Chapter 66: What τ Earns

the relevant chapter catalogued Mode D: theorems that orthodox mathematics cannot prove. This chapter addresses Mode E: results where the content is the same but the proof status differs.

Mode E (Earned vs. Axiomatized) identifies constructions where τ derives what orthodox mathematics takes as axiom or obtains via a fundamentally different proof. The mathematical content is identical; the epistemic status is upgraded. “Proved from fewer assumptions” outranks “postulated”—even when the conclusion is the same.

The chapter catalogues 13 earned results across Books I–II, develops the number tower and Tarski geometry as case studies, and formulates the Epistemic Upgrade Principle that governs all Mode E assessments.