Pedagogical derivation of the Chern-Simons formula in TQNT: from the action SCS to the Jones polynomial, the level shift k -> k+hᵛ, and alphaEM = 1/137

Within Topological Knot Quantum Theory (TQNT V5), the fine-structure constant emerges from a single Chern-Simons level: alphaEM = 1/ (k + hᵛ) = 1/137, where hᵛ = 2 is the dual Coxeter number of SU (2) and k = 135 is the bare level appearing in the Chern-Simons action. This pedagogical note presents the complete derivation, without shortcuts, of the Chern-Simons formula underlying TQNT V5. Five levels are exposed successively: Level 1. The classical action SCSA = (k/4 pi) integral Tr (A wedge dA + (2/3) A³), its geometric setting (principal SU (2) -bundle), gauge transformations, and the Dirac quantization k in Z. Level 2. The functional integral Z (M) = int DA exp (i SCS), Wilson loops, and Witten's central theorem (1989): ₒ℃, ₒₔ (₂) 䂵 = VK (q) with q = exp (2 pi i / (k+2) ), recovering the Jones polynomial at a root of unity. Level 3. The origin of the shift k -> k + hᵛ, derived independently via three equivalent routes: (A) one-loop renormalisation of the Chern-Simons action; (B) the Chern-Simons / Wess-Zumino-Witten correspondence and the central charge cWZW = k dim g / (k + hᵛ) ; (C) the quantum group Uq (sl₂) at q a root of unity of order k + hᵛ. Level 4. The TQNT identification alphaEM = 1/ (k + hᵛ) (postulate P10), with the unique consistent solution k + hᵛ = 137 (prime, required for non-degenerate fusion rules) and therefore k = 135. Level 5. Coherence with the V5 oscillation programme: the quantum dimension d₁/₂ (135) = 2 cos (pi/137) = 1. 99947 appears directly in the V5p31 first-principles derivation of delta = -4 ln d₁/₂ (k=135) = -2. 7715; SU (2) ₁₃₅ fusion rules truncate at j <= 67, comfortably accommodating the Rolfsen catalogue. Answer to a frequent question. Why not k = 137 directly? Because the physical coupling is k + hᵛ = 137, not k; to obtain k = 137 alone one would need hᵛ = 0, i. e. an abelian group U (1), but U (1) Chern-Simons is trivial for knots (no Jones polynomial, no non-abelian braiding). The shift of 2 is Witten's 1989 theorem, verified by three independent routes. Bundle contents. A single French LaTeX source and PDF (8 pages) presenting the complete derivation with explicit theorems, proofs, and references to Witten 1989, Reshetikhin-Turaev 1991, Verlinde 1988, Axelrod-Singer 1991, Kac 1990 and Bar-Natan 1995.