V5p31 — First-principles derivation of delta = -4 ln(d_{1/2}(k=135)) from SU(2)_135 modular structure (TQNT)

Within Topological Knot Quantum Theory (TQNT V5), the V5p29 closure (Sum|log| = 7 × 10-4 dex on the four neutral-meson oscillations {K0, D0, B0, Bs0}) uses a chiral inertia I_chi = I_V5.34_F3 × (1 + beta · A_rel)_pairs × prod_{i<j<k}(1 + delta · f_seq), with two parameters: beta = -1.30 (still calibrated, see V5p30 negative result) and delta = -2.775. V5p31 settles the closed form of delta. Two STRONG candidates had been flagged during the V5p30 survey within 0.3 %: D1: delta = -ln 16 = -4 ln 2 = -2.7726 and D2: delta = -4 ln d_{1/2}(k = 135) = -2.7715, where d_{1/2}(k) = 2 cos(pi/(k+2)) is the SU(2)_k quantum dimension of the spin-1/2 primary and k = 135 is the unique Chern-Simons level of TQNT (alpha_EM = 1/(k+2) = 1/137). The exact closed form is D2. D1 is its k → infinity limit. The Taylor expansion delta_exact = -4 ln 2 + 2(pi/137)^2 + (1/3)(pi/137)^4 + O(x^6) shows that D1 and D2 are not competing identifications but two terms of the same asymptotic series. Substituting delta_exact = -2.7715369 into V5p29 without recalibration yields Sum|log| = 0.0041 dex on {K0, D0, B0, Bs0}, still well below the V5.35b plateau (0.563 dex) and inside the closure window. Physical interpretation. The factor 4 admits a natural topological reading: four Wilson loops on the modular sphere, i.e. the three crossing-count directions {c_min, c_mid, c_max} of an asymmetric link triplet plus the collective linking reference. The minus sign carries the chiral Z_2 (mirror tunnelling). The factor 4 itself was the only remaining conjecture at V5p31 — its rigorous derivation via Bar-Natan v_4 + Reshetikhin-Turaev Wilson 4-loop is the content of V5p32. Programme status. TQNT V5 now uses 10 global parameters instead of 11 — delta is no longer free, and V5p29's 0.1 % closure on K, D, B0, Bs0 is preserved. The catalogue and the V5p7 monograph (concept DOI 10.5281/zenodo.19131538) are unchanged; only the parameter count drops. Bundle contents. EN + FR LaTeX sources and PDFs (4 pages each) of the V5p31 derivation note; references to V5p29 (where delta enters), V5p30 (failed beta derivation), V5p32 (factor-4 derivation), V5p33 (weight system).