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Neural networks reconstruct full two-dimensional CFT torus and annulus partition functions from modular or open/closed crossing, a spectral gap, and a single interior anchor.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · grok-4.5

2026-07-31 05:51 UTC pith:J2RVOPZY

load-bearing objection Solid numerical pipeline extending their neural bootstrap to modular/Cardy constraints; the selection-principle claim is weaker than the abstract once you inspect the ansatz and the bimodal WZW runs.

arxiv 2607.24913 v1 pith:J2RVOPZY submitted 2026-07-27 hep-th

Neural Spectral Bias and Conformal Correlators II: Modular and Annulus Bootstrap

Kausik Ghosh , Sidhaarth Kumar , Vasilis Niarchos , Andreas Stergiou This is my paper
classification hep-th
keywords neural bootstrapmodular invarianceannulus partition functionCardy conditionspectral biastwo-dimensional CFTtwist fieldsdefect-changing operators
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.

This paper shows that modular invariance of a torus partition function and Cardy open/closed duality of an annulus partition function can be rewritten as ordinary four-point crossing equations, then solved by the same anchored neural method used for correlators. A lightweight network is trained only on that crossing equation, the leading gap exponents, and the value of the reduced correlator at one interior point. The problem is under-determined: many smooth crossing-symmetric functions satisfy the same sparse data. Yet across unitary and non-unitary minimal models, WZW models, and non-compact examples such as Liouville, the network consistently recovers the physical partition function to sub-percent accuracy on the diagonal line. The selection is attributed to spectral bias in the lazy-training regime. If this bias continues to pick physical solutions, sparse consistency data become enough to rebuild entire partition functions without searching spectra or OPE coefficients directly.

Core claim

When modular S-invariance and the Cardy condition are cast as four-point crossing for twist fields or defect-changing operators, an anchored neural bootstrap—crossing loss plus a gap-weighted ansatz plus one interior anchor—reconstructs the full diagonal reduced correlator, and therefore the corresponding torus or annulus partition function, with high accuracy. Spectral bias in the lazy-training regime systematically selects the physical crossing-symmetric configuration among the many functions allowed by the same sparse input.

What carries the argument

Anchored neural bootstrap for modular and annulus crossing: the partition function is rewritten as a reduced four-point correlator on a line; a gap-dependent prefactor is factored out; a small feed-forward network learns the remainder under a crossing loss and a single anchor value; spectral bias in lazy training supplies the selection principle that picks the physical solution.

Load-bearing premise

The method assumes that neural spectral bias reliably prefers the partition function of a real consistent CFT over the large space of other smooth functions that obey the same crossing equation, gap, and single anchor.

What would settle it

Train the same architecture on a known rational CFT with fixed gap and anchor; if the ensemble mean of the reconstructed reduced correlator systematically deviates from the exact modular-invariant answer by more than the reported sub-percent errors (or lands on a stable wrong peak even after early-stopping filters), the selection claim fails.

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

If this is right

  • Full diagonal torus and annulus partition functions can be rebuilt from modular or Cardy crossing plus minimal spectral input without positivity or unitarity.
  • The same sparse protocol applies to non-compact theories, including free non-compact bosons and Liouville ZZ/FZZ annuli, once scales are balanced in the loss or ansatz.
  • Modular and annulus constraints become instances of the same anchored four-point neural search used for ordinary correlators.
  • Higher-genus sewing constraints and mixed defect systems become natural next targets for the same function-space search.
  • When ensembles are bimodal, early stopping can serve as a practical filter that retains the physical low-loss peak.

Where Pith is reading between the lines

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

  • If spectral bias is the real selector, one could deliberately vary network depth, width, or activation to map which function classes are preferred and turn the bias into a diagnostic of ‘physical’ smoothness.
  • The method may extend to off-diagonal modular parameter by learning concentric circles in the cross-ratio plane, yielding the full complex-structure dependence from the same sparse anchors.
  • Combining the neural reconstruction with independent spectral bounds could produce hybrid bootstraps that output complete partition functions rather than only exclusion plots.
  • Failure modes at large central charge, where correlators span many orders of magnitude, suggest that adaptive rescaling of the loss will be needed before the method is routine for holographic or high-c theories.

Editorial analysis

A structured set of objections, weighed in public.

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

Circularity Check

3 steps flagged

Mild circularity only: exact anchor plus self-cited spectral-bias premise; full-curve match is still a real test, but WZW early-stop filtering is answer-aware.

specific steps
  1. fitted input called prediction [§3, Eqs. (3.7)–(3.10) and (3.22)–(3.25)]
    "Lanc = (ẽG(z0)−Bt(z0)Gexact(z0))2, with one anchor at z0=0.3. … L(ann)anc=(ẽG(o)(z0)−Ba(z0)G(open)exact(z0))2+(ẽG(c)(z0)−Ba(z0)G(closed)exact(z0))2, with a single anchor at z0=0.3 for each of the two channels."

    The sole interior data point supplied to the loss is the exact target value of the reduced correlator (or both open/closed channels). That number is therefore fitted by construction; only the off-anchor shape is a genuine reconstruction test. Calling the overall procedure reconstruction from “sparse data” is fair for the curve, but any claim that the method predicts the anchored value itself would be circular.

  2. self citation load bearing [Abstract; §1; §7 (citing [9,10])]
    "A key ingredient of this approach is the spectral bias of the neural networks in the lazy training regime, which selects specific crossing-symmetric configurations. … our previous study [9,10] revealed a remarkable fact. In all examples studied there, the network always selected a solution close to a physical correlator. The main goal of the present paper is to present experimental evidence that this bias towards physical correlators extends to modular-invariant partition functions as well."

    The central selection principle—that lazy-training spectral bias picks the physical CFT partition function among many crossing-symmetric functions obeying the same gap and anchor—is not derived in this paper. It is load-bearing narrative imported from the same authors’ prior neural-bootstrap papers and then re-illustrated on modular/annulus examples. Without that self-cited premise, crossing+gap+anchor alone do not uniquely determine the reported curves.

  3. other [§4.4 ŝu(2)2 (and App. A.1 ŝu(2)1)]
    "In this case, we observed that the full ensemble of runs is bimodal, exhibiting two peaks at z=0.5 cleanly separated. Among the seed-level diagnostics we surveyed, the only feature that correlated with the peak a run landed on was whether the run triggered early stopping. Runs that terminated via the 5×10^3-epoch stagnation criterion … fell almost exclusively into the peak tracking the exact answer, while runs that ran out the full epoch budget clustered around the spurious peak. … Retaining only the early-stopped runs selected 92 of 1000 seeds."

    When selection is nontrivial, the majority of seeds miss the physical peak. The early-stopping cut that retains the physical minority is chosen because it correlates with the already-known exact answer. That is post-hoc, answer-aware filtering: the reported “NN prediction” for these models is the filtered sub-ensemble that matches the target, not an unsupervised output of spectral bias alone. For an unknown partition function the same cut would lack independent justification.

full rationale

The modular/Cardy-to-crossing reformulation and the gap exponents are independent external inputs, not defined from the target curves. The network is trained on crossing plus one interior anchor that is literally the exact reduced correlator at z0 (Eqs. 3.10, 3.25), so that single number is fitted, not predicted; the nontrivial claim is recovery of the rest of the function. That recovery is checked against closed-form or character-sum partition functions the authors already know, which is a legitimate reconstruction benchmark rather than a self-definitional loop. Load-bearing selection language (“spectral bias … selects specific crossing-symmetric configurations”) is imported from the authors’ companion papers [9,10] and treated as an empirical fact, not re-derived here—self-citation that supports the narrative but does not algebraically force the curves. In the ŝu(2)1/2 cases the unfiltered ensemble is bimodal and only the early-stopped minority is retained because it tracks the known answer; that filter is answer-aware and would not be independently justified for an unknown theory. None of these steps make the reported full-interval accuracy equivalent to the inputs by construction, so the circularity remains mild (score 3).

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The load-bearing content is empirical: spectral bias plus sparse CFT input yields physical partition functions. Almost all CFT structure (modular S, Cardy states, Verlinde, twist-field map, characters) is standard. Free choices are network/hyperparameter and anchor location; the unproved selection principle is the main domain assumption imported from the authors’ previous work.

free parameters (4)
  • anchor weight λ_anc = 100
    Fixed to 100 in the composite loss for all torus and annulus runs; controls how hard the single interior point is enforced versus crossing.
  • anchor location z0 = 0.3 (default)
    Chosen by hand (usually 0.3; 0.7 for Lee–Yang and some Liouville runs); not derived.
  • MLP architecture and training schedule = 2×64 GELU; schedule as in §3
    Two hidden layers width 64, GELU, Adam lr 5e-4, weight decay 1e-6, StepLR γ=0.98/500 epochs, max 2e5 epochs, early stop after 5e3 stagnant epochs—copied from companion paper and held fixed.
  • gap Δ_gap and leading channel exponents = theory-dependent
    Read from the target theory’s spectrum (or Cardy/Verlinde decompositions) and hard-coded into the ansatz prefactors; required input, not learned.
axioms (5)
  • domain assumption Modular S-invariance of the torus partition function is equivalent to crossing of four identical Z2 twist fields on CP1, reducing on the diagonal to G(z)=(z/(1-z))^{c/4} G(1-z).
    Standard branched-cover / symmetric-orbifold map; invoked in §2.1 and used as the training constraint.
  • domain assumption Open/closed duality of the annulus equals crossing of a mixed four-point function of defect-changing operators with Δ=c/16 endpoints.
    Standard interface-CFT reformulation; §2.2.
  • ad hoc to paper In the lazy-training regime, feed-forward nets exhibit spectral bias that preferentially selects smooth, physically realized crossing-symmetric correlators among many solutions compatible with gap and anchor data.
    Central working hypothesis imported from [9,10] and tested empirically here; not proved.
  • domain assumption Cardy boundary states and Verlinde open-channel multiplicities correctly give the annulus spectra for diagonal rational CFTs.
    Standard RCFT boundary CFT; §3 and §5.
  • ad hoc to paper Crossing-symmetric functions satisfying the same gap exponents and single anchor are highly non-unique, so successful full-interval reconstruction is a nontrivial selection effect.
    Stated in the Introduction and §7; underpins the claim that spectral bias is doing real work, but the size of the solution space is not characterized.

pith-pipeline@v1.2.0-grok45-kimik3 · 36459 in / 3532 out tokens · 83188 ms · 2026-07-31T05:51:47.735007+00:00 · methodology

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read the original abstract

We develop a neural network bootstrap framework for reconstructing partition functions of two-dimensional conformal field theories (CFTs) based on modular invariance and the Cardy condition, which are recast as crossing equations for four-point correlators. For torus partition functions, we use the twist-field representation in the symmetric-orbifold description to map modular S-invariance to four-point crossing and focus on the diagonal kinematics of four insertions on a line. For annulus partition functions, we formulate open/closed channel duality as crossing symmetry for mixed four-point functions of defect-changing operators in interface CFT. In both cases, the reconstruction problem is formulated in the anchored-bootstrap form, where the crossing constraints are supplemented by minimal spectral input (a gap) and anchor data. We solve this under-determined problem by using lightweight feed-forward neural networks to parametrise the correlators and their corresponding partition functions. A key ingredient of this approach is the spectral bias of the neural networks in the lazy training regime, which selects specific crossing-symmetric configurations. This reformulation unifies standard modular and annulus constraints in two dimensions with the anchored neural approach for CFT correlators, providing a new way to reconstruct full partition functions from sparse data with remarkable accuracy.

Figures

Figures reproduced from arXiv: 2607.24913 by Andreas Stergiou, Kausik Ghosh, Sidhaarth Kumar, Vasilis Niarchos.

Figure 1
Figure 1. Figure 1: The torus Στ presented as a two-sheeted branched cover of CP1 with branch points {0, λ(τ ), 1, ∞}. The two sheets (top and bottom) are pairwise identified along the cuts [0, λ(τ )] (blue) and [1, ∞) (red). The blue and red loops illustrate the Z2 monodromy at the two cuts; encircling a branch point once on sheet 1 (upper arc) crosses the corresponding cut and continues on sheet 2 (lower arc), so a single r… view at source ↗
Figure 2
Figure 2. Figure 2: The annulus partition function Zαβ in the open and closed channels. The open channel traces over Hαβ, while the closed channel propagates between boundary states |α⟩ and |β⟩. The appearance of 1/t reflects the exchange of the two directions of the annulus. In the open channel, Euclidean time runs around the annulus. In the closed channel, Euclidean time runs across the annulus. Thus, the aspect ratio is in… view at source ↗
Figure 3
Figure 3. Figure 3: , with z = λ(it) , t(z) = K(1 − z) K(z) . (2.16) β α annulus z = λ(it) Dα Dβ 0 λ(it) 1 ∞ CP1 with defect lines [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: NN-predicted reduced correlator Ge(z) for the M(4, 5) tricritical Ising model (c = 7/10, ∆gap = 3/40). The NN prediction at z = 0.5 is Gepred(0.5) = 1.13700 ± 0.00181 against Geexact(0.5) = 1.13426. Three-state Potts model: M(5, 6)D, c = 4/5, ∆D gap = 2/15. The canonical D-series example is the three-state Potts CFT, obtained from M(5, 6) by the (A4, D4) modular invariant. Its torus partition function is Z… view at source ↗
Figure 7
Figure 7. Figure 7: NN-predicted reduced correlator Ge(z) for the three-state Potts CFT (M(5, 6)D, c = 4/5, ∆D gap = 2/15). The NN prediction at z = 0.5 is Gepred(0.5) = 1.11342 ± 0.00091 against Geexact(0.5) = 1.11029. 4.4. Wess–Zumino–Witten Models WZW models are rational CFTs whose chiral algebra is the affine Kac–Moody algebra bgk, with g simple and k ∈ Z>0. The basic input for the torus reconstruction is fixed by the rep… view at source ↗
Figure 8
Figure 8. Figure 8: NN-predicted reduced correlator Ge(z) for the level-two sub (2)2 WZW model, on the filtered ensemble (92/1000 seeds after the early-stopping cut). The NN prediction at z = 0.5 is Gepred(0.5) = 0.84119 ± 0.01430 against Geexact(0.5) = 0.83776. 5. Annulus Reconstructions in Compact CFTs 5.1. Minimal Model Boundary Conditions The diagonal minimal model M(m, m + 1), with Kac table (4.10), has modular S-matrix … view at source ↗
Figure 9
Figure 9. Figure 9: NN-predicted reduced annulus correlators Ge(o)(z) (open, blue) and Ge(c)(z) (closed, red) for the Ising (α, β) = (1, 1) annulus over 100 seeds. At z = 0.5, Geexact = 0.7689 vs ensemble means 0.7685 (open) and 0.7681 (closed). 21 [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: NN-predicted reduced annulus correlators Ge(o)(z) (open, blue) and Ge(c)(z) (closed, red) for the Ising (α, β) = (1,σ) (fixed-free) annulus over 100 seeds. At z = 0.5, Geexact = 0.5202 vs ensemble means 0.5214 (open) and 0.5228 (closed). 22 [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: NN-predicted reduced annulus correlators Ge(o)(z) (open, blue) and Ge(c)(z) (closed, red) for the tricritical Ising (α, β) = (σ,σ ′ ) annulus over 100 seeds. At z = 0.5, Geexact = 0.4289 vs ensemble means 0.4288 (open) and 0.4302 (closed). 23 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: NN-predicted reduced annulus correlators Ge(o)(z) (open, blue) and Ge(c)(z) (closed, red) for the tricritical Ising (α, β) = (σ ′ , ε ′ ) annulus over 100 seeds (equivalent to (1, σ)). At z = 0.5, Geexact = 0.6089 vs ensemble means 0.6099 (open) and 0.6110 (closed). 24 [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: NN-predicted reduced annulus correlators Ge(o)(z) (open, blue) and Ge(c)(z) (closed, red) for the sub (2)1 (λ0, λ0) annulus over 100 seeds. At z = 0.5, Geexact = 0.7740 vs ensemble means 0.7683 (open) and 0.7666 (closed). 26 [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: NN-predicted reduced annulus correlators Ge(o)(z) (open, blue) and Ge(c)(z) (closed, red) for the sub (2)1 (λ0, λ1) annulus over 100 seeds. At z = 0.5, Geexact = 0.3206 vs ensemble means 0.3195 (open) and 0.3204 (closed). 27 [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: NN-predicted reduced annulus correlators G (o) red(z) (open, blue) and G (c) red(z) (closed, red) for the Liouville ZZ[1, 1]-FZZ[s = 0.2] annulus at c = 31.5 over 100 seeds. At z = 0.5, G (o),exact red = 2.63×10−4 vs ensemble mean 3.08 × 10−4 ; G (c),exact red = 2.64 × 10−4 vs ensemble mean 2.45 × 10−4 . 31 [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: NN-predicted reduced annulus correlators Ge(o)(z) (open, blue) and Ge(c)(z) (closed, red) for the Liouville ZZ[1, 1]–ZZ[1, 1] annulus at c = 31.5 over 100 seeds. At z = 0.5, Ge(o),exact = 0.0688 vs ensemble mean 0.0689; Ge(c),exact = 0.0696 vs ensemble mean 0.0697. 32 [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: NN-predicted reduced correlator Ge(z) for the M(13, 14) minimal model (c = 88/91, ∆gap = 3/364). The NN prediction at z = 0.5 is Gepred(0.5) = 3.72515 ± 0.01330 against Geexact(0.5) = 3.72224. M(15, 16)D9 : c = 39/40, ∆D gap = 1/60. The corresponding D-series invariant at m = 15 is (A14, D9) with lightest non-vacuum primary at ∆D gap = 1/60. Over 100 seeds, the MS training loss was (1.80 ± 1.01) × 10−6 , … view at source ↗
Figure 18
Figure 18. Figure 18: NN-predicted reduced correlator Ge(z) for the (A14, D9) D-series modular invariant of M(15, 16) (c = 39/40, ∆D gap = 1/60). The NN prediction at z = 0.5 is Gepred(0.5) = 2.56242 ± 0.00595 against Geexact(0.5) = 2.56008. M(11, 12)E6 : c = 21/22, ∆E gap = 5/88. The smallest exceptional invariant is (A10, E6) with lightest non-vacuum primary at ∆E gap = 5/88. Over 100 seeds, the MS training loss is (1.57 ± 1… view at source ↗
Figure 19
Figure 19. Figure 19: NN-predicted reduced correlator Ge(z) for the (A10, E6) E-series modular invariant of M(11, 12) (c = 21/22, ∆E gap = 5/88). The NN prediction at z = 0.5 is Gepred(0.5) = 1.68098 ± 0.00360 against Geexact(0.5) = 1.67959. suc(2)1: c = 1, ∆gap = 1/2. At level one the spectrum of affine primaries contains only the vacuum and the ℓ = 1 affine primary, with h1 = h¯ 1 = 1/4 and hence ∆gap = 1/2. The unfiltered l… view at source ↗
Figure 20
Figure 20. Figure 20: NN-predicted reduced correlator Ge(z) for the level-one sub (2)1 WZW model, on the filtered ensemble (184/1000 seeds after the early-stopping cut). The NN prediction at z = 0.5 is Gepred(0.5) = 0.70671 ± 0.00292 against Geexact(0.5) = 0.70183. suc(3)1: c = 2, ∆gap = 2/3. For sub (3)1 there are three affine primaries: the vacuum and the two fundamentals 3, ¯3, both with h = 1/3. Therefore, ∆gap = 2/3. Over… view at source ↗
Figure 21
Figure 21. Figure 21: NN-predicted reduced correlator Ge(z) for the level-one sub (3)1 WZW model (c = 2, ∆gap = 2/3). The NN prediction at z = 0.5 is Gepred(0.5) = 0.77315 ± 0.01010 against Geexact(0.5) = 0.77671. 39 [PITH_FULL_IMAGE:figures/full_fig_p040_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: NN-predicted reduced annulus correlators Ge(o)(z) (open, blue) and Ge(c)(z) (closed, red) for the Ising (α, β) = (σ,σ) annulus over 100 seeds. At z = 0.5, Geexact = 0.8022 vs ensemble means 0.7981 (open) and 0.7981 (closed). 41 [PITH_FULL_IMAGE:figures/full_fig_p042_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: NN-predicted reduced annulus correlators Ge(o)(z) (open, blue) and Ge(c)(z) (closed, red) for the Ising (α, β) = (1, ε) annulus over 100 seeds. At z = 0.5, Geexact = 0.0333 vs ensemble means 0.0317 (open) and 0.0307 (closed). We also collect here the remaining six WZW reconstructions of Section 5.2 listed in [PITH_FULL_IMAGE:figures/full_fig_p043_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: NN-predicted reduced annulus correlators Ge(o)(z) (open, blue) and Ge(c)(z) (closed, red) for the sub (2)2 (λ0, λ0) annulus over 100 seeds. At z = 0.5, Geexact = 0.7747 vs ensemble means 0.7690 (open) and 0.7673 (closed). 43 [PITH_FULL_IMAGE:figures/full_fig_p044_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: NN-predicted reduced annulus correlators Ge(o)(z) (open, blue) and Ge(c)(z) (closed, red) for the sub (2)2 (λ0, λ1) annulus over 100 seeds. At z = 0.5, Geexact = 0.4770 vs ensemble means 0.4775 (open) and 0.4798 (closed). 44 [PITH_FULL_IMAGE:figures/full_fig_p045_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: NN-predicted reduced annulus correlators Ge(o)(z) (open, blue) and Ge(c)(z) (closed, red) for the sub (2)2 (λ0, λ2) annulus over 100 seeds. At z = 0.5, Geexact = 0.1001 vs ensemble means 0.0977 (open) and 0.0969 (closed). 45 [PITH_FULL_IMAGE:figures/full_fig_p046_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: NN-predicted reduced annulus correlators Ge(o)(z) (open, blue) and Ge(c)(z) (closed, red) for the sub (3)1 (λ0, λ0) annulus over 100 seeds. At z = 0.5, Geexact = 0.7827 vs ensemble means 0.7761 (open) and 0.7747 (closed). 46 [PITH_FULL_IMAGE:figures/full_fig_p047_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: NN-predicted reduced annulus correlators Ge(o)(z) (open, blue) and Ge(c)(z) (closed, red) for the sub (3)1 (λ0, λ1) annulus over 100 seeds. At z = 0.5, Geexact = 0.2865 vs ensemble means 0.2854 (open) and 0.2862 (closed). 47 [PITH_FULL_IMAGE:figures/full_fig_p048_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: NN-predicted reduced annulus correlators for the sub (2)2 (λ1, λ1) pair over 100 seeds. The open spectrum N11 jχj = χ0 + χ2 is S-invariant, so Ge(o) = Ge(c) and both channels coincide. At z = 0.5, Geexact = 0.8748 vs mean 0.8651. 48 [PITH_FULL_IMAGE:figures/full_fig_p049_29.png] view at source ↗

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