BFG Universal Reclosure Unification: A Unified Mathematical Architecture from Level-0 Formation to Matter, Life, Conscious Subjectivity, and Qualitative Experience

This whitepaper presents a unified mathematical formulation of the Balance-Field Framework (BFG) in which physical, atomic, molecular, biological, organismic, cognitive, conscious, and phenomenal regimes are constructed as recursive realizations of one universal closure architecture rather than as sectors governed by independently retuned fundamental laws. The framework begins from Level-0 admissible formation and develops a positive closure geometry with the universal neutral resolvent pair CN(Y)=(I+Y)−1,BN(Y)=Y(I+Y)−1,C_N(Y)=(I+Y)⁻¹, B_N(Y)=Y(I+Y)⁻¹, together with recursive persistence, retention/complement partition, order-indexed mediation, and a universal reclosure operator. The resulting hierarchy is expressed in the compact form Xn+1=U[Xn],Xn=Un[X0],Xₙ₊₁= U[X_n], X_n= U^n[X_0], where successive regimes are identified through invariant closure properties rather than by introducing new sector-specific response laws. The paper develops a continuous structural sequence from Level-0 formation through local physical closure, atomic and molecular organization, prebiotic and protocellular closure, cellular and organismic integration, cognitive recursion, conscious subjectivity, phenomenal unity, qualia, and selfhood. Numerical and three-dimensional synthetic simulations are included to illustrate neutral partitioning, order-feedback stability, generativity conditions, cognitive integration, differentiation, and bounded recursive trajectories. The manuscript distinguishes mathematical definitions, theorem candidates, conditional theorems, physical principles, model completions, spectral prescriptions, and empirical hypotheses. It does not claim that BFG is established fundamental physics or that the proposed identifications with life, consciousness, or qualitative experience have already been empirically confirmed. Its central contribution is a no-hidden-retuning unification program in which later structural regimes are derived from a single recursive closure architecture and subjected to explicit falsification and transfer criteria.