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REVIEW 3 major objections 6 minor

The rate of seed-normalized Krylov-Wigner negativity matches Krylov variance growth if and only if the operator dimension saturates the AdS3 BF bound, where it becomes the product of proper radial position and momentum and therefore the rat

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2026-07-11 21:54 UTC pith:HT55IA3D

load-bearing objection Clean analytic second-moment extension of the Caputa dictionary that correctly isolates Δ=1; the seed-normalization caveat is real but already flagged by the author and does not break the calculation. the 3 major comments →

arxiv 2607.04065 v2 pith:HT55IA3D submitted 2026-07-05 hep-th cond-mat.stat-mechquant-ph

The Normalised Wigner Negativity Rate as a Second-Moment Probe of Infall in AdS₃

Ritam Basu This is my paper
classification hep-th cond-mat.stat-mechquant-ph
keywords spread complexityKrylov-Wigner negativityseed-normalized distributionSU(1,1) coherent statesBreitenlohner-Freedman boundtidal stretchAdS3negative binomial statistics
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.

Spread complexity recovers the proper radial momentum of an infalling particle in AdS, but only as a first-moment (centre-of-mass) diagnostic. This paper proposes a second-moment probe: the rate of a seed-normalized Krylov-Wigner negativity that tracks how the wavepacket spreads away from the classical trajectory. By stripping the decaying return amplitude from the amplitudes before taking the Wigner transform, the author obtains an analytic Bessel form for the seed-normalized distribution and shows that its total negativity grows as sinh to the power 4 Delta of pi t over beta, while the raw state negativity saturates. Matching this rate against the exact growth rate of the Krylov variance (from negative-binomial statistics) holds at late times if and only if Delta equals 1, the Breitenlohner-Freedman saturating dimension in AdS3. At that value the negativity rate is proportional to the product of proper radial position and momentum, i.e., the rate of tidal stretching of neighbouring geodesics falling into the horizon. A sympathetic reader cares because the construction supplies a concrete boundary observable for quantum spreading about the classical infall trajectory, and isolates a geometrically distinguished dimension where second-moment and tidal quantities coincide.

Core claim

The late-time rate of the seed-normalized Krylov-Wigner negativity scales as the growth rate of the Krylov variance if and only if Delta equals 1. At that Breitenlohner-Freedman-saturating value the rate is proportional to C times P_rho, the product of proper radial position and proper radial momentum, which is the rate of tidal stretch of neighbouring infalling geodesics.

What carries the argument

The seed-normalized Krylov-Wigner distribution: the discrete Wigner transform of the descendant amplitudes with the return amplitude divided out, which in the macroscopic limit takes the closed Bessel form proportional to |A|^{2Q} Q^{2 Delta} J_Delta(Z)/Z^Delta. Its total negativity grows as sinh^{4 Delta}(pi t / beta) and supplies the rate that is matched to the variance rate.

Load-bearing premise

That the physically preferred diagnostic of geometric spreading is the seed-normalized (return-amplitude-divided) negativity rather than the raw saturating negativity or another conditioning; the paper itself notes that the leading late-time growth of this quantity coincides with the inverse survival probability, so the matching is largely a statement about the two-point-function exponent 2 Delta.

What would settle it

Compute the exact discrete seed-normalized Krylov-Wigner negativity rate and the exact Krylov variance rate for a primary of dimension Delta not equal to 1, and check whether their late-time ratio remains constant; the paper predicts it scales as sinh^{4 Delta - 4}(pi t / beta) and is flat only at Delta = 1.

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

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 / 6 minor

Summary. The paper constructs a seed-normalized Krylov–Wigner distribution for a local operator quench in a 2d thermal CFT (dividing out the return amplitude from the SU(1,1) coherent-state amplitudes on the Krylov chain), obtains its macroscopic Bessel form, and evaluates the total negativity. It finds that the seed-normalized negativity grows as sinh^{4Δ}(πt/β) while the raw (normalized-state) negativity saturates. Using exact negative-binomial moments and Caputa et al.’s proper-momentum dictionary, the late-time negativity rate tracks the Krylov variance rate if and only if Δ=1 (BF-saturating in AdS3), at which point R∝C P_ρ is interpreted as the rate of tidal stretching of neighbouring infalling geodesics. A speculative bulk-string operator identification via a shared SU(1,1) Casimir is sketched as future work.

Significance. If the boundary matching and tidal rewriting are accepted, the work supplies a clean, parameter-free second-moment companion to the first-moment spread-complexity/radial-momentum dictionary, and singles out the BF edge in AdS3 by a higher-moment diagnostic. Strengths that should be credited: (i) exact SU(1,1)/negative-binomial control of the Krylov statistics (Lemma 3.1, eqs. 3.4–3.6); (ii) an analytic continuum evaluation of the seed-normalized negativity that retains the Bessel variable Z=Qθ̃ through the phase-space integral (eqs. 3.21, 4.7–4.11 and Apps. A–C); (iii) an explicit asymptotic (not exact) matching condition 4Δ−1=3 that is algebraic and falsifiable; (iv) careful separation of the speculative bulk-string conjecture from the boundary results. The main limitation on novelty is that the leading growth of the seed-normalized negativity coincides with that of the inverse survival probability, so the power matching is largely a statement about the two-point exponent 2Δ.

major comments (3)
  1. §3.3, Remark 4.4, and §5 (eqs. 4.8–4.11, 5.3–5.4): After seed normalization, the leading late-time growth of N is that of 1/P=(1−|A|^{2})^{-2Δ}∼sinh^{4Δ}(πt/β). The power-matching condition that selects Δ=1 is therefore essentially a matching of the two-point-function exponent 2Δ against the variance’s sinh^4 growth, not an independent diagnostic of Wigner magic or of second-moment structure beyond the survival amplitude. Remark 4.4 already concedes this. The abstract, title, and §§1, 5–6 still present N as a genuine second-moment/magic probe of geometric spreading. Either (a) give a sharper operational justification for preferring the seed-normalized quantity over the raw saturating negativity (or another conditioning), or (b) reframe the central claim throughout as a structural relation between boundary two-point data and the variance rate that happens to select the BF edge. Leaving th
  2. §6, eqs. (6.3)–(6.4): The rewriting R∝C^{2Δ−1}P_ρ and the identification with tidal stretch at Δ=1 rest on the same seed-normalized N and on the Caputa et al. first-moment dictionary. Given Remark 4.4, the tidal reading inherits the same caveat: at leading order one is rewriting the survival-probability decay dressed by the complexity factor, not an independent bulk tidal observable. The geometric-deviation language should be stated as an interpretation of the matched boundary rates under the seed-normalization choice, not as a derived bulk dual of Wigner negativity.
  3. §7.3 and Conjecture 7.4: The bulk-string section is long relative to its logical status. The only firm input is Casimir compatibility (Lemma 7.2); the operator identification N̂_str≃N̂_Krylov is a conjecture, and Remark 7.1 correctly notes a spectral mismatch (rational vs integer eigenvalues). The tidal-activation heuristic (App. D) is further undercut by the frame caveat the paper itself records. For the present manuscript this material should be substantially shortened and clearly demoted (e.g. a short outlook paragraph), or else the conjecture should be given a concrete, testable criterion beyond shared Casimir. As written it risks diluting the solid boundary results.
minor comments (6)
  1. Abstract and §1: “raw, seed-normalized state negativity” is confusing; the text means the raw (normalized-state) negativity. Align terminology with §4.1 (N_raw vs N).
  2. §4.3 (i)–(iii): The three discrete-to-continuum factors of two are carefully motivated, but a one-line cross-check against the exact discrete Wigner sum for a small prime D (as in Fig. 2) would make the continuum measure less opaque.
  3. Fig. 1 caption and Prop. 4.3: “N_raw(∞) is 2 i.e. lim_{Δ→∞} N_∞=2” is stated without derivation in the main text; either derive the large-Δ limit of c_Δ Γ(2Δ)/[2^{Δ−2}Γ(Δ)] or drop the claim.
  4. §5, Remark 5.1: The asymptotic (not exact) status of (5.4) is well stated; consider adding the same caveat once in the abstract so that “if and only if Δ=1” is not read as an all-time identity.
  5. Typos/notation: “necessarilyspread” (p.2); “theseed-normalized” spacing; occasional missing spaces after commas in displayed math; “AdS 3” vs “AdS_3” inconsistency in headings.
  6. References: Caputa et al. is cited as [20] in the abstract and as Caputa:2024 in the text; ensure the arXiv identifier (2410.23334) is uniformly linked. The concurrent extended-probe work [46] is appropriately flagged.

Circularity Check

1 steps flagged

Leading late-time matching of seed-normalized negativity rate to Krylov variance rate reduces, by the paper's own Remark 4.4, to matching the two-point survival exponent 2Δ (via N ~ 1/P by seed-normalization construction).

specific steps
  1. self definitional [Remark 4.4 (and eqs. 4.1, 4.8–4.12, 3.13–3.17)]
    "Comparing(4.8) with1 /P = (1−|A|2)−2Δ, the two share the same leading late-time scalingsinh4Δ and differ only at subleading order (2|ln|A||=−ln(1−ξ)=ξ+ 1 2ξ2 +···versus1−|A|2 =ξ). Thus the leading growth ofN coincides with that of the inverse survival probability, and the matching identified in section 5 is, at leading order, a statement about the survival-probability decay exponent2Δrather than about the magnitude of the magic."

    Seed-normalization defines N(t) ≡ N_raw(t)/P(t). Once N_raw is shown to saturate (Proposition 4.3), N grows exactly as 1/P by construction. The late-time rate R therefore inherits the exponent 4Δ of the return amplitude. Matching R to dVar/dt (itself fixed by the same |A(t)| via the negative-binomial identity) then reduces to the algebraic condition that the two-point exponent 2Δ equal the variance power, i.e., Δ=1. The claimed second-moment diagnostic is, at leading order, a rephrasing of the survival probability rather than an independent probe of magic or spreading.

full rationale

The boundary derivations of the negative-binomial moments (Lemma 3.1), the macroscopic Bessel form of the seed-normalized Wigner function (3.21), the continuum integral for N (4.7–4.11), and the exact variance rate (5.1) are self-contained and non-circular; they follow from the SU(1,1) coherent-state structure and Stirling/Euler–Maclaurin asymptotics without fitting or self-referential uniqueness theorems. Caputa et al. is an external citation for the first-moment dictionary. The sole load-bearing circularity is the diagnostic choice itself: seed-normalization defines N ≡ N_raw/P, N_raw saturates to a Δ-dependent constant (Proposition 4.3), and therefore the leading growth of N is identical to that of the inverse return probability 1/P ~ sinh^{4Δ}. The power-matching condition 4Δ−1 = 3 that selects Δ = 1 (and the subsequent rewriting R ∝ C P_ρ) is then an algebraic comparison of two consequences of the same |A(t)| extracted from the thermal two-point function. The paper itself states this limitation in Remark 4.4; the magic content (c_Δ, N_∞) enters only subleadingly. This is partial circularity of the central claim, not of the intermediate calculations, hence score 5 rather than 0 or 8.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 2 invented entities

The central boundary claim rests on the standard SU(1,1) structure of the thermal Krylov chain for a 2d CFT primary, the Caputa et al. first-moment dictionary, the definition of the seed-normalized Wigner function, and the macroscopic continuum limit. No free parameters are fitted to data; Δ, β, ε are physical inputs. The bulk-string operator identification is an invented entity with no independent evidence and is cleanly separated as future work.

axioms (4)
  • domain assumption Lanczos coefficients of a thermal 2d CFT primary realize the discrete series D+_Δ of SU(1,1) with b_n^{2} ∝ n(n+2Δ-1)
    Lemma 2.1; taken from the CFT two-point function and standard in the Krylov-complexity literature.
  • domain assumption Spread-complexity rate equals proper radial momentum of an infalling particle (dC/dt ∝ P_ρ)
    Eq. (2.19); imported from Caputa et al. (2024) and used as the first-moment dictionary.
  • standard math Macroscopic continuum limit Q ≫ 1 converts the discrete Wigner sum into the Bessel integral via Stirling + Euler-Maclaurin
    Section 3.4 and Appendices A–B; standard asymptotic analysis.
  • ad hoc to paper Seed normalization (division by the return amplitude P(t)) isolates geometric spreading rather than probability leakage
    Definition (3.14)–(3.16) and Remarks 3.3, 4.4; a modeling choice that makes N grow while raw negativity saturates.
invented entities (2)
  • Seed-normalized Krylov-Wigner distribution W = W_raw / P no independent evidence
    purpose: To obtain a growing negativity that tracks wavepacket spreading after the return amplitude is divided out
    Introduced in §3.3; its leading growth coincides with 1/P, so independent evidence that it measures 'magic content per surviving amplitude' rather than pure survival is limited.
  • Operator identification N̂_str ≃ N̂_Krylov via shared SU(1,1) Casimir no independent evidence
    purpose: To give a bulk-string reading of the Krylov variance as transverse string-area fluctuations
    Conjecture 7.4; supported only by Casimir equality (necessary but not sufficient) and explicitly labeled future work.

pith-pipeline@v1.1.0-grok45 · 30986 in / 3133 out tokens · 29095 ms · 2026-07-11T21:54:55.648666+00:00 · methodology

0 comments
read the original abstract

In spread complexity, the average position of an operator along its Krylov chain, recovers the right radial momentum of an infalling particle in AdS, yet it is a measure of the first moment, irrespective of the spread of the wavepacket away from its classical trajectory. The rate of a normalized Krylov-Wigner negativity can be proposed as a diagnostic of the second moment of the boundary state that captures this spreading. Starting with the seed-normalized Krylov-Wigner distribution -- that is, the Wigner transform of the descendant cloud, with the decaying return amplitude divided out -- we obtain an analytic Bessel form in the macroscopic limit and compute its total negativity explicitly. Retaining the Bessel variable all the way through, we find that the negativity goes as $\sinh^{4\Delta}(\pi t/\beta)$, while the raw, normalized-state negativity saturates, as dictated by the $O(\sqrt{D})$ bound. Using the exact negative binomial statistics of the Krylov chain, the normalized negativity in late times a fixed power of the second moment of the Krylov wavepacket, $\mathcal{N}(t)\propto\bigl[\mathrm{Var}(n)\bigr]^{\Delta}$, for every dimension $\Delta$. The relation linearizes precisely at $\Delta=1$: only at this dimension does the negativity rate track the growth rate of the Krylov variance, and there, through the momentum dictionary of Caputa et al. [arXiv:2410.23334], the rate becomes the product of the proper radial position and momentum, $\dot{\mathcal{N}}\propto\mathcal{C} P_\rho$, i.e., the rate of the tidal stretch of nearby geodesics falling into the horizon. We comment on the direction for future research, in particular the interpretation of the transverse string size operator in terms of the Krylov number operator through the common $\text{SU}(1,1)$ discrete series.

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