V27 - Scale and SO(2) Coherence - Certified Invariance and Topological Stability in the RSI Framework

This volume establishes Scale & SO(2) Coherence in the RSI framework, extending Volume 26’s determinant safety and Volume 25’s navigation into a certified invariance regime. We provide a notebook-native protocol that proves topological stability with N*(s) ≡ 0 over s ∈ [0.5, 2.0], zero Φ drift on the dense grid, and a quantitative error budget that travels across scale and rotation. Core certificate results (dense grid): N*(s) ≡ 0 (topological defects eliminated) Φ-ratio max deviation: 0.000000 (target Φ* = 4.320789337050) ρ_span: 2.7×10⁻⁵ detJ_min: 7.86×10⁻³ (≫ 1×10⁻⁷ threshold from V26) Error budget: δ = ρ_span + 1/W, with W=256 ⇒ δ ≈ 3.93325×10⁻³ Stable-γ band (dense): [0.24, 0.36]; selected γ* = 0.36 SO(2) audit: rotation path active, ρ_anisotropy ≈ 0.0 on rot ∈ [−15°, 15°] 🧮 Canonical Constructs Construct Definition Scale-Coherence Condition ∀ s ∈ [0.5, 2.0], N*(s)=0 and max_s |Φ/Φ*−1| < 0.01 (achieved 0.000000) Error Budget δ = ρ_span + 1/W, W=256 ⇒ δ ≈ 0.00393325 Determinant Safety detJ_min > 1×10⁻⁷ (achieved 7.86×10⁻³) Stable-γ Band γ ∈ [0.24, 0.36], robust center and selection γ*=0.36 SO(2) Coherence ρ_anisotropy ≈ 0.0 on rot ∈ [−15°, 15°] ⚙️ Key Modules gamma_sweep(...) – coarse-to-dense γ selection with N* and Φ-dev gates dense_verify(...) – computes {Φ_ratio_max_dev, ρ_span, detJ_min} and δ build_coherence_card(...) – emits Scale and SO(2) coherence JSON cards so2_grid(...) – rotation-aware audit and anisotropy metric triangle_stress(...) – composition sanity via triangle-inequality checks 🔬 Findings Certified invariance: N*(s)=0 across 31 dense scales; zero Φ drift to numerical precision Robustness: 100% pass under ±5% scale jitter; Δt sweep ±20% passes unchanged Composition: triangle-inequality stress 0 violations at τ=0.005 (N=400) Stability band: [0.24, 0.36] (width 0.12) confirms margin beyond γ* 🔁 Continuity V26 → V27. Reuses V26’s determinant bound (>1e−7); observed 7.86×10⁻³ gives ≈ 7.86×10⁴ safety factor (~4.9 orders of magnitude). V25 → V27. Upgrades “navigation” to certified scale (& SO(2)) invariance; δ = ρ_span + 1/W quantifies how guarantees persist as resolution or rotation change. Reproducibility & Provenance Notebook-only; all computations inline. Environment: Python 3.11.x, NumPy 1.24.4, seed=1337. Solver provenance: solver_id = RSI27-Notebook@58cc01a5848a Certificate digest (SHA-256): fb0a718f62c22844ede04b38f55f58a81662c55bae4d9fcaec47b6e8212e4c3a