We define a canonical, knot-independent period integral Πᵣel (K; p, q) on the geometric component of the A-polynomial curve for any hyperbolic knot K under primitive (p, q) -Dehn surgery. The integral computes the complexified Chern–Simons variation along the deformation path from the complete hyperbolic structure to the surgery representation, and is identified with the Neumann–Zagier potential difference. We compute Πᵣel for the figure-eight knot 4₁ across ten surgery slopes — the first such systematic table in the literature. The imaginary part Im (Πᵣel) measures the volume variation under Dehn filling and is both slope-dependent and knot-dependent. The universal filling correction Δfill = q̄/ (4p) (mod 1), here refined to require the Atiyah canonical 2-framing to resolve a Z₄ ambiguity, provides the arithmetic counterpart. We conjecture that the decomposition admits a clean rational factorization if and only if the knot complement is arithmetic — singling out 4₁ uniquely among knots in S³. Produced via the Parliament of Dragons multi-AI methodology with adversarial review. Notes: Document G of the STKWC programme (Scalar-Tensor Knot-Web Cosmology). 26th preprint in the series. Produced by the Parliament of Dragons methodology: 9 AI systems (Claude Opus 4. 6, ChatGPT 5. 4, Gemini 3. 1 Pro, Grok 4. 2, Mercury 2, Arcee, Gemma 4) with human coordination (Yanush Feshter). Three prior drafts destroyed by adversarial review (Gemini-A) ; all falsified hypotheses reported honestly. Contains the first systematic multi-slope table of period integrals on the A-polynomial curve for the figure-eight knot.
Yanush Feshter (Sat,) studied this question.