{"id":"9f7e2a04-1b40-4447-9578-d62aafee8dad","arxiv_id":"2608.13139","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every maximally monotone operator of type (NI) on an arbitrary real Banach space, the Haraux function is at least half the weighted squared graph distance, with 1/2 optimal, for every positive weight.","lead":"This paper proves that a sharp lower bound on the Haraux function, previously known only in reflexive spaces, also holds for every maximally monotone operator of type (NI) on any real Banach space. The result rests on a simple energy identity and removes reflexivity as an obstacle to the sharp 1/2 constant.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: internal proof is sound; the only load-bearing step is the cited type (NI)-quasidensity equivalence.","rationale":"The reader's ACCEPT verdict is consistent with my reading. The proof is short, explicit, and algebraically sound; the sharpness and separation examples are correct. The only point where an unstated condition could enter is the cited type (NI)-quasidensity equivalence, which is exactly the reader's weakest assumption. Because that equivalence is a published theorem and no countervailing evidence appears in the manuscript, I do not regard it as a demonstrated flaw, so the verdict should remain unchanged. The proposed check is a verification of the citation's scope rather than a correction to the proof.","tokens_in":10849,"tokens_out":27280,"duration_ms":236096,"concrete_test":"Consult Simons's theorem in [16] and verify its exact hypotheses: it must state the equivalence between type (NI) and quasidensity for maximally monotone operators on arbitrary real Banach spaces with no additional geometric conditions, and its quasidensity definition must match Definition 2.4. Then re-run Lemma 2.1 with the scaled operator B from (24), checking that (28) follows; if [16] turns out to require an extra hypothesis, Theorem 3.1 would have to be restricted accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The internal argument of Theorem 3.1 checks out. Proposition 3.1 is an exact algebraic identity; Corollary 3.1's defect inequality (34) follows by a routine epsilon argument; Lemma 2.1 correctly constructs the scaled operator B in (24), proves it is maximal monotone and of type (NI), and then uses quasidensity of gra B to obtain the weighted zero-infimum (22). The passage from (22) to (39) needs only a residual below epsilon and no attainment. Example 4.1 gives exact equality for every gamma, so the constant 1/2 cannot be improved, and Example 4.2 correctly separates exact certificate attainment from zero defect. The one load-bearing step not proved in this paper is the external equivalence between type (NI) and quasidensity for maximally monotone operators on arbitrary real Banach spaces, cited as [16]. I found no internal inconsistency and no evidence that the cited theorem fails or needs additional hypotheses; the concern is therefore only that the central claim would collapse if the citation were wrong or narrower than stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a sharp lower bound for the Haraux function of maximally monotone operators of type (NI) on arbitrary real Banach spaces. Theorem 3.1 states that for every such operator A, every (x,u*) in X times X*, and every gamma > 0, one has H_A(x,u*) >= (1/2) d^2_{gra A, gamma}(x,u*), and that 1/2 is the best uniform constant. The proof is built on an exact local decomposition, Proposition 3.1, expressing the sum of the Haraux contribution and a weighted residual as half the squared weighted graph displacement. Lemma 2.1 supplies the key zero-infimum property of the residual via the published equivalence between type (NI) and quasidensity. Corollary 3.1 records a defect-corrected bound and a Haraux-side characterization of type (NI). Example 4.1 proves sharpness for every weight, Example 4.2 separates exact-certificate attainment from zero defect on c0, and Corollary 4.2 transfers the bound to Fenchel-Young gaps of lower semicontinuous convex functions.","tokens_in":11052,"tokens_out":10047,"duration_ms":86547,"significance":"If the cited type (NI)-quasidensity equivalence is valid in the stated generality, the paper fully answers the question raised by Combettes and Mayrand: the sharp 1/2 constant survives beyond reflexivity, without any attainment assumption. The main strengths are the exact algebraic energy decomposition, the circulation-free and explicit epsilon arguments, the clean separation of the three logical levels in Remark 4.1, and the concrete c0 example showing that approximate certificates can exist where exact ones do not. The authors are also careful to state that the proof does not assert equality between the Haraux function and the squared graph distance. The only nonlocal input is the cited equivalence from Simons [16]; this is a standard published result, not an internal gap, but because the main theorem rests on it, the manuscript should state its precise scope explicitly. Overall, this is a focused, correct, and useful contribution to monotone operator theory.","major_comments":[],"minor_comments":[{"comment":"The nonnegativity of R_gamma is proved in (23) under the hypotheses of Lemma 2.1, but Definition 2.3 defines delta_{A,gamma} for an arbitrary operator with nonempty graph and refers to (23). Since the norm inequality R_gamma(a,a*) >= (1/2)(||a||/sqrt(gamma) - sqrt(gamma)||a*||_*)^2 holds for all (a,a*) independently of monotonicity or type (NI), the nonnegativity statement should be moved before Definition 2.3 and stated in that generality.","section":"Definition 2.3 / Lemma 2.1"},{"comment":"The zero-infimum step is the single load-bearing external input: it applies the type (NI)-quasidensity equivalence of [16] to the scaled operator B. The authors should state the equivalence explicitly as a named known theorem or a clearly identified premise, and confirm that it holds for maximally monotone operators on arbitrary real Banach spaces. As written, the main theorem is conditional on the exact scope of [16].","section":"Lemma 2.1, Eq. (22)"},{"comment":"The display defining z^{(N)} contains a typographical artifact in the underbrace; it should read z^{(N)}=(1,...,1,0,0,...) with N leading ones.","section":"Example 4.2"},{"comment":"The citation chain should be made fully explicit: Corollary 4.2 invokes Theorem 3.1 after using [17] for quasidensity of the subdifferential, and the quasidensity-to-type-(NI) step depends on the equivalence quoted in Lemma 2.1. A one-sentence reminder would make the logic easier for readers to verify.","section":"Corollary 4.2"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is squarely within the scope of math.FA and is a natural follow-up to recent work on the Haraux function. I have no reason to doubt the cited type (NI)-quasidensity equivalence, and I found no internal inconsistency in the proofs. The requested revision is limited to making the external premise and a few presentation points explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper settles the open question from Combettes and Mayrand by proving the constant 1/2 Haraux lower bound for every maximally monotone operator of type (NI) on an arbitrary real Banach space, for every positive weight. The proof is transparent: an exact algebraic identity splits half the weighted squared graph displacement into the Haraux contribution plus a weighted residual; type (NI) makes the residual infimum zero via the known equivalence with quasidensity. That is essentially the whole argument, and it is correct.\n\nWhat is genuinely new: the best nonreflexive bound in the literature was 1/4 (Voisei–Zălinescu). Here it is 1/2 for the type (NI) class, and the paper shows the constant is sharp for every weight using the identity operator. The defect-corrected inequality for any operator with nonempty graph (Corollary 3.1) is a useful byproduct, and the c0 example cleanly separates zero defect from attained certificates, which is a nice conceptual clarification. The Fenchel–Young corollary is a straightforward application rather than a new theorem, but it is honest about that.\n\nSoft spots: the load-bearing step is the cited equivalence between type (NI) and quasidensity on arbitrary real Banach spaces (Simons). The paper does not prove it, and if that equivalence were wrong or narrower than stated, the main theorem would collapse. I checked the internal logic: Lemma 2.1 transfers type (NI) to the scaled operator correctly, the scaling argument looks valid, and nothing is circular. Still, a referee should verify that the cited theorem applies to the scaled operator without extra hypotheses; that is the hinge.\n\nThe paper is small and clean. It does not overclaim: it identifies the right mechanism (zero defect rather than attainment) and proves the sharp bound. My recommendation: send it out. It is a solid within-subfield contribution that deserves a serious referee, and with the citation checked it should be publishable essentially as is.","headline":"A small, honest paper that answers Combettes–Mayrand by moving the sharp 1/2 Haraux bound to type (NI) operators on arbitrary Banach spaces; the internal proof is sound, and the only risky hinge is a cited equivalence the authors do not prove.","tokens_in":11564,"tokens_out":2148,"would_cite":true,"duration_ms":18938,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47H05","47N10","46A20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The sharp 1/2 Haraux lower bound holds for type (NI) monotone operators on any real Banach space.","keywords":["Haraux function","maximally monotone operator","type (NI)","quasidensity","graph distance","nonreflexive Banach space","Fenchel–Young inequality","certificate hierarchy"],"falsifier":"A concrete refutation would be a maximally monotone operator of type (NI) on a nonreflexive Banach space with a point $(x,u^*)$ and weight $\\gamma>0$ for which the weighted residual infimum $\\delta_{A,\\gamma}(x,u^*)$ is positive, since Lemma 2.1 asserts it is always zero; equivalently, any pair with $H_A(x,u^*) < \\tfrac12 d^2_{\\mathrm{gra}\\,A,\\gamma}(x,u^*) - \\varepsilon$ would overturn Theorem 3.1. One could search for this by evaluating the residual infimum on the explicit $c_0$ operator $A=0$ at shifted targets and checking whether the computed value ever exceeds a positive $\\varepsilon$.","tokens_in":10670,"feed_emoji":"📐","tokens_out":5536,"duration_ms":36763,"temperature":0.7,"pith_summary":"This paper proves that the sharp lower bound $H_A(x,u^*) \\ge \\tfrac12 d^2_{\\mathrm{gra}\\,A,\\gamma}(x,u^*)$ for the Haraux function remains true for every maximally monotone operator of type (NI) on an arbitrary real Banach space, for every positive weight $\\gamma$. In the reflexive case the bound was obtained from an exact metric-resolvent witness; the paper shows that beyond reflexivity the same geometry follows from approximate certificates alone. The engine is an exact energy decomposition in which the Haraux contribution and a nonnegative weighted residual are complementary terms summing to half the weighted squared displacement. With the known equivalence between type (NI) and quasidensity, the residual infimum vanishes, so the sharp constant emerges without any attainment assumption and without reflexivity. A sympathetic reader should care because this turns the earlier $1/4$ bound for nonreflexive spaces into a non-sharp estimate and identifies reflexivity as an attainment principle rather than a geometric requirement.","feed_headline":"Sharp 1/2 Haraux bound survives beyond reflexivity","feed_subtitle":"An exact energy split shows type (NI) monotone operators keep the reflexive-space constant on every Banach space.","key_machinery":"The load-bearing object is the weighted residual $R_\\gamma(a,a^*) = \\|a\\|^2/(2\\gamma) + (\\gamma/2)\\|a^*\\|^2_* + \\langle a,a^*\\rangle$, together with the exact local identity $h_A(x,u^*;y,y^*) + R_\\gamma(y-x,y^*-u^*) = \\tfrac12 D^2_\\gamma((x,u^*),(y,y^*))$. This identity decomposes the half squared graph displacement into a Haraux coupling and a quasidensity mismatch for every graph point. The zero-infimum step is supplied by Lemma 2.1, which scales the operator by $\\gamma$ and applies the type-(NI)-quasidensity equivalence to conclude that the residual infimum vanishes. The defect $\\delta_{A,\\gamma}(x,u^*) = \\inf_{(y,y^*)\\in\\mathrm{gra}\\,A} R_\\gamma(y-x,y^*-u^*)$ packages this residual, and Corollary 3.1 shows that $\\delta=0$ is equivalent to a Haraux-side asymptotic exhaustion condition and yields the defect-corrected bound.","core_discovery":"The central claim is Theorem 3.1: for every maximally monotone operator $A$ of type (NI) on an arbitrary real Banach space $X$, every $(x,u^*)\\in X\\times X^*$ and every $\\gamma>0$, $H_A(x,u^*) \\ge \\tfrac12 d^2_{\\mathrm{gra}\\,A,\\gamma}(x,u^*)$, and $\\tfrac12$ is the best uniform constant. The proof does not produce a point where the graph-distance infimum, the residual infimum, or the Haraux supremum is attained. Instead it uses the exact identity $h_A(x,u^*;y,y^*) + R_\\gamma(y-x,y^*-u^*) = \\tfrac12 D^2_\\gamma((x,u^*),(y,y^*))$, valid for every graph point. Because type (NI) is equivalent to quasidensity for maximally monotone operators, the infimum of the residual $R_\\gamma$ is zero, and $\\varepsilon$-level certificates give the bound by taking $\\varepsilon\\to 0$. The paper also derives a defect-corrected bound for arbitrary operators with nonempty graph and shows that the zero-defect regime characterizes type (NI) within the maximally monotone class.","pith_inferences":["For a maximally monotone operator that is not type (NI), the defect $\\delta_{A,\\gamma}(x,u^*)$ is positive at some targets, so the decomposition suggests a quantitative hierarchy $H_A \\ge \\tfrac12 d^2 - \\delta$ in which computing or bounding the defect becomes a natural numerical target.","The same identity may transfer to other graph-displacement energies built from different gauges or moduli, producing analogous sharp constants tied to the gauge's modulus rather than to the Hilbertian square norm.","In optimization over nonreflexive spaces, Corollary 4.2 gives a checkable lower bound on the Fenchel\\,–\\,Young gap that could certify when a candidate pair is far from optimality.","A natural stress test is to seek a maximally monotone type (NI) operator on $\\ell^\\infty$ or another nonreflexive space where the old $\\tfrac14$ bound is attained; the theorem says none exists, so any apparent counterexample would expose an error in the type-(NI)-quasidensity equivalence or in the scaling step."],"forward_implications":["For every weight $\\gamma>0$, $\\tfrac12$ is the best uniform constant in the Haraux graph-distance bound for maximally monotone operators of type (NI) on arbitrary real Banach spaces.","Reflexivity is demoted from a geometric hypothesis to an attainment device: reflexive operators admit exact certificates, but the sharp bound only needs approximate certificates.","Every proper lower semicontinuous convex function $\\varphi$ on an arbitrary Banach space satisfies $\\varphi(x)+\\varphi^*(u^*) - \\langle x,u^*\\rangle \\ge \\tfrac12 d^2_{\\mathrm{gra}\\,\\partial\\varphi,\\gamma}(x,u^*)$, by the Fenchel\\,–\\,Young bridge.","The earlier $\\tfrac14$ nonreflexive estimate is not an intrinsic obstruction; it is superseded for the type (NI) class.","The defect-corrected inequality gives a quantitative lower bound even for operators with nonempty graph that are not type (NI)."],"supporting_citations":[{"why":"Provides the sharp reflexive-space bound and raises the extension question that this paper answers.","marker":"[1]"},{"why":"Supplies the prior general Banach-space bound with constant $1/4$ that Theorem 3.1 improves for type (NI) operators.","marker":"[10]"},{"why":"Gives the equivalence between type (NI) and quasidensity for maximally monotone operators, the zero-infimum step in Lemma 2.1.","marker":"[16]"},{"why":"Establishes quasidensity as a Banach-space approximation property and supplies the fact that subdifferentials are quasidense, used for the $c_0$ example and Corollary 4.2.","marker":"[17]"},{"why":"Develops quasidense monotone multifunctions, anchoring Definition 2.4 and the approximation class behind the proof.","marker":"[18]"}],"fun_headline_variants":["Sharp 1/2 Haraux bound holds on all Banach spaces","Type NI monotone operators keep the 1/2 Haraux bound","Exact energy split yields sharp Haraux bound for type NI","Defect-corrected Haraux bound: 1/2 for every type NI operator","Beyond reflexivity: sharp 1/2 Haraux bound proven"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the published equivalence, for maximally monotone operators on arbitrary real Banach spaces, between type (NI) and quasidensity: Lemma 2.1 uses it to conclude that the weighted residual infimum is zero, and that zero-infimum step is what turns the local decomposition into the global sharp bound. If the equivalence fails in this generality, the proof of Theorem 3.1 collapses.","fun_headline_variants_meta":{"raw":{"variants":["Sharp 1/2 Haraux bound holds on all Banach spaces","Type NI monotone operators keep the 1/2 Haraux bound","Exact energy split yields sharp Haraux bound for type NI","Defect-corrected Haraux bound: 1/2 for every type NI operator","Beyond reflexivity: sharp 1/2 Haraux bound proven"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000845,"raw_usage":{"total_tokens":3710,"prompt_tokens":1005,"completion_tokens":2705,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":621,"completion_tokens_details":{"reasoning_tokens":2608}},"tokens_in":621,"tokens_out":2705,"duration_ms":372025,"temperature":1.0,"reasoning_tokens":2608,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:53:50.891602+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete refutation would be a maximally monotone operator of type (NI) on a nonreflexive Banach space with a point $(x,u^*)$ and weight $\\gamma>0$ for which the weighted residual infimum $\\delta_{A,\\gamma}(x,u^*)$ is positive, since Lemma 2.1 asserts it is always zero; equivalently, any pair with $H_A(x,u^*) < \\tfrac12 d^2_{\\mathrm{gra}\\,A,\\gamma}(x,u^*) - \\varepsilon$ would overturn Theorem 3.1. One could search for this by evaluating the residual infimum on the explicit $c_0$ operator $A=0$ at shifted targets and checking whether the computed value ever exceeds a positive $\\varepsilon$.","supporting_citations":[{"cited_title":"Lower bounds on the Haraux function,","cited_arxiv_id":null,"evidence_quote":"Provides the sharp reflexive-space bound and raises the extension question that this paper answers."},{"cited_title":"Strongly-representable monotone op- erators,","cited_arxiv_id":null,"evidence_quote":"Supplies the prior general Banach-space bound with constant $1/4$ that Theorem 3.1 improves for type (NI) operators."},{"cited_title":"A stand-alone analysis of quasidensity,","cited_arxiv_id":null,"evidence_quote":"Gives the equivalence between type (NI) and quasidensity for maximally monotone operators, the zero-infimum step in Lemma 2.1."},{"cited_title":"“Densities","cited_arxiv_id":null,"evidence_quote":"Establishes quasidensity as a Banach-space approximation property and supplies the fact that subdifferentials are quasidense, used for the $c_0$ example and Corollary 4.2."},{"cited_title":"Quasidense monotone multifunctions,","cited_arxiv_id":null,"evidence_quote":"Develops quasidense monotone multifunctions, anchoring Definition 2.4 and the approximation class behind the proof."}],"review_version":1}