Document K: Resolving the H-R3 Integral at the Degeneracy Surface via Discrete Boundary Capacity

We resolve the H-R3 integral at the degeneracy surface ε = 0 for the (5, 2) -Dehn filling of the figure-eight knot 4₁ within the discrete ontology C₀ of the STKWC programme. The continuous paradigm (E₁), as instantiated by smooth character varieties and numerical hyperbolic solvers such as SnapPy, collapses on the singular locus ε = 0. Document K demonstrates that the collapse is not a numerical accident but an ontological signal, and that a finite, well-typed C₀ value can be assigned to the integral once an honest boundary-capacity axiom is imposed. We prove one No-Go theorem and introduce one selection axiom: • Theorem 3. 1 (C₀ No-Go Theorem for denominator selection). The sealed displayed data 𝒟₅, ₂ — the surgery vector (5, 2), the meridional support SM = -4, -2, 0, 2, 4, the Newton-polygon vertices V (N), and the reversal involution τ — admit several distinct admissible C₀ denominator candidates (20, 40, 16). The displayed data do not by themselves canonically select one. • Axiom 4. 1 (Boundary Capacity Axiom). The total boundary capacity is defined as C_∂ (p, q;SM): = |p| · max₌∈ₒ₌ |m|. Imposed as an exogenous selection principle, it picks the surgery-weighted meridional-radius functional. • Filling invariant. For the (5, 2) -filling on 4₁, the discrete invariant evaluates to Δfill (5, 2) = 3/20. Under Axiom 9 (Honest Labelling), 3 is post-definition C₀-evaluated, 20 is axiom-selected, and 3/20 is post-axiom. • Universality (Test P3a). Axiom 4. 1 applies uniformly to two independent surgery data — (5, 1) on 4₁ and (3, 1) on the trefoil 3₁ — yielding well-defined C₀ filling invariants Δfill (5, 1) = 3/10 and Δfill (3, 1) = 2/3 via an identical chain of inference. The axiom is a structural selection principle of universal scope within the displayed-data category, not a parameter tuned to a single filling. The relation of these C₀ outputs to canonical E₁ invariants, where the latter are well-defined, is left as an open problem. The document is the output of the Parliament of Dragons multi-AI collaborative pipeline under the Anti-Ptolemy Protocol v1. 2. Lead Mathematician: ChatGPT 5. 5. Red Censor v2. 0 audit complete: no E₁-smuggling detected. Every numerical claim carries an explicit provenance tag (C₀-evaluated / Axiom-selected / Post-axiom) and no smooth, hyperbolic, or otherwise E₁ structure is imported into the chain of inference. This work is part of the STKWC programme series and directly builds upon Documents D, D Supplement, F, I, and J. It is the discrete-boundary resolution complementing Document J's structural impossibility result (Theorem 8. 1, the fourth No-Go in the D/I/F/J sequence), and the entry point for the open problems of the H-series (H-R3 closure, non-perturbative completion at ε = 0).