Φ-Catalog: A Systematic Notation for Impossibility-Possibility Extensions in Mathematics — v0.2 (Addendum: Case 6 + Forward Application Results + Hodge / Φ_v3 / 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 — Φ_N ≡ 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 (X_6^4, X_7^4 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). Φ_v1 cannot be tested because dim(H^{p,p}_alg) requires solving Hodge itself — Forward generativity ceiling exposed.
• ★ §6.2 Collatz Φ_v3 STABLE — σ-cascade peak invariant via Büchi-25 basin membership. All 25 mod-96 atomic cores reach peak 9232 universally (Lean 4 native_decide verified in PeakMergeInvariant.lean). 25 control non-Büchi odd seeds: 18 distinct peaks, none reaching 9232. Refines the WEAK Φ_v2 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: andrica_from_gap_squared_le_four_p, andrica_from_cramer_bound, andrica_sufficient_conditions 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). Φ_v3 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 — buchi25_all_peak_9232, n27_peak_9232 (native_decide)
• CollatzVerifiedFacts.lean — 12 native_decide trajectory facts for n ∈ {1,3,7,15,27,31,63,127,255}
• AndricaConditional.lean — 8 theorems incl. andrica_from_cramer_bound 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.