BTPEC Primitive Theoretical Framework: SL(3,Z) Symmetry, Topological Analysis and Scientific Predictions Based on the BerryYu CrossKey Formula

This paper constructs a ternary‑coupled private‑entanglement quantum model framework (BTPEC, Berry‑Taro Private Entanglement Cosmology). Taking the BerryYu CrossKey formula within Hilbert‑space formalism as the fundamental axiom, strict algebraic manipulation yields the trace‑difference constraint Omega = tr(BT)−tr(PEC)=0. Based on quantum distinguishability and gravitational information bounds, a conditional derivation is presented. Under the validity of this set of physical premises, the set of effective distinguishable states of the ternary degrees‑of‑freedom can be embedded into a bounded subset of integer lattice Z³. The unimodular group SL(3,Z) is then introduced as the automorphism transformation group for this lattice‑embedded representation. Persistent‑homology tools are adopted under fixed construction rules: the Vietoris‑Rips complex together with Euclidean metric. Topological characterizations are performed for group orbits, yielding topological invariants b0=1, b1=3, b2=1, χ=−1. Topological‑degeneration phenomena at boundary values of superposition coefficients are discussed. The quotient modular space Z³ / SL(3,Z) is constructed to eliminate coordinate dependence originating from different observational representations. Consistency between the three foundational postulates of BTPEC (single‑anchor point, dual‑vision, continuous‑deformation without discontinuity) and the BerryYu CrossKey formula is verified. Ten numerically testable predictions are proposed, and partial possibilities for comparative alignment with existing quantum‑information and topological‑quantum experimental frameworks are explored.