Scoped negative result: rank-two obstruction for 78 CK cleanup pairs

[thakrarsagar]

The selected #626 CK cleanup contains 78 repeated target/source addition pairs. Every pair has an exact rank-two defect relative to its unchanged middle word, so it cannot be replaced by a single addition, scaling, or arbitrary rank-one update at any position among those middle gates. Status: scoped negative result / research note; no new kappa.

This strengthens a failed commuting-cancellation search into a finite obstruction on the complete register map. The package records all 156 forward/dual minors and a short rank proof. A separate implementation reconstructs all 624 minor entries by replaying 312 basis columns through the inverse middle and original window. Neither implementation treats a commutation blocker alone as an impossibility proof.

Only research/complex-ck-cleanup-fusion/ is added. The selected supplier, certificates, accounting, and upstream/ are unchanged.

Reproduce

python3 -B research/complex-ck-cleanup-fusion/verify.py

Expected: PASS_CK_CLEANUP_RANK_TWO pairs=78 rank=2 columns=312 controls=8.

Python 3.11+, standard library, offline, no writes. Validation passes on macOS arm64 with Python 3.14.8 and 3.11.15, including optimized execution, a minimal relocated checkout, gzip recompression, changed-source/certificate rejection, five semantic controls, and three invalid-certificate controls. make selected-record-check passes. Two additional Codex workers checked the narrowly claimed mathematics and implementation; no independent human review is claimed.

Scope and provenance

• Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Assignment seed 20, complex supplier. Exact source and dependency pins are checked, including the CK word's binding to the selected Conditional κ = 0.001635527380375392: coordinated owner repair on PR623 #626 SPEC.
• Complete domain: all 78 pairs of additions with identical ordered target/source in the fixed 8,309-operation CK cleanup. No sampled search or timeout. Pairs with different endpoints are outside the domain.
• The obstruction requires equality on arbitrary register inputs. It does not exclude reachable-subspace rewrites, multiple replacement gates, altered middle gates, different frames/helpers, or broader resynthesis. It is not primitive-cost optimality or a lower bound on kappa.
• No pinned coordination issue exists; the claim is local. The overlap refresh through Research: optimal disjoint R420 target pairs save 3,250 XORs (no new kappa) #716 includes concurrent Research: shorten R420 cleanup by two XORs with certified conjugation (no new kappa) #710's bit-cleanup conjugation construction, which is credited. Its joint two-sandwich rewrite over F2 is compatible with this rational, fixed-middle pair obstruction. Consolidate six circuit PRs: 31.72% fewer bit ADDs and 22–25% fewer complex primitives #681 factory resynthesis, Scoped negative result: selected CK B frames are optimal at fixed boundaries #698 B-frame optimization, and Research: lower selected complex scalar primitives by 22–25% (no new kappa) #696 primitive lowering are distinct and are not dependencies.
• Credit: Jeremy Kothe's Conditional κ = 0.00163542231500688: K-triangle skeleton atom (492 helpers per call) on the PR617 cold-ended witness #622 construction, Chafik Boukhalfa's Conditional κ = 0.001635527380375392: coordinated owner repair on PR623 #626 package, DreamingOfClouds' verifier/cold-helper work, icekylinx's factory/direct-delivery lineage, Dugongue, eumemic, and the retained predecessors. Related Cancel cleanup sandwiches of 99 helpers on #272: conditional κ = 7.1366215638961e-4 #271 and κ = 1.017629458064773e-3: exact target-prefix resynthesis on sequential206 #411 are credited in the note.

Prepared for Sagar Thakrar (@thakrarsagar) with substantial OpenAI Codex assistance, Apache-2.0. All inherited compiler, realization, hosting, precision/tape, recovery and analytic assumptions remain. This is an exact finite circuit statement, not a formalization of the multiplication theorem.

The full repository suite, selected cold/global replay, Lean and remote Linux CI were not run for this isolated research addition.

[Mister-Meeseeks]

100-PR batch review at a3d9c0aa0a16b68b1f4dd064671ad018013e6dd4.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: 78 CK duplicate-run contractions have rank-two defects; excludes rank-one replacement with the fixed middle map, not arbitrary compensation rewrites.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.