{"id":"224d3dd8-a578-4912-8c8e-01b8741bfa4f","arxiv_id":"2608.12770","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"At quantum critical points, static susceptibility (causation) is controlled by the lowest quasiprimary, not the lowest descendant, so decay can be suppressed by up to fifteen powers and new CFT data such as a corner primary Δ≈8.8 become visible.","lead":"Static susceptibility, called causation, decays much faster than correlation functions at quantum critical points because time-derivative operators cannot contribute to static response. This gives a practical tool to measure hidden primary operator dimensions in higher-dimensional CFTs and explains the sharp localization of edge modes in gapless topological phases.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The d≥3 primary sieve rests on an unproven parity-genericity assumption: an indefinite-parity lattice operator need not contain the primary when a spatial descendant appears, so causation could be descendant-dominated; the corner demonstrations do not test this.","rationale":"The reader's weakest assumption correctly identifies the parity-genericity assertion in the d≥3 primary sieve as the least secure step. The exact argument that time-derivative operators have vanishing static response is rigorous, and the (0+1)D defect or corner case follows without any parity assumption, which is why the Fig. 3 demonstrations support but do not test the broader claim. The broader claim, stated in the abstract and Discussion as 'leading causation arises from primaries' in higher-dimensional CFTs, requires that whenever a spatial descendant of a primary contributes to a lattice operator's expansion, the primary itself is also present. The paper only provides a no-symmetry genericity argument for this; it is not proven, and the parity-odd component test I propose would directly show whether a definite-parity descendant operator behaves as the assumption predicts. If the odd component gives the descendant exponent, the sieve relies on the nonzero primary component and the paper should either prove that generic indefinite-parity operators always have such a component or explicitly restrict the claim. This does not warrant rejection: the 1+1D results and the corner extraction are valuable, and the missing proof and error quantification are addressable. The verdict should remain conditional, unchanged from the reader's assessment.","tokens_in":16092,"tokens_out":18384,"duration_ms":216897,"concrete_test":"Run the same DMRG corner computation as Fig. 3 with O₁ replaced by its spatial-parity-odd component, O₁^odd = σ^y_{(1,1)}σ^x_{(1,2)} − σ^x_{(1,1)}σ^y_{(1,2)}, keeping O₂ fixed. This operator has the same Z₂ and T charges but definite parity, so by the paper's own reasoning its continuum expansion need not contain the corner primary φ_T, only a spatial descendant; the predicted causation exponent is then −(2Δ_T+1) ≈ −18.6 instead of the −16.6 found for the indefinite O₁. If the measured exponent is ≈ −18.6, the sieve is not automatic and the indefinite-parity assumption is load-bearing; if it is ≈ −16.6, the concern is refuted and the primary is present regardless of parity. The comparison should include bond-dimension convergence and finite-size error estimates.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The d≥3 extension of the central claim has a load-bearing gap. The exact vanishing of time-derivative causation (Eq. (2)) and the (0+1)D defect case are solid, and the 1+1D numerics are convincing. But the claim that in the bulk, or on extended defects, 'spatial derivatives survive but do not spoil the sieve' depends on the assertion in the section 'Application: Primary sieve in d≥3 CFTs' that a lattice operator of indefinite spatial parity 'flows to fields of every spatial parity, so no parity rule forbids its primary φ whenever a spatial descendant ... contributes.' This is a genericity assumption, not a consequence of conformal invariance. In a unitary CFT there is no theorem that a local operator containing a spatial descendant ∂φ must also contain φ; one can construct an operator with indefinite spatial parity whose primary overlap is zero, e.g., a lattice finite-difference of the spin field combined with an opposite-parity higher-primary piece. For such an operator, the leading causation would be set by the descendant, with scaling L^{-2(Δφ+k)+1} instead of L^{-2Δφ+1}, so the sieve would fail. The numerical demonstrations in Fig. 3 use corner and point-on-defect operators, which are (0+1)D and therefore do not exercise this parity-genericity assumption; no bulk or extended-defect test is provided. The claimed new dimensions from fits over L=2–6 without error bars add to this fragility but are secondary.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces 'causation' as the static susceptibility ⟨A G B⟩, with G = (E0 − H)^{−1}, and proves that operators which are total time derivatives have vanishing Hermitian causal response. It then argues that at criticality the leading decay of causation is controlled by the lowest-dimension quasiprimary/primary in the operator expansion that is not a time derivative, in contrast with correlation, which is controlled by the lowest-dimension field. In 1+1D boundary CFT this yields explicit exponents (L^{−14}, L^{−21}, L^{−17}, exponential) for the Ising, tricritical Ising, and free-fermion chains, with numerical confirmation. The same logic is applied to gapless SPT edge splittings, explaining prior L^{−14} results and predicting L^{−18} and L^{−25} splittings in new spin chains. In d≥3, the authors claim a 'primary sieve' and use DMRG on the (2+1)D Ising model to extract a corner primary Δ≈8.8 and a magnetic-defect primary Δ≈4.6. The End Matter contains a proof of exponential decay of causation for BDI Majorana chains and derivations of the SPT splittings.","tokens_in":16442,"tokens_out":20929,"duration_ms":214068,"significance":"Equation (2) is a clean, rigorous identity, and the 1+1D exponent bookkeeping from Virasoro characters is convincing; Figs. 1 and 2 provide explicit numerical confirmation, including the new L^{−18} and L^{−25} splittings. The free-fermion theorem in the End Matter is a genuine proof and a useful result in its own right. If the d≥3 'primary sieve' can be put on firm footing, this would be a valuable new numerical tool for probing heavy primary operators in higher-dimensional CFTs; the defect dimension Δ≈4.6 matching the independent fuzzy-sphere value 4.64(14) is encouraging. However, the d≥3 sieve rests on an unproven parity-genericity assumption, and the numerical extractions are based on small system sizes without error bars. These issues do not affect the 1+1D results or the central identity, but they do limit the strength of the third key result as currently stated.","major_comments":[{"comment":"The claim that a lattice operator of indefinite spatial parity 'flows to fields of every spatial parity, so no parity rule forbids its primary φ whenever a spatial descendant ... contributes' is not a consequence of conformal invariance. Indefinite parity only ensures that the expansion contains fields of both parities; it does not ensure that the parent primary of a contributing spatial descendant is present. For example, in a unitary CFT, O = ∂_x φ + ψ, with φ and ψ both even under spatial parity, has indefinite parity (odd component ∂_x φ, even component ψ), contains no φ, and yet contains a spatial descendant of φ; if Δψ > Δφ+1, the leading causation of O is set by the descendant ∂_x φ rather than by a primary. This is a load-bearing point for the third key result: the bulk and extended-defect versions of the sieve require either a proof from lattice locality and translation invariance, or a reformulation that restricts the sieve to (0+1)D defects and corners and states the bulk sieve as a conjecture. The numerics in Fig. 3 use corner operators and a point on the defect; only the latter, if the defect is genuinely extended, would test the parity-genericity assumption, and a single example does not establish the general sieve.","section":"Application: Primary sieve in d≥3 CFTs"},{"comment":"The extraction of the corner primary Δ≈8.8 and the defect primary Δ≈4.6 is based on power-law fits over L=2–6 (and up to L=10 for the defect) at a single bond dimension, with no error bars or convergence checks reported. The correlation fit for the light corner operator already gives Δ_Z≈1.8 versus the Monte Carlo value β_2/ν≈2.03, which the authors attribute to finite-size effects; the same or larger systematic uncertainties could affect the heavy-primary exponents. To make the 'previously unresolved' primary claims convincing, the paper should report bond-dimension dependence, fit-range variation, or a scaling collapse, and state explicit error estimates.","section":"Application: Primary sieve in d≥3 CFTs / Figure 3"}],"minor_comments":[{"comment":"Clarify whether the 'magnetic line defect' is implemented by pinning a single site or a line of sites; the text says 'the center spin pinned downwards' while the caption and references describe a line defect, and the phrase '(0+1)D magnetic line defect' is internally inconsistent, since a line defect in a 2D spatial lattice has one spatial dimension. The distinction matters for the comparison with the fuzzy-sphere line-defect value.","section":"Application: Primary sieve in d≥3 CFTs / Figure 3c"},{"comment":"The statement 'anti-hermiticity forces ⟨γ_a γ_L⟩=δ_{aL}' is incomplete: for a≠L, γ_a γ_L is anti-Hermitian, but the conclusion that its expectation vanishes also uses T-symmetry of the ground state; please spell this out explicitly.","section":"End Matter, Theorem 1 proof"},{"comment":"The phrase 'fifteen additional orders in x' (and 'fifteen orders of magnitude') is a statement about a difference in power-law exponents; at a generic L it is not fifteen orders of magnitude in the value of the function. Rephrase to avoid overstatement.","section":"Abstract and Introduction"},{"comment":"Specify the symmetry, for example reflection across the diagonal, that relates O1 at the bottom-left corner to O2 at the top-right corner, and state how the operators are defined for even L in the defect geometry.","section":"Figure 3a"}],"recommendation":"major_revision","confidential_remarks":"The 1+1D core is solid and the new SPT constructions are nice. My main reservation is the unproven parity-genericity assumption for the d≥3 primary sieve; without it, the 'primary sieve' is not established beyond (0+1)D defects and corners. The numerical extraction of the two new dimensions is also thin. I recommend major revision to either prove or explicitly qualify the sieve and to add error estimates, or to reframe the d≥3 claims as conjectural with supporting numerics."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Ryan—quick read of 2608.12770. The thing to know: the central mechanism is right and clean. Static response cannot see time-derivative operators, Eq. (2), and that cleanly explains the anomalously small gSPT edge splittings and gives a practical sieve for primary dimensions. The paper earns its keep in 1+1D and at point defects. The d≥3 bulk/extended-defect sieve is oversold, though, and a careful referee should push on it.\n\nWhat is actually new: the framing of static susceptibility as a causation probe, the Virasoro character counting for boundary towers (L^-14, L^-21, L^-17), the two spin-chain constructions with L^-18 and L^-25 splittings, and the corner/defect primary extraction. The End Matter free-fermion theorem is a real proof, and the numerics in Figs. 1–2 are convincing: free-fermion and exact-diagonalization data match predicted exponents over a good range. The defect primary Δ≈4.6 matching the fuzzy sphere value 4.64(14) is a nice external check. They also cite the prior L^-14 gSPT work honestly.\n\nSoft spots, in order of importance. The d≥3 sieve claim leans on the statement that a lattice operator of indefinite spatial parity flows to fields of every spatial parity. Indefinite parity does not imply the specific primary of a given descendant is present. You can write an operator that is a finite difference of the spin plus an opposite-parity higher primary: σ absent, ∂σ present, and causation becomes descendant-dominated. The paper’s own demonstrations are at corners and at a pinned-spin point defect, where the (0+1)D argument is exact, so they do not test the bulk or extended-defect version. Second, the 3D Ising fits are over L=2–6 with no error bars; the defect value has the fuzzy sphere anchor, but the corner Δ≈8.8 rests on a handful of points. Third, a minor terminology slip: calling the pinned-spin defect a “(0+1)D magnetic line defect” conflates a point defect with a line defect.\n\nNone of this kills the paper. Eq. (2) is rigorous, the 1+1D story is fully confirmed, and the d≥3 method has a clear path to being made solid: prove the genericity assumption, or state it explicitly as an assumption and test it in a genuine bulk or line-defect geometry. The paper is for people doing numerical CFT spectroscopy and gSPT edge physics; it deserves a serious referee. I would accept it for review and ask for error bars and a bulk/line-defect test before publication.","headline":"Right mechanism, real results in 1+1D and at point defects; the d≥3 bulk sieve is overclaimed and needs one honest assumption stated or a proof.","tokens_in":16981,"tokens_out":5960,"would_cite":true,"duration_ms":65602,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Causation can decay fifteen powers faster than correlation at quantum critical points, and the mechanism exposes hidden primary operators.","keywords":["quantum criticality","static susceptibility","causation function","conformal field theory","primary operators","gapless symmetry-protected topological phases","edge mode splitting","time reversal symmetry"],"falsifier":"In the (2+1)D Ising model on an $L\\times L$ lattice, measure the corner-to-corner static response of a parity-even operator (a symmetric linear combination of two mirror-related local operators). If the decay exponent matches a spatial descendant of the lowest primary instead of that primary itself, the claimed primary sieve is false.","tokens_in":15875,"feed_emoji":"⚛️","tokens_out":7627,"duration_ms":68543,"temperature":0.7,"pith_summary":"This paper argues that static susceptibility — the change in one observable when the Hamiltonian is perturbed by another, which the authors call 'causation' — behaves very differently from correlation at quantum critical points. In time-reversal-symmetric systems, any operator that is a time derivative of another contributes nothing to static response, so causation is controlled by the lowest-dimension quasiprimary (in 1+1D) or primary (in three or more dimensions) in the operator expansion, not by the lowest-dimension field overall. The result is that causation can decay far faster than correlation, up to fifteen powers of distance faster in the Ising CFT, and can expose primary operators that correlations hide. The authors use this to identify a corner primary of dimension about 8.8 and a magnetic line defect primary of dimension about 4.6 in the (2+1)-dimensional critical Ising model, and to explain and engineer extremely small edge-mode splittings in gapless symmetry-protected topological phases.","feed_headline":"Causation can beat correlation by 15 powers at critical points","feed_subtitle":"Time-derivative operators give zero static response, so susceptibility filters descendants and reveals hidden primaries.","key_machinery":"The carrying identity is Eq. (2): if $A = i[H,C]$ is a time derivative of $C$, then the causation function $\\langle A G B\\rangle$ equals $i\\langle [C(x),B(y)]\\rangle$, which vanishes for spatially separated operators (or, under time-reversal symmetry and same $T$-charge, vanishes generally). Here $G=(E_0-H)^{-1}$ is the resolvent, so the causation function is the static Green's function of perturbation theory. This identity kills every descendant that is a time derivative; in $(0+1)$-dimensional defects or boundaries every descendant is either a time derivative or a quasiprimary, so the leading causation comes from the lowest quasiprimary, and in $d\\ge 3$ from the lowest primary. Virasoro character counting supplies the quasiprimary levels in the 1+1D examples.","core_discovery":"The central claim is that in a critical system with time-reversal symmetry, the static response (causation) between two lattice operators is governed by the lowest-dimension field in each operator's continuum expansion that is not a time derivative. Because time-derivative fields have vanishing static response, causation decays as $L^{-2\\Delta+1}$ where $\\Delta$ is the dimension of that lowest non-time-derivative field, whereas correlation decays as $L^{-2\\Delta_{\\rm min}}$ with the absolute lowest dimension. In one spatial dimension this means causation is controlled by the lowest quasiprimary on the boundary; in three or more dimensions, by the lowest primary. This explains why causation can be dramatically more suppressed than correlation, and turns causation into a sieve that filters out descendant fields. The paper demonstrates the mechanism in the critical Ising and tricritical Ising chains and free-fermion chains, extracts previously invisible primary dimensions in the (2+1)D Ising CFT, and shows that edge-mode splittings in gapless symmetry-protected topological phases are causation functions, leading to spin chains with splittings as small as $1/L^{18}$ and $1/L^{25}$.","pith_inferences":["If the primary sieve holds generically, causation could complement the conformal bootstrap as a numerical source of conformal data, especially for heavy operators that are difficult to access through correlation.","The same principle that time-derivative fields are invisible to static response may apply to other symmetry constraints, such as spatial parity or rotation symmetry, potentially providing analogous sieves for other representation-theoretic sectors; the paper leaves this open.","The exponential localization of free-fermion causation suggests a broader design principle: irrelevant perturbations convert algebraic cancellations into exponential localization, which could be useful for protecting edge qubits in noisy settings.","Since causation is a linear-response quantity, it may be directly measurable in ultracold-atom or trapped-ion quantum simulators through the response to a local perturbation, giving experimental access to primary dimensions."],"forward_implications":["Causation functions become practical probes of primary operator spectra in higher-dimensional CFTs, since they filter out descendants that dominate correlation.","Computing causation by symmetry sector isolates the lowest primary with given quantum numbers, as demonstrated for the corner and defect primaries in the 3D Ising CFT.","Edge-mode splittings in gapless symmetry-protected topological phases are governed by causation, explaining previously observed anomalously small splittings and predicting how to engineer even smaller ones.","The mechanism extends to other boundaries, defects, and critical models; any ground-state method that computes correlation functions can compute causation.","Because time-derivative fields vanish from static response, the suppression is a generic feature of time-reversal-symmetric critical points, not a fine-tuned accident."],"supporting_citations":[{"why":"Defines static susceptibility as a derivative of the ground-state expectation value, which is the causation function studied in this paper.","marker":"[4]"},{"why":"Supplies the Virasoro characters and descendant/quasiprimary structure used to locate the first T-odd quasiprimary in the Ising and tricritical Ising towers.","marker":"[58]"},{"why":"Computed a $1/L^{14}$ edge-mode splitting in the interacting Majorana CFT whose origin the causation mechanism explains and generalizes.","marker":"[11]"},{"why":"Established topological edge modes in critical BDI Majorana chains and provided the Laurent-polynomial winding-number formalism used in the exponential-decay proof.","marker":"[16]"},{"why":"Provides the boundary CFT statement that on a (0+1)D defect every descendant is either a time derivative or a quasiprimary.","marker":"[61]"},{"why":"Gives the classical corner magnetization exponent used as a check on the extracted corner primary dimension.","marker":"[80]"},{"why":"Reports the heavy magnetic line defect primary $\\Delta=4.64(14)$ from fuzzy-sphere regularization, which the extracted $\\Delta\\approx 4.6$ matches.","marker":"[84]"}],"fun_headline_variants":["Causation beats correlation by 15 powers at criticality","Static response beats correlation by 15 decay orders","Causation filters descendants, wins by 15 powers","Causation reveals hidden primaries, 15 powers sharper than correlation","At critical points, causation is 15 orders more suppressed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The sieve in higher dimensions rests on the assumption that a generic lattice operator of indefinite spatial parity flows to fields of every spatial parity, so no parity selection rule can prevent the operator's own primary from being the lowest non-time-derivative contributor; if a conserved spatial parity excluded that primary, a spatial descendant could dominate causation and the sieve would fail.","fun_headline_variants_meta":{"raw":{"variants":["Causation beats correlation by 15 powers at criticality","Static response beats correlation by 15 decay orders","Causation filters descendants, wins by 15 powers","Causation reveals hidden primaries, 15 powers sharper than correlation","At critical points, causation is 15 orders more suppressed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000556,"raw_usage":{"total_tokens":2688,"prompt_tokens":1026,"completion_tokens":1662,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":642,"completion_tokens_details":{"reasoning_tokens":1579}},"tokens_in":642,"tokens_out":1662,"duration_ms":11097,"temperature":1.0,"reasoning_tokens":1579,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:40:43.697456+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the (2+1)D Ising model on an $L\\times L$ lattice, measure the corner-to-corner static response of a parity-even operator (a symmetric linear combination of two mirror-related local operators). If the decay exponent matches a spatial descendant of the lowest primary instead of that primary itself, the claimed primary sieve is false.","supporting_citations":[{"cited_title":"Di Francesco, P","cited_arxiv_id":null,"evidence_quote":"Supplies the Virasoro characters and descendant/quasiprimary structure used to locate the first T-odd quasiprimary in the Ising and tricritical Ising towers."},{"cited_title":"Verresen, R","cited_arxiv_id":null,"evidence_quote":"Computed a $1/L^{14}$ edge-mode splitting in the interacting Majorana CFT whose origin the causation mechanism explains and generalizes."},{"cited_title":"Pleimling and W","cited_arxiv_id":null,"evidence_quote":"Gives the classical corner magnetization exponent used as a check on the extracted corner primary dimension."},{"cited_title":"Hu, Y.-C","cited_arxiv_id":null,"evidence_quote":"Reports the heavy magnetic line defect primary $\\Delta=4.64(14)$ from fuzzy-sphere regularization, which the extracted $\\Delta\\approx 4.6$ matches."}],"review_version":1}