A Lean certified proof of the Twin Prime Conjecture

In this paper it is shown that the Twin Prime Conjecture follows from a quotient-weighted Chen--Richert crossing in which both sides can be kept on the same smoothed dyadic scale.  On the lower side, the exact-$P_3$ contribution is divided into three rigorously controlled regions: a factor-placement-good region, a special-factor region covered by Pascadi's distribution theorem, and a remaining triple-prime region covered by a Bombieri--Vinogradov induction principle followed by the ordinary upper sieve.  Directed fixed-point interval certificates for this $P_3$ term and for the elementary $P_4$ residual give
\(A_3(X)\ge(0.7541+o(1))H_X\).
On the upper side, the balanced-semiprime main coefficient is geometric, the completed second moment is estimated at its natural scale on the fixed interior, and a fixed-$q$ Selberg estimate treats the last $0.002$ of exponent range, giving
\(E(X)\le(0.74938+o(1))H_X\).
After the negligible prime-power contribution is removed, the resulting genuine twin-prime count is eventually positive.

The accompanying Lean~4 development https://github.com/lieur-wq/twin-prime-v9-lean , is a publication-boundary certification of the manuscript-specific deductions, normalisations and theorem applications.  Standard published analytic theorems and the explicitly archived numerical interval certificates remain mathematical inputs rather than being re-proved from foundations.  The decisive Lean certificates report only \texttt{[propext, Classical.choice, Quot.sound]}, with no \texttt{sorryAx} or project-specific axiom.  The term \emph{Lean-certified} is used in this precise interface sense and does not mean that Lean independently proves the external analytic source statements.

This is a slightly modified version to address the criticisms raised in https://zenodo.org/records/23177600.