Φ-Catalog: A Systematic Notation for Impossibility-Possibility Extensions in Mathematics (9 Historical Cases + 5 Rei-Stack Instances)

We propose the Φ-Catalog, a descriptive notation (impossibility, ctx-extension, Φ) that records a recurring template in the history of mathematics and physics: a previously-impossible statement becomes well-defined when (a) the domain of discourse is extended (notation / value range / connection rule), and (b) a "correction term" Φ accounts for the cost of that extension. We catalog 9 historical instances spanning analysis, special and general relativity, quantum field theory, modular forms, and number theory — Tachyon (m² < 0 quantization), Alcubierre (FTL via metric extension), mock theta functions (Ramanujan / Zwegers 2002), Dirac equation (negative-energy completion), Maxwell displacement current, Einstein cosmological constant, Yang-Mills mass gap conjecture, Ramanujan 1/π series, and Σn = -1/12 via ζ(-1) analytic continuation. We additionally re-read 5 Rei-AIOS papers (61, 63, 89, 145, 152) as Φ-Catalog instances, making the Rei design pattern auditable from outside the stack.

Honest scope (NOT claims): We do NOT claim "We have broken any classical impossibility theorem." All historical cases preserve the original impossibility; they extend the domain so something formally analogous becomes well-defined. We do NOT claim "We have invented Φ as a new mathematical object" — Φ is a uniform name for a family of correction terms over a century of mathematics. We do NOT claim "world-first impossibility-resolution framework" — Lakatos 1976 (Proofs and Refutations), Wilder 1981 (Mathematics as a Cultural System), Bourbaki structuralism, category theory universal properties, and Homotopy Type Theory all formalize related ideas. The Φ-Catalog is a lighter-weight, narrative-oriented view tailored to teaching and historical commentary; it is NOT a competitor to these foundational frameworks.

Differentiators in to-our-knowledge form: (D1) uniform (impossibility, ctx-extension, Φ) notation applied across 9 historically distinct mathematical episodes; (D2) re-reading 5 specific Rei-AIOS papers (61 ZCSG, 63 SNST, 89 D-FUMT₈ × Hodge, 145 D-FUMT₈ silicon, 152 Collatz σ-cascade) as Φ-Catalog instances. Three-party co-authorship per OUKC charter v1.0 (Nobuki Fujimoto / Rei / Claude). DRAFT v0.1 — first publication of this notation; feedback welcome at fc0web/rei-aios GitHub Discussions.