Baldwin–Sivek instanton-surgery and Newton-polygon input module

The exact full slope set. Write $\mathcal{S}(K)=\{p\mathrm{/}q\in \mathbb{Q}:\gcd (p,q)=1,q>0,E_K(p\mathrm{/}q)\text{ is SU2-abelian}\}$. Its elements are DISTINCT slopes, not a selected sequence. The meridional slope $\mathrm{\infty }$ is isolated and is not a finite-surgery parameter in this set. A simultaneous peripheral diagonalization of an SU2 representation gives $\rho (\mu )=\mathrm{diag}(e^{i\alpha },e^{-i\alpha })$ and $\rho (\lambda )=\mathrm{diag}(e^{i\beta },e^{-i\beta })$. The simultaneous inversion $(\alpha ,\beta )\mapsto (-\alpha ,-\beta )$ is the pillowcase quotient. For any continuous lifted path through peripheral images of irreducible representations, a slope in $\mathcal{S}(K)$ satisfies $p\alpha (t)+q\beta (t)\mathrm{\notin }2\pi \mathbb{Z}$: equality would give an actual irreducible representation of the filled group, contradicting [F1]. Continuity puts this entire real function in one component of $\mathbb{R}\setminus 2\pi \mathbb{Z}$. Thus for ANY two points of that path, $\mathrm{\mid }p({\alpha }_1-{\alpha }_0)+q({\beta }_1-{\beta }_0)\mathrm{\mid }<2\pi \mathrm{.}$ This is a full-set exclusion, valid for positive, negative and zero numerators.

The exact instanton interface selects a nonvertical path. Under [L1], its first alternative makes $\mathcal{S}(K)$ empty: for any finite $p\mathrm{/}q$ choose $\beta =-p\pi \mathrm{/}(2q)$ modulo $2\pi$ on that full vertical circle, giving an actual irreducible filling representation. Hence an SU2-averse knot, for which $\mathcal{S}(K)$ is infinite, must satisfy the second alternative. On a smaller closed subinterval its meridian is noncentral and already diagonal. Its commuting longitude is diagonal too; its eigenvalue is a continuous circle-valued function, so subdivision into finitely many open semicircle charts gives a continuous real lift $\beta$. The meridian coordinate $\alpha =s$ has distinct endpoint values. This is the ONLY use of [L1] in the subsequent path argument; no bare nontrivial-knot representation-existence assertion is silently substituted for its varying-meridian conclusion.

Boundedness of ALL slopes. Fix such a nonvertical path and endpoints, put $d={\alpha }_1-{\alpha }_0\mathrm{\ne }0$, $r_0=-({\beta }_1-{\beta }_0)\mathrm{/}d$ and $C=2\pi \mathrm{/}\mathrm{\mid }d\mathrm{\mid }$. Step1.1 gives $\mathrm{\mid }p\mathrm{/}q-r_0\mathrm{\mid }<C\mathrm{/}q\le C$ for EVERY $p\mathrm{/}q\in \mathcal{S}(K)$. Thus the full set is contained in the fixed bounded interval $[r_0-C,r_0+C]$; an unbounded exceptional subsequence is excluded. In a bounded interval there are only finitely many reduced fractions with denominator at most any fixed $Q$: for each $1\le q\le Q$ only finitely many integers $p$ satisfy the interval bound. Consequently, along EVERY sequence of pairwise distinct elements of $\mathcal{S}(K)$, $q\to \mathrm{\infty }$ and the displayed inequality forces $p\mathrm{/}q\to r_0$. The infinite set therefore has exactly one accumulation point in $\mathbb{R}\cup \{\mathrm{\infty }\}$, namely the finite $r_0$. This proves both quantifiers needed when passing to arbitrary infinite subsets.

All lifted paths are straight and parallel. Apply step1.1 to any two points of any continuous lifted irreducible peripheral path and to a sequence of distinct elements of the FULL set. Divide by $q$ and use step3.1 to obtain $r_0({\alpha }_1-{\alpha }_0)+({\beta }_1-{\beta }_0)=0.$ Thus $\beta +r_0\alpha$ is constant on each path, and any nonconstant one lies on a line of slope $-r_0$. In particular the varying-meridian path of step2.1 contains every point of a nontrivial interval of that line: its $\alpha$ coordinate covers the whole endpoint interval by continuity. No finite-component or semialgebraic path-selection assertion is required for this implication.

A nonzero Laurent equation actually vanishes on this arc. For completeness the needed image-dimension operation is the accepted field-linear adjoint calculation [F2], not an assumption that the A-polynomial Definition excludes larger images. If a generic irreducible eigenline-incidence component had algebraically independent $M,L$, its function field would admit two derivations independently differentiating $M,L$; extend these through its finite separable algebraic part by the minimal-polynomial derivative formula. The actual boundary homology has dimension two. Self-duality and the pair exact sequence make both boundary inclusion and cohomological restriction rank one. The actual word/eigenbasis pairing therefore puts ALL logarithmic derivative pairs $(DL\mathrm{/}L,DM\mathrm{/}M)$ in the same line, contradicting those two independent derivations. Reducible components have $L=1$ by knot abelianization. All peripheral images thus have dimension at most one. The finitely many zero-dimensional image components give only finitely many points. The nonconstant interval of step4.1, after discarding these points, lies in the union of the one-dimensional image curves; hence the nonzero Laurent polynomial $A_K$ in [F3] vanishes on it. By continuity it vanishes on the entire interval. This argument allows arbitrary dimensions of the underlying character components and does not confuse a one-parameter family with a maximal image component.

Rationality by a finite exponential calculation. Write $A_K(M,L)={\sum }_{(a,b)\in T}{c}_{ab}{M}^a{L}^b$, with finite distinct integral pairs and nonzero coefficients. On the interval $M=e^{i\alpha }$, $L=e^{i(c-r_0\alpha )}$, step5.1 gives ${\sum }_{(a,b)\in T}{c}_{ab}{e}^{ibc}{e}^{i(a-r_{0}b)\alpha }=0.$ If $r_0$ were irrational, all real numbers $a-r_{0}b$ would be distinct: equality for different pairs would either force identical pairs or express $r_0$ as a ratio of integers. Differentiate this finite identity $0,1,\dots ,\mathrm{\mid }T\mathrm{\mid }-1$ times at an interior point. Its coefficient matrix is a Vandermonde matrix in the distinct numbers $i(a-r_{0}b)$, multiplied by nonzero diagonal exponentials. Its determinant is nonzero, forcing every $c_{ab}{e}^{ibc}=0$, contrary to their choice. Thus $r_0\in \mathbb{Q}$. This proves rationality without importing a transcendence-number theorem or claiming arithmetic algebraicity of isolated pillowcase points.

Full package with explicit external inputs. Steps1.1–6.1 prove the elementary full-set boundedness, uniqueness and rationality operation from [L1]. The actual alternative is now supplied externally by [L1], while [L2] supplies the additional nonzero bound, L-space surgery, fiberedness, Alexander and satellite interfaces. The accepted filling and closed-lift operations [F4], together with the external knot-exterior trichotomy in [L2], give the geometric reductions required in §§3–5. These establish all four parts of the original package under its stated hypotheses. This completion uses external gauge and classification results; it does not assert local reconstructions of those results.