{"id":"de73a320-faf8-4b5b-8277-5060d4d1288b","arxiv_id":"2507.15002","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Myers theorem, Laplacian comparison, and volume comparison are proved for Hermitian manifolds with lower bounds on the Strominger-Bismut holomorphic Ricci curvature.","lead":"This paper proves comparison theorems for Hermitian manifolds using the Strominger-Bismut connection, a torsionful cousin of the Levi-Civita connection. It establishes a Myers-type diameter bound, a Laplacian comparison, and a volume comparison under lower bounds on Bismut Ricci curvature.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Myers argument hinges on Proposition 2.6, whose proof is too incomplete to certify the bridge from the holomorphic Bismut-Ricci bound to the real Ricci bound; a separate inequality reversal in (4.9) also blocks the volume comparison as written.","rationale":"The reader correctly identified Proposition 2.6 as the weakest bridge: every part of Theorem 1.3 depends on converting (1.10), a condition on the holomorphic Bismut-Ricci form, into the real Ricci lower bound used in the variational argument. The manuscript's proof of Proposition 2.6 is not sufficiently detailed to verify this conversion: it contains an index mismatch in (2.23), omits the derivation of (2.24)-(2.27), calls the key Hermitian-matrix claim a 'similar computation', and invokes prior work rather than proving the needed identity. This is a genuine correctness risk, though not evidence of falsehood. I also independently noticed the inequality reversal in (4.9), which the reader flagged as a sign error. That error is concrete and blocks the volume comparison as written, but it is likely fixable and does not undermine the Myers or fundamental-group parts once Proposition 2.6 is secured. Because the reader's conditional verdict already reflects exactly these unresolved gaps, my stress-test does not change the verdict. The paper's main ideas are plausible and the cited prior work suggests the underlying identities may be true, but the manuscript should not be accepted without a complete, self-contained proof of Proposition 2.6 and correction of (4.9).","tokens_in":18454,"tokens_out":16307,"duration_ms":171253,"concrete_test":"Produce a complete coordinate proof of Proposition 2.6: using only (2.17), compute h^{i bar l}R^SB_{i bar j k bar l} explicitly, verify that it is a Hermitian matrix for a general Hermitian metric, and verify that h^{i bar l}R^SB_{i j k bar l} vanishes when d(omega^{n-1})=0, without invoking [WY25] or an omitted 'similar computation'. As a numerical cross-check on a nontrivial balanced non-Kahler manifold (e.g. the Iwasawa manifold with its standard balanced metric), evaluate both sides of (2.21) for a generic real vector field; if they differ, Proposition 2.6 is false and Theorem 1.3 collapses. Separately, confirm whether (4.9) should read 'Delta r <= (2n-1) sn'_K(r)/sn_K(r)'; with the corrected inequality the monotonicity of the volume-density ratio and the volume bound (1.11) follow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.3 is that the holomorphic Ricci lower bound (1.10) implies the real Strominger-Bismut Ricci lower bound Ric^SB(X,X) >= (2n-1)K|X|^2, which drives the Myers and Laplacian comparison arguments. This implication depends entirely on Proposition 2.6. The proof of Proposition 2.6 is a sketch with several unstated steps: (2.23) contains an index inconsistency (the first term pairs a holomorphic index j in R^SB_{i j k bar l} with X^{bar j}, while the second term pairs R^SB_{i bar j k bar l} with X^j); (2.24)-(2.27) are asserted without derivation from the Christoffel symbols (2.17); the balanced condition is used to claim the trace identity (2.29), but the computation is not shown; and the crucial claim that (h^{i bar l}R^SB_{i bar j k bar l}) is a Hermitian matrix is delegated to a 'similar computation' and to [WY25, Corollary 1.8]. If any of these steps fails, then (1.10) does not imply Ric^SB(gamma',gamma') >= (2n-1)K, so the index-form contradiction in the proof of Theorem 1.3(1) collapses. Even granting Proposition 2.6, the volume comparison proof contains a sign error in (4.9): the Laplacian comparison (4.6) gives Delta r <= (2n-1) sn'_K(r)/sn_K(r), but (4.9) asserts Delta r >= (2n-1) sn'_K(r)/sn_K(r). Taken literally, this inequality makes the volume-density ratio increasing, which would imply Vol(M) >= Vol(S^{2n}(1/sqrt K)), the opposite of the claimed (1.11). The error is likely a typographical sign flip, but as written the proof of Theorem 1.3(3) is not valid.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops second-variation formulas and index forms for the Strominger-Bismut connection on Hermitian manifolds, and then uses them to prove Myers-type diameter and volume comparison theorems. Theorem 1.3 is the main result: on a complete balanced Hermitian manifold, a lower bound of the form ℜ𝔦𝔠^{SB}(V,V) ≥ (2n−1)K|V|² implies compactness, diam(M) ≤ π/√K, finiteness of π₁(M), and Vol(M) ≤ Vol(S^{2n}(1/√K)). The paper also proves a holomorphic-sectional-curvature Myers theorem (Theorem 1.4) and a Weinstein-type fixed-point theorem (Theorem 1.5).","tokens_in":18868,"tokens_out":6330,"duration_ms":77715,"significance":"If Theorem 1.3 is correct, it is a substantial extension of the classical Myers and volume comparison theorems from the Kähler setting to balanced non-Kähler Hermitian manifolds, and it would establish a genuinely useful bridge between holomorphic Bismut-Ricci curvature and Riemannian geodesic analysis. The paper is honest in relying on explicit prior formulas from [LY17] and [WY25]; there are no fitted parameters or ad hoc assumptions, and the target theorems are deductions from stated curvature hypotheses. The second-variation framework itself is natural and potentially reusable. However, the decisive link between the holomorphic curvature hypothesis and the real Ricci bound is contained in a proof sketch with several unjustified steps, and the volume comparison proof as written contains an inequality with the wrong direction. These issues are load-bearing for the central claims.","major_comments":[{"comment":"This proposition is the bridge that turns the holomorphic condition (1.10) into the real curvature bound Ric^{SB}(γ′,γ′) ≥ (2n−1)K used in the index-form argument, so it must be proved completely. The present proof is a sketch with several unverified identities. First, equation (2.23) has an index inconsistency: the first term contracts R^SB_{i j k \\bar l} with X^k X^{\\bar j}, while the second contracts R^SB_{i \\bar j k \\bar l} with X^k X^j; it is not explained how either contraction is hermitian or how (2.21) follows from this expression. Second, equations (2.24)–(2.27) are asserted without derivation from the Christoffel symbols in (2.17), and the passage from ∂/∂z derivatives to the displayed ∂/∂x derivatives is not justified. Third, the 'Bochner-Kodaira formula' in (2.28) is garbled: the formula [∂*,L]=√−1(∂+[Λ,∂ω]) is not the standard identity, and the subsequent equality ∂*ω = √−1Λ(∂ω) is stated without proof; the step from balancedness to SB T^s_{sk}=0 and then to (2.29) is therefore not established, especially because the sentence 'Here we assume M is compact' leaves the complete noncompact case unresolved. Finally, the assertion that (h^{i\\bar l}R^SB_{i \\bar j k \\bar l}) is a Hermitian matrix is delegated to a 'similar computation' and to [WY25, Corollary 1.8]. Because Theorem 1.3 collapses if this proposition fails, the proof needs to be completed or the proposition should be proved in full.","section":"§2, Proposition 2.6 (Eqs. (2.22)–(2.29))"},{"comment":"The displayed inequality in (4.9) has the wrong direction for the claimed monotonicity. The Laplacian comparison in (4.6) gives Δg r ≤ (2n−1) sn′_K(r)/sn_K(r), but (4.9) states Δg r ≥ (2n−1) sn′_K(r)/sn_K(r). Consequently the displayed derivative ∂_r log(r^{2n−1}√det g) ≥ ∂_r log(sn_K^{2n−1}) makes the volume-density ratio λ(ρ,ω) increasing in ρ, which is the opposite of the 'decreasing' assertion used immediately afterwards and would lead to Vol(M) ≥ Vol(S^{2n}), contradicting the claimed (1.11). The error is likely a sign flip, but as written the proof of Theorem 1.3(3) does not go through.","section":"§4, Eq. (4.9)"},{"comment":"In the index-form contradiction, the sum over the 2n−1 parallel fields is replaced by Ric^{SB}(γ′,γ′) without comment. Since Proposition 2.6 is stated with an explicit factor of 2 and a contracted holomorphic expression, the trace identity ∑_{i=2}^{2n} R^{SB}(e_i,γ′,γ′,e_i) = Ric^{SB}(γ′,γ′) needs to be verified explicitly with the conventions of (1.8) and (2.21). This is a small but necessary step: if the factor or the ordering of arguments is different, the constant (2n−1)K in the final inequality changes. Please spell out the trace computation.","section":"§4, proof of Theorem 1.3(1)"}],"minor_comments":[{"comment":"There are typos: 'mainfold' in Theorems 1.1 and 1.2, 'Sygne' should be 'Synge', and the reference list contains a duplicated entry [FZ19]. These should be corrected.","section":"§1"},{"comment":"The notation for curvature components is confusing: the ordering of holomorphic and anti-holomorphic indices in R^SB_{i j k \\bar l} versus R^SB_{i \\bar j k \\bar l} is not defined explicitly relative to the coordinate expression in (2.3). A short table of conventions would improve readability and would help verify equations such as (2.20) and (2.23).","section":"§2"},{"comment":"In the simple-connectivity part of the proof, the variation α(t,s)=exp_{γ(t)}(sJγ′(t)) needs a little more justification: one must verify explicitly that α(0,s)=α(ℓ,s) for all s and that the boundary term in (1.4) vanishes for this loop variation. The argument is likely correct, but it is only sketched.","section":"§4, Theorem 1.4"},{"comment":"The volume comparison argument uses the cut-locus indicator χ_{Σ(p)} and the normal-coordinate volume element without a precise definition of Σ(p). Since the comparison is an important part of Theorem 1.3(3), this notation should be pinned down, for example by writing the injectivity-radius domain explicitly.","section":"§4, volume comparison"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a natural and potentially important problem, and the announced results would be of interest if the proofs are completed. My recommendation is driven by the two unresolved pillars—Proposition 2.6 and the inequality direction in (4.9)—rather than by skepticism about the overall program. The author should be encouraged to supply full proofs for Proposition 2.6 and the trace/volume computations; with those in place, the paper is likely suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea is genuine: rewrite the index form using the Strominger-Bismut connection and then run the standard Myers/Laplacian/volume arguments under a holomorphic Bismut-Ricci lower bound. Theorems 1.1 and 1.2 are clean and likely correct, and the Myers-type conclusion for balanced non-Kähler manifolds would be a real advance if the bridge holds. The paper also honestly notes that simple connectivity remains open in the balanced case, which is the right amount of caution.\n\nThe problem is the bridge. Proposition 2.6 is the entire mechanism converting the holomorphic Ricci condition (1.10) into the real Bismut-Ricci bound used along geodesics. Its proof is not convincing. The computation in (2.23) has index inconsistencies, the steps from (2.24) to (2.27) are asserted without derivation, the Bochner-Kodaira identity in (2.28) is garbled, and the claim that the relevant matrix is Hermitian is delegated to the author's prior work. This is not a stylistic quibble: if Proposition 2.6 fails, Theorem 1.3 collapses. I cannot certify that proposition from the manuscript as written.\n\nThere is also a clear sign error in (4.9). The Laplacian comparison gives Δr ≤ (2n−1) sn'/sn, but (4.9) asserts the reverse inequality. Taken literally, the volume density ratio is increasing, which would imply Vol(M) ≥ Vol(S^{2n}), the opposite of the theorem. This looks like a typographical sign flip, and it is easy to repair, but as written the proof of Theorem 1.3(3) is invalid.\n\nThe rest is more solid. The second variation computation in Theorem 1.1 follows the classical template with torsion terms that vanish for the chosen variations, and the free-homotopy argument in Theorem 1.4 is standard. The citation pattern is also fine: the author leans on his own prior work, but as tools, not as the target conclusions, and there are no fitted parameters or invented entities.\n\nWho is this for? Anyone working on Hermitian comparison geometry or on applications of the Bismut connection in non-Kähler settings. It deserves a serious referee, but the referee should insist on a complete proof of Proposition 2.6 and a corrected (4.9). If those pieces land, this is a worthwhile paper. My recommendation is to send it to peer review with a request for major revision.","headline":"A plausible and useful extension of Riemannian comparison theorems to balanced Hermitian manifolds via the Bismut connection, but the main bridge (Proposition 2.6) is underproved and a sign error in the volume comparison needs fixing.","tokens_in":740,"tokens_out":781,"would_cite":true,"duration_ms":78056,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C21","53C55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes a Myers-type theorem for complete balanced Hermitian manifolds: a lower bound on the holomorphic Strominger-Bismut Ricci curvature forces compactness, diameter $\\pi/\\sqrt{K}$, finite fundamental group, and…","keywords":["Hermitian geometry","Strominger-Bismut connection","Myers theorem","balanced manifold","comparison theorem","volume comparison","holomorphic sectional curvature","Bismut-Ricci curvature"],"falsifier":"A direct geometric test is to search for a complete balanced Hermitian manifold satisfying $\\mathfrak{Ric}^{SB}(V,V) \\ge (2n-1)K|V|^2$ that contains a minimizing geodesic longer than $\\pi/\\sqrt{K}$, has infinite fundamental group, or has volume larger than the round sphere $S^{2n}(1/\\sqrt{K})$; any one of these would refute Theorem 1.3. A narrower coordinate check is to compute both sides of the Proposition 2.6 identity, $\\mathrm{Ric}^{SB}(X,X)=2(h^{i\\ell}R^{SB}_{i j k \\ell})X^k X^j$, on an explicit balanced non-Kähler metric; a single point where the identity fails would break the bridge from the holomorphic Ricci hypothesis to the geodesic bound used in the proof.","tokens_in":18249,"feed_emoji":"📐","tokens_out":20285,"duration_ms":207349,"temperature":0.7,"pith_summary":"This paper tries to extend the classical comparison package---Myers' diameter theorem, Laplacian comparison, and Bishop-Gromov volume comparison---from Kähler manifolds to arbitrary Hermitian manifolds by replacing the Levi-Civita connection with the Strominger-Bismut connection. The main result states that on a complete balanced Hermitian manifold, a lower bound of the form $\\mathfrak{Ric}^{SB}(V,V) \\ge (2n-1)K|V|^2$ on the holomorphic Bismut-Ricci curvature forces compactness, diameter at most $\\pi/\\sqrt{K}$, finite fundamental group, and volume at most that of the round $2n$-sphere of radius $1/\\sqrt{K}$. If correct, this gives full Myers-type control over a large class of non-Kähler manifolds, where previously such conclusions were known only in the Kähler or nearly-Kähler setting. The paper also proves a Hermitian analogue of the positive-holomorphic-sectional-curvature theorem: $HSC^{SB} \\ge K > 0$ implies compactness, the same diameter bound, and simple connectivity, together with a fixed-point theorem for isometries when $HSC^{SB} > 0$. Because geodesics of the Levi-Civita and Bismut connections coincide, the classical proof strategy can be rerun with Bismut curvature and torsion terms.","feed_headline":"A Bismut-Ricci bound now yields Myers' theorem beyond Kähler","feed_subtitle":"It forces complete balanced Hermitian manifolds to be compact, with diameter, volume, and fundamental group all controlled.","key_machinery":"The machinery is the Strominger-Bismut connection $\\nabla^{SB}$, the unique metric-compatible Hermitian connection with totally skew-symmetric torsion, characterized by $g(\\nabla^{SB}_X Y, Z) = g(\\nabla^{LC}_X Y, Z) + \\tfrac{1}{2}(d\\omega_g)(JX,JY,JZ)$; its torsion tensor is $T^{SB}(X,Y,Z) = (d\\omega_g)(JX,JY,JZ)$. Because $\\nabla^{SB}_X X = \\nabla^{LC}_X X$, the geodesics of the two connections agree, so the paper rewrites the energy second variation and the index form in terms of $\\nabla^{SB}$: for a unit-speed geodesic $\\gamma$, $I_\\gamma(V,W) = \\int (\\langle \\hat{\\nabla}^{SB}_{d/dt} V, \\hat{\\nabla}^{SB}_{d/dt} W\\rangle + T^{SB}(V,\\gamma',\\hat{\\nabla}^{SB}_{d/dt} W) - R^{SB}(V,\\gamma',\\gamma',W))\\,dt + \\tfrac{1}{2}T^{SB}(V,W,\\gamma')\\big|_a^b$. On a balanced Hermitian manifold ($d\\omega^{n-1}=0$), Proposition 2.6 identifies the real Bismut-Ricci curvature along a real vector $X$ with $2(h^{i\\ell}R^{SB}_{i j k \\ell})X^k X^j$, converting the assumed holomorphic lower bound into $\\mathrm{Ric}^{SB}(\\gamma',\\gamma') \\ge (2n-1)K$ along every geodesic. This bridge lets the Myers sine-variation argument, the Jacobi-field Laplacian comparison with Strominger-Bismut parallel frames, and the monotone volume-density estimate run without a Kähler assumption.","core_discovery":"The central discovery is that comparison-theoretic consequences of positive curvature survive in non-Kähler Hermitian geometry when the curvature is read through the Strominger-Bismut connection $\\nabla^{SB}$. Theorem 1.3 is the main instance: for a complete balanced Hermitian manifold $(M,\\omega)$ of complex dimension $n$, if the holomorphic Ricci curvature satisfies $\\mathfrak{Ric}^{SB}(V,V) \\ge (2n-1)K|V|^2$ for every $V \\in T^{1,0}M$ and some $K > 0$, then $M$ is compact, $\\mathrm{diam}(M,\\omega) \\le \\pi/\\sqrt{K}$, $\\pi_1(M)$ is finite, and $\\mathrm{Vol}(M,\\omega) \\le \\mathrm{Vol}(S^{2n}(1/\\sqrt{K}), g_{\\mathrm{can}})$. The proof gives a Laplacian comparison theorem and a local Bishop-Gromov-style volume comparison under the same curvature condition, and the global sphere bound follows by letting the radius reach $\\pi/\\sqrt{K}$. Remark 4.1 notes that the same conclusions hold if the holomorphic Ricci condition is replaced by the real condition $\\mathrm{Ric}^{SB}(X,X) \\ge (2n-1)K|X|^2$ for $X \\in T_{\\mathbb{R}}M$. The paper further proves that $HSC^{SB} \\ge K > 0$ on a complete Hermitian manifold forces compactness, $\\mathrm{diam} \\le \\pi/\\sqrt{K}$, and simple connectivity, and that a compact Hermitian manifold with positive $HSC^{SB}$ has the property that every isometry of the metric has a fixed point.","pith_inferences":["The balanced hypothesis appears to enter exactly where Proposition 2.6 converts a holomorphic curvature bound into a real one; this suggests testing a complete non-balanced Hermitian metric with $\\mathfrak{Ric}^{SB}(V,V) \\ge (2n-1)K|V|^2$ and infinite diameter to see whether balance is essential.","The index form now contains a first-order torsion term $T^{SB}(V,\\gamma',\\hat{\\nabla}^{SB}_{d/dt}V)$, so the conjugate-point and Morse-index estimates for this index form may differ from the Levi-Civita ones even for the same metric; locating the first conjugate point on a torsion-dominated example would show how the torsion shifts the comparison.","The monotone volume density ratio opens an equality case the paper does not discuss: if $\\mathrm{Vol}(M,\\omega)$ equals the sphere volume, the density ratio is constant along radial directions, which should force a rigidity statement for balanced metrics that the paper leaves implicit."],"forward_implications":["A complete balanced Hermitian manifold satisfying $\\mathfrak{Ric}^{SB}(V,V) \\ge (2n-1)K|V|^2$ is compact with finite fundamental group, so it admits only finite-sheeted covers that keep the same curvature bound.","The Laplacian comparison $\\Delta r \\le (2n-1)\\mathrm{sn}'_K(r)/\\mathrm{sn}_K(r)$ holds away from the cut locus, hence every metric ball has volume no larger than the corresponding ball in the constant-curvature model.","A complete Hermitian manifold with $HSC^{SB} \\ge K > 0$ has diameter at most $\\pi/\\sqrt{K}$ and is simply connected, so no free homotopy class can be represented by a shortest closed geodesic.","A compact Hermitian manifold with $HSC^{SB} > 0$ has the isometry fixed-point property, so it cannot carry a free isometric action by any nontrivial group.","The real Ricci version, $\\mathrm{Ric}^{SB}(X,X) \\ge (2n-1)K|X|^2$ for $X \\in T_{\\mathbb{R}}M$, yields exactly the same compactness, diameter, fundamental-group, and volume conclusions."],"supporting_citations":[{"why":"It introduces the Strominger connection with skew torsion in the physical setting; the paper's curvature and geodesic arguments are built on this connection.","marker":"[Str86]"},{"why":"It defines the Bismut connection and its torsion, giving the curvature and torsion objects used throughout the variational formulas.","marker":"[Bis89]"},{"why":"It supplies the complexified Ricci-curvature computation used in the proof of Proposition 2.6, the key bridge to the real Bismut-Ricci bound.","marker":"[LY17]"},{"why":"Its Corollary 1.8 is invoked to show the matrix $h^{i\\ell}R^{SB}_{i j k \\ell}$ is Hermitian, completing Proposition 2.6.","marker":"[WY25]"},{"why":"It provides the Kähler comparison and vanishing theorems whose framework Theorem 1.3 extends to balanced Hermitian manifolds.","marker":"[NZ18]"},{"why":"It is the classical theorem on Kähler manifolds with positive holomorphic sectional curvature that Theorem 1.4 generalizes.","marker":"[Tsu57]"},{"why":"It supplies the axis-of-isometry lemma used in the proof of the isometry fixed-point theorem.","marker":"[Pet16]"}],"fun_headline_variants":["Bismut-Ricci bound yields Myers' theorem beyond Kähler","Positive Bismut-Ricci forces compactness in Hermitian manifolds","Myers' theorem survives in non-Kähler Hermitian geometry","Bismut-Ricci curvature controls diameter and volume in Hermitian case"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that on a balanced Hermitian manifold a lower bound on the holomorphic Bismut-Ricci form is genuinely a lower bound on the real Bismut-Ricci curvature along geodesic directions; if that numerical bridge gives way, the diameter, volume, and fundamental-group conclusions do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Bismut-Ricci bound yields Myers' theorem beyond Kähler","Positive Bismut-Ricci forces compactness in Hermitian manifolds","Myers' theorem survives in non-Kähler Hermitian geometry","Bismut-Ricci curvature controls diameter and volume in Hermitian case"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000192,"raw_usage":{"total_tokens":1336,"prompt_tokens":925,"completion_tokens":411,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":541,"completion_tokens_details":{"reasoning_tokens":331}},"tokens_in":541,"tokens_out":411,"duration_ms":5180,"temperature":1.0,"reasoning_tokens":331,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:45:36.889026+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct geometric test is to search for a complete balanced Hermitian manifold satisfying $\\mathfrak{Ric}^{SB}(V,V) \\ge (2n-1)K|V|^2$ that contains a minimizing geodesic longer than $\\pi/\\sqrt{K}$, has infinite fundamental group, or has volume larger than the round sphere $S^{2n}(1/\\sqrt{K})$; any one of these would refute Theorem 1.3. A narrower coordinate check is to compute both sides of the Proposition 2.6 identity, $\\mathrm{Ric}^{SB}(X,X)=2(h^{i\\ell}R^{SB}_{i j k \\ell})X^k X^j$, on an explicit balanced non-Kähler metric; a single point where the identity fails would break the bridge from the holomorphic Ricci hypothesis to the geodesic bound used in the proof.","supporting_citations":[],"review_version":1}