# Verification of the irreducible extension

The seven-page version 2.0 of Hidden Constants after Free Semicircular
Regularization retains version 1 and adds an explicit irreducible II1-factor
example with bounded conjugate variable and enlarged closed-derivative kernel.

The written proof covers Bernoulli shift outerness, Fourier-coefficient
irreducibility, the free-product relative commutant, twisted-pinching
recovery, algebraic freeness, closability, both kernel inclusions and the
sharp gap relative to the intermediate factor. Its exact squared failing
ratios are 2^m+1+epsilon^2. The unwanted opposite Fourier mode is retained;
its squared norm is 2^{-m}. The proof does not assume equality of full
closed-operator domains.

The original exact checker performs 2,190 predicates. The new checker
performs 421, using exhaustive binary-cylinder enumeration and a separate
constraint-graph count for 1<=m<=12, plus 60 rational noise models. Both
are replayed in normal and optimized Python modes. These 2,611 finite
checks supplement the written infinite-dimensional proof; they are not
formal verification or an independent mathematical review.

The package is accepted only after a clean source build, inspection of all
seven page images, extraction of source.zip into an isolated directory,
rebuilding there and replaying both checkers. Detailed hashes, execution
outputs and acceptance records are retained in the research repository.

Prior results of Ito and Lee-Nelson are credited. This strengthens the same
negative answer to Ito v3 Question 7.4, not a characterization of all
positive cases in AIM Section 0.1. No additional manuscript count, complete
source closure or absolute-priority claim follows from this version.