[1337isnot1337]

The conditional three-field controller in #4698 leaves open how much useful work two homogeneous chain calls can deliver. This package proves and attains the exact capacity r = 2 floor((n-1)/2) for every chain length n >= 2 in the explicitly restricted scalar/global-P ring, where P=Q^2. It includes an all-odd construction and a separate weighted cyclic middle boundary attaining n-2 jobs for every even n. Both restore all arbitrary spectator inputs. The matching rank/parity proof allows arbitrary restored dirty workspace, arbitrary invertible boundaries in that ring, and either inverse/transpose orientation.

The explicit reversible boundaries have linear field-macro counts. The proof states Q^T=Q, Q^4=I and the faithful-width restriction for the upper bound; it does not extend that bound to arbitrary physical address/monomial operations or every isolated small width. The three-field identity, native lineage and source attribution remain credited to #4698 and its predecessors.

The complete conditional moment is 2 H_t + r(2b+1)^t + (n-r)(2b)^t - 2 P_t. Direct synthesis from n-1 unary queries loses at least 2(n-1)-r moment units throughout 0<=t<=1, even with favorable transverse pricing. Physical chain acquisition, actual charts, helper autonomy/restoration, movement/precision, completion, compiler, BIT and outer funding remain unresolved. No native supplier, new kappa, selected-frontier change or full multiplication theorem is claimed. The next physical gate is a named positive complete provider invoice; the package stops direct unary synthesis.

Validation on CPython 3.12.15:

• Exact cyclic-ring replay on all 527 independent input columns for every length 2 through 32; complete boundary counts; thirteen active semantic mutations; 256 finite dirty-workspace rank fixtures; faithful tensor-width qualification.
• Registered package replay and isolated optimized direct replay passed. The finite checks supplement the written all-length proof.
• Exact committed-byte/source-history preflight, whitespace check, official seven-file export/integrity and fresh Git-free optimized export replay all passed at f2d83999edccdad625ea686ff3f0d03b7d208965.
• Canonical main refreshed at 20a1e31bc6009b9b3f4d084bbaadc6350e34fcba; priority revision 7 and selected conditional kappa 0.00302015 unchanged. Live overlap refresh covered 318 open PRs and 4,290 Discussion records with complete pagination, plus complete local-note inventory; this is not a novelty guarantee. Hosted CI is pending publication.

Reproduce: python3.12 -B scripts/research_package.py --package research/homogeneous-chain-extraction.

Investigation: #4840 (comment). Two substantive commits retain the initial odd result and the even attaining construction. Prepared for Cinna Davis with substantial OpenAI Codex assistance; Apache-2.0. No independent human or proof-kernel acceptance is claimed.
Hosted CI update: the planning job failed eight onboarding assertions, so the hosted package replay was skipped. A separate detached checkout of exact canonical base 20a1e31bc6009b9b3f4d084bbaadc6350e34fcba, materializing only the unchanged test/support inputs, reproduces the same eight failures in eleven targeted tests (CPython 3.12.15, 2.757 seconds). Seven failures encounter the pre-existing scripts/check_research_commit.py source-pin drift in the historical descendant fixture; the eighth encounters that fixture's absent research_package import when copying find_overlap.py alone. These failures predate this package. Local registered replay and all mathematical/export checks above passed; hosted research replay remains unrun, not passed. Shared baseline/test fixtures are unchanged by this PR.