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: Aₛ = (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 (γBI = 0. 2375 from Loop Quantum Gravity) and the mathematical constant π. All dependence on MU, rₛ, ℏ, c, G, kB, and lP cancels exactly in the Hawking exponent βH ω₁ = 4π√2. Part II establishes that Paper 19's late-time clustering theorem does not authorize ωb, 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 (ωb, eff = 0. 02910). Part III proposes an effective optical damping parameter τₑff, IO = Kgauge/2 = (1/2) ln (1+γ²) = 0. 02744 from the inverse of the tangential horizon operator. The combined effective amplitude Aₑff = Aₛ × exp (−Kgauge) = 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/CONDITIONALVERIFIED on the reduced centered Gaussian source-covariance class; result OA = exp (Kgauge) O_Γ, GA⁽¹⁾ = exp (−Kgauge) G_Γ⁽¹⁾, τₑff, IO = Kgauge/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/PREMISEGAP. BBN scorecard rows aligned with Paper 24 v3. 0 Pastore QGS branch (χ² = 1. 17). Master table updated per Paper 17 v1. 5 R4FIRAS 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/CONDITIONALVERIFIED, DERIVED/NO-GO, VERIFIED, IMPORTED/EMPIRICAL, RECONSTRUCTION, OPEN/PREMISEGAP, 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.