Quantitative Bridges in Divisive Foundation Theory: Dimensionless Invariants, Variational Structure, and Cosmological Predictions

AbstractThis paper develops the quantitative bridges deferred in Paper 2 v7.1 (D´ıaz Mu˜noz, 2026b).Three interconnected programs are executed. First, the Recursive Division Algorithm (RDA)of Paper 1 v3.9 generates a natural dimensionless clock R(t) whose value at the currentcosmic epoch is consistent with the observed ratio ρ = ℓp/ℓHMBBE ≈ 10−61, yielding astructural prediction for αeff ≈ 1−1.76×10−60. Second, the fixed-point condition B(P) ∼=Pis shown to be formally equivalent to the stationarity condition δSP = 0. Theorem 3.1.1(Extended) establishes the L2 norm identification unconditionally. Theorems 3.4.1 and 3.6.1—the Maxwell and Yang-Mills equations — are unconditional consequences of DFT axioms.Third, residual tension τ(x,t) is inserted into Buchert’s backreaction formalism, yieldingquantitative predictions E1–E9.Version 2.0 adds three results. (i) The Minkowski Bridge is established (CompanionPhysics Paper, Theorem 1), making all gravitational theorems in Paper 2 unconditional andpromoting E8 (tensor suppression r ∼ 10−28) to a formal derivation. (ii) New Section 3.7aidentifies QFT vacuum modes as physical primes — irreducible atomic patterns of thesub-threshold background — and gives a structural account of the cosmological constantunconformity as a mismatch between two orientations of the constitutive axis, with theself-dual midpoint Re(s) = 1/2 identified as the structural target for quantum gravity Stage 2.(iii) Priority 5 is marked COMPLETE.