Backward-Certified TPOP Transitions: -- A Ten-Digit Numerical Demonstration

We describe and numerically realize a backward search procedure that,
starting from a certified prime endpoint $p\ge 11$, locates the last
twin-prime pair whose center precedes the center associated with $p$.
The procedure exploits the fact that every prime greater than $3$ lies on
one of the two wings of a multiple of $6$, so that the search reduces to
scanning centers rather than integers. We report absolute, relative and
center-based distance statistics, all linked by wing-dependent identities
that furnish free consistency checks, and we distinguish two structurally
different endpoint cases. When the endpoint is itself a twin-prime pair,
the backward search yields a \emph{backward-certified TPOP transition},
subject to explicit verification of the sandbox inequality
$q\le p''q''$. When the endpoint is a solitary prime, the computation
remains a valid backward-search experiment but does \emph{not} constitute
a TPOP transition, because no successor pair exists. Both cases are
demonstrated at ten digits with fully reproducible exact arithmetic.