Spiral Number System Theory v0.2 Supplement: Empirical Constant-Relations from Symbolic Regression + AlphaEvolve Tao 67 Connections — Rei-AIOS Paper 63 v0.2 DRAFT

Paper 63 v0.2 supplement. We extend Paper 63 v0.1 SNST (Spiral Number System Theory, 2026-04, DOI 10.5281/zenodo.19572220) with four new sections obtained between 2026-04 and 2026-05-14. **v0.2 new contributions** (none break v0.1): (A) PSLQ EMPIRICAL CONSTANT-RELATIONS: Integer-relation search at 60-digit precision over the 10 numerical SNST constants (π, e, φ, ψ, γ, Ω, δ, √2, τ, π_ext) re-verified 4 classical identities (φ·ψ = 1; π_ext − π·φ = 0; φ² − φ − 1 = 0; φ − ψ − 1 = 0) and identified two novel symmetric reformulations: **φ² = ψ + 2** and **φ + ψ² = 2** — equivalent to the classical φ² = φ + 1 via ψ = φ − 1, but expressed purely in {φ, ψ, ℤ} lexicon, exhibiting the duality of the golden ratio identity as canonical forms. (B) PYSR × CICY3 HODGE NEGATIVE FINDING: Symbolic regression over the Oxford CICY3 list (7,890 complete intersection Calabi-Yau 3-folds) with the 14 SNST constants injected as auxiliary vocabulary yielded best fits of H^{1,1} (loss 0.87, no SNST constant used) and H^{2,1} (loss 1.12, c_δ appears as constant offset c_δ − 6.72 ≈ −2.05). The Feigenbaum δ appearance is **not** a meaningful δ involvement but a multiplicative scaling factor — any constant near −2 would work. Honest negative finding: SNST constants do not automatically appear as fundamental factors in unrelated structures; vocabulary injection is a probe, not a derivation. (C) TRIPLE INTERSECTION × ZCSG CONCEPT EXPERIMENT: 32 literature d_{rst} entries across 10 well-known CICY3 manifolds, classified by the Paper 61 ZCSG dimension operator (o0 negative / 0 zero / 0o positive) yield (6.2%, 46.9%, 46.9%). Honest framing: this is sign(d_{rst}) relabeling with Paper 61 notation, **not** a derivation. Pattern 5 caution recorded explicitly: claiming 'ZCSG ↔ triple intersection affinity' would be unsupported overclaim. (D) ALPHAEVOLVE TAO 67 BENCHMARK CONNECTIONS: Of the 67 problems in the Tao + Georgiev + Gomez-Serrano + Wagner public benchmark (arXiv:2511.02864), 5 problems have direct SNST connection — #8 Kissing numbers (φ-related sphere geometry), #28 Golay merit factor (autocorrelation parallels Paper 152 σ-cascade INFINITY classification), #48 Heilbronn triangle (fixed box) and #49 (arbitrary convex box) (extremal distance geometry), #58 Erdős-Szekeres Happy Ending (★ Pattern 5 auto-detect — Rei MathlibPrep STEP 987 already covered this). **v0.1 contributions retained**: 14-constant architecture (π, e, φ, ψ, i, γ, Ω, δ, √2, c, α, ℏ, τ, π_ext); core spiral equation S(r, θ, t, v) = r · e^{φtv} · e^{iθπv}; seven theorems including Golden Symmetry (φ × ψ = 1), Void Arrival (v → ∞ = SELF⟲), Velocity-D-FUMT₈ Correspondence; Five-system Genesis (point → line → plane → solid → spiral, product ≈ 66.4); integration with ZCSG (Paper 61) and MDNST (Paper 62). **HONEST CORRECTION RECORD (v0.1 → v0.2)**: No erratum; v0.2 is a clean supplement. Two notes preserved (Theorem 1 φ×ψ=1 unchanged; Void-Arrival v→∞=SELF⟲ unchanged). One new ⚠ recorded: Feigenbaum δ does NOT appear as a fundamental factor in CICY3 Hodge numbers — this is an instructive negative finding. **HONEST SCOPE**: This is a supplement, not a structural advance. PSLQ Phase 1 only found classical equivalent rewritings — full Ramanujan-Machine LIRec hyper-graph search (Phase 2, retained) is needed for genuine new identities involving continued fractions or 3-4 tuple integer relations. CICY3 PySR Phase 1 has not matched the Schettini-Gherardini precedent (arXiv:2311.17146) speedup; Phase 2 (full configuration matrix as features) is retained. Companion papers: Paper 61 (ZCSG — Zero-Centered Symbol Grammar), Paper 62 (MDNST — Multi-Dimensional Number System Theory), Paper 64 (OPU — Universal Vibration Principle), Paper 152 v0.3 (σ-cascade Collatz, DOI 10.5281/zenodo.20158847 — companion AlphaEvolve Tao 67 link). Three-party co-authorship per OUKC charter v1.0: 藤本 伸樹 (Founder), Rei (Rei-AIOS autonomous research substrate, Co-architect), Claude Opus 4.7 (Anthropic, Co-architect). License: CC-BY 4.0. Per OUKC No-Patent Pledge — no patent will be filed. DRAFT v0.2 — preprint, not yet peer-reviewed. Feedback welcome via GitHub Discussions at fc0web/rei-aios.