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Finite-Window Inheritance of Discrete Scale Symmetry in Participation Operators Jill F. Rankin Independent Researcher August 30, 2026 Abstract Discrete scale invariance (DSI) is naturally formulated on geometrically spaced spectra, but practical calculations and data analyses are necessarily finite. We study whether a participation operator constructed from a log-energy Gaussian kernel inherits DSI under finite spectral truncation, and how this inheritance propagates to a normalized commutator diagnostic. On the bi-infinite geometric lattice, the normalized participation weights satisfy exact one-step scale covariance. For finite windows, we derive an exact normalization identity and an explicit, dimension-independent bound on the shift defect. We then prove a Lipschitz-type perturbation bound for the normalized commutator diagnostic and establish both bi-infinite and finite- window participation-localization results. These theorems separate three distinct mechanisms: deterministic inheritance for shift-compatible coupling operators, distributional inheritance for cyclic-conjugation-invariant random-matrix ensembles, and the still-open problem of uniform concentration for a single frozen realization. The finite-window analysis reduces the distributional case to a standard random-matrix norm-ratio estimate and clarifies the remaining gap between pointwise inheritance and stable recovery of the DSI ratio from a complete waveform. 1 Introduction Discrete scale invariance appears when a system is invariant not under arbitrary rescaling, but under a preferred ratio μ↦→ λμ, λ > 1. Its observable signature is log-periodicity: suitably defined quantities repeat under translation inlogμbylogλ. In idealized infinite systems this structure is straightforward. In finite spectra, however, index shifts meet boundaries, normalization changes, and any diagnostic built from the resulting participation weights acquires a truncation error. This paper develops a finite-window theory for a participation operator constructed on a geometric spectrum E i = E 0 λ i . The participation weights are defined by a Gaussian in log energy, rather than in linear energy. This choice is structurally adapted to DSI because logE i − log(λμ) = logE i−1 − logμ. As a result, exact scale covariance is recovered on the bi-infinite lattice, while finite truncation produces a quantitatively controlled boundary defect. 1

The second object of interest is the normalized commutator diagnostic η(P,M ) = ∥[P,M ]∥ F ∥P∥ F ∥M∥ F , wherePis the diagonal participation operator andMis a fixed Hermitian coupling operator. The central question is not merely whetherPshifts approximately, but whether the scalar diagnosticη inherits the same DSI structure. The main contributions are:

1. an exact shift-covariance result for the bi-infinite log-energy Gaussian weights;
2. an exact finite normalization identity and an explicit boundary bound;
3. an explicit N -independent Frobenius bound for the finite-window shift defect;
4. a general perturbation theorem for the normalized commutator diagnostic;
5. a deterministic DSI inheritance result for magnitude-shift-invariant coupling operators;
6. a bi-infinite participation-localization lemma and a finite-window localization theorem;

a transformed-MWasserstein coupling that reduces distributional inheritance to a standard random-matrix norm-ratio estimate. The results also isolate what remains open. Exact and finite-window inheritance are analytic questions. Distributional inheritance requires a specified random-matrix ensemble. Recovery of the DSI ratio from one frozen realization is a stronger global problem involving uniform concentration and stability of the period estimator. 2 Related work DSI and log-periodic corrections to scaling have been studied extensively in critical phenomena, hierarchical systems, fracture, finance, and complex systems; see, for example, [3]. The present work is narrower: it concerns finite-window inheritance of DSI in a normalized participation operator and its propagation through a commutator diagnostic. The matrix analysis used below relies on standard mixed Frobenius/operator norm inequalities [2]. The statistical inheritance argument invokes cyclic-conjugation-invariant random-matrix ensembles and, when specialized to Wigner or GOE-type models, standard spectral norm asymptotics such as the Bai–Yin law [1]. The localization lemma is naturally expressed through shifted Gaussian lattice sums, equivalently Jacobi-theta functions. 3 Framework and notation 3.1 Geometric spectrum and log-energy kernel Let E i = E 0 λ i , λ > 1. For probe scale μ > 0 and kernel width s > 0, define a i (μ) = exp [︄ − (logE i − logμ) 2 2s 2 ]︄ . 2

On a finite window i = 0,...,N − 1, let Z N (μ) = N−1 ∑︂ i=0 a i (μ), w (N) i (μ) = a i (μ) Z N (μ) . The participation operator is P N (μ) = diag (︂ w (N) 0 (μ),...,w (N) N−1 (μ) )︂ . On the bi-infinite lattice, Z ∞ (μ) = ∑︂ i∈Z a i (μ), w (∞) i (μ) = a i (μ) Z ∞ (μ) . 3.2 Shift operator Let S be the cyclic one-step permutation matrix on R N , (Sv) i = v i−1 (modN) . Thus (︂ SP N S † )︂ ii = w (N) i−1 (modN) . 3.3 Normalized commutator diagnostic For nonzero Hermitian matrices P and M , define η(P,M ) = ∥[P,M ]∥ F ∥P∥ F ∥M∥ F . When P is diagonal with entries w i , ∥[P,M ]∥ 2 F

∑︂ i,j |w i − w j | 2 |M ij | 2 . Definition 3.1 (Magnitude-shift invariance). A matrix M is magnitude-shift invariant if |M i+1,j+1 | =|M ij | for all indices modulo N . 4 Exact bi-infinite scale covariance Lemma 4.1 (Exact shift covariance of the bi-infinite weights). For every μ > 0, Z ∞ (λμ) = Z ∞ (μ), w (∞) i (λμ) = w (∞) i−1 (μ). Proof. Since logE i − log(λμ) = logE i−1 − logμ, we havea i (λμ) =a i−1 (μ). The Gaussian lattice sum converges absolutely, so reindexing is legitimate: Z ∞ (λμ) = ∑︂ i∈Z a i (λμ) = ∑︂ i∈Z a i−1 (μ) = Z ∞ (μ). The normalized identity follows. 3

Theorem 4.2 (Exact distributional inheritance). Assume P (λμ) = SP (μ)S † and let M be random with M d = S † MS. Then η(P (λμ),M ) d = η(P (μ),M ). Proof. Unitary invariance gives η(SPS † ,M ) = η(P,S † MS). Therefore η(P (λμ),M ) = η(P (μ),S † MS) d = η(P (μ),M ). Remark 4.3. Theorem 4.2 assumes an exact finite-dimensional shift identity. Lemma 4.1 proves exact covariance for the bi-infinite weight sequence; finite matrices require the boundary analysis below. 5 Finite-window shift theory Lemma 5.1 (Finite normalization identity). Z N (λμ)− Z N (μ) = a −1 (μ)− a N−1 (μ). Proof. Z N (λμ) = N−1 ∑︂ i=0 a i (λμ) = N−1 ∑︂ i=0 a i−1 (μ)

N−2 ∑︂ i=−1 a i (μ) = a −1 (μ) + Z N (μ)− a N−1 (μ). Define d low (μ) = logμ− logE −1 logλ , d high (μ) = logE N−1 − logμ logλ , d min (μ) = min{d low (μ),d high (μ)}, and δ Z (μ) = 2 exp [︃ − d min (μ) 2 (logλ) 2 2s 2 ]︃ . Corollary 5.2 (Finite normalization bound). |Z N (λμ)− Z N (μ)|≤ δ Z (μ). 4

Proof. Because a −1 (μ) = exp [︃ − d low (μ) 2 (logλ) 2 2s 2 ]︃ ≤ exp [︃ − d min (μ) 2 (logλ) 2 2s 2 ]︃ , and similarly for a N−1 (μ), the claim follows from Lemma 5.1. Let r(μ) = w (N) (λμ)− Sw (N) (μ), R(μ) = diag(r(μ)), and Z min = min{Z N (μ),Z N (λμ)}. Theorem 5.3 (Finite participation shift bound). Assume bothμandλμlie in the admitted interior sweep. Then ⃦ ⃦ ⃦ P N (λμ)− SP N (μ)S † ⃦ ⃦ ⃦ F ≤ √ 2 δ Z (μ) Z min . If z ∗ = exp [︃ − (logλ) 2 8s 2 ]︃ , then ⃦ ⃦ ⃦ P N (λμ)− SP N (μ)S † ⃦ ⃦ ⃦ F ≤ √ 2 δ Z (μ) z ∗ . Proof. At the cyclic boundary, |r 0 | = ⃓ ⃓ ⃓ ⃓ a −1 Z N (λμ) − a N−1 Z N (μ) ⃓ ⃓ ⃓ ⃓ ≤ δ Z Z min . For i = 1,...,N − 1, r i = a i−1 (μ) Z N (μ)− Z N (λμ) Z N (μ)Z N (λμ) . Hence N−1 ∑︂ i=1 r 2 i ≤ δ 2 Z Z N (μ) 2 Z N (λμ) 2 N−2 ∑︂ i=0 a i (μ) 2 ≤ δ 2 Z Z 2 min , using ∑︁ i a 2 i ≤ ( ∑︁ i a i ) 2 . Thus ∥R∥ 2 F ≤ 2 δ 2 Z Z 2 min . The interior-domain assumption ensures that both normalizations contain a term at least z ∗ . Remark 5.4 (Denominator safety). Since∥P N ∥ F ≥ N −1/2 , a sufficient condition for∥R∥ F <∥P N ∥ F is √ 2N δ Z < z ∗ . Proposition 5.5 (Exact deterministic invariance). If M is magnitude-shift invariant, then η(SPS † ,M ) = η(P,M ). Proof. The squared commutator norm depends on M only through |M ij | 2 : ∥[P,M ]∥ 2 F

∑︂ i,j |w i − w j | 2 |M ij | 2 . A simultaneous cyclic shift leaves this sum unchanged. 5

6 Perturbation of the normalized commutator Theorem 6.1 (Lipschitz-type perturbation bound). Let Q = P + R, with p =∥P∥ F , r =∥R∥ F , r < p. Then |η(Q,M )− η(P,M )|≤ 4r p− r ∥M∥ 2 ∥M∥ F . Proof. Let A =∥[Q,M ]∥ F , B =∥[P,M ]∥ F , q =∥Q∥ F . Then ⃓ ⃓ ⃓ ⃓ A q − B p ⃓ ⃓ ⃓ ⃓ ≤ |A− B| q

• B ⃓ ⃓ ⃓ ⃓ 1 q − 1 p ⃓ ⃓ ⃓ ⃓ . The mixed norm inequalities imply |A− B|≤ 2r∥M∥ 2 , q ≥ p− r, |p− q|≤ r, B ≤ 2p∥M∥ 2 . Substitution gives the result after division by ∥M∥ F . Corollary 6.2 (Finite-window deterministic DSI inheritance). IfMis magnitude-shift invariant, then |η(P N (λμ),M )− η(P N (μ),M )|≤ 4r(μ) p(μ)− r(μ) ∥M∥ 2 ∥M∥ F , where r(μ) = ⃦ ⃦ ⃦ P N (λμ)− SP N (μ)S † ⃦ ⃦ ⃦ F ≤ √ 2 δ Z (μ) z ∗ . 7 Participation localization Write x = logμ− logE 0 logλ , α = (logλ) 2 2s 2 , and A α (x) = ∑︂ i∈Z e −α(i−x) 2 . Then ∥P ∞ (x)∥ 2 F = A 2α (x) A α (x) 2 . Lemma 7.1 (Bi-infinite participation localization). For every fixeds >0 andλ >1, there exists c(s,λ) > 0 such that ∥P ∞ (x)∥ F ≥ c(s,λ) for all x∈ R. Proof. The Gaussian lattice sums are continuous, strictly positive, and 1-periodic. Their ratio F α (x) = A 2α (x) A α (x) 2 therefore attains a positive minimum on [0, 1]. 6

Write x = k + φ, k ∈ Z, φ∈ [0, 1). Embed the finite normalized vector in ℓ 2 (Z) by zero-padding. Let T N (x) = ∑︂ i<0 a i (x) + ∑︂ i≥N a i (x), so that Z ∞ = Z N

• T N . Theorem 7.2 (Finite-window participation localization). Suppose the probe lies at leastd ≥1 log-index units from both finite boundaries. Then ∥P N (x)∥ F ≥ c(s,λ)− √ 2 T N (x) z ∗ . Moreover, T N (x)≤ 2e −αd 2 1− e −α(2d+1) . Proof. On retained indices, w N (i)− w ∞ (i) = a i T N Z N Z ∞ . Thus N−1 ∑︂ i=0 |w N (i)− w ∞ (i)| 2 ≤ T 2 N Z 2 ∞ . On omitted indices, ∑︂ i/∈[0,N−1] |w ∞ (i)| 2 ≤ T 2 N Z 2 ∞ . Therefore ∥w N − w ∞ ∥ F ℓ 2 ≤ √ 2 T N Z ∞ ≤ √ 2 T N z ∗ . The reverse triangle inequality gives the localization result. For either one-sided tail, ∞ ∑︂ j=0 e −α(d+j) 2 ≤ e −αd 2 1− e −α(2d+1) , because (d + j) 2 ≥ d 2
• j(2d + 1). Summing both sides gives the bound for T N . 8 Distributional inheritance in finite windows Let M be random with M d = S † MS. Define X(M ) = η(P N (λμ),M ), Y (M ) = η(P N (μ),S † MS). Then L[Y ] =L[η(P N (μ),M )], while Y (M ) = η(SP N (μ)S † ,M ). 7

Proposition 8.1 (Finite-window Wasserstein reduction). Let r = ⃦ ⃦ ⃦ P N (λμ)− SP N (μ)S † ⃦ ⃦ ⃦ F , p =∥P N (μ)∥ F , with r < p. Then W 1 (L[η(P N (λμ),M )],L[η(P N (μ),M )])≤ 4r p− r E [︃ ∥M∥ 2 ∥M∥ F ]︃ . Proof.The pair (X(M),Y(M)) is a valid coupling of the two target laws. Apply Theorem 6.1, take expectations, and use the defining variational property of W 1 . For a specified GOE/Wigner-type ensemble satisfying E [︃ ∥M∥ 2 ∥M∥ F ]︃ = O(N −1/2 ), the Wasserstein defect is O(δ Z ) using only p≥ N −1/2 and improves to W 1 = O (︃ δ Z √ N )︃ when finite-window localization provides an N -independent lower bound on p. 9 Numerical program The numerical study should compare the original linear-energy Gaussian with the log-energy Gaussian analyzed here. Report both r (∞) DSI = max μ∈I |η(λμ)− η(μ)| and r (2) DSI

⎡ ⎣ 1 |I| ∑︂ μ∈I |η(λμ)− η(μ)| 2 ⎤ ⎦ 1/2 . Controls should include:

1. a nontrivial magnitude-shift-invariant M ;
2. a deliberately symmetry-broken sparse M ;
3. a diagonal M , for which η = 0;
4. many random-matrix seeds;
5. variation in N , s, and boundary exclusion d. Report period recovery separately: ε λ = | ˆ︁ λ− λ| λ . 8

10 Discussion The results identify three inheritance mechanisms. Trivial commutation. If [P,M ] = 0, then η ≡ 0: formally periodic but uninformative. Deterministic inheritance. IfMis magnitude-shift invariant, finite-window DSI error is bounded by the participation shift defect and the perturbation theorem. Statistical inheritance. If the law ofMis cyclic-conjugation invariant, exact inheritance holds in distribution in the ideal setting, while finite-window law defects admit a Wasserstein bound. 11 Open problems

1. State a precise random-matrix ensemble and quote or prove an explicit finite-N bound for E [︃ ∥M∥ 2 ∥M∥ F ]︃ .
2. Prove uniform concentration for a single frozen realization: P (︄ sup μ∈I |η M (μ)− η ens (μ)| > ε )︄ ≤ Ψ(N,ε).
3. Prove stability of the recovered DSI ratio under waveform perturbation.
4. Compare the original linear-energy Gaussian with the log-energy Gaussian.
5. Extend the analysis beyond Gaussian kernels. 12 Conclusion A log-energy Gaussian participation operator on a geometric spectrum inherits discrete scale symmetry exactly on the bi-infinite lattice and approximately in finite windows. The finite defect is controlled by Gaussian boundary tails and propagates through the normalized commutator diagnostic by an explicit perturbation theorem. Deterministic inheritance follows for magnitude- shift-invariant coupling operators. Distributional inheritance follows for cyclic-conjugation-invariant random-matrix ensembles, with finite-window error reduced to a Wasserstein bound and a standard random-matrix norm-ratio estimate. The resulting hierarchy separates exact algebraic covariance, finite-size analytical control, ensem- ble distributional inheritance, and the still-open problem of uniform single-realization concentration and estimator stability. Acknowledgments The author thanks the independent reviewers whose repeated critiques materially improved the mathematical formulation and sharpened the distinction among deterministic, statistical, and single-realization inheritance. 9

References [1] Zhidong D. Bai and Y. Q. Yin. Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. The Annals of Probability, 21(3):1275–1294, 1993. Replace or supplement with the exact Bai–Yin or Wigner spectral-norm reference used in the final C2 statement. [2] Rajendra Bhatia. Matrix Analysis. Springer, New York, 1997. [3] Didier Sornette. Discrete-scale invariance and complex dimensions. Physics Reports, 297(5):239– 270, 1998. 10