What question did this study set out to answer?

This work aims to derive the closed form of delta in Topological Knot Quantum Theory's framework.

June 20, 2026Open Access

V5p31 — First-principles derivation of delta = -4 ln (d₁/₂ (k=135) ) from SU (2) ₁35 modular structure (TQNT)

Key Points

This work aims to derive the closed form of delta in Topological Knot Quantum Theory's framework.
Utilized first-principles calculations and Taylor expansions to derive delta.
Identified candidates through a survey and validated them by analyzing their quantum dimensions.
Confirmed the exact closed form of delta = -4 ln d_{1/2}(k=135) = -2.7715.
Established that two candidates are part of the same asymptotic series, not in competition.
Maintained a closure within the required precision for neutral-meson oscillations, with results still below the plateau.

Abstract

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 Ichi = IV5. 34F3 × (1 + beta · Aᵣel) ₚairs × prod₈<₉<₊ (1 + delta · fₛeq), 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 d₁/₂ (k = 135) = -2. 7715, where d₁/₂ (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 (alphaEM = 1/ (k+2) = 1/137). The exact closed form is D2. D1 is its k → infinity limit. The Taylor expansion deltaₑxact = -4 ln 2 + 2 (pi/137) ² + (1/3) (pi/137) ⁴ + O (x⁶) shows that D1 and D2 are not competing identifications but two terms of the same asymptotic series. Substituting deltaₑxact = -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 cₘin, cₘid, cₘax of an asymmetric link triplet plus the collective linking reference. The minus sign carries the chiral Z₂ (mirror tunnelling). The factor 4 itself was the only remaining conjecture at V5p31 — its rigorous derivation via Bar-Natan v₄ + 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).

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