Weighted Selection Functionals

• Weighted selection functionals are mathematical constructs that combine a deterministic weighting mechanism with a selection rule to isolate key contributions in varied structures.
• They are employed in diverse settings such as spectral analysis of Gaussian fields, temporal weighting in CUSUM processes, empirical distribution estimation, network dynamics, and nonlocal phase energies.
• Their design governs asymptotic limits, normal approximations, and extremal behavior, providing precise insights into aggregation phenomena and phase transitions.

Searching arXiv for recent and relevant papers on “weighted selection functionals” and closely related formulations. Weighted selection functionals are functionals in which a deterministic weighting mechanism is combined with a selection, filtering, aggregation, or interface-cost rule on an underlying mathematical object, and the resulting quantity is then studied through asymptotic, extremal, or variational analysis. Across the arXiv literature, the phrase does not denote a single standardized formalism; instead, it appears in several technically distinct settings, including weighted linear functionals of isotropic Gaussian random fields with spectral singularities, weighted $L^p$ functionals of CUSUM processes, weighted averages of Poisson avoidance indicators, smooth functionals of weighted empirical distributions, fixation-probability mappings on weighted networks, convex functionals on weighted Bergman and Hardy spaces, and weighted nonlocal area energies for multiphase interfaces (Olenko, 2013, Horváth et al., 2020, Barrio, 2015, Withers et al., 2010, Bhaumik et al., 2024, Melentijević, 3 Aug 2025, Dipierro et al., 23 Jun 2025).

1. Formal scope and recurring structure

A recurrent pattern is that a weight profile modifies which parts of a stochastic field, path, point configuration, empirical measure, graph, analytic function, or phase partition contribute most strongly to the functional. In some cases the weight acts in physical space, in some cases in frequency space, in some cases through a measure on parameter sets, and in some cases through edge weights or interfacial coefficients. The selected object may be a spectral band, a temporal region, an avoidance event, a subset of observations, a fixation trajectory, a reproducing kernel, or an intermediate phase.

Domain	Representative functional	Selected feature
Random fields	$\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$	singular frequency band $|\lambda|=a_j$
CUSUM asymptotics	$\int_0^1 |Z_N(t)|^p / w(t)\,dt$	temporal regions via $1/w(t)$
Poisson geometry	$$\int_{A\times B}\mathbf{1}\{\eta(x+\lambda^{-1/d}Q(s))=0\}\,d(x\otimes v)$$	avoidance events over space and shapes
Weighted empirical distributions	$T(\hat F_n)$, $\hat F_n(x)=n^{-1}\sum_i w_i\mathbf 1\{X_i\le x\}$	observation-specific weighted contributions
Weighted networks	$r\mapsto \rho_G(r;W)$	fixation under weighted neighbor replacement
Weighted Bergman spaces	$$\Phi_G(f)=\int_{\mathbb D} G(|f(z)|^p(1-|z|^2)^{\alpha+2})\,d\mu(z)$$	reproducing-kernel extremizers
Nonlocal three-phase energies	$\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$0	interface types and intermediate phases

This comparison suggests a broad but precise organizing principle: a weighted selection functional couples a weighting device with a structural criterion that determines which configurations dominate. What differs across fields is the ambient space, the meaning of “selection,” and the notion of limit or optimality.

2. Spectral filtering and weighted functionals of random fields

In the theory of cyclical long-range dependent random fields, weighted selection functionals are weighted linear functionals of a real-valued Gaussian random field

$\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$1

assumed homogeneous, isotropic, mean-square continuous, and centered. The field is represented spectrally by an isotropic spectral density $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$2, and the central regime is one in which $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$3 is unbounded at some frequencies. Under Assumption A, the density has a singularity at zero of order $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$4 and may also have singularities at non-zero radii $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$5 of order $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$6. For $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$7, each non-zero singularity lies on the sphere $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$8, so cyclical long-range dependence is geometrically distributed over all directions with that radius (Olenko, 2013).

The functionals studied are

$\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$9

together with the normalized processes $|\lambda|=a_j$0. The weights $|\lambda|=a_j$1 are radial, belong to $|\lambda|=a_j$2 for $|\lambda|=a_j$3, are essentially nonzero, and satisfy

$|\lambda|=a_j$4

where $|\lambda|=a_j$5 is even and obeys the decay bound $|\lambda|=a_j$6 for large $|\lambda|=a_j$7. In frequency space, the weight is therefore concentrated near the sphere $|\lambda|=a_j$8. The decomposition

$|\lambda|=a_j$9

is unique under these assumptions.

The decisive distinction is between the Donsker line and general schemes. For the Donsker-type weight

$\int_0^1 |Z_N(t)|^p / w(t)\,dt$0

the Fourier transform is a Bessel kernel centered at zero, and the scaling limit depends only on the zero-frequency singularity. Non-zero singularities do not affect the limit for a wide class of radial weights of type $\int_0^1 |Z_N(t)|^p / w(t)\,dt$1. By contrast, for general frequency-selective weights with $\int_0^1 |Z_N(t)|^p / w(t)\,dt$2, the corresponding normalized functionals converge in finite-dimensional distributions to Gaussian limits

$\int_0^1 |Z_N(t)|^p / w(t)\,dt$3

and the limit depends on $\int_0^1 |Z_N(t)|^p / w(t)\,dt$4, $\int_0^1 |Z_N(t)|^p / w(t)\,dt$5, and $\int_0^1 |Z_N(t)|^p / w(t)\,dt$6. The central conclusion is that singularities at non-zero frequencies can matter even for linear functionals, provided the weights are matched to those singularities. This directly contrasts with the Donsker scheme, where cyclical peaks are asymptotically invisible.

3. Temporal weighting in $\int_0^1 |Z_N(t)|^p / w(t)\,dt$7 functionals of CUSUM processes

In change-point asymptotics, the weighted selection functional is an integral functional applied to the CUSUM path rather than a pointwise modification of the process itself. Under the null hypothesis $\int_0^1 |Z_N(t)|^p / w(t)\,dt$8, the centered CUSUM process is

$\int_0^1 |Z_N(t)|^p / w(t)\,dt$9

and its rescaled version is

$1/w(t)$0

The weighted $1/w(t)$1 functional is

$1/w(t)$2

where the deterministic weight $1/w(t)$3 is positive on compact subintervals away from the endpoints and satisfies

$1/w(t)$4

Under Assumptions 1.1–1.3 and $1/w(t)$5,

$1/w(t)$6

where $1/w(t)$7 is a Brownian bridge (Horváth et al., 2020).

This framework makes the weight a temporal selection device. Small values of $1/w(t)$8 magnify the contribution of the corresponding region through the factor $1/w(t)$9, while large values downweight it. The paper emphasizes that the weighting acts at the level of the functional, not through a process of the form $$\int_{A\times B}\mathbf{1}\{\eta(x+\lambda^{-1/d}Q(s))=0\}\,d(x\otimes v)$$0.

A critical regime is the standardizing choice

$$\int_{A\times B}\mathbf{1}\{\eta(x+\lambda^{-1/d}Q(s))=0\}\,d(x\otimes v)$$1

for which the integrand becomes $$\int_{A\times B}\mathbf{1}\{\eta(x+\lambda^{-1/d}Q(s))=0\}\,d(x\otimes v)$$2. In this case the functional no longer converges to a fixed Brownian-bridge integral. Instead, after normalization, it exhibits a Darling–Erdős-type behavior with leading growth $$\int_{A\times B}\mathbf{1}\{\eta(x+\lambda^{-1/d}Q(s))=0\}\,d(x\otimes v)$$3 and Gaussian fluctuations of order $$\int_{A\times B}\mathbf{1}\{\eta(x+\lambda^{-1/d}Q(s))=0\}\,d(x\otimes v)$$4. For heavier endpoint emphasis, the paper adopts Rényi-type truncation to intervals $$\int_{A\times B}\mathbf{1}\{\eta(x+\lambda^{-1/d}Q(s))=0\}\,d(x\otimes v)$$5 satisfying Assumption 1.4 and proves convergence of suitably rescaled integrals with weights $$\int_{A\times B}\mathbf{1}\{\eta(x+\lambda^{-1/d}Q(s))=0\}\,d(x\otimes v)$$6, $$\int_{A\times B}\mathbf{1}\{\eta(x+\lambda^{-1/d}Q(s))=0\}\,d(x\otimes v)$$7. The left and right endpoints contribute separate Wiener-process terms in the limit.

The methodological basis is a strong approximation of the error partial sums by Wiener processes, construction of a coupled Brownian bridge $$\int_{A\times B}\mathbf{1}\{\eta(x+\lambda^{-1/d}Q(s))=0\}\,d(x\otimes v)$$8, and control of the difference $$\int_{A\times B}\mathbf{1}\{\eta(x+\lambda^{-1/d}Q(s))=0\}\,d(x\otimes v)$$9 in weighted sup-norms. In this setting, a weighted selection functional is best understood as a path functional whose asymptotic law is highly sensitive to how the weight distributes mass near the endpoints of $T(\hat F_n)$0.

4. Avoidance averages and smooth functionals of weighted empirical distributions

Two statistically distinct literatures formulate weighted selection functionals through weighted aggregation over events or observations. In stochastic geometry, the object is a Poisson avoidance functional

$T(\hat F_n)$1

where $T(\hat F_n)$2 is bounded and open, $T(\hat F_n)$3 is a Borel shape indexed by $T(\hat F_n)$4, and $T(\hat F_n)$5 provides the deterministic weighting. This functional averages the selection event “no points of $T(\hat F_n)$6 fall in $T(\hat F_n)$7” over locations and shapes. In the homogeneous case, its expectation is

$T(\hat F_n)$8

its variance satisfies $T(\hat F_n)$9, and the standardized functional obeys a central limit theorem with Berry–Esseen bounds of order $\hat F_n(x)=n^{-1}\sum_i w_i\mathbf 1\{X_i\le x\}$0 in Wasserstein distance and again $\hat F_n(x)=n^{-1}\sum_i w_i\mathbf 1\{X_i\le x\}$1 in Kolmogorov distance. The same Malliavin–Stein analysis yields applications to the volume of the union of balls around Poisson centers and to quantization error, while the empirical-measure analogue has variance of order $\hat F_n(x)=n^{-1}\sum_i w_i\mathbf 1\{X_i\le x\}$2 and Wasserstein bound $\hat F_n(x)=n^{-1}\sum_i w_i\mathbf 1\{X_i\le x\}$3 for every $\hat F_n(x)=n^{-1}\sum_i w_i\mathbf 1\{X_i\le x\}$4 (Barrio, 2015).

The key technical simplification is that the difference operator and inverse Ornstein–Uhlenbeck difference admit closed-form pathwise expressions on avoidance indicators. For the centered elementary avoidance functional $\hat F_n(x)=n^{-1}\sum_i w_i\mathbf 1\{X_i\le x\}$5,

$\hat F_n(x)=n^{-1}\sum_i w_i\mathbf 1\{X_i\le x\}$6

and

$\hat F_n(x)=n^{-1}\sum_i w_i\mathbf 1\{X_i\le x\}$7

This converts normal approximation into explicit geometric intersection integrals.

A separate, more classical statistical formulation starts from independent non-identically distributed observations $\hat F_n(x)=n^{-1}\sum_i w_i\mathbf 1\{X_i\le x\}$8 with deterministic weights $\hat F_n(x)=n^{-1}\sum_i w_i\mathbf 1\{X_i\le x\}$9 satisfying

$r\mapsto \rho_G(r;W)$0

The weighted empirical distribution is

$r\mapsto \rho_G(r;W)$1

which is a signed measure of total mass $r\mapsto \rho_G(r;W)$2. A smooth functional $r\mapsto \rho_G(r;W)$3 is then analyzed by extending von Mises derivatives to signed measures of total measure $r\mapsto \rho_G(r;W)$4. For $r\mapsto \rho_G(r;W)$5, the cumulants admit expansions

$r\mapsto \rho_G(r;W)$6

which feed into Edgeworth and Cornish–Fisher expansions through order $r\mapsto \rho_G(r;W)$7. The framework covers the weighted mean, central moments, Studentized mean, coefficient of variation, and in particular the sample variance, for which the required cumulant coefficients are worked out explicitly (Withers et al., 2010).

These two traditions share a formal theme: one first builds a weighted empirical or geometric selection mechanism, and only afterwards applies asymptotic machinery. In the Poisson case the emphasis is on normal approximation with explicit geometric constants; in the non-iid empirical-distribution case the emphasis is on higher-order distributional and quantile expansions.

5. Selection on weighted networks

In constant-selection evolutionary dynamics, the selection functional is the mapping from a fitness ratio and a weighted graph to fixation probability. The model consists of $r\mapsto \rho_G(r;W)$8 individuals on an undirected, static weighted network with weighted adjacency matrix $r\mapsto \rho_G(r;W)$9, weighted degree

$$\Phi_G(f)=\int_{\mathbb D} G(|f(z)|^p(1-|z|^2)^{\alpha+2})\,d\mu(z)$$0

resident fitness $$\Phi_G(f)=\int_{\mathbb D} G(|f(z)|^p(1-|z|^2)^{\alpha+2})\,d\mu(z)$$1, mutant fitness $$\Phi_G(f)=\int_{\mathbb D} G(|f(z)|^p(1-|z|^2)^{\alpha+2})\,d\mu(z)$$2, and a single uniformly random initial mutant. Under birth–death updating, a reproducing node is chosen proportional to fitness and then a neighbor is chosen proportional to edge weight. The resulting transition probabilities depend explicitly on the ratios $$\Phi_G(f)=\int_{\mathbb D} G(|f(z)|^p(1-|z|^2)^{\alpha+2})\,d\mu(z)$$3, so the “selection environment” is weighted at the neighbor-replacement stage. The paper summarizes this as

$$\Phi_G(f)=\int_{\mathbb D} G(|f(z)|^p(1-|z|^2)^{\alpha+2})\,d\mu(z)$$4

or equivalently as the fixation-probability functional

$$\Phi_G(f)=\int_{\mathbb D} G(|f(z)|^p(1-|z|^2)^{\alpha+2})\,d\mu(z)$$5

(Bhaumik et al., 2024).

Amplifiers and suppressors are defined relative to the Moran fixation probability

$$\Phi_G(f)=\int_{\mathbb D} G(|f(z)|^p(1-|z|^2)^{\alpha+2})\,d\mu(z)$$6

A network is an amplifier if $$\Phi_G(f)=\int_{\mathbb D} G(|f(z)|^p(1-|z|^2)^{\alpha+2})\,d\mu(z)$$7 for $$\Phi_G(f)=\int_{\mathbb D} G(|f(z)|^p(1-|z|^2)^{\alpha+2})\,d\mu(z)$$8 and $$\Phi_G(f)=\int_{\mathbb D} G(|f(z)|^p(1-|z|^2)^{\alpha+2})\,d\mu(z)$$9 for $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$00; it is a suppressor if the inequalities are reversed. The weighted setting preserves these definitions but makes them depend on both topology and the matrix $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$01.

The most systematic numerical evidence concerns all $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$02 non-isomorphic connected undirected graphs with $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$03, each equipped with $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$04 independent $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$05 edge-weight realizations, producing $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$06 weighted networks. Under birth–death updating, $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$07 are amplifiers, $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$08 suppressors, and $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$09 neither; under death–birth updating, $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$10 are amplifiers, $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$11 suppressors, and $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$12 neither. For comparison, in the unweighted $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$13 birth–death case there are $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$14 amplifiers, $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$15 suppressor, and the remainder are neither. The weighted graphs are therefore often less amplifying than their unweighted counterparts, and in many cases become suppressors. The same qualitative conclusion is reported for larger empirical weighted networks, where weighted versions are suppressors or much less amplifying in five out of six examples.

The paper also analyzes semi-analytic families of weighted complete graphs and weighted stars. On complete graphs with two communities, the isothermal line $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$16 yields $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$17; for symmetric community sizes, $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$18 gives amplifiers, whereas $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$19 gives suppressors. Weighted stars, despite the unweighted star being a classic strong amplifier under birth–death updating, are typically less amplifying than the unweighted star, except in restricted parameter ranges.

A frequent misconception is that adding weights necessarily strengthens selection. In this literature the opposite is often true: random or empirical weight heterogeneity tends to flatten the mapping $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$20 and to bias it toward suppression.

6. Extremal selection in weighted analytic spaces and weighted nonlocal geometry

In weighted Bergman spaces on the unit disk,

$\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$21

convex weighted selection functionals take the form

$\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$22

Kulikov’s extremal result states that if $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$23 is convex, then among $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$24 with $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$25, the maximum of $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$26 is attained at the constant function $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$27, which is the normalized reproducing kernel at the origin. The stability result sharpens this by showing that for $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$28,

$\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$29

with explicit $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$30, and the defect is controlled by the squared normalized distance to the nearest reproducing kernel. The same kernel-selection phenomenon has a Hardy-space analogue, and power choices $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$31 yield quantitative hypercontractive stability for embeddings $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$32 (Melentijević, 3 Aug 2025).

A geometrically different variational use of weighted selection functionals appears in three-phase nonlocal area energies

$\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$33

for measurable partitions $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$34. When the coefficients violate the triangle inequality in the form

$\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$35

equivalently $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$36 in the perimeter representation $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$37, the energy can be lowered by inserting a thin strip of phase $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$38 between phases $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$39 and $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$40. Under suitable interface regularity, the resulting competitor $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$41 satisfies

$\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$42

This is a genuine selection effect: the weighted nonlocal functional favors an intermediate phase because direct $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$43 interaction is too expensive. Yet, unlike the local counterpart, the nonlocal functional remains lower semicontinuous for every fixed $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$44, even without the triangle inequality. As $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$45, the renormalized energies $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$46 $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$47-converge to a relaxed local functional $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$48 in which

$\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$49

so the limiting direct interfacial cost is replaced by the cheaper two-step path through phase $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$50 (Dipierro et al., 23 Jun 2025).

These two areas are mathematically remote, but they share a rigid extremal logic: the weight determines which configurations are selected. In weighted Bergman spaces the selected states are reproducing kernels; in weighted nonlocal geometry they are layered phase configurations.

7. Comparative interpretation and recurring misconceptions

Several claims that are valid in one class of weighted selection functionals fail in another. In spectral asymptotics, it is not correct to say that non-zero singularities are always negligible: they disappear under Donsker-type weights but govern the limit under frequency-matched weights. In evolutionary dynamics, it is not correct to say that weights simply intensify fitness-based selection: random and empirical edge weights often produce suppressors. In multiphase geometry, it is not correct to say that failure of the triangle inequality necessarily destroys lower semicontinuity: the nonlocal functional remains lower semicontinuous for $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$51, and the relaxation only appears in the $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$52 limit. In weighted Bergman and Hardy spaces, near-maximality is not structurally ambiguous: the convex-functional deficit quantitatively controls the distance to reproducing kernels (Olenko, 2013, Bhaumik et al., 2024, Dipierro et al., 23 Jun 2025, Melentijević, 3 Aug 2025).

A broad comparative reading suggests three recurring components. First, there is always a deterministic weighting device: spectral kernels, temporal weights, parameter measures, observation weights, edge weights, hyperbolic/Bergman weights, or surface-tension coefficients. Second, there is a selection rule: spectral localization, temporal emphasis, emptiness detection, weighted empirical aggregation, biased replacement, kernel extremization, or interface avoidance. Third, there is a criterion that reveals the effect of weighting: a limit theorem, a Berry–Esseen bound, a higher-order cumulant expansion, a fixation-probability comparison, an extremal theorem with stability, or a $\int_{\mathbb{R}^n} f_{j,r}(x)\xi(x)\,dx$53-limit.

This suggests that “weighted selection functional” is best treated as a family resemblance concept rather than a single formal definition. What unifies the examples is not a common formula, but a common mechanism: weights do not merely rescale an existing quantity; they determine which configurations, frequencies, times, neighborhoods, observations, graph transitions, analytic states, or interfaces become asymptotically or variationally dominant.