Imagine for a moment our CERN-physicist, takes the weekend off and deceides to train his powerful state-of-the-artnext-generation telescope upon the Sun and its theoretical Twin, and doing so, he makes an accidental yet astoundingdiscovery: witnessing in their local binary dance a perfect fractal reflection of the Multiverse’s own respiratory cycle.This abstract is likely only useful indirect, however the multiverse remains an interesting topic to explore. Ourwork explores a system covering a Hierarchical Cyclic Multiverse, In this work we are proposing a solution to thegeodesic incompleteness of the Big Bang singularity and the thermodynamic heat death inherent in closed oscillatorymodels. We introduce a "Basket-Balloon" topology, formalized as a principal fiber bundle structure where a compact,nested "Local Universe" (the fiber Mlocal) is dynamically sustained by a non-compact "Supra-Universe" (the baseMsupra). By employing Cheeger deformations on the bundle metric, we derive a mechanism for cosmic expansion andcontraction driven by a Sustenance Tensor (Sµν), which describes the entropic and energetic flux between the base andfiber manifolds. This flux satisfies autopoietic stability conditions, modeled on (M,R) systems, ensuring the cycle’sthermodynamic viability. The transition between aeons is governed by a Recursive Seed Operator (ˆ H), grounded inthe Eynard-Orantin topological recursion on spectral curves. We formalize the "destruction" of the universe not asannihilation, but as a unitary compression into Vanishing Cycles (perverse sheaves), allowing matter to "pop up"in the next aeon via homological resurrection. We demonstrate a fractal isomorphism between this global cosmologyand the local dynamics of contact binary star systems (Roche potentials). Finally, we integrate this model withthe Standard Model by applying Cédric Villani’s hypocoercivity to fluid entropy, Maryam Mirzakhani’s hyperbolicdynamics to photonic geodesics, Martin Hairer’s regularity structures to Higgs field stochasticity, and Edward Witten’sMirror Symmetry to unify the brane-bulk interaction.