{"id":"2b3d1a4e-eebd-4188-972a-5a31709f1d26","arxiv_id":"2607.24319","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Compressed CDFCI converges linearly to a restricted eigenproblem whose eigenvalue error is O(τ²) under spectral-gap and exponential-decay assumptions.","lead":"Compressed coordinate-descent FCI is proved to converge linearly to a restricted eigenproblem, with ground-state energy error scaling as the square of the compression threshold. That bound justifies practical truncation and simple τ² extrapolation for large quantum-chemistry Hamiltonians.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"Assumption 3.9 is indeed the weakest premise, but the O(τ²) rate is structurally more robust than the reader claims; what genuinely hinges on exponential decay is the dimension-independence of the prefactor, not the quadratic rate itself.","rationale":"I agree with the reader that Assumption 3.9 is the weakest premise in the chain and that the CONDITIONAL verdict is appropriate; I disagree only on the sharpness of the consequence attributed to it. The quadratic energy-error scaling is the generic second-order behavior of eigenvalue perturbation under a spectral gap, and the paper already contains the ingredients (the ‖H_{21}c^{k-opt}_1‖₂ ≤ √k n_z τ counting bound in (5.15)–(5.16), plus gap(H) from Assumption 3.8) to obtain O(τ²) via a purely ℓ₂ route. The exponential-decay assumption is what keeps the prefactor free of factors of n or k — decisive for practical meaning at FCI dimensions, but a different statement than \"no quadratic accuracy.\" This distinction matters for how the result should be read and cited: the rate is likely robust; the dimension-free constant is not. The empirical support for the assumption (one equilibrium-geometry H2O-STO-3G plot) is thin precisely where it could fail — stretched-bond, multireference systems — so the assumption's scope remains genuinely open. The constant inconsistencies the reader noted ((3.3) vs (5.6) differ by a factor of 4; the m in (3.4) vs (5.16) by a factor of 16 inside the square root) are real but cosmetic relative to this structural question. None of this moves the verdict off CONDITIONAL: the logical spine holds, the main theorem is believable under its stated assumptions, and the needed work is (a) the constant-level cleanup the reader already required and (b) either an ℓ₂-only re-derivation clarifying what 3.9 buys, or broader numerical evidence of exponential decay across correlation regimes. Hence UNCHANGED with partial agreement.","tokens_in":16504,"tokens_out":11271,"duration_ms":895931,"concrete_test":"Re-derive Theorem 3.13 with Assumption 3.9 removed, replacing Lemma 3.10's ℓ₁/ℓ∞ control by the ℓ₂ perturbation bound ‖c_2‖₂ ≤ ‖H_{21}c^{k-opt}_1‖₂/gap(H) (which uses only Assumption 3.8 and the compression counting bound already in §5.2). Track exactly which powers of n and k enter the prefactor of (5.25). If the O(τ²) rate survives with only polynomial k-, n-dependence, the assumption is confirmed as necessary only for a dimension-free constant, and the reader's \"no quadratic accuracy without 3.9\" should be softened; if the rate itself degrades to O(τ), the assumption is fully load-bearing and the CONDITIONAL verdict's emphasis is right. Either outcome directly determines what the paper's main theorem actually guarantees for large FCI problems.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I traced the proof chain for the headline claim |λ − λ_{k-opt}| = O(τ²) (Thm 3.13). The spine is sound: Lemma 3.5's ℓ∞ gradient control via the compression rule (4.3) is defensible — a limit-unselected coordinate i must have had every contribution |c_j H_ij| ≤ τ discarded, else b_i ≠ 0 would eventually outcompete the vanishing selected-coordinate gradients and i would be selected. Lemma 3.12's pseudo-inverse argument is also coherent: since −∆c solves (5.13), the minimum-norm solution satisfies ‖x‖₂ ≤ ‖∆c‖₂ < d, and the contradiction argument for ‖c_2‖∞ < 2‖x‖∞ closes correctly. The exponential-decay assumption enters at two points: Lemma 3.10 (‖c‖₁ ≤ Q‖c‖∞) feeding Lemma 3.11's m-constant, and (5.23), where Σ_{j>k}|(c_2)_j| ≤ Q‖c_2‖∞ converts the ℓ∞ tail bound into an ℓ₁ bound without an (n−k) factor. However, the reader's statement that \"without it the paper only controls the residual in ℓ∞ and does not obtain quadratic energy accuracy\" overstates the case. The O(τ²) scaling is second-order eigenvalue perturbation and should follow from the ℓ₂ bound alone: ‖H_{21}c^{k-opt}_1‖₂ ≤ √k n_z τ (the paper's own counting bound, no decay needed) plus gap(H) gives ‖c_2‖₂ = O(τ) by standard eigenvector perturbation, and |c_{k-opt,1}⊤H_{12}c_2| ≤ ‖H_{21}c^{k-opt}_1‖₂‖c_2‖₂ = O(k n_z² τ²/gap²). What exponential decay actually buys is removal of the k/n-dependence from the prefactor — which matters enormously here because FCI dimensions are astronomical (Table 1: up to 10^14), so an n-dependent constant would make the bound vacuous in practice. The genuinely open empirical question is whether FCI ground eigenvectors obey Assumption 3.9 with system-size-independent P in strongly correlated (multireference) regimes, where several determinants have comparable weight and the single dominant-determinant intuition behind Fig. 1 breaks down. The paper supports the assumption with one H2O-STO-3G plot (R² = 0.95) at equilibrium geometry — the regime least likely to stress it. So: same l","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper analyzes CDFCI with threshold compression within the unconstrained formulation min ‖H+ccᵀ‖²_F of [12]. Three results are claimed: (i) under local strong convexity on D±, the compressed iteration converges linearly to ±c^{k-opt}, the ground eigenvector of the principal submatrix H_{VV} on the cumulative support V (Thm 3.7); (ii) the compression-induced ℓ∞ residual is O(n_z τ) (Lemma 3.5, Eq. (4.3)); (iii) under a spectral gap (Assump. 3.8) and exponential decay of the sorted ground eigenvector (Assump. 3.9), the eigenvalue error obeys |λ−λ_{k-opt}| = O(τ²) with an explicit prefactor (Thm 3.13). Numerics on synthetic matrices and H2O/C2 cc-pVDZ confirm linear convergence, the τ² scaling, and support a τ²-extrapolation scheme.","tokens_in":17022,"tokens_out":21774,"duration_ms":553497,"significance":"This is, to my knowledge, the first convergence theory for CDFCI as actually implemented (with compression), a method in practical use for FCI. The τ² result gives a principled basis for the energy-extrapolation heuristic, and the paper ships a falsifiable scaling prediction that is tested on both synthetic and realistic systems, with honest reporting of the bound's looseness (ratios 10¹–10⁴, §6.2). The proof architecture (residual control → restricted-problem identification → pseudoinverse tail bound → quadratic form) is sound in outline, and the extrapolation protocol of §6.3 is a useful practical deliverable. The theory is explicitly conditional (local assumptions, realized support), which is appropriate for the problem.","major_comments":[{"comment":"Eq. (4.3) (and the same issue in (3.2)): the residual accounting is not rigorous as written. 'b_i has been compressed at most k times' is unjustified — selected coordinates are revisited many times, so the number of discarded contributions per (i,j) pair is unbounded a priori. The salvageable argument is: b_i^{(l)}=0 forces every contribution to have been discarded at the most recent selection of each j, giving |H_ij c_j^{(l)}| ≤ τ directly. Two more gaps: (b) coordinates with b_i≠0 but c_i=0 are untreated, though the argmax rule (2.3) uses the proxy b; (c) on selected coordinates b differs from Hc by the frozen pre-acceptance error (bounded by n_z τ via the same argument), so the claim above (4.2) that |∇_{j_l}f| attains the max over selected coordinates holds only up to O(n_z τ) slack. Lemma 3.5 feeds Lemma 3.6, Thm 3.7 and Lemma 3.11, so this needs a careful rewrite; constants, not ex","section":"§4.1, Eq. (4.2)-(4.3)"},{"comment":"The proof conflates two index orderings. Assumption 3.9 is formulated in the magnitude-sorted order of c, while 'the first k coordinates are selected' is the algorithm-determined ordering. The bound Σ_{j>k}|(c_2)_j| ≤ Q‖c_2‖∞ used in (5.23)–(5.24) holds only if the unselected set is (a subset of) the magnitude tail of c; this does not follow from the stated assumptions, since the dynamics need not select coordinates in magnitude order. Without it, the dimension-independent prefactor in (3.5) is not established. Please either add an assumption/lemma that unselected entries are small (e.g. derived from (3.2) via a resolvent bound) or restate Thm 3.13 accordingly. Relatedly, note that the τ² rate itself does not require Assumption 3.9: by Cauchy interlacing and (5.15), λ^{k-opt}−λ = ⟨H_21c_1^{k-opt}, (H_22−λI)^{-1}H_21c_1^{k-opt}⟩/(c^{k-opt⊤}c) ≤ 2kn_z²τ²/((−λ)gap(H)). Stating this wouldcl","section":"§5.3, Eq. (5.23) / Assumption 3.9"}],"minor_comments":[{"comment":"Lemma 3.12 states m = √(−λ^{k-opt}‖c‖₁²/(4kλ²)+1), but the proof (5.16) derives m = √(−4λ^{k-opt}‖c‖₁²/(kλ²)+1) — a factor-16 discrepancy inside the root that propagates into the constant of Thm 3.13. Please reconcile.","section":"Lemma 3.12 vs Eq. (5.16)"},{"comment":"d in (5.4) is written with √(−λ) in the numerator, but the radius of B± in §3.1 has it in the denominator; only the latter is consistent with the subsequent bounds d < ½√(−λ) and 2d < ½√(−λ).","section":"Eq. (5.4) vs definition of B±"},{"comment":"'As a direct consequence of Theorem 3.9' and 'Under Theorem 3.8 and Theorem 3.9' should refer to Assumptions 3.8/3.9. Also in Lemma 3.11's statement the premise should read |(Hc^{k-opt})_i| < n_z τ, matching (3.2), not < τ.","section":"§3.2, statements of Lemmas 3.10–3.12"},{"comment":"As proved, V is finite merely because V ⊆ {1,…,n}; this gives no effective control. Since k = |V| enters the final bound as √k in (3.5) and |V| grows as τ decreases, please state explicitly that the theory is conditional on the realized support, and that the limit point in Thm 3.7 is defined by that support (i.e., the algorithm converges to the optimum on its own limit support) — this should also temper the abstract's phrasing.","section":"Lemma 3.4"},{"comment":"The system (5.13) is singular; solvability requires RHS ⊥ ker(λI−H) = span(c) (which holds since cᵀ(H−λI)=0), and the solution-set argument requires λ simple (implied by gap(H)>0). Please add these checks; they are one line each.","section":"§5.2, Eq. (5.13)-(5.14)"},{"comment":"By Cauchy interlacing λ ≤ λ^{k-opt}, so computed energies are variational upper bounds. This sign information justifies E(τ)−E_0 > 0 in the log-ansatz (6.2) and the bisection direction; worth stating explicitly.","section":"§3.2 and §6.3"},{"comment":"§2.1 asserts exponential decay of the FCI ground eigenvector as fact, supported only by one H2O/STO-3G log-linear fit (R²=0.95, Fig. 1, showing ~5×10⁴ of 2×10⁶ entries). Please temper the wording and/or cite broader evidence, since Assumption 3.9 is the theory's key empirical input.","section":"§2.1 / Fig. 1"},{"comment":"State the reference ('exact') energies used for the error columns, the τ samples, and the synthetic matrices' parameters (n, n_z, gap, decay constants — Fig. 3 suggests n=1000 but the text does not say). Please also add a code/data availability statement for the numerics.","section":"Table 2 and §6"},{"comment":"Typos/formatting: 'T echnical proof', 'V erification' headings; 'local maximizer' → 'maximizers' (§2.2); 'each is supported' (§3.1); 'adjustsb' (§2.2); missing period after the reference list in §1; Ref. [7] lacks venue/year; Ref. [4] title in all caps.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a natural follow-up to [12] and [18] (overlapping authorship); the incremental contribution — the compression analysis — is real but builds heavily on the earlier framework, which the authors acknowledge appropriately. I see no circularity: the main theorems are genuine implications from stated assumptions, and the τ² extrapolation is an application, not a definition, of the proved scaling. The two major issues (residual accounting in (4.3); the ordering conflation in (5.23)) are, in my judgment, repairable within the paper's scope — I sketched fixes — but they are gaps in the proof chain of the headline results, not presentation issues, hence major rather than minor revision. Fit with the journal is good."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new piece is the compressed dynamics. Prior CDFCI theory covered the uncompressed iteration; this paper shows that with thresholding on b the iterates still converge linearly to ±c_{k-opt} on the cumulative support, and that under a spectral gap plus exponential decay of the true ground vector the eigenvalue error is O(τ²). That is exactly the gap practitioners needed closed.\n\nThe architecture is standard and mostly clean: exact line search plus local strong convexity on D±, support eventually fixed, then residual control from the compression rule into a restricted eigenproblem, then a residual-to-eigenvalue argument. The spine of Theorems 3.7 and 3.13 holds. Numerics on synthetic matrices and H2O/C2 match the linear rate and the quadratic scaling, and the extrapolation ansatz is a direct, useful application of the proved scaling rather than circular fitting.\n\nSoft spots, in proportion. Assumption 3.9 (sorted exponential tails) is the load-bearing premise for a dimension-independent prefactor. Without it you still get quadratic scaling from standard ℓ₂ perturbation (their own counting already gives ‖H₂₁ c₁‖₂ = O(√k n_z τ)), but the constant picks up factors that are fatal at FCI dimensions. They only show one H2O-STO-3G plot at equilibrium—the easy regime. Multireference cases are untested. There are also minor constant mismatches between lemma statements and proofs, and the theoretical bounds on real FCI are loose by 10³–10⁴, which is expected but worth tightening in revision. No code or data release.\n\nThis is for people who run or analyze selected/compressed FCI and large sparse eigenproblems. It will not move pure spectral theory, but it organizes and justifies practice. I would send it to peer review; the contribution is real and the math is honest enough to deserve referee time. Clean the constants, state more carefully what exponential decay buys (prefactor vs rate), and preferably ship artifacts.","headline":"Solid local theory for an already-used compressed CDFCI solver: linear rate to a restricted eigenproblem and a usable O(τ²) energy error, with the usual caveats on assumptions and constants.","tokens_in":17962,"tokens_out":515,"would_cite":false,"duration_ms":19502,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65F15","65K10","81Q05"],"pacs":[],"model":"grok-4.5","headline":"Compressed coordinate descent for full configuration interaction still converges linearly, and the ground-state energy error is only quadratic in the compression threshold.","keywords":["full configuration interaction","coordinate descent","compression","eigenvalue error","linear convergence","exponential decay","sparse Hamiltonian"],"falsifier":"On a molecular Hamiltonian whose sorted ground eigenvector is known not to decay exponentially, measure |λ − λ_{k-opt}| versus τ; if the observed scaling is slower than quadratic (or the linear rate fails near a Hartree–Fock start), the central error claim is false.","tokens_in":17538,"feed_emoji":"⚛️","tokens_out":908,"duration_ms":22387,"temperature":0.7,"pith_summary":"Full configuration interaction finds the exact ground-state energy of a molecule inside a finite basis, but the Hamiltonian is so large that one cannot store full vectors. Coordinate Descent FCI updates one coordinate at a time and compresses tiny entries to keep the iterate sparse. This paper proves that, near a good starting guess, the compressed iteration still converges linearly—not to the true ground state, but to the ground state of the principal submatrix on the coordinates that ever become active. Under the extra assumption that the true eigenvector’s entries decay exponentially when sorted by size, the energy error between that restricted eigenvalue and the true ground-state energy is only on the order of the square of the compression threshold. The result justifies the practical compression rule used in large FCI calculations and supplies a concrete error model that can be used for extrapolation.","feed_headline":"Compression error in CDFCI is only O(τ²)","feed_subtitle":"Compressed coordinate descent still converges linearly and the energy error scales with the square of the threshold.","key_machinery":"The unrestricted nonconvex objective f(c) = ∥H + ccᵀ∥_F², whose local minimizers are exactly the scaled ground eigenvectors, together with the compression rule that only discards updates to coordinates that are still zero in c; the argument reduces the compressed dynamics to ordinary coordinate descent on the growing but finite support and then controls the residual of the unselected block by n_z τ.","core_discovery":"Under local strong-convexity assumptions the compressed CDFCI iteration converges linearly to ±c_{k-opt}, the suitably normalized ground eigenvector of the principal submatrix on the cumulative support; when the true ground eigenvector also has a spectral gap and exponential tail decay, the eigenvalue error satisfies |λ − λ_{k-opt}| = O(τ²).","pith_inferences":["The same O(τ²) residual-to-energy conversion should apply to other selected-CI or compressed subspace methods whenever the target vector has an exponential tail and the discarded residual is controlled in ℓ_∞.","If a cheaper certificate of exponential decay (or a weaker polynomial tail with a correspondingly weaker power of τ) could be proved for general FCI Hamiltonians, the error bound would become unconditional for chemistry applications.","The extrapolation procedure already demonstrated on H₂O and C₂ suggests a practical workflow: run several moderately compressed CDFCI solves and fit rather than drive τ all the way to machine precision."],"forward_implications":["Compression can be used safely inside CDFCI without destroying the linear rate once the iterate is inside the local strong-convexity neighborhood.","The energy error is theoretically O(τ²), so energies computed at several thresholds can be extrapolated to τ = 0 by a quadratic fit.","The limiting point is exactly the ground eigenpair of the principal submatrix on the coordinates that ever become nonzero, giving an explicit characterization of what the algorithm actually solves.","Only coordinates that remain zero in c are ever discarded, so the Rayleigh quotient computed from the auxiliary vector b stays exact throughout."],"fun_headline_variants":["Compressed CDFCI converges linearly with O(τ²) eigenvalue error","CDFCI compression keeps linear rates; energy error is O(τ²)","Compressed coordinate descent FCI error scales as τ squared","Linear convergence of compressed CDFCI to restricted ground state","Eigenvalue error after CDFCI compression is order τ²"],"cache_read_input_tokens":128,"weakest_assumption_plain":"After the true ground-state coefficients are sorted by size, their magnitudes must fall off exponentially; without that tail the paper only obtains a linear residual bound and loses the quadratic energy error.","fun_headline_variants_meta":{"raw":{"variants":["Compressed CDFCI converges linearly with O(τ²) eigenvalue error","CDFCI compression keeps linear rates; energy error is O(τ²)","Compressed coordinate descent FCI error scales as τ squared","Linear convergence of compressed CDFCI to restricted ground state","Eigenvalue error after CDFCI compression is order τ²"]},"model":"grok-4.5","effort":"low","cost_usd":0.00389,"raw_usage":{"total_tokens":1100,"prompt_tokens":623,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":38904000,"prompt_tokens_details":{"text_tokens":623,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":409,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":623,"tokens_out":68,"duration_ms":7948,"temperature":1.0,"reasoning_tokens":409,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T18:14:08.688636+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"On a molecular Hamiltonian whose sorted ground eigenvector is known not to decay exponentially, measure |λ − λ_{k-opt}| versus τ; if the observed scaling is slower than quadratic (or the linear rate fails near a Hartree–Fock start), the central error claim is false.","supporting_citations":[],"review_version":1}