# Verification of the eventual five point theorem

The seven-page manuscript proves the eventual large-exponent theorem stated
in its Theorem 1. The threshold is existential. This is a self-audited analytic
proof, not a formal verification or an independently reviewed result.

## Analytic dependencies and delicate estimates

Bondarenko-Hardin-Saff Theorem 7 supplies qualitative convergence modulo
congruence and labels. No localization rate is imported. Positivity and
q times the minimum height tending to infinity are proved before any
logarithmic upper bound is used.

The height equation is estimated relative to each neighboring sum S_i.
The parity-class bound on every adjacent height product gives a correction
of order T squared times S_i plus T times epsilon. Dividing by S_i is
justified because q S_i tends to infinity. This avoids an unsupported
additive-error argument when the two parity classes have unequal sizes.

The resulting localization is o(1/q) around the moving pyramid, not around
the fixed limiting packing. Positivity is proved throughout a convex box:
the spoke Hessian terms of size q squared times t_q dominate the negative
terms of order q. Center-only positivity would not prove global uniqueness.

## Finite exact checks

The standard-library checker compares automatic differentiation with an
independent vector-geometry implementation using exact fractions. It checks
16 rational configurations, 5,472 distance derivative entries, 2,736 energy
derivative entries, 192 height identities and 192 angular identities.
The limiting contact rank is seven, cycle rank three, with stress vector
(0,0,0,0,1,1,1,1) up to scale.

Three historical fresh-process runs in two Python runtimes agree exactly.
The publication archive is additionally tested after extraction using both
runtimes, with assertions enabled. These finite checks do not substitute
for the general argument.

## Floating regression and limits

The optional NumPy 2.3.5 program samples eleven real q values and 128 box
corners per value. It retains negative corners even at q=10,000. The sampled
corners at q=1,000,000 are positive, but neither observation certifies any
continuum interval or threshold. There is no numerical global search
certificate.

## Document checks

The built-in LaTeX compiler reports success. The exported PDF has seven
pages, all visually inspected, with no missing glyphs, undefined references,
overfull or underfull box warnings. Exact hashes and portable-run receipts
are retained with the publication package and the local manuscript checkpoint.