Fractal Equivalence Principle

• Fractal Equivalence Principle is a concept defining equivalence via the transfer of key invariants (e.g., singular-value spectra, Hausdorff dimension) across different representations.
• It distinguishes between exact, quasi-, and quasi-unitary equivalence, enabling easier computation and interpretation by choosing the most tractable formalism.
• The principle bridges abstract fractal geometry with practical applications in image analysis, structural mechanics, and spacetime metric deformation.

Searching arXiv for the cited papers to ground the article in current records. The expression Fractal Equivalence Principle is used in several technically distinct ways across contemporary research. In the strongest instances, it denotes an exact transfer of a fractal object into another formalism while preserving a decisive invariant such as a singular-value spectrum, a Hausdorff dimension, a stiffness law, or an energy form. In other instances it denotes a compensation principle, a renormalized equivalence, or a quasi-equivalence rather than literal identity. This suggests a family of equivalence statements centered on self-similarity, scale hierarchy, and invariant transfer, rather than a single canonical theorem (Lee et al., 2014, Epstein, 3 Apr 2025, Svozil, 2017, Moroz, 3 Oct 2025, Zhang et al., 5 Apr 2025, Post et al., 2018).

1. Terminological scope and recurring structure

A common pattern across the literature is that a fractal system is not treated as sui generis, but as equivalent to another representation in which the relevant invariant becomes easier to compute or interpret. The preserved object varies by domain: a spectrum in image analysis and quantum many-body theory, stiffness contributions in structural mechanics, intrinsic geometry in speculative spacetime models, Hausdorff dimension in symbolic expansions, metric structure in self-similar sets, or Laplacian data in graph and manifold approximations.

Domain	Equivalent descriptions	Preserved object
Fractal images and 1D free fermions	Fractal singular-value data and entanglement spectrum	Exact multiplicative spectrum
Fractal stiffness modeling	Total stiffness and additive self-similar stiffness modes	Mechanism-specific scaling
Fractal gravity	Fractal topology with standard metric and integer-dimensional support with nonstandard metric	Intrinsic observed continuum
Perron-type expansions	Positive and alternating digit systems	Hausdorff dimension
Self-similar set classification	Strict Hölder equivalence and dimension-rescaled Lipschitz equivalence	Metric classification invariant
pcf fractals, graphs, manifolds	Fractal, weighted graph, metric graph, graph-like manifold	Energy forms and spectral data

The literature also distinguishes sharply between exact equivalence, quasi-equivalence, and approximation in a controlled topology. This distinction is essential. In the Perron setting the correspondence is an exact Hausdorff-dimension transfer; in the Engel and Pierce settings only quasi-equivalence is proved. In structural mechanics, the total object is not self-similar under one universal scaling, but becomes equivalent only after decomposition into additive constituents. In graph-manifold approximation of pcf fractals, the relevant notion is not identity but $\delta$-quasi-unitary equivalence of energy forms.

2. Exact spectral equivalence between fractal images and quantum entanglement

The most explicit use of the idea is the exact mapping between the singular value spectrum of tensor-product fractal images and the entanglement spectrum of one-dimensional free fermions. For an image $M(x,y)$, the singular-value decomposition is written as

$$M(x,y)=\sum_{l=1}^{L} M^{(l)}(x,y),\qquad M^{(l)}(x,y)=U_l(x)\sqrt{\Lambda_l}\,V_l(y),$$

with normalized singular values

$\lambda_l=\frac{\Lambda_l}{\sum_l\Lambda_l},$

and snapshot entropy

$S_\chi=-\sum_{l=1}^{\chi}\lambda_l\ln\lambda_l.$

The coarse-grained snapshot is

$M_\chi=\sum_{l=1}^{\chi}M^{(l)}(x,y).$

For self-similar fractals generated by repeated tensor products,

$$M = H\otimes H\otimes \cdots \otimes H \qquad (N\ \text{copies},\ L=h^N),$$

if the nonzero eigenvalues of $H$ are $\gamma_1,\dots,\gamma_r$, then the nonzero eigenvalues of the full fractal matrix are

$\Gamma_{\mathbf a}=\prod_{j=1}^{N}\gamma_{a_j}.$

In the rank-two case, which includes the white-centered Sierpinski carpet and Sierpinski triangle, this yields a binary multiplicative spectrum identical in form to the reduced-density-matrix spectrum of free fermions,

$M(x,y)$0

Accordingly, the paper’s central claim is that the singular values of these fractals are mapped exactly to the occupation probabilities of entanglement modes in a 1D free-fermion system (Lee et al., 2014).

The entropy structure follows directly from this multiplicative factorization. For tensor-product self-similar fractals,

$M(x,y)$1

with

$M(x,y)$2

For the Sierpinski carpet the explicit value is

$M(x,y)$3

The logarithmic growth matches the Calabrese–Cardy form

$M(x,y)$4

The paper interprets this as a holographic relation: the fractal level $M(x,y)$5 acts as a scale direction, and the entropy counts information distributed across nested scales.

A central contrast appears at finite $M(x,y)$6. Because the singular values are highly degenerate, the coarse-grained entropy obeys

$M(x,y)$7

The mechanism is that the leading singular values satisfy $M(x,y)$8, so each newly retained mode contributes the same amount. This is unlike the usual near-critical one-dimensional quantum scaling

$M(x,y)$9

The paper therefore distinguishes a global logarithmic law in $$M(x,y)=\sum_{l=1}^{L} M^{(l)}(x,y),\qquad M^{(l)}(x,y)=U_l(x)\sqrt{\Lambda_l}\,V_l(y),$$0 from a linear finite-$$M(x,y)=\sum_{l=1}^{L} M^{(l)}(x,y),\qquad M^{(l)}(x,y)=U_l(x)\sqrt{\Lambda_l}\,V_l(y),$$1 law, interpreting the latter as a signature of scale invariance without full conformal symmetry.

The result is not claimed for arbitrary visually self-similar patterns. The exact mapping depends on the algebraic form $$M(x,y)=\sum_{l=1}^{L} M^{(l)}(x,y),\qquad M^{(l)}(x,y)=U_l(x)\sqrt{\Lambda_l}\,V_l(y),$$2. A counterexample is the black-centered Sierpinski carpet, written as $$M(x,y)=\sum_{l=1}^{L} M^{(l)}(x,y),\qquad M^{(l)}(x,y)=U_l(x)\sqrt{\Lambda_l}\,V_l(y),$$3, where the additive shift disrupts tensor-product factorization. The paper also states that the snapshot entropy is asymmetric under white/black exchange. A common misconception is therefore that any fractal image automatically carries a quantum-entanglement spectrum; the exact correspondence is restricted to a strong tensor-product self-similarity.

3. Mechanism-specific equivalence in fractal stiffness self-similarity

In structural mechanics, the relevant principle is not spectral identity but scaling equivalence of stiffness contributions. The generalized principle of fractal stiffness self-similarity states:

“Each additive contribution to the total stiffness matrix will separately abide by the principle of stiffness self-similarity, with its own stiffness-scaling factor.”

The starting point is the earlier stiffness self-similarity idea that “a change of geometrical scale brings about a concomitant change in the scale of the stiffness matrix.” For the Euler-Bernoulli beam, self-similarity reduces the stiffness scaling to

$$M(x,y)=\sum_{l=1}^{L} M^{(l)}(x,y),\qquad M^{(l)}(x,y)=U_l(x)\sqrt{\Lambda_l}\,V_l(y),$$4

so doubling the beam length reduces stiffness by a factor of $$M(x,y)=\sum_{l=1}^{L} M^{(l)}(x,y),\qquad M^{(l)}(x,y)=U_l(x)\sqrt{\Lambda_l}\,V_l(y),$$5. The generalized paper argues that this single-factor principle is too restrictive when the total stiffness arises from several physical mechanisms. For a triangular frame,

$$M(x,y)=\sum_{l=1}^{L} M^{(l)}(x,y),\qquad M^{(l)}(x,y)=U_l(x)\sqrt{\Lambda_l}\,V_l(y),$$6

and the axial and bending parts scale differently under geometric enlargement. The consequence is that self-similarity should not be imposed on the total matrix as a whole when several mechanisms coexist (Epstein, 3 Apr 2025).

The main application is a Sierpiński gasket with in-plane displacements and in-plane rotational “drilling” modes at each vertex, yielding 9 degrees of freedom for the triangular element. Symmetry and equilibrium restrict the total stiffness matrix to the block form

$$M(x,y)=\sum_{l=1}^{L} M^{(l)}(x,y),\qquad M^{(l)}(x,y)=U_l(x)\sqrt{\Lambda_l}\,V_l(y),$$7

with equilibrium condition

$$M(x,y)=\sum_{l=1}^{L} M^{(l)}(x,y),\qquad M^{(l)}(x,y)=U_l(x)\sqrt{\Lambda_l}\,V_l(y),$$8

After assembling three smaller copies of side $$M(x,y)=\sum_{l=1}^{L} M^{(l)}(x,y),\qquad M^{(l)}(x,y)=U_l(x)\sqrt{\Lambda_l}\,V_l(y),$$9 into a larger gasket of side $\lambda_l=\frac{\Lambda_l}{\sum_l\Lambda_l},$0, static condensation is performed via

$\lambda_l=\frac{\Lambda_l}{\sum_l\Lambda_l},$1

The ordinary one-factor self-similarity principle fails at this stage, and the paper instead identifies additive constituents that are separately self-similar.

The numerical solution yields two distinct stiffness modes: an axial mode and a bending mode. Their reported scaling ratios are

$\lambda_l=\frac{\Lambda_l}{\sum_l\Lambda_l},$2

The interpretation is that a self-similar fractal structure may be equivalent to a sum of mechanism-specific self-similar components rather than to one monolithic self-similar law. This is the paper’s deeper modeling message and the basis for its stated motivation toward fractal shells and related engineered structures.

A plausible implication is that, in mechanics, a “Fractal Equivalence Principle” is best understood as equivalence after decomposition, not literal scale-invariant identity of the full stiffness matrix. The paper also states its limits explicitly: the method is tailored to self-similar fractals and linear behavior, and it does not yet provide a general treatment for arbitrary fractals or nonlinear constitutive laws.

4. Fractal topology and metric deformation as interchangeable descriptions

A different use of the idea appears in the speculative proposal of fractal gravity. The paper hypothesizes that physical spacetime may be an embedded fractal continuum in a higher-dimensional ambient space, and that an intrinsic observer may nevertheless perceive an ordinary integer-dimensional continuum if local variations of fractal dimension are compensated by metric variation. Conversely, deviations from Euclidean or Minkowskian metric form may be shifted into nontrivial fractal topology. The operative trade is stated as:

fractal topology + standard metric can be re-described as integer-dimensional support + nonstandard metric

The paper does not give a formal theorem called exactly “Fractal Equivalence Principle,” but it clearly proposes an analogous compensation principle in which variations in dimension or topology are exchanged with variations in metric or curvature (Svozil, 2017).

The mathematical embodiment of this idea is a volume-matching relation

$\lambda_l=\frac{\Lambda_l}{\sum_l\Lambda_l},$3

where $\lambda_l=\frac{\Lambda_l}{\sum_l\Lambda_l},$4 is the extrinsic fractal dimension, $\lambda_l=\frac{\Lambda_l}{\sum_l\Lambda_l},$5 the outer curvature scale, $\lambda_l=\frac{\Lambda_l}{\sum_l\Lambda_l},$6 the intrinsic target dimension, and $\lambda_l=\frac{\Lambda_l}{\sum_l\Lambda_l},$7 the intrinsic curvature scale. The paper imposes

$\lambda_l=\frac{\Lambda_l}{\sum_l\Lambda_l},$8

with $\lambda_l=\frac{\Lambda_l}{\sum_l\Lambda_l},$9 the dimension of the embedding space. The Cantor set serves as the canonical example, with Hausdorff dimension

$S_\chi=-\sum_{l=1}^{\chi}\lambda_l\ln\lambda_l.$0

In the paper’s interpretation, the difference between noninteger extrinsic dimensionality and integer intrinsic dimensionality is absorbed into a modified metric scale.

The physical picture is framed in terms of “holes” or “gaps” in spacetime. Removing pieces of a continuum and stitching the remainder together changes the effective geometry; the claim is that such punctured or fractalized support could give rise to curvature. This is presented as a possible extension or alternative perspective on general relativity, not as an established gravitational theory. The embedded-observer perspective is central: an observer inside the system may be unable to distinguish between a space with fractal support and standard metric and a space with integer-dimensional support and nonstandard metric.

A recurrent misconception would be to read this as a proved equivalence theorem. The paper instead advances a speculative reformulation principle. Its significance lies in the explicit operational claim that extrinsic fractality and intrinsic metric deformation may be two descriptions of the same observed geometry.

5. Hausdorff-dimension transfer and strict Hölder classification

In symbolic and metric fractal geometry, “equivalence” is formulated with precise dimension-theoretic and metric criteria. For Perron expansions, the exact fractal equivalence principle states that if the positive and alternating Perron expansions are determined by the same sequence $S_\chi=-\sum_{l=1}^{\chi}\lambda_l\ln\lambda_l.$1, then corresponding digit-defined sets have the same Hausdorff dimension. Equivalently, the digit-preserving map

$S_\chi=-\sum_{l=1}^{\chi}\lambda_l\ln\lambda_l.$2

satisfies

$S_\chi=-\sum_{l=1}^{\chi}\lambda_l\ln\lambda_l.$3

A structural reason is that the corresponding $S_\chi=-\sum_{l=1}^{\chi}\lambda_l\ln\lambda_l.$4- and $S_\chi=-\sum_{l=1}^{\chi}\lambda_l\ln\lambda_l.$5-cylinders have the same diameter. The proof uses faithful families of coverings $S_\chi=-\sum_{l=1}^{\chi}\lambda_l\ln\lambda_l.$6 and $S_\chi=-\sum_{l=1}^{\chi}\lambda_l\ln\lambda_l.$7, together with covering theorems asserting that every interval or interval-minus-exceptional-set can be covered by at most three sets from the corresponding family. The same paper proves only quasi-equivalence for the classical and modified Engel expansions and for the two notations of Pierce expansions, because the relevant digit correspondences involve shifts rather than exact digit preservation (Moroz, 3 Oct 2025).

The distinction between exact equivalence and quasi-equivalence is substantive. In the Engel case, the correspondence shifts digits by $S_\chi=-\sum_{l=1}^{\chi}\lambda_l\ln\lambda_l.$8, and the dimension equality is recovered only under the lower-growth hypothesis

$S_\chi=-\sum_{l=1}^{\chi}\lambda_l\ln\lambda_l.$9

In the Pierce case, the shift is constant, and the condition becomes

$M_\chi=\sum_{l=1}^{\chi}M^{(l)}(x,y).$0

Thus small-digit behavior can obstruct a full equivalence principle even when the large-scale fractal behavior is transferable.

A related but distinct classification problem appears in strict Hölder equivalence of self-similar sets. Two metric spaces $M_\chi=\sum_{l=1}^{\chi}M^{(l)}(x,y).$1 and $M_\chi=\sum_{l=1}^{\chi}M^{(l)}(x,y).$2 are strictly Hölder equivalent if there exist a bijection $M_\chi=\sum_{l=1}^{\chi}M^{(l)}(x,y).$3, an exponent $M_\chi=\sum_{l=1}^{\chi}M^{(l)}(x,y).$4, and $M_\chi=\sum_{l=1}^{\chi}M^{(l)}(x,y).$5 such that

$M_\chi=\sum_{l=1}^{\chi}M^{(l)}(x,y).$6

for all pairs. For totally disconnected fractal cubes $M_\chi=\sum_{l=1}^{\chi}M^{(l)}(x,y).$7 and $M_\chi=\sum_{l=1}^{\chi}M^{(l)}(x,y).$8, with $M_\chi=\sum_{l=1}^{\chi}M^{(l)}(x,y).$9 and $$M = H\otimes H\otimes \cdots \otimes H \qquad (N\ \text{copies},\ L=h^N),$$0, the paper proves

$$M = H\otimes H\otimes \cdots \otimes H \qquad (N\ \text{copies},\ L=h^N),$$1

For self-similar sets under the strong separation condition, if

$$M = H\otimes H\otimes \cdots \otimes H \qquad (N\ \text{copies},\ L=h^N),$$2

and $$M = H\otimes H\otimes \cdots \otimes H \qquad (N\ \text{copies},\ L=h^N),$$3, then

$$M = H\otimes H\otimes \cdots \otimes H \qquad (N\ \text{copies},\ L=h^N),$$4

In the two-branch case, the generic irrational regime is classified by

$$M = H\otimes H\otimes \cdots \otimes H \qquad (N\ \text{copies},\ L=h^N),$$5

while the rational case admits only the exceptional pairs $$M = H\otimes H\otimes \cdots \otimes H \qquad (N\ \text{copies},\ L=h^N),$$6 and $$M = H\otimes H\otimes \cdots \otimes H \qquad (N\ \text{copies},\ L=h^N),$$7 inherited from the Lipschitz classification (Zhang et al., 5 Apr 2025).

These results sharpen the meaning of equivalence in fractal geometry. Exact Hausdorff-dimension preservation, quasi-equivalence under growth hypotheses, and strict Hölder equivalence after dimension renormalization are all precise but nonidentical notions. The broader lesson is that fractal equivalence is frequently governed by rigid symbolic, arithmetic, or dimension-theoretic invariants.

6. Quasi-unitary and spectral forms of fractal equivalence

In analysis on pcf fractals, the relevant principle is operator-theoretic. The paper on approximation by manifolds and graph-like spaces defines a distance between energy forms on different Hilbert spaces using identification operators $$M = H\otimes H\otimes \cdots \otimes H \qquad (N\ \text{copies},\ L=h^N),$$8. Two forms are $$M = H\otimes H\otimes \cdots \otimes H \qquad (N\ \text{copies},\ L=h^N),$$9-quasi-unitarily equivalent if the identification maps nearly intertwine the Hilbert structures and the quadratic forms. The analytic payoff is norm-resolvent control: for suitable continuous functions $H$0 on the spectrum,

$H$1

Consequently, resolvents, heat operators, and spectral projections are close; the transformed spectra approach each other; eigenvalues satisfy

$H$2

and corresponding eigenfunctions converge in the energy norm. The paper then applies transitivity of quasi-unitary equivalence to pass from a symmetric pcf fractal $H$3 to weighted graphs, from weighted graphs to metric graphs, and from there to graph-like manifolds. For the fractal-to-graph step,

$H$4

where $H$5 is the number of IFS maps and $H$6 is the energy renormalization parameter. For the Sierpiński triangle,

$H$7

hence

$H$8

for the metric-graph approximation. For graph-like manifolds with longitudinal scale $H$9 and transversal scale $\gamma_1,\dots,\gamma_r$0 satisfying

$\gamma_1,\dots,\gamma_r$1

the error is

$\gamma_1,\dots,\gamma_r$2

and the Sierpiński-triangle choice

$\gamma_1,\dots,\gamma_r$3

gives the rate

$\gamma_1,\dots,\gamma_r$4

Here the “equivalence principle” is that the spectral and analytic data of the fractal Laplacian can be recovered from graph or manifold models with explicit convergence rates (Post et al., 2018).

A related spectral manifestation appears in open quantum maps with Cantor trapped sets. The paper on open quantum baker’s maps does not use the phrase “Fractal Equivalence Principle,” but it demonstrates that fractal geometry of the trapped set controls quantum spectral behavior through a fractal uncertainty principle. The trapped Cantor set has dimension

$\gamma_1,\dots,\gamma_r$5

and the fractal uncertainty exponent is defined by

$\gamma_1,\dots,\gamma_r$6

The resulting spectral gap satisfies

$\gamma_1,\dots,\gamma_r$7

and the annular counting law is

$\gamma_1,\dots,\gamma_r$8

This supports a weaker but still precise principle: fractal classical geometry is reflected quantitatively in quantum resonance distribution, spectral gaps, and Weyl exponents (Dyatlov et al., 2016).

Taken together, these operator-theoretic and spectral results show that “equivalence” in fractal research often means preservation or asymptotic transfer of a functional calculus, a resonance law, or a Laplacian-based analytic structure. The principle is therefore domain-dependent, but the recurrent theme is stable: self-similar or fractal organization is not merely descriptive geometry; it can be recoded into another formal language without losing the governing invariant.