The Interior Observer Cosmological Framework: Paper 9 - The Gauge Projection: Completing the Dark Energy Derivation via the Spectral Norm of the Ashtekar-Barbero Connection

Paper 9 in the Interior Observer (IO) cosmological framework. We complete the derivation of the IO dark energy constant K_Λ by identifying the missing factor √ (1+γBI²) as the spectral norm ratio of the Ashtekar-Barbero connection to the spin connection on the Schwarzschild horizon. In Painlevé-Gullstrand slicing at the equator (θ = π/2), the extrinsic curvature K and the spin connection Γ occupy orthogonal subspaces of the connection matrix space: K is purely diagonal, Γ is purely off-diagonal (antisymmetric 2-3 block). Their spectral norms (operator norms = largest singular values) are exactly equal: ||K||ₛpec = ||Γ||ₛpec = 1/rₛ. The Ashtekar-Barbero connection A = Γ + γBI K therefore satisfies ||A||ₛpec/||Γ||ₛpec = √ (1+γBI²) by the Pythagorean theorem — an exact analytical result for any γBI. Verified independently by three AI systems (Claude Code, Wolfram/ChatGPT 5. 3, Gemini 2. 5 Pro). This completes the K_Λ derivation chain with three factors: (1) Bare EC-Holst torsion: Λbare = 9π²/4rₛ² (1+γ²) from the contact interaction (Papers 1, 4) (2) Spatial projection: ×x² = (rₛ/RU) ² from observer-scale spin density (Paper 7) (3) Gauge projection: ×√ (1+γ²) = ||A||/||Γ|| from boundary-to-bulk connection translation (this paper) Result: K_Λ = 9π²x²/4√ (1+γBI²) = 49. 85, matching the Friedmann closure value 49. 82 to 0. 054% with zero free parameters. The Rosetta Stone identity from Paper 8 is thereby explained: Δ = x⁴ (1+γ²) = x² × √ (1+γ²) ² = (spatial projection × gauge projection) ². The temporal decoupling is the square of the total projection factor. The equatorial evaluation (θ = π/2) is justified by the Hairy Ball Theorem: spherical coordinates introduce coordinate singularities at the poles (cot θ divergence in Γ), and the equator is the unique point where these artifacts vanish. PG slicing is the natural frame for LQG variables (flat spatial 3-metric produces the orthogonality between K and Γ that makes the Pythagorean result exact). Caveats: The spectral norm ratio is an exact geometric identity in the PG frame, not a fully covariant spacetime invariant. The physical mechanism by which one power of ||A||/||Γ|| enters the effective Λ (the "one-leg" boundary-bulk ansatz) is presented as a phenomenological postulate, not a rigorous QFT derivation. A fully covariant derivation via the boundary-to-bulk torsion propagator remains open. The IO framework's dark energy derivation is complete: from the mass of the universe MU, through the Schwarzschild radius rₛ, Einstein-Cartan-Holst torsion, and the Ashtekar-Barbero gauge projection, to the observed cosmological constant — connecting general relativity, Loop Quantum Gravity, Einstein-Cartan gravity, and observational cosmology through the Pythagorean theorem applied to gauge connections on a black hole horizon. Multi-AI research: Claude Code (Anthropic) for full numerical and symbolic verification; Wolfram/ChatGPT 5. 3 (OpenAI) for independent analytical verification and frame-dependence analysis; Gemini 2. 5 Pro (Google DeepMind) for proposing the spectral norm approach, identifying a coordinate error in v1. 0, and adversarial review; Claude (Anthropic) for orchestration and the boundary-bulk projection framework. https: //dfife. github. io/index. html v1. 5 (April 2026): Post-publication review corrections. (2) Appendix A added: 8 theorem-grade steps (9. 1–9. 8). (3) §5 intro clarified: two active factors, Step 3 withdrawn. (4) §6 redundancy reduced. (5) §8. 2 and §8. 4 updated with Paper 10 v2. 0 projected values (H₀ = 67. 58, h = 0. 6758). (6) §9 conclusion sharpened: matter-energy projection; dark energy derivation independent. (7) Paper 10 version reference corrected from v1. 8 to v2. 0 throughout. (8) Stale page header removed. v1. 4 (April 2026): Paper 10 v1. 8 systematic rebuild. (1) §4. 2 one-leg phenomenological ansatz REMOVED — the IO framework carries zero fitted parameters, and a phenomenological ansatz is a fitted assumption regardless of its physical motivation. (2) §5 Step 3 gauge projection withdrawn from the vacuum K_Λ chain. The corrected effective vacuum uses only Paper 1 bare torsion × x² position correction: K_Λ = 9π²x²/4 (1+γ²) = 48. 50. (3) §6 K_Λ (eff) /K_Λ (bare) = √Δ interlock withdrawn — the correct ratio is x² (position correction only). (4) The spectral norm identity ||A||/||Γ|| = √ (1+γ²) remains DERIVED (Pythagorean theorem, exact). Its application to matter-sector projection √Δ = x²√ (1+γ²) in Paper 10 remains DERIVED. (5) Title updated: the spectral norm completes the matter projection, not the dark energy derivation. The dark energy derivation is Paper 1 §6 + x² position correction only. Website: https: //dfife. github. io v1. 3 (March 2026): Inline revision markers removed. Content is now clean and currently accurate; version history documents all changes. No content changes. v1. 2 (March 2026): Inherited temporal rhetoric removed. §6 retitled from "The Rosetta Stone Explained" to "The Spatial Decoupling Explained. " Temporal decoupling references replaced with spatial Δgeometric. "Three anomalies" downgraded to "two. " §6 anomaly table updated: Age duality row marked WITHDRAWN, Δ row marked EXACT (spatial). §8. 1 Age Duality withdrawn. The spectral norm derivation ||A||/||Γ|| = √ (1+γ²) and all K_Λ results are convention-independent. Title page reformatted to series standard. See Paper 21 v1. 1 for the full audit. Companion to Papers 1-8. DOIs: 10. 5281/zenodo. 18854813 (P1), 10. 5281/zenodo. 18868612 (P2), 10. 5281/zenodo. 18876346 (P3), 10. 5281/zenodo. 18883069 (P4), 10. 5281/zenodo. 18889865 (P5), 10. 5281/zenodo. 18891475 (P6), 10. 5281/zenodo. 18894611 (P7), 10. 5281/zenodo. 18895980 (P8).