From Mathematical Problems to the Generation of Problems: Generative Audit, Methodology, and an Open Worldview in GHT

Generative History Theory (GHT) begins not by adding another property to already constituted mathematical objects, but by auditing how an object acquires the admissibility to count as an object at all. Bare occurrence is marked by the boundary notation A=0 but does not enter mathematics. Mathematics begins when an occurrence that has already ended leaves a re-addressable trace; subsequent directedness allows that trace to re-enter the present relation, through which distinction, re-identification, relation, composition, invariance, and scale acquire stable expression. The Generative History Law N → N+n further requires every new act of directedness, interpretation, proof, or retrospection to occur within an already accumulated history and to enter the background available for subsequent inquiry. On this foundation, the paper synthesizes the methodological and worldview consequences of the first two papers in the GHT Foundations of Mathematics Series. It elevates identity restoration into the more general framework of GHT generative audit. Generative audit contains two complementary movements: starting-point audit, which asks where the objects, identities, operational backgrounds, scales, and equivalence relations of a proposition acquired their admissibility; and generative abduction, which traces the genealogy of objectification backward from a mature proposition in order to determine whether a present difficulty reflects genuinely new mathematical structure, a real change of identity, a gap in cross-background transport, or proof debt generated by objectification itself. The paper develops the notions of identity recession, identity re-proof loops, over-splitting, over-merging, operation-type audit, provenance continuity, and the Preserved/Hidden/Destroyed (P/H/D) ledger. It also examines the multiplicative provenance of Prime and asks how proof obligations may arise when mature representations obscure their common generative history. The Riemann Hypothesis serves as the principal high-pressure example inherited from Paper II. Goldbach-type questions are retained explicitly as an open direction rather than presented as a completed proof. The Yang–Mills mass-gap problem and other difficult mathematical problems are likewise treated as exploratory interfaces for generative audit, not as automatic consequences of GHT. Finally, the paper extends generative audit toward a broader open worldview. Mathematics, physics, computation, life, cognition, language, the humanities, and social knowledge all depend in different ways on addressability, directedness, re-identification, objectification, and stable expression. GHT therefore proposes interfaces rather than annexation: whatever can become a stable object may be asked for its addressable provenance, relational history, mode of directedness, and cost of objectification. This is Paper III and the methodological synthesis of the initial GHT Foundations of Mathematics three-paper series.