Perfect Bubble Space Theory III: Set-Theoretic Bubble Spaces, Transfinite Return, and Forcing Semantics

We construct a first-order algebraic-geometric presentation of set theory by canonical marked incidence schemes. Every set-sized relation is encoded by a squarefree monomial algebra and recovered, up to definable isomorphism, from distinguished monomial nonvanishing tests. An explicit bridge theory yields\[ABF_2defBST_2MoritaZFC,\]with effective translations of formulas, proofs, models, and axiom schemes. We distinguish absolute finite algebraic generation from power-set-sensitive geometry, separate increasing fusion stabilization from decreasing return collapse, and realize forcing through complete Boolean algebras and Stone spectra. Incidence translation preserves Boolean truth and commutes with generic quotients and Boolean-valued Mostowski collapse. The effective presentation preserves theoremhood and model-theoretic degree spectra. Explicit systems realize \(Σ^0_1\)-, \(Σ^0_2\)-, and \(Π^1_1\)-complete problems, while bounded hereditary-finite truth remains decidable. All equivalences concern decorated incidence geometry and add no consistency or large-cardinal strength to ZFC. Keywords Bubble space; algebraic foundation; incidence scheme; Morita equivalence; set theory; forcing; transfinite return; computability; decidability; Boolean-valued semantics.