Vietoris--Rips Coindex Thresholds for Round Spheres: Spherical Joins and Chromatic Obstructions for Gromov--Hausdorff Distances

For $0\le m\le n$, let $c_{m,n}$ be the infimum of scales at which the Vietoris–Rips filtration of the round sphere ${\mathbb{S}}^m$ admits a continuous odd map from ${\mathbb{S}}^n$. A quantitative Borsuk–Ulam theorem gives $c_{m,n}\mathrm{/}2\le d_{\mathrm{G}\mathrm{H}}({\mathbb{S}}^m,{\mathbb{S}}^n)$, and it was asked whether equality always holds. Using a synchronized product-measure lift of the spherical join, we construct continuous odd maps between Vietoris–Rips metric thickenings with target scale equal to the maximum input scale. Iteration gives $c_{m+d,n+d}\le c_{m,n}$ for every integer $d\ge 0$. More generally, the finite join law adds $m_i+1$ and $n_i+1$ separately and bounds the resulting $c$-value by ${\max }_i{c}_{m_i,n_i}$. For $0\le r<\pi$, the pairs $(k,\mathrm{\ell })$ with $c_{k-1,\mathrm{\ell }-1}\le r$ are closed under addition, so Fekete’s lemma gives a limit for the maximal admissible $\mathrm{\ell }\mathrm{/}k$ as $k\to \mathrm{\infty }$. This structure, exact values, and projective-code estimates give finite and asymptotic bounds. If $1\le m_j<n_j$, $m_j\to \mathrm{\infty }$, and $\log (n_j+1)=o(m_j)$, then $c_{m_j,n_j}\to \pi \mathrm{/}2$. At fixed $r\in (\pi \mathrm{/}2,\pi )$, projective-code and projective-covering estimates give lower and upper exponential bounds for the largest $n$ with $c_{m,n}\le r$. Together with exact values, these yield a sharp transition: the largest $n$ admitting an odd map is $m$ at scale $\pi \mathrm{/}2$ and grows exponentially with $m$ at each fixed $r\in (\pi \mathrm{/}2,\pi )$. We finally return to the conjectural equality. We introduce a Gromov–Hausdorff-stable invariant from chromatic numbers of Borsuk-graph filtrations. Combining projective-code upper bounds for $c_{m,n}$ with chromatic lower bounds from spherical-cap volumes, we show that if $m_j<n_j$, $m_j\to \mathrm{\infty }$, and $n_j\mathrm{/}{m}_j\to \lambda \ge 9$, then $d_{\mathrm{G}\mathrm{H}}({\mathbb{S}}^{m_j},{\mathbb{S}}^{n_j})>c_{m_j,n_j}\mathrm{/}2$ for all sufficiently large $j$, producing infinitely many counterexamples.