{"id":"0c79d2eb-1395-4a23-85eb-e38c3f8e6f77","arxiv_id":"2605.04893","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Any spectral diagnostic that depends only on singular values or the symmetric part of a degree-normalized attention matrix is invariant under transpose, so it cannot see the direction of information flow.","lead":"This paper proves that spectral diagnostics based on the symmetric part or singular values of a degree-normalized attention matrix cannot detect the direction of information flow, then builds a two-axis diagnostic (capacity + asymmetry) with moderate length-controlled hallucination signal. The value is a clear structural boundary for a popular family of attention-based detectors.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 1/5 Cheeger floor is proved for the exact conductance φ on the raw graph A_bip, but the empirical estimator bφ is computed on the degree-normalized dilation N_bip; the paper's own degree-preserving nulls show bφ is largely degree-reducible, so floor-violation fractions may not be the architectur","rationale":"The core algebraic claims — orientation blindness of singular-value and Msym-spectrum functionals, and the Lipschitz transpose bound — are correct as stated, so the strongest mathematical result is not the weak point. The load-bearing concern sits where the theory is operationalized: the paper's novel empirical descriptor, the fraction of heads piercing the 1/5 Cheeger floor, requires transferring a theorem about exact conductance on the raw attention graph to a spectral-sweep estimator on the degree-normalized graph. The paper itself provides the ingredients for the concern: Theorem 3 only controls this transfer under near-regular column degrees, and the degree-preserving nulls show bφ is largely degree-reducible, so the empirical estimator may not be measuring the theoretical object. This matches the reader's weakest assumption, and the recommendation remains CONDITIONAL: the issue should be addressed by validating the transfer or by reinterpreting floor-violation fractions as degree-related statistics rather than Cheeger-landscape signatures.","tokens_in":40416,"tokens_out":17980,"duration_ms":169119,"concrete_test":"On the same real attention heads used in Table 3, compute (i) the spectral-sweep estimator bφ on N_bip and (ii) an exact or near-exact conductance φ on the raw bipartite graph A_bip (e.g., by applying the same temporal-cut sweep directly to the edge weights B_ij, or by solving the Cheeger minimization exactly for small n). Stratify by degree ratio κ = d_max/d_min. If the rank correlation between bφ and φ drops below 0.8 for heads with κ > 2, or if the floor-violation fraction changes materially when computed from φ instead of bφ, then the architectural-signature interpretation in Table 3 is not supported. A complementary synthetic check: generate degree-preserving rewired attention matrices with identical column-degree distribution but scrambled coupling; if bφ's floor-violation fraction is preserved under rewiring, it is a degree artifact, not a Cheeger-landscape signature.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's empirical bridge is the use of the 1/5 Cheeger floor (Theorem 20) as an 'idealized benchmark' against which real heads are scored (Table 3). Theorem 20 is a statement about the exact conductance φ of the raw bipartite graph A_bip = H(B) under uniform causal attention. The empirical quantity bφ is a spectral-sweep estimate computed on the degree-normalized dilation N_bip = H(M). These two objects coincide only under near-regular column degrees: Theorem 3 bounds the discrepancy by √κ−1, but for real heads κ can be large, and the paper's own degree-preserving null analysis shows bφ's discriminative power is largely accounted for by degree heterogeneity alone (z-AUROC 0.51–0.57, Sec. 7.1). If bφ does not track the exact φ on A_bip outside the near-regular regime, then the 36–42%/55–70%/79–82% floor-violation fractions in Table 3 are not measuring violation of the theorem's floor; they may be measuring degree-distribution shape. The paper acknowledges the floor is 'not an empirical attractor' (Sec. 3, Table 1), but the architectural-signature claim depends on the transfer, and the transfer is unvalidated exactly where it is needed.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies what spectral diagnostics of the degree-normalized attention operator M = D_Q^{-1/2} B D_K^{-1/2} can and cannot infer about information-flow orientation. It proves that every functional of the singular values of M or of the spectrum of the symmetric part M_sym is transpose-invariant (Theorem 5), so such diagnostics are orientation-blind; a Lipschitz converse bounds the transpose sensitivity of any Lipschitz functional by 2L||M_asym||_F (Proposition 8). On the capacity axis, it derives closed-form conductance bounds for canonical causal attention: uniform causal attention satisfies an n-independent temporal-cut floor phi >= 1/5 (Theorems 19, 20), while window attention pierces it as O(w/n) (Lemma 16). Empirically, under length-controlled AUROC, the conductance/spectral features are reported to retain 0.62--0.84 LC-AUROC, with a polarity reversal between HaluEval (bottleneck) and MedHallu (diffuse), and the fraction of heads whose empirical estimator falls below 1/5 is claimed as an architectural signature (Table 3). The mathematical core is elementary and mostly correct, but the bridge from the exact theoretical object (conductance phi on the raw graph H(B)) to the empirical estimator (spectral sweep on the degree-normalized dilation H(M)) is not adequately validated, and the polarity 'prediction' is partly circular as presented.","tokens_in":40769,"tokens_out":14617,"duration_ms":129355,"significance":"If the empirical bridge were validated, the paper would provide a clean identifiability boundary for a class of attention diagnostics and a useful closed-form Cheeger benchmark for causal architectures. The formal results are sound as stated: the transpose-invariance of singular-value and symmetric-spectrum functionals is correctly proved, and the 1/5 floor for uniform causal attention is a correct closed-form computation. The paper also ships a Lean 4 formalization of the core structural lemmas, which is a strength. However, the central empirical claim that the floor-violation fraction is an architectural descriptor depends on the unexamined transfer from phi on H(B) to the estimator on H(M), and the paper's own null analysis shows the estimator bphi is largely degree-reducible. The contribution is therefore a solid theoretical clarification with empirical claims that currently outrun the evidence.","major_comments":[{"comment":"The empirical floor-violation fractions are computed with the spectral-sweep estimator bphi on the degree-normalized dilation N_bip = H(M), while Theorem 20's floor phi >= 1/5 is proved for the exact conductance of the raw bipartite graph A_bip = H(B). The paper acknowledges this distinction but justifies the transfer by Theorem 3, which only controls the discrepancy under near-regular column degrees (kappa ~ 1). The paper's own degree-preserving nulls (Sec. 7.1) show bphi is largely degree-reducible (z-AUROC 0.51--0.57), so the reported 36--42% / 55--70% / 79--82% violation fractions may be measuring degree-distribution shape rather than violation of the temporal-cut floor. This is load-bearing for the architectural-signature claim. Please report the kappa distribution for the evaluated heads, and/or compute exact phi on A_bip for a subset of heads, or otherwise demonstrate that the swe","section":"Sec. 3, Table 3, Theorem 3"},{"comment":"The 'falsifiable polarity prediction' is not tested as stated. HaluEval is labeled bottleneck-dominated and MedHallu diffuse-dominated after the tercile analysis (Sec. 7.1), and the LC-AUROC is reported as flipped, i.e., label-informed polarity (Sec. 6.2). The paper also notes that polarity is bin-consistent in only 7 of 12 model--dataset pairings (Sec. 8). To support the claim, the authors should specify an a priori criterion (e.g., the sign of the length-label correlation, r=+0.70 vs -0.21, or an independent failure-mode taxonomy) and evaluate unflipped directional AUROC. As written, the reversal is a descriptive pattern, not a confirmation of the prediction.","section":"Secs. 1, 7.1, 8.2"},{"comment":"The set of methods claimed to be limited by Theorem 5 is inconsistent. Corollary 7 and the Introduction list EigenTrack, but Sec. 8.1 states that the orientation-blindness theorem 'does not bear directly on hidden-state-based detection,' and EigenTrack operates on hidden-state covariance spectra, not on the degree-normalized attention operator. For LapEigvals, the main text asserts that its raw Laplacian eigenvalues are functions of symmetrized spectra but defers the proof to the Online Supplement. Since naming the affected detectors is part of the claimed contribution, the coverage claim should be reconciled, and the explicit reduction for each named method should be shown in the main text or the names removed.","section":"Intro, Corollary 7, Sec. 8.1"}],"minor_comments":[{"comment":"The phrase 'every transpose-invariant spectral diagnostic is orientation-blind' is definitional; the substantive statement is that singular-value and symmetric-spectrum diagnostics are transpose-invariant. Corollary 7 already notes that eigenvector summaries reading query/key blocks separately can break invariance, so the abstract should be qualified to avoid overstatement.","section":"Abstract, Corollary 7"},{"comment":"The manuscript refers to 'Theorem 8' in the proof structure of Theorem 5 and in Sec. 8.1, but the displayed result is Proposition 8. Similarly, Appendix B's roadmap refers to 'Theorem 9' for the asymmetry-coefficient characterization while the main text states Proposition 9. Numbering should be unified.","section":"Sec. 4, Sec. 8.1"},{"comment":"The text says the 1/5-floor violation fractions are '36--42% for GPT-2, 55--70% for Pythia-160M, and 79--82% for Flan-T5 decoder across HaluEval, MedHallu, and TruthfulQA.' Table 3 omits the Flan-T5 decoder/TruthfulQA row because no samples passed the n>=32 filter; the phrasing 'across ... TruthfulQA' is therefore inaccurate for Flan-T5.","section":"Sec. 3, Table 3"},{"comment":"The caption does not state that the AUROC values are flipped (label-informed polarity), although Sec. 6.2 says 'unless stated otherwise, reported LC-AUROC values are flipped.' Please make this explicit in the table caption.","section":"Table 5 caption"},{"comment":"The notation phi(M), phi_ref, delta_ref, and epsilon is introduced informally. The displayed bound 'phi(M)-phi_ref <= sqrt(2(delta_ref+epsilon)) - delta_ref/2' can be negative (trivial), and the derivation of epsilon from the spectral-gap difference should be stated precisely.","section":"Appendix B, Conductance transfer corollary"}],"recommendation":"major_revision","confidential_remarks":"The theoretical core is sound and the paper is well organized, but the empirical claims currently outrun the evidence: the floor-violation architectural signature depends on an unvalidated transfer from the exact conductance of the raw graph to the degree-normalized spectral-sweep estimator, and the polarity 'prediction' is partly circular due to label-informed flipping and post-hoc dataset labeling. The manuscript would be suitable after major revision if the authors validate the transfer (e.g., report kappa distributions or exact phi on a subset) and reposition the polarity claim as descriptive, or provide an a priori prediction rule. The inconsistent attribution of the blindness result to EigenTrack and LapEigvals should also be fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a serious paper, not a desk reject. The core mathematical claim—any diagnostic built from singular values or the spectrum of the symmetric part of the degree-normalized attention operator is invariant under transpose, hence blind to the orientation of information flow—is elementary but correctly identified, and the authors are upfront that the algebra is classical. What is new is the application: naming the affected methods (LLM-Check, EigenTrack, LapEigvals) and giving a quantitative converse through the asymmetry coefficient G. The closed-form Cheeger landscape for uniform causal attention (phi >= 1/5) and window attention (O(w/n)) is the strongest genuinely new piece, and it's proved cleanly.\n\nI don't fully share the reader's worry about the abstract. Saying \"every transpose-invariant spectral diagnostic is orientation-blind\" is close to tautological, but the substantive claim is that the named methods are transpose-invariant, and Corollary 7 supports that while explicitly carving out eigenvector summaries that read query and key blocks separately. The abstract could be sharper, but it isn't misleading once you read the corollary.\n\nThe soft spot is the empirical bridge. The 1/5 floor is a theorem about exact conductance on the raw graph; the experiments use the spectral-sweep estimator on the degree-normalized operator. Theorem 3 bounds the gap only under near-regular degree conditions, and the paper's own degree-preserving nulls show bphi is largely degree-reducible. The authors hedge appropriately—\"idealized benchmark, not an empirical attractor\"—but the floor-violation fractions in Table 3 still lean on the transfer. I'd want a sensitivity analysis or a frank statement that the violation fraction is largely a degree-distribution descriptor. That's a moderate, fixable issue.\n\nThe polarity prediction is also weakened by label-informed flipping of LC-AUROC; the scatter and tercile analysis carry some sign evidence that is not fully circular, but the headline AUROC numbers should not be sold as unsupervised detection. And the Lean formalization claim is unverifiable from the text because no artifacts or links are present.\n\nOverall: honest, mathematically sound, moderately useful. The theorems will age well; the empirical claims need tightening. This deserves serious peer review.","headline":"A solid structural result in attention diagnostics—symmetric spectral methods provably cannot see flow direction—plus a genuinely new Cheeger floor for uniform causal attention; the empirical bridge to real heads is softer than the theorems, but the paper is honest and worth refereeing.","tokens_in":41222,"tokens_out":3554,"would_cite":true,"duration_ms":39814,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Symmetric spectral diagnostics of attention operators are provably blind to the direction of information flow, and only the antisymmetric residual G can recover it.","keywords":["orientation blindness","attention transport operator","spectral diagnostics","Cheeger inequality","conductance floor","hallucination detection","length-controlled evaluation","asymmetry coefficient"],"falsifier":"Compute exact graph conductance by brute force on real attention heads with n <= 20 tokens and compare with the spectral-sweep estimator bphi; if bphi deviates beyond the degree-ratio bound even on near-regular heads, the floor-transfer assumption fails. Alternatively, a benchmark with known bottleneck-type hallucinations that does not reverse polarity on low-conductance terciles would falsify the two-sided diagnostic.","tokens_in":40297,"feed_emoji":"","tokens_out":3984,"duration_ms":33049,"temperature":0.7,"pith_summary":"The paper treats each attention head as a degree-normalized bipartite transport operator and asks what spectral diagnostics can and cannot infer. It proves that every transpose-invariant diagnostic—any function of singular values or of the spectrum of the symmetrized operator—returns the same value for M and its transpose, so methods that read singular values or symmetric spectra cannot distinguish an operator from its reversed flow. It then proves a converse: any diagnostic sensitive to direction must pay a cost proportional to the antisymmetric component's norm G. On the capacity axis, it derives a closed-form conductance floor for uniform causal attention (phi >= 1/5) that window attention pierces, and shows empirically that the polarity of the conductance signal reverses across benchmarks as predicted.","feed_headline":"Spectral attention diagnostics can't see which way info flows","feed_subtitle":"Transpose-invariant diagnostics can't distinguish an operator from its reverse; the asymmetry coefficient G is the only lever.","key_machinery":"The degree-normalized bipartite transport operator M = D_Q^{-1/2} B D_K^{-1/2}, its orthogonal split into symmetric part M_sym (capacity) and antisymmetric part M_asym (direction), the Hermitian dilation H(M) that converts the rectangular operator into a symmetric eigenvalue problem whose spectrum is the signed singular values, and the closed-form cut/volume identities for temporal cuts in uniform causal attention that yield the 1/5 floor.","core_discovery":"Every attention head defines a degree-normalized transport operator M = D_Q^{-1/2} B D_K^{-1/2}. The paper's central claim is a structural identifiability limit: any spectral diagnostic that depends only on singular values of M or on the spectrum of the symmetric part M_sym is invariant under transpose, hence blind to the orientation of information flow. The proof uses the Hermitian dilation H(M) and a block-swap similarity showing H(M) and H(M^T) share a spectrum. A quantitative converse (Prop. 8) bounds the transpose sensitivity of any L-Lipschitz diagnostic by 2L||M_asym||_F, so the asymmetry coefficient G = ||M_asym||_F / (||M||_F + eps) is the unique control parameter for directional se","pith_inferences":["The orientation-blindness result likely extends to any diagnostic built on symmetrized representations of the same operator, even if it reads eigenvectors, as long as the summary is block-swap symmetric; hidden-state methods are not covered by the theorem but face a similar structural limitation if they symmetrize.","The closed-form conductance template (row-sum/column-sum decomposition) can probably be applied to other causal masks such as exponential decay or RoPE-style decay to produce per-architecture floors; the paper sketches this but leaves the closed forms open.","The empirical status of the 1/5 floor suggests a testable design rule: architectures with high floor-violation fractions should exhibit bottleneck polarity on datasets with concentrated hallucinations, which could guide dataset construction.","The degree-sufficiency result implies future attention diagnostics should separate degree-based signal from coupling-based signal, or risk misattributing hallucination signal to routing structure when it is largely a degree artifact."],"forward_implications":["Symmetric spectral diagnostics such as LLM-Check, EigenTrack, and LapEigvals are structurally unable to detect the direction of information flow; any improvement must explicitly use the antisymmetric residual.","The asymmetry coefficient G provides a guaranteed bound on directional sensitivity: if G is small, no Lipschitz diagnostic can have large transpose sensitivity.","The conductance floor phi >= 1/5 distinguishes failure modes by shape: uniform causal attention can never be a global bottleneck, while window attention provably pierces the floor on balanced cuts.","Polarity of the conductance signal is regime-dependent and reverses between bottleneck-dominated and diffuse-dominated datasets, so a universal sign is not expected.","Length-controlled evaluation is required; raw AUROC inflates by up to 0.28 points due to length confounding."],"fun_headline_variants":["Spectral attention diagnostics are orientation-blind","G is the only lever for attention direction","Transpose-invariant spectra: no flow direction","Spectral polarity needs G, not eigenvalues","Attention spectra miss which way info flows"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The measured spectral-sweep estimate bphi on the degree-normalized graph tracks the exact conductance phi of the raw attention graph closely enough that the closed-form 1/5 floor, proved for the idealized object, can serve as a fixed benchmark for real attention heads.","fun_headline_variants_meta":{"raw":{"variants":["Spectral attention diagnostics are orientation-blind","G is the only lever for attention direction","Transpose-invariant spectra: no flow direction","Spectral polarity needs G, not eigenvalues","Attention spectra miss which way info flows"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000274,"raw_usage":{"total_tokens":1545,"prompt_tokens":879,"completion_tokens":666,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":623,"completion_tokens_details":{"reasoning_tokens":601}},"tokens_in":623,"tokens_out":666,"duration_ms":6447,"temperature":1.0,"reasoning_tokens":601,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T14:46:29.841964+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute exact graph conductance by brute force on real attention heads with n <= 20 tokens and compare with the spectral-sweep estimator bphi; if bphi deviates beyond the degree-ratio bound even on near-regular heads, the floor-transfer assumption fails. Alternatively, a benchmark with known bottleneck-type hallucinations that does not reverse polarity on low-conductance terciles would falsify the two-sided diagnostic.","supporting_citations":[],"review_version":3}