The Interior Observer Cosmological Framework: Paper 26 - The Primordial Scalar Amplitude from the Hawking Boundary State, Toward IO-Native Replacements for ΛCDM-Borrowed Inputs, Source-Side Reduction, the CMB Baryon Class Diagnostic, and the Effective Optical Damping Parameter

Paper 26 of the Interior Observer (IO) Cosmological Framework. Beginning from two premises — (1) the observable universe exists inside a Schwarzschild black hole, and (2) the physics inside the horizon is the same as outside — this paper proposes IO-native replacements for ΛCDM-borrowed inputs in the framework's confrontation with Planck 2018 CMB data, with zero fitted parameters. The headline result is a conditional forward prediction for the primordial scalar amplitude from the Hawking boundary state on the S² horizon: A_s = (25/9) × [γ²/(1+γ²)] × [1/√2] × 1/(exp(4π√2) − 1) = 2.007 × 10⁻⁹ (Planck: 2.100 ± 0.030 × 10⁻⁹, deviation −4.4%). The formula contains one external input (γ_BI = 0.2375 from Loop Quantum Gravity) and the mathematical constant π. All dependence on M_U, r_s, ℏ, c, G, k_B, and l_P cancels exactly in the Hawking exponent β_H ω₁ = 4π√2. Part II establishes that Paper 19's late-time clustering theorem does not authorize ω_b,clustering in CMB perturbation slots, resolving a catastrophic χ² failure. The Thomson Kernel Lemma proves that the visibility function and acoustic oscillations share the same primitive opacity factor. The visibility readout is conditionally assigned to the acoustic baryon class (ω_b,eff = 0.02910). Part III proposes an effective optical damping parameter τ_eff,IO = K_gauge/2 = (1/2)ln(1+γ²) = 0.02744 from the inverse of the tangential horizon operator. The combined effective amplitude A_eff = A_s × exp(−K_gauge) = 1.900 × 10⁻⁹ matches the ΛCDM TT extraction (1.885 × 10⁻⁹) to 0.8%. Part IV shows CLASS reionization shape defaults are numerically irrelevant for TT high-ℓ (Δχ² < 0.4). Four conditional framework-internal identifications (C1, C2c, AV1, C3) are required, each explicitly documented with assumptions, limitations, and promotion paths. Includes Lemma C2.1 (background/perturbation channel separation) and Lemma C2.2 (carrier identification via Hopf lift), both derived, resolving the radial-vs-angular mode counting tension. Full cumulative Appendix A catalog (Steps 1–398) spanning Papers 1–26. https://dfife.github.io/index.html v2.0 (May 2026): New Theorem 26.C3 (Reduced Source-Covariance Propagator, §4.5) closes the broad C3 propagator-identification premise of v1.1 as DERIVED/CONDITIONAL_VERIFIED on the reduced centered Gaussian source-covariance class; result O_A = exp(K_gauge) O_Γ, G_A⁽¹⁾ = exp(−K_gauge) G_Γ⁽¹⁾, τ_eff,IO = K_gauge/2 under Definition 26.C3.3. Original C2 decomposed in §2.9 into Lemmas C2.1 and C2.2 (both DERIVED/THEOREM); C2c remains OPEN/PREMISE_GAP. BBN scorecard rows aligned with Paper 24 v3.0 Pastore Q_GS branch (χ² = 1.17). Master table updated per Paper 17 v1.5 R4_FIRAS framing; stale acoustic and R4-era diagnostic rows removed. All body theorem and appendix Step labels canonicalized against the Claims Discipline taxonomy (DERIVED/THEOREM, DERIVED/CONDITIONAL_VERIFIED, DERIVED/NO-GO, VERIFIED, IMPORTED/EMPIRICAL, RECONSTRUCTION, OPEN/PREMISE_GAP, SUPERSEDED, Historical/SUPERSEDED), including inherited Papers 1–25 catalog rows. Appendix Steps 382, 387, 400, 401, 407, 408, 35 Piece 5, 88, and 383 corrected for body-appendix consistency. Cross-references synced to Paper 24 v3.0 and Paper 25 v2.0. Open Problems restructured. Reproducibility bundle posted as GitHub Release paper26-v2.0 (SHA256: 7aba6d13362150fdc987ec5fdc3d179570e85c778ca4b455157acfe893295749; validator 22/22 PASS). v1.1 (April 2026): Full appendix from Paper 25 v1.1 clean + Paper 26 enriched results (Steps 382–409). Open/Closed tracking added. Author block updated. Rosetta terminology retired.