ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck pending

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Affine filling and the exact reducible-locus repair step

Example

Assume ZF together with the Axiom of Choice. Let $M = S^{3}$ or $S^{4}$ , let $N \geq 2$ , take the trivial principal right $S U (N)$ -bundle $P = M \times S U (N)$ , and use its zero connection as affine origin. For any integer $j \geq 1$ and any continuous sphere family $f : S^{j} \to A_{\infty}^{*}$ of smooth connections on $P$ , the cone formula $F ([r, x]) = r f (x)$ extends $f$ continuously over $D^{j + 1}$ in the full affine connection space. The apex is the zero connection, whose holonomy group is the identity and which preserves every complex line, so this affine filling is reducible and does not lie entirely in $A_{\infty}^{*}$ . Relative avoidance changes the filling while fixing its boundary and produces a filling in $A_{\infty}^{*}$ ; contraction of the disk to any chosen boundary point gives a based nullhomotopy. The weak-contractibility theorem ensures that such families exist for the trivial bundle and that their based homotopy classes vanish.

Facts & Assumptions

Given: ZF and the Axiom of Choice; $M = S^{3}$ or $S^{4}$ ; $N \geq 2$ ; the trivial principal right $S U (N)$ -bundle; its product connection $0$ ; and a continuous map $f : S^{j} \to A_{\infty}^{*}$ for $j \geq 1$ .

[A1]

The Axiom of Choice means every family of nonempty sets has a choice function (The Axiom of Choice).

[F1]

In the trivialization $P = M \times S U (N)$ , the affine connection space is $Ω^{1} (M; s u (N))$ translated by the product connection, and the product connection has identity holonomy around every loop (Smooth and Sobolev configuration and reduced gauge spaces).

[F2]

The relative avoidance lemma homotopes a continuous finite-polyhedron family into $A_{\infty}^{*}$ while fixing its irreducible values on a subpolyhedron (Relative finite-parameter avoidance of reducible connections).

[F3]

The weak-contractibility theorem asserts nonemptiness and vanishing of all based homotopy groups for this irreducible locus, under the same bundle, dimension, and group hypotheses (Weak contractibility of the irreducible connection locus).

Proof

technique · explicit affine cone and relative repair

1.1F1givenalgebra

The zero connection has parallel transport equal to the identity in the product trivialization, so its based holonomy group is ${I_{N}}$ . Since $N \geq 2$ , every complex line in $C^{N}$ is a nonzero proper subspace invariant under $I_{N}$ . Thus $0 \notin A_{\infty}^{*}$ .

1.2F1givenalgebraconstruct

The cone $D^{j + 1} = C S^{j}$ identifies all $(0, x)$ to one apex and has boundary parameter $r = 1$ . The formula $F ([r, x]) = r f (x)$ is well-defined because it is independent of $x$ at $r = 0$ , and equals $f (x)$ at $r = 1$ . For every continuous section seminorm $p$ , compactness of $S^{j}$ gives $C_{p} = \sup_{x} p (f (x)) < \infty$ , so $p (F ([r, x])) \leq r C_{p} \to 0$ as $r \to 0$ . Away from the apex, continuity follows from continuity of $f$ and scalar multiplication in the Fréchet vector space. Hence $F$ is a continuous affine filling in $A_{\infty}$ .

2.1step 1.1step 1.2algebra

By step 1.1, the apex value lies outside $A_{\infty}^{*}$ ; therefore this explicit extension is not an irreducible filling, regardless of its irreducible boundary values. In particular, convexity of the full affine space does not imply convexity of its irreducible subset.

2.2F2givenstep 1.2constructalgebra

Apply [F2] to $F$ with $K = D^{j + 1}$ and $L = S^{j}$ . The lemma fixes the boundary and supplies a homotopic endpoint $\tilde{F} : D^{j + 1} \to A_{\infty}^{*}$ . If $x_{0} \in S^{j}$ is the based point with $f (x_{0}) = A$ , the maps $x \mapsto \tilde{F} ((1 - t) x + t x_{0})$ stay in the disk, remain in $A_{\infty}^{*}$ , and fix $x_{0}$ . Their restriction to the boundary is a based nullhomotopy of $f$ .

3.1A1F3step 2.2algebra∎

Theorem [F3] supplies such irreducible sphere families, for example the constant family at any irreducible connection, and identifies the conclusion with vanishing of $π_{j} (A_{\infty}^{*}, A)$ . The example therefore isolates the precise role of relative avoidance: it upgrades a continuous affine filling, which may hit a reducible connection, to an irreducible filling without moving the sphere family.

Affine filling and the exact reducible-locus repair step

Example

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources