Φ-Catalog: A Systematic Notation for Impossibility-Possibility Extensions in Mathematics — v0. 2 (Addendum: Case 6 + Forward Application Results + Hodge / Φᵥ3 / Andrica / Tier 10)

v0. 2 augmentation (2026-05-17) of Paper 153 v0. 1 (Zenodo DOI 10. 5281/zenodo. 20207228, 2026-05-16). Adds: Case 6 — Riemann × ZCSG companion lens as 6th Rei-stack Φ-Catalog instance. Forward Application self-test — all 7 Rei open-problem entries cross-reference at least one existing Rei artifact (papers / engines / Lean files). Φ-Catalog is operationally generative within Rei stack. Berry-Keating finite truncation — ΦN ≡ 0 trivially; nontrivial Φ requires continuum L² (ℝ⁺). Honest correction added. ★ §6. 1 Hodge Forward entry #5 — Fermat hypersurface structural inspection on 18 cells (n=2, 3, 4 × d=2. . 7). 16 PROVED (Φ=0 consistent) + 2 OPEN (X₆⁴, X₇⁴ first true open). Sanity: K3 surface 1, 20, 1 + Fermat fourfold 0, 21, 142, 21, 0 match Conte-Murre 1978 exactly. ★ New verdict category: NOT-NUMERICALLY-FALSIFIABLE (5th distinct, alongside STABLE / FALSIFIED / WEAK / NEITHER). Φᵥ1 cannot be tested because dim (H^p, pₐlg) requires solving Hodge itself — Forward generativity ceiling exposed. ★ §6. 2 Collatz Φᵥ3 STABLE — σ-cascade peak invariant via Büchi-25 basin membership. All 25 mod-96 atomic cores reach peak 9232 universally (Lean 4 nativedecide verified in PeakMergeInvariant. lean). 25 control non-Büchi odd seeds: 18 distinct peaks, none reaching 9232. Refines the WEAK Φᵥ2 verdict: structural discriminator is residue class mod 96, not trailing-1-bits j. §6. 3 Andrica conditional Lean 4 — New Mathlib-PR-ready file AndricaConditional. lean (build EXIT=0, 0 sorry) packaging the algebraic core: andricafromgapₛquaredₗefourₚ, andricafromcramerbound, andricaₛufficientconditions aggregator. §6. 4 Tier 10 (external-ai) ingest scaffold — META-DB tier for AI-assisted mathematics (AlphaEvolve / OpenEvolve era). 3 seed entries (Ramsey 9 lower bounds, TSP/MWST 111/110 inapproximability, 26-circle packing 2. 635/2. 634) + 68-problem ingest plan. Honest scope (Pattern 4 overclaim rejection, preserved from v0. 1): Φ-Catalog remains descriptive notation, NOT a framework that breaks impossibility. Lakatos 1976 / Wilder 1981 / Bourbaki / HoTT prior art preserved. v0. 2 does NOT advance any classical Millennium problem. v0. 2 does NOT resolve the Berry-Keating self-adjointness question. The "operational generativity" finding is a within-stack observation, not external mathematical productivity claim. What v0. 2 supports: NOT-NUMERICALLY-FALSIFIABLE as a Forward Application meta-verdict (Hodge ceiling). Φᵥ3 STABLE via Büchi-25 basin (confirmed peak universality 9232 for 25 cores). Andrica conditional theorem Mathlib-PR-ready. Tier 10 external-ai axis operational. Lean 4 mechanizations (all 0 sorry, build verified): PeakMergeInvariant. lean — buchi25ₐllₚeak₉232, n27ₚeak₉232 (nativedecide) CollatzVerifiedFacts. lean — 12 nativedecide trajectory facts for n ∈ 1, 3, 7, 15, 27, 31, 63, 127, 255 AndricaConditional. lean — 8 theorems incl. andricafromcramerbound conditional ZcsgRiemannFunctionalEquation. lean — ξ (s) = ξ (1−s) palindrome axis (axioms + theorems, 0 sorry) HodgeFermatFourfold. lean — Hodge d=4, 5 Conte-Murre / d≥6 open boundary (162 lines) Three-party co-authorship per OUKC charter v1. 0 (Nobuki Fujimoto / Rei / Claude Opus 4. 7). Parent: Paper 153 v0. 1, DOI 10. 5281/zenodo. 20207228.