The Interior Observer Cosmological Framework: Paper 7 - Big Bang Nucleosynthesis, the Unified Tension, and the c³ Hypothesis

Paper 7 in the Interior Observer (IO) cosmological framework, which proposes the observable universe exists inside a Schwarzschild black hole. Big Bang Nucleosynthesis: Primordial element abundances are computed with IO parameters (HIO/H_ΛCDM = 0. 955, ωb = 0. 01694, η = 5. 0×10⁻¹⁰). Helium-4 is consistent with observations (Yₚ = 0. 239, 1. 5σ). The cosmological lithium problem is reduced by 42% compared to ΛCDM. Deuterium is overproduced by 38% (D/H = 3. 5×10⁻⁵ vs observed 2. 53×10⁻⁵), driven by the low baryon-to-photon ratio. All results independently verified by analytical computation (Wolfram/ChatGPT 5. 3). Unified Tension Diagnosis: The deuterium excess, CMB first peak shift (ℓ₁ = 180, Paper 6), and H₀ gap (58. 2 vs 67. 4) trace to a single root cause: h = 0. 582. A systematic investigation eliminates all candidate corrections: the FRW lapse is identically 1 (confirmed by Wolfram), Einstein-Cartan torsion-fermion coupling does not modify the metric (verified by three AI systems), holographic x^ (1/3) scaling lacks a derivation, and Buchert backreaction is four orders of magnitude too small. Age Prediction: The IO Friedmann equation with reference parameters gives age = 19. 2 Gyr — not 13. 8 Gyr. This is consistent with all direct stellar age measurements (lower bounds ~12-13 Gyr) and provides the extended temporal runway needed to explain JWST observations of unexpectedly mature high-redshift galaxies. The derivation chain is self-consistent: MU → rₛ → ΛIO → H (a) → age = 19. 2 Gyr → RU = 4. 40×10²⁶ m. The c³ Hypothesis: In the Λ-dominated IO expansion (Ω_Λ = 0. 933), the dark energy term provides 93. 3% of H₀². The Hubble constant scales as H₀ ≈ c³√K_Λ/ (2√3 × GMU), an algebraic consequence of deriving Λ from the Schwarzschild horizon (ΛIO = K_Λ/rₛ²). The sensitivity d ln H₀/d ln c = 2. 67 (→ 3. 0 in the Λ-only limit). H₀ and TIO move in opposite directions with MU (d ln H₀/d ln M = −0. 83, d ln T/d ln M = +0. 41), proving the framework is structurally un-tunable. K_Λ Gap: The bare EC torsion formula gives K_Λ = 9π²/4 (1+γBI²) = 21. 02. The Friedmann closure requires K_Λ = 49. 82 — a factor of 2. 37, close to x² = (rₛ/RU) ² = 2. 31. Papers 1 and 4 used an x² correction, which this paper identifies as phenomenological, not derived. If the x² factor is physical, Λ ∝ 1/RU² (epoch-dependent, w ≠ −1). This gap is identified as the framework's most critical open problem. The IO framework predicts H₀ = 58. 2 km/s/Mpc and age = 19. 2 Gyr — specific, falsifiable values derived from c³/ (GMU) with zero free parameters. Multi-AI research: Claude Code (Anthropic) for BBN computation, lapse investigation, speculative path analysis, and MU sensitivity; Wolfram/ChatGPT 5. 3 (OpenAI + Wolfram) for analytical verification of all results; Gemini 2. 5 Pro (Google DeepMind) for unified tension diagnosis, torsion-fermion proposal, and adversarial review; Claude (Anthropic) for orchestration. Companion to Paper 1 (DOI: 10. 5281/zenodo. 18854813), Paper 2 (DOI: 10. 5281/zenodo. 18868612), Paper 3 (DOI: 10. 5281/zenodo. 18876346), Paper 4 (DOI: 10. 5281/zenodo. 18883069), Paper 5 (DOI: 10. 5281/zenodo. 18889865), and Paper 6 (DOI: 10. 5281/zenodo. 18891475). v1. 2 correction (Paper 12, DOI: 10. 5281/zenodo. 18936508): Baryon sector annotations per Paper 12 (Baryon Dictionary Principle). Paper 12 derived fb = 2γ/x = 0. 313, superseding the BAO-optimized fb = 0. 254 used for the BBN computation in this paper. Under the BDP: ωb, geom = 0. 02108, η₁₀ = 6. 19, D/H = 2. 49 × 10⁻⁵ (−1. 2σ), Yₚ = 0. 2471 (+0. 6σ). The deuterium crisis reported in this paper (D/H 38% high, +31. 6σ) is resolved by the BDP. The unified tension diagnosis (§4) is substantially modified: the "single root cause h = 0. 582" framing remains correct for the ℓ₁ and H₀ tensions, but the deuterium tension is no longer part of the triad. The c³ hypothesis, age prediction (19. 2 Gyr), and K_Λ gap analysis are unaffected.