# Verification of the radial barrier results

The seven-page manuscript proves an auxiliary counterexample and two
conditional radial criteria for positive discrete Newtonian potentials.
The full AMR-036-0023 force-summability question is not resolved.

## Mathematical checks

The scanned original Lemma 8.1 was checked against its printed dimension,
mass and summability hypotheses. The unit lattice satisfies those
hypotheses in dimensions at least four, but its radial liminf is
2(2^p-1)zeta(p), not zero. Its equilibrium points were also counted to
show explicitly that it is not a counterexample to the existence theorem.

The replacement proof checks local convergence and differentiation,
strict kernel clipping, both dimension-dependent integrals, uniform far
tails, the subsequence argument and its weighted corollaries. The
gradient-flow step uses an explicitly compact strip and does not assume
Morse nondegeneracy. The finite-mass grids are locally finite; their
potential lower bounds and entropy/root-mass counts are proved for all
blocks, including the effect of a partial final block.

The originating researcher performed a detailed self-audit. No known
mathematical gap remains in these stated scopes. There was no independent
peer review and no proof-assistant verification.

## Reproducible finite regressions

The included checker uses only Python's standard library and exact integer
and rational arithmetic. It checks 56,395 conditions and produces identical
output in ordinary Python and with optimization enabled. The domains are:

- lattice powers 2 through 12, rational offsets with denominators 2 through
  19, and certified axial brackets for dimensions 4 through 12;
- finite clipping lists, including a negative control for replacing the
  strict inequality by a non-strict one;
- exact clipped-kernel integral identities and midpoint-grid estimates;
- harmonic dyadic bounds up to 4,096 terms, with symbolic large-block,
  entropy and root-mass identities for block indices 1 through 64.

These are regression tests, not a proof for arbitrary infinite
configurations. The manuscript's analytic arguments establish the results.

## Source and publication limits

The 1993 deformation method is credited. A bounded primary-source search
on 11 October 2026 found no identical correction or the exact displayed
replacement conditions; it does not certify absolute novelty. The
manuscript remains unrefereed. Its completion does not increment the
number of fully resolved frozen source problems.

Both the built-in LaTeX compiler and Tectonic completed successfully.
All seven rendered pages were inspected; formulas, citations, author
footnote and page breaks are legible, without clipping or overflow.