Document F: Renormalization of Topological Knowledge in Contrast-Native Architectures

We define a renormalization operator T on abelian contrast systems within the C0 (Contrast Calculus) framework, where contrast is ontologically prior to objects and coordinates. The operator uses Čech cohomology on coverings of the underlying graph, avoiding the information-destroying aggregation that naive coarse-graining would produce. Main results: (1) Čech preservation theorem: The operator T preserves the first cohomology H¹(G; A) via the discrete Čech–de Rham isomorphism. Nonzero cycles in the original system map faithfully to nontrivial transition functions in the coarsened system. (2) Stabilization conjecture (F.1): Iterated renormalization stabilizes in at most ⌈diam(G)/L⌉ steps to a minimal Čech resolution. (3) Worked example on 4₁: The abelian shadow of the figure-eight knot carries a universal triangle holonomy dκ = π (proven from z·z'·z'' = −1), verified numerically via SnapPy. After (5,2)-surgery, the Čech cocycle in the surviving H₁ direction is computed explicitly. A systematic scan of 25 functionals confirms that no abelian bridge to Δ_top = 3/20 exists — the value lives in the ℚ/ℤ fiber of the extended Bloch group. (4) Non-abelian extension and canonical projection (Theorem 18): The projection from H¹(T²; SL(2,ℂ)) to the filling invariant Ω_res = q̄/(4p) ∈ ℚ/ℤ factors through a four-step chain where every arrow is algebraically forced: peripheral abelianization (by boundary commutativity), logarithmic lift, Cheeger–Chern–Simons regulator (by integrality of the symplectic pairing), and torsion reduction after surgery. The endpoint is basis-independent. This resolves Open Problem S2 from the Document D Supplement. (5) Three open problems: stabilization bound (F.1), surgery-dependent relative period connecting analytic and arithmetic derivations (F.3), and Two-Path verification for knots 5₂, 6₁, pretzel(-2,3,7) (F.4). Computational verification performed within the Parliament of Dragons multi-AI methodology: Claude Opus 4.6 (Anthropic) as Validator/Auditor and document author, ChatGPT (OpenAI) as Generator, Gemini (Google DeepMind) for initial LaTeX, Mercury 2 (Inception AI) for gauge-invariance confirmation, Claude Sonnet 4.6 (Anthropic) for final Axiom-9 audit. Notes Part of the STKWC preprint series. Document F v1.2.