Generative Mechanism Atlas A Structural Taxonomy of Generators Behind Canonical Values

This paper reframes mathematical and physical constants not as primitive objects but as canonical values selected by underlying generative mechanisms. It introduces the Generative Mechanism Atlas, a mechanism-first framework that records how structures, reusable generators, explicit generative maps, and canonicalization rules produce distinguished outputs. The Atlas proceeds in three stages: Stage 1 records constant-to-generator entries, including maps, selection rules, generator motifs, structures, and invariance claims; Stage 2 clusters generators into families such as closure, flow, recursion, branching, endpoint compression, and invariant structure; and Stage 3 proposes qualitative universality indices to assess recurrence and compression strength. Focused case studies, including endpoint compression producing the trace 4/3 and fixed-point recursion producing phi, illustrate how numerical traces can arise from explicit maps, pipelines, and re-encoding conventions. The work offers a conceptual and practical program for cataloguing canonical values by their generative origins rather than by numerical coincidence.