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This work did not originate from a deliberate searchfor new geometric laws of quantum entropy. Instead, itemerged from a pragmatic attempt to amplify the thermodynamic footprint of Photon Number Splitting (PNS)attacks in BB84 protocols using chaotic sensitivity. Unexpectedly, we observed that information loss acts as a catalyst for fractal roughness rather than geometric smoothing — a counterintuitive behavior that persisted acrossdistinct decoherence channels and multipartite configurations.We formalize this structural phenomenon as theFractal–Entropic Scaling Law. By mapping the Linear Entropy of a quantum state, SL(ρ) = 1 − Tr(ρ2),onto the parameter space of the quadratic complex family(zn+1 = z2n + c), we demonstrate that the Box-CountingFractal Dimension Db of the associated Julia set exhibits astrict monotonic dependence on entropy and entanglementmonotones such as Negativity. This reveals a visually robust “Topological Gap” that explicitly separates entangledfrom separable states.Conceptually, this monotonic fractal response alignswith a deeper entropy–dimension correspondence established via functional box-counting theory, where normalized entropy equals normalized fractal dimension,MqMmax=DqDmax,indicating that entropic growth can be equivalently interpreted as dimensional amplification [16]. Within thisbroader mathematical framework, our construction provides a concrete quantum-mechanical instantiation inwhich an operational entropy measure (SL) is nonlinearlytransduced into geometric complexity through chaotic embedding.Our results reveal that topological complexity can serveas an efficient indirect observable for quantum purity, bypassing full state tomography. More broadly, the worksuggests that entropy may not merely quantify disorder,but may geometrically manifest as fractal dimensional expansion, positioning chaotic geometry as a natural amplifier of quantum informational structure