What question did this study set out to answer?

April 1, 2026Open Access

The Complex Chern-Simons Level of STKWC: A Zero-Parameter Derivation from the Surgery-Spectral Intersection on the SL (2, C) Character Variety of 4₁

Key Points

The aim is to derive the surgery-spectral triad for (5,2)-Dehn surgery on the figure-eight knot 4_1 without free parameters.
Derivation of the complete surgery-spectral triad
Calculation of period integral and topological displacement
Analysis of critical amplitude from eigenvalue projections
Application of insights to LIGO candidate in gravitational waves
Period integral calculated as Π_+(1) = 2π²/15
Topological displacement Δ_top found to be 3/20
Critical amplitude ε_* identified as cos(π/5) = φ/2
Identification of a pre-merger frequency plateau in gravitational waves

Abstract

Version 2. 0. Major upgrade: RevTeX4-2 (APS) format. Published on the birthday of Stefan Banach (30 March 1892), whose fixed-point theorem underlies the convergence structure of this work. We derive, with zero free parameters, the complete surgery-spectral triad for (5, 2) -Dehn surgery on the figure-eight knot 4₁ within Scalar-Tensor Knot-Web Cosmology (STKWC). Three hard numbers from one intersection on the SL (2, C) character variety: (1) Period integral: Π_+ (1) = 2π²/15, computed on the real oval of the A-polynomial. (2) Topological displacement: Δₜop = 3/20, proved in the companion paper (DOI: 10. 5281/zenodo. 18894178). (3) Critical amplitude: ε_* = cos (π/5) = φ/2 (golden ratio over two), emerging as the eigenvalue projection of the (5, 2) -surgery Dehn twist. The three quantities are bonded by surgical quadratic reciprocity: Δₜop = (q') ² × Δₗocal, where q' = 2⁻¹ mod 5 = 3 and Δₗocal = Π_+/ (8π²) = 1/60. The square of the modular inverse closes the loop: 3² × 1/60 = 3/20. No 4D bulk metric enters at any stage. Applied to LIGO candidate S251112cm: the gravitational chirp is interpreted as A-polynomial anharmonicity approaching the discriminant locus. A falsifiable prediction — a pre-merger frequency plateau — is identified. Changelog v1. 0 → v2. 0: RevTeX4-2 format (APS PRD style, two-column reprint). Added references to Volume Fibrations paper (DOI: 10. 5281/zenodo. 19193955) and Document A v3 (DOI: 10. 5281/zenodo. 19163002). Removed non-academic affiliation from author footnote. Streamlined acknowledgments. Document D v2. 0. Eighth preprint in the STKWC series.

The Complex Chern-Simons Level of STKWC: A Zero-Parameter Derivation from the Surgery-Spectral Intersection on the SL (2, C) Character Variety of 4₁

Key Points

Abstract

Cite This Study