Fusion Density Under Ambiguity.
Preserving useful density under unresolved fringe counts
A density-risk decision for two-color interferometer–polarimeter systems in magnetic-confinement fusion.
The numbers on this page come from a Gaussian model with assumed priors and noise. They demonstrate a mechanism; operational reliability is not claimed and nothing here has been measured on a machine.
An interferometer measuring the line-integrated electron density of a confined plasma sees a wrapped phase. A single wrapped observation does not determine the whole number of fringes accumulated since the start of the discharge; that integer has to be supplied from somewhere else. When refraction, pellets or disruptions interrupt the signal, the restored count can jump, and the density handed to plasma control inherits the error.
The correction approaches EN-003 reviews treat that as integer recovery: decide the count, then report the density that follows. But the control loop consumes the density, and in a two-color instrument the count and the density can part company. This note moves the decision onto the quantity that is actually used.
Engineering Note EN-003 · v1.0 · 2 October 2026 · CC BY 4.0
The decision, in one line
qt is a conservative bound on the probability that the density is wrong by more than a stated tolerance ε. The fringe count may still be unresolved when the density is released.
01 · Position, stated plainly
A decision rule for one diagnostic, not a claim about fusion.
EN-003 works within established measurement physics and leaves the burn, confinement, materials and magnets of fusion devices untouched. Its closest relatives come from satellite navigation, where integer least squares, integer-equivariant estimation and integer-aperture estimation have handled carrier-phase ambiguities for decades. What is proposed here is the placement of the acceptance criterion: on a tolerance for the density, rather than on the resolution of the count.
IS What the approach is
- A posterior over integer hypotheses and a marginalized density estimate, for two-color interferometer–polarimeter systems of the kind ITER's primary density-control diagnostic is designed around.
- A density risk with a conservative bound, and an availability rule that releases the density when the bound stays inside a chosen risk level.
- An accounting discipline: count accuracy, density accuracy and vibration-state accuracy reported separately, branch-induced displacement kept apart from continuous error, search bounds stated beside every rate.
- An evaluation protocol that others can run: replay, baselines at matched availability, named falsifiers, and a retirement criterion.
IS NOT What the approach is not
- Not a statement about fusion performance. The claims end at the density measurement. Plasma behaviour, confinement, burn, materials and magnets lie outside.
- Not a measured result. The numbers are model-conditional, computed from Gaussian priors and an assumed phase noise. Reliability on real records is exactly what the note asks others to test.
- Not demonstrated in real time. Feasibility at TIP rates and any benefit to density control remain open.
- Not a universal recipe. Dispersion interferometers and other topologies have their own lattices and need their own models.
Three principles shape the design
A wrapped observation does not determine the integer.
The likelihood of a wrapped measurement is invariant under integer shifts, so no single observation picks out a count. Every recovery supplies it from somewhere else: continuity from a known start, congruence between channels of different scale factor, a fiducial, or an external prior.
The decision belongs on the consumed quantity.
Some count errors move the density by a small fraction of a fringe while moving the inferred optical path by tens of micrometres. An unresolved count can then coexist with a trustworthy density, and a rule that waits discards usable measurements.
Reliability is a measured number.
A claim of reliability names its data, its baseline, its tolerance and the number at which it fails. Nothing on this page meets that standard yet; Section 06 states how it would be met.
02 · Substrate
What the approach consumes.
An instrument returns the wrapped phase, missing an unknown integer number of cycles. Four classes of assumption can restore it, and practical systems combine a prior with continuity tracking. Without noise, nearest-branch tracking recovers increments strictly below half a fringe in magnitude. At the limit, rising and falling become indistinguishable. Larger increments alias modulo one fringe: a positive increment between half a fringe and one fringe appears reversed.
Claim tags run through this page, as they do in the note. Hover any of them.
[D] definition ·
[A] assumption ·
[M] model-conditional ·
[N] computed number ·
[R] external result ·
[C] conjecture, with a stage attached.
Known-start continuity
Track the phase from a start inside a known band and assume the change between samples stays under half a cycle.
Two-rate congruence
Compare channels with different scale factors. The two-color pair is exactly this, and it is why the problem becomes two-dimensional.
Fiducial or absolute reference
An independent channel that bypasses the wrapped stream. Here, the Faraday polarimeter, whose angle stays far below a wrap.
External coarse prior
A second modality accurate to within half a wrap. Thomson scattering, other chords through a profile model, or closure at the pulse end.
The joint decision margin
A branch decision compares candidate solutions against priors. Each integer hypothesis fixes a solution; two hypotheses differ by a lattice vector δ, and with the covariance Σ of the decision residual their separation is a Mahalanobis distance.
For Gaussian errors and equal prior weights, two hypotheses are confused with probability ½ erfc(d / 2√2), and summing that over all competitors bounds the wrong-count probability. The familiar half-fringe rule is the special case of one unresolved branch with everything else fixed.
A competitor that differs mainly in a nuisance direction can stay unresolved while the target quantity stays accurate. That sentence is the whole argument of this note, and the next section shows the instrument in which it is true.
Risk over a discharge
For n repairs with wrong-repair probabilities pi, the union bound holds under arbitrary dependence. The independent form needs evidence of independence, which clustered jumps during a disruption contradict. For p = 10−3 and n = 50 the two give 5.0 × 10−2 and 4.9 × 10−2. A simulation with zero failures in n independent Bernoulli trials bounds the rate at 95 % confidence by about 3/n; that approximation does not carry over to correlated repairs within a discharge.
Parity, with priors counted
The count of unknowns includes every continuous state free at that instant, nuisance drifts included. A prior enters as a pseudo-measurement and adds one parity dimension, whose residual is only as reliable as the prior. With one parity dimension every fault signature lies on one line: a fault with a nonzero signature that exceeds the noise is detectable, but a single residual generally cannot say which channel produced it, since sources with arbitrary unknown amplitudes are not distinguishable along one line.
03 · The instrument
Two colours, two unknowns, one lattice of errors.
The topology combines a 10.59 μm CO₂ interferometer, a second colour for vibration compensation, and a 10.59 μm Faraday polarimeter. The DIII-D prototype of ITER's Toroidal Interferometer and Polarimeter uses a 5.22 μm quantum-cascade laser as its second colour; the ITER design uses 4.6 μm. That one difference changes the entire picture.
In the cold-plasma approximation, an interferometer at wavelength λ measures a phase proportional to the line-integrated density:
One fringe corresponds to 2.11 × 1020 m−2 at 10.59 μm, 4.27 × 1020 at 5.22 μm and 4.85 × 1020 at 4.6 μm. Densities below are quoted in fringes at 10.59 μm.
Vibration moves the optics, so the two phases carry both the density and an optical path change. The vibration-free combination removes the path, and in doing so turns a pair of count errors into a density step and a path step together. The two phases determine density and path exactly for every pair of integers, so the integers follow only from priors on both continuous quantities.
Where the selected count errors land, in density and in optical path
Source · EN-003 §3.2 · [N]Scroll the figure horizontally →
| Slip (k₁, k₂) | Prototype δN (fringe) | Prototype δL (μm) | ITER δN (fringe) | ITER δL (μm) |
|---|---|---|---|---|
| (1, 2) | +0.0187 | +10.39 | +0.1618 | +8.88 |
| (3, 7) | −0.5950 | +38.07 | −0.0500 | +32.30 |
| (10, 23) | −1.7663 | +124.60 | +0.0116 | +105.78 |
| (1, 3) | −0.6324 | +17.29 | −0.3736 | +14.55 |
| (0, 1) | −0.6511 | +6.90 | −0.5354 | +5.67 |
| (1, 1) | +0.6698 | +3.50 | +0.6972 | +3.21 |
| (1, 0) | +1.3209 | −3.40 | +1.2326 | −2.46 |
Each step comes with its negative. The near-null direction, the lattice vector whose ratio best approximates the wavelength ratio within the path range of the search set, depends on that range. For the prototype it is (1, 2), since 10.59 μm is close to twice 5.22 μm. Its 0.0094 fringe decision boundary lies well inside the assumed 0.05 fringe prior spread, so density alone provides weak branch discrimination; the 10.4 μm path separation makes a path prior informative. Integer multiples follow the same near-null direction, with density and path displacements increasing proportionally. The table lists the selected vectors of §3.2, not every slip inside the search bounds: (2, 4) sits at +0.0374 fringe and +20.78 μm for the prototype. Marked in magenta is each pair's near-null density step. The mark means near-null, not admitted by the tolerance: the ITER (3, 7) step shown here as −0.0500 is −0.050047, just outside it.
The polarimeter is an absolute channel
Its Faraday angle stays far below a wrap, so it supplies a density prior without an integer of its own. Accuracy rests on the density-weighted parallel field from the equilibrium reconstruction. For an assumed 1.5–2 T, the prototype's angle noise corresponds to 0.036–0.047 fringe.
At reactor temperature the electrons turn relativistic
To first order the interferometric phase carries a factor 1 − 3τe/2 and the Faraday rotation 1 − 2τe; at 25 keV these correspond to reductions of about 7.3 % and 9.8 % respectively. Left uncorrected they offset a polarimeter-based prior by about 0.12 fringe at 1021 m−2, comparable to the lattice steps themselves.
Windows drift chromatically
Temperature, pressure and humidity change the optical path differently at the two wavelengths. A chromatic path difference enters the density at 0.012 fringe per 0.1 μm for both pairs, the size of the prototype's near-null step. The estimator carries this drift as a slow state with its own prior.
04 · The density-risk decision
Marginalize over the integer. Decide on the density.
Each integer hypothesis in a bounded search set gets a weight and a Gaussian posterior. The density risk is the posterior probability that the reported density is wrong by more than the tolerance, summed over those components. Evaluating only the heaviest of them and counting the rest as failure gives a bound that is conservative by construction.
The bound holds for every subset K and every estimate, and counts the out-of-set hypothesis in full. Normalizing only over an incomplete search set can produce unjustified confidence and underestimate the density-error risk when the true count lies outside it, so a rising out-of-set probability triggers search expansion, and abstention follows when the expanded set still fails to explain the data.
The estimator reports three quantities separately: the probabilities of the integer hypotheses, for count recovery; the density with its bound, for density accuracy; and the path state with its own uncertainty, for vibration-state accuracy. An unresolved count can therefore coexist with an available density.
Two proposed changes, kept apart.
The proposal changes both the estimate and the gate, and their potential benefits have to be evaluated separately: an evaluation that reports one number cannot tell them apart. Four comparisons separate them, run at matched availability with identical inputs and latency.
| # | Comparison | What it reports | What it measures |
|---|---|---|---|
| 1 | Hard-branch estimate | The density of the most probable integer hypothesis, for every decision | The integer-recovery baseline |
| 2 | Marginalized estimate | The marginalized density, for every decision | Against 1: what the estimate gains |
| 3 | Density-risk gate | The marginalized density, only when the bound stays within α | Against 2: what withholding reaches at reduced availability |
| 4 | Count gate, same estimate | The marginalized density, only for the highest count confidence, at the availability of 3 | Against 3: what a criterion on the density adds over one on the count |
Comparing 2 and 3 alone leaves the value of the density criterion open, because a count gate can reach a similar rate. Hard branch selection gated by count confidence mixes both effects and serves as the integer-recovery baseline.
05 · A model-conditional illustration
Where the benefit actually is.
Two million simulated decisions per configuration, with Gaussian priors, 1.9° of phase noise per colour, a tolerance of 0.05 fringe, and a search set bounded at 3 fringes of density and 60 μm of path. The true count lies inside the search set by construction. These are model numbers, and the configuration decides the answer.
| Configuration and availability | Hard-branch estimate, count gate | Marginalized estimate, count gate | Marginalized estimate, density-risk gate | Mean bound |
|---|---|---|---|---|
| Prototype, 5 μm path prior, 100 % | 6.6 × 10−5 (132) | 1.7 × 10−5 (33) | 1.7 × 10−5 (33) | 1.7 × 10−5 |
| Prototype, 5 μm, 81.9 %, α = 2 × 10−5 | 3.8 × 10−5 (63) | 1.5 × 10−5 (24) | 1.5 × 10−5 (24) | 1.6 × 10−5 |
| ITER, 2 μm path prior, 100 % | 6.2 × 10−3 (12,334) | 6.2 × 10−3 (12,334) | 6.2 × 10−3 (12,334) | 6.2 × 10−3 |
| ITER, 2 μm, 86.1 %, α = 10−3 | 6.3 × 10−5 (108) | 6.3 × 10−5 (108) | 6.3 × 10−5 (108) | 7.4 × 10−5 |
| ITER, 2 μm, 71.0 %, α = 10−4 | 1.2 × 10−5 (17) | 1.2 × 10−5 (17) | 1.2 × 10−5 (17) | 1.4 × 10−5 |
Rate of total density errors above 0.05 fringe among the reported densities, with event counts in parentheses. Every entry draws on the same 2 × 106 decisions per configuration, so the columns form paired comparisons, and at each availability the gates report the same number of decisions. Each mean bound lies inside the corresponding exact Poisson 95 % interval in this simulation.
The estimate earns its place
For the prototype the marginalized estimate cuts the error rate fourfold with every density available: 112 errors of the hard-branch estimate disappear and 13 new ones appear. A rule that waits for a resolved count would have withheld every prototype density, because the count confidence stayed below 0.99 in all two million decisions.
For the ITER pair, the estimate gains nothing
Both estimates fail on the same 12,334 decisions. There the useful mechanism is withholding instead: at 86.1 % availability the error rate drops about a hundredfold. Which mechanism helps is decided by the configuration, the wavelength pair together with the assumed priors, phase noise, tolerance and search bounds, and not by the method alone.
The gate proved nothing yet
With the estimate held fixed, the density-risk gate and a count-confidence gate accepted identical sets of decisions at every tested availability, for both pairs. Neither configuration establishes an advantage for the density criterion. That is reported as the open question it is, not buried.
The demonstrated benefit is one of estimation, not of gating.
EN-003 §5.3 · the result the note declines to overstate
06 · Evaluation protocol
How someone else would prove this wrong.
The note is offered to groups with access to interferometer–polarimeter records. Offline evaluation on archived data needs no change to an operating machine.
Replay has to act where the errors arise
On detector waveforms before demodulation, or on complex I/Q signals before phase extraction, by imposing amplitude fades, phase noise, fast ramps and interruptions with reacquisition; or on the tracker's count state, by corrupting it directly.
A multiple of 2π added to an already wrapped phase leaves it unchanged, so replay at that level tests nothing. That is listed among the invalidators, not left to chance.
Ground truth comes from evidence independent of the estimator's own inputs: Thomson scattering, other chords through a profile model, closure at the pulse end, or bench optics where the density is zero and a displacement sensor records the mirror motion. Natural events count only when identified independently of the two-color inversion, because a path step and a density step inferred from the same inversion confirm each other by construction.
Requested from partner groups
- Detector waveforms or complex I/Q signals, with time stamps and tracker states
- Equilibrium reconstructions for the parallel field along each chord
- Thomson scattering and other chords, for independent truth
- Vibration or path sensor records where they exist
- Wavelength calibration and window temperature logs
- Event annotations for pellets, ELMs, disruptions and mitigation
- Bench access for zero-density optical tests
Baselines: the deployed correction algorithm of the data source, continuity unwrapping, integer least squares or state-space estimation, integer-aperture acceptance at a fixed fail rate, the best integer-equivariant estimate, and recent machine-learning methods. Every method receives the same sensor inputs and the same latency budget, and all are compared on risk–coverage curves at matched availability.
07 · Validation ladder and falsifiers
This note is Stage 0, and says so.
Validation proceeds in stages, each gated by the previous one. Everything on this page sits in the first row of the table below.
| Stage | Status | Purpose | Primary outputs | Failure condition |
|---|---|---|---|---|
| 0 | This note | Model-conditional mechanism | Lattices, margins, parity accounting, synthetic comparisons, reproduction script | Mismatch between text and reproduction script |
| 1 | Next | Implementation tests and raw-signal replay | Count-state injection checks; replay with controlled injections; the reference calculation beside the bound | Lattice residuals missed in the uncorrected inversion; F1 fires on replay |
| 1b | Pending | Bench optics | Zero-density tests with independent path truth | F3 fires |
| 2 | Pending | Prototype records, offline | The four comparisons at matched availability against all baselines | F1 or F2 fires; retirement criterion met |
| 3 | Pending | Real-time shadow operation | Latency within the control budget; calibration stable across campaigns | Calibration drift; latency over budget |
Every number regenerates
A single script recomputes every value the note reports, in the order of the stored reference output. The note's Appendix B embeds the same code.
Pinned environment
CPython 3.11.15, NumPy 2.4.4, SciPy 1.17.1. The reported run took about four minutes on two cores, with a peak memory use of about 4 GB.
Checksums ship with it
SHA-256 sums for every release file, so a reader can verify the package before running it.
Platform caveats stated
Deterministic values are platform-independent. On other BLAS builds, rounding can move decisions that sit on a boundary, so event counts may differ by a few. The note says so rather than claiming bit-identity it cannot hold.
Model falsifiers
- F1, calibration and bound validity. Tested separately. Calibration: the reported probabilities must match observed frequencies within their statistical uncertainty. Bound validity: among released densities, a prespecified one-sided test must find no significant excess of observed errors over the reported bounds, accounting for sampling uncertainty and for dependence between decisions. A finite sample can exceed a valid bound by chance, so the verdict is a test, not an inequality. Finding no significant excess means the bound was not falsified by that test, not that its validity was established. A conservative bound can also stay valid while the probabilities are miscalibrated, so the two verdicts are reported apart.
- F2, advantage at matched availability. F2a, the estimate: marginalization must beat the hard-branch estimate under the same gate. F2b, the gate: the density-risk gate must beat gating by count confidence. Section 05 supports F2a in one synthetic configuration. The synthetic comparisons establish no F2b advantage in either tested configuration; evaluation on measured records remains pending.
- F3, the two-color model. On the bench, with zero density and an independent displacement sensor, the uncorrected inversion must show the lattice residuals and the final estimate must stay within tolerance at the rate its bound allows.
- F4, controlled aliasing. Injected phase ramps must produce the wrapped apparent change: ramps between π and 2π per step appear reversed, and a ramp of 2π per step appears as stillness.
Retirement criterion
If, on prototype records at matched availability with identical inputs and latency, the marginalized estimate fails to lower density errors relative to the hard-branch estimate in the event classes where unresolved counts occur, the decision rule retires.
If the estimate passes and F2b fails, the density-risk gate retires on its own, and the marginalized estimate is gated by count confidence or integer-aperture acceptance instead.
The accounting discipline remains useful in any case: count, density and vibration accuracy reported separately; displacement and continuous error kept apart; search bounds stated; zero-event results bounded; every continuous state and every prior counted.
Invalidators
Each of these invalidates the affected comparison: ground truth derived from the estimator's own inputs; training and test sets split within discharges; replay applied to wrapped phases; baselines given different inputs, latency or coverage; gates compared with different estimates; rates reported without their search bounds; zero-event results reported as zero rates; natural events identified through the same inversion that they test; and probabilities normalized over a search set without an out-of-set hypothesis.
08 · Scope and limits
Where it would pay, if the baselines are beaten.
Discharge phases in which fringe jumps cluster, such as disruption mitigation and high-density fuelling, where a marginalized estimate released under a valid bound keeps the control loop supplied while a rule that waits for a resolved count would blank it. And long pulses, in which discharge-level risk accumulates.
That is a conditional, and the condition is the whole evaluation protocol of Section 06.
Inherited limits
Gaussian priors and noise, with prior errors independent of measurement errors; an assumed per-colour covariance; a true integer state inside the search set by construction, with zero prior weight on the out-of-set hypothesis; a cold-plasma model in the decision simulation, with the temperature corrections evaluated separately; a single chord without a profile model; and a candidate-based estimate with a conservative bound, checked against a numerical reference calculation.
09 · Release
Download the release.
The engineering note and its reproduction package: the script that recomputes every reported value, its pinned dependencies, and the complete output of the reported run.
Fusion Density Under Ambiguity, EN-003 v1.0
The full note: substrate, the TIP measurement model and its slip lattices, the density-risk decision, the model-conditional illustration, the evaluation protocol, the validation ladder with falsifiers and a retirement criterion, and the reproduction script as Appendix B.
Download the paper PDF · 1.3 MB ZIP · Reproduction packThe reproduction package
Everything needed to recompute the note: the note itself, the reproduction script, pinned dependencies, the complete reference output, the licence, and SHA-256 checksums for every file in the release.
Download the pack ZIP · 1.2 MBCite this note
Dupke, A. (2026). Fusion Density Under Ambiguity: Preserving Useful Density Under Unresolved Fringe Counts. A Density-Risk Decision for Two-Color Interferometer–Polarimeter Systems in Magnetic-Confinement Fusion (Engineering Note EN-003, Version 1.0). Scale-Time Dynamics LLC. Zenodo. https://doi.org/10.5281/zenodo.23105860
SCOPE NOTE · All results on this page are model-conditional and depend on stated search bounds. The illustration uses Gaussian priors and an assumed phase noise; the true integer state lies inside the search set by construction, with zero prior weight on the out-of-set hypothesis. Operational reliability, real-time feasibility and any consequence for fusion performance remain open, and the note is offered for evaluation by groups with access to interferometer–polarimeter records.
10 · Contact
Offered for evaluation
EN-003 asks for records, not agreement. Correspondence from groups with interferometer–polarimeter data, or with bench optics, is welcome.