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

Version 1. 3 introduces a foundational typographic and epistemological overhaul. The document has been transitioned to a one-column layout utilizing the proprietary stkwc. sty package. This update formally integrates the Anti-Ptolemy Protocol and Axiom-9 rules into the document's DNA, utilizing color-coded macros to visually enforce the absolute epistemological status of every claim (avoiding continuous parameter approximations). Furthermore, the conclusion formally sets the stage for the non-perturbative completion problem on the degeneracy surface (Open Problem H-R3), which is directly addressed in the forthcoming Document H. Abstract 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 confirms that no abelian bridge to Δₜop = 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 Ωᵣes = q̄/ (4p) ∈ ℚ/ℤ factors through a four-step chain where every arrow is algebraically forced: peripheral abelianization, logarithmic lift, Cheeger–Chern–Simons regulator, and torsion reduction after surgery. The endpoint is basis-independent. (5) Open Problems Formulation: Identifies stabilization bounds (F. 1), the surgery-dependent relative period (F. 3), and sets the crucial Open Problem H-R3 regarding the non-perturbative exact arithmetic of the integral ∫v du. Methodology: Computational verification and document compilation performed within the Parliament of Dragons distributed AI architecture: Claude Opus 4. 7 (Anthropic) for typographic identity (stkwc. sty) and final Axiom-9 PDF compilation, Claude Sonnet 4. 6 (Anthropic) for structural coordination, Gemini 3. 1 AI Studio (Google DeepMind) as Red Censor for Anti-Ptolemy logic verification, and ChatGPT (OpenAI) / Mercury (Inception AI) as generation/gauge-invariance nodes.