{"id":"4c5a0620-7379-48fa-8eef-fdb3f68711d8","arxiv_id":"2507.18434","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For even n, a carefully chosen sequence of vectors makes the spectrahedral relaxation bound for Eulerian polynomial roots exceed the univariate bound by asymptotically (3/8)(9/8)^{n/2}.","lead":"New test vectors, guessed from numerical eigenvectors, are used to extract root bounds from a spectrahedral relaxation of multivariate Eulerian polynomials. For even degrees, the improvement over the previous univariate bound now grows exponentially.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 13.2's (9/8)^m growth rests on hand-checked cancellations in chained conjugations, a failure mode the paper itself flags; the claim needs independent recomputation before acceptance.","rationale":"Agree with the reader that Lemma 13.2 is the weakest link. The main advertised result is that a better vector yields a bound whose difference from the univariate bound grows as (9/8)^m, and that is exactly the content of the lemma. The proof is not machine-checked and is described as a lengthy manual search for surviving terms under chained conjugations. The paper's own warnings (11.3, 11.4, 11.6, 11.8) make the failure mode explicit: ignoring a surviving term in a difference of radicals can change the first growth term. No ancillary code, notebook, or independent derivation is supplied, and the paper states that the vector was found by numerical guessing. These facts do not make the claim false, but they make the central asymptotic computation the right place to probe. The proposed test is direct: because D and N are written out in Proposition 12.1, one can recompute the asymptotic expansion from those explicit formulae with exact arithmetic, bypassing the manual conjugation steps. A successful recomputation would remove the concern and support the claimed exponential improvement; a mismatch would invalidate the main quantitative claim. The reader's verdict of CONDITIONAL is therefore appropriate and unchanged.","tokens_in":28238,"tokens_out":5065,"duration_ms":52739,"concrete_test":"Use a CAS to recompute the bound from Proposition 12.1: define D and N as given, take y = (b+sqrt(b^2-4ac))/(2a) with a, b, c from Lemma 9.1 at n=2m, and form mult_v = N/D. Compute the asymptotic expansion of mult_v - un, with un from Lemma 9.1, in the scale a^n for rational a, using exact arithmetic and e.g. Maple's asympt or Mathematica with Collect[expr, 2^n] followed by series. Verify that the leading term is exactly (3/8)(9/8)^m and that no term with base larger than 9/8 survives. Also independently check the intermediate orders stated in the proof of Lemma 13.2: k, v, u, w, (k+u)-(v+w), s-t, gamma-delta*sqrt(g), and the final numerator and denominator. If any order is off by a power of 2 or 3, the main claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that mult_v(n)-un(n) ~ (3/8)(9/8)^m, Lemma 13.2. This is the entire evidence for the explosive improvement of the multivariate relaxation. The proof is a hand-managed asymptotic computation: it defines k, v, u, w, asserts their dominant terms, notes that they cancel, multiplies by the conjugate (k+u)-(v+w), then by s-t, and states the surviving leading terms. None of these asymptotic orders is derived in the text; the expressions for D and N in Proposition 12.1 are substituted into y and then analyzed. The paper explicitly reports that Mathematica could not determine these growth terms automatically and that they were checked manually (Procedure 11.4, Warning 11.6). More importantly, Warning 11.8 states that a mistake in the chained conjugations could imply that we ignore asymptotic growth terms at some point in the chain process that actually survive, and that this may significantly and wrongly alter the whole computation. That is exactly the risk here: if any of the asserted cancellations is incomplete, the leading term of the difference changes and the (9/8)^m bound is no longer established. The vector v is found by numerical guessing (Experiment 10.5) and y is carried over from [7] rather than re-optimized; these choices affect the value of the bound but are not themselves unsound once the arithmetic is fixed. The load-bearing step is the unverified asymptotic expansion in Lemma 13.2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies spectrahedral relaxations of Eulerian rigidly convex sets, building on the author's earlier work [7]. The main new claim is that, for even n=2m, the bound obtained by linearizing the multivariate spectrahedral relaxation along the vector (y,(-2^{m-i})_{i=3}^m,0,1/2,(1)_{i=1}^m) satisfies mult_v(n)-un(n) ~ (3/8)(9/8)^m. This would show that the certified advantage of the multivariate relaxation over the univariate bound grows exponentially, in contrast to the vanishing improvement obtained in [7]. The paper constructs the vector through numerical experiments, derives the form of the bound in Proposition 12.1, and computes the asymptotic difference in Lemma 13.2 by a hand-managed chain of conjugations.","tokens_in":28518,"tokens_out":2315,"duration_ms":27877,"significance":"If Lemma 13.2 is correct, the result is significant: it provides the first certificate that the multivariate spectrahedral relaxation of [21] yields an exponentially growing diagonal accuracy improvement over the univariate relaxation, thereby substantially strengthening the main quantitative claim of [7]. The paper also gives a transparent and honest account of the numerical guessing procedure and explicitly warns about the fragility of the chained-conjugation computation (Warning 11.8). However, the central asymptotic claim rests on a lengthy hand-verified computation that is not independently checked, so the significance is conditional on verification.","major_comments":[{"comment":"The proof of Lemma 13.2 is the load-bearing step: it asserts that after writing mult_v-un=(k+v+u+w)/(s(gamma+delta sqrt(g))) the dominant terms k, v, u, w all grow like 2^{22m+19}3^{2m+1}m^4 and annihilate, then asserts the subdominant terms survive with the claimed leading growth. None of these orders is derived in the text; the steps are described as 'as happened in [7]' and 'computing we can see'. The paper itself, in Warning 11.8, states that a mistake in the chained conjugations could cause one to ignore asymptotic growth terms that actually survive and 'may significantly and wrongly alter the whole computation'. Since the entire (9/8)^m improvement is the first growth term of this difference, this is exactly the risk. I request an independent verification of the cancellation and of the surviving terms, ideally by (a) writing out the explicit asymptotic expansions of each of k, u, v, w after the first conjugation, and (b) providing a reproducible symbolic or numeric check of the final limit mult_v-un over (9/8)^m -> 3/8.","section":"Lemma 13.2 (and Warning 11.8)"},{"comment":"The proof of Proposition 12.1 consists solely of displaying the sums for D and N and concluding 'Computing these last expressions finishes the proof.' The resulting closed forms for D and N are extremely long, and they are the input to Lemma 13.2. Without an indication of how the sums over the L_p entries (using Computation 6.4) are simplified, or a machine-checked derivation, the reader cannot verify that the displayed D and N are free of transcription errors. Please provide the intermediate summation steps or a verifiable computer algebra script, and state explicitly the indexing correspondence between the vector entries and the variables x_i (the expressions use x_{i-2} and x_m, but the vector is written as (y,(-2^{m-i})_{i=3}^m,(0,1/2),(1)_{i=1}^m); this indexing should be spelled out).","section":"Proposition 12.1"},{"comment":"The proof that the chosen y from [7] gives N>0 and D>0 is outlined only for N, and for D it says 'the result is immediate' after referring to [7]. Since the positivity of D is used to identify the bound as -D/N (equivalently N/D), this step should be expanded or replaced by a direct computation, especially because y contains a radical and the asymptotic argument for N involves comparing 6^{n+1} terms against 8^{n+1} terms. This is less central than Lemma 13.2, but it is still part of the chain establishing the bound.","section":"Proposition 13.1"}],"minor_comments":[{"comment":"The phrase 'because of the following three seasons' should be 'three reasons'.","section":"Section 3, Remark 3.1"},{"comment":"The text refers to 'the Figure 10.5' but the caption reads 'Figure 1'; renumber the cross-reference.","section":"Experiment 10.5 and Figure 1"},{"comment":"The vector v is written as (y,(-2^{m-i})_{i=3}^m,(0,1/2),(1)_{i=1}^m) but the sums in Proposition 12.1 use indices i-2 and m, which suggests an offset between the two notations. Clarify the indexing consistently: for example, state that the entries correspond to variables (x_1,x_2,...,x_n) with x_1 mapped to y, x_2 to 0, x_3 to 1/2, etc.","section":"Notation, Sections 10-13"},{"comment":"The univariate bound un(n) is used throughout Sections 13 and 14 but is not defined in this paper; a formal definition or reference to the exact statement in [7] would improve readability.","section":"Section 9 and Lemma 9.1"},{"comment":"The abstract and conclusion state the exponential improvement without always specifying that it is proven only for even n=2m; since the odd case is only claimed to follow 'similarly', state this restriction more prominently in the abstract.","section":"Abstract and Section 17"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claim is plausible and the Rayleigh-quotient mechanism is sound once the vector is fixed, but the exponential improvement rests on a hand-checked asymptotic computation that the paper itself flags as risky. I would like the editor to require either a fully written-out derivation of the surviving terms in Lemma 13.2 or an independent machine-checked verification before considering acceptance. I also note the result depends heavily on the author's own prior work [7] for the value of y and the definition of un; this is not a problem per se, but it makes the manuscript difficult to assess without that companion paper at hand."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: Lemma 13.2 is new and worth knowing. For even n=2m, the multivariate relaxation bound linearized by the vector (y,(-2^{m-i})_{i=3}^m,0,1/2,(1)_{i=1}^m) beats the univariate bound by ~(3/8)(9/8)^m. That replaces the vanishing (1/2)(3/4)^n improvement from [7] with a growing exponential one. The result is narrower than the abstract makes it sound—even n only, diagonal only, and it is a certificate of accuracy rather than a new relaxation framework—but it is a genuine step.\n\nWhat is good: the method is coherent. Once the vector is fixed, the inequality a+xb>=0 is a real Rayleigh-quotient bound; the comparison with un is a well-defined asymptotic statement. The author is unusually candid about the computational pain. Warning 11.8 spells out exactly the failure mode that worries me: a missed surviving term in the chained conjugations would change or destroy the (9/8)^m estimate. Procedure 11.4 says Mathematica could not do the growth-term analysis and it was done manually. That honesty is valuable, and it tells the referee where to look.\n\nThe soft spot is exactly there. Lemma 13.2 is the load-bearing result, and its proof is a hand-managed asymptotic computation: k,v,u,w are asserted with dominant terms, the cancellations are asserted, and the surviving leading term is computed. Proposition 12.1's proof is just 'computing these last expressions.' No code, no formal proof, no independent check. I don't see an obvious error, and the vector does not look fitted to the final difference—it was guessed from numerics, which is legitimate—but the arithmetic is not something I would certify by reading once. The inherited y from [7] is suboptimal and the author says so; that is a minor limitation, not a flaw.\n\nWho it is for: specialists in spectrahedral relaxations and Eulerian polynomials. A serious referee should spend time on Section 13 and ask for a reproducible computation or formal verification of the asymptotics. I'd send it to review.","headline":"A real but narrow asymptotic improvement, with the load-bearing proof resting on hand-checked cancellations that need independent verification before I'd stake anything on it.","tokens_in":29054,"tokens_out":2054,"would_cite":false,"duration_ms":23013,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A15","14P10","26C10","90C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes that a chosen linearizing vector makes the multivariate spectrahedral relaxation beat the univariate bound by a gap growing like $\\frac{3}{8}(\\frac{9}{8})^m$ for even $n=2m$, reversing a previously vanishing…","keywords":["Eulerian polynomials","spectrahedral relaxation","rigidly convex sets","real zero polynomials","linear matrix polynomials","generalized eigenvalues","descent top sets","asymptotic root bounds"],"falsifier":"Evaluate the exact closed-form expressions from Proposition 12.1 and Lemma 9.1 at even $n=2m$ using exact rational arithmetic and check whether $\\mathrm{mult}_v(n)-\\mathrm{un}(n)$ divided by $(\\frac98)^m$ tends to $\\frac38$; any drift or different constant would show that a surviving term was missed in the conjugation chain.","tokens_in":27998,"feed_emoji":"📈","tokens_out":11129,"duration_ms":114554,"temperature":0.7,"pith_summary":"This paper claims that adding variables helps in a specific, measurable way: a spectrahedral relaxation built from multivariate Eulerian polynomials—an outer approximation of a convex set by linear matrix inequalities—can certify a bound on the extreme roots of the univariate Eulerian polynomials (the polynomials counting permutations by descents) that is exponentially better than the best univariate bound. Earlier work had already shown an improvement, but the certified gap between the multivariate and univariate bounds shrank to zero as the degree grew, at the rate $(\\frac12)(\\frac34)^n$. By feeding the relaxation a carefully guessed sequence of vectors, the paper proves that for even $n=2m$ the gap grows like $\\frac38(\\frac98)^m$, i.e. it diverges exponentially. If correct, this turns “more variables help” from a negligible correction into a decisive asymptotic effect, and it gives a template for extracting accurate root bounds from linearizations of spectrahedra.","feed_headline":"Root bound gap grows as (9/8)^m","feed_subtitle":"A structured eigenvector guess turns a vanishing advantage into exponential accuracy in Eulerian root bounds.","key_machinery":"The machinery is the linear matrix polynomial (LMP) produced by the spectrahedral relaxation of the real-zero multivariate Eulerian polynomial, restricted to the diagonal $x_1=\\cdots=x_n=x$. For any vector $v$, the quadratic form $v^T(M_{n,0}+xM_{n,\\mathrm{sum}})v=D+xN$ is nonnegative on the spectrahedron, so whenever $N>0$ it yields the bound $x\\ge -D/N$. The work of the paper is choosing $v$: the entries $(-2^{m-i})_{i=3}^m$, the single $0$ and $\\tfrac12$, and the tail of ones are a numerical guess at the shape of the true generalized eigenvector. The asymptotic estimate of $N/D$ is carried out by conjugation, rewriting $c-\\sqrt{b}$ as $(c^2-b)/(c+\\sqrt{b})$ so that surviving terms below the leading order are not discarded; the surviving term is $\\frac38(\\frac98)^m$.","core_discovery":"The central result is a comparison of two bounds for the leftmost root of the $n$-th univariate Eulerian polynomial. The univariate relaxation gives a bound $\\mathrm{un}(n)$; the multivariate relaxation, linearized with the vector $v=(y,(-2^{m-i})_{i=3}^m,0,\\tfrac12,(1)_{i=1}^m)$ where $n=2m$, gives a bound $\\mathrm{mult}_v(n)$. Lemma 13.2 states that $\\mathrm{mult}_v(n)-\\mathrm{un}(n)\\sim \\frac38(\\frac98)^m$ along even indices. Since $(\\frac98)^m\\to\\infty$, the certified advantage of the multivariate relaxation no longer vanishes; it grows exponentially. The proof writes the diagonal linear matrix polynomial as $D+xN$, uses $x\\ge -D/N$ for any vector, and then computes the asymptotics of $N/D$ by repeated conjugation to remove radicals, after the leading candidate terms cancel exactly.","pith_inferences":["The same guessing strategy could be tested on other real-stable multivariate liftings, such as W-Eulerian polynomials, to see whether their spectrahedral relaxations also certify exponentially growing gaps rather than vanishing ones.","Because the difference between bounds is measured along the diagonal, the exponential gap likely reflects genuinely deeper asymptotic terms of the extreme roots; an inner bound from a dual construction would sandwich the roots and expose those terms.","A concrete extension is to build the analogous vector for odd $n$; the paper says the odd case follows similarly, but does not compute the base of the exponential growth, so the exact rate for odd $n$ remains open.","The vector's entry profile (powers of $-2$ up to a $0,\\tfrac12$ step, then a tail of ones) may correspond to a weighting of descent-top statistics; if so, the optimal vector could be derived combinatorially rather than guessed numerically."],"forward_implications":["For every even $n=2m$ large enough, the multivariate spectrahedral relaxation gives a strictly better extreme-root bound than the univariate one, with the gap growing like $\\frac38(\\frac98)^m$.","The previously established vanishing improvement is an artifact of the simple vector $(y,1,-1)$; structured vectors can certify that multivariability changes the asymptotic accuracy of the relaxation.","Because univariate Eulerian polynomials are palindromic, the same exponentially growing gap also translates into an exponentially improved bound for the rightmost root.","The bound is proven with the $y$ optimized for the earlier vector; the author leaves as an exercise the optimization of $y$ for this new vector, which should only increase the certified gap.","The success of this guessed vector suggests a concrete design principle for future linearizations: mimic the observed shape of the actual generalized eigenvector instead of using a generic constant tail."],"supporting_citations":[{"why":"Defines the previous vector $(y,1,-1)$, the univariate bound $\\mathrm{un}(n)$, and the optimized $y$ reused here; the new bound's difference is measured against this work.","marker":"[7]"},{"why":"Supplies the spectrahedral relaxation construction, the L-forms, and the proof that the spectrahedron contains the rigidly convex set, which is the object being relaxed.","marker":"[21]"},{"why":"Introduces stable multivariate Eulerian polynomials, the source of the real-zero multivariate Eulerian polynomials used to build the relaxation.","marker":"[8]"},{"why":"Establishes real stability of the relevant multivariate Eulerian liftings, needed to guarantee the polynomials are real zero and thus eligible for the relaxation.","marker":"[23]"},{"why":"Provides the equivalence between real stability and hyperbolicity in positive directions used to pass from stable polynomials to real zero polynomials.","marker":"[17]"}],"fun_headline_variants":["Root bound gap grows as (9/8)^m","Eigenvector guess yields exponential root bound gains","Sequenced eigenvectors spike root bound accuracy exponentially","Multivariate trick turns root bound gap exponential","Exponential root bound improvement via eigenvector sequence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole claim of exponential improvement rests on the manual asymptotic bookkeeping in the proof of Lemma 13.2: after repeated conjugation, every surviving term has been identified correctly, and no neglected subdominant term changes the first growth term.","fun_headline_variants_meta":{"raw":{"variants":["Root bound gap grows as (9/8)^m","Eigenvector guess yields exponential root bound gains","Sequenced eigenvectors spike root bound accuracy exponentially","Multivariate trick turns root bound gap exponential","Exponential root bound improvement via eigenvector sequence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000686,"raw_usage":{"total_tokens":3160,"prompt_tokens":1043,"completion_tokens":2117,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":659,"completion_tokens_details":{"reasoning_tokens":2046}},"tokens_in":659,"tokens_out":2117,"duration_ms":17592,"temperature":1.0,"reasoning_tokens":2046,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:32:33.126256+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the exact closed-form expressions from Proposition 12.1 and Lemma 9.1 at even $n=2m$ using exact rational arithmetic and check whether $\\mathrm{mult}_v(n)-\\mathrm{un}(n)$ divided by $(\\frac98)^m$ tends to $\\frac38$; any drift or different constant would show that a surviving term was missed in the conjugation chain.","supporting_citations":[{"cited_title":"Spectrahedral relaxations of Eulerian rigidly convex sets","cited_arxiv_id":"2507.03800","evidence_quote":"Defines the previous vector $(y,1,-1)$, the univariate bound $\\mathrm{un}(n)$, and the optimized $y$ reused here; the new bound's difference is measured against this work."},{"cited_title":"Spectrahedral relaxations of hyperbolicity cones","cited_arxiv_id":"1907.13611","evidence_quote":"Supplies the spectrahedral relaxation construction, the L-forms, and the proof that the spectrahedron contains the rigidly convex set, which is the object being relaxed."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces stable multivariate Eulerian polynomials, the source of the real-zero multivariate Eulerian polynomials used to build the relaxation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes real stability of the relevant multivariate Eulerian liftings, needed to guarantee the polynomials are real zero and thus eligible for the relaxation."},{"cited_title":"Hyperbolicity and stable polynomials in combinatorics and probability","cited_arxiv_id":"1210.3231","evidence_quote":"Provides the equivalence between real stability and hyperbolicity in positive directions used to pass from stable polynomials to real zero polynomials."}],"review_version":1}