Why we tried it
Turn qualitative low-particle Coulomb-inequality structure into inspectable explicit gap and stability conditions.
What the evidence shows
Symbolic and finite checks support their local identities only; the manuscript depends on attainment, foundational reconstruction and localization inputs.
Where the work stands
The supplement does not supply the missing uniform optimizer localization or improve the established universal numerical constant.
Possible next steps
Verify imported assumptions and foundational analytic lemmas before pursuing uniform localization; do not turn a conditional recursion into an unconditional all-particle result.
Suggested next steps are untested research questions, not new experimental results. The archive records manuscript provenance, licensing and status.