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

Saurabh Rai; Sanmay Ganguly

hep-phOct 27, 2026

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

Authors: Saurabh Rai, Sanmay Ganguly

Organizations: Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India

Abstract

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).

Figures & tables

	ParT	Lund-ParT (primary)	Lund-ParT (full tree)
AUC	$0.9838\pm 0.0003$	$0.9759\pm 0.0004$	$0.9822\pm 0.0001$
rej. at $\varepsilon_{S}=0.5$	$291\pm 13$	$136\pm 4$	$248\pm 5$
rej. at $\varepsilon_{S}=0.3$	$926\pm 46$	$495\pm 10$	$895\pm 58$

Table 1: Standalone test performance, mean

\pm

standard deviation over 5 seeds, 50% data scale. The full-tree column anticipates Sec. V .

Figure 1: Schematic of the stitching construction. (a) Both ParT and Lund-ParT instantiate the same six-block backbone with a two-block class-attention readout; the dashed line marks where a stitch cuts at depth

d

. (b) At depth

d

(shown for

d=3

), source network

A

contributes its first

d

blocks and class attention readout, frozen at standalone trained values, producing a single class token

z_{A}^{(d)}\in\mathbb{R}^{128}

; a trained

128\times 128

linear map

W

carries it, broadcast as an unit length sequence, into target network

B

’s remaining blocks, readout, and head, likewise frozen. (c) The four conditions run at every depth and seed: both cross-architecture directions and the two same-architecture self stitching controls;

d=6

is excluded for the reason given below.

	$d$
direction	1	2	3	4	5
lund $\to$ part	0.991	0.991	0.992	0.992	0.992
part $\to$ lund	0.977	0.996	1.000	1.004	1.007
part-self	0.983	0.989	0.994	0.996	0.999
lund-self	0.998	0.999	0.999	1.000	1.000

Table 2: Retained AUC, target-normalized, primary-branch run, mean over 5 seeds.

Figure 2: Absolute background rejection at

\varepsilon_{S}=0.5

and

0.3

for all four stitching conditions, primary-branch Lund-ParT, versus depth. Horizontal lines mark each network’s own standalone value. The lund-to-part curve (solid red) sits at or above Lund-ParT’s own self-stitching curve (dashed red) at every depth.

$d$	lund $\to$ part	lund-self	part $\to$ lund	part-self
	(/136)	(/136)	(/291)	(/291)
1	0.942	0.878	0.202	0.290
2	0.969	0.910	0.384	0.407
3	0.977	0.947	0.504	0.555
4	0.990	0.977	0.692	0.709
5	0.998	0.986	0.913	0.926

Table 3: Retained rej. at

\varepsilon_{S}=0.5

, primary-branch run, source-normalized (

R^{\mathrm{src}}

), mean over 5 seeds. lund

\to

part and lund-self both divide by Lund-ParT’s own standalone value (136); part

\to

lund and part-self both divide by ParT’s own standalone value (291). Self-stitching values coincide with

R^{\mathrm{tgt}}

Figure 3: Linear CKA and SVCCA mean canonical correlation between ParT’s and Lund-ParT’s class tokens versus depth.

Figure 4: Jet mass and

p_{T}

for all background jets, the subset rejected by standalone ParT at its own

\varepsilon_{S}=0.5

threshold, and the subset the lund-to-part stitch (

d=1

) fails to reject that standalone ParT would have. The affected population concentrates at and above the top quark mass; there is no corresponding effect in

p_{T}

Figure 5: Lund-to-part absolute background rejection at

\varepsilon_{S}=0.5

and

0.3

, primary-branch Lund-ParT versus the full-declustering-tree rebuild, same ParT target throughout. The horizontal line marks ParT’s own standalone value.

Figure 6: The Cambridge/Aachen declustering tree of one real top jet. The primary branch is the horizontal spine; the subtrees hanging below it are what the primary Lund plane discards and the full tree keeps. The jet was chosen to be small enough to draw and to have a primary fraction (

0.26

) close to the population’s

12.0/48.0

, but is otherwise unremarkable.

Figure 7: Lund-plane density $\rho=N_{\mathrm{jet}}^{-1}\,\mathrm{d}n/\mathrm{d}\ln(1/\Delta R)\,\mathrm{d}\ln k_{T}$ of the two node populations, over all 40,000 test jets. Dotted lines mark $k_{T}=1$ GeV and $\Delta R=0.1$ . The colour scales are independent, since the secondary population is three times as numerous and a shared scale leaves the left panel unreadable; the per-jet node counts in the titles fix the relative normalisation.

condition	nodes	rej. at $\varepsilon_{S}=0.5$	AUC
all nodes	48.0	$230.6\pm 8.2$	$0.9818$
drop 12 secondary	36.2	$186.5\pm 16.3$	$0.9789$
drop 12 random	35.9	$53.9\pm 3.5$	$0.9292$
drop 12 primary	35.9	$9.2\pm 5.6$	$0.8307$
keep 12 primary	12.0	$34.7\pm 12.4$	$0.9229$
keep 12 random	12.0	$4.4\pm 1.8$	$0.6845$

Table 4: Background rejection and AUC under each masking condition, full-tree Lund-ParT, mean

\pm

standard deviation over 5 seeds, 40,000 test jets. Conditions are grouped into matched pairs: the same number of surviving nodes, differing only in which nodes those are.

Figure 8: Left: background rejection under each masking condition, with the matched pairs adjacent. Right: rejection as a growing fraction of each jet’s secondary nodes is removed, ending at the primary branch alone. The dashed line marks the complementary condition, the primary branch removed and every secondary node kept.

dataset	condition	rej. at $\varepsilon_{S}{=}0.5$	loss	excess vs. random
top tagging	unablated	$136.28\pm 4.26$	—	—
	remove first	$40.13\pm 0.83$	$-70.6\%$	$+35.2\%$ ( $t{=}55.8$ )
	remove random	$88.07\pm 2.00$	$-35.4\%$	—
	remove last	$128.12\pm 1.93$	$-6.0\%$	$-29.4\%$ ( $t{=}-24.0$ )
quark/gluon	unablated	$30.62\pm 0.17$	—	—
	remove first	$22.32\pm 0.26$	$-27.1\%$	$+11.5\%$ ( $t{=}21.3$ )

Table 5: One node removed per jet from the standalone primary-branch Lund-ParT, mean

\pm

standard deviation over 5 seeds. “Excess” is the paired difference against the random control, as a percentage of the unablated rejection, with its paired

t

statistic.

Figure 9: Left: loss in background rejection when one node is removed. Right: the same conditions under the threshold-free per-jet score shift. Error bars are the seed standard deviation.

Figure 10: The measured importance placed on the angular scale it corresponds to. Circles mark the median

\Delta R

of successive declustering steps; filled circles are the two steps actually ablated, with area proportional to their excess damage over the random control, and hollow circles are the intermediate steps, which were not measured. Dotted lines mark the top and

W

decay opening angles

2m/p_{T}

p_{T}=600

GeV, drawn on the top-tagging lane only since they are a statement about top jets. The first splitting of a top jet falls close to these scales; quark/gluon jets have no such scale, and their importance profile is correspondingly flatter.

representation	hypergraph	this paper’s network
	rej. at $\varepsilon_{S}=0.5$	rej. at $\varepsilon_{S}=0.5$
particles	133	$208\pm 12$
Lund, primary branch	116	$112\pm 7$
Lund, full tree	181	$201\pm 11$

Table 6: Hypergraph taggers against the corresponding stitching-study networks, all at 10% of the dataset. The hypergraph column is a single seed and carries no uncertainty; the right-hand column is the mean over 5 seeds at the same scale.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

	ParT	Lund-ParT (primary)	Lund-ParT (full tree)
AUC	$0.9110\pm 0.0011$	$0.8905\pm 0.0003$	$0.9006\pm 0.0007$
rej. at $\varepsilon_{S}=0.5$	$39.44\pm 0.73$	$30.59\pm 0.17$	$33.54\pm 0.52$
rej. at $\varepsilon_{S}=0.3$	$98.14\pm 2.52$	$82.13\pm 0.92$	$88.12\pm 1.55$

Appendix

Table 7: Standalone test performance on quark/gluon tagging, mean

\pm

standard deviation over 5 seeds.

	$d$
representation	1	2	3	4	5
primary branch	$15.1\pm 8.2$	$21.9\pm 7.0$	$25.0\pm 4.6$	$23.6\pm 12.4$	$30.1\pm 0.3$
full tree	$7.8\pm 7.1$	$15.2\pm 7.9$	$20.8\pm 6.0$	$30.6\pm 0.8$	$32.8\pm 0.7$

Appendix

Table 8: Absolute background rejection at

\varepsilon_{S}=0.5

for the lund-to-part stitch on quark/gluon tagging, both Lund representations, against the same ParT target. ParT’s own standalone value is

39.44

	$d$
direction	1	2	3	4	5
lund $\to$ part	$0.930\pm 0.053$	$0.963\pm 0.016$	$0.969\pm 0.008$	$0.858\pm 0.263$	$0.977\pm 0.002$
part $\to$ lund	$0.941\pm 0.066$	$0.839\pm 0.341$	$0.989\pm 0.022$	$1.004\pm 0.007$	$1.016\pm 0.006$
part-self	$0.931\pm 0.045$	$0.953\pm 0.035$	$0.964\pm 0.021$	$0.981\pm 0.008$	$0.993\pm 0.004$
lund-self	$0.970\pm 0.026$	$0.981\pm 0.018$	$0.994\pm 0.002$	$0.997\pm 0.001$	$0.998\pm 0.003$

Appendix

Table 9: Retained AUC, target-normalized (

R^{\mathrm{tgt}}

), primary-branch run, quark/gluon tagging.

	$d$
direction	1	2	3	4	5
lund $\to$ part	$0.494\pm 0.268$	$0.715\pm 0.228$	$0.818\pm 0.149$	$0.772\pm 0.405$	$0.983\pm 0.011$
lund-self	$0.547\pm 0.259$	$0.655\pm 0.234$	$0.858\pm 0.097$	$0.944\pm 0.038$	$0.967\pm 0.039$
part $\to$ lund	$0.524\pm 0.188$	$0.542\pm 0.300$	$0.679\pm 0.166$	$0.799\pm 0.077$	$0.918\pm 0.042$
part-self	$0.460\pm 0.225$	$0.582\pm 0.176$	$0.587\pm 0.170$	$0.790\pm 0.081$	$0.900\pm 0.082$

Appendix

Table 10: Retained rejection at

\varepsilon_{S}=0.5

, source-normalized (

R^{\mathrm{src}}

), primary-branch run, quark/gluon tagging. lund

\to

part and lund-self divide by Lund-ParT’s standalone

30.59

; part

\to

lund and part-self divide by ParT’s

39.44

. Uncertainties propagate the numerator’s seed spread only; the denominators are known to

0.6\%

and

1.9\%

respectively.

Figure 11: Absolute background rejection at

\varepsilon_{S}=0.5

and

0.3

for all four stitching conditions on quark/gluon tagging, primary-branch run, versus depth. Horizontal lines mark each network’s own standalone value. Compare to Fig. 2 for top tagging.

	CKA	SVCCA
primary branch, $d{=}1$	$0.639\pm 0.118$	$0.769\pm 0.091$
primary branch, $d{=}6$	$0.884\pm 0.008$	$0.925\pm 0.034$
full tree, $d{=}1$	$0.379\pm 0.119$	$0.670\pm 0.111$
full tree, $d{=}6$	$0.919\pm 0.005$	$0.948\pm 0.026$

Appendix

Table 11: Geometric alignment between ParT’s and Lund-ParT’s class tokens, quark/gluon tagging.

Figure 12: Linear CKA and SVCCA mean canonical correlation between ParT’s and Lund-ParT’s class tokens versus depth, quark/gluon tagging. Compare to Fig. 3 for top tagging.

representation	$d$	leaked	% of rejected set	newly rejected
primary branch	1	$2607\pm 51$	$2.7\%$	643
primary branch	5	$1306\pm 36$	$1.3\%$	486
full tree	1	$5539\pm 74$	$5.7\%$	655
full tree	5	$897\pm 30$	$0.9\%$	400

Appendix

Table 12: Leakage relative to standalone ParT’s rejected set, quark/gluon tagging.

Figure 13: Jet mass and

p_{T}

for all gluon jets, the subset standalone ParT rejects at its own

\varepsilon_{S}=0.5

threshold, and the subset the lund-to-part stitch at

d=1

fails to reject that standalone ParT would have.

Figure 14: As Fig. 13 , at

d=5

	primary/ParT	full tree/ParT	gap closed by tree
quark/gluon	$77.6\%$	$85.0\%$	$33\%$
top tagging	$46.9\%$	$85.2\%$	$72\%$

Appendix

Table 13: Standalone rejection at

\varepsilon_{S}=0.5

as a fraction of ParT’s, on the two datasets.

Figure 15: Lund-to-part absolute rejection on quark/gluon tagging, primary branch versus full declustering tree, against the same ParT target throughout. The horizontal line marks ParT’s own standalone value. Compare to Fig. 5 for top tagging.

Figure 16: Primary branch against full declustering tree across the stitch sweep, quark/gluon tagging.

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