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.
Hiroyuki Shioiri (2026) studied this question.