[CubeJerry]

Every exact five-call binary scalar-idempotent word with nonzero torsion-free abelian frames, relation rank at most two, and a whole diagonal monomial endpoint has at most two high registers, at every finite scalar width. High means neither identity nor a signed individual frame; repeated endpoints count separately. Arbitrary diagonal auxiliary outputs are included.

This combines Nick Petalas's retained image restriction in #2900 with Josie Elliston's stronger finite census in #6055. It removes the census's width cap; it does not claim a new width method, a census-free classification, or the six-call target floor already proved by JoyboyBrian in #5481. #6760's sharper paired-helper bound is credited as an alternative.

From the repository root:

python3.12 -B scripts/research_package.py --package research/bit-high-width-transfer
python3.12 -I -B research/bit-high-width-transfer/verify.py --full-census

Expected: PASS_REGISTERED_RESEARCH_PACKAGE; the long replay reports 11 rows, 65,677,982 diagonal words, max_high_registers=2, candidates=0 and not_replayed=0. The complete long replay ran successfully this session. Normal/optimized endpoint checks with four failure controls, source authentication, committed-byte/mode checks and an unchanged Git-free export also passed. Separate source-aware AI proof review is retained.

No new κ, native supplier, paid invoice or selected-frontier change. A census-free proof remains open; the five inherited framework assumptions are unchanged. One existing blank line in a frozen source is preserved byte-for-byte. No complete-history preflight or selected cold replay is claimed.

Prepared for Jerry Chen / @CubeJerry with substantial OpenAI Codex assistance. Source authors and notices are retained; no human or proof-kernel acceptance is claimed. Base 7c49fe3. Exact dependency pins: ORIGINS.json. Exact source pins, proof, receipts and review: research package. The published complete tree matches the tested tree.

Selected-baseline source bindings and metadata passed the required orientation check; this was not a construction cold replay.