nuclear interpretations, proposed mechanisms, and theoretical models all depend on whether that one number survives scrutiny. The strongest published case that it does not survive scrutiny is a statistics argument, not a physics argument — a claim that the calibration cannot be trusted to better precision than the size of the reported effect.¹

Section I — A Measurement Is a Number and an Uncertainty

The heat surplus is a claim about a small difference, and a claim is only as good as its error budget.

Every physics lab course establishes but rarely presses the key point: a measurement is a number together with a statement of how far that number can be trusted. An international framework, the Guide to the Expression of Uncertainty in Measurement (GUM), formalizes the procedure. Write down the model relating what you measure to what you want to know. Identify every input: temperatures, voltages, calibration coefficients, correction factors. Attach a standard uncertainty to each. Propagate them into a combined uncertainty on the final result. When the model is too nonlinear for first-order propagation, you sample the input distributions and push them through by Monte Carlo.²

That propagation is the whole game. "Is there excess heat?" stated carefully becomes: "Is the output power separated from the input power by more than the combined standard uncertainty of the difference?" A claimed excess of a few hundred milliwatts on a few watts of input is a claim that the books balance to better than a few percent, with every source of drift included. Whether that claim holds is settled by the error budget, not by any theory of what might be generating the heat.

The calibration is the weakest link in that budget. A calorimeter does not measure heat directly. It measures temperature at a handful of points and infers the heat source through a model of how thermal energy moves through the apparatus. Reconstructing an internal source from boundary temperature readings is a classic inverse problem, and this particular variety (inverse heat conduction) is formally ill-posed: small errors in the temperature data can produce large errors in the inferred source unless the inversion is stabilized through regularization.³ Most SSF experiments sidestep the full inversion with simpler calorimeter designs and a calibration curve, but the lesson carries. The reported heat output is a derived quantity, sensitive to the calibration model, and its uncertainty propagates through that model. Treating the calorimeter as an inference engine — with an explicit forward model, stated assumptions, and tracked parameter uncertainties — is the only way to see clearly where a claimed excess stands on solid ground and where it leans on an unexamined assumption.

Section II — Is the Calibration the Culprit?

The strongest published objection names a specific mechanism and a specific magnitude.

The most prominent statistical objection to SSF excess heat names a concrete mechanism. During a long electrochemical run, the location where heat is generated inside the cell can shift relative to the geometry assumed during calibration. Since heat originating from different internal locations propagates differently to the thermometers, a calibration that was accurate at the start of the run can develop an error of a few percent over hours. That error would appear in the data as an apparent excess.¹

The response from the other side is equally concrete: the proposed drift is larger than independent recalibrations actually allow, and it would leave specific fingerprints in the temperature residuals that are not observed.⁴ This exchange has the right form. A candidate systematic error is advanced with a magnitude; it is met with a magnitude. Whether the calibration actually drifts that much is, in principle, testable: run the calibration protocol with the heat source placed at different geometries inside the cell, and measure the spread in the calibration constant directly. That single experiment would adjudicate the main objection. The fact that it has not been done to general satisfaction, after more than thirty years, says something about how this field has operated. It also makes plain why measurement science, rather than nuclear theory, is the right discipline to engage here.

Section III — Three Places Where Numerical Methods Do Real Work

Calibration is one source of trouble. There are others, and each has a mathematical address.

Sporadic Signals and the Look-Elsewhere Effect

Excess-heat events, when they appear, tend to be sporadic: bursts that come and go rather than a steady offset. Intermittent, low-signal data are exactly where naive significance testing misleads. If you monitor many cells across many time windows and report the most striking excursion you find, the probability of finding something by chance somewhere in the full dataset is much larger than the p-value at any single cell and window would imply. High-energy physics formalized this as the trial factor, also called the look-elsewhere effect, and built correction procedures for precisely this kind of large-scan experiment.⁵

The fix is pre-registration: before seeing the data, commit in writing to which cells, which time windows, and which threshold will count as a positive. Then the computed probability is actually the right one. Applied to SSF, this is cheap methodological bookkeeping with significant epistemic payoff. It converts a debatable "striking excursion" into a test with a computable false-positive rate. Its routine absence is part of why outside scientists discount the reported bursts.

Loading as a Coupled Partial Differential Equation

The experimental variable that most consistently separates active cells from inactive ones is the loading ratio: the number of deuterium atoms absorbed per palladium atom in the cathode. The threshold claimed for any chance of observing an effect is high, with the ratio pushed toward one deuterium per palladium. Reaching that ratio, and knowing that you have reached it, is a transport problem before it is a physics problem.

Deuterium in palladium obeys a nonlinear diffusion equation. Both the diffusivity (how fast deuterium moves through the lattice) and the solubility (how much the material can hold) depend on the local concentration. On top of this, absorbed deuterium swells the lattice and generates mechanical stress, which feeds back into the diffusivity and can drive cracking. The governing equations are a coupled PDE system: diffusion and elasticity, linked through concentration-dependent constitutive relations, on a material whose properties near full loading are not fully characterized.⁶ For a reader comfortable with finite-element or finite-difference simulations, this is a recognizable class of problem, with the added complication that the material parameters in the extreme-loading regime carry substantial uncertainty.

Why does the simulation matter? A cell that nominally reached a loading ratio of 0.95 may have sustained that value briefly at the surface while never achieving it at the core. Without a model of the loading dynamics, a reported ratio is a point measurement on a nonuniform, time-varying field. The most careful recent attempt to reproduce SSF, a multi-institution program reported in Nature in 2019, pushed loading as high as about 0.96 deuterium atoms per palladium and found no excess heat across hundreds of samples.⁷ The program concluded, nonetheless, that reliably reaching and sustaining extreme loading is genuinely hard and that the high-loading region of parameter space remains thinly explored. That conclusion is the PDE speaking: the target state is hard to certify, and a null is only as strong as the loading it can demonstrate.

Deciding What to Run Next

SSF experiments are slow, expensive, and individually uncertain. That is exactly the regime where the choice of what to run next carries the most information per experiment. Optimal experimental design, the decision-theoretic framework for choosing settings that maximize expected information gain, was built for this situation.⁸ Applied across loading ratio, current density, and material batch, it would let a program map the shape of the parameter space from far fewer cells than a one-variable-at-a-time sweep, in a form an outside analyst could audit afterward. The experiment is expensive; the design step is cheap relative to what it buys.

Section IV — What the Missing Signatures Say

A physics fact with direct consequences for the measurement prior.

A nuclear heat source of the reported magnitude should produce characteristic by-products. Evaluated nuclear data (the ENDF/B database) fix the expected rates: a deuteron-deuteron source running fast enough to warm a calorimeter by milliwatts should also throw off a large neutron flux, along with gamma rays and energetic charged particles.¹⁰ In SSF cells, essentially none of these appear in quantitative proportion to the energy. Only helium has been reported in correlation with heat, in a single-group result that remains contested on calorimetry and atmospheric-contamination grounds.⁹

The absence of radiation is usually treated as a puzzle for nuclear theory: whatever mechanism generates the heat must somehow suppress the expected by-products. But it also feeds back into the measurement question. A heat reading of a given size, with no commensurate radiation, is on any reasonable prior more likely to be a measurement artifact than the same reading arriving alongside the expected nuclear signature. The missing by-products lower the probability that the heat is real, and a complete error budget should carry that prior rather than treat it as another field's problem. The line between "is the heat real" and "is it nuclear" is cleaner in principle than in practice.

One question scientists have not yet answered is whether any physical mechanism could route significant nuclear energy into lattice heat while suppressing all other by-products. Addressing it requires theoretical modeling of energy transfer in the relevant quantum systems, well outside what measurement alone can say, and outside the scope of this piece.

Section V — Decidable, but Not Symmetrically

A clean positive — a reproducible excess that clears a full error budget under blinded analysis at a loading ratio verified rather than assumed — would be close to decisive. It would be very difficult to explain away with calibration-drift arguments or look-elsewhere corrections if those analyses had been carried out prospectively and in public.

A clean negative is weaker by definition, and the reason comes back to loading. The 2019 multi-institution program was careful and well-resourced, but the program itself concluded that the decisive experimental conditions — sustained extreme loading — were never reliably demonstrated even in its own cells. A null is only as strong as the loading it can certify. That asymmetry routes directly back to the coupled PDE: what keeps a null from closing the question is a control problem. Did you reach the target ratio, and can you prove it? Making that provable is, again, work for numerical methods.

The 2004 Department of Energy review divided eighteen independent reviewers roughly evenly on whether evidence for excess power was compelling. On the further question of nuclear reactions, about two-thirds found the evidence unconvincing, one found it compelling, and the remainder were partially convinced.¹¹ An even split on the heat with a majority against the nuclear reading is a fair description of evidence that is suggestive on the measurement and thin on the mechanism, underpowered on both counts.

A student coming to this from mathematical physics or simulation would find several tractable open problems here on their own technical merits. The loading dynamics in extreme-concentration palladium hydride, near the phase transition where material behavior is least characterized, is a well-posed coupled PDE problem with genuine gaps in the validated parameter set. A full Monte Carlo uncertainty budget for a long electrochemical run, propagated from thermometry to reported excess, has never been published for the highest-profile experiments. Whether the heat–helium correlation survives careful contamination controls is an open experimental question even the field's proponents acknowledge. None of these problems requires a prior view on whether the effect is real. They are good problems on their own, and they happen to sit exactly where a real result, if there is one, would first need to be demonstrated.

Editorial note: This article presents a scholarly synthesis of SSF's relationship to numerical methods and applied mathematics. The underlying nuclear claims of SSF/LENR remain scientifically contested. Evidence claims are tiered as established, contested, or reported-but-unconfirmed as noted inline. Readers are directed to primary experimental literature for empirical evaluation.

Notes & References

Kirk L. Shanahan, "A Systematic Error in Mass Flow Calorimetry Demonstrated," Thermochimica Acta 382 (2002): 95–101. Proposes the "calibration constant shift" mechanism, by which an unrecognized change in the calibration constant during a run can produce an apparent excess of the size reported in heavy-water electrolysis cells. ↩
JCGM 100:2008, Evaluation of Measurement Data — Guide to the Expression of Uncertainty in Measurement (GUM) (Joint Committee for Guides in Metrology, 2008); and JCGM 101:2008, Supplement 1 — Propagation of Distributions Using a Monte Carlo Method (JCGM, 2008). The international reference framework for stating and propagating measurement uncertainty. ↩
James V. Beck, Ben Blackwell, and Charles R. St. Clair Jr., Inverse Heat Conduction: Ill-Posed Problems (New York: Wiley-Interscience, 1985); 2nd ed., Wiley, 2023. The standard treatment of reconstructing surface heat flux from interior temperature measurements and of the regularization needed because the problem is ill-posed. ↩
Edmund Storms, "Comment on Papers by K. Shanahan That Propose to Explain Anomalous Heat Generated by Cold Fusion," Thermochimica Acta 441, no. 2 (2006); and Kirk L. Shanahan, "Reply to 'Comment on Papers by K. Shanahan…,'" Thermochimica Acta 441, no. 2 (2006): 210–214, doi:10.1016/j.tca.2005.11.029. The published exchange over whether the calibration-shift mechanism can quantitatively account for reported excess heat. ↩
Eilam Gross and Ofer Vitells, "Trial Factors for the Look Elsewhere Effect in High Energy Physics," European Physical Journal C 70 (2010): 525–530, doi:10.1140/epjc/s10052-010-1470-8. Defines the trial factor and gives a practical estimation method for correcting p-values in large-scan experiments. ↩
Yuh Fukai, The Metal-Hydrogen System: Basic Bulk Properties, 2nd ed., Springer Series in Materials Science 21 (Berlin: Springer, 2005), doi:10.1007/3-540-28883-X. Covers hydrogen diffusion, solubility, and the phase behavior of palladium hydride, including the concentration dependence and lattice-stress coupling that govern loading. ↩
Curtis P. Berlinguette et al., "Revisiting the Cold Case of Cold Fusion," Nature 570 (2019): 45–51, doi:10.1038/s41586-019-1256-6. A multi-institution program that did not reproduce excess heat but identified the extreme-loading materials regime as genuinely difficult to reach and thinly explored. ↩
Kathryn Chaloner and Isabella Verdinelli, "Bayesian Experimental Design: A Review," Statistical Science 10, no. 3 (1995): 273–304, doi:10.1214/ss/1177009939. The standard review of decision-theoretic optimal experimental design for choosing settings that maximize expected information gain. ↩
Melvin H. Miles, R. A. Hollins, B. F. Bush, et al., "Correlation of Excess Power and Helium Production During D₂O and H₂O Electrolysis Using Palladium Cathodes," Journal of Electroanalytical Chemistry 346 (1993): 99–117. Reports the heat–helium correlation and an energy-per-helium ratio roughly consistent with deuterium fusing to helium-4; a single-group result, criticized on calorimetry and atmospheric-helium-contamination grounds, and presented here as contested. ↩
For the branching-ratio expectation that a deuteron–deuteron source running fast enough to warm a calorimeter should produce a large neutron flux, see D. A. Brown et al., "ENDF/B-VIII.0," Nuclear Data Sheets 148 (2018): 1–142. The "missing neutrons" are treated at length in the companion Nuclear Physics primer in this series. ↩
U.S. Department of Energy, Report of the Review of Low Energy Nuclear Reactions (Washington, DC: DOE, December 2004). Of the eighteen reviewers, opinion divided roughly evenly on whether the evidence for excess power was compelling; on whether nuclear reactions were occurring, about two-thirds found the evidence unconvincing, one found it compelling, and the remainder were somewhat convinced. ↩

Introduction: The Measurement at the Center of the Dispute

Picture the measurement at the center of the whole dispute. A small electrolytic cell sits in a temperature-controlled bath. Current goes in, the cell warms, and a calibration curve turns that warming into a figure for the power coming out. Then you run the books: electrical power in, heat out, ordinary chemistry accounted for. In most cells the two sides balance. In a stubborn few, the heat side comes out a little high. Not by orders of magnitude. By a few percent, sometimes a few hundred milliwatts, sometimes more, often intermittently, almost never on command.

That surplus is the entire empirical core of solid-state fusion (SSF). Strip away three decades of argument and what is left is a number with an error bar: an excess power that is either real or an artifact. Everything else — the helium and tritium and the nuclear interpretations — hangs downstream of whether that one measurement survives scrutiny.

Here is the part that should catch an applied mathematician's attention. The strongest published case that the heat reading itself is an artifact is not a physics argument at all. It is a statistics argument. One critic has argued, in the peer-reviewed calorimetry literature, that a few-percent change in the calibration constant — the kind that can creep in when the place heat is generated inside a cell shifts over the hours of a run — is by itself enough to reproduce an apparent excess of the size reported.¹ That proposal is a single-author hypothesis, and it is contested: its critics say it has not been shown to occur at the magnitude claimed, and that it does not explain why the effect tracks helium or shows up only at high loading. Notice what kind of dispute that is. It is a quantitative claim about a measurement, met by quantitative objections, and it reduces to one sharp question. Can the calibration be trusted to better than the size of the claimed effect? That is a question about uncertainty quantification, and the field equipped to answer it is the one named at the top of this page.

This article makes a narrow, deliberate claim. Applied mathematics cannot tell you whether cold fusion is real, and it is not being asked to. What it can do is settle the prior question — the one that actually sits on the table: whether there is a signal here to explain at all. The calibration objection is the strongest case that the reading specifically is an artifact, but it is not the strongest case against cold fusion. For most physicists that distinction goes to the near-absence of the nuclear by-products a real nuclear heat source should throw off, and this article does not bracket that objection away. It returns to it in Section II, because it bears on the measurement question and not only on the mechanism. The prior question can be settled with tools built for exactly this kind of small-signal, drifting-baseline, hard-to-repeat measurement. The anomaly is contested. It is not undecidable.

Section I — Where the Home Field Stands

A measured quantity means nothing without the budget of everything that could be wrong with it.

Begin with what the home field already knows, on its own ground. A measurement is not a number. It is a number together with a statement of how far that number can be trusted, and there is an international convention for writing the statement down. The Guide to the Expression of Uncertainty in Measurement sets out the procedure plainly: write the model relating what you measure to what you want to know, identify every input quantity, attach a standard uncertainty to each, and propagate them into a combined uncertainty on the result. Where the model is too nonlinear for the textbook propagation formula, you sample the input distributions and push them through by Monte Carlo.²

This is patient, unglamorous work, and it is the whole game. "Is there excess heat?" stated properly becomes "Is the output power separated from the input power by more than the combined standard uncertainty of the difference?" A reported excess of a few hundred milliwatts on a few watts of input is, in plain terms, a claim that the books balance to better than a few percent with every source of drift included. Whether that claim holds is decided by the error budget, not by any theory of what might be generating the heat. The home field's first contribution is to insist the budget be written out in full, and then to check the arithmetic.

Section II — What the Mathematics Can Actually Decide

Five places where a numerical method does real work on a real question.

Calorimetry is an inference problem

A calorimeter never measures heat. It measures temperature, at a few points, and infers a heat source through a model of how heat moves through the apparatus. Reconstructing an internal source from boundary temperatures is a well-studied and notoriously delicate class of problem — the inverse heat conduction problem — and it is formally ill-posed: small errors in the temperature data can swing the inferred source wildly unless the inversion is regularized.³ Most SSF calorimetry sidesteps the full inversion with simpler closed or flow designs and a calibration curve, but the underlying lesson carries. The output power is a derived quantity, sensitive to the model and to the calibration, and its uncertainty has to be propagated through that model rather than read off a thermometer. Treating the calorimeter as an inference engine, with a stated forward model and stated priors, is the cleanest way to see exactly where a claimed excess holds and where it leans on an assumption.

The calibration dispute is a worked example

Return to the calibration-constant objection. It deserves to be taken seriously precisely because it is quantitative. It does not say the experiments are sloppy. It names a mechanism by which a specific, bounded error reproduces a specific, bounded effect.¹ The replies are quantitative in turn: that the proposed shift is larger than independent recalibrations actually allow, and that it would leave fingerprints in the residuals that are not seen.⁴ That exchange — a candidate systematic error advanced with a number and met with a number — is the form every excess-heat result ought to be made to survive. Applied math supplies the form. A calibration is a fitted model carrying its own parameter uncertainties; those uncertainties propagate into the excess; the question is whether the excess clears them. Most reported results have never been put through that wringer in public. Doing it, result by result, is tedious and decisive.

Sporadic signals need honest statistics

Several SSF observables refuse to hold still. Excess-heat bursts come and go, and neutron and tritium reports tend to be intermittent and close to background. Intermittent, low signal-to-noise data are exactly where naive significance testing misleads, and exactly where the home field has sharp instruments. Scan many cells and many time windows, then report the most striking excursion, and the probability of a chance excursion somewhere is far larger than the probability at any fixed point. High-energy physics formalized this as the trial factor, or look-elsewhere effect, and built procedures to correct for it.⁵ A program reporting rare, hunted-for bursts needs that correction, and its routine absence is one reason outside scientists discount the reports. The remedy is cheap. Pre-register which cells, which windows, and which thresholds will count as a positive, fix them before the data are seen, and a look-elsewhere problem becomes a clean test.

Loading is a partial differential equation

The one experimental knob that most consistently separates active cells from dead ones is loading: how many deuterium atoms occupy the metal per host atom. The reported threshold for any chance of an effect is high, with the deuterium-to-palladium ratio pushed toward one-to-one. Reaching that, and knowing you have reached it, is a transport problem before it is anything else. Hydrogen in palladium follows a nonlinear diffusion equation coupled to the stress the absorbed hydrogen itself generates, with both the diffusivity and the solubility depending on the local concentration.⁶ Predicting where a cathode will load, and where it will crack and shed its hydrogen, is finite-element work of an ordinary kind on a material that behaves in an unusual way. It matters because the most careful modern attempt to reproduce the effect pushed loading as high as about 0.96 deuterium atoms per palladium — close to the theoretical limit of one — found no excess heat across hundreds of samples, and still concluded that reaching and holding extreme loading reliably is genuinely hard and that the high-loading region of the parameter space remains thinly explored.⁷ A field that cannot reliably hit and certify its own most important variable has a numerics problem standing in front of any physics problem. Hold onto that result, because it decides what a null is worth.

Few expensive runs reward good design

SSF experiments are slow, costly, and individually uncertain, which is the regime where the choice of what to run next carries the most information. Optimal experimental design — the decision-theoretic machinery for choosing settings that maximize expected information gain — was built for precisely this situation.⁸ Applied across loading ratio, current density, and material batch, it would let a program map the shape of the parameter space from far fewer cells than a one-variable-at-a-time march, and it would do so in a way an outsider could audit afterward.

Through all five, one distinction has to stay sharp, because keeping it sharp is what makes the case honest. Deciding to high confidence that an excess power is real is a measurement result. Deciding what produces it is a separate question, and a nuclear one. Applied math can carry the first the whole distance and has nothing privileged to say about the second. The most-cited bridge between them — that excess power has been reported to track helium-4 production near the roughly 24 MeV per atom expected if deuterium were fusing to helium — is itself contested on calorimetry and contamination grounds and belongs in the second category, not the first.⁹ There is one place the separation leaks, though, and honesty means naming it. A heat source of nuclear magnitude should be accompanied by nuclear by-products: neutrons, gamma rays, energetic charged particles. In these cells essentially none are seen in quantitative correlation with the energy, only helium, and that without the kinetic signature a fusion event should impart.¹⁰ That absence is usually filed under mechanism, but it also moves the measurement prior. A watt-scale anomalous heat reading with no commensurate radiation is, on any reasonable prior, more likely to be a measurement error than the same reading arriving with the expected glare. So the missing signatures pull down the prior that the heat is real, and a complete error budget has to carry that prior rather than wave it off as another field's business. The split between "is the heat real" and "is it nuclear" is clean in principle and a little leaky in practice. Conflating the two is still the move that has let boosters and skeptics talk past the same data for thirty years.

Section III — The Return Traffic

The bridge carries cargo both ways.

The exchange is not charity. A calibration constant that drifts over a long electrochemical run is a non-stationary systematic of a sort metrology usually engineers away rather than models. Here it cannot be engineered away, so it has to be modeled, and modeling it well is a contribution to measurement science on its own. The intermittent, hunted-for signal is a live testbed for blinded and pre-registered analysis outside the well-funded arenas — particle physics and clinical trials — where those protocols matured. The loading-and-cracking behavior is a genuinely hard coupled diffusion-mechanics problem on a material whose response near full loading is not fully characterized. None of these needs SSF to be real to be worth the time. They are good problems on their own terms, and they happen to sit at the spot where a real result, if there is one, would first show itself.

There is a larger draw, the kind that pulls a discipline forward. A thirty-five-year-old dispute that has resisted resolution, in which the live question is not "what is the mechanism" but "is the signal real," is the kind of problem careful inference exists to close. Closing it would demonstrate something about what rigorous uncertainty quantification can do that few tidier problems could match.

Section IV — The Part That Is Actually About Rigor

Reproducibility is not a failing to confess. It is a property of a protocol, and protocols can be fixed.

The honest description of SSF's record is that the effect seems present in some hands and absent in others, and that this irreproducibility is the field's central obstacle. The instinct from outside is to read irreproducibility as proof of nothing there. Sometimes it is exactly that. But irreproducibility is also a measurable property of a procedure, and metascience has a standard prescription for it: standardize the measurement so results compare across labs, blind the analysis so the person computing the excess does not know which runs are supposed to work, pre-register the criteria for a positive, publish the raw data alongside the analysis code, and replicate across independent groups before anyone claims anything.

This is the same prescription psychology and parts of biomedicine adopted after their own replication crises, and it works for the same reason: it removes the small freedoms through which an experimenter's expectations leak into the result. The last formal federal look, a 2004 Department of Energy review, split almost evenly on whether the evidence for excess power was convincing. On the further question of whether nuclear reactions were occurring, the same panel tilted the other way: of its eighteen reviewers, about two-thirds found that evidence unconvincing, one found it compelling, and the rest were somewhat convinced.¹¹ An even split on the heat with a clear majority against the nuclear reading is neither a refutation nor an endorsement. It is a fair description of evidence that is suggestive on the measurement, weak on the mechanism, and underpowered on both. The applied-math contribution to that picture is not a verdict. It is the apparatus that would let the next round of experiments be powered, comparable, and hard to fool, so that a future panel could split less.

Section V — A Decidable Question

The reflex that files solid-state fusion under settled error has skipped a step. A settled error, if that is what this is, would be a measurement error, and a measurement error is something you demonstrate with an error budget, not something you assert. The reflexive belief in a real effect has skipped the very same step from the other direction. Both positions are claims about a number and its uncertainty, and neither has been carried to the standard the number deserves.

That standard is within reach, though it is not symmetric, and saying so plainly is part of being honest about it. A clean positive — a reproducible excess that clears a full error budget under blinding and pre-registration, at a loading shown to meet the proponents' own stated target — would be close to decisive. A clean null is worth only as much as the loading it can certify. The record since 2004 shows why. A careful, well-resourced null already exists — the 2019 program — and the field read it not as a verdict but as a sign that the decisive conditions were never reliably reached. That reading is fair only if the loading target is specified in advance and then demonstrably hit, and reaching and certifying it is the hard part. So the asymmetry routes straight back to the spine of this article. What keeps a null from closing the question is a problem of measurement and control: did you reach the target loading, and can you prove it?

An applied mathematician who took up the excess-heat measurement — the calibrated error budget, the blinded analysis, the loading verified rather than assumed — would not be endorsing a nuclear claim or signing on to a cause. They would be doing what the field is for, on a problem that has stayed open in large part because the people best equipped to close it have stayed away. A positive result could close the question. A null could close it too, but only once the loading has stopped being the thing in doubt, and making it stop is, once again, work for this field.

Notes

Kirk L. Shanahan, "A Systematic Error in Mass Flow Calorimetry Demonstrated," Thermochimica Acta 382 (2002): 95–101. Proposes the "calibration constant shift" mechanism, by which an unrecognized change in the calibration constant during a run can produce an apparent excess of the size reported in heavy-water electrolysis cells. ↩
JCGM 100:2008, Evaluation of Measurement Data — Guide to the Expression of Uncertainty in Measurement (GUM) (Joint Committee for Guides in Metrology, 2008), bipm.org; and JCGM 101:2008, Supplement 1 — Propagation of Distributions Using a Monte Carlo Method (JCGM, 2008). The international reference framework for stating and propagating measurement uncertainty. ↩
James V. Beck, Ben Blackwell, and Charles R. St. Clair Jr., Inverse Heat Conduction: Ill-Posed Problems (New York: Wiley-Interscience, 1985); 2nd ed., Wiley, 2023. The standard treatment of reconstructing surface heat flux from interior temperature measurements and of the regularization needed because the problem is ill-posed. ↩
Edmund Storms, "Comment on Papers by K. Shanahan That Propose to Explain Anomalous Heat Generated by Cold Fusion," Thermochimica Acta 441, no. 2 (2006); and Kirk L. Shanahan, "Reply to 'Comment on Papers by K. Shanahan…,'" Thermochimica Acta 441, no. 2 (2006): 210–214, doi:10.1016/j.tca.2005.11.029. The published back-and-forth over whether the calibration-shift mechanism can quantitatively account for reported excess heat; cited here as a model of how such a dispute is adjudicated, not as a settled outcome. ↩
Eilam Gross and Ofer Vitells, "Trial Factors for the Look Elsewhere Effect in High Energy Physics," European Physical Journal C 70 (2010): 525–530, doi:10.1140/epjc/s10052-010-1470-8. Defines the trial factor quantifying the increased chance of a spurious excess when many points are searched, and gives a practical estimation method. ↩
Yuh Fukai, The Metal-Hydrogen System: Basic Bulk Properties, 2nd ed., Springer Series in Materials Science 21 (Berlin: Springer, 2005), doi:10.1007/3-540-28883-X. Covers hydrogen diffusion, solubility, and the phase behavior of palladium hydride, including the concentration dependence and lattice-stress coupling that govern loading. ↩
Curtis P. Berlinguette et al., "Revisiting the Cold Case of Cold Fusion," Nature 570 (2019): 45–51, doi:10.1038/s41586-019-1256-6. A multi-institution program that did not reproduce excess heat but identified the extreme-loading materials regime as genuinely difficult to reach and thinly explored. ↩
Kathryn Chaloner and Isabella Verdinelli, "Bayesian Experimental Design: A Review," Statistical Science 10, no. 3 (1995): 273–304, doi:10.1214/ss/1177009939. The standard review of decision-theoretic optimal experimental design for choosing settings that maximize expected information gain. ↩
Melvin H. Miles, R. A. Hollins, B. F. Bush, et al., "Correlation of Excess Power and Helium Production During D₂O and H₂O Electrolysis Using Palladium Cathodes," Journal of Electroanalytical Chemistry 346 (1993): 99–117. Reports the heat–helium correlation and an energy-per-helium ratio roughly consistent with deuterium fusing to helium-4; a single-group result, criticized on calorimetry and atmospheric-helium-contamination grounds, and presented here as contested. ↩
For the branching-ratio expectation that a deuteron–deuteron source running fast enough to warm a calorimeter should produce a large neutron flux, see the evaluated nuclear data: D. A. Brown et al., "ENDF/B-VIII.0," Nuclear Data Sheets 148 (2018): 1–142. The 2004 DOE review material (note 11) records that in these cells essentially no energetic nuclear products are observed in quantitative correlation with the energy, only helium, and that without significant kinetic energy. The "missing neutrons" are treated at length in the companion Nuclear Physics primer in this series. ↩
U.S. Department of Energy, Report of the Review of Low Energy Nuclear Reactions (Washington, DC: DOE, December 2004). Of the eighteen reviewers, opinion divided roughly evenly on whether the evidence for excess power was compelling; on whether nuclear reactions were occurring, about two-thirds found the evidence unconvincing, one found it compelling, and the remainder were somewhat convinced. Atmospheric helium contamination was a stated concern. ↩

Solid State Fusion & Numerical Methods