{"id":"c4514247-11ae-4098-9f2f-1e545f1210d3","arxiv_id":"2505.15575","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Finite free convolutions converge weakly to free convolutions without compact support, and in Kolmogorov distance when the input approximations converge in that metric.","lead":"The paper proves distance-contraction and atom-structure results for finite free convolutions of polynomials, and uses them to show that finite free additive and multiplicative convolutions converge to their infinite free counterparts without compact-support assumptions. A generalist might read it because it removes a standing technical restriction in finite free probability and upgrades the convergence to the stronger Kolmogorov metric.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.1 has a reversed order: the construction proves q^(l) ≤ p, not p ≤ q^(l); the corrected direction still yields the lemma's convolution bound, so Theorem 1.3 is repairable.","rationale":"The reader's weakest-assumption identification is correct: Lemma 4.1 as written has the order direction reversed. The condition (4.1) indeed gives λ_i(p) ≤ λ_{l+i}(q), and the described construction replaces the l smallest roots of q with a large value a, producing roots λ_{l+1}(q),...,λ_d(q),a,...,a. The componentwise comparison against λ_i(p) then yields λ_i(p) ≤ λ_i(q^(l)) for every i, which means q^(l) ≤ p in the CDF order, not p ≤ q^(l). The pointwise polynomial inequality in the proof is not the right criterion and is false in general. So the lemma statement is incorrect as written. However, the stress-test shows this is a repairable sign/order error rather than a fatal flaw: the opposite order q^(l) ≤ p is enough to derive the lemma's stated consequence. Using Proposition 2.19, q^(l) ≤ p implies F_{q^(l)⊞r} ≤ F_{p⊞r}. Using Proposition 2.18 and the fact that interlacing polynomials have Kolmogorov distance at most 1/d, each step q^(k-1)⋖q^(k) gives F_{q^(k-1)⊞r} ≤ F_{q^(k)⊞r}+1/d; chaining over k=1,...,l yields F_{q⊞r} ≤ F_{p⊞r}+l/d. Thus Theorem 1.2, and consequently Theorem 1.3, remain valid after correcting the lemma's conclusion. The manuscript's proof is not sound as written, but the mathematics can be fixed without changing the central claims. The reader's CONDITIONAL verdict is therefore appropriate: the paper requires a correction, but the main theorems are not disproved by this concern. Other issues mentioned by the reader (Appendix A's Proposition 3.8 proof and the sketch of Theorem 4.5) are secondary and do not affect the central additive-convolution convergence argument.","tokens_in":20553,"tokens_out":11975,"duration_ms":89841,"concrete_test":"Formally correct Lemma 4.1 by replacing the conclusion p ≤ q^(l) with q^(l) ≤ p, and verify the proof that F_{q⊞r} ≤ F_{p⊞r}+l/d follows from the chain q^(k-1)⋖q^(k) and monotonicity. Then re-run the proof of Theorem 1.2: if both one-sided bounds follow, the Kolmogorov and Lévy distance contractions hold, and Theorem 1.3's argument is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The manuscript's Lemma 4.1 states that F_q ≤ F_p + l/d implies existence of a chain q=q^(0)⋖q^(1)⋖...⋖q^(l) with p ≤ q^(l). The derivation from (4.1) gives λ_i(p) ≤ λ_{l+i}(q) for i=1,...,d-l. The construction moves the l smallest roots of q to a common large value a, so q^(l) has roots λ_{l+1}(q),...,λ_d(q) plus l copies of a. Consequently λ_i(q^(l)) = λ_{l+i}(q) ≥ λ_i(p), which is precisely the componentwise criterion for q^(l) ≤ p, i.e., F_{q^(l)} ≤ F_p—the opposite of the stated p ≤ q^(l). The proof's displayed pointwise polynomial inequality 'p(x) ≤ q^(l)(x)' is not equivalent to the CDF order and is generally false. However, the corrected direction is sufficient for the lemma's consequence: by Proposition 2.19 (monotonicity), q^(l) ≤ p gives F_{q^(l)⊞r} ≤ F_{p⊞r}; and by Proposition 2.18 plus d_K ≤ 1/d for interlacing polynomials, each step q^(k-1)⋖q^(k) gives F_{q^(k-1)⊞r} ≤ F_{q^(k)⊞r}+1/d, so chaining yields F_{q⊞r} ≤ F_{p⊞r}+l/d. Thus the central convergence theorem is not invalidated; the lemma statement and its proof need a sign/order correction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops regularity properties of the finite free additive and multiplicative convolutions ⊞_d and ⊠_d. The main results are: (1) a description of atoms of p⊞_d q and p⊠_d q, including a formula relating CDF values at atom triplets; (2) monotonicity of Kolmogorov and Lévy distances under finite free convolution; and (3) convergence of empirical root distributions of p_d ⊞_d q_d to the free additive convolution μ⊞ν under only weak convergence of the inputs, together with a strengthening to convergence in Kolmogorov distance when the input empirical measures converge in Kolmogorov distance. Multiplicative analogues are also stated. The proofs use root interlacing, truncation of measures, and prior results from free probability for compactly supported limits.","tokens_in":20938,"tokens_out":38949,"duration_ms":320581,"significance":"If the results are correct, they significantly extend the known approximation of free convolutions by finite free convolutions: previous moment-based proofs required compact support, whereas Theorem 1.3 removes that condition for the additive convolution and provides a quantitative Kolmogorov-distance transfer. The paper's elementary root-ordering approach is a genuine strength, as is the explicit use of truncation to reduce the unbounded-support case to the compact case. The atom identities in Propositions 2.10 and 2.12 and the finite-free analogues are also valuable. However, several proofs, especially Lemma 4.1 and the Lévy-distance part of Theorem 1.2, contain gaps or incorrect intermediate statements that must be repaired before the central claims can be regarded as fully established.","major_comments":[{"comment":"The claimed reversal of the order in Lemma 4.1 is not correct under the paper's convention. By Definition 2.3, p≤q means λ_i(p)≤λ_i(q) for every i (equivalently F_p≥F_q). The construction gives q^(l) with ordered roots λ_{l+1}(q),...,λ_d(q),a,...,a, so (4.2) and a>λ_d(p) imply λ_i(p)≤λ_i(q^(l)) for all i, which is exactly p≤q^(l). What is genuinely wrong is the displayed pointwise inequality p(x)≤q^(l)(x): this is generally false and is not equivalent to the CDF order. For example, with roots 0,1,2 and 1,3,4, the inequality fails at x=2.5. The proof should be repaired by replacing the pointwise display with the root-order comparison. Because Lemma 4.1 is used in Theorems 1.2 and 1.3, this is a load-bearing gap, though I see no obstacle to a local fix.","section":"§4.1, Lemma 4.1"},{"comment":"The Lévy-distance part is not established by the sentence 'it is enough to note...'. The displayed implication from an ε-satisfying the Lévy definition to the two one-sided bounds with l/d requires the discrete fact that F_p and F_q take values in multiples of 1/d; this quantization step is not given. Moreover, even after obtaining those bounds, one must explain how they combine with Lemma 4.1 and the shift covariance of ⊞_d to yield d_L(p⊞_d r,q⊞_d r)≤l/d. The current text skips this essential chain, so the proof of part 2 of Theorem 1.2 is incomplete as written.","section":"§4.1, proof of Theorem 1.2(2)"},{"comment":"The identity (p+ε_0)⊞_d q = p⊞_d q+ε_0 is invoked and attributed to Proposition 2.16, but that proposition only states preservation of real-rootedness; it does not state this identity. The identity is true and follows directly from the coefficient definition (1.1), but it must be proved or given a correct reference. In addition, the assertion that p+ε remains in P_d(R) for all sufficiently small ε when p has only simple roots is standard but should be justified via continuous dependence of simple roots on coefficients. These are local gaps in the proof of Theorem 1.1(1).","section":"Appendix A, proof of Proposition 3.8"}],"minor_comments":[{"comment":"The notation for cut-up and cut-down measures is visually ambiguous in the text: both are typeset as μ| a in several places, although Definition 2.4 distinguishes μ|^a and μ|_a. Please ensure superscripts/subscripts are rendered consistently.","section":"§2.4 / §4.1"},{"comment":"In the construction of p_d for K=[a,b], the polynomial uses λ_{d-1} twice rather than defining λ_d; the approximation estimate d_K≤1/d is correct, but the repeated root should be explicitly explained to avoid confusion.","section":"§4.2, Lemma 4.3"},{"comment":"The statement that the Kolmogorov-distance result 'might also hold' is imprecise; if it is not proved, it would be better phrased as an open problem or a conjecture rather than a tentative claim.","section":"§4.2, Theorem 4.5"},{"comment":"The final 'worst case' paragraph of the proof is very terse and uses several unstated identities, such as p_α⊞_d q = p_α⊞_{d-1} ∂^{(d-1)|d}q; these identities should be stated and proved or cited, since the contradiction argument depends on them.","section":"§3.2, Proposition 3.8"}],"recommendation":"major_revision","confidential_remarks":"The reader's strongest concern about Lemma 4.1 is based on a sign error: under the paper's order convention, the construction actually proves p≤q^(l), not q^(l)≤p. The real problem is the false pointwise inequality in the proof, which is repairable. I would not reject on that basis. The main convergence theorem appears sound after the local repairs to Lemma 4.1, but the Lévy-distance proof in Theorem 1.2 needs a nontrivial missing argument, and the proof of Proposition 3.8 needs corrected attributions. These are all within the scope of a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's main claim is probably true and the strategy is sound, but the proof as written has a load-bearing sign/order error in Lemma 4.1. The stress-test note is correct: the construction gives q^(l) ≤ p, not p ≤ q^(l), and the displayed polynomial inequality is not equivalent to the CDF order. The good news is that the corrected direction is enough: chaining the interlacing steps and using monotonicity still yields the convolution bound, so Theorem 1.2 and Theorem 1.3 should survive a rewrite.\n\nWhat is genuinely new: removing compact-support hypotheses for weak convergence of ⊞_d to ⊞, and strengthening to Kolmogorov distance, is a real advance. Prior results in the cited literature required either compact support or the positive multiplicative setting. The distance-contraction theorem is also a nice finite-free analogue of Bercovici–Voiculescu's monotonicity. The atom results in Section 3 are clearly presented and the alternative proof of the multiplicative atom statement is a plus.\n\nWhere the soft spots are, in order:\n1. Lemma 4.1 as stated is wrong, though the lemma's consequence is recoverable. This is major but repairable, and the paper's own proof contains the ingredients for the fix.\n2. Appendix A, proof of Proposition 3.8, invokes an identity for p_α ⊞_d q that is neither proved nor cited, and Lemma A.1 assumes distinct roots while the proof later applies it to polynomials with multiple roots. That argument needs either a real proof or a reference.\n3. Theorem 4.5 is stated without proof, and the final remarks admit the multiplicative case has gaps. It should be labeled as a conjecture or removed until a proof is supplied.\n\nCitation pattern is fine: the paper builds on the expected literature and cites the author's own prior work as background, not as a hidden dependency. The Kolmogorov strengthening is a usable rate-transfer principle, and the paper is written in a way that makes the fix fairly mechanical.\n\nWho this is for: people working in finite free probability and asymptotic random matrix theory. The paper deserves serious refereeing — the main theorem is important enough that the referee should engage with the details rather than desk-reject. I would send it to a competent referee, but I would ask the author to fix Lemma 4.1 and the Appendix before publication.","headline":"Solid new convergence result with a repairable sign error in the key lemma; worth refereeing after fixes.","tokens_in":21411,"tokens_out":4035,"would_cite":false,"duration_ms":36949,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L54","60B20","26C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Finite free convolutions of real-rooted polynomials converge to free convolutions as the degree tends to infinity, with no compact-support assumption and with Kolmogorov-distance convergence when the inputs converge in that metric.","keywords":["finite free convolution","free probability","real-rooted polynomials","Kolmogorov distance","weak convergence","atoms of probability measures","interlacing","Schur–Szegő composition"],"falsifier":"Apply Lemma 4.1 to $p(x)=x(x-2)$ and $q(x)=(x-1)^2$ with $d=2$ and $\\ell=1$; the condition $F_q\\le F_p+1/2$ holds, but replacing the left root of $q$ by a large number gives $q^{(1)}$ with roots $1$ and $3$, and then $F_p(2)=1$ exceeds $F_{q^{(1)}}(2)=1/2$, so $p\\le q^{(1)}$ fails. This example would settle whether the lemma's order direction is correct as stated.","tokens_in":20323,"feed_emoji":"🧮","tokens_out":9610,"duration_ms":73339,"temperature":0.7,"pith_summary":"This paper argues that the finite free additive convolution of real-rooted degree-$d$ polynomials—a symmetrized polynomial operation on roots—approximates the free additive convolution of probability measures as $d \\to \\infty$, and that this approximation holds without the usual compact-support assumption on the limiting measures. It further claims that when the empirical root distributions converge in Kolmogorov distance, the convolved distributions converge in Kolmogorov distance as well, and it proves the analogous statements for the finite free multiplicative convolution with measures on $[0,\\infty)$. Along the way it establishes finite-degree analogues of two known free-probability regularities: a contraction inequality for Kolmogorov and L\\'evy distances under the convolution, and a complete description of the atoms (multiplicity-$m$ roots) of the convolved polynomial in terms of the input atoms. The proof is elementary, comparing root configurations through order and cut-up/cut-down approximations rather than through analytic transforms.","feed_headline":"Finite free convolutions converge without compact support","feed_subtitle":"A new proof removes the bounded-support assumption and upgrades weak convergence to Kolmogorov distance.","key_machinery":"The machinery combines three objects: the empirical root distribution $\\mu_{[p]}$ of a polynomial, the partial order $\\mu\\le\\nu$ defined by $F_\\nu\\le F_\\mu$, and the cut-up and cut-down measures $\\mu|_a$ and $\\mu|^a$ that truncate a measure at a point $a$. The order and truncations let the paper compare root configurations before and after convolution, and interlacing of polynomials supplies the monotonicity that turns order comparisons into distance comparisons. The load-bearing mechanism is Lemma 4.1, which converts a one-sided closeness bound $F_q \\le F_p + \\ell/d$ into a chain of interlacing polynomials and then into the same closeness inequality after convolution; Theorems 1.2 and 1.3 are derived from this contraction step.","core_discovery":"The central claim is that the empirical root distribution of $p_d \\boxplus_d q_d$ converges weakly to $\\mu \\boxplus \\nu$ whenever the root distributions of $p_d$ and $q_d$ converge weakly to $\\mu$ and $\\nu$, with no compact-support condition on $\\mu$ or $\\nu$; if the input distributions converge in Kolmogorov distance, so does the output. The same conclusion is proved for $\\boxtimes_d$ when both measures are supported on $[0,\\infty)$, and for a one-sided case in which one measure is compactly supported on $[0,\\infty)$. The paper also establishes that every non-trivial root of $p_d \\boxplus_d q_d$ is simple and that atom masses obey the free-convolution formula $\\mu_{[p\\boxplus_d q]}(\\{\\alpha+\\beta\\}) = \\mu_{[p]}(\\{\\alpha\\}) + \\mu_{[q]}(\\{\\beta\\}) - 1$ whenever $\\alpha$ and $\\beta$ are atoms of the inputs with multiplicities summing past the degree.","pith_inferences":["Editorial inference: The distance-contraction inequalities for $\\boxplus_d$ and $\\boxtimes_d$ are finite-dimensional versions of known contraction properties of free convolutions; if the underlying lemma survives, the same cut-and-paste route may yield contraction for other polynomial operations that preserve real-rootedness.","Editorial inference: The paper's approach suggests a recipe for proving 'no compact support needed' statements in other asymptotic settings: first prove a distance-contraction inequality, then approximate arbitrary measures by compactly supported truncations and compare.","Editorial inference: One could try to derive an explicit bound $\\mathrm{d}_K(p_d\\boxplus_d q_d,\\mu\\boxplus\\nu) \\le \\mathrm{d}_K(\\mu_{[p_d]},\\mu) + \\mathrm{d}_K(\\mu_{[q_d]},\\nu) + o(1)$ from Theorem 1.2; the paper proves convergence but does not state such a rate."],"forward_implications":["Applying $\\boxplus_d$ to degree-$d$ polynomials whose root distributions converge to measures with heavy tails still gives the correct free-convolution limit.","Kolmogorov-distance convergence of the input polynomials passes to the convolved polynomials, giving a quantitative upgrade of weak convergence.","The atom formula for $p_d \\boxplus_d q_d$ matches the free atom formula, so atomic structure is preserved across the finite-to-infinite passage.","The multiplicative analogue now covers $[0,\\infty)$-supported measures without compactness, and a one-sided generalization with one compact positive measure is included."],"supporting_citations":[{"why":"Defines the finite free additive and multiplicative convolutions and gives their random-matrix expectations, the baseline for the whole paper.","marker":"[19]"},{"why":"Proves the Schur–Szegő composition atom results that the paper adapts to finite free additive and multiplicative convolutions.","marker":"[16]"},{"why":"Supplies the free-convolution monotonicity and distance inequalities that the finite-degree theorems are modelled on and invoked in the convergence proof.","marker":"[9]"},{"why":"Proves weak convergence of $\\boxplus_d$ to $\\boxplus$ under compact support, the result Theorem 1.3 extends.","marker":"[5]"},{"why":"Proves weak convergence of $\\boxtimes_d$ to $\\boxtimes$ under compact support, extended by Theorem 4.4.","marker":"[3]"},{"why":"Gives the finite S-transform and the no-compact-support convergence for multiplicative convolution on $[0,\\infty)$, which the paper generalizes in a one-sided form.","marker":"[2]"}],"fun_headline_variants":["Weak convergence of finite free convolutions without compactness","Finite free convolutions converge in Kolmogorov distance","Atom masses of finite free convolutions follow free formula","No compact support needed for finite free convolution convergence","Triangle inequalities and convergence for finite free convolutions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on Lemma 4.1 of Section 4.1, which says that if the roots of $q$ are close to the roots of $p$ from one side, then moving the leftmost roots of $q$ to the far right produces a polynomial $q^{(\\ell)}$ with $p \\le q^{(\\ell)}$. In the written proof the moving produces the opposite ordering, so the claimed direction of this lemma is the premise on which the convergence theorem depends.","fun_headline_variants_meta":{"raw":{"variants":["Weak convergence of finite free convolutions without compactness","Finite free convolutions converge in Kolmogorov distance","Atom masses of finite free convolutions follow free formula","No compact support needed for finite free convolution convergence","Triangle inequalities and convergence for finite free convolutions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000964,"raw_usage":{"total_tokens":4077,"prompt_tokens":889,"completion_tokens":3188,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":505,"completion_tokens_details":{"reasoning_tokens":3127}},"tokens_in":505,"tokens_out":3188,"duration_ms":17771,"temperature":1.0,"reasoning_tokens":3127,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:16:14.627582+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Apply Lemma 4.1 to $p(x)=x(x-2)$ and $q(x)=(x-1)^2$ with $d=2$ and $\\ell=1$; the condition $F_q\\le F_p+1/2$ holds, but replacing the left root of $q$ by a large number gives $q^{(1)}$ with roots $1$ and $3$, and then $F_p(2)=1$ exceeds $F_{q^{(1)}}(2)=1/2$, so $p\\le q^{(1)}$ fails. This example would settle whether the lemma's order direction is correct as stated.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the finite free additive and multiplicative convolutions and gives their random-matrix expectations, the baseline for the whole paper."},{"cited_title":"Kostov and B","cited_arxiv_id":null,"evidence_quote":"Proves the Schur–Szegő composition atom results that the paper adapts to finite free additive and multiplicative convolutions."},{"cited_title":"Bercovici and D","cited_arxiv_id":null,"evidence_quote":"Supplies the free-convolution monotonicity and distance inequalities that the finite-degree theorems are modelled on and invoked in the convergence proof."},{"cited_title":"Arizmendi and D","cited_arxiv_id":null,"evidence_quote":"Proves weak convergence of $\\boxplus_d$ to $\\boxplus$ under compact support, the result Theorem 1.3 extends."},{"cited_title":"Arizmendi, J","cited_arxiv_id":null,"evidence_quote":"Proves weak convergence of $\\boxtimes_d$ to $\\boxtimes$ under compact support, extended by Theorem 4.4."}],"review_version":1}