{"id":"3a04b544-5fc5-440b-8154-44b3d162e63e","arxiv_id":"2607.24913","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Lightweight neural nets with spectral bias reconstruct full 2d CFT torus and annulus partition functions from crossing, a gap, and a single interior anchor to sub-percent accuracy on known theories.","lead":"Neural networks reconstruct 2d CFT torus and annulus partition functions from modular or Cardy crossing, a spectral gap, and one anchor value. The method turns classical consistency conditions into four-point crossing and uses network spectral bias to pick physical solutions among many allowed functions.","discovery_kind":"new_application","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"The \"spectral bias selects the physical solution\" claim is confounded: the ansatz already encodes most of the answer, and in the one hard test (WZW) the majority of seeds select the wrong solution, rescued only by a post-hoc filter.","rationale":"The reader identified the same load-bearing issue — the empirical, underived selection principle and the post-hoc early-stopping filter in the bimodal WZW cases — and weighted it correctly in landing on CONDITIONAL with medium correctness risk. My pass sharpens rather than replaces that concern: (i) the confound between spectral-bias selection and determination by the prefactor ansatz + exact anchor is not isolated by any control in the paper, and (ii) the WZW numbers (92/1000 and 184/1000 seeds retained) mean that in the only regime where selection is non-trivial, the raw procedure majority-fails, with the rescue justified only by consistency with previously known answers. This does not push the verdict to REJECT: the paper is transparent about all of this (§4.4 states the bimodality and filtering openly, §7 lists the missing theoretical understanding as future work), the code is public, and as a methods-and-experiments paper demonstrating accurate reconstruction under stated inputs it delivers what it promises. The adjustment I would want is presentational, not verdict-level: the abstract and conclusion should scope the claim to \"reconstruction given exact endpoint data, an exact anchor, and (for WZW) a validated filter,\" and the proposed ablation would clarify what is actually doing the work. Hence UNCHANGED — the reader's CONDITIONAL with medium correctness risk already prices this in.","tokens_in":31928,"tokens_out":2280,"duration_ms":86238,"concrete_test":"Two controls on the torus setup. (a) Ablation: replace NNθ in (3.6) with a comparably smooth generic basis (e.g., 20-term Chebyshev or cubic-spline fit) minimising the identical loss (3.7) with the same anchor on Ising and tricritical Ising; if it reproduces eG_exact to similar accuracy, the ansatz+anchor, not spectral bias, is selecting the solution. (b) Blind holdout: run the NN pipeline on a withheld case (e.g., ŝu(2)_3 torus, same c/gap inputs only), apply the early-stopping filter before any comparison to characters; if the filtered ensemble is not unimodal around eG_exact(0.5), the selection principle fails out-of-sample.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that spectral bias in the lazy regime \"selects the physical crossing-symmetric solution among many allowed ones.\" Two things make this weaker than stated. First, the selection is confounded by construction. The ansatz (3.5)-(3.6) fixes the z→0 endpoint analytically via L(z) (vacuum-character asymptotics, with c0 and ∆gap taken from the exact theory), crossing then pins z→1, and the anchor Lanc uses the exact value G_exact(z0) at an interior point. The NN only supplies a correction multiplied by z^{2∆gap+1/6}(1−z)^{(2−3c)/12}, which vanishes at both ends. So the function space actually explored is a narrow neighborhood around an already nearly-determined curve; \"many allowed crossing-symmetric solutions\" is true of the bare equation (1.2) but not obviously of the constrained problem the network actually solves. No control experiment (e.g., a non-neural smooth basis fitted to the same loss) is offered to show spectral bias, rather than the factorization plus exact anchor, is doing the selecting. Second, where selection is genuinely nontrivial it mostly fails: for ŝu(2)_2 (§4.4) the unfiltered ensemble is bimodal and only 92/1000 seeds — the early-stopped minority — track the exact answer; for ŝu(2)_1 it is 184/1000. The filter is defensible only because the correlation \"early stopping ⇒ physical peak\" was observed against known answers; for an unknown CFT there is no independent reason given to trust that the minority peak is physical. Thus the paper demonstrates accurate reconstruction conditional on knowing the answer well enough to supply exact endpoint data, an exact anchor, and a validated filter — which is a real but narrower result than the headline selection principle.","agreement_with_reader":"agree"},"referee_report":null,"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is Part II of their neural spectral-bias programme, now aimed at torus modular S and annulus open/closed duality by rewriting both as four-point crossing (twist fields; defect-changing operators). Those geometric maps are standard. What is actually new is the worked anchored pipeline—gap exponents, single interior anchor, lightweight MLP—and a broad reconstruction survey on known targets: Ising, Lee–Yang, ADE minimal models, WZW, free boson, Liouville ZZ/FZZ, with public code and ensemble diagnostics.\n\nThe numerics are real. On most compact examples the ensemble mean tracks the exact reduced correlator to sub-percent level with tiny MS losses. They are honest that there are no rigorous error bars, and they handle non-unitary and some non-compact cases without positivity. That is useful methods work inside 2d bootstrap, especially where SDP is awkward.\n\nThe soft spot is the load-bearing story that lazy-training spectral bias “selects the physical solution among many.” The stress-test lands. The ansatz already builds in vacuum asymptotics, the gap, and crossing-friendly endpoint powers; the anchor is the exact G(z0). The network only fits a correction that vanishes at the ends, so the explored function space is a narrow neighbourhood of an almost-determined curve. No non-neural control (smooth basis, same loss) is shown. Where selection is nontrivial—su(2)_1 and su(2)_2—the unfiltered ensemble is bimodal and only a minority of seeds (early-stopped) hit the physical peak. That filter is validated against known answers; for an unknown CFT you would not know which peak to trust. So the paper demonstrates accurate reconstruction conditional on exact spectral input, an exact anchor, and sometimes a post-hoc filter—not a free-standing selection principle.\n\nStill a serious, reproducible methods paper. For people already in neural bootstrap or function-space modular methods it is worth reading; for pure modular-bootstrap theorists the novelty is narrower. I would send it to peer review and expect referees to demand controls and a clearer statement of what is assumed versus discovered.","headline":"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.","tokens_in":33509,"tokens_out":537,"would_cite":false,"duration_ms":25155,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"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.","keywords":["neural bootstrap","modular invariance","annulus partition function","Cardy condition","spectral bias","two-dimensional CFT","twist fields","defect-changing operators"],"falsifier":"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.","tokens_in":33293,"feed_emoji":"⚛️","tokens_out":986,"duration_ms":27563,"temperature":0.7,"pith_summary":"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.","feed_headline":"Nets rebuild 2d CFT partition functions from sparse data","feed_subtitle":"Modular crossing, a gap, and one anchor recover the full torus or annulus answer via spectral bias","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Neural nets rebuild torus partition functions from modular crossing","Spectral bias recovers full 2d CFT partition functions from a gap","Anchored bootstrap reconstructs annulus answers via defect crossing","Nets turn modular S-invariance into four-point crossing for CFTs","Lazy-training nets select physical partition functions from sparse data"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Neural nets rebuild torus partition functions from modular crossing","Spectral bias recovers full 2d CFT partition functions from a gap","Anchored bootstrap reconstructs annulus answers via defect crossing","Nets turn modular S-invariance into four-point crossing for CFTs","Lazy-training nets select physical partition functions from sparse data"]},"model":"grok-4.5","effort":"low","cost_usd":0.004314,"raw_usage":{"total_tokens":1286,"prompt_tokens":796,"num_sources_used":0,"completion_tokens":69,"cost_in_usd_ticks":43144000,"prompt_tokens_details":{"text_tokens":796,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":421,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":796,"tokens_out":69,"duration_ms":7283,"temperature":1.0,"reasoning_tokens":421,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T05:51:47.735007+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"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.","supporting_citations":[],"review_version":1}