Chapter map

1. 1. Space X
2. 2. Space Y
3. 3. Weights
4. 4. Coupling
5. 5. Pair distortions
6. 6. Squared objective
7. 7. Two-point X
8. 8. Two-point Y
9. 9. Identity coupling
10. 10. Off-diagonal terms
11. 11. Distance convention
12. 12. Scale normalization
13. 13. Entropic regularization
14. 14. Initializations
15. 15. Semantic audit
16. 16. Claim

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

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1. Space X

The useful question is not only how to calculate, but what the calculation assumes. Use path distances [[0,1,2],[1,0,1],[2,1,0]].

A concrete situation

Three points have uniform weights 1/3.

What the mathematics does not license

Distances must match their row labels. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record X. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. Space Y

A small example makes the mechanism visible. Relabel points to get [[0,2,1],[2,0,1],[1,1,0]].

A concrete situation

The same geometry uses another ordering.

What the mathematics does not license

No direct feature match is assumed. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record Y. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. Weights

This is where modelling choices begin to affect evidence. Assign each point mass 1/3.

A concrete situation

Both marginals sum to one.

What the mathematics does not license

Unequal weights change feasible couplings. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record a,b. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. Coupling

The central mathematical idea is concrete. Place 1/3 on the three correct relabelled matches.

A concrete situation

Rows and columns satisfy the marginals.

What the mathematics does not license

A coupling is a mass allocation. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record π. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Pair distortions

Here the geometry becomes operational. Compare every dX(i,j) with dY(k,l) weighted by πikπjl.

A concrete situation

All supported comparisons agree.

What the mathematics does not license

Use within-space distances, not cross-space coordinates. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Audit terms. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. Squared objective

This step deserves a careful pause. Sum the weighted squared differences.

A concrete situation

GW²=0 for the relabelled path.

What the mathematics does not license

Zero reflects isomorphic measured geometry. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record value. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Two-point X

The useful question is not only how to calculate, but what the calculation assumes. Use nonzero distance one with weights 1/2.

A concrete situation

The tiny case is fully auditable.

What the mathematics does not license

Keep diagonal zero. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record matrix. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Two-point Y

A small example makes the mechanism visible. Use nonzero distance two with weights 1/2.

A concrete situation

Geometry has the same pattern but another scale.

What the mathematics does not license

Ordinary GW keeps scale. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record matrix. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. Identity coupling

This is where modelling choices begin to affect evidence. Match the two points with masses 1/2.

A concrete situation

The coupling satisfies both marginals.

What the mathematics does not license

A swap gives the same result. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record π. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. Off-diagonal terms

The central mathematical idea is concrete. Two supported pair comparisons each have weight .25 and squared error one.

A concrete situation

Total squared distortion=.5.

What the mathematics does not license

Diagonal comparisons contribute zero. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Show ledger. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Distance convention

Here the geometry becomes operational. Take sqrt(.5) if reporting the square-root distance.

A concrete situation

GW=.707107.

What the mathematics does not license

Papers and software may report the squared objective. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Name convention. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. Scale normalization

This step deserves a careful pause. Normalize distances only if the application makes scale irrelevant.

A concrete situation

The answer changes.

What the mathematics does not license

Report the normalization. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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13. Entropic regularization

The useful question is not only how to calculate, but what the calculation assumes. Add a declared entropy term for computation.

A concrete situation

Couplings become softer and faster to optimize.

What the mathematics does not license

Regularized value is not the unregularized distance. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Report epsilon. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. Initializations

A small example makes the mechanism visible. Run the nonconvex solver from several starts.

A concrete situation

Different local solutions may appear.

What the mathematics does not license

Keep objective histories. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Summarise. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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15. Semantic audit

This is where modelling choices begin to affect evidence. Inspect high-mass matches with domain knowledge.

A concrete situation

Geometric fit can still join unlike meanings.

What the mathematics does not license

A coupling is evidence, not a label oracle. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Validate. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. Claim

The central mathematical idea is concrete. State distances, weights, loss, regularization and solver.

A concrete situation

Relabelled paths match exactly; scaled two-point spaces do not.

What the mathematics does not license

Do not infer common cause. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Bound conclusion. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

Gromov–Wasserstein comparison matches relational geometry when pointwise features or one-to-one labels are unavailable. It searches over couplings between two weighted spaces and penalizes disagreement between within-space distances. The method can compare tables with different numbers of points because the coupling transports probability mass rather than requiring row equality.

Consider a three-point path with distance matrix [[0,1,2],[1,0,1],[2,1,0]]. Relabel its points to obtain [[0,2,1],[2,0,1],[1,1,0]]. A permutation coupling with mass 1/3 on the correct three matches reproduces every within-space distance, so the squared Gromov–Wasserstein distortion is zero despite different row labels.

Now compare two equally weighted two-point spaces whose nonzero distances are one and two. An identity or swap coupling places mass 1/2 on each match. The squared distortion is .25(1-2)²+.25(1-2)²=.5. Under the square-root convention the distance is sqrt(.5)=.707107. The example also shows that ordinary GW is not scale invariant.

The optimization is generally nonconvex, so initialization, entropic regularization, stopping rules and solver accuracy affect computed solutions. A coupling is a soft relational match, not automatically a semantic correspondence. Distances, probability weights, normalization and structural coverage must be justified, and low distortion does not prove common cause.

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Frequently asked questions

What is Gromov–Wasserstein distance?

A comparison of weighted spaces based on how well a coupling preserves their internal pairwise relations.

Does it require the same number of points?

No. Couplings can transport mass between spaces of different sizes.

Why is optimization difficult?

The objective couples pairs of matches and is generally nonconvex.

What does entropic regularization do?

It smooths the coupling and helps computation but changes the objective and solution.

Does a low value prove semantic equivalence?

No. Similar relational geometry can describe different meanings or causes.

What must be reported?

Distances or kernels, weights, loss, normalization, squared or square-root convention, regularization, solver, starts, convergence, coupling and domain validation.

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Primary source and fact-check note

Gabriel Peyré, Marco Cuturi and Justin Solomon, “Gromov-Wasserstein Averaging of Kernel and Distance Matrices,” Proceedings of Machine Learning Research 48, 2664–2672 (2016). Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• Classical multidimensional scaling
• Procrustes analysis
• Representational Similarity Analysis
• Vector quantile regression and optimal transport
• How Mathematics Works

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: dependence, alignment and relational evidence

Geometry audit

Verify within-space distances, triangle assumptions where relevant and distance units.

Marginal audit

Check nonnegative weights and exact row and column sums of every coupling.

Solver audit

Record starts, objective history, tolerance, iterations and whether the value is regularized.

Meaning audit

Inspect high-mass matches against independent domain evidence before assigning semantic labels.

Mechanism workshop: calculate, challenge and transfer

Manual reconstruction

Rebuild the smallest example and label every row, matrix and normalization used for relational transport between unaligned spaces.

One-row experiment

Change one observation while holding preprocessing and formula fixed. Recalculate all affected pairwise entries and the final statistic.

Geometry experiment

Change the distance, kernel or representation scale for a stated reason. Explain which question changed.

Permutation experiment

Shuffle only valid exchangeable units. Compare one illustrative shuffle with a complete or Monte Carlo null distribution.

Invariance experiment

Apply translation, rotation, isotropic scaling or relabelling where relevant. Predict the statistic before calculating it.

Stability experiment

Repeat across resamples, seeds, bandwidths, regularization values or solver starts and report a distribution.

Student transfer

Apply the method to handwriting, sensor readings, study patterns, image features or map distances using real identifiers.

Parent explanation

Explain why a straight-line correlation can miss a curve, why similarity kernels need a scale, or why two maps can match without shared coordinates.

Limit statement

Write one supported claim, one overclaim and one repaired claim that names data, estimator and uncertainty.

Next measurement

Choose a complementary predictive, local, causal or substantive check and explain the separate evidence it supplies.

Evidence notebook: reconstruct every mathematical step

Evidence notebook 1: Space X

Step 1 isolates space x. Use path distances [[0,1,2],[1,0,1],[2,1,0]]. The reproducible result is: Three points have uniform weights 1/3. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: Distances must match their row labels. Record X. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 2: Space Y

Step 2 isolates space y. Relabel points to get [[0,2,1],[2,0,1],[1,1,0]]. The reproducible result is: The same geometry uses another ordering. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: No direct feature match is assumed. Record Y. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 3: Weights

Step 3 isolates weights. Assign each point mass 1/3. The reproducible result is: Both marginals sum to one. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: Unequal weights change feasible couplings. Record a,b. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 4: Coupling

Step 4 isolates coupling. Place 1/3 on the three correct relabelled matches. The reproducible result is: Rows and columns satisfy the marginals. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: A coupling is a mass allocation. Record π. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 5: Pair distortions

Step 5 isolates pair distortions. Compare every dX(i,j) with dY(k,l) weighted by πikπjl. The reproducible result is: All supported comparisons agree. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: Use within-space distances, not cross-space coordinates. Audit terms. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 6: Squared objective

Step 6 isolates squared objective. Sum the weighted squared differences. The reproducible result is: GW²=0 for the relabelled path. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: Zero reflects isomorphic measured geometry. Record value. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 7: Two-point X

Step 7 isolates two-point x. Use nonzero distance one with weights 1/2. The reproducible result is: The tiny case is fully auditable. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: Keep diagonal zero. Record matrix. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 8: Two-point Y

Step 8 isolates two-point y. Use nonzero distance two with weights 1/2. The reproducible result is: Geometry has the same pattern but another scale. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: Ordinary GW keeps scale. Record matrix. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 9: Identity coupling

Step 9 isolates identity coupling. Match the two points with masses 1/2. The reproducible result is: The coupling satisfies both marginals. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: A swap gives the same result. Record π. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 10: Off-diagonal terms

Step 10 isolates off-diagonal terms. Two supported pair comparisons each have weight .25 and squared error one. The reproducible result is: Total squared distortion=.5. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: Diagonal comparisons contribute zero. Show ledger. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 11: Distance convention

Step 11 isolates distance convention. Take sqrt(.5) if reporting the square-root distance. The reproducible result is: GW=.707107. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: Papers and software may report the squared objective. Name convention. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 12: Scale normalization

Step 12 isolates scale normalization. Normalize distances only if the application makes scale irrelevant. The reproducible result is: The answer changes. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: Report the normalization. Compare. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 13: Entropic regularization

Step 13 isolates entropic regularization. Add a declared entropy term for computation. The reproducible result is: Couplings become softer and faster to optimize. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: Regularized value is not the unregularized distance. Report epsilon. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 14: Initializations

Step 14 isolates initializations. Run the nonconvex solver from several starts. The reproducible result is: Different local solutions may appear. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: Keep objective histories. Summarise. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 15: Semantic audit

Step 15 isolates semantic audit. Inspect high-mass matches with domain knowledge. The reproducible result is: Geometric fit can still join unlike meanings. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: A coupling is evidence, not a label oracle. Validate. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Evidence notebook 16: Claim

Step 16 isolates claim. State distances, weights, loss, regularization and solver. The reproducible result is: Relabelled paths match exactly; scaled two-point spaces do not. This is evidence about weighted couplings and within-space distortion. Keep observation identities beside every row and write the exact formula before entering values. The safeguard is: Do not infer common cause. Bound conclusion. Then vary one input while holding all other decisions fixed. Recalculate the matrix cells, trace, norm, coupling or statistic that must move. Explain whether the change came from data, centring, geometry, regularization or matching. A result that another reader cannot reconstruct is not ready for a scientific or practical claim. End with the narrow conclusion and name the extra evidence still needed.

Failure cases and responsible recovery

Cross-space distance mistake

GW compares internal distances through a coupling; it does not require coordinates from both spaces in one metric. Rebuild a small counterexample, repeat under another defensible setting and state what evidence would change the conclusion.

Weight neglect

Uniform and empirical mass weights answer different questions. Report and justify marginals. Rebuild a small counterexample, repeat under another defensible setting and state what evidence would change the conclusion.

Scale surprise

Ordinary GW changes when every distance in one space is scaled. Normalize only when scientifically justified. Rebuild a small counterexample, repeat under another defensible setting and state what evidence would change the conclusion.

Regularization confusion

Entropic GW is computationally useful but not numerically identical to unregularized GW. Rebuild a small counterexample, repeat under another defensible setting and state what evidence would change the conclusion.

Local optimum

One nonconvex solver run can settle on a poor coupling. Repeat starts and inspect objectives. Rebuild a small counterexample, repeat under another defensible setting and state what evidence would change the conclusion.

Semantic hallucination

A high-mass match preserves geometry, not necessarily meaning. Validate with external labels or experts. Rebuild a small counterexample, repeat under another defensible setting and state what evidence would change the conclusion.

Solver opacity

Tolerance, iterations and stabilization choices can materially affect results. Preserve the computation record. Rebuild a small counterexample, repeat under another defensible setting and state what evidence would change the conclusion.

Responsible conclusion

Claim relational geometric fit under stated distances, weights and solver settings, not shared cause or identity. Rebuild a small counterexample, repeat under another defensible setting and state what evidence would change the conclusion.