V5p30 — Attempted first-principles derivation of beta from c_J(3_1): an honest negative result (TQNT)

Within Topological Knot Quantum Theory (TQNT V5), V5p29 closes the four neutral-meson oscillations {K0, D0, B0, Bs0} to 8 × 10-4 dex using two parameters: beta = -1.30 (complexity-asymmetry coupling) and delta = -2.7715 (three-body multiplicative renormalisation, derived from first principles in V5p31). V5p30 reports an honest negative result. A conjecture (B3) anticipated that beta = -ln c_J(3_1), where c_J(3_1) is the Jones-coloured factor of the trefoil knot, with target value c_J(3_1) = exp(1.30) ≈ 3.66. Ten distinct derivation methods were tested (Wilson loops on the modular sphere, Verlinde fusion hierarchy applied at k=135, framing anomalies, Kashaev-style volume prefactor, Reshetikhin–Turaev unknot normalisation, twist tower expansion, surface-graph counting, multi-loop contraction, fusion-channel weighting, modular S-matrix). None reached the required precision (< 1 %) — the closest value (3.43) still misses by 6 %, and no consistent scheme produces a unique answer. Conclusion. beta cannot be derived in the present framework. It remains a calibrated parameter of the V5p29 closure formula. We document the ten attempts and their failure modes so future work knows what has already been ruled out. Comparison with V5p31 (delta). The parallel conjecture for delta — that delta = -4 ln d_{1/2}(k=135) — does succeed (proved exactly in V5p31). The structural difference is that delta couples to the modular-sphere quantum dimension of the spin-1/2 primary, an object with a closed-form expression at k=135, whereas beta would need a closed form for c_J(3_1) which is presently unavailable. Bundle contents. EN + FR LaTeX sources and PDFs of the V5p30 negative-result note: list of methods tried, target value, deviations achieved, structural reason for failure. References to V5p29 (where beta enters), V5p31 (successful delta derivation), V5p32 (factor-4 derivation), V5p33 (weight system).