Prove nine calls are necessary for three abelian bit registers

[aliromman]

The pinned version of #288 leaves the abelian three-register bit cases with seven or eight calls open. This proves that nine calls are necessary and sufficient when paid frames are nonzero rational rank-one matrices and all three target frames are invertible and lower triangular.

The proof reduces the remaining span-five case to 2,276 scalar families and checks all 21,317,520 orders. Two separate exact engines agree on every family: 415,398 diagonal terminals, zero survivors. The argument covers unbounded rational frame coefficients; it is not a bounded coefficient search.

The seven-file package contains the proof, catalogue generator, two engines, controls, and a replay command. It requires Python and a C++17 compiler and includes no generated search data. Attribution and AI assistance are stated in the proof.

Validation: python3 research/bit-three-register-lower-bound/replay.py passes. The result does not change κ and does not cover extra registers or arbitrary free address maps between paid calls.

The existing gen4-bank-verify target also passes. make verify stops at the existing joint-dual-verify source-hash assertion for Makefile; the same failure was reproduced on unchanged base 12a4934. This PR does not change the affected files. The remaining verify-pair and verify-tests groups pass when run separately.