Mathematical Foundations and Carefully Delimited Extensions of TCCT

Source abstract

This technical companion note strengthens the Transformation Carrying Capacity Theory (TCCT) as published on SSRN by closing the remaining structural gaps in its mathematical foundations. The note provides: (i) a unified cusp parametrization linking the phenomenological parameters of Chapter 6 to the algebraic parameters derived in Appendix B via an explicit Bridge Lemma, including a full first-order Taylor expansion; (ii) a complete dimensional analysis assigning units to every variable appearing in an exponent, logarithm, cubic, or survival integral; (iii) a justification of the cusp choice via Thom's classification theorem; (iv) an explicit debt servicing equation reconciling Axiom A3 with Axiom A5; (v) a direction-of-collapse theorem explaining why the system jumps downward; (vi) an existence theorem for the critical repertoire size R*; (vii) a dimensionally consistent reformulation of the cusp SDE; (viii) a consolidated reading guide aligning the SSRN manuscript with the technical results derived here; and (ix) an optional simulation protocol distinguishing cusp from fold dynamics. Every statement is labelled as axiomatic, derived, assumed, or empirical. The note preserves the manuscript's declaration that the cusp model is phenomenological and that the bridge propositions are additional structural assumptions, not deductive consequences of the axioms.