Scale-Dilation Operator (SDO)

• Scale-Dilation Operator (SDO) is a family of scale-changing operators that adjust scale while preserving critical structures like commutativity, tight frames, and reversibility.
• In operator theory and harmonic analysis, SDO leverages optimized scaling factors—such as the sharp scale factor θ(d) and isotropic transformations—to dilate matrices and functions effectively.
• SDO underpins multiscale methods, from generating needlet spectral windows on the sphere to enabling reversible Fourier scaling in AI solvers for differential equations, thus enhancing stability and efficiency.

Searching arXiv for the cited SDO-related papers and adjacent terminology to ground the article in current literature. Scale-Dilation Operator (SDO) denotes a family of scale-changing constructions rather than a single universally standardized operator. In the cited literature, the term appears in at least four technically distinct senses: as a sharp scaling preceding commutative dilation of matrix tuples in operator theory; as a function-space dilation associated with a matrix action $A$; as a scale-dependent operator generating multiscale spectral windows on the sphere; and as a local or Fourier-based dilation used to regularize multiscale PDEs and their AI surrogates. The common thread is the controlled modification of scale while preserving a structural constraint such as commutativity, frame tightness, homogenized behavior, or exact reversibility (Helton et al., 2014, Zakharov, 2013, Durastanti, 7 Jul 2025, Chen et al., 28 Jun 2025, Gong et al., 30 Jul 2025).

1. Terminological scope and core definitions

The literature does not attach a unique canonical meaning to “Scale-Dilation Operator.” In operator theory, the underlying concept is an optimal scale factor that permits dilation of finite-dimensional symmetric contractions to commuting self-adjoint contractions. In harmonic analysis, the phrase refers directly to the map

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$

with the Fourier-domain action

$$\widehat{D_A f}(\xi)=|\det A|^{-1/2}\widehat f((A^T)^{-1}\xi).$$

On the sphere, the SDO is encoded by a dilation sequence $\{S_j\}$ and a difference-of-scales construction for needlet windows. In multiscale elliptic PDEs, it is a local coordinate shrinkage map applied to oscillatory coefficients. In AI solvers, it is a reversible Fourier scaling

$$D_Nu(x)=N^d\,\mathcal F^{-1}[\hat u(Nk)](x)=u(x/N),$$

used to compress high-frequency content into a lower-frequency regime (Zakharov, 2013, Durastanti, 7 Jul 2025, Chen et al., 28 Jun 2025, Gong et al., 30 Jul 2025).

This suggests that SDO is best understood as a problem-dependent scale transform equipped with an invariant or recovery mechanism. Depending on context, that invariant may be exact compression by an isometry, preservation of an ellipse under rotation-similarity, partition of unity for spectral windows, commutation with homogenization, or reversibility under inverse dilation.

Setting	SDO form	Structural role
Operator theory	scale $\vartheta(d)^{-1}$ before dilation	commuting self-adjoint dilation
Matrix dilation on functions	$(D_A f)(x)=|\det A|^{1/2}f(Ax)$	scale and rotation in adapted coordinates
Flexible needlets	$D_j:a\mapsto a_j$ with $b_j^2=a_{j+1}-a_j$	multiscale spectral window generation
Elliptic multiscale PDEs	$D_{L,m,\nu}B(x)=B(\phi_{L,m,\nu}(x))$	local microscale relaxation
AI solvers for DEs	$(D_A f)(x)=|\det A|^{1/2}f(Ax),$0	reversible frequency compression

2. Operator-theoretic SDO: scaled commutative dilation and free convexity

In the operator-theoretic setting, dilation means the following. An operator $(D_A f)(x)=|\det A|^{1/2}f(Ax),$1 on a Hilbert space $(D_A f)(x)=|\det A|^{1/2}f(Ax),$2 dilates to an operator $(D_A f)(x)=|\det A|^{1/2}f(Ax),$3 on a Hilbert space $(D_A f)(x)=|\det A|^{1/2}f(Ax),$4 if there is an isometry $(D_A f)(x)=|\det A|^{1/2}f(Ax),$5 such that

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$6

For a tuple $(D_A f)(x)=|\det A|^{1/2}f(Ax),$7 of symmetric matrices, one seeks commuting self-adjoint operators $(D_A f)(x)=|\det A|^{1/2}f(Ax),$8 with

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$9

The central theorem establishes that for each positive integer $$\widehat{D_A f}(\xi)=|\det A|^{-1/2}\widehat f((A^T)^{-1}\xi).$$0 there is a Hilbert space $$\widehat{D_A f}(\xi)=|\det A|^{-1/2}\widehat f((A^T)^{-1}\xi).$$1, a family $$\widehat{D_A f}(\xi)=|\det A|^{-1/2}\widehat f((A^T)^{-1}\xi).$$2 of commuting self-adjoint contraction operators on $$\widehat{D_A f}(\xi)=|\det A|^{-1/2}\widehat f((A^T)^{-1}\xi).$$3, and an isometry $$\widehat{D_A f}(\xi)=|\det A|^{-1/2}\widehat f((A^T)^{-1}\xi).$$4 such that for each symmetric $$\widehat{D_A f}(\xi)=|\det A|^{-1/2}\widehat f((A^T)^{-1}\xi).$$5 contraction matrix $$\widehat{D_A f}(\xi)=|\det A|^{-1/2}\widehat f((A^T)^{-1}\xi).$$6 there exists $$\widehat{D_A f}(\xi)=|\det A|^{-1/2}\widehat f((A^T)^{-1}\xi).$$7 with

$$\widehat{D_A f}(\xi)=|\det A|^{-1/2}\widehat f((A^T)^{-1}\xi).$$8

The factor $$\widehat{D_A f}(\xi)=|\det A|^{-1/2}\widehat f((A^T)^{-1}\xi).$$9 is sharp: if $\{S_j\}$0, there exist $\{S_j\}$1 and a $\{S_j\}$2-tuple of $\{S_j\}$3 symmetric contractions for which $\{S_j\}$4 does not dilate to any commuting self-adjoint contractions (Helton et al., 2014).

The scale factor $\{S_j\}$5 has an exact optimization characterization,

$\{S_j\}$6

and the minimization reduces to two-point spectra. For even $\{S_j\}$7,

$\{S_j\}$8

hence

$\{S_j\}$9

For odd $$D_Nu(x)=N^d\,\mathcal F^{-1}[\hat u(Nk)](x)=u(x/N),$$0, $$D_Nu(x)=N^d\,\mathcal F^{-1}[\hat u(Nk)](x)=u(x/N),$$1 is determined implicitly by a balance of regularized incomplete beta functions. The asymptotic relation

$$D_Nu(x)=N^d\,\mathcal F^{-1}[\hat u(Nk)](x)=u(x/N),$$2

places the scale in a precise high-dimensional regime, and small-dimensional values include $$D_Nu(x)=N^d\,\mathcal F^{-1}[\hat u(Nk)](x)=u(x/N),$$3, $$D_Nu(x)=N^d\,\mathcal F^{-1}[\hat u(Nk)](x)=u(x/N),$$4, and $$D_Nu(x)=N^d\,\mathcal F^{-1}[\hat u(Nk)](x)=u(x/N),$$5 (Helton et al., 2014).

The same scale appears in free spectrahedra and LMIs. For a monic linear pencil

$$D_Nu(x)=N^d\,\mathcal F^{-1}[\hat u(Nk)](x)=u(x/N),$$6

the classical spectrahedron is

$$D_Nu(x)=N^d\,\mathcal F^{-1}[\hat u(Nk)](x)=u(x/N),$$7

while the free spectrahedron is the graded set of tuples $$D_Nu(x)=N^d\,\mathcal F^{-1}[\hat u(Nk)](x)=u(x/N),$$8 satisfying

$$D_Nu(x)=N^d\,\mathcal F^{-1}[\hat u(Nk)](x)=u(x/N),$$9

If $\vartheta(d)^{-1}$0 is bounded, any $\vartheta(d)^{-1}$1 dilates, up to a uniform scale, to a commuting self-adjoint tuple $\vartheta(d)^{-1}$2 whose joint spectrum lies in $\vartheta(d)^{-1}$3. The commutability index $\vartheta(d)^{-1}$4 equals the inclusion scale $\vartheta(d)^{-1}$5, so the dilation factor is exactly the worst-case error incurred when a classical inclusion $\vartheta(d)^{-1}$6 is relaxed to a tractable free inclusion $\vartheta(d)^{-1}$7 (Helton et al., 2014).

This same framework quantifies how a positive unital map can fail to be completely positive. If $\vartheta(d)^{-1}$8 is positive between the relevant operator systems, then scaling the images of the generators by $\vartheta(d)^{-1}$9 makes the map completely positive. In this sense, the operator-theoretic SDO is not a single formula but the sharp universal scaling that upgrades compression into commuting dilation and positivity into complete positivity.

3. Matrix dilation, isotropy, and rotation-similarity

A more classical meaning of scale-dilation operator arises from dilation matrices acting on functions. For a real $(D_A f)(x)=|\det A|^{1/2}f(Ax)$0 dilation matrix $(D_A f)(x)=|\det A|^{1/2}f(Ax)$1, the operator

$(D_A f)(x)=|\det A|^{1/2}f(Ax)$2

has Fourier action

$(D_A f)(x)=|\det A|^{1/2}f(Ax)$3

When $(D_A f)(x)=|\det A|^{1/2}f(Ax)$4 is isotropic and similar, up to a scalar, to a rotation, the operator combines uniform scaling and angular rotation in coordinates adapted by a symmetric positive definite matrix $(D_A f)(x)=|\det A|^{1/2}f(Ax)$5 (Zakharov, 2013).

In the bivariate case, if

$(D_A f)(x)=|\det A|^{1/2}f(Ax)$6

with $(D_A f)(x)=|\det A|^{1/2}f(Ax)$7 symmetric positive definite, $(D_A f)(x)=|\det A|^{1/2}f(Ax)$8, and $(D_A f)(x)=|\det A|^{1/2}f(Ax)$9 the standard rotation matrix, then in the coordinates $D_j:a\mapsto a_j$0 the action becomes a pure rotation-scaling. The paper formulates a sharp spectral test. Writing $D_j:a\mapsto a_j$1 and $D_j:a\mapsto a_j$2, the normalized matrix $D_j:a\mapsto a_j$3 is similar to a rotation if and only if

$D_j:a\mapsto a_j$4

Under this condition,

$D_j:a\mapsto a_j$5

so the eigenvalues lie on the unit circle as $D_j:a\mapsto a_j$6 (Zakharov, 2013).

This rotation-similarity modifies the interpretation of the two-scale refinement equation

$D_j:a\mapsto a_j$7

After the SPD change of coordinates, the refinement relation becomes a relation among rotated copies of the scaling function. If $D_j:a\mapsto a_j$8, then for $D_j:a\mapsto a_j$9 one obtains formulas of the form

$b_j^2=a_{j+1}-a_j$0

and, in space,

$b_j^2=a_{j+1}-a_j$1

If $b_j^2=a_{j+1}-a_j$2 is irrational, $b_j^2=a_{j+1}-a_j$3 is dense in $b_j^2=a_{j+1}-a_j$4, producing a dense family of rotated refinement relations (Zakharov, 2013).

The associated geometry is encoded by the quadratic form

$b_j^2=a_{j+1}-a_j$5

The ellipse

$b_j^2=a_{j+1}-a_j$6

is preserved up to homogeneous scaling: $b_j^2=a_{j+1}-a_j$7 so $b_j^2=a_{j+1}-a_j$8. In $b_j^2=a_{j+1}-a_j$9-adapted coordinates, the invariant ellipse becomes a circle. This geometric invariant shows that the scale-dilation operator does not merely resize functions; it can conjugate anisotropy into isotropic rotation-scaling.

4. Spectral-window SDO in flexible bandwidth needlets

On the sphere, the SDO is encoded by a strictly increasing sequence of center scales $D_{L,m,\nu}B(x)=B(\phi_{L,m,\nu}(x))$0 with $D_{L,m,\nu}B(x)=B(\phi_{L,m,\nu}(x))$1 and dilation factors $D_{L,m,\nu}B(x)=B(\phi_{L,m,\nu}(x))$2 defined by

$D_{L,m,\nu}B(x)=B(\phi_{L,m,\nu}(x))$3

The operator acts on a spectral template through the affine map

$D_{L,m,\nu}B(x)=B(\phi_{L,m,\nu}(x))$4

so that

$D_{L,m,\nu}B(x)=B(\phi_{L,m,\nu}(x))$5

Equivalently, one may view the $D_{L,m,\nu}B(x)=B(\phi_{L,m,\nu}(x))$6-th band as $D_{L,m,\nu}B(x)=B(\phi_{L,m,\nu}(x))$7, centered at $D_{L,m,\nu}B(x)=B(\phi_{L,m,\nu}(x))$8 with window width

$D_{L,m,\nu}B(x)=B(\phi_{L,m,\nu}(x))$9

The relative bandwidth ratio

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$00

summarizes the geometry of scale and overlap (Durastanti, 7 Jul 2025).

These windows generate flexible-bandwidth needlets through the projector

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$01

where

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$02

The windows satisfy compact support on $(D_A f)(x)=|\det A|^{1/2}f(Ax),$03, smoothness bounds

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$04

and the partition of unity

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$05

At cubature points $(D_A f)(x)=|\det A|^{1/2}f(Ax),$06 with weights $(D_A f)(x)=|\det A|^{1/2}f(Ax),$07, the needlets are

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$08

and they form a tight frame: $(D_A f)(x)=|\det A|^{1/2}f(Ax),$09

The dilation sequence induces three asymptotic regimes. The shrinking regime has $(D_A f)(x)=|\det A|^{1/2}f(Ax),$10, $(D_A f)(x)=|\det A|^{1/2}f(Ax),$11, and $(D_A f)(x)=|\det A|^{1/2}f(Ax),$12. The stable regime has $(D_A f)(x)=|\det A|^{1/2}f(Ax),$13, $(D_A f)(x)=|\det A|^{1/2}f(Ax),$14, and $(D_A f)(x)=|\det A|^{1/2}f(Ax),$15, recovering classical $(D_A f)(x)=|\det A|^{1/2}f(Ax),$16-needlets with $(D_A f)(x)=|\det A|^{1/2}f(Ax),$17. The spreading regime has $(D_A f)(x)=|\det A|^{1/2}f(Ax),$18, $(D_A f)(x)=|\det A|^{1/2}f(Ax),$19, and $(D_A f)(x)=|\det A|^{1/2}f(Ax),$20 (Durastanti, 7 Jul 2025).

These regimes control overlap, localization, and decorrelation. Because $(D_A f)(x)=|\det A|^{1/2}f(Ax),$21, windows with $(D_A f)(x)=|\det A|^{1/2}f(Ax),$22 do not overlap, while adjacent windows overlap on $(D_A f)(x)=|\det A|^{1/2}f(Ax),$23 and $(D_A f)(x)=|\det A|^{1/2}f(Ax),$24. The overlap fraction

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$25

is constant in the standard regime and tends to $(D_A f)(x)=|\det A|^{1/2}f(Ax),$26 in spreading regimes. Spatial localization follows from bounds such as

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$27

Under a power-spectrum condition $(D_A f)(x)=|\det A|^{1/2}f(Ax),$28 with the derivative control stated in Condition (Cl), same-scale correlations of needlet coefficients obey decay bounds in terms of $(D_A f)(x)=|\det A|^{1/2}f(Ax),$29 and $(D_A f)(x)=|\det A|^{1/2}f(Ax),$30. The shrinking subregimes $(D_A f)(x)=|\det A|^{1/2}f(Ax),$31 and $(D_A f)(x)=|\det A|^{1/2}f(Ax),$32 with $(D_A f)(x)=|\det A|^{1/2}f(Ax),$33 guarantee the one-step separation condition

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$34

for large $(D_A f)(x)=|\det A|^{1/2}f(Ax),$35, yielding asymptotic uncorrelation (Durastanti, 7 Jul 2025).

5. Local and partial SDOs in seamless multiscale elliptic problems

For elliptic equations with oscillatory coefficients, the SDO is introduced as a device for increasing the effective microscale without resolving the original fine scale. The model problem is

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$36

with $(D_A f)(x)=|\det A|^{1/2}f(Ax),$37 symmetric, uniformly elliptic, and bounded. Two related operators are defined: a partial dilation acting in the fast variable and a local dilation acting directly in physical space (Chen et al., 28 Jun 2025).

The partial operator is defined on the class

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$38

using the wrapping map

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$39

For $(D_A f)(x)=|\det A|^{1/2}f(Ax),$40,

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$41

and the dilated coefficient satisfies

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$42

This formulation requires access to the underlying two-scale representation.

The local SDO avoids that inversion. With

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$43

the scale-dilation operator is

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$44

Applied componentwise in $(D_A f)(x)=|\det A|^{1/2}f(Ax),$45, it locally shrinks coordinates by the factor $(D_A f)(x)=|\det A|^{1/2}f(Ax),$46 inside each mesoscopic window of length $(D_A f)(x)=|\det A|^{1/2}f(Ax),$47, anchored at $(D_A f)(x)=|\det A|^{1/2}f(Ax),$48. The bound

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$49

implies that $(D_A f)(x)=|\det A|^{1/2}f(Ax),$50 remains close to $(D_A f)(x)=|\det A|^{1/2}f(Ax),$51 when $(D_A f)(x)=|\det A|^{1/2}f(Ax),$52 is Lipschitz (Chen et al., 28 Jun 2025).

The resulting seamless approximation solves

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$53

This creates a middle ground between the fully oscillatory operator and the homogenized limit. The error splits into discretization, homogenization, and dilation components: $(D_A f)(x)=|\det A|^{1/2}f(Ax),$54 Under the periodic setting with effective scale $(D_A f)(x)=|\det A|^{1/2}f(Ax),$55,

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$56

while the local dilation error obeys

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$57

In 1D aligned cases with discontinuous dilated coefficients,

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$58

A structure-preserving variant decomposes

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$59

and dilates only the oscillatory part: $(D_A f)(x)=|\det A|^{1/2}f(Ax),$60 This preserves macroscopic structures such as channels while relaxing only the small-scale oscillations. On box domains, the paper further shows

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$61

so dilation and homogenization commute in the stated setting (Chen et al., 28 Jun 2025).

6. Reversible Fourier SDO and AI solvers for differential equations

A recent AI-oriented usage defines SDO as a reversible linear automorphism acting through Fourier scaling. With the Fourier transform

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$62

the SDO with dilation factor $(D_A f)(x)=|\det A|^{1/2}f(Ax),$63 is

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$64

and its inverse is

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$65

The Fourier scaling identity

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$66

shows that $(D_A f)(x)=|\det A|^{1/2}f(Ax),$67 compresses spectral support by a factor of $(D_A f)(x)=|\det A|^{1/2}f(Ax),$68, shifting content from wavenumber magnitude $(D_A f)(x)=|\det A|^{1/2}f(Ax),$69 to $(D_A f)(x)=|\det A|^{1/2}f(Ax),$70 (Gong et al., 30 Jul 2025).

The intended application is the approximation of high-frequency components (AHFC) in neural solvers for differential equations. Because

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$71

the dilated field is smoother in the sense of reduced pointwise gradient magnitude. The paper proves an upper bound for the condition number of the Gauss-Newton approximation of the loss Hessian: $(D_A f)(x)=|\det A|^{1/2}f(Ax),$72 where $(D_A f)(x)=|\det A|^{1/2}f(Ax),$73 is determined by network Lipschitz constants, $(D_A f)(x)=|\det A|^{1/2}f(Ax),$74 is the maximum magnitude of the spatial gradients of the solution and source terms, and $(D_A f)(x)=|\det A|^{1/2}f(Ax),$75 is the minimum eigenvalue of $(D_A f)(x)=|\det A|^{1/2}f(Ax),$76. Under dilation, $(D_A f)(x)=|\det A|^{1/2}f(Ax),$77, so the bound decreases by a factor $(D_A f)(x)=|\det A|^{1/2}f(Ax),$78. This is the paper’s formal mechanism for claiming a smoother loss landscape and improved training behavior (Gong et al., 30 Jul 2025).

The SDO is embedded into a spatiotemporally coupled, attention-based Transformer of the form

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$79

The preprocessing step applies $(D_A f)(x)=|\det A|^{1/2}f(Ax),$80 to the source, initial condition, and boundary data; the network predicts the solution in the dilated domain; and the output is mapped back by $(D_A f)(x)=|\det A|^{1/2}f(Ax),$81. Training uses the $(D_A f)(x)=|\det A|^{1/2}f(Ax),$82 loss

$(D_A f)(x)=|\det A|^{1/2}f(Ax),$83

The same work couples SDO with first-principles data generation: one synthesizes solution fields $(D_A f)(x)=|\det A|^{1/2}f(Ax),$84, substitutes them into the governing PDE or ODE, and derives the corresponding forcing terms and initial or boundary conditions by balancing the equations. This produces arbitrarily large first-principles-consistent datasets at low cost (Gong et al., 30 Jul 2025).

Empirical results are reported for three PDEs and two ODEs. Adding generated data reduced the average relative $(D_A f)(x)=|\det A|^{1/2}f(Ax),$85 error across models by $(D_A f)(x)=|\det A|^{1/2}f(Ax),$86 to $(D_A f)(x)=|\det A|^{1/2}f(Ax),$87. For Navier-Stokes, the combined dataset yielded an average relative $(D_A f)(x)=|\det A|^{1/2}f(Ax),$88 error of about $(D_A f)(x)=|\det A|^{1/2}f(Ax),$89 for the proposed solver, compared to $(D_A f)(x)=|\det A|^{1/2}f(Ax),$90 for FNO, $(D_A f)(x)=|\det A|^{1/2}f(Ax),$91 for U-NO, $(D_A f)(x)=|\det A|^{1/2}f(Ax),$92 for Transolver, and $(D_A f)(x)=|\det A|^{1/2}f(Ax),$93 for LSM. Additional reported values are approximately $(D_A f)(x)=|\det A|^{1/2}f(Ax),$94 for steady Navier-Cauchy, approximately $(D_A f)(x)=|\det A|^{1/2}f(Ax),$95 on generated-data elastic wave tests and approximately $(D_A f)(x)=|\det A|^{1/2}f(Ax),$96 on simulation-data elastic wave tests, approximately $(D_A f)(x)=|\det A|^{1/2}f(Ax),$97 and approximately $(D_A f)(x)=|\det A|^{1/2}f(Ax),$98 for generated and simulation EoM tests, and approximately $(D_A f)(x)=|\det A|^{1/2}f(Ax),$99 and approximately $$\widehat{D_A f}(\xi)=|\det A|^{-1/2}\widehat f((A^T)^{-1}\xi).$$00 for generated and simulation Lorenz tests (Gong et al., 30 Jul 2025).

A recurring misconception is that this AI SDO is merely a preprocessing heuristic. In the cited formulation it is an exact reversible map, with explicit inverse and a theorem relating dilation to the condition number bound. A different misconception is that SDO has a single cross-disciplinary definition. The literature instead supports a broader encyclopedic interpretation: SDO names a class of operators that alter scale while preserving a specified analytical, geometrical, statistical, or computational structure.