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Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read 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

desk verdict 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. read the letter →

arxiv 2607.05294 v2 pith:VFR4UYJY submitted 2026-07-06 hep-th math-phmath.MPquant-ph

Abhishek Chowdhury , Ajit Prasad Mahapatra This is my paper · ORCID
classification hep-thmath-phmath.MPquant-ph MSC 42C0533C4581Q10
keywords KrylovcomplexityspreadsubspacesChristoffeltransformsorthogonalpolynomialsJacobichainsCharlierKrawtchouk
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [D.3, Eq. (D.69)] 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.
  2. [D.3, Eq. (D.71)] 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.
  3. [D.3, Eqs. (D.72)–(D.86)] 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.
minor comments (4)
  1. [Section 2, Eq. (2.1)] 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.
  2. [Section 5.1, footnote 9] 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.
  3. [General readability] 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.
  4. [Section 5.3, Eq. (5.61)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained.

full rationale

The paper's central chain — polynomial seed Q(H)|K0⟩, Christoffel measure dν_Q = |Q|²/N_Q dµ, finite-band connector Γ, projected Christoffel–Darboux kernels, and the Charlier/Krawtchouk/Chebyshev applications — is derived from reference Jacobi data and Fourier–OP moments without fitting any parameter to the target quantities. The finite-band transfer is a proven orthogonality identity (Prop. 3.1), the shifted Jacobi data follow algebraically from connector rows (Prop. 3.2), and cumulative probabilities are obtained by finite-rank projection of the reference kernel (Prop. 4.1); none of these steps defines the input in terms of the output. The solvable examples use physical parameters (λ, ρ, ω, γ) and are checked against independent computations, including direct Lanczos calculations and the separately derived sine-transform dynamics of ref. [4]. The one notable weakness is the compressed large-index estimates in Appendix D.3, especially (D.71)–(D.73), but that is a rigor/completeness issue, not circularity: the asserted estimates are not fitted constants, and the finiteness conclusion follows from them rather than being assumed. No load-bearing self-citation chain, no fitted input renamed as prediction, and no known result merely relabeled as a new one are present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The central claims rest on the standard Lanczos/OP dictionary for self-adjoint operators, on the determinacy/density hypothesis for infinite-support chains, on support-compatibility for sequential filter composition, and on the self-adjoint domain assumption for the Liouville-space extension. No numerical constants are fitted: λ, ρ, ω, γ are physical inputs. The constructed entities are mathematical (connectors, remainder algebras, parent measure); none is a new physical degree of freedom, and all have checkable outputs against direct computation or prior results.

assumptions (5)
  • domain assumption For complete infinite chains: |K0⟩ ∈ ∩_{n≥0} Dom(H^n) and polynomials are dense in L²(µ); determinacy via Carleman's criterion Σ b_n^{-1} = ∞.
    §2 'Reference cyclic data'. Needed so the polynomial cyclic subspace coincides with the spectral cyclic subspace and the Jacobi data determine µ uniquely; satisfied by the solvable examples but a genuine hypothesis for general application.
  • domain assumption Relative composition (Prop 2.1) requires ν_j ≪ ν_i (support compatibility); filters that delete an atom remove that component from the relative problem.
    §2 discussion after Prop 2.1. The cocycle F_{i→j} exists only on the common surviving support; the direct transform µ→ν_j always exists, but sequential routes need the compatibility condition. This is an assumption about the seed family, satisfied except at explicit resonances.
  • domain assumption Liouville-space extension: a positive operator inner product exists for which L = [H, ·] is self-adjoint on the relevant cyclic domain.
    §1 and App H. The operator-space results (Q(L)O seeds, transition-resolved endpoint-energy data) are conditional on this domain hypothesis.
  • standard math Charlier polynomials have simple real zeros, so the CRT identifies R_r and R_r^(2) with evaluation and first-jet data at those zeros.
    §5.1, D.2. Standard for orthogonal polynomials of a positive measure; used to build the root-free remainder calculus.
  • standard math Finite cyclic chains: T_d is both the characteristic and minimal polynomial of the finite Jacobi matrix; polynomial functions on the support are represented by A_d = C[E]/⟨T_d⟩.
    App A, eqs. (A.4)–(A.7). Used for all terminal-quotient constructions; lift independence is proved in Prop B.1.
invented entities (2)
  • Finite-band connector Γ_{n,m}^Q independent evidence
    purpose: Carries all seed dependence in the amplitude transfer; converts reference Fourier–OP moments into shifted Krylov amplitudes and Jacobi data.
    Mathematical object, not a physical entity. Independently anchored by explicit consistency checks (r=1 vs the (i∂_t)^2 rule; spin-3/2 vs direct 4-d Jacobi; tight-binding vs ref [4]), rather than by a predicted observable.
  • Remainder coordinate algebras R_r and R_r^(2) independent evidence
    purpose: Root-free encoding of the divisibility and first-derivative constraints for Charlier number-state jumps.
    Mathematical replacement for the zeros of P_r. Its outputs (connector coefficients, eq. (D.46)) are checked against the determinant formula at r=1 and r=3 and against direct Hilbert-space construction of shifted vectors (D.61).

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Pith. "Pith review of Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity." pith.science (2026). https://pith.science/paper/VFR4UYJY

@misc{pith2026260705294,
  author       = {Pith},
  title        = {Pith review of: Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VFR4UYJY}},
  note         = {Machine review of arXiv:2607.05294}
}
abstract

State Krylov, or spread, complexity belongs to the cyclic pair $(H,|K_0\rangle)$, so changing the initial state at fixed-$H$ reorganizes the Lanczos chain within the reference cyclic subspace. For normalized polynomial descendants $|\psi_Q\rangle \propto Q(H)|K_0\rangle$, this reorganization is the positive Christoffel reweighting of the reference measure by $|Q|^2$. Orthogonality gives a finite-band transfer from reference Fourier-orthogonal-polynomial moments to shifted amplitudes, while a finite-rank projection of its Christoffel-Darboux kernel yields cumulative probabilities and spread complexity. Complex superpositions, confluent roots, deletion of spectral atoms and terminal closure enter the same construction. In the Heisenberg-Weyl/Charlier oscillator, root-free remainder recurrences govern arbitrary number-state jumps. Their large-index behavior distinguishes generic shifts from resonant deletion of Poisson atoms, determines the Jacobi asymptotics, and proves the finiteness of the complexity at finite time for every fixed jump. In finite $SU(2)$/Krawtchouk and tight-binding/Chebyshev chains, product identities and product-Gram factorizations in the terminal quotient determine all weight-state and localized-site connectors through the terminal edge, while Weyl reflection pairs opposite spin weights. The first-jump Charlier-Hermite scaling carries it to continuous spectral support. Finite seed families admit a matrix-valued parent measure, and the relative calculus extends to polynomial operator descendants whenever the Liouvillian has a self-adjoint realization for the chosen inner product. A solved cyclic problem determines a family of fixed-$H$ dynamics and separates preparation dependence from changes of the generator or Hilbert-space dimension.

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