Are the topological-existence and constructive-arithmetic halves one phenomenon or two tools?

Explicitly speculative. Nothing in the source chat tests this. Filed so the question stops being re-asked from scratch, not because there is evidence either way.

The statement. One body of work reaches its results by topological existence — a Borsuk–Ulam-type argument that a fair split exists, non-constructive, indifferent to arithmetic. Another reaches results about the same objects by constructive arithmetic — an explicit greedy construction with a deficit invariant and an exactness theorem at a checkpoint. They produce compatible statements about the same objects.

The friction. Compatible statements from disjoint methods admit two readings, and nothing currently distinguishes them:

1. One phenomenon. The discrete invariant the constructive half tracks is derivable from the continuum/Borsuk–Ulam side — the constructive proof is the topological one, discretised, and the deficit invariant is a shadow of the continuum statement.
2. Two tools. They are genuinely disjoint instruments that happen to solve related problems. The agreement is agreement about the objects, not evidence of a shared mechanism.

Why it is open rather than answered. The obvious bridge was tried and died. The hypothesis that k-thief difficulty tracks the availability of free ℤ/k actions on spheres — which would have tied the two halves together through the division-algebra ladder — does not survive PPA-completeness at k = 2 (disagreements.js, Free ℤ/k group actions are not the source of k-thief necklace difficulty). That refutation removed one candidate bridge; it did not answer the question, and the changed: field of that entry says so explicitly. This probe is that sentence, given a home.

What would settle it. Derive the deficit invariant, or its checkpoint-exactness, from a continuum statement — that is branch 1. Or exhibit a problem where the two halves give incompatible answers, or where one applies and the other provably cannot — branch 2.

What would not settle it. More instances of agreement. Agreement is what both branches predict, which is exactly why this has stayed open.