FMU: Fractal-Mirror Unification of Frequency--Scale--Information

FMU: Fractal-Mirror Unification of Frequency–Scale–Information Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 19, 2025 Abstract Within weighted Mellin–Hilbert space L2(R+, dt/t), we establish rigorous theory of fractal mirror (FM) signal families generated by mother function Mthrough multiplicative scale replication. Around three main threads “frequency–scale– information”, we provide equivalent characterizations and complete proofs: (A) Under Mellin–Calder´on condition, multiplicative self-similarity ⇐⇒ Mellindomain quasi-periodicity (critical line exhibits equidistant frequency-shift array); (B) spectral power-law ⇐⇒ (in logarithmic scale) entropy slope firstorder approximate linearity, and in self-affine model derive classical relation D= (3 −β)/2 for image box dimension; (C) With Nyquist–Poisson–Euler– Maclaurin (three-fold decomposition) achieve non-asymptotic error closure and separable budget. Further, with spectral density weight measure dµ = (1/π)ℑm(ω+i0) dω, isometrically embed FM subspace into Paley–Wiener type bandlimited space, thereby deriving Landau type sampling/interpolation density thresholds,Wexler–Raz tight/dual criteria, and Balian–Low impossibility at critical density; prove this spectral density weight consistent with Herglotz representation of Weyl–Titchmarsh m-function of one-dimensional self-adjoint canonical systems. Above scales and criteria compatible with established standards including Mellin isometry, Paley–Wiener, Poisson summation, Euler–Maclaurin, Landau density, Wexler–Raz and Balian–Low, Herglotz representation, and de Branges inverse spectral theorem. Keywords: Fractal mirror; Mellin transform; Quasi-periodicity; Power-law spectrum; Entropy-slope coupling; Nyquist–Poisson–EM three-fold; Landau density; Wexler–Raz; Balian–Low; Herglotz; de Branges 0. Notation & Baseplates (0.1) Mellin Isometry and Critical Line For x∈L2(R+, dt/t), Mellin transform Mx(s) = Z∞ 0 x(t)ts−1dt, (1) 1 on critical line s=1 2+iω isometric with “logarithmic Fourier” unit: |x|2 L2(dt/t)=1 2πZRMx(1 2+iω)2dω. (2) Thus f M:x7→ (2π)−1/2Mx(1 2+i·) is unitary isometry on L2(R+, dt/t)→L2(R). (0.2) Logarithmic Variable and Scaling Take u= log t, let m(u) = M(eu). Scale t7→ 2ktbecomes logarithmic translation u7→ u+klog 2. Mellin scaling law yields MM(2k·)(s) = 2−ks MM(s) (s=1 2+iω),(3) i.e., amplitude factor 2−k/2and phase modulation e−ikω log 2. (0.3) Spectral Density Measure and Phase Coordinate Notation and transform convention: Mdenotes Mellin transform with respect to t evaluated at s=1 2+iω;b·universally denotes Fourier transform with respect to u= log t (equivalent to ω-domain operator along critical line). For systems associated with Weyl–Titchmarsh m-function, define spectral density weight measure dµ(ω) = 1 πℑm(ω+i0) dω. (4) When ℑm≥0, µis positive measure. Under special normalization (e.g., certain boundary conditions making ℜm(λ+i0) ≡0 a.e.), can use phase representation dµφ(ω) = (1/π)d(arg m(ω+i0)); general case uses spectral density weight dµ. Consistency with spectral measure of self-adjoint systems see § 6. Convention (u-domain Fourier): b f(ω) = (2π)−1/2RRf(u)e−iωu du,f(u) = (2π)−1/2RRb f(ω)eiωu dω. Notation Convention (Avoiding Conflict): Throughout fix u:= log tas logarithmic time variable; spectral density weight coordinate denoted separately as vµ.u-domain Fourier only acts on uvariable; isometric maps and density criteria related to spectral density weight stated in vµcoordinate. (0.4) Finite-Order Euler–Maclaurin (EM) Throughout use only finite-order EM, decomposing “sum–integral” difference into endpoint Bernoulli layer and remainder: b X n=a f(n) = Zb a f(x)dx +f(a) + f(b) 2+ p−1 X r=1 B2r (2r)!f(2r−1)(b)−f(2r−1)(a)+Rp,(5) with control |Rp|≲2ζ(2p) (2π)2pRb a|f(2p)|. 2 1 Fractal Mirror (FM) Signal Family: Definition and Basic Properties Definition 1.1 (FM Generation and Mellin–Calder´on Condition).Given mother function M∈L2(R+, dt/t), weight sequence {ak}k∈Z∈ℓ2, phase sequence {ϕk} ⊂ Rwith supk|ϕk|<∞. Define x(t) = X k∈Z akM(2kt)eiϕk, t > 0.(6) Assumption (H0) Weighted ℓ2Consistency. Let bk:= ak2−k/2eiϕk, require {bk}k∈Z∈ℓ2(equivalently Pk|ak|22−k<∞). Assumption (H1) Mellin–Calder´on Boundedness. Write G(ω) := MM(1 2+iω), P:= 2π/ log 2, require Calder´on sum CG(ω) := X n∈ZG(ω+nP)2∈L∞([0, P]).(7) Proposition 1.2 (L2Unconditional Convergence and Frequency Structure, Under (H0)–(H1)). Under assumptions (H0)–(H1), series PkakM(2k·)eiϕkunconditionally converges in L2(R+, dt/t), and Mx1 2+iω=G(ω)X k∈Z bke−ikω log 2, bk:= ak2−k/2eiϕk.(8) With Bessel bound |x|2 L2(dt/t)≤P 2π∥CG∥L∞([0,P]) X k∈Z |bk|2, P =2π log 2.(9) Proof. By (H1)’s Calder´on upper bound and Plancherel–Mellin isometry, obtain |x|2 L2(dt/t)≤P 2π∥CG∥L∞([0,P]) X k∈Z |bk|2, P =2π log 2,(10) thus series Cauchy converges; frequency expression follows directly from scaling law. 2 Main Theorem A: Mellin-Quasi-Periodic Characterization of Multiplicative Self-Similarity Theorem 2.1 (Self-Similarity ⇐⇒ Quasi-Periodicity).For xfrom Definition 1.1, following equivalent: (i) x(t) = PkakM(2kt)eiϕk(multiplicative self-similar superposition); (ii) Exists envelope G(ω) = MM(1 2+iω)∈L2(R)such that Mx1 2+iω=G(ω)·X k∈Z bke−ikω log 2 | {z } Bohr quasi-periodic; frequency-shift lattice spacing 2π/ log 2 .(11) Proof. “⇒” by Proposition 1.2. “⇐” Apply inverse Mellin (σ=1 2) to quasi-periodic part and use “logarithmic-domain translation ↔Mellin frequency-shift” duality, immediately obtain superposition of logarithmic translation family M(2k·). 3 3 Main Theorem B: Spectral Power-Law–Entropy Slope and Self-Affine Dimension 3.1 Power-Law Spectrum and Logarithmic Binned Entropy Linear Coupling (Approximate Relation) Proposition 3.1 (Entropy–Slope Approximate Linearity, First-Order Regime).Let power spectrum over wide frequency ratio Λ = fmax/fmin ≫1satisfy S(f)≍Cf−β, logarithmic uniform binning Ij= [eyj, eyj+1 ],δy := yj+1 −yj≪1fixed, bin number J∼(log Λ)/δy, probability Pj∝RIjS(f)df normalized. Then in first-order approximation H:= −X j Pjlog Pj=c1(δy) + c2(δy)β+Oδy+O(log Λ)−1,(12) where c1, c2depend only on binning step δy and window overlap constants, explicitly computable; as δy →0,Λ→ ∞, main term linearity holds. Proof Sketch. Let y= log f, have df =f dy, thus Pj∝Ryj+1 yje(1−β)ydy. Uniform ybinning makes {Pj}approximately exponentially distributed; substitute back into entropy and approximate by Riemann sum, δy discretization error yields O(δy) term, endpoints/overlap yield O((log Λ)−1) term; both negligible when δy ≪1/log Λ. 3.2 Self-Affine Spline and Image Dimension (Canonical Model) Theorem 3.2 (D= (3 −β)/2).If spline satisfies x(λt)d =λHx(t)(e.g., fBm), then S(f)∼f−(2H+1) =⇒Dgraph = 2 −H=3−β 2.(13) Proof. Self-affine processes like fBm satisfy S(f)∝f−(2H+1); their spline graph’s Hausdorff/box dimension D= 2 −His classical result, jointly eliminate Hto obtain above formula. Related conclusions see Flandrin’s analysis of fBm spectrum and Xiao’s rigorous measure results for image dimension. 4 Main Theorem C: Nyquist–Poisson–EM Non-Asymptotic Error Closure Let target frequency-domain quantity written as F(ω) = bw(ω)b h(ω)X(ω), X(ω) := (2π)−1/2Mx1 2+iω,(14) where bw(ω), b h(ω) are analysis window and interpolation kernel frequency responses after u-domain Fourier transform per § 0.3 convention, X(ω) is Mellin image along critical line (normalized by § 0.1’s unitary isometry). Theorem 4.1 (Nyquist Condition and Three-Fold Decomposition Error).Suppose exists B > 0such that supp F⊂[−B, B], where B:= min{Ωw,Ωh}(if Xnon-bandlimited, take effective bandwidth as intersection supp( bw)∩supp(b h)). If sampling step ∆≤π B,(15) 4 then aliasing energy is zero. In general case, aliasing term written as εalias =X ℓ=0 F·+2πℓ/∆L2([−π/∆, π/∆]),(16) for bandlimited linear reconstruction operator E, total error decomposes as |x−Ex|L2≤εalias |{z} periodic superposition +ε(p) EM |{z} O(|B2p|∆2p) +εtail |{z} band-edge/truncation ,(17) and ε(p) EM =O|B2p|∆2p. Proof. Use Poisson summation to convert discretization into spectrum periodization; Nyquist threshold eliminates inter-band overlap. Sum–integral difference decomposed by finite-order EM into endpoint Bernoulli layer and remainder, remainder bound see § 0.4. 5 Sampling–Interpolation–Stability: Landau–Wexler– Raz–Balian–Low in Spectral Density Coordinate 5.1 Isometric Embedding of Spectral Density Weight Coordinate In region ℑm(ω+i0) >0, define spectral density weight coordinate vµ(ω) = 1 πZω −∞ ℑm(ω′+i0) dω′, dvµ=1 πℑm(ω+i0) dω, (18) then ZR |X(ω)|21 πℑm(ω+i0) dω =ZR |X(ω(vµ))|2dvµ,(19) thus isometrically embed spectral density weight weighted FM-subspace into unit bandwidth Paley–Wiener type space. Under special normalization, can simplify to phase coordinate vµ=φ(ω)/π. Paley–Wiener and Hardy/Mellin-Hardy structure in Mellin context see literature. 5.2 Landau Type Necessary Condition (FM Version) Theorem 5.1 (Necessary Density Threshold, vµ-Domain).Let Ω = {ωn}be sampling sequence (respectively: interpolation sequence). In vµcoordinate (defined in § 5.1), its Beurling lower (respectively upper) density satisfies Dµ(Ω) ≥1 (respectively , Dµ(Ω) ≤1).(20) Remark: Above are necessary conditions. Sufficiency generally requires additional separation/stability structural conditions; in practice can design stable sampling/reconstruction via § 5.3’s WR/Parseval conditions. Proof Sketch. By 5.1’s isometry, problem reduces to non-uniform sampling of unit bandwidth Paley–Wiener space, directly invoke Landau necessary density theorem. 5 5.3 Wexler–Raz and Parseval Tight Frame (Critical Nyquist) Theorem 5.2 (WR/Parseval Condition).Under Nyquist (non-aliasing) condition, system generated by multi-windows {wα}r α=1 is Parseval tight frame if and only if 1 ∆ r X α=1 cwα(ξ)2≡1(a.e.).(21) With aliasing present, Parseval condition becomes 1 ∆ r X α=1 X m∈Zcwα ξ+ 2πm/∆2≡1.(22) Reconstruction with dual windows {ewα}satisfies corresponding biorthogonality formula. Proof Sketch. WR identity yields frequency-domain pointwise orthogonality necessary and sufficient condition; through u= log tand phase coordinate transformation, losslessly transplants to Mellin/logarithmic model. 5.4 Balian–Low Type Impossibility (Critical Density) Theorem 5.3 (BLT–Mellin Version).At critical density D= 1, if single window w“welllocalized” on both logarithmic time uand frequency ωsides (e.g., finite second moment), then system generated by wand critical lattice in vµcoordinate cannot be Riesz basis; to obtain basis, must relax at least one side’s localization or employ oversampling. Proof Sketch. Through u= log tand § 5.1’s isometric embedding, problem equivalent to standard Gabor lattice BLT; immediately follows from BLT’s Riesz/ONB version. 6 Consistency with de Branges–Kre˘ın / Weyl–Titchmarsh m-Function Proposition 6.1 (Spectral Density Weight and Herglotz Representation).In one-dimensional self-adjoint canonical system/Schr¨odinger type operator case, Weyl–Titchmarsh mis Herglotz–Nevanlinna function, exists spectral measure µsuch that m(z) = az +b+ZR1 λ−z−λ 1 + λ2dµ(λ),(23) boundary value satisfies ℑm(λ+i0) = π ρ(λ)(a.e.), where ρis spectral density. Thereby define spectral density weight measure dµ(ω) = 1 πℑm(ω+i0) dω =ρ(ω)dω, (24) consistent with absolutely continuous spectral measure. Under special normalization (e.g., certain boundary conditions making ℜm(λ+i0) ≡0a.e.), can simplify to phase representation dµφ= (1/π)d(arg m); general case must use above spectral density weight. Remark 6.2. This consistency ensures spectral density scale defined by dµ in this paper seamlessly interfaces with spectral theory of de Branges spaces/canonical systems, becoming natural coordinate for density and frame criteria in § 5; phase derivative φ′=d(arg m)/dω generally also depends on ℜmand m′, reduces to πρ only in special cases. 6 7 Reproducible Experimental Paradigm (Verification and Engineering) P1 — Power-Law/Entropy Coupling. Over sufficiently wide logarithmic bandwidth, fit S(f)∝f−βand verify with logarithmic binned entropy Hthat regression slope is linear within error band ( § 3.1’s first-order approximation). P2 — Mellin Peak Array. Compute Mx(1 2+iω), verify equidistant peak array and relative phase stability ( § 2). P3 — Sampling/Window Design. Select lattice per § 5.2’s density threshold; tune windows per § 5.3’s WR condition to obtain Parseval; perform separable budget for error per § 4’s three-fold decomposition (Nyquist margin, EM order, band-edge tail term). 8 Information Trinity (i+, i0, i−)and Model Selection  i+: Cross-scale overflow benefit (βtrending red ⇒low-frequency concentration ⇒ compression/prediction benefit).  i0: Intra-layer rearrangement (phase–coherence “neutral” redistribution).  i−: Sparsity and complexity penalty (avoid overfitting/over-dense lattice). Proposition 8.1 (Strategy in Approximate Linear Regime).In § 3.1’s first-order approximation regime, ∂βH≈c2, effective layer number Neff ≍(1 + β) log Λ. Accordingly jointly select “window/lattice density–model complexity” to maximize i+−i−satisfying § 4’s three-fold decomposition budget. 9 Interface with S-series / WSIG-QM / UMS 9.1 Interface with S24–S26  S24’s fiber Gram characterization and Wexler–Raz biorthogonality provide concrete implementation framework for this paper’s § 5.3 WR condition.  S25’s non-stationary Weyl–Mellin framework shares mathematical structure with this paper’s Mellin isometry ( § 0.1) and logarithmic translation–frequency shift duality ( § 0.2).  S26’s spectral density scale consistent at Herglotz representation level with this paper’s § 0.3 and § 6’s spectral density weight measure dµ = (1/π)ℑm(ω+i0) dω; S26’s Landau necessary density, Balian–Low impossibility directly correspond to this paper’s Theorems 5.1, 5.3. 9.2 Interface with WSIG-QM  WSIG-QM’s axiom A2 (finite window readout) shares Nyquist–Poisson–EM threefold decomposition framework with this paper’s § 4 windowed reconstruction. 7  WSIG-QM’s axiom A5 (phase–density–delay scale) consistent at spectral theory level with this paper’s § 0.3, § 6’s spectral density weight measure.  WSIG-QM’s theorem T6 (window/kernel optimization) shares frame theory criteria with this paper’s § 5.2–5.3’s Landau density threshold, WR condition. 9.3 Interface with UMS  UMS’s core unification formula dµ =1 2πtr QdE =ρrel dE in Mellin context corresponds to this paper’s § 0.3, § 6’s spectral density weight measure; under special normalization can simplify to phase representation.  UMS’s axiom A2 (finite window readout) shares framework at numerical implementation level with this paper’s Theorem 4.1’s three-fold decomposition error closure.  UMS’s axiom A6 (sampling–frame threshold) completely aligns with this paper’s § 5’s Landau–Wexler–Raz–Balian–Low criteria. 9.4 Interface with Windowed Path Integral Theory  Path integral theory’s window–kernel duality (Theorem 2.1) can be rewritten in Mellin domain as this paper’s Theorem 2.1’s quasi-periodic formulation.  Path integral theory’s Nyquist–Poisson–EM error closure consistent in discretization framework with this paper’s Theorem 4.1’s three-fold decomposition. 9.5 Interface with Quantum Gravity Field Theory  Quantum gravity field theory’s spectral density scale consistent in spectral shift context with this paper’s § 0.3, § 6’s spectral density weight measure dµ = (1/π)ℑm(ω+ i0) dω.  Quantum gravity field theory’s windowed sampling ( § 6.1) shares frame theory foundation with this paper’s § 5’s Landau–Wexler–Raz criteria. 9.6 Maintaining “Poles = Principal Scales” Finite-Order EM Discipline  Throughout all discrete–continuous exchanges, this paper employs finite-order EM ( § 0.4, Theorem 4.1), ensuring no new singularities introduced.  Consistent with S15–S26, WSIG-QM, UMS, path integral theory, quantum gravity field theory: EM remainder serves only as bounded perturbation. Appendix A: Proof Details and Tools A.1 Mellin–Hardy and Isometry. f Mis unitary isometry on σ=1 2; construction of Mellin–Paley–Wiener and Mellin–Hardy spaces see Bardaro–Butzer–Mantellini– Schmeisser. 8 A.2 Poisson and Aliasing. Dirac comb of sampling step ∆ in frequency domain is Dirac comb of period 2π/∆; aliasing energy equals periodized side-spectrum superposition energy in main band. A.3 Euler–Maclaurin Remainder. Employ DLMF version EM formula and remainder bound, ensuring finite-order approximation introduces no additional singularities; error given by Bernoulli numbers and step size. A.4 Landau Density. Sampling (interpolation) sequences of Paley–Wiener space PWBmust satisfy D≥B/π (D≤B/π); in this paper isometric to threshold 1 at unit bandwidth in vµcoordinate. A.5 Wexler–Raz and BLT. WR identity yields frequency-domain pointwise necessary and sufficient condition for tight/dual frames; BLT shows at critical density “good double-sided localization + non-redundancy” incompatible. A.6 Herglotz Representation and Spectral Density. mHerglotz function ⇒ exists spectral measure representation; boundary imaginary part ℑm(λ+i0) = πρ(λ) (a.e.). Thereby define spectral density weight measure dµ = (1/π)ℑm dω; phase derivative φ′=d(arg m)/dω generally also depends on ℜmand m′, reduces to πρ only under special normalization. Conclusion 1. Multiplicative self-similarity via Mellin transform equivalent to quasi-periodic frequency-shift array, controlled by envelope G(Theorem 2.1); under weighted ℓ2consistency (H0) and Mellin–Calder´on condition (H1), series unconditionally converges (Proposition 1.2). 2. Power-law spectrum and logarithmic binned entropy linearly coupled in first-order approximation; in self-affine limit D= (3 −β)/2 holds (Proposition 3.1, Theorem 3.2). 3. Employing Nyquist–Poisson–EM three-fold decomposition, can stably decompose total error into “aliasing/Bernoulli layer/tail term” three-term budget (Theorem 4.1). 4. Spectral density weight coordinate vµmakes FM subspace isometric with Paley– Wiener space, thereby inheriting Landau density threshold, Wexler–Raz tight/dual criteria, and Balian–Low impossibility; its spectral density weight measure consistent with Herglotz representation of m-function ( § 5– § 6). References [1] Bardaro, Butzer, Mantellini, Schmeisser, On the Paley–Wiener theorem in the Mellin transform setting (2015) and sequel (2017). [2] Butzer & Jansche, A Direct Approach to the Mellin Transform (1997). [3] NIST DLMF § 1.8(iv); Cand`es lecture notes (2021). [4] NIST DLMF § 2.10(i), § 24.17. 9