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What the Primary Lund Plane Omits: A Model-Stitching Diagnosis of a Jet-Tagger Representation Gap

Saurabh Rai, Sanmay Ganguly

arXiv:2610.37137v1hep-phcs.AIhep-ex

TL;DR

Standalone comparisons leave open whether a jet-tagger performance gap comes from its representation or its downstream reader, while AUC can obscure rejection differences. The paper uses model stitching to separate those causes and finds that the primary-branch representation is weaker regardless of the downstream reader, with the deficit repaired by restoring omitted branches.

  • Problem

    Standalone performance cannot distinguish a weaker representation from a weaker downstream reader, and AUC can be nearly blind to background-rejection differences.

    Source: §I · p. 1, §VIII · p. 11

  • Method

    Model stitching connects a frozen source network to a frozen target through a trained map, using the primary-branch representation as a positive control for a known omission.

    Source: §I · p. 2, §VIII · p. 11

  • Results

    The primary-branch representation is weaker regardless of downstream reader, and restoring omitted branches repairs the deficit; the full-tree advantage appears only at depths 4–5.

    Source: §VIII · p. 11, §7 · p. 16

  • Takeaways & Limitations

    The full tree’s added information is distributed and requires several rounds of attention before it becomes discriminating.

    Source: §7 · p. 16

    The sweep excludes d = 6, so the paper cannot speak to behavior at that depth.

    Source: A. A hypergraph reading of the same… · p. 13

Abstract

from arXiv · show

Lund-plane and Particle Transformer (ParT) jet taggers can differ in performance because one representation is weaker, or because a weaker downstream computation reads an adequate one. Standalone performance cannot separate these. Model stitching can: it links the frozen early layers of one network to the frozen late layers of another through a single trained linear map. As a positive control we use an incomplete Lund representation, the primary declustering branch, whose deficit is known independently. Stitched into ParT's downstream, it reaches a background rejection at least as much as its own downstream recovers, yet far below ParT's background rejection: the deficit is representational, not computational. A hypergraph reader over splitting triplets does no better, ruling out higher-order correlations as the cause. The gap is invisible in AUC, which stays within 1\% of its target value, and appears only in background rejection at low signal efficiency. It localizes to 0.6\% of background jets, at and above the top mass, with more constituents but fewer primary splittings than typical background. Rebuilding the representation on the full Cambridge/Aachen declustering tree repairs the deficit as predicted: standalone rejection rises, closing 72\% of the gap. Masking nodes at inference shows the repaired network still relies almost entirely on the primary branch; the secondary splittings make the representation learnable rather than supplying information it reads. Along the primary branch, the widest-angle splitting, near the top and $W$ opening angles, is worth nearly twice a random one. For quark/gluon tagging the primary branch is far less deficient, retaining 77.6\% of ParT's rejection (46.9\% for top), and the full tree closes only 33\% of the gap (72\% for top).

I. INTRODUCTION

Standalone performance cannot distinguish a weaker representation from a weaker downstream reader, motivating a functional comparison that holds architecture constant. The paper introduces a primary-branch Lund-ParT positive control and compares it with ParT on a shared top-tagging dataset.

I. INTRODUCTION

  • Model stitching tests whether a performance gap comes from representation content or downstream computation, a distinction standalone accuracy cannot resolve.The primary Lund plane provides a positive control because its harder-branch-only representation is independently known to omit radiation captured by the full declustering tree.Source: §I · pp. 1–2
  • AUC can conceal substantial rejection differences: a cited benchmark differs by 0.004 in top-tagging AUC but by a factor of 2.5 in rejection at εS = 0.5.The paper reports its own representational deficit moves AUC by less than one percent while halving background rejection.Source: §I · p. 1

A. Dataset

  • The study uses 50% random subsamples of the Kasieczka et al. top-tagging dataset, with boosted tops against light-quark and gluon jets at pT ∈[550, 650] GeV.Each split contains 605,500/201,500/202,000 train/validation/test jets; ParT reaches background rejection 291 at εS = 0.5 on this scale.Source: A. Dataset · p. 2

B. Architectures

  • ParT and Lund-ParT share the ParticleTransformer architecture, but use constituent four-momenta versus primary-branch Lund coordinates as their inputs.The primary branch follows the harder subjet at each Cambridge/Aachen declustering; background jets average 43.3 particles and 12.6 primary-plane splittings.Source: B. Architectures · pp. 2–3

C. Stitching

Stitching connects a frozen source network to a frozen target network through one trained linear map at a selected depth. The construction uses a single injected class-token vector and compares cross-network stitches with self-stitching controls.

C. Stitching

  • At depth d, the source’s frozen embedding, first d blocks, and class-attention readout produce a class token; only a 128 × 128 map is trained before the frozen target’s remaining blocks and head.The map is trained for up to 15 epochs on a fixed 20,000-jet subsample, while test metrics use the full 50% test split.Source: C. Stitching · p. 3
  • The target’s remaining blocks receive the mapped source vector as a one-token sequence, which computes the same function as broadcasting it to any sequence length.The paper treats the injection cost as common to cross-architecture and same-architecture self-stitch comparisons at the same depth and source.Source: C. Stitching · p. 3
  • Depth d = 6 is excluded because the target has no self-attention blocks left, making the measurement different from the depth sweep’s intended question.At that depth the construction instead maps a completed class token directly into the target readout and head.Source: C. Stitching · p. 3
  • Four conditions run at each depth and seed: both cross-architecture directions and self-stitch controls for ParT and Lund-ParT.Standalone test performance is summarized in Table I, whose full-tree column anticipates the later analysis.Source: C. Stitching · p. 3, Table I · p. 4

FROZEN TRAINABLE FROZEN

The schematic shows frozen source and target portions joined by a trained map, with cross-architecture and self-stitch controls run across depths and seeds. The sweep omits depth six because no target self-attention block remains there.

FROZEN TRAINABLE FROZEN

  • At depth d, a frozen source produces a class token that passes through a trained linear map into the frozen target network.The figure depicts d = 3 as an example and shows the same six-block backbone and two-block class-attention readout for both architectures.Source: Figure 1 · p. 4
  • Both cross-architecture directions and both self-stitch controls run at every depth and seed, using seed-matched standalone backbones.The sweep excludes d = 6 because the target would contribute no self-attention block, and AUC-based early stopping cannot detect the resulting degradation.Source: Figure 1 · p. 4, FROZEN TRAINABLE FROZEN · p. 4

D. Metrics and normalization

The paper reports AUC and background rejection at fixed signal efficiencies, alongside normalized retained fractions. These normalizations distinguish whether information survives a foreign reader from whether the target’s downstream computation is limiting.

D. Metrics and normalization

  • The reported metrics are AUC and background rejection 1/εB at fixed signal efficiencies εS ∈{0.5, 0.3}.Absolute rejection is reported alongside each fraction so its denominator remains visible.Source: D. Metrics and normalization · pp. 4–5
  • Source-normalized retention asks how much of a source’s information survives a foreign downstream, relative to what its own downstream recovers.The target-normalized fraction instead asks whether the fixed target downstream is limiting.Source: D. Metrics and normalization · p. 4
  • Table II reports target-normalized retained AUC for the primary-branch run, averaged over five seeds.Across depths and stitching conditions, retained AUC lies between 0.977 and 1.007.Source: Table II · p. 5

III. THE GAP IS REPRESENTATIONAL, NOT DOWNSTREAM

Standalone performance leaves open whether the primary-branch Lund-ParT gap comes from its representation or its downstream reader. Stitching indicates the representation is limiting: ParT’s downstream does at least as well with it as Lund-ParT’s own, while geometric similarity does not establish functional interchangeability.

A. AUC does not distinguish the representations

  • 0.977–1.007: retained AUC stays in a narrow band across all four stitching conditions, making the representations appear nearly interchangeable by this metric.For lund-to-part, retained AUC moves only from 0.991 at d = 1 to 0.992 at d = 5.Source: A. AUC does not distinguish the… · p. 5

B. Background rejection under both normalizations

  • ParT’s downstream has a suggestive, but not conventionally significant, mean advantage of +8±7 rejection units over Lund-ParT’s across directions and depths (t = 2.56, p = 0.062).The advantage has the same direction in all ten per-depth comparisons, but the five-seed test is not significant at the conventional threshold.Source: B. Background rejection under both… · p. 5
  • 136 versus 291: Lund-ParT’s primary-branch rejection is lower than ParT’s, and ParT’s downstream does at least as well with that representation at every depth.The comparison holds the representation fixed and changes only the reader, supporting a representational rather than downstream explanation for the shortfall.Source: B. Background rejection under both… · p. 5, Figure 2 · p. 6

C. Geometric alignment between the two representations

  • 0.759 to 0.934: CKA rises with depth, but lund-to-part retained AUC remains 0.991–0.992 and rejection gains only 8 units, showing geometric alignment need not track functional transfer.The paper distinguishes what the representations geometrically share from what a particular downstream computation can extract.Source: C. Geometric alignment between the two… · p. 6, Figure 3 · p. 6

IV. WHAT THE REPRESENTATION IS MISSING

The primary-branch deficit is concentrated in a small, atypical background-jet population: these jets are massive and constituent-rich but have fewer primary splittings. Rebuilding Lund-ParT on the full declustering tree raises its rejection and closes most of the gap to ParT.

IV. WHAT THE REPRESENTATION IS MISSING

  • 557±24 leaked jets at d = 1 and 515±23 at d = 5 represent only 0.6% and 0.5% of ParT’s rejected background, respectively.The score rankings otherwise agree closely, with Spearman ρ = 0.955 and 0.956 across all background jets.Source: §IV · p. 6
  • The missed population is concentrated near top-mass scales and has more constituents but fewer primary splittings than typical background.It averages 67.2 constituents and 11.3 primary splittings at d = 1, versus 43.3 constituents and 12.6 splittings for typical background; its mean mass is 180 GeV versus 95 GeV.Source: §IV · pp. 6–7

V. CONFIRMATION: THE FULL DECLUSTERING TREE

  • 136±4 to 248±5: full-tree Lund-ParT closes 72% of the rejection gap to ParT’s 291±13; its stitched rejection also rises from 128 to 242 between d = 1 and d = 5.The full-tree representation has a mean background sequence length of 42.3, versus 12.6 for the primary branch.Source: §V · p. 7
  • The repaired representation reduces leakage from 557±24 to 399±20 jets at d = 1 and from 515±23 to 177±13 at d = 5.At d = 5, the shrinkage is a factor of 2.9, compared with 28% at d = 1.Source: §V · p. 7
  • After full-tree repair, the remaining leakage stays near the top mass, but the primary-branch signature of extra constituents with fewer splittings is gone.The full-tree splitting count tracks constituent count, as expected when every splitting is recorded; the residual leakage is consistent with ordinary classifier disagreement.Source: §V · p. 7

VI. WHICH NODES CARRY THE REPAIR

Inference masking shows that the trained full-tree network relies much more on primary than secondary nodes, while the secondary-node contribution is spread across the tree. The results distinguish what the network reads at inference from what full-tree training provides.

VI. WHICH NODES CARRY THE REPAIR

  • A pre-order traversal places the primary branch in the leading contiguous block; across 40,000 test jets, the averages are 12.0 primary and 35.9 secondary nodes.The leading block was bit-identical to primary-branch features on 2,000 randomly drawn test jets.Source: §VI · p. 7
  • 51% of primary nodes lie in the hard region kT > 1 GeV, versus 13% of secondary nodes; 59% of secondary nodes cluster in a soft, collinear quadrant.The text relates the primary branch’s hard, wide-angle structure to top-decay discrimination from QCD jets.Source: §VI · pp. 7–8
  • The masking probe deletes selected nodes from frozen full-tree weights, with size-matched controls and compaction of surviving nodes before inference.This holds token counts fixed within matched conditions and avoids silently discarding scattered survivors.Source: §VI · p. 8
  • 230.6 → 9.2 after removing twelve primary nodes, versus 230.6 → 186.5 after removing twelve secondary nodes.The matched comparison holds the number of deleted nodes fixed; the table reports background rejection and AUC for the masking conditions.Source: §VI · p. 9, Table IV · p. 9
  • 34.7 with only twelve primary nodes exceeds 9.2 with all 36 secondary nodes, while removing secondary nodes progressively lowers rejection without a cliff.Rejection falls from 230.6 with all nodes to 194.9, 118.8, 58.9, and 34.7 as successive quarters of secondary nodes are removed.Source: §VI · p. 10
  • Full-tree training raises standalone rejection from 136 to 248, while inference masking indicates secondary nodes shape the learned representation rather than supply information read at test time.The authors conclude that no small subset of secondary nodes accounts for the repair.Source: §VI · p. 10
  • A modest selection of secondary splittings is not supported as a shortcut to the repair, because their contribution is not concentrated in a small subset.The authors caution that absolute values across masking conditions still reflect the cost of shorter sequences unseen during training.Source: §VI · p. 10

VII. WHERE ALONG THE PRIMARY BRANCH INFORMATION SITS

The primary branch carries more discrimination at wide angles, especially for top tagging, where the first splitting lies near the top and W opening-angle scales. The node-removal comparison uses random ablation as a control because removing any node also incurs a distribution-shift penalty.

VII. WHERE ALONG THE PRIMARY BRANCH INFORMATION SITS

  • The first splitting is the widest-angle step, and its scale lies near the top and W opening angles.Cambridge/Aachen declustering is angular ordered; the median ∆R falls monotonically along the chain.Source: §VII · pp. 10–11
  • The comparison removes the first, last, or a random step from every eligible jet, using the random removal as the control.The frozen network is evaluated without retraining; paired differences against random ablation, rather than absolute drops, are interpretable because removal itself changes the sequence.Source: §VII · p. 10
  • +35.2% of unablated rejection is lost against the random-control loss of +11.5% when removing the first splitting in top tagging.The widest-angle splitting is also more important than a random one for quark/gluon tagging, while importance falls toward the most collinear step.Source: §VII · pp. 10–11
  • Ablation measures what the trained network uses, not all information physically available in the representation.Removing a node costs more than its naive 1/N share: a random node costs 35% for top tagging and 16% for quark/gluon, so only paired excess over random is reported.Source: §VII · p. 11

VIII. DISCUSSION AND CONCLUSION

The primary-branch deficit is representational, and restoring the full declustering tree repairs it, although the repaired network relies mostly on primary-branch nodes at inference. A hypergraph reader over the primary branch does not recover the full-tree result, while quark/gluon tagging shows a smaller primary-branch deficit and less repair from the full tree.

VIII. DISCUSSION AND CONCLUSION

  • The primary-branch representation is weaker than ParT’s regardless of which downstream reads it; the deficit is concentrated in identifiable high-mass, high-multiplicity background jets and is repaired by restoring omitted branches.The deficit is invisible in AUC and appears in background rejection at fixed, low signal efficiency.Source: §VIII · p. 11
  • The repaired full-tree network relies almost entirely on the primary branch at inference; secondary nodes make the representation learnable rather than supplying information it reads.Removing twelve secondary nodes costs a fifth of rejection, while removing twelve primary nodes costs almost all of it.Source: §VIII · p. 12

A. A hypergraph reading of the same representations

  • The primary-branch hypergraph reaches rejection 116, agreeing within one standard deviation with Lund-ParT’s 112 ± 7, while restoring the full tree raises rejection to 181.The full-tree gain is 55%, compared with the 79% gain measured in the stitching study; the hypergraph comparison used 10% of the dataset and one seed.Source: A. A hypergraph reading of the same… · p. 13
  • The hypergraph test is eliminative: it removes a higher-order-reader explanation but improves neither the diagnosis nor the repair.It identifies no jet population beyond the one already identified in Section IV.Source: A. A hypergraph reading of the same… · p. 13
  • For quark/gluon tagging, the primary branch retains 77.6% of ParT’s standalone rejection versus 46.9% for top tagging, and the full tree closes 33% of the gap versus 72%.The full declustering tree reaches about 85% of ParT’s rejection on both tasks; shallow-depth quark/gluon stitch measurements are noisier.Source: A. A hypergraph reading of the same… · p. 13

VIII. DISCUSSION AND CONCLUSION

  • The paper’s contribution includes a class-token stitch point, a same-architecture Lund-ParT, dual normalization, and a positive control with an independently known deficit.The hypergraph tagger cannot be incorporated into stitching without reintroducing an architecture confound; the study also excludes depth d = 6 and leaves the consumer-side fine-tuning check unresolved at this scale.Source: A. A hypergraph reading of the same… · pp. 13–14

AI USAGE STATEMENT

The authors acknowledge model use for cross-checking and code refactoring, and state that they wrote the manuscript themselves.

AI USAGE STATEMENT

  • Anthropic’s Opus-5 was used to cross-check the SVCCA analysis and refactor code; the authors state that they manually wrote the manuscript.Source: AI USAGE STATEMENT · p. 14

Appendix A: Generalization to quark/gluon tagging

The appendix applies the same tagging pipeline to quark/gluon jets, using primary-branch and full-tree Lund-ParT alongside ParT. Restoring the full tree improves standalone rejection but closes less of the gap than for top tagging; the consumer-asymmetry test is not significant.

1. Setup

  • The quark/gluon setup uses EnergyFlow jets and compares ParT, primary-branch Lund-ParT, and full-tree Lund-ParT in the same ParticleTransformer architecture.The primary-branch and full-tree inputs average 13.6 and 42.3 nodes per jet, respectively.Source: Appendix A · p. 14, §1 · p. 14

2. Standalone performance

  • 33.54 versus 30.59 standalone rejection for full-tree versus primary-branch Lund-ParT, closing 33.3% of the gap to ParT’s 39.44.The full-tree closure is 33.3%, compared with 72% for top tagging.Source: §2 · p. 14, Table VII · p. 15

3. Stitched performance

  • The mean consumer-side advantage is −1.1 ± 2.7 rejection units and is not significant (p = 0.395, n = 5 seeds).This contrasts with the top-tagging test, which gave +8 ± 7 (p = 0.062).Source: §3 · p. 15

4. Geometric alignment

  • The appendix computes linear CKA and SVCCA between standalone ParT and Lund-ParT class tokens on a fixed probe sample, independently of stitching and classification objectives.The class-token depth index remains defined at d = 6 for this alignment analysis.Source: §4 · p. 15

5. Jet-level localization

The localization analysis defines the missed-jet population relative to standalone ParT and reports mixed depth-by-depth behavior. Shallow-depth stitch comparisons are noisy, so individual entries do not establish a resolved ordering.

5. Jet-level localization

  • 97,575 of 100,016 test-split gluon jets are rejected by standalone ParT at εS = 0.5; the leakage count tracks jets the stitch fails to reject.The passage introduces Table XII as counting these jets but does not provide the stitched-model count.Source: §5 · p. 15
  • Spearman rank correlation with standalone ParT is 0.812 (d = 1) and 0.836 (d = 5) for the primary branch, versus 0.661 and 0.907 for the full tree.Source: §5 · p. 15

6. Reading the numbers

  • The depth-by-depth spreads are 20–55% for d ≤ 4, so only d = 5 is measured sharply and shallow non-monotonic entries should not be read as structure.Examples are 15.1 ± 8.2 at d = 1 and 23.6 ± 12.4 at d = 4 for the primary branch.Source: §6 · p. 15
  • The consumer-asymmetry test returns −1.1 ± 2.7 (p = 0.395), consistent with zero, while the full tree leads standalone but not at every stitch depth.The full tree is behind at d = 1–3 and ahead only at d = 4–5; the five-seed asymmetry test contrasts with +8.0 ± 7.0 (p = 0.062) in top tagging.Source: §6 · p. 15
  • At d = 1, the full tree leaks 5539 jets versus 2607 for the primary branch; at d = 5, it leaks 897 versus 1306.Standalone rejection is 33.54 for the full tree and 30.59 for the primary branch.Source: §6 · p. 15

7. Physical interpretation

The primary branch retains much more of ParT’s rejection for quark/gluon tagging than for top tagging, and the full tree recovers less of the gap. The full tree’s advantage requires depth, while its residual shortfall from ParT is similar across the two tasks.

Physical interpretation

  • 85.0% and 85.2% of ParT’s rejection are reached by the full tree on the two datasets, respectively.The passage interprets the similar residual shortfall as a cost of the Lund encoding rather than a particular omitted branch.Source: §7 · p. 16
  • The consumer-asymmetry test returns −1.1 ± 2.7 (p = 0.395) for quark/gluon tagging, unlike +8.0 ± 7.0 (p = 0.062) for top tagging.The authors caution that the five-seed test is underpowered, so no detected asymmetry is not evidence of its absence.Source: §7 · p. 16
  • Shallow-depth ordering is unresolved on its own, and the two task comparisons use different fractions of their reference samples with differing jet kinematics.The interpretation of depth relies on consistent direction across d = 1, 2, 3 together with leakage and CKA evidence, not any single entry.Source: §7 · p. 16
  • 77.6% of ParT’s rejection is retained by the primary branch on quark/gluon tagging, versus 46.9% on top tagging.The passage attributes the task difference to quark/gluon separation tracking emissions along the hardest chain, while top tagging requires a three-prong structure partly on discarded branches.Source: §7 · p. 15, Table XIII · p. 18
  • 33% of the quark/gluon gap is closed by the full tree, versus 72% for top tagging.Source: §7 · p. 15, Figure 15 · p. 20
  • The full tree trails the primary branch at d = 1, 2, 3 and leads only at d = 4, 5; its added nodes require several attention rounds to aggregate.At d = 1, its CKA is 0.379 versus 0.639; by d = 6, it is 0.919 versus 0.884.Source: §7 · p. 16, Figure 16 · p. 20
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