{"id":"9c742c1b-ab7c-4a02-bb69-26a1dc76dd8a","arxiv_id":"1908.06753","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"There exists an entire free holomorphic function in more than one variable that is unbounded on the row ball.","lead":"A short math paper constructs a noncommutative holomorphic function that is defined everywhere yet grows without bound on the unit ball of matrices. This answers a question from Agler, McCarthy, and Shamovich about the limits of the 'free topology' in several variables.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unboundedness rests on an unproved assertion that the q_k sum has infinite H² norm; an orthogonality/multiplicity argument is missing, as is the claimed H² multiplicativity.","rationale":"The reader's CONDITIONAL verdict is appropriate. The central construction is a known style: use matrix identities p_n vanishing on finite dimensions, multiply with controlled degree. The main missing justification in the paper is the transition from individual q_k to infinite H² norm; the proof as printed only says the degrees grow linearly. I checked the underlying mechanism: the monotonicity of the exponent vectors makes same-degree q_k identical, and the degree grows at least linearly from the n=1 term, so f really has infinitely many orthogonal homogeneous components of norm equal to their multiplicity. Thus no fatal flaw is apparent, but the omitted argument is load-bearing because if the q_k did cancel, the 'unbounded' conclusion would fail. Similarly, H² multiplicativity for homogeneous products is true, but the proof does not say why. The existence of p_n is a genuine PI-algebra fact rather than a trivial exercise, though standard. All of these gaps are fillable, so the verdict should remain CONDITIONAL rather than move to ACCEPT or REJECT.","tokens_in":2097,"tokens_out":25507,"duration_ms":275461,"concrete_test":"Encode p_n explicitly for d=2, n=1,2 (e.g. [x,y] and a normalized two-variable identity for M_2, such as [[x,y]^2,x] normalized). Compute q_k for k up to 100 from the formula. Verify (i) ||q_k||_{H²}=1 by exact monomial enumeration; (ii) the degree sequence D(k)=Σ_n deg(p_n)⌊k/(2^n deg p_n)⌋ is nondecreasing, tends to infinity, and any two indices with the same D have identical exponent vectors; (iii) the H² norms of partial sums S_N tend to infinity. If any check fails, the unboundedness argument in Proposition 1 collapses; if all pass, the missing steps are confirmed routine.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive step in the proof of Proposition 1 is the line '∞ = ||f||_{H²} ≤ ||f||_{C_d}'. From the preceding text this does not yet follow. Knowing only that each q_k has H²-norm 1 does not make the infinite sum have infinite H²-norm: the q_k could cancel in the Hilbert space. The proof must show that the homogeneous components of f are nonzero in infinitely many distinct degrees and that repetitions do not cancel. This is true, but only because the exponent vectors e_n(k)=⌊k/(2^n deg p_n)⌋ are monotone in k: if two q_k have the same degree, the weighted degree equality forces e_n(k)=e_n(l) for all n, so q_k=q_l and repetitions only increase the norm; and the n=1 term makes deg q_k ≥ deg p_1⌊k/(2 deg p_1)⌋, so infinitely many distinct degrees occur. This argument is absent. The companion assertion ||q_k||_{H²}=1 is also unproved; it is valid because p_n are homogeneous and H² norms multiply for homogeneous products, but that fact is not stated. Finally, the existence of p_n vanishing on M_d^n is deferred to a 'delightful combinatorial exercise'; for d=2 this is a genuine nonzero-PI-theorem (e.g. normalized two-variable polynomial identities for M_n), not a one-line check.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to construct, for d > 1, an entire free holomorphic function f on the matrix universe M_d that is locally bounded in the free topology but unbounded on the row ball C_d. The construction chooses, for each n, a nonzero homogeneous free polynomial p_n of H^2 norm 1 that vanishes on M_d^n, forms products q_k of powers of the p_n, and defines f as the infinite sum of the q_k. The author argues that f is unbounded on C_d because its H^2 norm is infinite, and that f is locally bounded by estimating the q_k on specially chosen basic sets.","tokens_in":2384,"tokens_out":10125,"duration_ms":97111,"significance":"If the construction is made fully rigorous, the result is significant: it answers a question attributed to Shamovich and shows that local boundedness in the free topology does not imply boundedness on the row ball, even for entire free functions. It also sharpens the picture around the Agler-McCarthy Oka-Weil theorem and the paucity of compact sets in the free topology. The paper is refreshingly brief and the underlying idea is attractive, but the proof as written is a sketch with several load-bearing gaps; the good news is that these gaps appear fillable within the scope of a revision.","major_comments":[{"comment":"The assertion that for d > 1 there exists a nonzero homogeneous polynomial p_n with H^2 norm 1 vanishing on M_d^n is deferred to a 'delightful combinatorial exercise.' This is not a negligible detail: for d = 2 it is essentially a polynomial identity theorem for M_n (for instance, a two-variable PI or a specialization of the Amitsur-Levitzki theorem). Since the entire construction depends on the existence of these p_n, a proof or an exact reference must be supplied.","section":"Proposition 1, proof, existence of p_n"},{"comment":"The equality ||q_k||_{H^2}=1 is stated without justification. It is true for this construction, but only because the p_n are homogeneous: for homogeneous free polynomials, the H^2 norm of a product equals the product of the H^2 norms, since each word of the appropriate length has a unique split at the degree boundary. This fact must be stated explicitly, because for general norm-one polynomials the H^2 norm of a product is not automatically one.","section":"Proposition 1, proof, claim ||q_k||_{H^2}=1"},{"comment":"This line is the decisive step for unboundedness, and as written it is unjustified. Since f is an infinite sum, one must first prove that the infinite H^2 norm is actually infinite; in particular, one must rule out cancellation among the q_k in the same degree. This can be done by noting that the exponent vectors e_n(k)=floor(k/(2^n deg p_n)) are monotone in k, so equal weighted degrees force e_n(k)=e_n(l) for all n, and the n=1 term forces infinitely many distinct degrees. The proof must also justify applying the classical H^2-to-C_d norm inequality to a formal power series with possibly infinite H^2 norm: if f were bounded on C_d, then its H^2 norm would be finite. Neither of these points appears in the manuscript.","section":"Proposition 1, proof, line '∞ = ||f||_{H^2} ≤ ||f||_{C_d}'"},{"comment":"The sentence 'The function f well-defined for all inputs as the terms in the series are eventually zero' is false as stated. For X of size N, the factors p_n(X) with n < N generally do not vanish, and they appear in q_k(X) with exponents growing in k, so the summands q_k(X) are not eventually zero. The later local-boundedness estimate, once corrected, would give uniform convergence and hence well-definedness, but the manuscript does not present the argument in that form.","section":"Proposition 1, proof, well-definedness of f"},{"comment":"The displayed estimate for ||q_k||_{Bδ} contains sign and direction errors. With Bδ defined using B_{(2r)^{2m deg p_m}} p_m (positive exponent), the bound ||p_m||_{Bδ} ≤ (2r)^{2m deg p_m} does not yield the reciprocal (1/(2r))^{...} used in the next line; the neighborhood would need a negative exponent to make the estimate produce geometric decay. In addition, the final inequality r^k( ... ) ≤ r^{2m deg p_m}/2^k has the wrong direction for r ≥ 1, since (2r)^N ≥ r^N. These appear to be typographical, but the local boundedness proof is invalid as printed.","section":"Proposition 1, proof, local boundedness estimate"}],"minor_comments":[{"comment":"There are several typographical slips in the displayed text, including 'TH E ROW BALL' in the title and 'functio n' in the first paragraph; these should be cleaned up in revision.","section":"Throughout"},{"comment":"The notation M_d^n and C_d^m is used without definition; in particular, the phrase 'X ∈ rC_d^m' should be explained, since the row ball C_d is a union over matrix sizes.","section":"Notation"},{"comment":"The concluding remarks connecting the example to the Augat-Balasubramanian-McCullough compactness theorem and the Agler-McCarthy Oka-Weil theorem are suggestive but not proved; consider marking them explicitly as observations or adding a precise statement.","section":"Final paragraph"}],"recommendation":"major_revision","confidential_remarks":"The paper is best viewed as a research announcement or proof sketch. The central construction is plausible and likely correct after filling several gaps, but the proof as written has load-bearing omissions and at least one displayed inequality that is wrong as printed. I would encourage the editor to send it back for a careful revision with full proofs of the p_n existence, the H^2 multiplicativity, the infinite H^2 norm / non-cancellation argument, and the corrected local boundedness estimate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a two-page note that answers an open question from Agler, McCarthy, and Shamovich: for d>1 there is an entire free function (locally bounded in the free topology) that is unbounded on the row ball. That is genuinely new and worth knowing. The construction is simple: take homogeneous noncommutative polynomials p_n that vanish on M_d^n, form infinite products q_k with carefully chosen exponents, and sum them. The idea works.\n\nWhat the paper does well: it states the question clearly, gives a short proof that the constructed f is locally bounded in the free topology, and connects the example to the Augat–Balasubramanian–McCullough compact-set results and the Agler–McCarthy Oka–Weil theorem. That context is useful.\n\nThe soft spots are exactly where the reader and stress-test note point. The line '∞ = ||f||_{H2} ≤ ||f||_{C_d}' is not justified in the text. You need to know that the q_k don't cancel in H2 and that infinitely many distinct degrees occur. That's true: the exponents e_n(k) are monotone in k, the n=1 term forces the degree of q_k to grow at least linearly, and any repeated degree just makes the homogeneous component norm larger. But the argument is absent. Similarly, ||q_k||_{H2}=1 is asserted without proof; it's true because each p_n is homogeneous, so the H2 norm of the product is the product of the H2 norms, but that fact should be stated. And the existence of the p_n is not merely a 'delightful combinatorial exercise': for d=2 it is the Amitsur–Levitzki theorem (standard identity), which should be cited or at least acknowledged. None of these gaps sinks the construction; a competent referee can fill them all in an afternoon.\n\nAlso minor: the paper says 'entire free holomorphic function' but defines entire through local boundedness in the free topology. That is the right definition in this setting, but a footnote or sentence clarifying that would help.\n\nBottom line: this is a real counterexample, probably correct, and useful for people working in noncommutative function theory. As written it is a proof sketch, not a completed proof. I would send it to peer review because the result is significant enough to warrant referee time, but I would ask the author to expand the proof of Proposition 1 and cite Amitsur–Levitzki before accepting.","headline":"A short, credible counterexample answering an open question, but the proof as written is a sketch with two or three fillable gaps.","tokens_in":2799,"tokens_out":3888,"would_cite":true,"duration_ms":38242,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47L25","46L52","32A70"],"pacs":[],"model":"deepseek-v4-flash","headline":"A free holomorphic function can be entire yet unbounded on the row ball in more than one variable.","keywords":["free holomorphic function","row ball","free topology","noncommutative function","row contractions","entire function","H^2 norm","unboundedness"],"falsifier":"For $d=2$, write out the promised polynomials $p_n$ explicitly, then compute the $H^2$ norm of $q_1$ and $q_2$ directly; if any $\\|q_k\\|_{H^2}\\neq 1$ or if the series $\\sum_k q_k$ has finite $H^2$ norm, the proof's unboundedness argument fails. Alternatively, evaluate the first few partial sums on a sequence of row contractions that concentrate on growing matrix sizes and check numerically whether the norms diverge.","tokens_in":1866,"feed_emoji":"♾️","tokens_out":12013,"duration_ms":89839,"temperature":0.7,"pith_summary":"The paper answers a question about whether every free holomorphic function on the row ball is bounded on smaller concentric balls. It constructs, for $d>1$, an entire free holomorphic function $f:\\mathbb{M}_d\\to\\mathbb{M}$ that is unbounded on the row ball $\\mathcal{C}_d$ but is locally bounded in the free topology, meaning around every matrix $X$ there is a basic open set on which $f$ is bounded. The construction is a formal power series $f=\\sum_{k=1}^{\\infty}q_k$ built from homogeneous noncommutative polynomials that vanish on all $d$-tuples of $n\\times n$ matrices. The paper shows $\\|f\\|_{H^2}=\\infty$, and since the $H^2$ norm is bounded above by the sup norm on the row ball, $f$ must be unbounded there.","feed_headline":"Entire free function is unbounded on the row ball","feed_subtitle":"In several variables, local boundedness in the free topology does not imply boundedness on row contractions.","key_machinery":"The construction is carried by three ingredients: (1) the free topology, whose basic open sets are $B_\\delta=\\{X:\\|\\delta(X)\\|<1\\}$ for noncommutative polynomial matrices $\\delta$; (2) the $H^2$ norm on noncommutative polynomials, defined as the square root of the sum of squared coefficient norms, which is bounded above by the sup norm on the row ball; and (3) a sequence of homogeneous polynomials $p_n$ with $\\|p_n\\|_{H^2}=1$ that vanish identically on $d$-tuples of $n\\times n$ matrices. The function is $f=\\sum_{k=1}^{\\infty}q_k$, where $q_k=\\prod_n p_n^{\\lfloor k/(2^n\\deg p_n)\\rfloor}$. The degree of $q_k$ is at most $k$, and the proof estimates the sup norm of $q_k$ on a carefully chosen $B_\\delta$ by $r^{2m\\deg p_m}/2^k$, yielding a convergent geometric bound for $f$ on that neighborhood.","core_discovery":"The central discovery is a counterexample: for $d>1$, there is an entire free holomorphic function $f\\colon \\mathbb{M}_d\\to\\mathbb{M}$ such that for every $X\\in\\mathbb{M}_d$ there is a basic open set $B_\\delta$ in the free topology containing $X$ on which $f$ is bounded, yet $f$ is unbounded on the row ball $\\mathcal{C}_d$. The function is given explicitly as an infinite sum of products of homogeneous polynomials $p_n$ that vanish on $M_d^n$; the construction forces $\\|f\\|_{H^2}=\\infty$, and since the $H^2$ norm is bounded above by the sup norm on the row ball, $f$ must be unbounded there. The paper also notes that the same example shows the limits of the free uniform approximation theorem, since $r\\mathcal{C}_d$ is not compact in the free topology and $f$ cannot be uniformly approximated by polynomials on it.","pith_inferences":["The paper's deferred combinatorial step could be made explicit for small $d$; doing so would let one test numerically whether the $H^2$ norm of each $q_k$ is exactly 1, which is the proof's main unverified assertion.","The same product-and-sum technique may generate free holomorphic functions with arbitrarily prescribed unbounded growth on other free-semialgebraic sets, provided one can find polynomials vanishing on the appropriate finite-level sets.","A natural open question is whether the constructed $f$ is also unbounded on the row ball when restricted to matrices of a single fixed size $n$; the paper's argument only shows unboundedness across all sizes.","If a more quantitative proof is found, it might yield concrete sequences of row contractions on which $\\|f(X)\\|$ grows, giving explicit witnesses of the failure of local-to-global boundedness."],"forward_implications":["For $d>1$, the row ball is not a set on which every entire free holomorphic function is bounded, even though it is bounded in norm.","Local boundedness in the free topology is strictly weaker than boundedness on the row ball; the free topology does not control growth along the row ball.","The scaling $r\\mathcal{C}_d$ of the row ball is not compact in the free topology for $r\\ge 1$, as shown by the failure of uniform polynomial approximation for this $f$.","The construction yields an entire free holomorphic function with infinite $H^2$ norm, so the $H^2$-to-sup-norm inequality cannot be reversed."],"supporting_citations":[{"why":"Defines the free topology and the class of free holomorphic functions, and supplies the classical $H^2$-norm inequality $\\|f\\|_{H^2}\\le \\|f\\|_{\\mathcal{C}_d}$ used to infer unboundedness.","marker":"[1]"},{"why":"Establishes the scarcity of compact sets in the free topology; the example's behavior on row-ball scalings contrasts with that compactness picture.","marker":"[2]"},{"why":"Provides a uniform approximation theorem for free holomorphic functions on compact sets in the free topology, whose limitations the example exposes.","marker":"[3]"}],"fun_headline_variants":["Free holomorphic function defies row ball boundedness","Counterexample: entire free function unbounded on row ball","Local boundedness fails to imply global on row ball","Entire free function: locally bounded, unbounded on row ball","Row ball escape: entire free function unbounded"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the unverified assertions that homogeneous polynomials $p_n$ of $H^2$ norm 1 vanishing on $M_d^n$ exist, and that the products $q_k$ built from them each have $H^2$ norm 1; the latter is not generally true for products of unit-norm homogeneous polynomials.","fun_headline_variants_meta":{"raw":{"variants":["Free holomorphic function defies row ball boundedness","Counterexample: entire free function unbounded on row ball","Local boundedness fails to imply global on row ball","Entire free function: locally bounded, unbounded on row ball","Row ball escape: entire free function unbounded"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000763,"raw_usage":{"total_tokens":3300,"prompt_tokens":775,"completion_tokens":2525,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":391,"completion_tokens_details":{"reasoning_tokens":2446}},"tokens_in":391,"tokens_out":2525,"duration_ms":18896,"temperature":1.0,"reasoning_tokens":2446,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:59:32.832173+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $d=2$, write out the promised polynomials $p_n$ explicitly, then compute the $H^2$ norm of $q_1$ and $q_2$ directly; if any $\\|q_k\\|_{H^2}\\neq 1$ or if the series $\\sum_k q_k$ has finite $H^2$ norm, the proof's unboundedness argument fails. Alternatively, evaluate the first few partial sums on a sequence of row contractions that concentrate on growing matrix sizes and check numerically whether the norms diverge.","supporting_citations":[{"cited_title":"McCarthy","cited_arxiv_id":null,"evidence_quote":"Defines the free topology and the class of free holomorphic functions, and supplies the classical $H^2$-norm inequality $\\|f\\|_{H^2}\\le \\|f\\|_{\\mathcal{C}_d}$ used to infer unboundedness."},{"cited_title":"Augat, S","cited_arxiv_id":null,"evidence_quote":"Establishes the scarcity of compact sets in the free topology; the example's behavior on row-ball scalings contrasts with that compactness picture."},{"cited_title":"William Helton, Igor Klep, and Scott McCu llough","cited_arxiv_id":null,"evidence_quote":"Provides a uniform approximation theorem for free holomorphic functions on compact sets in the free topology, whose limitations the example exposes."}],"review_version":1}