Interaction-Operator Theory of Generated-Class Universality

The uploaded corpus develops a layered mathematical theory of learnability, structure formation, asymptotic comparison, and controlled universality. At its foundation, File 01 defines learnability through an effective interaction operator A = K^(1/2) C K^(1/2), and distinguishes local learnability, bridge-valid learnability, and renormalized weak-sector learnability. It is deliberately a bounded foundational volume: it establishes the operator core, bridge inheritance, tractable verification, and a first renormalized weak-spectrum layer, while explicitly refusing unrestricted global universality claims. File 02 then extends this foundation into a reusable technical spine: finite-route bridge concatenation, handoff algebra, controlled broadened transfer, descriptor closure, phase/threshold geometry, and restricted universality on declared regimes. Together, Files 01–02 provide the formal transport-and-synthesis backbone of the whole stack. Beyond that backbone, the theory shifts from packet-level structure to class-level structure. File 03A shows how organizing classes can be generated lawfully from lower-tier data through discovery seeds, pre-class signatures, pre-equivalence, generated classes, canonical representatives, passports, and canonical comparison coordinates. File 03B asks whether those generated classes remain coherent when the mechanism regime is broadened to stronger stochastic, nonlinear, or mixed settings; it introduces classwise synthesis data, slotwise defect ledgers, and a five-way outcome grammar: persistence, residual growth, bifurcation, invalidation, or reopen-required. File 03C then studies whether such generated and mechanism-tested classes can be continued across enlarging or moving domains, using enlargement certificates, moving-domain transport, finite-prefix continuation, exhaustion atlases, horizon packets, and descendant extraction—while still refusing any silent jump to unrestricted infinity claims. The capstone layer is File 03D, which turns finite-prefix and horizon-visible continuation into a more ambitious asymptotic architecture. Its burden is to construct certified tail objects, asymptotic class comparison at infinity, descendant exhaustiveness, lawful merge and quotient doctrines, threshold compression, universality-coordinate completion, and finally a ledgered universality verdict. A crucial idea of the corpus is that even the strongest positive result must remain regime-explicit, residual-visible, exclusion-visible, and stop/reopen-disciplined. Universality is therefore never treated as a slogan; it is a typed terminal classification supported by a full burden stack. After the capstone, the theory does not simply broaden claims; it sharpens the detector and exhaustiveness interface. File 03E adds hidden-tail detector records, hidden-branch detector records, omitted-branch irrelevance certificates, and stronger exhaustiveness ledgers, with the explicit rule that detector success is not already universality. File 03F then focuses on a particularly subtle frontier: whether weaker detector windows can still support stronger global exhaustiveness. It builds an observability hierarchy, an exhaustiveness ladder, a local-to-global gluing architecture, defect-controlled and exact globalization corridors, and a non-upgrade law showing where stronger closure is impossible or only conditional. Finally, P1–P3 reorganize the manuscript stack into a forward program. P1 is the mathematical control layer: it studies compression of ledger syntax, stronger exhaustiveness from weaker observability, and replacement of schematic witnesses by explicit model families. P2 is the machine-learning bridge: it converts theorem-bearing mathematical outputs into typed ML-facing packets, with benchmark, calibration, detector, and audit discipline, but without allowing informal AI interpretation to outrun theorem inheritance. P3 is the productization layer: it turns those typed packets into workflows, reports, gates, and software objects, while preserving lineage, burden visibility, calibration status, and stop/reopen structure. In short, the entire uploaded corpus is a rigorously staged program that moves from foundational mathematics to classes, mechanisms, domains, asymptotics, detector sharpening, ML-facing packetization, and finally claim-governed operationalization—always under anti-drift discipline.