Syntropy: A Formal Outer-Domain Principle for the Constitution of Relational Domains

AbstractThis paper introduces syntropy as the formal outer-domain principle that specifies howa relational domain is constitutively generated from a pre-relational ground. It proves thatevery division sequence of the Divisive Foundation Theory (DFT) satisfying the axiomsof Paper 1 v3.8 is a syntropic sequence (Theorem 1, Realization Theorem). As a directconsequence, the non-collapse condition is derived as a theorem of syntropic irreversibility(Corollary 1), thereby reducing the axiomatic basis of the entire DFT program. Definition 2 introduces the syntropic depth measure Ht, which supplies a quantitative link to thestructural entropy of Papers 1, 2, and 2B.Version 9.3 establishes the complete threshold structure of syntropic division: (i) symmetric division is the degenerate case of collapse; (ii) αeff is simultaneously the noise floor,the zero-point vacuum equivalent, and the asymptotic attractor; (iii) the excess-asymmetryspectrum ∆α classifies all division events from vacuum fluctuations to the perfect storm;(iv) rational asymmetries generate locked futile cycles while irrational αeff guarantees inexhaustibility; (v) large-∆α events produce the Ulysses spiral and identify black holes asterminal local cascade states; and (vi) the Minkowski Bridge derives the Minkowski metricand the full Riemann curvature tensor directly from the structural directed quasi-metricds and the balance attractor. Consequently, all gravitational theorems in Papers 2 and 2Bbecome unconditional formal consequences of the outer-domain principle