{"id":"543b2d77-b124-4cda-a7f0-34967b825ade","arxiv_id":"2607.05294","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Changing the initial state to Q(H)|K0⟩ is exactly a Christoffel reweighting of the spectral measure by |Q|²; Krylov complexity then transfers from the reference problem through finite-band connectors and finite-rank kernel projections, with closed forms in Charlier, Krawtchouk and Chebyshev chains.","lead":"Starting a quantum system from a 'polynomial-filtered' version of a reference state is shown to be exactly a Christoffel reweighting of the spectral measure, so all Krylov-complexity data transfer from the original calculation through finite-band connectors and finite-rank kernel projections. A generalist might care because Krylov (spread) complexity is a standard probe of quantum chaos and state growth, and this provides an exact toolkit for how it depends on the initial sta","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Finiteness of Charlier jump complexity rests on compressed large-index estimates (D.71)-(D.73); if the asserted decay is off, the advertised K_r(τ)<∞ can fail.","rationale":"The reader's verdict CONDITIONAL with low correctness risk is appropriate. The mathematical structure of the Christoffel transform, finite-band connectors, and projected Christoffel–Darboux kernels is standard and the proofs in Props 3.1–4.2 are solid; no fatal flaw appears there. The solvable examples (Charlier, Krawtchouk, Chebyshev) are internally consistent and contain multiple independent checks. The one substantive gap is the large-index control in Appendix D.3: the asserted uniform bounds on moving-basis coordinates and the Casoratian asymptotics are exactly what turn the exact finite-row formulas into the infinite complexity series and the advertised finiteness theorem. The derivation is sketched rather than fully written, and the constants C_*, σ_*, L_0 are asserted without explicit construction. This is not an ad hominem or a matter of external consensus; it is an internal completeness issue in a load-bearing proof. A numerical test using the exact finite-dimensional remainder recurrences can verify whether the asymptotic intercepts and the O(n^{-1}) corrections actually hold, thereby checking whether the concern lands. If the estimates were wrong, the central claim about finite-time finiteness for Charlier jumps would collapse; if they pass, the paper still would benefit from a fully detailed proof, but the current CONDITIONAL verdict remains the right call. The reader's weakest assumption and mine coincide; no other part of the argument raises a comparable risk.","tokens_in":63802,"tokens_out":30517,"duration_ms":273311,"concrete_test":"For r=1 with λ=0.7, r=3 with λ=3 (the paper's resonant example, s_3=1), and r=2 with λ=1.5, generate the shifted Jacobi coefficients ã_n, β̃_n using the exact remainder recurrences of §D.2 (companion matrices of size r, no root extraction) for n up to 10^4. Check that n(ã_n − n − s_r(λ)) and β̃_n − λ(n+2r−s_r(λ)) stay bounded or converge, and that the truncated complexity K_{r,M}(τ) at τ=1 stabilizes as M grows. If the sequences diverge or the intercepts differ from (5.15), the finiteness theorem fails; if they pass, the D.71/D.73 scaling is validated numerically, though a complete analytic proof remains desirable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's core relative-calculus theorems (finite-band transfer, projected kernels, terminal-quotient connectors) are well founded and internally consistent. The load-bearing weak spot is the proof of finiteness for Charlier number-state jumps, which relies on the large-index asymptotics (D.66) and on the C_hop bound in the differential inequality. These are derived from the moving-basis estimates (D.71) and Casoratian determinant (D.73), but the text only sketches the derivation, asserting constants C_*, σ_*, L_0 after Cramer's-rule/Vandermonde arguments. It does not display the determinant expansion or justify uniformity over 1≤ℓ≤L/2. If the off-lattice coordinates decayed only as L^{-ℓ/4} rather than L^{-(ℓ+j-1)/2}, the tail sum of the correction series would not be superalgebraically small and K_r(τ) could diverge. Since Corollary 5.1 and the intercept formula (5.15) explicitly cite these estimates, this is a genuine omitted proof step, not mere stylistic compression.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops an exact 'relative calculus' for changing the initial state in Krylov/spread complexity without rerunning Lanczos, for seeds of the form |ψ_Q⟩ ∝ Q(H)|K0⟩. The central construction identifies the new spectral measure as the Christoffel reweighting dν_Q = |Q|²/N_Q dµ. For a degree-r seed it proves a finite-band transfer (Prop 3.1) expressing shifted Krylov amplitudes as a finite linear combination of reference Fourier–OP moments, a connector dictionary for shifted Jacobi data (Prop 3.2), and a finite-rank projection of the Christoffel–Darboux kernel (Prop 4.1) giving cumulative probabilities and spread complexity. The results are applied to solvable chains: Heisenberg–Weyl/Charlier number-state jumps, with root-free remainder recurrences and large-index asymptotics claimed to prove finiteness of K_r(τ) and a vacuum lower bound (Cor 5.1); finite SU(2)/Krawtchouk weight-state jumps with terminal-quotient product-Gram connectors and Weyl reflection; tight-binding/Chebyshev localized-site jumps matching known Bessel dynamics; and a Charlier–Hermite continuous-spectrum endpoint. A matrix-valued parent measure organizes finite seed families; mixed-state and Liouville-space extensions are sketched.","tokens_in":63900,"tokens_out":21824,"duration_ms":212946,"significance":"If the stated results hold, this is a significant contribution to the Krylov-complexity literature: it gives a parameter-free, exact transfer from a solved reference problem to a whole family of fixed-H initial-state problems, with applications to separating preparation dependence from Hamiltonian/dimension changes. The paper contains many explicit propositions with proofs and strong internal consistency checks: the r=1 Charlier connector matches the (i∂_t)^2 rule and local sum rules; the spin-3/2 connector matches a direct four-dimensional Lanczos calculation; the tight-binding connector agrees with the sine-transform dynamics of ref [4]; and the Hermite endpoint reproduces β̃1=3. These checks give confidence in the core finite-band and projected-kernel machinery. The main reservation is that the headline finiteness result for Charlier jumps rests on the large-index analysis of Appendix D.3, which contains a likely erroneous determinant identity and only sketched uniform estimates; this needs a full correction before the result can be considered established.","major_comments":[{"comment":"The Casoratian identity stated in (D.69) is incorrect as written. For r_off=2, the left-hand side det Γ(n−j−x_a) equals Γ(n−x1)Γ(n−x2)(x1−x2)/[(n−x1−1)(n−x2−1)], not the product of Γ(n−x_a) with the Vandermonde alone. The missing row-dependent denominator factors are of order n^{-j} for columns j≥1, so they change the L-power in (D.73) and hence the consecutive determinant ratio d^{(r)}_L/d^{(r)}_{L-1} used in (D.79) and (D.86). Since the intercept asymptotics (5.15) and Corollary 5.1 depend on this ratio, the finiteness proof is not established as it stands. Please supply a correct determinant evaluation, or state explicitly if a different matrix (e.g., the normalized evaluation matrix with prefactors removed) is meant, and re-derive the L-power.","section":"D.3, Eq. (D.69)"},{"comment":"The uniform large-index bounds (D.71) are asserted after a Cramer's-rule/Vandermonde argument, but the proof is not shown. What is needed is a componentwise bound uniform over 1≤ℓ≤L/2, with constants C_*, σ_*, L_0, and the superalgebraic tail sum in the second line. These bounds are load-bearing: they control the convergence of the correction series ∑ C_L^{(r)} in (D.53) and thus the entire finiteness claim for K_r(τ). If the decay of the off-lattice coordinates were only L^{-ℓ/4} rather than L^{-(ℓ+j−1)/2}, the tail sum would not be superalgebraically suppressed and K_r(τ) could diverge. Please provide a complete proof, including the form of the generalized Vandermonde bound and the uniformity in ℓ, or replace the argument with an alternative summability proof.","section":"D.3, Eq. (D.71)"},{"comment":"The derivation of the shifted Jacobi asymptotics (D.66) is compressed at several points beyond the determinant identity. In particular, the treatment of the lattice sector via the Schur complement (D.74)–(D.78) states that the mixed block is suppressed by Z_L^{-1} 'faster than every inverse power of L', but the relevant cancellation between a polynomially growing mixed block and the interpolation-coordinate decay is not demonstrated. The same applies to the endpoint interpolation (D.80)–(D.82), where the leading O(L) term is quoted with little derivation. Since eq. (D.66) and Corollary 5.1 are advertised as principal results, this section needs to be rewritten at a level that allows the reader to verify the uniformity in ℓ and the constants.","section":"D.3, Eqs. (D.72)–(D.86)"}],"minor_comments":[{"comment":"The definition of the polynomial cyclic subspace uses the condition 'p(H)|K0> exists', which is tautological for polynomial functions of a self-adjoint operator. It would be clearer to say that the span is taken over polynomials for which the spectral-theoretic action is defined, or to state the domain assumption on |K0> once and then use it.","section":"Section 2, Eq. (2.1)"},{"comment":"The formula for e^{-iHt}|1> is stated without derivation; a one-line derivation from [H,a†]=ωa†+\\bar g would make the check more transparent.","section":"Section 5.1, footnote 9"},{"comment":"The paper introduces a very large number of symbols (Γ, G, H, M, Q, W, and many superscripts). A consolidated notation table, especially for the different Gram matrices and kernel notions, would substantially improve readability.","section":"General readability"},{"comment":"The infinite sum defining K_1(t) is written without an explicit convergence statement. The Bessel asymptotics make it finite for all finite t, but a brief parenthetical would help the reader.","section":"Section 5.3, Eq. (5.61)"}],"recommendation":"major_revision","confidential_remarks":"The core Christoffel-transfer formalism is novel and well-executed, with excellent internal consistency checks. The blocking issue is the rigor and apparent correctness of the large-index asymptotics in Appendix D.3, which underpins the advertised finiteness theorem for Charlier jumps. If the authors can supply a correct, complete proof of (D.69)–(D.73) and the derived asymptotics, the paper should be acceptable for publication. The scope and novelty appear appropriate for the journal; no citation concerns were identified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know up front. The central construction—polynomial seed jumps as Christoffel reweightings, with shifted cyclic data recovered from the reference problem through finite-band connectors and projected kernels—is new, precisely stated, and mostly holds up. The weak point is the Charlier finiteness theorem: it rests on large-index estimates in Appendix D.3 that are sketched with constants asserted, not proven in full.\n\nWhat is actually new: the finite-band transfer (Prop 3.1), the connector-to-Jacobi dictionary (Prop 3.2), the projected Christoffel–Darboux kernels (Props 4.1–4.2), and the root-free remainder recurrences for Charlier jumps. The solvable examples are not decorative. The Charlier, Krawtchouk and Chebyshev chains exercise three different closures—infinite discrete support with atom deletion, finite quotient with support loss, continuous band with site-localized seeds—and each has closed-form connectors. The internal checks are genuine: spin-3/2 against a direct four-dimensional Lanczos run, tight-binding against ref. [4] (modulo the noted time-conjugation), Hermite endpoint β̃1=3. No fitted parameters appear anywhere.\n\nWhere it gets soft, in proportion. The load-bearing bounds (D.71)–(D.73), which underlie Corollary 5.1 and the intercept asymptotics (5.15), are asserted after a Cramer's-rule/Vandermonde argument; the constants C_*, σ_*, L_0 are not displayed and uniformity over 1 ≤ ℓ ≤ L/2 is not justified. That is an omitted proof step, not stylistic compression. I doubt the stress-test's pessimistic scenario—it would require the off-lattice asymptotics (D.68) to produce much weaker decay than they manifestly do—but the argument should be written out or the claims weakened. Secondary: Apps G and H are brief and domain-conditional, and the paper itself concedes that the numerical error propagation in App F is unfinished. Minor by comparison.\n\nWho gets value: anyone using spread complexity as a diagnostic (chaos, quenches, operator growth), because reference Lanczos data become reusable for whole polynomial families of seeds. It is a long paper with heavy appendices, but the main body is readable and the claims are specific enough to check.\n\nRecommendation: serious refereeing, yes. The one concrete demand: a complete derivation of the D.3 estimates, or a scoped restatement of the finiteness claims.","headline":"The Christoffel/connector toolkit is new and mostly sound, but the Charlier finiteness theorem rests on a sketched large-index estimate that a referee should demand be written out.","tokens_in":64564,"tokens_out":4227,"would_cite":true,"duration_ms":46228,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42C05","33C45","81Q10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Changing the initial state to a polynomial descendant Q(H)|K0⟩ is not a new Krylov problem: it is an exact Christoffel reweighting of the reference measure, with finite-band transfer of amplitudes and a finite-rank kernel projection giving","keywords":["Krylov complexity","spread complexity","Krylov subspaces","Christoffel transforms","orthogonal polynomials","Jacobi chains","Charlier polynomials","Krawtchouk polynomials"],"falsifier":"Compute high-precision Charlier shifted Jacobi coefficients for a fixed jump (e.g., r = 3, λ = 3) at large n and test Eq. (D.66): if the O(n⁻¹) corrections fail to decay or the asymptotic intercepts drift, the large-index estimates are wrong. More directly, evaluate the moving-basis coordinate bound (D.71) and the Casoratian asymptotics (D.73) numerically for n up to 10⁵; a violation would falsify the finiteness proof.","tokens_in":63504,"feed_emoji":"🌀","tokens_out":4844,"duration_ms":54314,"temperature":0.7,"pith_summary":"This paper establishes that for any normalized polynomial seed |ψQ⟩ ∝ Q(H)|K0⟩ at fixed H, the entire Krylov complexity problem is encoded in the already-solved reference Lanczos data. In the spectral representation, the new seed's measure is dνQ = (|Q|²/NQ)dµ, a positive Christoffel reweighting. Orthogonality makes each shifted Krylov amplitude a finite linear combination of reference Fourier–orthogonal-polynomial moments, with bandwidth 2 deg Q, and a finite-rank projection of the reference Christoffel–Darboux kernel gives cumulative probabilities and spread complexity. The paper carries this through the Charlier/Heisenberg–Weyl, Krawtchouk/SU(2), and Chebyshev/tight-binding chains, proving in particular that every fixed number-state jump in the Charlier oscillator has finite spread complexity at finite time and is bounded below by the vacuum complexity. If correct, reference Lanczos data never need to be recomputed for any polynomial-related seed.","feed_headline":"One Lanczos run gives every polynomial-seed spread complexity","feed_subtitle":"A Christoffel reweighting turns polynomial state changes into finite-band transfers; seed changes no longer require a new Krylov solve.","key_machinery":"The load-bearing object is the Christoffel transform of the scalar spectral measure, dνQ = (|Q|²/NQ)dµ. The transfer identity WQ(E)Rn^Q(E) = Σ_{m=n}^{n+2r} Γn,m Pm(E) (Prop. 3.1) expresses the shifted Krylov amplitude as a finite combination of reference Fourier–orthogonal-polynomial moments, with the connector Γ fixed by root/confluent constraints, Gram factorization, or reduced-Jacobi data. The second mechanism is the projected Christoffel–Darboux kernel: multiplication by Q̂ embeds the shifted degree-ℓ polynomial space into the reference space of degree ℓ+r, so the shifted cumulative probability is the reference kernel minus a rank-r (or derivative-jet) projector.","core_discovery":"The central discovery is that a polynomial initial-state jump reorganizes the Krylov chain through the Christoffel transform dνQ = (Q♯Q/NQ)dµ, and this reorganization is exactly carried by reference data. Multiplication by WQ maps the shifted monic orthogonal polynomial Rn^Q into the finite reference window span{Pn, …, Pn+2r} with connector coefficients Γn,m; the shifted amplitudes are Γ-weighted sums of reference Fourier moments Im(t). Cumulative probabilities are obtained by projecting the evolved seed onto the multiplication image Q̂Pℓ, equivalently by subtracting a rank-r (or derivative-jet) Gram correction from the reference Christoffel–Darboux kernel; spread complexity then follows fro","pith_inferences":["Inference: the paper's formalism makes the marginal cost of a polynomial seed family nearly zero once the reference Lanczos data exist, so the practical bottleneck in many-body applications shifts from Lanczos diagonalization to controlling truncation and rank errors in the reference chain.","Inference: the Charlier result that fixed polynomial degree preserves the linear slopes of the Jacobi coefficients suggests a dynamical diagnostic: a seed deformation whose shifted Jacobi coefficients develop different large-index asymptotics is genuinely non-polynomial, and the intercepts s_r(λ) could be used to quantify atom-deletion resonances.","Inference: the reference calculus for rational or resolvent-dressed filters would not be finite-band; approximating such filters by polynomials would turn the present exact construction into a controlled large-degree asymptotic problem, likely coupling to the thermodynamic limit in many-body settings.","Inference: the matrix-valued parent measure organizes a family of nonlinear scalar problems and shows that block or multiseed complexity cannot be obtained by linearly superposing scalar complexities; a genuine block-Lanczos complexity for the parent would require a separate definition and analysis."],"forward_implications":["Given any solved cyclic problem (H, |K0⟩), the Krylov dynamics of every polynomial descendant Q(H)|K0⟩ is obtained from finite reference data: connector rows of width 2 deg Q and the reference moments Im(t).","Shifted Lanczos coefficients ãn^Q and b̃n^Q are determined algebraically from connector ratios, so no fresh Lanczos pass in the full Hilbert space is needed for polynomial seeds.","Cumulative probabilities and spread complexity follow from a finite-rank projection of the reference Christoffel–Darboux kernel; for a degree-r seed only a rank-≤r correction is subtracted.","In the Heisenberg–Weyl/Charlier oscillator, every fixed number-state jump has finite spread complexity at finite time and Kr(τ) ≥ K0(τ), with strict inequality for r ≥ 1 away from revival times.","In finite SU(2)/Krawtchouk chains, Weyl reflection pairs weights r and N−r with equal complexity; in tight-binding/Chebyshev chains polynomial seeds become localized-site jumps with explicit Bessel amplitude sums."],"fun_headline_variants":["One Krylov solve covers all polynomial seed shifts","Christoffel transform makes seed changes a finite-band transfer","Polynomial seed jumps: reuse one Krylov chain","Christoffel reweighting: all polynomial seeds from one Lanczos run","Spread complexity for any polynomial seed from one Lanczos chain"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is the large-index control of the Charlier root-Gram sequences — the asserted uniform bounds on moving-basis coordinates and the Casoratian asymptotics in Eqs. (D.71) and (D.73) — since finiteness of spread complexity for every fixed jump and the asymptotic intercepts depend on those estimates; a milder secondary premise is the regularity/domain hypothesis |K0⟩ ∈ ∩ Dom(H^n) with polynomial density in L²(µ), automatic for the solvable examples but not","fun_headline_variants_meta":{"raw":{"variants":["One Krylov solve covers all polynomial seed shifts","Christoffel transform makes seed changes a finite-band transfer","Polynomial seed jumps: reuse one Krylov chain","Christoffel reweighting: all polynomial seeds from one Lanczos run","Spread complexity for any polynomial seed from one Lanczos chain"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0008,"raw_usage":{"total_tokens":3412,"prompt_tokens":859,"completion_tokens":2553,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":603,"completion_tokens_details":{"reasoning_tokens":2471}},"tokens_in":603,"tokens_out":2553,"duration_ms":17279,"temperature":1.0,"reasoning_tokens":2471,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T08:27:48.130857+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute high-precision Charlier shifted Jacobi coefficients for a fixed jump (e.g., r = 3, λ = 3) at large n and test Eq. (D.66): if the O(n⁻¹) corrections fail to decay or the asymptotic intercepts drift, the large-index estimates are wrong. More directly, evaluate the moving-basis coordinate bound (D.71) and the Casoratian asymptotics (D.73) numerically for n up to 10⁵; a violation would falsify the finiteness proof.","supporting_citations":[],"review_version":2}