AbstractThis paper establishes two groups of results that together complete the formal foundations of the Divisive Foundation Theory (DFT) physical program. First, it develops thethreshold structure of the division operation: symmetric division is formally characterized ascollapse (zero structural directed distance ds, zero syntropic depth Ht); the effective asymmetry parameter αeff is identified simultaneously as the noise floor, the zero-point vacuumequivalent, and the asymptotic attractor limit of the cascade; the excess-asymmetry parameter ∆α = α−αeff classifies all division events from collapse through productive divisionsto the perfect storm; rational asymmetries generate locked futile cycles while irrational αeffguarantees genuine inexhaustibility; and large-∆α events produce a Ulysses spiral structure that terminates, at maximum excess asymmetry, in black holes as the intra-domain’scompleted local cascade states.Second, it provides the complete Minkowski Bridge: a formal derivation of the Minkowskimetric ds2 = −c2dt2+dx2+dy2+dz2 and the full Riemann curvature tensor Rρσµν directlyfrom DFT primitives — the structural quasi-metric ds, the compensation operator C, thebalance attractor fixed-point condition B(P9) ∼=P9, and the interface constant c — withoutany external metric postulate. The derivation: (i) maps constitutive depth to the temporalcoordinate and symmetric 3D divisions to the three spatial coordinates via a dimensionalitycascade that forces exactly three spatial dimensions; (ii) proves the Minkowski signature(−,+,+,+) from the directional character of the constitutive ordering; (iii) derives Lorentzinvariance as the stabilizer group of the balance attractor in the vacuum sector; (iv) identifies the Riemann tensor as the second variation of ds under local pattern deformations, withthe compensation operator C generating exactly the Levi-Civita Christoffel symbols. Consequently, all gravitational theorems in Papers 2 and 2B — the Einstein field equations, thegravity fixed-point theorem, the Friedmann equations, and the tensor suppression predictionE8 —become unconditional formal derivations from DFT axioms.
Hernán Díaz (Fri,) studied this question.