Chapter 65: What τ Gains

the relevant chapter catalogued what Category τ structurally refuses. This chapter catalogues the opposite: what τ structurally gains—theorems, properties, and constructions that are provably impossible, independent, or false in orthodox mathematics (ZFC, PA).

Mode D (Gained) objects have no orthodox counterpart. They are not improvements of existing theorems; they are new mathematical facts that arise only because τ’s axioms differ from orthodox axioms. The chapter identifies 23 gains across Books I–II, develops the Euclidean-from-hyperbolic resolution as a case study, and explains why each gain is structurally impossible in orthodox foundations.