The Interior Observer Cosmological Framework — Paper 17: The Modular Projection Theorem: Operator-Level Closure of the Gauge Thermal Transfer Principle via Shared Hilbert Space Construction and Fiberwise KMS Rigidity

Paper 17 of the Interior Observer (IO) Cosmological Framework series. The Gauge Thermal Transfer Principle (GTTP), previously classified as a semiclassical principle (Papers 13–14), is upgraded to DERIVED/THEOREM within the reduced thermal-plus-gauge sector. A shared Hilbert space HIO is constructed containing both the thermal photon sector and the reduced SU (2) horizon gauge sector. An explicit A-vacuum GNS state is built from fiberwise KMS data, proved positive, normalized, faithful, and normal. The Modular Projection Theorem (Theorem 17. 1) proves that the Tomita-Takesaki modular flow of the A-vacuum restricted to the thermal photon sector satisfies Δₚhys^it = exp (it (D ⊗ K̂g) ), giving σ = Kgauge = ln (1+γ²) and Tₒbs = TIO × x^Kgauge = 2. 7253 K (0. 3σ from FIRAS). A framework-constructible uniqueness result confirms Kgauge is the sole structurally meaningful transfer candidate in the FIRAS band. Thirteen investigation rounds are documented. Multi-AI adversarial collaboration: Claude (Anthropic), Codex/ChatGPT (OpenAI), Wolfram/ChatGPT (OpenAI), Gemini (Google DeepMind). v1. 1 - Paper 19 correction annotations. Abstract updated with two founding premises. Core results entirely unaffected: HIO Hilbert space construction, GNS representation, A-vacuum state, modular flow, coupled generator, product-flow no-go, all operator identities. Paper 19 uses Paper 17's HIO as the foundation for the IO Friedmann equation derivation. Appendix entries for background-dependent quantities (projection map, Nₑff = Δ, Ωₖ) superseded.