[1337isnot1337]

The eight owned center lifetimes in Cao Yifan's PR4610 round-three word can be made reversible: replace each reset/copy by an addition from a distinct arbitrary field and retain its output. This constructs an exact 640-field coupled interface with the unchanged recursive paid profile. No higher K or independent-center productive credit is claimed. The selected conditional K remains .00302015.

After stripping the actual endpoint charts, the new map is M = [[A,B],[R,I8]], where A(X,Y,H)=(X,Y+X,H). Both B and R have exact rational rank eight, RB=0, and for K=M-I we have K² nonzero and K³=0. The complete sparse ABI is provided. All 640 input columns and the actual gatewise raw inverse transpose replay exactly with the same bill. The source backward word is not substituted for this new dual.

A second exact kernel removes all eight center transforms and has operator N = [[A-BR,B],[0,I8]], with (N-I)²=0. Its original 632 outputs agree with A precisely under the explicit caller relation c=Rx. The actual cache-kernel raw dual also replays, and all 106 helper contributions in R are proved to start cold. The direct cache acquisition adds 112 rank-one calls against eight removed rank-eight calls: a surcharge 112-8*8^(1-C) >= 48, so it loses. A separate exact rational Laurent screen finds no causal scalar reconstruction at any of the p16 controller's 18 full stages from its previous query demands and external main snapshots. This is a scoped model test, not a universal physical cache obstruction. See CACHE.md and CACHE-SPAN.json. No cache-acquisition cost is silently set to zero.

The new inputs contaminate all 112 Y outputs, and the center outputs contain 100–104 old-input terms each. Those couplings are recorded explicitly. The tempting stripped-coordinate block elimination gives A-BR rather than A and needs physical chart conversions; it is not a free decoder. A controller consuming the coupled interface, independent Gaussian targets, completion/compiler, BIT and outer integration remain open. The draft preserves a changed architecture and its exact remaining obligation rather than adding unsupported target credit.

Validation completed on Linux CPython 3.12.15:

• Registered metadata/source-pin check and package replay pass.
• Fresh 640-column primal, raw dual, exact two-sided inverse and pairing pass, as do the new cache kernel and its raw dual.
• Every scalar common chart and actual nested paid rank is checked; the whole source profile is recounted.
• Wrong center injection/dual signs and a false center rank reject; arbitrary-input leakage and contamination are explicit controls.
• Both runtime verifiers pass under isolated optimized Python from a fresh Git-free export. Registered replay and committed-byte/source-history checks pass at final head 2add42d1edf14a263abe90a3bcde67fc1ba30768 (13 package/input files).
• Hosted bounded package replay and unchanged-input steps succeeded at final head 2add42d. The planning check passed; final replay-job aggregation was still pending when inspected.
• No fresh physical phase/all-width, selected cold replay, uniform compiler, outer theorem or independent human verification is claimed.

Only research/retained-center-extension/ changes. The exact PR4610 source commit is retained through a tree-preserving provenance merge; its original compressed blob is pinned and unchanged. Canonical main was refreshed before submission at b58ad9e, priority revision 7. Fresh overlap coverage: 460 open PRs and 3,619 Discussion records with complete remote pagination; sparse local note coverage is incomplete. PR4359's logger/dirty-helper work is a credited predecessor; this change retains all eight centers on the round-three word without adding recursive calls and supplies its complete coupled ABI and raw dual. It does not replace PR4655's quotient budget or PR4665's phase certificate.

Reproduce:

python3.12 -B scripts/research_package.py --package research/retained-center-extension
python3.12 -I -O -B research/retained-center-extension/verify.py
python3.12 -I -O -B research/retained-center-extension/cache_span.py
python3.12 -B scripts/check_research_commit.py --package research/retained-center-extension --check-source-history

Credit Cao Yifan, Cinna Davis's PR4440/4441/4447/4470 lineage, zxv's native/productive/compiler work, Jacob Alan Sussman, and all retained scalar, E8, Clifford, precision and outer contributors. Prepared for Cinna Davis with substantial OpenAI Codex assistance; no subagents. Apache-2.0.