DefinitionDefinition: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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.

Standard orientation of the affine simplex

Definition

For $k \geq 1$ , orient the affine span of the standard simplex $Δ^{k} = [v_{0}, \dots, v_{k}]$ by the ordered basis $E_{1} = v_{1} - v_{0}, \dots, E_{k} = v_{k} - v_{0} .$ The ordered face opposite $v_{i}$ has the remaining vertices in their original order. Its outward-normal-first boundary orientation is $(- 1)^{i}$ times that ordered orientation. Thus the oriented boundary convention is $\partial [v_{0}, \dots, v_{k}] = \sum_{i = 0}^{k} (- 1)^{i} [v_{0}, \dots, {\hat{v}}_{i}, \dots, v_{k}]$ .

Orient $Δ^{0}$ as a positive point. It has no faces. For an interval, the displayed boundary is its positive terminal point minus its positive initial point. These coefficients record determinant-line signs on zero-dimensional faces; they do not assert that a one-vertex abstract simplex has two vertex orderings. Boundary orientation is computed at the relative interior of each face, where the simplex is locally a half-space; no smooth structure on general manifolds with corners is used.

Facts & Assumptions

[F1]

The standard topological simplex and its affine face maps gives barycentric coordinates, vertices and the zero-insertion affine face maps.

[F2]

An orientation of a simplex identifies ordered vertex lists up to even permutation.

[F3]

Determinant-line orientations of finite-dimensional real vector spaces supplies determinant-line rays, including the two rays in dimension zero.

[F4]

Induced boundary orientation uses an outward vector first, followed by a positive boundary determinant.

Verification

Given: The standard simplex and its ordered vertices. For positive dimension write $Ω = E_{1} \land \dots \land E_{k}$ .

1.1F1F2F3given

The affine parametrization $x \mapsto v_{0} + \sum_{j = 1}^{k} x_{j} E_{j}$ identifies the simplex with $x_{j} \geq 0$ and $\sum_{j} x_{j} \leq 1$ . The $E_{j}$ are independent: their coordinates in positions $1, \dots, k$ form the identity matrix. Swapping two vertices other than $v_{0}$ swaps two basis columns and changes the wedge sign. Swapping $v_{0}, v_{1}$ replaces the basis by $- E_{1}, E_{2} - E_{1}, \dots, E_{k} - E_{1}$ , whose wedge is $- Ω$ . These swaps generate the vertex permutations, so a permutation changes the ray by its permutation sign. The affine convention therefore agrees with [F2] for $k \geq 1$ .

2.1F3F4step 1.1

For $1 \leq i \leq k$ , the face $x_{i} = 0$ has its ordered basis $E_{1}, \dots, {\hat{E}}_{i}, \dots, E_{k}$ . At a relative interior point, $- E_{i}$ points outward, since the interior has $x_{i} > 0$ . Moving this vector from the first position to position $i$ gives $(- E_{i}) \land E_{1} \land \dots \land {\hat{E}}_{i} \land \dots \land E_{k} = (- 1)^{i} Ω .$ Multiplying the face determinant by $(- 1)^{i}$ makes its wedge after that outward vector positive. Thus [F4] gives exactly the claimed boundary sign on this face.

2.2F3F4step 1.1

On face zero, $\sum_{j} x_{j} = 1$ , the remaining ordered vertices are $v_{1}, \dots, v_{k}$ , with basis $E_{2} - E_{1}, \dots, E_{k} - E_{1}$ . The vector $E_{1}$ points outward, since it increases the coordinate sum. Its wedge with this face basis is $Ω$ , because all terms selecting another $E_{1}$ vanish by alternation. Hence the ordered face already has the boundary orientation, giving sign $(- 1)^{0} = 1$ . The outward vectors in this calculation need not be perpendicular: their strict transverse directions are exactly what [F4] requires.

3.1F1F2F3F4step 2.1step 2.2∎

For $k = 1$ , the face bases in steps 2.1–2.2 are empty determinants, namely $1 \in Λ^{0} {0} = R$ . The outward vectors at $v_{0}, v_{1}$ are $- E_{1}, E_{1}$ , so their induced determinant-line signs are respectively minus and plus. This gives $[v_{1}] - [v_{0}]$ , despite the unique vertex ordering of each abstract point in [F2]. For $k = 0$ , the chosen ray is positive and [F1] gives no face maps or negative-dimensional simplex. The simplex is never empty; zero coefficients or degenerate maps into a target do not change this domain orientation. Every vector and sign was specified explicitly, so no choice principle is used.

Depends on

Dependency tree · 2 levels

Sources

Peter S. Park, Proof of de Rham's Theorem