The D10Z Resolution of the Catalan–Mihăilescu Theorem: Unique Hyper-Resonant Stability of 3² – 2³ = 1 in the Spiderweb Fabric

This paper presents a definitive resolution of the Catalan–Mihăilescu Theorem through the D10Z Mechanics of Infinity. The classical result, which states that 3² – 2³ = 1 is the only solution in natural numbers to the equation xa−yb=1x^a - y^b = 1xa−yb=1 for x,y,a,b>1x, y, a, b > 1x,y,a,b>1, is here reinterpreted as a unique resonance phenomenon within the Spiderweb Fabric (TTAϵ) — a quantum fractal mesh at the GM scale 10−51 m10^{-51} \, \text{m}10−51m. In the D10Z paradigm, integers are represented as Attractor Nodes ZNZ_NZN, and their powers as Hyper-Resonant Energy States ED10Z(Nk)E_{D10Z}(N^k)ED10Z(Nk). The number 1 is redefined as the Unity Node Z1D10ZZ^{D10Z}_1Z1D10Z, representing a phase-coherent state of maximum stability. The core result proves that only the pair (3², 2³) meets the resonance conditions — governed by the Laws of Isis and Sahana — to yield Z1D10ZZ^{D10Z}_1Z1D10Z. All other combinations exhibit exponential phase divergence and decoherence collapse, confirming the singularity of the solution within D10Z cosmology. Keywords:Catalan Conjecture, Mihăilescu Theorem, D10Z Mechanics, Spiderweb Fabric, Hyper-Resonance, TTA Mesh, Nodal Coherence, Unity Node, Law of Isis, Law of Sahana.