bibkey: he2025operatorconvexity
authors: Kerry He, James Saunderson, and Hamza Fawzi
year: 2025
title: Operator convexity along lines, self-concordance, and sandwiched Rényi entropies
doi: 10.1007/s10107-025-02314-0
url: https://arxiv.org/abs/2502.05627v1
claim: Conjecture 6.2 in arXiv v1, renumbered Conjecture 1 in the journal article, proposes the sharp compatibility parameter (2 alpha - 1)/3 for the sandwiched trace function in every finite dimension.
strata_touched:
  - D5/S3/Quantum/Matrix/HsfCubicRadial
  - D5/S3/Quantum/Matrix/RpowPencilTaylor
  - D5/S3/Quantum/Matrix/HsfBalancedSpectralCertificate
  - D5/S3/Quantum/Matrix/HsfBalancedCfcBridge
license: citation-only
triage: anchor

Sandwiched trace compatibility

For positive definite complex Hermitian matrices, the source considers

[ \Psi_\alpha(X,Y) =\operatorname{Tr}!\left[ \left(Y^{(1-\alpha)/(2\alpha)}X Y^{(1-\alpha)/(2\alpha)}\right)^\alpha \right]. ]

The conjecture covers every positive integer dimension and every (\alpha\ge2). In directional derivative form, for Hermitian directions with (X\pm H,Y\pm K\succeq0), its proposed sharp estimate is (|g’‘’(0)|\le(2\alpha-1)g’’(0)), where (g(t)=\Psi_\alpha(X+tH,Y+tK)). The journal article’s Proposition 3 proves the scalar result and its sharpness; Conjecture 1 asks for the matrix extension.

The repository’s HsfCubicRadial proves the normalized polynomial-trace estimate at (\alpha=3): for Hermitian (A,B) with (A\pm B\succeq0), the actual function (f(t)=\operatorname{Re}\operatorname{Tr}[(A+tB)^3]/(1+kt)^2) satisfies (f’‘(0)\ge0) and (|f’‘’(0)|\le(1+4|k|)f’’(0)). Over (|k|\le1), the least uniform constant is 5. Its proof is repository-derived and allows noncommuting matrices. Its physical radial application takes (K=kY), (S=Y^{-1/3}), (A=SXS), and (B=SHS). For positive definite (X,Y) and Hermitian (H) with (X\pm H\succeq0), positive-scalar CFC scaling identifies the original trace function with (f) near zero. Thus its actual second and third derivatives satisfy the same bound, including singular endpoints. General second-variable directions remain outside this radial application; other exponents are outside the existing FORMAL module scope, although the analytic radial extension below covers every real (\alpha\ge2).

Analytic radial extension (external measure theorem; not formalized)

Let (X,Y>0) be finite-dimensional complex Hermitian matrices, (\alpha\ge2) be real, (H=H^*) with (X\pm H\succeq0), and (K=kY) with (|k|\le1) (\bigl(Y\pm K\succeq0\bigr)), and put (s=(1-\alpha)/(2\alpha)), (A=Y^sXY^s), and (B=Y^sHY^s). Then (A>0), (A\pm B\succeq0), and for (|t|<1) [ g(t)=(1+kt)^{1-\alpha}\operatorname{Tr}(A+tB)^\alpha. ] Heinävaara, Tracial Joint Spectral Measures, arXiv:2310.03227v1, Theorem 1.4 and the power identity following it, supplies a positive measure (\mu_{A,B}) with (\operatorname{Tr}(A+tB)^\alpha =\alpha(\alpha+1)\int|a+tb|^\alpha,d\mu_{A,B}(a,b)). Lemma 1.6 gives half-plane support for a positive first coordinate, and Proposition 3.8(1) gives the basis-change pushforward. Applying these to (A), (A+B), and (A-B) yields the support containment (a\ge|b|) (\mu_{A,B})-almost everywhere. This is containment; no literal equality with the whole cone is claimed.

For each (\alpha), push this measure forward with the weight (a^\alpha): [ \nu_\alpha(E)=\alpha(\alpha+1)\int_{a>0}a^\alpha 1_E(b/a),d\mu_{A,B}(a,b),\qquad \nu_\alpha([-1,1])=\operatorname{Tr}A^\alpha. ] The measure is finite because its mass is (\operatorname{Tr}A^\alpha); no unweighted finiteness of (\mu_{A,B}) is assumed. The resulting (\nu_\alpha) is alpha-indexed; no single measure for different exponents is asserted. Therefore [ g(t)=\int_{[-1,1]}(1+tu)^\alpha(1+kt)^{1-\alpha},d\nu_\alpha(u). ] On every compact subinterval of ((-1,1)), the integrand and its first three derivatives are uniformly bounded (also for noninteger (\alpha) and (\alpha=2)); dominated differentiation gives (g\in C^3((-1,1))). Writing (q_u(t)=(1+tu)^\alpha(1+kt)^{1-\alpha}), scalar differentiation gives [ q_u’‘(0)=\alpha(\alpha-1)(u-k)^2,\qquad q_u’‘’(0)=q_u’‘(0)\bigl[(\alpha-2)u-(\alpha+1)k\bigr]. ] Consequently, including singular endpoints and zero Hessian, [ g’‘(0)\ge0,\qquad |g’‘’(0)|\le[(\alpha-2)+(\alpha+1)|k|]g’‘(0) \le(2\alpha-1)g’‘(0). ] If (g’‘(0)=0), positivity of the integral forces (u=k) (\nu_\alpha)-almost everywhere, hence (g’‘’(0)=0). This is an analytic consequence of the external measure theorem. The measure prerequisite and this all-alpha corollary have no Lean proof here; arbitrary independent (H,K) and the full HSF conjecture remain OPEN.

The current formal library extends this record with a separate, explicitly scoped two-level supplier. D5/S3/Quantum/Matrix/RpowPencilTaylor proves for an arbitrary finite-dimensional positive definite complex Hermitian center Y and Hermitian direction K that the actual CFC cube-root pencil is smooth near zero (contDiffAt_cubeRoot), that its first three factorial-normalized jets satisfy the ordered noncommutative cubic recurrences (cubeRoot_jet_recurrences), and that the actual two-thirds power is smooth (contDiffAt_twoThirds) with the corresponding ordered jet formula (normalized_twoThirds_jet). Its fixed spectral-coordinate interface (cubeRoot_jet_spectral) has strictly positive entrywise divisors, including repeated eigenvalues; it does not assume commutation, a spectral floor, or a derivative premise.

D5/S3/Quantum/Matrix/HsfBalancedSpectralCertificate supplies the finite weighted PSD Gram certificate for the displayed two-level frame. The two diagonal blocks may have arbitrary finite multiplicities and the off-diagonal block is an arbitrary rectangular complex matrix, subject to positivity of the full block direction. The diagonal blocks are therefore consequences of this single full-block PSD hypothesis, not independent arbitrary-PSD inputs. Singular directions and r=1 are included, and the certificate keeps all matrix multiplication orders. Its supplier_of_cubic_recurrence theorem is the repository’s balanced cubic algebraic certificate, not a claim about an arbitrary spectrum.

D5/S3/Quantum/Matrix/HsfBalancedCfcBridge.actual_twoLevel_supplier consumes the actual CFC jets and gives the stated weighted inequality in that supplied two-level frame. Its D5/S3/Quantum/Matrix/HsfBalancedCfcBridge.transported_twoLevel_supplier version permits only the supplied finite reindexing, unitary conjugation, and common positive scaling of the center and direction. It does not extract a frame from a bare eigenvalue-count premise and does not provide nonunitary congruence or arbitrary PSD diagonal blocks outside the displayed hypothesis. There is no formal three-level or arbitrary-spectrum theorem here, and this library update does not claim the full all-alpha HSF conjecture, a novelty result, or an official open-problem resolution. The primary HSF source and the original radial alpha=3 exposition above remain the citation anchor.

Verified locators

• arXiv:2502.05627v1, Conjecture 6.2 and the preceding scalar discussion.
• DOI 10.1007/s10107-025-02314-0, Proposition 3 and Conjecture 1.
• arXiv:2310.03227v1, Theorem 1.4 and its power identity, Lemma 1.6, and Proposition 3.8(1).

These sources identify the target, scalar predecessor, and analytic radial measure implication. This note does not establish bibliographic priority or exclude a later resolution.