Simplex integrals are independent of affine coordinate identification

Suppose $k\ge 1$. Each of $K_a,K_b$ is an invertible affine image of the standard compact Jordan simplex from [F1], hence compact Jordan by [F2]. Put $h=a^{-1}\circ b$. It is an affine diffeomorphism of ${\mathbb{R}}^k$, it maps $K_b$ onto $K_a$, and its constant derivative has determinant $d>0$ because both parametrizations are positive.

For a single smooth extension write $a^{\ast }{\bar{\sigma }}^{\ast }\omega =f(x)d{x}_1\wedge \cdots \wedge d{x}_k$. The coefficient is smooth near $K_a$ and bounded on $K_a$. By [F4], its expression in the $b$ parametrization is $f(h(y))\det Dhd{y}_1\wedge \cdots \wedge d{y}_k$: evaluating the alternating form on the $k$ columns of $Dh$ gives exactly this determinant. Thus the coefficient in $b$ coordinates is $d(f\circ h)$.

Apply [F3] to the open set ${\mathbb{R}}^k$, injective $C^1$ map $h$, compact Jordan set $K_b$ and continuous function $f$ on $h(K_b)=K_a$. Every derivative is invertible and $\mathrm{\mid }\det Dh\mathrm{\mid }=d$. Therefore ${\int }_{K_a}f(x)dx={\int }_{K_b}df(h(y))dy,$ which is precisely equality of the two form integrals for a fixed extension. If two neighbourhood extensions are used, they agree on the relative interior of the affine simplex. Their pullback coefficients therefore agree there; every boundary point is a limit of relative-interior points, so continuity makes the coefficients agree on the whole compact simplex. Their integrals consequently coincide in either coordinate system.

For $k=0$ both parametrizations of the affine point are the unique point map and both definitions are evaluation at $\sigma (v_0)$; no positive-dimensional change-of-variables theorem is used. For $k=1$, step 3.1 includes all positive affine rescalings of the closed interval, with endpoints included in the compact set. Zero coefficients and rank-deficient simplex maps require no division by a form value, so the equality still holds. Empty targets supply no simplex. All maps $h$ and their coefficients are determined by the two given parametrizations; no choice is used.