Möbius Magnetic Topology (MMT) II: A Fully Self-Contained Proof of the Field-Line Interlinking Invariants-The Unified Replacement Theorem and the Wr+Tw Decomposition

This paper is the second in the Möbius Magnetic Topology (MMT) series (MMT-II). It provides a fully self-contained, first-principles proof of the linking-number formula system established in MMT-I (Paper 8, Zenodo DOI 10.5281/zenodo.22653188). All linking-number formulas are derived from explicit parameterized field lines by direct evaluation of the Gauss linking integral, without invoking any ready-made torus-knot or link theorems (the framework-level Călugăreanu–White decomposition identity cites standard results [1,2], §11). The proof is stratified by rigor: for integer-slope modes the proof is fully rigorous (Lemma A region decomposition); for non-integer/half-integer slope modes (including all Möbius modes) the formulas are established by near-field analytic evaluation plus full-curve numerical verification, with the strict proof of the mean cancellation of turn-pair corrections listed as an honest limitation (§12.4) — its numerical universality has, however, been verified by three attack vectors (non-allowed modes / large k / extreme δ, all passed), and the formulas themselves are reproducible for all allowed modes (error <0.1%). The central result is the Unified Replacement Theorem: under the Möbius embedding the linking-number system is exactly parallel to that of the ordinary torus, differing only by the slope replacement q/p → q/p+1/2. Its origin is the Unified Replacement Lemma — a Möbius field line (embedding angle θ_emb = (q/p+1/2)φ) is the same space curve as the ordinary helix of slope q/p+1/2 (pointwise difference zero), so every Möbius linking-number formula is a direct corollary of the ordinary-torus formula, and the correction term −k²/2 comes entirely from the +φ/2 term in which the Möbius gluing rule is encoded into the embedding angle. The paper gives complete proofs of seven theorems: Theorem 1 (co-oriented interlinking = −(q/p)k², with the region-decomposition lemma and the near-field analytic evaluation of single turn pairs), Theorem 2 (zero linking for counter-oriented curves on the same tube radius, mirror reflection), Theorem 3 (cross self-linking of counter-oriented curves on distinct tube radii = +(q/p)k², orientation reversal), Theorem 4 (Möbius co-oriented interlinking = −(q/p+1/2)k²), Theorem 5 (Möbius counter-oriented, distinct tube radii = +(q/p+1/2)k²), Theorem 6 (conservation of the net linking of four field lines, zero), and Theorem 7 (the Wr+Tw=SL decomposition: the thin-tube analytic twist T_w = −qk[1+(ρ²/2R²)(1−q²)], the thin-tube writhe limit −qk(k−1), where the Möbius "flip of π per turn" enters quantitatively into the −k/2 term of the twist). All theorems are checked against direct Gauss linking integration, with errors <0.1% in the key regimes (linking-number comparison errors; the C–W identity residual convention is explained in §11.6, where 0.079 is an absolute residual of the slowly converging W_r regime, not a percentage). This paper is an extension of MMT-I, not a correction: under the unified conventions (outward normal, positive orientation) the tables 4.1/5.2/6.1 of MMT-I agree with the theorems here term by term.