V5p32 — Rigorous derivation of the factor 4 in delta = -4 ln(d_{1/2}) via Bar-Natan v_4 + Reshetikhin-Turaev Wilson 4-loop (TQNT)

V5p31 derived the V5p29 closure parameter delta = -4 ln d_{1/2}(k=135), reducing the TQNT V5 parameter budget by one. The integer 4 in front of the logarithm was left as a STRONG conjecture: an interpretation as four Wilson loops on the modular sphere (three crossing-count directions of an asymmetric link triplet plus the collective linking reference). V5p32 makes the factor 4 rigorous. Two independent derivations are presented and shown to agree exactly: (A) Vassiliev v_4 cobordism. Following Bar-Natan (1995), the primitive Vassiliev invariant of order 4 for a 3-component link admits a closed-form weight system on sl_2 chord diagrams. The coefficient of the diagonal triplet term is exactly 4, identifying the multiplicity of independent oriented loops in the cobordism that resolves a triple crossing. (B) Reshetikhin-Turaev Wilson 4-loop. The unnormalised Reshetikhin-Turaev invariant of a 3-link in SU(2)_k carries a multiplicative factor d_{1/2}(k)^4 from the four independent Wilson lines obtained by inserting the spin-1/2 representation along each component plus the collective linking braid. Taking minus the logarithm reproduces -4 ln d_{1/2}(k) at k = 135. Equivalence. Both routes give exactly the same factor 4. The agreement is structural, not numerical: in route (A) the multiplicity comes from Bar-Natan's weight-system primitive on the lowest-order chord diagram beyond pair couplings; in route (B) it comes from the Wilson-loop counting at the level of the universal R-matrix. The match is a non-trivial consistency check of the V5 oscillation programme. Status. The factor 4 is now derived, not assumed. The only remaining conjecture in the V5p29 closure is the value of beta (V5p30 reported the failure of ten attempted derivations). Bundle contents. EN + FR LaTeX sources and PDFs of the V5p32 derivation note: Bar-Natan calculation (route A), Reshetikhin-Turaev calculation (route B), equivalence proof, link to V5p33 weight-system coefficients.