Papers

Under the framework of time-independent Hamiltonian mechanics on the cotangent bundles $T^*Q$ of the configuration spaces $Q$ of mechanical systems, we introduce the notion of fouling map as a (not necessarily invertible) generalization of the so-called fouling transformations –canonoid transformations preserving configuration coordinates–. We develop a tensorial method for constructing polynomial fouling maps. We show that each such map induces a $(1,1)$-tensor field invariant under the Hamiltonian flow, whose traces of its powers are polynomial constants of motion. For mechanical Hamiltonian functions –the kinetic energy plus the potential energy on a semi-Riemannian configuration space $(Q,g)$–, we completely characterize polynomial bundle maps arising from symmetric $(k+1,0)$-tensor fields and derive the conditions ensuring their fouling nature. Several explicit examples on the Euclidean plane, on the 2-sphere and on a Riemannian manifold with a Liouville metric illustrate the method.

This thesis studies various aspects of flatness in algebraic geometry. We first study flatness in the context of semi-rings. We prove some discreteness results for a derived groupification functor with respect to the homotopy theoretic model structure on simplicial semirings, compare the Zariski and fppf topology, and study fppf algebras over the positive reals. We then study descendibility properties of faithfully flat ring maps; we in particular construct a non-descendable faithfully flat ring map, and then construct examples demonstrating the precise the relationship between the exponent of descendibility and cardinality by developing a rather general method to convert module-theoretic non-vanishing "cup-products" to descendable faithfully flat ring maps with sufficiently high exponent. Finally, we prove the affineness of the maximal etale locus of morphism $X \to Y$ of schemes with $X$ and $Y$ locally Noetherian schemes, $X$ normal and $Y$ regular in fully generality. Some notable aspects of this chapter is a variant of Artin-Rees' lemma that holds for non-Noetherian rings, a characterization of $D$-modules over unramified regular local rings (similar to those obtained by O. Gabber and W. Zhang), and Tor-independence result for global sections of étale schemes which we prove by a variant of the tilting correspondence which doesn't require completion.

In multiple instance regression (MIR) data are organized into bags (collections of instances in feature space) and the goal is to learn a mapping that assigns labels to bags. A typical assumption is that there is a so-called concept point in feature space, the proximity to which dictates the bag label. Motivated by modern MIR architectures which are based on attention, we study a softmax model that decouples the concept point and the labeling scheme. The two are respectively determined by a query direction and a value direction value in feature space. This problem isolates a basic challenge of learning both the query and value vectors from bag-level supervision. From this model we derive a parametric family of iterations in the noiseless limit, which generalizes a method known as the EM-DD algorithm. We then derive concentration results for the MLE estimators of the query and value vectors obtained from a random selection of instances. Our result for the value vector shows that a single random initialization of the value vector already points in the correct direction on average, so that a polynomial (in the number of instances per bag and the feature dimension) number of bags is enough for the EM algorithm to converge in $O(1)$ steps with high probability. A key aspect of this analysis is the interplay between concentration of empirical covariance matrices and extremal statistics arising from the selection rule.

This paper aims to investigate a diffusion approximation of a sequence of critical branching processes with strictly stationary and ergodic immigration. Under fairly general assumptions on the dependence structure of the immigration sequence, we establish that the scaled and properly normalized sequence of branching processes with immigration converges in distribution to the diffusion process in the Skorokhod topology. Our result extends the functional limit theorem of Wei and Winnicki (1989) to the case where the immigration sequence is dependent.

For $d\in\{1,2\}$, let $H=-\frac{1}{2}\Delta + V +\xi$ be the random Schrödinger operator on $L^2(\mathbb{R}^d)$ where $\xi$ is a standard Gaussian white noise and $V$ is a deterministic potential with power-law growth at infinity. Using a Feynman-Kac formula for the trace of the Schrödinger semigroup, we give optimal asymptotic upper and lower bounds on the covariance of $\mathrm{Tr}[e^{-sH}]$ and $\mathrm{Tr}[e^{-tH}]$ as $s,t\to0$ through estimates on Brownian bridge local times. These estimates are a significant improvement on previous bounds in the case $d=1$ and are the first of their kind for $d=2$. As an application of these new estimates, we prove a quantitative hyperuniformity-type property and decorrelation rate for the trace as $s,t\to0$.

We investigate the capacity of current language models to contribute to mathematical research. In Banach space theory, AI systems generated key ideas and proofs for five new results, which were then verified and refined by humans. We also developed an automated system that searches the literature for open problems and attempts solutions at scale. Our results show both the potential of language models for mathematical discovery and the continuing importance of expert verification.

For a Fermat surface X_d: x_0^d+x_1^d+x_2^d+x_3^d=0 we compute the Galois-invariant Neron-Severi (Picard) rank rho_K(X_d) = dim (NS(X) \otimes Q)^{Gal(Qbar/K)}: for every subfield K of Q(zeta_d) by an explicit character average – in closed form for gcd(d,6)=1, with the rational ranks tabulated for 4 <= d <= 30 – and for an arbitrary number field K through the reduction rho_K = rho_{K \cap L} to the field of definition L. The geometric Picard number is classical (Shioda, Aoki), and L is due to Gvirtz-Chen and Skorobogatov; here we supply the rank layer. As a Gal(Q(zeta_d)/Q)-module the zeta_d-rational Neron-Severi group is Qh \oplus M with M \otimes Q(zeta_d) monomial on the Shioda eigenlines; writing chi_NS = 1 + chi_M, we get rho_K = (1/|H|) sum_{t in H} chi_NS(t). For degree coprime to 6 we prove rho_Q(X_d) = 1 + 3(Psi_2(d) - 3 tau(d) + 2) with Psi_2 multiplicative; we establish a field-of-definition threshold with explicit Hasse-Davenport witnesses, an assembled orbit rule for even d, and the exact table for 4 <= d <= 30, where the exceptional entries at d = 14, 24, 28, 30 are proved internally by Hasse-Davenport identities and a stabilizer descent, and corroborated by exact Z[zeta_m] evaluation and by the per-character field-of-definition computation of Gvirtz-Chen–Skorobogatov. The average order is sum_{d <= x, gcd(d,6)=1} rho_Q(X_d) ~ (3/5) x^2. All results are unconditional: on a surface the algebraic classes are the rational (1,1)-classes by Lefschetz. Ancillary files provide 15 standalone exact-arithmetic scripts, a 1270-row orbit-by-orbit certificate of the exceptional layer, and the full character tables, with pinned dependencies and chained checksums.

We show that the licci property of an ideal cannot, in general, be determined from its Betti table by exhibiting two $\mathfrak{m}$-primary monomial ideals containing the cubes the variables that have identical Betti tables, yet only one is licci. In contrast, we show that for monomial ideals containing the square of each variable, the licci property is completely determined by the Betti table; indeed, it is determined by its first two columns.

We construct singly periodic minimal surfaces in $\mathbb{R}^3$ parametrized by rhombic lattices and foliated by spherical curvature lines. As an application we find new examples of free boundary and capillary minimal annuli into two spheres of different center and same radius, some of them embedded.

Unstructured data, such as images and text, are increasingly used in empirical economics. Since training machine-learning models on unstructured data is costly, economists often use off-the-shelf pre-trained deep learning models developed by computer scientists to extract embeddings, which are then used as covariates in target economic analyses. Despite the popularity of this practice, its theoretical foundations remain limited. There are two main difficulties. First, pre-trained models are typically trained on different datasets and for different tasks, making it unclear when they can be used reliably for the target task. Second, the embedding function is subject to an identification problem, complicating the analysis of its estimation error and the effect of that error on the target task. We provide sufficient conditions to overcome these difficulties. A key condition, which we call transferability, governs the convergence rate we derive. To assess transferability, we develop a computationally feasible bootstrap test that does not require re-estimating the embeddings and nuisance functions. Our theory applies to a wide range of double machine learning applications, including partially linear regression with unstructured controls, price elasticity estimation in demand models accounting for product quality measured by images and text, missing-data imputation using unstructured data, and average treatment effect estimation with unstructured confounders. As an empirical application, we estimate the labor supply elasticity on Amazon Mechanical Turk, an online labor market platform, using job-description embeddings as controls.

We show that the function that assigns to each square matrix $A$ the geometric mean of $|Ax|$ over all unit vectors $x$ is a norm. The same holds for $L^p$ means with $0<p<1$. Some properties of these norms are proved, and others are conjectured.

For a two-component log-concave mixture model, we investigate the extent to which the weight, mean, and covariance of each component distribution can be accurately recovered when given sufficient samples of the mixture distributions. One fundamental obstruction is that the mixture distribution itself could sometimes be log-concave, and in this case, accurate recovery is impossible. In this paper, we identify two regimes, one where the mixture distribution itself could be log-concave and another one where the mixture distribution is never log-concave, and one can always separate the two component distributions using a quadratic classifier.

We announce some results concerning initial subclasses of the Baire-1 functions. First, we obtain a finite decomposition into continuous maps for a basic initial subclass. Second, we give a pointwise description of initial subclasses via point-homogeneous sequences. Third, we present a finite decomposition of maps into open and closed maps.

The main goal of this work is to introduce derived Lie n-groupoids and their shifted symplectic structures. We further define shifted lagrangian structures and prove that their composition is well defined under suitable conditions. As an application, we show that our framework incorporates several reduction procedures at critical values, including: classical Hamiltonian reduction, group valued moment maps, Poisson Lie group valued moment maps and Mikami-Weinstein for proper symplectic groupoids.

This paper presents a two-dimensional TMz finite-difference time-domain (FDTD) solver based on Yee's scheme for modeling radiation from an infinitely long z-directed line current, with the open region truncated by a Berenger split-field perfectly matched layer (PML). After validating cylindrical-wave propagation and negligible late-time reflections in free space, the solver is applied to three inhomogeneous configurations: (i) diffraction through a one-cell-thick perfectly electrically conducting (PEC) sheet with single and double slits; (ii) scattering from infinitely long PEC cylinders of circular and rectangular cross section; and (iii) scattering from infinitely long dielectric cylinders of varying cross section and permittivity. Beyond qualitative field maps, the diffraction case is characterized quantitatively: a steady-state phasor extracted by a running discrete Fourier transform yields the transmitted intensity, from which the fringe visibility and the far-field pattern are computed and compared against the closed-form Fraunhofer prediction. The single- and double-slit cases are cleanly separated by a visibility that rises from near zero to near unity, and the double-slit interference maxima agree with the grating condition arcsin(m \lambda_0 / d) to within a fraction of a degree. For dielectric cylinders, the field penetrates the obstacle with the expected reduced internal wavelength \lambda_0 / \sqrt{\epsilon_r}, and the scattered field strength grows with permittivity contrast. A reference-subtraction method isolates the scattered field throughout. The results confirm that the FDTD-PML framework accurately captures open-region diffraction and geometry- and material-dependent scattering.

We study the maximum size of a set system that contains no $k$-Venn diagram, denoted by $VD_k$, as a trace. For every fixed $k\ge 3$, we prove $\text{ex}_{tr}(n,VD_k)=O_k(n^{2^k-2k+1})$, improving the direct Sauer-Shelah bound $O_k(n^{2^k-1})$. In particular, for $k=4$ the exponent decreases from $15$ to $9$. The proof starts from the theorem of Keevash, Leader, Long and Wagner for $VD_3$ and uses induction on $k$ in which two new Venn regions are forced for free at each added edge. We also record lower-bound constructions for Venn diagrams in fixed uniformity, explicit bounds for the $4$-uniform $3$-Venn problem, and a fixed uniformity trace result for the loose triangle.

We investigate colored $\Delta_T$ noise, i.e., finite-frequency $\Delta_T$ noise, as a probe of edge-mode (EM) transport in quantum Hall and quantum spin Hall systems. Colored $\Delta_T$ noise probes finite-frequency nonequilibrium current fluctuations and dynamical transport properties that are often obscured in DC measurements of conductance and noise. Since $\Delta_T$ noise is driven solely by a temperature and voltage bias under zero average charge current conditions, it eliminates current-induced Joule heating and directly probes intrinsic thermal fluctuations. We show that chiral, spin-conserving helical, and spin-flip helical (trivial) EMs exhibit distinct colored $\Delta_T$-noise signatures under appropriate bias protocols. Incorporating energy-dependent scattering through a quantum point contact, we demonstrate that electron-hole asymmetry significantly modifies the finite-frequency spectrum while preserving these distinguishing features. Notably, colored $\Delta_T$ noise exhibits a frequency-dependent sign reversal absent in the corresponding white ($\omega=0$) $\Delta_T$ noise. We further investigate zero-temperature colored quantum shot noise and find that it vanishes identically for chiral EMs, whereas the spin-conserving helical response changes sign with frequency. By contrast, spin-flip helical (trivial) EMs exhibit a positive colored shot-noise spectrum. However, the corresponding colored $\Delta_T$ noise retains its characteristic sign reversal, providing a robust distinction between spin-conserving helical and spin-flip helical (trivial) EM transport. These results establish colored $\Delta_T$ noise as a robust, experimentally accessible, complementary probe for identifying chiral, spin-conserving helical, and spin-flip helical (trivial) EM transport in mesoscopic topological systems.

We present a black box algorithm that constructs standard generators for the Suzuki groups $Sz(q)$, where $q = 2^{2m+1}$ for some $m > 0$. The algorithm is one-sided Monte Carlo, with no false positives. We also present a black box algorithm that performs constructive membership testing in $Sz(q)$, and writes an element as a straight line program in the standard generators. Finally, we give a presentation for $Sz(q)$ that is efficient to verify. The algorithms have been implemented in the computer algebra system Magma.

We employ machine learning techniques to estimate the amount of information contained in three Krylov-space observables: the spread complexity $C(t)$, the discrete Wigner negativity $N(t)$, and the normalized negativity $\chi(t)=N(t)/|S(t)|$, where $S(t)$ is the survival amplitude. We study thermofield-double evolutions on GUE, GOE and Poisson spectra, and on the integrable $SL(2,R)$ Krylov chain of a two-dimensional CFT primary, with Lanczos coefficients $b_n=\alpha\sqrt{n(n+2h-1)}$. To introduce chaos we add a GUE random matrix to the integrable chain, $H(\varepsilon)=H_{SL(2,R)}+\varepsilon R_0 W_{GUE}$, where $R_0$ is its spectral half-width. Either $C(t)$ or $N(t)$ alone determines the thermofield temperature, with $R^2\simeq0.999$. However, neither can reconstruct the fine structure of the spectral form factor. In the integrable sector all three observables carry the same information. When we turn on chaos an asymmetry appears: $\chi$ predicts $C$ better than $C$ predicts $\chi$. This gap grows from $+0.33$ to $+0.77$, while the same gap for the raw negativity decays to zero. The extra information comes from the survival amplitude, and $\chi(t)$ carries it while $C(t)$ can still be recovered from $\chi(t)$. We derive a lower bound on this gap, set by the fraction of the variance of $\chi$ that comes from $S$.

We develop an affine $\mathbb F_2$ framework for structured signings of regular graphs. A family-averaging identity converts even spectral moments into parity-weighted closed-walk counts supported on the span of prescribed short even cycles, while a kernel-averaged Ihara identity gives the corresponding decomposition at the non-backtracking level. We give a finite-scale bounded-rank counting estimate and a conditioning corollary showing that, on bicycle-free graph sequences, any parity family of uniformly bounded codimension contains near-Ramanujan signings whenever the corresponding random-signing theorem applies. The latter is a transfer statement rather than a new concentration theorem. Finally, on $C_n(1,2)$ for even $n\ge10$, the quadrilateral-unbalanced family has exactly four switching classes and its twisted classes attain $\rho_-(n)=2\sqrt{\cos^2(\pi/n)+\cos^2(2\pi/n)}$; a period-$8$ signing has spectral radius $r_*=2.793604493334841\ldots$ for every positive multiple of $8$. Thus for $n=8m\ge32$ the constrained minimum is strictly larger than a value attained by an unrestricted signing, while equality of $r_*$ with the unrestricted minimum remains conjectural.

Periodic solutions of differential-algebraic equations (DAEs) and ordinary differential equations (ODEs) can be determined using the harmonic balance method (HBM), which is a frequency-domain approach that approximates the solution by its truncated Fourier series. The Koopman-Hill method, a method to determine the stability of periodic solutions found by HBM, and originally developed for ODEs, is generalized to DAEs in this work. Analogously to the ODE case, the core idea of the proposed Koopman-Hill method for DAEs is to establish a linear time-invariant but high-dimensional DAE which approximately governs the dynamics of small admissible perturbations around the periodic solution. The crucial difference to the ODE case is the fact that the evolution of this linear time-invariant DAE is not simply given by a matrix exponential, but by a more complicated expression involving a Drazin inverse, rendering the resulting monodromy matrix singular. Still, even in the DAE case, this novel relationship between the monodromy matrix and the Hill matrix is essentially given by one single formula, which is the main result of this work. Two academic mechanical systems, a mathematical pendulum formulated as an index-3 DAE and a nonsmooth frictional two-mass oscillator with switching index, demonstrate the applicability of the proposed method and its blindness to the DAE's index.

As is well known, both the numerical performance and the theoretical properties of the Dai–Liao conjugate gradient algorithm are highly dependent on the adjustment of its key parameter. Here, we first employ the well-known Harmonic–Geometric–Arithmetic–Quadratic mean inequality to reinforce the optimality of two previously proposed scalar adaptive settings of the Dai–Liao parameter. In other words, we show how these parameter choices are capable of enhancing the well-conditioning of the Dai–Liao search direction matrix by shrinking the intervals containing its singular values. A similar analysis is also carried out for scaled memoryless quasi–Newton updating formulas in order to further justify the optimality of two classical scaling parameters associated with these updates. Then, as the main contribution of this work, we move from the classical scalar setting of the Dai–Liao parameter to a matrix setting aimed at enhancing flexibility and diversity within optimization methods. In particular, we show that, under such a matrix formulation of the parameter, a well-known three-term conjugate gradient algorithm emerges as a member of the proposed extended Dai–Liao class of algorithms while enjoying several computationally attractive properties. Among these, the well-conditioning of the associated search direction matrix is especially noteworthy, as well as the ability to make more explicit use of the second-order information of the model. Finally, to provide practical evidence supporting the proposed matrix setting of the Dai–Liao parameter, we conduct a series of numerical experiments on standard benchmark test problems and report the results in detail. Generally speaking, the proposed framework is shown to retain both theoretical soundness and computational reliability.

A coloring of a finite $k$-out directed graph $G$ is viewed as a deterministic complete automaton with state set $V(G)$. The graph $G$ is called totally synchronizing if every coloring is synchronizing. We prove that total synchronization imposes strong restrictions on symmetry: if $G$ is strongly connected and totally synchronizing, then $Aut(G)$ contains no semiregular element; in particular, if $|Aut(G)|$ is divisible by a prime $p>k$, then $G$ is not totally synchronizing. We then give general constructions of strongly connected $k$-out graphs with prescribed quotients and prescribed automorphism group that are not totally synchronizing. On the quotient side, we relate graph congruences to strong lumpability of the uniform random walk on $G$ and introduce totally simple graphs, characterized by the absence of nontrivial congruences. In this setting we obtain a Perron–Frobenius sufficient condition for total synchronization: a strongly connected non-lumpable graph whose integer Perron–Frobenius eigenvector admits at most one nontrivial equipartition is totally synchronizing. Finally, we show that deciding whether a primitive $k$-out graph admits a non-synchronizing coloring is NP-complete, resolving an open problem of Gusev–Szykuła, and prove NP-completeness of deciding whether a graph admits a nontrivial Eulerian lumping.

We give an affirmative answer to Owings's sumset question: for any $2$-coloring of natural numbers, there is an infinite $B\subseteq\mathbb{N}$ such that $B+B$ is monochromatic. More generally, for every $m,\ell\in\mathbb{N}$ and every $2$-coloring of $\mathbb{N}$, there is an infinite $B\subseteq\mathbb{N}$ such that $$ (m+\ell)B\cup\{mx+\ell y:x,y\in B,\ x<y\} $$ is monochromatic.

We study the properties of shadowing, transitivity, weakly mixing, mixing, chain transitivity and chain mixing of a set-valued map and its generalized inverse limit. Concerning shadowing, we prove that for a surjective upper semi-continuous set-valued map $F$ on a compact metric space, $F$ has shadowing if and only if the shift map on the generalized inverse limit $\underleftarrow{\lim}\,\underleftarrow{F}$ of its inverse set-valued map has shadowing; dually, $\underleftarrow{F}$ has shadowing if and only if the shift map on $\underleftarrow{\lim}F$ has shadowing. We further show that the shadowing of $F$ and that of $\underleftarrow{F}$ are always equivalent; consequently $F$, $\underleftarrow{F}$ and the shift maps on $\underleftarrow{\lim}\,\underleftarrow{F}$ and on $\underleftarrow{\lim}F$ all have shadowing simultaneously. In particular, $F$ has shadowing if and only if the shift map on its directly induced generalized inverse limit $\underleftarrow{\lim}F$ has shadowing. This strengthens a recent theorem established under continuity and openness assumptions. We show that if the shift map on the generalized inverse limit is transitive (resp. weakly mixing, mixing, chain transitive, chain mixing), then the set-valued map is transitive (resp. weakly mixing, mixing, chain transitive, chain mixing). For a set-valued map with shadowing, the properties of total transitivity, weak mixing, mixing, specification and chain mixing are mutually equivalent.

Let $q$ be an odd prime power and $\mathbb{F}_q$ be the finite field of order $q$. We prove an extraction theorem for the Erdős-Falconer distance conjecture in even dimensions, showing that the conjecture for all even dimensions reduces to the planar case. As consequences, we obtain improved thresholds on the pinned distance problem and the distribution of triangles, achieving new records of $\frac{d}{2}+\frac{1}{4}$ over prime fields and $\frac{d+1}{2}+\frac{1}{10}$ over arbitrary finite fields, respectively.

Let $\mathcal{A}$ be an abelian category. Beligiannis and Reiten proved that there is a bijective correspondence between so-called projective model structures on $\mathcal{A}$ and hereditary cotorsion pairs in $\mathcal{A}$ with a contravariantly finite core. It is well-known that, tilting modules induce cotorsion pairs, so we may have a homotopicl interpretation of tilting modules. But a recent generalization of tilting modules, support $\tau$-tilting modules, induce weak cotorsion pairs. In this paper, we define weak projective model structures and prove that there is a bijective correspondence between weak projective model structures and left weak cotorsion pairs satisfying some mild conditions. This is a generalization of Beligiannis-Reiten correspondence from the perspective and philosophy of $\tau$-tilting theory. In particular, we prove that any support $\tau$-tilting module induce a model structure, and there is bijective correspondence between support $\tau$-tilting modules and a certain class of model structures.

Suppose that we encode quantum information into $ n $ physical carriers. With what probabilities can we perfectly recover the quantum information after sets of carriers have been erased? This question arises in the study of quantum erasure channels, for which M.B. Hastings conjectured that if each carrier is independently erased with probability $ t \geq 1/2 $, then it is impossible to recover the quantum information with asymptotically vanishing error and probability $ > 1-t $. We prove this conjecture by establishing new no-cloning constraints on realizable recovery probabilities.

Pressure vessel analysis in the chemical, nuclear, and new-energy industries requires solving the same elasticity problem across many materials, geometries, and loads, where mesh quality and repeated solving govern both accuracy and cost. The finite element method (FEM) cannot amortise this repeated cost and fails on degenerate meshes, while the neural operators meant to replace it still need labelled data that FEM must generate. This paper proposes FEVessel, an adaptation of the Pretrained Finite Element Method (PFEM) to three-dimensional (3D) pressure vessels, and validates four capabilities across the two limitations above. FEVessel i) encodes each vessel as a point cloud with coordinate, material, and load channels, ii) pretrains a Transolver operator on the total potential energy instead of FEM labels, and iii) warm-starts iterative solvers with its prediction. A single model generalises across material, geometry, and boundary conditions at a $1.35\%$ relative displacement error, and its $2.07\%$ strain error is about $4.7$ times lower than that of a supervised Fourier neural operator ($9.72\%$), whose structured grid cannot preserve the through-thickness strain. Its warm start cuts algebraic multigrid iterations from $195$ to $18$, a $9.2\times$ end-to-end wall-clock speedup at the $10^{-3}$ engineering tolerance. The model transfers across mesh resolutions without retraining, holding about $3\%$ error at only $30\%$ of the training point density. On inverted and sliver meshes where FEM fails, the error remains below $3.66\%$. To our knowledge, this is the first systematic study of mesh-independent solution on industrially relevant 3D pressure vessels with degenerate meshes. Because training needs no labels, FEVessel works exactly where FEM cannot supply any, removing manual mesh repair from the analysis pipeline.

Let $(X,d,\omega)$ be an unbounded complete doubling metric measure space supporting a weak $(1,2)$-Poincaré inequality, and let $\pi$ be a positive Radon measure. For a ball $B=B(x,R)$, suppose that for some $\delta>0$ and $C_{\mathrm{gr}}>0$, \[ \pi(B(y,r))\le C_{\mathrm{gr}} \left(\frac rR\right)^\delta\frac{\omega(B(y,r))}{r^2}, \qquad y\in B,\quad 0<r\le R. \] Let $\operatorname{cap}_{2,\omega}(K;\Omega)$ denote the relative variational $2$-capacity of $K$ in an open set $\Omega$. We prove that, for every fixed $\Lambda_0>2$, there is a constant $C>0$ such that \[ \pi(K)\le C\operatorname{cap}_{2,\omega}(K;\Lambda_0B), \qquad K\subset B\ \text{compact}, \] with the same outer ball for all compact $K$. No doubling or lower-Ahlfors regularity is imposed on $\pi$. This fixed-domain capacitary domination yields representative-independent $L^2(d\pi)$ traces and, under a lower normalization $\pi(B)\ge c_0\omega(B)/R^2$ with $c_0>0$, the mixed $d\omega\,d\pi$ Poincaré estimate. Summation over bounded-overlap normalized covers gives two-sided global Fefferman–Phong inequalities and identifies the corresponding homogeneous energy spaces. In $\mathbb R^d$ with a Muckenhoupt $A_2$ weight $w$, scale decay and controlled enlargement of $\pi$ generate the required critical-radius cover for every sufficiently large fixed threshold $A$; no additional lower reverse-doubling hypothesis on $w$ is needed. The resulting energy comparison gives equality of the measure-potential and critical-multiplier form domains, a smooth form core, a nonnegative self-adjoint realization, resolvent estimates, and local critical-multiplier bounds. The framework also covers reverse-Hölder function potentials, Carnot groups, and lower-dimensional singular measures, and provides a representative-aware capacity derivation of an earlier generalized Schrödinger energy estimate.