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This paper tries to establish that the d=5 magic state cultivation circuit, whose 53 non-Clifford gates naively require a 6,377,292-term magic cat stabiliser decomposition, can be simulated as a weighted sum of about 8 pure Clifford ZX-diag

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A d=5 magic state cultivation circuit with 53 non-Clifford gates can be represented, with 0.1% edge noise, as about 8 Clifford diagrams on average instead of 6,377,292.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection The ~8-term noise-robustness result is real and worth knowing; the end-to-end sampling method ignores interference and overstates the simulation cost claim. the 3 major comments →

arxiv 2509.08658 v8 pith:IPSORCG6 submitted 2025-09-10 quant-ph

Simulating magic state cultivation with few Clifford terms

Kwok Ho Wan , Zhenghao Zhong , Ainhoa Zapirain This is my paper
classification quant-ph PACS 03.67.Lx03.67.Pp
keywords magic state cultivationstabiliser decompositionZX-calculusClifford simulationPauli errorslogical error ratemagic cat statescutting decomposition
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Magic state cultivation is a fault-tolerance protocol that grows the hard-to-manufacture T states needed for universal quantum computation; the d=5 version contains 53 non-Clifford gates, which is why prior simulations replaced them with cheaper Clifford gates or used a 6,377,292-term stabiliser sum. This paper reports a numerical recipe that, under per-edge depolarising noise at realistic rates (10^-4 to 10^-3), represents the errored cultivation circuit as an average of about 8 pure Clifford ZX-diagrams (about 4 for the smaller d=3 circuit), a reduction of more than 700,000 in term count. The recipe combines a cutting decomposition that splits non-Clifford phase spiders into Clifford pieces with a secondary step (magic cat, BSS, or closed-Pauli-web post-selection) that controls leftover non-Clifford parts. If the average stays this low, Monte-Carlo logical error rate simulation no longer needs to swap T for S gates, and the cultivated state can be ported into a larger Clifford circuit with overhead proportional to about 8 terms per shot. The result is presented as numerical evidence, not a proven bound.

Core claim

The central discovery is empirical: applying the cutting decomposition from the authors' companion work to Pauli-errored ZX-diagrams of the d=3 and d=5 cultivation circuits keeps the average number of pure Clifford terms near 4 and 8, respectively, for edge error rates 10^-4 to 10^-3, against a baseline of 6,377,292 pure Clifford terms from the magic cat state stabiliser decomposition of all 53 non-Clifford spiders. The reduction is not automatic: if measurement spiders are randomly flipped and the secondary decomposition is the magic cat expansion, the average jumps to 10^2 to 10^3 terms; the low average is restored by using the BSS stabiliser decomposition or by post-selecting on closed Pa

What carries the argument

The cutting stabiliser decomposition is the central mechanism: a ZX-calculus rewriting that turns selected non-Clifford phase spiders into small combinations of Clifford diagrams, applied to diagrams where every edge carries an independent Pauli X/Y/Z error. When the cutting step leaves behind spiders that are still non-Clifford, a secondary decomposition (magic cat, BSS, or additional cuts) finishes the job; closed Pauli webs and detecting regions serve as a post-selection filter that decides which error realisations even need to be decomposed. The work these pieces do is to keep the pure-Clifford term count near 4 or 8 across the operationally relevant noise range, while still preserving t

Load-bearing premise

The claim stands on the empirical observation—not a proof—that the chosen cutting recipe keeps about 8 terms on average when every wire is independently depolarised at rates near 10^-3; if a realistic circuit-level noise model or a different decomposition choice inflates that average, the near-Clifford simulation story collapses.

What would settle it

Run the same decomposition on the d=5 cultivation circuit with circuit-level SD6 noise (the noise model used in the original cultivation paper) instead of per-edge depolarising noise; if the average number of terms in a sufficiently large sample exceeds a few tens at p=10^-3, the claim does not transfer. Independently, contract the eight Clifford terms of the noiseless companion-paper decomposition and compare against direct state-vector simulation; any discrepancy beyond numerical precision would invalidate the base decomposition.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For d=3 and d=5 cultivation circuits, per-shot sampling can run nearly as fast as a stabiliser simulation: about 4×10^6 shots per second on a laptop at SD6-level noise, within about 1.1 times the speed of a fully Clifford proxy.
  • End-to-end logical error rate simulation of cultivation becomes feasible without the T-to-S approximation that caused an about 2× logical error rate discrepancy in the d=3 benchmark.
  • The final cultivated state is delivered as a sum of about 8 Clifford tableaus, so the escape stage—tomography or injection into a much larger near-Clifford circuit—can be handled with overhead proportional to the mean term count.
  • Low average term counts survive even when -1 measurement outcomes are allowed, provided the decomposition is supplemented with BSS cuts or the simulation post-selects on un-violated closed Pauli webs.
  • The magic-cat-only secondary route without post-selection is not practical at 10^-3 noise (100–1000 terms), so the choice of secondary decomposition and post-selection is load-bearing for the claim.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A likely broader lesson, not proven here: for structured non-Clifford circuits, simulability under realistic noise is governed less by raw T-count than by how the T gates are wired and by which measurement outcomes are accepted; circuits with a low post-selection discard ratio may inherit a low decomposition cost.
  • The rare 432-term tail implies that worst-case shots dominate runtime variance; a robust simulator should either count over-threshold shots as logical failures, as the paper suggests, or terminate and retry, so the average term count alone understates the cost of a long run.
  • The closed-Pauli-web approach could be tested directly as a noise filter: because it discards most rejected shots before decomposition, the cost per accepted shot may be even lower than the about-8 average, and the discard-ratio curves already show the expected trade-off.
  • A natural next experiment is to apply the same cutting-plus-post-selection recipe to other magic state cultivation variants; if the about-8-term average persists, near-Clifford simulation of 50-T-class circuits may be a general phenomenon rather than a feature of this one protocol.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper proposes that the d=5 magic state cultivation circuit can be simulated classically by representing its ZX-diagram, under per-edge depolarising noise, as a sum of on average about 8 Clifford ZX-diagrams, compared with 6,377,292 terms for a magic-cat-state stabiliser decomposition. It reports numerical experiments on the average and maximum number of terms for d=3 and d=5 circuits, with different secondary decompositions (magic cat, BSS) and with post-selection on closed Pauli webs, and sketches an end-to-end simulation procedure. The abstract additionally claims a tsim-based throughput of nearly 4e6 shots/s for d=3 at SD6 noise. The paper is an empirical sketch; no code or data are shipped.

Significance. If the ~8-term average were robust and if a valid sampling procedure existed, the result would be significant: it would bring a 53-T-gate cultivation circuit into a regime where near-Clifford simulation is feasible, and the explicit closed Pauli webs in Appendices D/E and the Appendix A observation that the flag-free double-checking circuit equals I^⊗n+(XS†)^⊗n are concrete, reusable contributions. However, the central end-to-end simulation claim is not justified by the current manuscript: the proposed sampling step ignores quantum interference, and the headline benchmark appears only in the abstract. The paper is currently a preliminary research sketch whose main quantitative evidence is not independently verifiable without code/data.

major comments (3)
  1. [§8.1, Algorithm 1; §8.3] The proposed sampling procedure is not a valid estimator for measurement probabilities. A decomposition of the form |ψ⟩ = Σ_j c_j |C_j⟩ (Eqs. (11)-(12)) gives p(m) = |Σ_j c_j ⟨m|C_j⟩|^2, which contains cross terms 2 Re Σ_{j<k} c_j^* c_k ⟨m|C_j⟩^*⟨m|C_k⟩ unless the Clifford terms are orthogonal on the measured algebra. Algorithm 1 samples one term and reads its measurement outcomes; §8.3 instructs to 'simulate χ different numerical experiments' per shot. Both estimate only the diagonal terms. The paper does not state or prove that the Clifford terms are orthogonal, and the phases in Eqs. (6)-(7) and (11)-(12) make orthogonality non-obvious. Consequently, the claim 'only track ≈8 Clifford terms per shot' does not imply a valid Monte Carlo estimate of logical error rates; at least the estimator's bias must be analysed or the procedure replaced by an exact amplitude-summing method.
  2. [Abstract vs §6 and §8-§9] The abstract promises 'numerical results for full non-Clifford stabiliser rank simulation based on tsim' and a throughput of 'nearly 4×10^6 shots per second', 'only ~1.1 times slower' than a stim proxy, at SD6 noise p=0.0005. The body does not contain this benchmark. Section 6 instead says a 'definitive test would be to develop and integrate quizx with the code from [18] and carry out the end-to-end simulations', and §8 is explicitly a sketch. Either the benchmark must be reported with methodology and data, or the abstract must be corrected. This is not a presentation issue: the practical payoff of the paper depends on this claim.
  3. [§5 and §7; Figures 1-6] The numerical evidence for the central '~8 terms' claim is not reproducible from the manuscript. Sample sizes differ (10^4 shots in §5, 10^5 in §7), no confidence intervals are reported, and the maximum term count at p=0.0015 (432 terms) is a single event that the authors attribute to sample size. The decomposition pipeline is underspecified: the trigger for secondary decompositions, the number/placement of extra cuts, and the exact BSS routine are described only qualitatively (§5-§6). Figure 3 shows that with random measurement flips and magic-cat secondary decomposition the average is 10^2-10^3, so the low average is not a property of the circuit alone but of a particular heuristic pipeline. Code/data or a precise algorithm description is needed to verify the main quantitative claim.
minor comments (3)
  1. [Section 5, Eqs. (11)-(12)] The 4- and 8-term base decompositions are taken from the authors' companion paper [2] and are not proved in this manuscript. If Eqs. (11)-(12) are incorrect, the noise-averaged results inherit the error. I am not claiming circularity, but since the current paper's central claim rests on these external decompositions, a concise proof or a machine-checkable certificate for Eqs. (11)-(12) should be included or explicitly cited with the relevant theorem.
  2. [Figure 6] The comparison of discard ratios mixes different error models (per-edge depolarising vs SD6 circuit-level noise). The text acknowledges this, but the figure could mislead; please label the models clearly and state that the comparison is qualitative.
  3. [Throughout] There are several informal remarks and unpolished sentences ('are need', 'Probably not useful', 'we wonder if'). These are presentation issues but should be cleaned for a journal submission.

Circularity Check

0 steps flagged

No circularity: term-count results are outputs of the decomposition algorithm, and the companion-paper decompositions are restated and checkable.

full rationale

The claimed quantities (average ~8 terms for d=5 and ~4 for d=3 under per-edge depolarising noise) are empirical outputs of the cutting decomposition applied to Pauli-errored ZX-diagrams under Eq. (13). They are not fit parameters reused as predictions, and no equation defines the result in terms of itself. The error-free 4/8-term decompositions are attributed to the authors' companion paper [2], but Eqs. (11)-(12) restate them as explicit ZX-diagram equalities, so the central identity is not accepted solely on the authority of a self-citation. The choice of secondary decomposition (magic cat vs BSS) and the post-selection on closed Pauli webs are disclosed algorithmic conditioning choices that affect the averages, and the paper reports the adverse regime (random measurement flips, Section 6/Fig. 3) rather than hiding it. The paper's own qualifications ('preliminary simulations', 'definitive test would be...') corroborate that the term-counts are exploratory outputs rather than definitional predictions. The potential neglect of interference cross-terms in Algorithm 1/Section 8.3 is a correctness question about whether term-sampling reproduces Born-rule probabilities; even if valid, it is not an input-output identity and, under the given instructions, is not scored as circularity. No circular step can be exhibited.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

No new physical entities are introduced. The paper's dependence is on algorithmic heuristics (cutting strategy, secondary decomposition choice), a specific depolarising error model, and ZX-calculus rules pulled from prior literature. The 4/8-term error-free decompositions come from the authors' own earlier preprint.

free parameters (2)
  • cutting decomposition heuristic settings (number/placement of cuts, choice of secondary decomposition: magic cat states,
    Hand-tuned algorithm parameters that strongly affect the headline ~8 average; Section 6 shows the average jumps to 10^2-10^3 with one secondary decomposition and returns to ~8 with another (Figures 3-4).
  • secondary decomposition trigger threshold
    The paper applies magic cat or BSS decompositions when a term contains more than one non-Clifford spider; this threshold is a chosen algorithm parameter (Sections 5-6).
axioms (4)
  • domain assumption ZX-calculus diagrammatic rules used for spider fusion, unfusion, and cuts are sound and complete
    The rewrites are taken as valid from [6,12,24]; no formal proof is given in this paper.
  • domain assumption The per-edge depolarising channel Eq. (13) with rate p is an adequate error model for simulating cultivation logical error rates
    Section 5; the authors note it has higher error density than SD6 circuit-level noise and corresponds to a different model, so the relevance to realistic devices is assumed.
  • ad hoc to paper The error-free d=3 and d=5 circuits admit cutting decompositions into exactly 4 and 8 Clifford terms as stated in [2] (Eqs. 11-12)
    Central input from the authors' own companion paper [2], not re-derived here; the noise study builds on this 4/8-term base.
  • domain assumption Detectors/closed Pauli webs from [18] are correctly translated into ZX-diagram subgraphs (appendices D/E)
    Required for the post-selected simulations in Section 7; only some are shown in the text, the rest are in supplementary tikz files not provided here.

reviewed 2026-08-04 · how reviews work

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Cite this review

Pith. "Pith review of Simulating magic state cultivation with few Clifford terms." pith.science (2026). https://pith.science/paper/IPSORCG6

@misc{pith2026250908658,
  author       = {Pith},
  title        = {Pith review of: Simulating magic state cultivation with few Clifford terms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IPSORCG6}},
  note         = {Machine review of arXiv:2509.08658}
}
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abstract

Building upon [arXiv:2509.01224], we present a few methods on how to simulate the non-Clifford $d=5$ magic state cultivation circuits [arXiv:2409.17595] with a sum of $\approx 8$ Clifford ZX-diagrams on average, at $0.1\%$ noise. Compared to a magic cat state stabiliser decomposition of all $53$ non-Clifford spiders ($6{,}377{,}292$ terms required), this is more than $7 \times 10^{5}$ times reduction in the number of terms. Our stabiliser decomposition has the advantage of representing the final non-Clifford state (in light of circuit errors) as a sum of Clifford ZX-diagrams. This will be useful in simulating the escape stage of magic state cultivation, where one needs to port the resultant state of cultivation into a larger Clifford circuit with many more qubits. Still, it's necessary to only track $\approx 8$ Clifford terms. Our result sheds light on the simulability of operationally relevant, high $T$-count quantum circuits with some internal structure. Finally, we provide numerical results for full non-Clifford stabiliser rank simulation based on $\mathtt{tsim}$ along with optimisations using our cutting decompositions. Nearly $4\times 10^{6}$ shots per second can be obtained on a laptop for the smaller $d = 3$ circuits at SD6 circuit level noise $p=0.0005$, making it only $\sim$$1.1$ times slower than its (circuit-unspecific and un-optimised) fully Clifford proxy simulation via $\mathtt{stim}$ using $S$ gates.

Figures

Figures reproduced from arXiv: 2509.08658 by Ainhoa Zapirain, Kwok Ho Wan, Zhenghao Zhong.

Figure 1
Figure 1. Figure 1: Average and maximum number of terms in the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Average and maximum number of terms in the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Average and maximum number of terms in the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Average and maximum number of terms in the cutting stabiliser decomposition followed by magic cat secondary decompositions, for the d = 5 circuit. The error realisation were post-selected such that all de￾tecting regions returned +1 parity. We also included the discard ratio at each error rate. We have repeated the simulations from the pre￾vious sections, this time post-selecting on all the detecting regio… view at source ↗
Figure 7
Figure 7. Figure 7: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
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Figure 9. Figure 9: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
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Figure 10. Figure 10: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
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Figure 11. Figure 11: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
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Figure 12. Figure 12: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
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Figure 13. Figure 13: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
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Figure 14. Figure 14: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
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Figure 15. Figure 15: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p015_15.png] view at source ↗
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Figure 16. Figure 16: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p016_16.png] view at source ↗
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Figure 17. Figure 17: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p016_17.png] view at source ↗
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Figure 18. Figure 18: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p016_18.png] view at source ↗
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Figure 19. Figure 19: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p016_19.png] view at source ↗
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Figure 20. Figure 20: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p017_20.png] view at source ↗
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Figure 21. Figure 21: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p017_21.png] view at source ↗
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Figure 22. Figure 22: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p017_22.png] view at source ↗
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Figure 23. Figure 23: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p017_23.png] view at source ↗
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Figure 24. Figure 24: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p018_24.png] view at source ↗
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Figure 25. Figure 25: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p018_25.png] view at source ↗
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Figure 26. Figure 26: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p018_26.png] view at source ↗
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Figure 27. Figure 27: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p018_27.png] view at source ↗
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Figure 28. Figure 28: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p019_28.png] view at source ↗
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Figure 29. Figure 29: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p019_29.png] view at source ↗
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Figure 30. Figure 30: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p019_30.png] view at source ↗
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Figure 31. Figure 31: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p019_31.png] view at source ↗
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Figure 32. Figure 32: Detecting region/closed Pauli web/detector [PITH_FULL_IMAGE:figures/full_fig_p020_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: Detector 81. 7π 4 7π 4 π 4 π 4 π 4 π 4 π 4 π 4 π 4 π 4 π 4 π 4 7π 4 7π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 π 4 7π 4 7π 4 7π 4 π 4 7π 4 π 4 7π 4 π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 π 4 π 4 π 4 7π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 7π 4 π 4 [PITH_FULL_IMAGE:figures/full_fig_p021_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: Detector 48. 7π 4 7π 4 π 4 π 4 π 4 π 4 π 4 π 4 π 4 π 4 π 4 π 4 7π 4 7π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 π 4 7π 4 7π 4 7π 4 π 4 7π 4 π 4 7π 4 π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 π 4 π 4 π 4 7π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 7π 4 π 4 [PITH_FULL_IMAGE:figures/full_fig_p021_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: Detector 55. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: Detector 33. 7π 4 7π 4 π 4 π 4 π 4 π 4 π 4 π 4 π 4 π 4 π 4 π 4 7π 4 7π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 π 4 7π 4 7π 4 7π 4 π 4 7π 4 π 4 7π 4 π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 π 4 π 4 π 4 7π 4 7π 4 7π 4 π 4 7π 4 7π 4 π 4 7π 4 π 4 [PITH_FULL_IMAGE:figures/full_fig_p022_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: Detector 88. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_37.png] view at source ↗

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Forward citations

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.