Exact replay

This directory checks the two pieces of arithmetic quoted by the problem notes:

1. the dimension threshold in Bourgain–Klein's proof; and
2. the local modulus power created by adding free Euclidean directions to a separable operator.

Run both independent implementations with:

./run_all.sh

The driver uses only Python's standard library and rustc. It checks the committed CSV certificates, compares the complete Python and Rust reports, and exits nonzero on any mismatch.

For dimension $d\geq2$, the checked quantities are

The method has room exactly when $g_d>0$. The certificate includes dimensions 2 through 8, including the equality at 4 and negative controls above it.

For $m\geq1$ free directions, the cumulative free DOS kernel has power $\alpha=m/2$. Its checked local continuity power is

These are algebraic replays of published or elementary formulas. They do not constitute a new IDS continuity bound.