Optimization Geometrodynamics:
A Framework for Dynamic Geometric Optimization
July 07, 2026
Most gradient-based optimization methods move parameters through a fixed background geometry, even when their internal states implicitly define changing notions of length, curvature, and preconditioning. We introduce optimization geometrodynamics, a benchmark language in which optimization is a coupled evolution of a parameter trajectory, a transported distribution of particles, and a controlled time-varying Riemannian metric. The language has three roles. First, it gives a coordinate-free language for dynamic geometric optimization through trajectory, transport, metric, and gauge equations. Second, it separates invariant obstructions from improvable geometric mismatch: positive metrics preserve critical points and Morse indices, and cannot remove global geodesic-convexity obstructions, but can alter conditioning, distributional transport, and flux away from exact critical points. Third, it introduces dynamic geometric complexity, the minimum geometric cost required to reduce an optimization difficulty observable. In the oracle benchmark model of strongly convex quadratic objectives with full positive-definite metric control, this complexity is exactly the affine-invariant distance from the relative log-spectrum to a low-condition-number set. We also analyze Hessian-matching flows, spectral Onsager relaxation, discrete exponential projection updates, gauge-invariant observables, and fixed-time local Morse-saddle flux. The paper is theory-only: its claims are formal statements with proofs, intended to provide invariants and benchmark costs against which implementable adaptive optimizers can be compared once their admissible metric families, curvature estimates, and discretization errors are specified.
Gradient methods are usually described as motion in a fixed Euclidean background. Yet many successful optimizers maintain internal states that change the geometry of the update: diagonal gradient histories in AdaGrad and Adam, Fisher or Gauss–Newton approximations in natural-gradient and K-FAC methods, Kronecker or tensor preconditioners in Shampoo, and Bregman potentials in mirror descent. These methods suggest a broader possibility: the optimizer state can be interpreted as an evolving geometry, with its own dynamics, costs, and observable effects.
This paper develops a formal benchmark language for that possibility. We call it optimization geometrodynamics. Its basic state is a triple \[(\theta_t,\mu_t,g_t),\] where \(\theta_t\) is a parameter trajectory, \(\mu_t\) is a transported distribution of particles or random initializations, and \(g_t\) is a time-varying Riemannian metric on parameter space. The objective \(f\) is a potential field. The trajectory follows the current geometry, while the geometry itself may respond to curvature, density, and energetic signals.
The goal is foundational and theoretical. We ask what a dynamic geometric theory of optimization must specify, what it can change, what it cannot change, and what it costs to improve geometric difficulty measures such as conditioning. This separation is necessary because changing a positive metric does not alter the differential \(df\). Critical sets, Morse indices, and global geodesic-convexification obstructions are therefore invariant. At the same time, metric evolution can change the rate and anisotropy of descent away from exact critical points, the transport of a distribution near saddles, and the geometric cost of moving from one preconditioning geometry to another.
The paper makes no empirical claim about a particular neural-network training recipe. Instead, it proposes a mathematical language and proves benchmark theorems inside explicitly stated model classes. This is deliberate: the first question is which claims about changing geometry are well-defined, invariant under coordinate changes, and supported by formal evidence. The full SPD model used for exact complexity is an oracle benchmark with unrestricted positive-definite metric control. Algorithmic restrictions such as diagonal, block-diagonal, Kronecker-factored, low-rank, stochastic, and finite-step metric updates are identified as follow-up specializations of the benchmark theory.
The paper makes five contributions.
We formulate optimization geometrodynamics as a benchmark trajectory–transport–metric system, with explicit positivity, locality, covariance, scale-gauge, and model-scope requirements.
We prove boundary theorems: dynamic positive metrics preserve critical sets and Morse indices, and global geodesic convexification is obstructed by non-global critical points.
We prove positive conditioning results. In strongly convex quadratic models, Hessian matching \(\dot{G}_t=H-G_t\) monotonically reduces the relative geometric condition number to one, without requiring \(G_0\) and \(H\) to commute.
We introduce dynamic geometric complexity: the minimum affine-invariant geometric cost required to reduce an optimization difficulty measure. In the oracle full SPD benchmark, the exact complexity is the distance from the relative log-spectrum to a low-width set. We also give a complete two-dimensional diagonal-family certificate showing how restricted metrics can have a finite or infinite gap relative to the full SPD benchmark.
We define gauge-invariant observables and analyze concrete geometric flows: optimal geodesic control, spectral Onsager relaxation, Hessian matching, discrete exponential projection, and fixed-time local Morse-saddle flux.
The resulting viewpoint gives a language for separating invariant obstructions from improvable geometric mismatch, and for measuring the cost of such improvement. The proofs are collected in the appendices; the main text states the framework, the key theorems, and the interpretation needed for use as benchmarks in future algorithm-specific work.
2 situates the framework relative to continuous-time optimization, Riemannian methods, adaptive preconditioning, SPD geometry, and geometric PDE ideas. 3 fixes notation. 4 defines optimization geometrodynamics. 5 proves structural limits. 6 gives conditioning mechanisms. 7 introduces exact dynamic geometric complexity. 8 compares metric flows. 9 treats observables and fixed-time local saddle flux.
Continuous-time analyses explain acceleration, momentum, and dissipation through ODE or variational models [1]–[7]. Convex and numerical optimization already treat geometry as central: oracle complexity separates information from method efficiency [8], mirror descent uses Bregman geometry [9], [10], and conditioning, preconditioning, quasi-Newton updates, and curvature approximations shape convergence analyses [11], [12]. Optimization geometrodynamics inherits this viewpoint and makes the metric itself a time-dependent object with an invariant deformation cost.
Natural gradient methods use Fisher geometry [13]; K-FAC uses Kronecker-factored curvature [14]; AdaGrad, Adam, and Shampoo maintain diagonal, moment, or tensor preconditioners [15]–[17]; and Online Newton Step and variable-metric splitting study changing matrix geometries under algorithmic restrictions [18], [19]. Riemannian optimization supplies the coordinate-free language of gradients, retractions, and Hessians [20], [21]. Our benchmark asks what invariant difficulty observable a given admissible metric family can change and how much affine-invariant deformation it spends.
Restricted families also connect to matrix equilibration and balancing, where diagonal scalings affect condition numbers [22]–[24], and to memory-efficient adaptive methods such as Adafactor and SM3 [25], [26]. The full SPD benchmark uses the standard affine-invariant geometry of positive definite matrices [27], [28]. Ricci flow, neural curvature, and Ricci–DeTurck gauge ideas provide structural analogies for metric evolution and gauge fixing [29]–[33].
The contribution is a benchmark separation among classical ingredients: invariant obstructions, reducible geometric observables, and a coordinate-invariant deformation cost. The exact full SPD complexity is an oracle lower bound; restricted certificates, such as the two-dimensional diagonal case in 6, indicate how algorithmic families can be compared against that benchmark.
This section fixes the conventions used throughout the paper. The goal is to make the framework unambiguous before introducing dynamic metric evolution.
Let \(\mathcal{M}\) be a smooth \(d\)-dimensional manifold and \(g\in\operatorname{Met}(\mathcal{M})\) a Riemannian metric. For a smooth objective \(f:\mathcal{M}\to\mathbb{R}\), the Riemannian gradient is the vector field \(\operatorname{grad}_g f\) satisfying \[g_\theta(\operatorname{grad}_g f(\theta),v)=df_\theta(v) \qquad \forall v\in T_\theta\mathcal{M}.\] The metric Hessian is the symmetric bilinear form \[\operatorname{Hess}_g f(X,Y)=X(Yf)-(\nabla^g_XY)f,\] where \(\nabla^g\) is the Levi-Civita connection. At a critical point \(df_\theta=0\), this Hessian equals the intrinsic second differential \(d^2f_\theta\), independently of the chosen positive metric.
In Euclidean coordinates with a constant metric \(G\in\mathbb{S}_{++}^d\) and a quadratic objective \[f(\theta)=\frac{1}{2}\theta^\top H\theta,\qquad H\in\mathbb{S}_{++}^d,\] the gradient flow is \[\dot{\theta}=-G^{-1}H\theta.\] The relevant linear operator is \(M=G^{-1}H\), or equivalently the generalized eigenproblem \[Hv=\lambda Gv .\] We call \[\kappa_g(f)=\frac{\lambda_{\max}(G^{-1}H)}{\lambda_{\min}(G^{-1}H)}\] the geometric condition number in the positive-curvature quadratic model. In nonlinear problems this is used locally with \(H=\nabla^2 f(\theta)\) or with a positive curvature proxy such as Fisher, Gauss–Newton, or another target tensor.
Given a positive target tensor \(Q\) and a current metric \(G\), the relative metric is \[S=Q^{-1/2}GQ^{-1/2}.\] In the quadratic benchmark \(Q=H\). The relative spectrum of \(S\) measures how far the current metric is from matching the target geometry. In particular, \(G=Q\) gives \(S=I\) and condition number \(1\).
The set \(\mathbb{S}_{++}^d\) is equipped with the affine-invariant metric whose path length is \[\operatorname{Len}(S_{[0,T]}) = \int_0^T \left\lVert S_t^{-1/2}\dot{S}_tS_t^{-1/2}\right\rVert_F\,\mathrm{d}t.\] Its geodesic distance is \[d_{\operatorname{AI}}(S_0,S_1) = \left\lVert \log(S_0^{-1/2}S_1S_0^{-1/2})\right\rVert_F.\] This distance is invariant under congruence transformations \(S\mapsto ASA^\top\) with \(A\) invertible, so it measures intrinsic deformation of positive definite metrics rather than a coordinate artifact.
Two geometric descriptions related by a time-independent diffeomorphism represent the same optimization geometry. Claims of improvement should therefore be formulated in terms of diffeomorphism-invariant or covariant quantities. There is also a scale freedom: multiplying a metric by a positive constant changes step speeds but not all shape quantities. We handle this either by using relative spectra and condition numbers, or by imposing a scale gauge such as \(\det S=1\) or fixed volume.
All formal claims in the paper are stated as definitions, theorems, propositions, or corollaries. The main text gives the mathematical objects and interpretation; complete proofs are placed in the appendix after the references.
Let \(\mathcal{M}\) be a smooth \(d\)-dimensional manifold and let \(f:\mathcal{M}\to\mathbb{R}\) be a smooth objective. A dynamic geometric optimization state is \[(\theta_t,\mu_t,g_t),\] where \(\theta_t\in\mathcal{M}\), \(\mu_t\) is a probability measure on \(\mathcal{M}\), and \(g_t\in\operatorname{Met}(\mathcal{M})\) is a Riemannian metric.
The path \(\theta_t\) is the primary optimizer state, \(\mu_t\) represents a transported population of particles or random initializations, and \(g_t\) converts \(df\) into a vector field. In algorithms, \(g_t\) may be a preconditioner, Fisher or Gauss–Newton proxy, Kronecker-factored geometry, diagonal accumulator, or another positive tensor. The objective fixes the critical set; the metric fixes descent directions, lengths, transport, and local conditioning.
The trajectory follows steepest descent in the current metric: \[\dot{\theta}_t=-\operatorname{grad}_{g_t} f(\theta_t). \label{eq:trajectory}\tag{1}\] The distribution is transported by the same velocity field \(v_t=-\operatorname{grad}_{g_t}f\). In weak form, \[\frac{\mathrm{d}}{\mathrm{d}t}\int_\mathcal{M}\varphi\,\mathrm{d}\mu_t = \int_\mathcal{M}\mathrm{d}\varphi(v_t)\,\mathrm{d}\mu_t, \qquad \forall \varphi\in C^\infty(\mathcal{M}). \label{eq:weak-transport}\tag{2}\] If \(\mu_t=\rho_t\,\mathrm{d}\operatorname{vol}_{g_t}\), then \[\partial_t\rho_t+ \operatorname{div}_{g_t}(\rho_t v_t) + \frac{1}{2}\rho_t\operatorname{tr}_{g_t}(\partial_tg_t)=0. \label{eq:density-transport}\tag{3}\] The final ingredient is a metric equation \[\partial_t g_t=\mathcal{G}(g_t,f,\rho_t), \label{eq:metric-field}\tag{4}\] where \(\mathcal{G}\) is a symmetric covariant tensor-valued operator.
Equations (1 )–(4 ) define a model class rather than a single optimizer. Different choices of \(\mathcal{G}\), admissible metric families, and discretizations produce different instances.
The equations are a formal language and benchmark template. The paper does not prove general well-posedness for arbitrary nonlinear \(\mathcal{G}\); each formal result below is stated under its own finite-dimensional, local, or fixed-time hypotheses. We use the full trajectory–transport–metric notation under the following minimal requirements:
The objective, metric path, density, and response law are smooth enough for the displayed identities. The weak transport equation is interpreted along \(v_t=-\operatorname{grad}_{g_t}f\), on compact domains, compactly supported tests, or regions with finite flux.
The metric remains in \(\operatorname{Met}(\mathcal{M})\), and \(\mathrm{d}f(\theta_t)/\mathrm{d}t\le0\). In the finite SPD examples, positivity is enforced by convex or exponential updates inside \(\mathbb{S}_{++}^d\).
Metric motion has finite energy, for example under an Ebin-type norm on a finite-volume domain, \[\int_0^T \int_\mathcal{M} \left\lVert g_t^{-1/2}(\partial_tg_t)g_t^{-1/2}\right\rVert_F^2 \,\mathrm{d}\operatorname{vol}_{g_t}\,\mathrm{d}t <\infty.\] The quadratic benchmark uses finite-dimensional affine-invariant SPD length; tensor-field benchmarks use a specified finite reference measure.
\(\mathcal{G}\) is built from tensorial data such as \(g_t\), \(df\), \(\operatorname{Hess}_{g_t}f\), \(\rho_t\), and \(\operatorname{Ric}_{g_t}\). Diffeomorphism covariance and scale freedom are handled through volume constraints, trace-free conditions, or gauge fixing.
A metric controller is evaluated relative to an admissible family, such as full SPD, diagonal, block, Kronecker, low-rank, or proxy-induced metrics. The full SPD family is an oracle lower bound for restricted controllers.
Proposition 1 (Local expressive completeness). Let \(U\subset\mathcal{M}\) be an open set on which \(df\neq0\). If \(v\) is a smooth vector field satisfying \(df(v)<0\) on \(U\), then locally on \(U\) there exists a smooth Riemannian metric \(g\) such that \[v=-\operatorname{grad}_g f.\]
This proposition, proved in 11, shows that away from critical points metric choice is locally expressive; the boundary theorems next show where that expressiveness stops.
Let \(\mathcal{N}\subset\operatorname{Met}(\mathcal{M})\) be a gauge submanifold, such as a fixed-volume or trace-free class. Given a geometric cost \(\mathcal{C}(g,f,\rho)\), define the Rayleighian over \(h\in T_g\mathcal{N}\): \[\mathcal{R}_g(h) = D_g\mathcal{C}[h] +\frac{1}{2}\left\lVert h\right\rVert_{\mathcal{H}_g}^2.\]
Theorem 1 (Constrained Onsager metric response). Assume \(T_g\mathcal{N}\) is a closed linear subspace under the Hilbert inner product \(\mathcal{H}_g\), and \(D_g\mathcal{C}\) is continuous. Then \(\mathcal{R}_g\) has a unique minimizer \[h^\star = -\operatorname{Proj}_{T_g\mathcal{N}}\operatorname{Grad}_{\mathcal{H}}\mathcal{C}(g,f,\rho).\] Consequently a principled metric law is \[\partial_tg = -\operatorname{Proj}_{T_g\mathcal{N}}\operatorname{Grad}_{\mathcal{H}}\mathcal{C}(g,f,\rho).\]
This theorem is an instantaneous variational selection rule. It assumes a specified cost, Hilbert metric, and gauge submanifold; positive definiteness must be preserved by the chosen admissible path. A minimal low-order trace-free local response template is \[\mathcal{G}_{\min} = a(df\otimes df)^\circ+ b(\operatorname{Hess}_g f)^\circ+ c\operatorname{Ric}_g^\circ , \label{eq:minimal-metric-response}\tag{5}\] where \(A^\circ=A-\operatorname{tr}_g(A)g/d\). A global potential whose Hilbert gradient produces this template may require additional integrability conditions; the paper uses it only locally and claims no general coupled PDE well-posedness.
Several familiar methods appear as special or approximate cases: \[g_t=I \quad\text{gives Euclidean gradient flow,}\] \[g_t=\nabla^2 h \quad\text{gives mirror or Bregman geometry,}\] \[g_t=F(\theta_t) \quad\text{gives natural-gradient geometry,}\] and adaptive preconditioners correspond to time-varying choices of \(g_t\). Optimization geometrodynamics begins when \(g_t\) is treated as a controlled state whose evolution is constrained, costed, and evaluated through invariant observables.
The first role of the theory is to prevent overinterpretation. Dynamic metrics can alter descent directions away from critical points, but positive metrics cannot change the basic critical structure determined by \(df\). The following theorems are simple, yet they are central for a credible dynamic-geometric theory: they identify which claims are mathematically impossible under positive metric evolution.
Theorem 2 (Critical set invariance). For any Riemannian metric \(g\), \[\operatorname{grad}_g f(\theta)=0 \quad\Longleftrightarrow\quad df_\theta=0.\] Thus the critical set of \(f\) is independent of \(g\).
Theorem 3 (Morse index invariance). Let \(\theta^\star\) be a critical point of \(f\). Then \(\operatorname{Hess}_g f(\theta^\star)\) equals the coordinate-free second differential \(d^2 f_{\theta^\star}\) as a quadratic form. Its inertia, and hence the Morse index of \(\theta^\star\), is independent of the positive metric \(g\).
Corollary 1. Positive metric evolution cannot turn a Morse saddle into a local minimum.
Theorem 4 (Geodesic-convexification obstruction). Let \((\mathcal{M},g)\) be geodesically connected. If \(f\) is geodesically convex and \(df_{\theta^\star}=0\), then \(\theta^\star\) is a global minimizer. Therefore, if \(f\) has a non-global critical point, no geodesically connected positive metric can make \(f\) globally geodesically convex.
These results rule out a strong but tempting claim: dynamic geometry cannot remove all nonconvex difficulty by changing the metric while keeping the objective fixed. Critical points and their local Morse type are properties of the objective, not of the descent geometry. The honest target of metric evolution is therefore reducible mismatch in regions where descent is already possible, together with distributional effects near invariant critical structures.
Any useful theorem about dynamic geometry should specify an observable that the metric can genuinely change. Examples include local condition numbers, transport rates, exit fluxes, curvature energies, and the geometric cost of changing from one metric to another. The remainder of the paper focuses on such quantities.
The boundary theorems say that metric evolution cannot alter critical structure. We next show that it can provably reduce a central optimization difficulty measure: the condition number in a positive-curvature region. The cleanest version is the strongly convex quadratic benchmark, where all objects are explicit and no commutativity assumption is needed.
Let \[f(\theta)=\frac{1}{2}\theta^\top H\theta,\qquad H\in\mathbb{S}_{++}^d,\] and let \(G_t\in\mathbb{S}_{++}^d\) be a spatially constant metric. The geometric Hessian operator is \(M_t=G_t^{-1}H\), with generalized eigenvalues given by \[Hv=\lambda G_t v.\] Define \(\kappa_t=\lambda_{\max}(M_t)/\lambda_{\min}(M_t)\).
Theorem 5 (Hessian matching monotonically improves conditioning). Let \(G_0\in\mathbb{S}_{++}^d\) and evolve \[\dot{G}_t=H-G_t .\] Then \(G_t\succ0\) for all \(t\ge0\), \(\kappa_t\) is monotonically nonincreasing, and \(\lim_{t\to\infty}\kappa_t=1\).
The proof uses the relative metric \[S_t=H^{-1/2}G_tH^{-1/2}=I+e^{-t}(S_0-I).\] Its eigenvalues are pulled linearly toward one, so the relative spectral width shrinks monotonically. This argument does not require \(G_0\) and \(H\) to commute in Euclidean coordinates.
Corollary 2 (Relative Hessian error controls conditioning). If \(H,G\in\mathbb{S}_{++}^d\) and \[\left\lVert H^{-1/2}(G-H)H^{-1/2}\right\rVert_{\operatorname{op}}\le\delta<1,\] then every eigenvalue of \(G^{-1}H\) lies in \[\left[\frac{1}{1+\delta},\frac{1}{1-\delta}\right],\] and \[\kappa(G^{-1}H)\le \frac{1+\delta}{1-\delta}.\]
The quadratic result becomes locally useful when a trajectory remains in a region where a positive curvature proxy changes slowly. Along such a trajectory, write \(H_t=\nabla^2 f(\theta_t)\succ0\) and consider \(\dot{G}_t=H_t-G_t\). With \(E_t=\frac{1}{2}\left\lVert G_t-H_t\right\rVert_F^2\), \[\dot{E}_t \le -E_t+\frac{1}{2}\left\lVert \dot{H}_t\right\rVert_F^2.\] Thus a slowly varying Hessian gives a controlled tracking error, which converts to a condition-number bound through 2. The proof in 12 also gives the explicit Gronwall estimate \[E_t \le e^{-t}E_0+ \frac{1}{2}\int_0^t e^{-(t-s)}\left\lVert \dot{H}_s\right\rVert_F^2\,\mathrm{d}s .\]
Hessian matching is not proposed here as a practical optimizer requiring exact Hessians. It is a positive theorem showing that dynamic geometry can reduce a real optimization difficulty observable while respecting the boundary theorems. Practical algorithms can replace \(H_t\) by Fisher, Gauss–Newton, diagonal, Kronecker-factored, or low-rank proxies; their quality can then be measured by tracking error and by the complexity gap introduced below.
Conditioning can be improved by changing the metric, but such improvement has a geometric cost. We now define this cost exactly in the full SPD oracle benchmark. The result is the paper’s main quantitative object: the minimum deformation needed to reach a target condition-number class when unrestricted positive-definite metric paths and the target curvature tensor are available.
For a strongly convex quadratic with Hessian \(H\), define the relative metric \[S=H^{-1/2}GH^{-1/2}.\] Use the affine-invariant SPD length \[\operatorname{Len}(S_{[0,T]}) = \int_0^T \left\lVert S_t^{-1/2}\dot{S}_tS_t^{-1/2}\right\rVert_F\,\mathrm{d}t,\] with distance \[d_{\operatorname{AI}}(S_0,S_1) = \left\lVert \log(S_0^{-1/2}S_1S_0^{-1/2})\right\rVert_F.\] Let \[\mathcal{C}_K=\{S\in\mathbb{S}_{++}^d:\kappa(S)\le K\}.\]
Definition 1 (Dynamic geometric complexity). For \(K\ge1\), the length-type complexity of reducing relative condition number to \(K\) is \[\mathfrak C_{\operatorname{AI}}^{\operatorname{Len}}(K;S_0) = \inf\{\operatorname{Len}(S_{[0,T]}):S_T\in\mathcal{C}_K\}.\] The action-type complexity over time \(T\) uses the squared speed integral \[\operatorname{Act}_T(S_{[0,T]}) = \int_0^T \left\lVert S_t^{-1/2}\dot{S}_tS_t^{-1/2}\right\rVert_F^2\,\mathrm{d}t.\]
Let \(s_1\le\cdots\le s_d\) be the eigenvalues of \(S_0\), and set \(y_i=\log s_i\). Let \(\ell=\log K\), and define \[D_K(S_0) = \min_{c\in\mathbb{R}} \left( \sum_{i=1}^d \operatorname{dist}(y_i,[c,c+\ell])^2 \right)^{1/2}.\]
Theorem 6 (Exact SPD dynamic geometric complexity). In the full SPD path class, \[\mathfrak C_{\operatorname{AI}}^{\operatorname{Len}}(K;S_0) = d_{\operatorname{AI}}(S_0,\mathcal{C}_K) = D_K(S_0).\] An optimal endpoint shares eigenvectors with \(S_0\) and projects the relative log-spectrum onto an interval of width \(\log K\). A constant-speed affine-invariant geodesic to that endpoint attains the optimum.
For \(d=2\), \[\mathfrak C_{\operatorname{AI}}^{\operatorname{Len}}(K;S_0) = \frac{1}{\sqrt2} \left[\log\frac{\kappa(S_0)}{K}\right]_+.\] In fixed time \(T\), the minimum action is \[\mathfrak C_{\operatorname{AI}}^{\operatorname{Act}}(K,T;S_0) = \frac{D_K(S_0)^2}{T}.\]
The cost of reducing condition number is exactly the cost of compressing the relative log-spectrum into a band of width \(\log K\); eigenvector rotation cannot lower it. The affine-invariant metric is used because congruent coordinate representations have the same cost. Computing \(D_K(S_0)\) requires a positive target tensor \(Q\), the Hessian \(H\) in the quadratic benchmark. For practical optimizers, where \(Q\) may be a Fisher, Gauss–Newton, diagonal, sampled, or other proxy, the full SPD value is an oracle lower bound and any algorithmic claim must also account for proxy-estimation error, restricted reachability, and discretization.
The full SPD result is a lower bound for implementable metric controllers. For an admissible metric family \(\mathcal{F}\subset\mathbb{S}_{++}^d\), define \[\mathcal{C}_K(\mathcal{F};H)= \{G\in\mathcal{F}:\kappa(G^{-1}H)\le K\}\] and let \(D_{K,\mathcal{F}}(G_0;H)\) be the infimum of affine-invariant lengths over paths that stay in \(\mathcal{F}\) and end in \(\mathcal{C}_K(\mathcal{F};H)\). Then \[D_{K,\mathcal{F}}(G_0;H)\ge D_K(H^{-1/2}G_0H^{-1/2}),\] with value \(+\infty\) if the restricted target is unreachable.
6 gives a complete example for the determinant-normalized two-dimensional diagonal family \[D(u)=\operatorname{diag}(e^{u/2},e^{-u/2}).\] For \(H\in\mathbb{S}_{++}^2\), diagonal reachability is an explicit interval in the one-dimensional log-scale \(u\), and the restricted complexity is the distance from \(u_0\) to that interval divided by \(\sqrt2\). This shows the intended benchmark-to-algorithm interpretation: the full SPD value is the best possible geometric cost, while diagonal restrictions can add finite excess cost or make a target \(K\) unreachable.
Appendix 12 also gives checkable block and Kronecker certificates (7 and 8). These diagnostics do not by themselves imply convergence of a finite-step stochastic optimizer.
The complexity theorem gives the optimal cost; flow laws can then be compared by how much length they spend before reaching \(\mathcal{C}_K\). The oracle geodesic to an optimal endpoint has length \(D_K(S_0)\) and action \(D_K(S_0)^2/T\) (12). Spectral Onsager relaxation contracts the log-spectrum distance to \(\mathcal{C}_K\) exponentially (13). Hessian matching is a local rule that monotonically improves condition number, while its path length may exceed the optimum (14). The gap can be summarized by \[\operatorname{Gap}_K(\mathcal{F};S_0) = \frac{\operatorname{Len}_{\mathcal{F}}(0,t_K)}{D_K(S_0)} \ge 1\] for a geometric flow \(\mathcal{F}\). The additive excess \[\operatorname{Excess}_K(\mathcal{F};S_0) = \operatorname{Len}_{\mathcal{F}}(0,t_K)-D_K(S_0)\ge0\] is also defined when \(D_K(S_0)=0\). A discrete exponential projection step contracts \(D_K\) exactly when the shared-eigenvector spectral projection is available (15); it is an idealized prototype rather than a practical optimizer. Together with the diagonal certificate in 6, these flows define a benchmark hierarchy for comparing restricted metric controllers before making empirical optimizer claims.
If \(\psi:\mathcal{N}\to\mathcal{M}\) is a diffeomorphism, the pulled-back system \[\bar f=f\circ\psi,\quad \bar g_t=\psi^\ast g_t,\quad \bar\mu_t=\psi^\ast\mu_t,\quad \psi(\bar\theta_t)=\theta_t\] is a coordinate representation of the same dynamics. Such transformations preserve objective values, metric gradient norms, transport integrals, Hessian operator spectra, curvature scalars, and geometric fluxes.
Definition 2 (Optimization-geometric observables). The observable algebra \(\mathfrak O\) is generated by gauge-invariant or gauge-covariant quantities such as \[\int_\mathcal{M}\left\lVert \operatorname{grad}_{g_t}f\right\rVert_{g_t}^2\,\mathrm{d}\mu_t,\qquad \mathcal{K}_{U,Q}(t)= \operatorname*{ess\,sup}_{\theta\in U} \kappa\!\left(g_t(\theta)^{-1}Q_t(\theta)\right),\] where \(Q_t\succ0\) is a specified positive Hessian or curvature proxy on \(U\); by the exit flux \[\mathcal{J}_t(\Sigma)= \int_\Sigma \rho_t\left[g_t(-\operatorname{grad}_{g_t}f,n_{g_t})\right]_+\,\mathrm{d}A_{g_t},\] and by curvature energies and dynamic geometric complexity.
For nonconvex regions the condition-number envelope is therefore a proxy-conditioned observable. If the Hessian is indefinite or singular on \(U\), the paper uses a positive proxy such as Fisher, Gauss–Newton, a positive Hessian restriction, or another explicitly stated tensor \(Q_t\). Without such a positive tensor, no condition-number improvement claim is made for \(\mathcal{K}_{U,Q}\).
Claims of improvement should be stated in terms of observables in \(\mathfrak O\). This prevents coordinate changes from being mistaken for optimization progress. Time-dependent gauge transformations introduce Lie-derivative directions \(\mathcal{L}_{X_t}g_t\). Ricci-type flows can be analyzed after DeTurck gauge fixing; 18 19 record the gauge direction and principal part used here.
The boundary theorems show that a positive metric cannot change the Morse index of a saddle. It can still change how a distribution leaves a saddle neighborhood.
Proposition 2 (Fixed-time local Morse-saddle flux). Fix a time \(t_0\), and let an index-one Morse saddle be centered at the origin. In local coordinates \((x,y)\in\mathbb{R}\times\mathbb{R}^{d-1}\), \[f(x,y)=f(0,0)-\frac{a}{2}x^2+\frac{1}{2} y^\top By+O(\left\lVert (x,y)\right\rVert^3), \qquad a>0,\;B\succ0.\] Assume \(g_{t_0}\) is \(C^1\) near the origin and satisfies \(g_{t_0}(0)=G_\gamma\), where \[G_\gamma=\operatorname{diag}(\gamma_-,\gamma_1,\ldots,\gamma_{d-1})\] is the frozen principal part. Let \(\rho_{t_0}\) be continuous, nonnegative, and satisfy \(\rho_{t_0}(0,0)>0\). For \(\Sigma_r^+=\{x=r,\;\left\lVert y\right\rVert\le cr\}\), the fixed-time outgoing flux satisfies \[\mathcal{J}_{t_0}(\Sigma_r^+) = \frac{ar\sqrt{\det G_\gamma}}{\gamma_-} \int_{\left\lVert y\right\rVert\le cr}\rho_{t_0}(r,y)\,\mathrm{d}y +O(r^{d+1}).\] Under the local volume normalization \(\det G_\gamma=1\), smaller unstable-direction metric weight \(\gamma_-\) gives strictly larger leading-order flux for sufficiently small \(r\), with \(f\) and \(\rho_{t_0}\) fixed.
This is a fixed-time local asymptotic statement. It does not assert long-time density regularity, well-posedness of the full coupled system, shorter escape time, or larger basin-transition probability without additional global estimates. It only isolates a local mechanism: metric changes can alter distributional flux while leaving the saddle and its Morse index intact.
Corollary 3 (Budget-controlled local flux factor). In the frozen diagonal principal model of 2, compare volume-normalized metrics \(G_\gamma\) to the identity using the affine-invariant budget \[\left\lVert \log G_\gamma\right\rVert_F\le B, \qquad \det G_\gamma=1.\] Then the leading outgoing flux multiplier \(1/\gamma_-\) is at most \[\exp\!\left(B\sqrt{\frac{d-1}{d}}\right).\] The bound is attained by \[\log\gamma_-=-B\sqrt{\frac{d-1}{d}}, \qquad \log\gamma_i=\frac{B}{\sqrt{d(d-1)}}\quad (i=1,\ldots,d-1).\]
Optimization geometrodynamics separates obstructions from reducible mismatch. Critical points, Morse indices, and global convexification barriers are invariant under positive metric changes; conditioning, distributional transport, exit flux, and the geometric cost of preconditioning are variable quantities.
The exact SPD complexity is a continuous-time, full-matrix benchmark. It should be read as a lower bound and comparison geometry; practical optimizers may only approximate this ideal object through restricted or noisy controllers. A diagonal, block, Kronecker-factored, low-rank, or stochastic controller introduces three additional errors: restriction error from the admissible metric family, estimation error from an inexact curvature proxy, and discretization error from finite steps. Bounding these terms relative to \(D_K(S_0)\) is the natural next step for turning the present framework into algorithm-specific convergence or complexity results. The diagonal two-dimensional certificate in 6 is included as a minimal worked restricted-family case: it gives an exact reachability interval and therefore exposes when the full SPD benchmark is unattainable by a diagonal controller.
In large models, the Hessian may be unavailable, indefinite, or too expensive to diagonalize. Then \(D_K(S_0)\) can be used only after choosing a positive target proxy and a spectral estimator, and should be reported together with the admissible metric family. The deterministic toy workflow sanity-checks these certificate quantities on small SPD examples; it is not neural-network training evidence.
The exact complexity theorem in 6 is a benchmark result under full SPD control. The two-dimensional diagonal result addresses only one restricted family; broader diagonal, block, Kronecker, low-rank, stochastic, finite-step, and high-dimensional neural-network regimes require additional approximation theory. In some regimes, computing or approximating the full benchmark may be more expensive than using a cheaper restricted preconditioner; the framework then serves as an evaluation baseline and a source of targets for approximation theory. The Ricci and DeTurck discussion gives the PDE entry point; a full well-posedness theorem for the complete coupled transport-metric system remains open. The saddle-flux result is a fixed-time, leading-order local asymptotic for an index-one Morse saddle with nondegenerate local quadratic structure and positive density near the critical point. It assumes local regularity of the frozen metric and density at the observation time. Degenerate saddles, flat regions, high-index saddles, and long-time density evolution under the coupled transport–metric system require separate estimates.
The framework gives a language for dynamic metric methods without overclaiming what geometry can accomplish. Positive metrics cannot change critical topology, yet they can reshape conditioning, transport, and flux in measurable ways. Dynamic geometric complexity makes this tradeoff quantitative.
This is a theory-only paper. Its reproducibility rests on explicit definitions, theorem statements, and proofs rather than empirical performance claims. The main claims are proved in [app:framework-boundary] [app:complexity] [app:flows-gauge-saddles]. No datasets, learned model checkpoints, or neural-network training experiments are used. The supplementary toy workflow contains a deterministic script and fixed configuration for computing small SPD certificates:
the two-dimensional diagonal formula in 6, checked against a brute-force grid;
the block Loewner certificate in 7, checked on a fixed block-Jacobi example; and
the Kronecker residual invariance in 8, checked under row–column spectral updates.
The script records the fixed seed and software versions in its summary output. These scripts support the benchmark definitions and do not constitute empirical optimizer evidence.
The paper develops mathematical foundations for optimization methods whose state is interpreted as evolving geometry. The immediate impact is conceptual: it clarifies which improvements are invariant, which are coordinate artifacts, and what geometric cost is spent to improve conditioning or transport. The work does not introduce a deployable learning system or a data collection process. Potential downstream risks are indirect and follow from the broad use of optimization in machine learning. A useful safeguard is the paper’s emphasis on gauge-invariant observables and explicit limitations, which discourages overstating empirical progress as a purely geometric effect.
This appendix gives the elementary geometric facts used in the main text. We state them for smooth finite-dimensional manifolds and smooth data. Boundary terms are omitted by assuming either compact support, no boundary, or the usual no-flux boundary condition.
Proposition 3 (Density form of transport). Let \(v_t\) be a smooth time-dependent vector field and let \(\mu_t=\rho_t\,\mathrm{d}\operatorname{vol}_{g_t}\). If \(\mu_t\) satisfies the weak transport equation \[\frac{\mathrm{d}}{\mathrm{d}t}\int_\mathcal{M}\varphi\,\mathrm{d}\mu_t = \int_\mathcal{M}\mathrm{d}\varphi(v_t)\,\mathrm{d}\mu_t\] for all smooth test functions \(\varphi\), then \(\rho_t\) satisfies \[\partial_t\rho_t+ \operatorname{div}_{g_t}(\rho_t v_t) +\frac{1}{2}\rho_t\operatorname{tr}_{g_t}(\partial_t g_t)=0\] in the weak sense.
Proof. The volume form of a time-varying metric satisfies \[\partial_t\,\mathrm{d}\operatorname{vol}_{g_t} = \frac{1}{2}\operatorname{tr}_{g_t}(\partial_tg_t)\,\mathrm{d}\operatorname{vol}_{g_t}.\] Therefore \[\frac{\mathrm{d}}{\mathrm{d}t}\int_\mathcal{M}\varphi\rho_t\,\mathrm{d}\operatorname{vol}_{g_t} = \int_\mathcal{M} \varphi \left( \partial_t\rho_t+ \frac{1}{2}\rho_t\operatorname{tr}_{g_t}(\partial_tg_t) \right) \mathrm{d}\operatorname{vol}_{g_t}.\] On the other hand, \[\int_\mathcal{M}\mathrm{d}\varphi(v_t)\rho_t\,\mathrm{d}\operatorname{vol}_{g_t} = -\int_\mathcal{M}\varphi\,\operatorname{div}_{g_t}(\rho_t v_t)\,\mathrm{d}\operatorname{vol}_{g_t}\] after integration by parts. Since this holds for all \(\varphi\), the claimed equation follows. ◻
Proposition 4 (Trajectory and distributional descent). For \(v_t=-\operatorname{grad}_{g_t}f\), the trajectory equation satisfies \[\frac{\mathrm{d}}{\mathrm{d}t}f(\theta_t) = -\left\lVert \operatorname{grad}_{g_t}f(\theta_t)\right\rVert_{g_t}^2\le0.\] If \(\mu_t\) is transported by the same velocity field, then \[\frac{\mathrm{d}}{\mathrm{d}t}\int_\mathcal{M}f\,\mathrm{d}\mu_t = -\int_\mathcal{M}\left\lVert \operatorname{grad}_{g_t}f\right\rVert_{g_t}^2\,\mathrm{d}\mu_t .\]
Proof. The first identity is the chain rule: \[\frac{\mathrm{d}}{\mathrm{d}t}f(\theta_t) = df_{\theta_t}(\dot{\theta}_t) = -df_{\theta_t}(\operatorname{grad}_{g_t}f) = -\left\lVert \operatorname{grad}_{g_t}f(\theta_t)\right\rVert_{g_t}^2.\] The second follows from the weak transport equation with test function \(f\). ◻
Proof of 1. It is enough to construct the metric in a coordinate neighborhood. Let \(\alpha=df\). Since \(\alpha(v)<0\), each tangent space decomposes as \[T_\theta\mathcal{M}=\operatorname{span}\{v_\theta\}\oplus\ker\alpha_\theta.\] Choose any smooth positive definite inner product \(q_\theta\) on \(\ker\alpha_\theta\). For \(a,b\in\mathbb{R}\) and \(w,z\in\ker\alpha_\theta\), define \[g_\theta(av_\theta+w,bv_\theta+z) = -\alpha_\theta(v_\theta)\,ab+q_\theta(w,z).\] This is positive definite because \(-\alpha(v)>0\). Moreover \[g_\theta(v_\theta,av_\theta+w) = -a\alpha_\theta(v_\theta) = -\alpha_\theta(av_\theta+w).\] Thus \(g(v,\cdot)=-df\), which is equivalent to \(-\operatorname{grad}_g f=v\). ◻
Proof of 2. By definition, \(\operatorname{grad}_g f(\theta)\) is the unique vector satisfying \[g_\theta(\operatorname{grad}_g f(\theta),w)=df_\theta(w) \qquad \forall w\in T_\theta\mathcal{M}.\] If \(\operatorname{grad}_g f(\theta)=0\), then \(df_\theta=0\). Conversely, if \(df_\theta=0\), nondegeneracy of \(g_\theta\) gives \(\operatorname{grad}_g f(\theta)=0\). ◻
Proof of 3. For any torsion-free connection compatible with \(g\), \[\operatorname{Hess}_g f(X,Y) = X(Yf)-(\nabla_XY)f .\] At a critical point \(\theta^\star\), \(df_{\theta^\star}=0\), so the second term vanishes at \(\theta^\star\). Hence \(\operatorname{Hess}_g f(\theta^\star)\) equals the coordinate-free second differential \(d^2f_{\theta^\star}\) as a quadratic form. The inertia of a quadratic form is intrinsic by Sylvester’s law of inertia, so the number of negative directions is independent of \(g\). ◻
Proof of 4. Let \(\gamma:[0,1]\to\mathcal{M}\) be any geodesic with \(\gamma(0)=\theta^\star\) and \(\gamma(1)=\theta\). Geodesic convexity of \(f\) implies that \(\phi(s)=f(\gamma(s))\) is convex on \([0,1]\). Since \(\theta^\star\) is critical, \(\phi'(0)=df_{\theta^\star}(\dot{\gamma}(0))=0\). For a convex one-dimensional function, \(\phi(1)\ge\phi(0)\). Thus \(f(\theta)\ge f(\theta^\star)\) for every \(\theta\), so \(\theta^\star\) is a global minimizer. Therefore the existence of a non-global critical point rules out global geodesic convexity for every geodesically connected positive metric. ◻
Proof of 1. By the Riesz representation theorem, there is a unique element \(\operatorname{Grad}_{\mathcal{H}}\mathcal{C}\) such that \[D_g\mathcal{C}[h] = \left\langle \operatorname{Grad}_{\mathcal{H}}\mathcal{C},h\right\rangle_{\mathcal{H}_g}.\] Restricted to the closed Hilbert subspace \(T_g\mathcal{N}\), the Rayleighian is the strictly convex quadratic \[\mathcal{R}_g(h) = \left\langle \operatorname{Grad}_{\mathcal{H}}\mathcal{C},h\right\rangle_{\mathcal{H}_g} +\frac{1}{2}\left\lVert h\right\rVert_{\mathcal{H}_g}^2 .\] The first-order optimality condition is \[\left\langle \operatorname{Grad}_{\mathcal{H}}\mathcal{C}+h^\star,\xi\right\rangle_{\mathcal{H}_g}=0 \qquad \forall \xi\in T_g\mathcal{N} .\] Thus \(h^\star\) is the negative orthogonal projection of the Hilbert gradient onto \(T_g\mathcal{N}\): \[h^\star = -\operatorname{Proj}_{T_g\mathcal{N}}\operatorname{Grad}_{\mathcal{H}}\mathcal{C}.\] ◻
Corollary 4 (Metric substep dissipation). If \(f\) and \(\rho\) are held fixed during the metric substep, the Onsager metric response satisfies \[\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{C}(g_t,f,\rho) = - \left\lVert \operatorname{Proj}_{T_{g_t}\mathcal{N}} \operatorname{Grad}_{\mathcal{H}}\mathcal{C}(g_t,f,\rho) \right\rVert_{\mathcal{H}_{g_t}}^2 \le0.\]
Proof. Substitute \(\partial_tg_t=-\operatorname{Proj}_{T_{g_t}\mathcal{N}} \operatorname{Grad}_{\mathcal{H}}\mathcal{C}\) into \[\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{C}(g_t,f,\rho) = D_g\mathcal{C}[\partial_tg_t].\] The projection is self-adjoint and idempotent, giving the stated negative square norm. ◻
This appendix proves the conditioning and dynamic geometric complexity results used in the main text. Throughout this appendix \(H,G,S\in\mathbb{S}_{++}^d\), and \[S=H^{-1/2}GH^{-1/2}\] denotes the relative metric.
Proof of 5. The linear matrix ODE has solution \[G_t=H+e^{-t}(G_0-H)=(1-e^{-t})H+e^{-t}G_0 .\] Thus \(G_t\succ0\) for all \(t\ge0\). In relative coordinates, \[S_t=H^{-1/2}G_tH^{-1/2} = I+e^{-t}(S_0-I).\] Since \(S_t\) is an affine function of \(S_0\), the matrices \(S_t\) share the same eigenvectors. If \(s_1(0)\le\cdots\le s_d(0)\) are the eigenvalues of \(S_0\), then \[s_i(t)=1+e^{-t}(s_i(0)-1).\] The generalized eigenvalues of \(G_t^{-1}H\) are the eigenvalues of \(S_t^{-1}\), so \[\kappa_t=\frac{s_d(t)}{s_1(t)}.\] Writing \(u=e^{-t}\), for \(a\ge b>0\), \[\frac{\mathrm{d}}{\mathrm{d}u} \frac{1+u(a-1)}{1+u(b-1)} = \frac{a-b}{(1+u(b-1))^2}\ge0.\] As \(t\) increases, \(u\) decreases, hence \(s_d(t)/s_1(t)\) is nonincreasing. Since every \(s_i(t)\to1\), \(\kappa_t\to1\). ◻
Proof of 2. The assumption is equivalent to the Loewner bounds \[(1-\delta)H\preceq G\preceq(1+\delta)H .\] For every nonzero \(v\), \[\frac{1}{1+\delta} \le \frac{v^\top Hv}{v^\top Gv} \le \frac{1}{1-\delta}.\] The Courant–Fischer characterization of generalized eigenvalues then gives the same interval for every eigenvalue of \(G^{-1}H\), and the condition-number bound follows. ◻
Proposition 5 (Slowly varying Hessian tracking). Let \(H_t\in\mathbb{S}_{++}^d\) be smooth and let \(\dot{G}_t=H_t-G_t\). Define \[E_t=\frac{1}{2}\left\lVert G_t-H_t\right\rVert_F^2 .\] Then \[\dot{E}_t = -\left\lVert G_t-H_t\right\rVert_F^2-\left\langle G_t-H_t,\dot{H}_t\right\rangle_F \le -E_t+\frac{1}{2}\left\lVert \dot{H}_t\right\rVert_F^2 .\] Consequently \[E_t \le e^{-t}E_0+ \frac{1}{2}\int_0^t e^{-(t-s)}\left\lVert \dot{H}_s\right\rVert_F^2\,\mathrm{d}s .\] If \(H_t\succeq mI\) and \(\delta_t=\sqrt{2E_t}/m<1\), then \[\kappa(G_t^{-1}H_t)\le\frac{1+\delta_t}{1-\delta_t}.\]
Proof. Differentiating \(E_t\) gives \[\dot{E}_t = \left\langle G_t-H_t,\dot{G}_t-\dot{H}_t\right\rangle_F = -\left\lVert G_t-H_t\right\rVert_F^2-\left\langle G_t-H_t,\dot{H}_t\right\rangle_F .\] Young’s inequality gives \[-\left\langle A,B\right\rangle_F\le\frac{1}{2}\left\lVert A\right\rVert_F^2+\frac{1}{2}\left\lVert B\right\rVert_F^2,\] hence \(\dot{E}_t\le -E_t+\frac{1}{2}\left\lVert \dot{H}_t\right\rVert_F^2\). Gronwall’s inequality gives the integral bound. Finally, \[\left\lVert H_t^{-1/2}(G_t-H_t)H_t^{-1/2}\right\rVert_{\operatorname{op}} \le \frac{\left\lVert G_t-H_t\right\rVert_{\operatorname{op}}}{m} \le \frac{\left\lVert G_t-H_t\right\rVert_F}{m} = \delta_t,\] and 2 applies. ◻
Lemma 1 (Spectral lower bound for affine-invariant distance). Let \(S,T\in\mathbb{S}_{++}^d\), and let \(y(S)\) and \(y(T)\) be their sorted log-eigenvalue vectors. Then \[d_{\operatorname{AI}}(S,T)\ge\left\lVert y(S)-y(T)\right\rVert_2 .\] If \(S\) and \(T\) are simultaneously orthogonally diagonalizable with aligned eigenvalue ordering, equality holds.
Proof. We prove the bound by showing that the sorted log-spectrum map is 1-Lipschitz from \((\mathbb{S}_{++}^d,d_{\operatorname{AI}})\) to Euclidean space. Let \(A_t\in\mathbb{S}_{++}^d\) be a \(C^1\) path and let \[y(t)=(\log\lambda_1(A_t),\ldots,\log\lambda_d(A_t))\] be the nondecreasing sorted log-eigenvalue vector. The eigenvalue functions of a \(C^1\) symmetric matrix path are locally Lipschitz and differentiable almost everywhere; this is the standard Hermitian perturbation theorem [34]. At a differentiability time, group the equal eigenvalues of \(A_t\). If \(\lambda_r\) is one such eigenvalue and \(U_r\) is an orthonormal basis for its eigenspace, then the derivatives of the eigenvalues in this cluster are the eigenvalues of \[U_r^\top \dot{A}_t U_r .\] Therefore the derivatives of the corresponding log-eigenvalues are the eigenvalues of \[\lambda_r^{-1}U_r^\top\dot{A}_tU_r .\] In an eigenbasis of \(A_t\), the matrix \[B_t=A_t^{-1/2}\dot{A}_tA_t^{-1/2}\] has diagonal blocks \(\lambda_r^{-1}U_r^\top\dot{A}_tU_r\). Hence \[\left\lVert \dot{y}(t)\right\rVert_2^2 = \sum_r \left\lVert \lambda_r^{-1}U_r^\top\dot{A}_tU_r\right\rVert_F^2 \le \left\lVert B_t\right\rVert_F^2 .\] This argument includes repeated eigenvalues; crossings only occur on a measure-zero set for the absolutely continuous sorted eigenvalue curves.
Consequently, for every piecewise \(C^1\) path from \(S\) to \(T\), \[\left\lVert y(T)-y(S)\right\rVert_2 \le \int_0^1\left\lVert \dot{y}(t)\right\rVert_2\,\mathrm{d}t \le \int_0^1\left\lVert A_t^{-1/2}\dot{A}_tA_t^{-1/2}\right\rVert_F\,\mathrm{d}t .\] Taking the infimum over paths gives \[\left\lVert y(T)-y(S)\right\rVert_2\le d_{\operatorname{AI}}(S,T),\] because the affine-invariant distance is the geodesic length distance on \(\mathbb{S}_{++}^d\) [27], [28]. This is equivalent to the logarithmic Lidskii/Hoffman–Wielandt spectral variation bound; the path proof above makes the noncommuting and repeated-eigenvalue cases explicit.
If \(S\) and \(T\) commute and their eigenvalues are aligned, then \[S^{-1/2}TS^{-1/2} = \operatorname{diag}(t_1/s_1,\ldots,t_d/s_d),\] so the distance is exactly \[\left(\sum_i(\log t_i-\log s_i)^2\right)^{1/2}.\] ◻
Proof of 6. Let \(y_i=\log s_i\) be the sorted log-spectrum of \(S_0\), and let \(\ell=\log K\). For any \(S\in\mathcal{C}_K\), the sorted log-spectrum \(z_i=\log\lambda_i(S)\) has width at most \(\ell\), so there is a \(c\in\mathbb{R}\) such that \(z_i\in[c,c+\ell]\) for every \(i\). By 1, \[d_{\operatorname{AI}}(S_0,S)\ge\left\lVert y-z\right\rVert_2 .\] Minimizing over all such \(z\) gives \[d_{\operatorname{AI}}(S_0,\mathcal{C}_K)\ge \min_{c\in\mathbb{R}} \left( \sum_{i=1}^d \operatorname{dist}(y_i,[c,c+\ell])^2 \right)^{1/2} = D_K(S_0).\] Conversely, choose a minimizing interval \([c_\star,c_\star+\ell]\), which exists because the objective is continuous and grows outside a bounded range containing the \(y_i\)’s. Let \[z_i=\operatorname{Proj}_{[c_\star,c_\star+\ell]}(y_i),\] and define \[S_\star=U\operatorname{diag}(e^{z_1},\ldots,e^{z_d})U^\top\] when \[S_0=U\operatorname{diag}(e^{y_1},\ldots,e^{y_d})U^\top .\] Then \(S_\star\in\mathcal{C}_K\), \(S_\star\) shares eigenvectors with \(S_0\), and 1 is an equality: \[d_{\operatorname{AI}}(S_0,S_\star)=D_K(S_0).\] The infimum over path lengths equals the distance to the target set because the affine-invariant metric is geodesic and the displayed geodesic from \(S_0\) to \(S_\star\) has length \(d_{\operatorname{AI}}(S_0,S_\star)\). ◻
The full SPD benchmark is a lower bound for restricted metric families. The following two-dimensional calculation gives a complete diagonal-family certificate in determinant gauge.
Proposition 6 (Two-dimensional diagonal exact complexity). Let \[H= \begin{pmatrix} h_{11} & h_{12}\\ h_{12} & h_{22} \end{pmatrix}\in\mathbb{S}_{++}^2\] and restrict metrics to \[D(u)=\operatorname{diag}(e^{u/2},e^{-u/2}),\qquad u\in\mathbb{R}.\] The diagonal affine-invariant length is \[\operatorname{Len}_{\operatorname{diag}}(u_{[0,T]}) = \frac{1}{\sqrt2}\int_0^T|\dot{u}_t|\,\mathrm{d}t .\] For \(K\ge1\), set \[R_K=\frac{K+1}{\sqrt K}\sqrt{\det H}.\] If \[R_K^2<4h_{11}h_{22},\] then no determinant-normalized diagonal metric reaches \(\kappa(D(u)^{-1}H)\le K\), and the restricted complexity is \(+\infty\). Otherwise define \[x_\pm(K) = \frac{R_K\pm\sqrt{R_K^2-4h_{11}h_{22}}}{2h_{22}}, \qquad u_\pm(K)=2\log x_\pm(K).\] The feasible set is the interval \[\mathcal{I}_K(H)=[u_-(K),u_+(K)],\] and the restricted diagonal complexity from \(D(u_0)\) is \[D^{2D}_{K,\operatorname{diag}}(u_0;H) = \frac{1}{\sqrt2}\operatorname{dist}(u_0,\mathcal{I}_K(H)).\]
Proof. Since the family is diagonal and commuting, \[d_{\operatorname{AI}}(D(u_0),D(u_1)) = \left\lVert \log D(u_1)-\log D(u_0)\right\rVert_F = \frac{|u_1-u_0|}{\sqrt2},\] which gives the line element. Let \[A(u)=D(u)^{-1/2}HD(u)^{-1/2}.\] Then \[\det A(u)=\det H\] and \[\operatorname{tr}A(u)=h_{11}e^{-u/2}+h_{22}e^{u/2}.\] If the eigenvalues of \(A(u)\) are \(0<\mu\le\nu\), then \(\nu/\mu\le K\) with fixed product \(\mu\nu=\det H\) is equivalent to \[\mu+\nu\le \left(\sqrt K+\frac{1}{\sqrt K}\right)\sqrt{\det H} = R_K .\] Writing \(x=e^{u/2}>0\), the condition becomes the scalar quadratic \[h_{22}x^2-R_Kx+h_{11}\le0 .\] If its discriminant is negative, the feasible set is empty. Otherwise the positive roots are \(x_-(K)\) and \(x_+(K)\), so the feasible \(u\)-values form \([2\log x_-(K),2\log x_+(K)]\). The distance formula follows from the one-dimensional line element. ◻
Proposition 7 (Block Loewner certificate). Let \(H,G\in\mathbb{S}_{++}^d\) and \(K\ge1\). Then \[\kappa(G^{-1}H)\le K\] if and only if there exists \(a>0\) such that \[aG\preceq H\preceq KaG .\] Equivalently, if \[\lambda_{\min} = \lambda_{\min}(G^{-1/2}HG^{-1/2}), \qquad \lambda_{\max} = \lambda_{\max}(G^{-1/2}HG^{-1/2}),\] then \(a=\lambda_{\min}\) gives the tight certificate \[aG\preceq H\preceq \frac{\lambda_{\max}}{\lambda_{\min}}aG .\] When \(G\) is constrained to a block-diagonal family, this provides a checkable block-restricted reachability certificate for the target condition class.
Proof. The Loewner sandwich \(aG\preceq H\preceq KaG\) is equivalent, after congruence by \(G^{-1/2}\), to \[aI\preceq G^{-1/2}HG^{-1/2}\preceq KaI .\] Thus every generalized eigenvalue of \(G^{-1}H\) lies in \([a,Ka]\), and the condition number is at most \(K\). Conversely, if the condition number is at most \(K\), choose \(a=\lambda_{\min}(G^{-1/2}HG^{-1/2})\). Then the largest eigenvalue is at most \(Ka\), which gives the displayed sandwich. ◻
Proposition 8 (Kronecker spectral residual invariance). Let \(Y\in\mathbb{R}^{m\times n}\), and define its double-centered residual \[R(Y)_{ij} = Y_{ij} -\bar Y_{i\cdot} -\bar Y_{\cdot j} +\bar Y_{\cdot\cdot},\] where \(\bar Y_{i\cdot}\), \(\bar Y_{\cdot j}\), and \(\bar Y_{\cdot\cdot}\) are the row, column, and grand means. For any \(\alpha\in\mathbb{R}^m\) and \(\beta\in\mathbb{R}^n\), \[R(Y+\alpha\mathbf{1}^\top+\mathbf{1}\beta^\top)=R(Y).\] Thus in a Kronecker log-spectral model, additive row–column updates cannot change the double-centered residual.
Proof. For \(Y'_{ij}=Y_{ij}+\alpha_i+\beta_j\), the row, column, and grand means are \[\bar Y'_{i\cdot} = \bar Y_{i\cdot}+\alpha_i+\bar\beta, \qquad \bar Y'_{\cdot j} = \bar Y_{\cdot j}+\bar\alpha+\beta_j, \qquad \bar Y'_{\cdot\cdot} = \bar Y_{\cdot\cdot}+\bar\alpha+\bar\beta .\] Substituting these identities into the definition of \(R(Y')\) cancels all terms involving \(\alpha\) and \(\beta\), leaving \(R(Y')=R(Y)\). ◻
Corollary 5 (Two-dimensional formula). For \(d=2\), \[d_{\operatorname{AI}}(S_0,\mathcal{C}_K) = \frac{1}{\sqrt2} \left[\log\frac{\kappa(S_0)}{K}\right]_+ .\]
Proof. Let \(w=y_2-y_1=\log\kappa(S_0)\). If \(w\le\ell=\log K\), the distance is zero. If \(w>\ell\), the closest interval of width \(\ell\) keeps its center at \((y_1+y_2)/2\), so the two endpoints move by \((w-\ell)/2\) in opposite directions. The squared distance is \[2\left(\frac{w-\ell}{2}\right)^2,\] which gives the formula. ◻
Proposition 9 (Volume-normalized complexity). Assume \(\det S_0=1\). In the submanifold \(\det S=1\), the minimum affine-invariant length needed to reach \(\kappa(S)\le K\) is \[\operatorname{dist}_{\ell^2}(y,\mathcal{P}_K), \qquad \mathcal{P}_K= \{z\in\mathbb{R}^d:\sum_i z_i=0,\;\max_i z_i-\min_i z_i\le\log K\}.\]
Proof. The constraint \(\det S=1\) is exactly \(\sum_i\log\lambda_i(S)=0\). Thus the admissible target log-spectra form the closed convex set \(\mathcal{P}_K\). The same spectral lower bound gives the lower bound, and the endpoint sharing eigenvectors with \(S_0\) and having log-spectrum equal to the Euclidean projection of \(y\) onto \(\mathcal{P}_K\) attains it. ◻
Proposition 10 (Fixed-time action complexity). Let \(D_K=d_{\operatorname{AI}}(S_0,\mathcal{C}_K)\). If action is the squared-speed integral \[\operatorname{Act}_T(S_{[0,T]}) = \int_0^T \left\lVert S_t^{-1/2}\dot{S}_tS_t^{-1/2}\right\rVert_F^2\,\mathrm{d}t,\] then the minimum action needed to reach \(\mathcal{C}_K\) by time \(T\) is \[\frac{D_K^2}{T}.\]
Proof. Every feasible path has length at least \(D_K\). By Cauchy–Schwarz, \[\operatorname{Len}(S_{[0,T]})^2 \le T\operatorname{Act}_T(S_{[0,T]}).\] Thus \(\operatorname{Act}_T\ge D_K^2/T\). A constant-speed affine-invariant geodesic from \(S_0\) to an optimal endpoint in \(\mathcal{C}_K\) attains equality. ◻
Proposition 11 (Tensor-field benchmark). Let \(U\subset\mathcal{M}\), let \(Q\) be a smooth positive definite target tensor on \(U\), and define the pointwise relative metric \[S_0(\theta)=Q(\theta)^{-1/2}g_0(\theta)Q(\theta)^{-1/2}.\] Let \(D_K(\theta)=d_{\operatorname{AI}}(S_0(\theta),\mathcal{C}_K)\). In the fully actuated benchmark that allows all measurable positive tensor paths and uses the \(L^2\) length \[\operatorname{Len}_U(g_{[0,T]}) = \int_0^T \left( \int_U \left\lVert g_t^{-1/2}\partial_tg_tg_t^{-1/2}\right\rVert_F^2 \,\mathrm{d}\nu \right)^{1/2} \mathrm{d}t,\] the minimum length required to make the pointwise condition-number envelope at most \(K\) is \[\left(\int_U D_K(\theta)^2\,\mathrm{d}\nu\right)^{1/2}.\] If the pointwise optimal endpoint can be chosen smoothly, the same value is attained within smooth tensor paths.
Proof. For every feasible path and every \(\theta\), \[D_K(\theta) \le \int_0^T \left\lVert g_t^{-1/2}\partial_tg_tg_t^{-1/2}\right\rVert_F(\theta)\,\mathrm{d}t .\] Taking the \(L^2(U,\mathrm{d}\nu)\) norm and applying Minkowski’s integral inequality gives the lower bound. The upper bound is obtained by moving each point along a constant-speed affine-invariant geodesic from \(g_0(\theta)\) to a pointwise optimal endpoint. This is a fully actuated benchmark; restrictions such as diagonal, low-rank, block, or smoothness-limited metric families can only increase the minimum. ◻
Proposition 12 (Optimal geodesic control). Let \(S_\star\in\mathop{\mathrm{arg\,min}}_{S\in\mathcal{C}_K}d_{\operatorname{AI}}(S_0,S)\), and set \[A_\star=\log(S_0^{-1/2}S_\star S_0^{-1/2}).\] The path \[S_t = S_0^{1/2}\exp\!\left(\frac{t}{T}A_\star\right)S_0^{1/2}\] has length \(D_K(S_0)\) and action \(D_K(S_0)^2/T\).
Proof. This is the affine-invariant geodesic from \(S_0\) to \(S_\star\). Its speed is constant and equal to \(\left\lVert A_\star\right\rVert_F/T\). Since \(\left\lVert A_\star\right\rVert_F=d_{\operatorname{AI}}(S_0,S_\star)=D_K(S_0)\), the length and action follow by integration. ◻
Proposition 13 (Spectral Onsager relaxation). Let \(\mathcal{Y}_K=\{y:\max_i y_i-\min_i y_i\le\log K\}\), and let \(p=\Pi_{\mathcal{Y}_K}(y_0)\). The gradient flow \[\dot{y}_t=-(y_t-\Pi_{\mathcal{Y}_K}(y_t))\] has solution \(y_t=p+e^{-t}(y_0-p)\). Consequently \[\operatorname{dist}(y_t,\mathcal{Y}_K)=e^{-t}\operatorname{dist}(y_0,\mathcal{Y}_K),\] \[\operatorname{Len}(y_{[0,T]})=(1-e^{-T})\operatorname{dist}(y_0,\mathcal{Y}_K),\] and \[\int_0^T\left\lVert \dot{y}_t\right\rVert_2^2\,\mathrm{d}t = \frac{1}{2}(1-e^{-2T})\operatorname{dist}(y_0,\mathcal{Y}_K)^2 .\]
Proof. For a closed convex set \(C\), if \(p=\Pi_C(y_0)\), then \(p=\Pi_C(p+\alpha(y_0-p))\) for every \(\alpha\ge0\). This follows from the normal-cone characterization \[\left\langle y_0-p,z-p\right\rangle\le0,\qquad z\in C.\] Thus the projection remains \(p\) along the ray, and the ODE reduces to \[\dot{y}_t=-(y_t-p), \qquad y_t=p+e^{-t}(y_0-p).\] The distance, length, and squared-speed integral are immediate. ◻
Proposition 14 (Hessian-matching complexity gap). For Hessian matching \(S_t=I+e^{-t}(S_0-I)\), let \[t_K=\inf\{t\ge0:\kappa(S_t)\le K\}.\] If \(t_K<\infty\), then \[D_K(S_0) \le \operatorname{Len}_{\operatorname{HM}}(0,t_K) \le \sum_i |\log s_i(0)|,\] where \(s_i(0)\) are the eigenvalues of \(S_0\).
Proof. The lower bound is optimality of \(D_K(S_0)\). The flow preserves the eigenvectors of \(S_0\), and \[s_i(t)=1+e^{-t}(s_i(0)-1).\] Writing \(y_i(t)=\log s_i(t)\), the affine-invariant length is \[\int_0^{t_K} \left(\sum_i \dot{y}_i(t)^2\right)^{1/2}\mathrm{d}t \le \sum_i\int_0^{t_K}|\dot{y}_i(t)|\,\mathrm{d}t.\] Each \(s_i(t)\) moves monotonically toward \(1\), hence each \(y_i(t)\) moves monotonically toward \(0\). Therefore \[\int_0^{t_K}|\dot{y}_i(t)|\,\mathrm{d}t \le |y_i(0)|=|\log s_i(0)|.\] ◻
Proposition 15 (Discrete exponential projection). Let \(P_K(S_k)\) be the shared-eigenvector spectral projection of \(S_k\) onto \(\mathcal{C}_K\), and let \(\alpha\in(0,1]\). Define \[S_{k+1} = S_k^{1/2} \exp\!\left( \alpha\log(S_k^{-1/2}P_K(S_k)S_k^{-1/2}) \right) S_k^{1/2}.\] Then \[D_K(S_{k+1})=(1-\alpha)D_K(S_k), \qquad d_{\operatorname{AI}}(S_k,S_{k+1})=\alpha D_K(S_k).\]
Proof. In the shared eigenbasis, write \(y_k\) for the log-spectrum of \(S_k\) and \(z_k=\Pi_{\mathcal{Y}_K}(y_k)\). The update is \[y_{k+1}=(1-\alpha)y_k+\alpha z_k.\] By the same normal-cone argument used in 13, \(z_k=\Pi_{\mathcal{Y}_K}(y_{k+1})\). Thus \[D_K(S_{k+1}) = \left\lVert y_{k+1}-z_k\right\rVert_2 = (1-\alpha)\left\lVert y_k-z_k\right\rVert_2.\] The affine-invariant single-step length is \[\left\lVert y_{k+1}-y_k\right\rVert_2 = \alpha\left\lVert y_k-z_k\right\rVert_2.\] ◻
Proposition 16 (Time-independent gauge invariance). Let \(\psi:\mathcal{N}\to\mathcal{M}\) be a diffeomorphism, and define \[\bar f=f\circ\psi,\qquad \bar g_t=\psi^\ast g_t,\qquad \bar\mu_t=\psi^\ast\mu_t,\qquad \psi(\bar\theta_t)=\theta_t .\] Then the pulled-back system preserves the gradient-flow equation, objective values, gradient norms, distributional descent integrals, Hessian-operator spectra, scalar curvature, volume, and geometrically defined fluxes.
Proof. The Riemannian gradient is natural under pullback: \[d\psi(\operatorname{grad}_{\bar g_t}\bar f) = \operatorname{grad}_{g_t}f .\] Therefore \[\dot{\bar\theta}_t=-\operatorname{grad}_{\bar g_t}\bar f(\bar\theta_t)\] is equivalent to \(\dot{\theta}_t=-\operatorname{grad}_{g_t}f(\theta_t)\). The identities \[\bar f(\bar\theta_t)=f(\theta_t), \qquad \left\lVert \operatorname{grad}_{\bar g_t}\bar f\right\rVert_{\bar g_t}^2 = \psi^\ast\left\lVert \operatorname{grad}_{g_t}f\right\rVert_{g_t}^2\] follow immediately. Integrals are preserved by change of variables and the definition of the pulled-back measure. Hessian operators, scalar curvature, unit normals, induced area measures, and volume forms are all natural tensorial objects under pullback, so their spectra and associated flux integrals are unchanged. ◻
Proposition 17 (Observable closure). Any scalar quantity generated from gauge-invariant scalar fields by finite algebraic operations, composition with scalar functions, time integration, spatial integration, suprema or infima over corresponding regions, and geometric flux integration is again invariant under time-independent gauge transformations.
Proof. Each listed operation preserves equality of corresponding scalar quantities under pullback. Region restrictions and hypersurface fluxes are matched by the diffeomorphism, and the induced measure and normal transform naturally. ◻
Proposition 18 (Time-dependent gauge direction). If \(\psi_t\) is a time-dependent diffeomorphism and \[X_t=\dot{\psi}_t\circ\psi_t^{-1},\] then \[\partial_t(\psi_t^\ast g_t) = \psi_t^\ast(\partial_tg_t+\mathcal{L}_{X_t}g_t).\] Thus \(\mathcal{L}_{X_t}g_t\) is the pure gauge direction introduced by moving coordinates.
Proof. This is the standard differentiation formula for time-dependent pullbacks. It follows by differentiating \(g_t(d\psi_t\cdot,d\psi_t\cdot)\) and identifying the terms arising from \(\dot{\psi}_t\) with the Lie derivative. ◻
Proposition 19 (Ricci–DeTurck principal part). Fix a smooth background metric \(\bar g\), and let \[W^k(g,\bar g)=g^{pq}\bigl(\Gamma^k_{pq}(g)-\Gamma^k_{pq}(\bar g)\bigr).\] Consider a gauge-fixed metric equation \[\partial_tg = -2\lambda\operatorname{Ric}_g+\lambda\mathcal{L}_{W(g,\bar g)}g+L(g,f,\rho), \qquad \lambda>0,\] using the displayed sign convention. If \(L\) contains only \(df\otimes df\), \(\operatorname{Hess}_g f\), \(\rho g\), and terms with at most first derivatives of \(g\), then the highest-order part is \[\partial_tg_{ij} = \lambda g^{pq}\partial_p\partial_q g_{ij} +\text{lower-order terms}.\]
Proof. In local coordinates, \(\operatorname{Ric}_g\) contains second derivatives of \(g\) with a degeneracy caused by diffeomorphism invariance. The DeTurck Lie-derivative term cancels the degenerate gauge part, leaving the Laplace-type principal symbol \(g^{pq}\partial_p\partial_q g_{ij}\), up to the positive factor \(\lambda\). The tensor \(df\otimes df\) contains no derivatives of \(g\), while \(\operatorname{Hess}_g f\) contains the Christoffel symbols and therefore at most first derivatives of \(g\). These terms do not alter the second-order principal symbol. ◻
Proof of 2. Fix \(t_0\) and write \(g=g_{t_0}\), \(\rho=\rho_{t_0}\). The statement is a fixed-time asymptotic calculation. In the chosen coordinates, \[g(z)=G_\gamma+O(\left\lVert z\right\rVert),\qquad z=(x,y),\] with the error controlled by the local \(C^1\) norm of \(g\). The constants in the \(O(\cdot)\) terms below depend on local bounds for \(g\), \(g^{-1}\), \(\nabla g\), the \(C^3\) norm of \(f\), and a local bound for \(\rho\).
From the Morse expansion, \[\partial_x f(x,y)=-ax+O(\left\lVert (x,y)\right\rVert^2).\] On \(\Sigma_r^+=\{x=r,\left\lVert y\right\rVert\le cr\}\), the normal flux density for the frozen principal metric is \[g(-\operatorname{grad}_g f,n_g)\,\mathrm{d}A_g = -\frac{\sqrt{\det G_\gamma}}{\gamma_-}\partial_x f(r,y)\,\mathrm{d}y +O(r^2)\,\mathrm{d}y .\] The \(O(r^2)\) term collects the quadratic remainder in \(f\), the first-order spatial variation of \(g\), and the corresponding variation of the unit normal and induced area element. Since the leading normal velocity is positive for sufficiently small \(r\), the positive part does not change the leading term. Integrating over \(\left\lVert y\right\rVert\le cr\), whose volume is \(O(r^{d-1})\), gives \[\mathcal{J}_{t_0}(\Sigma_r^+) = \frac{ar\sqrt{\det G_\gamma}}{\gamma_-} \int_{\left\lVert y\right\rVert\le cr}\rho(r,y)\,\mathrm{d}y +O(r^{d+1}).\] If \(\rho(0,0)>0\), the leading term is \[\frac{ar\sqrt{\det G_\gamma}}{\gamma_-} \rho(0,0)\operatorname{Vol}(B_{d-1}(cr)) +o(r^d).\] Under the same local volume normalization \(\det G_\gamma=1\), a smaller \(\gamma_-\) gives a strictly larger leading-order outgoing flux for sufficiently small \(r\). ◻
Proof of 3. Write \[x_-=\log\gamma_-, \qquad x_i=\log\gamma_i,\quad i=1,\ldots,d-1.\] The volume constraint is \[x_-+\sum_{i=1}^{d-1}x_i=0,\] and the affine-invariant budget from the identity is \[x_-^2+\sum_{i=1}^{d-1}x_i^2\le B^2.\] The leading flux multiplier in 2 is \(1/\gamma_-=\exp(-x_-)\), so maximizing it is the same as minimizing \(x_-\). Set \(x_-=-a\) with \(a\ge0\). For fixed \(a\), the stable-direction logs must sum to \(a\), and their squared norm is minimized when all are equal: \[x_i=\frac{a}{d-1}.\] Thus every feasible point satisfies \[B^2 \ge a^2+(d-1)\left(\frac{a}{d-1}\right)^2 = a^2\frac{d}{d-1}.\] Hence \(a\le B\sqrt{(d-1)/d}\), which gives \[\frac{1}{\gamma_-}\le \exp\!\left(B\sqrt{\frac{d-1}{d}}\right).\] The equal stable-direction choice with \[a=B\sqrt{\frac{d-1}{d}}\] satisfies both constraints and attains the bound. ◻