[Matarener]

Requested by Mark

Change and mathematical scope

Negative result, no new κ, no changed selected record. Adds research/fusion-pair-scalar-rewiring, answering the physical-pair request of #5243 and the fragmentation exchange table of #5226: in Cao Yifan's #4610 word (research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz) no pair of consecutive rank-1 climbs on one field can be fused into one rank-2 climb by free scalar rewiring (same-frame dyadic ADDs, owned COPY/CLEAR, any time, any field), in either orientation.

Exact structural certificate (verify.py, 2 s, standard library, both orientations): 3,290 climbs, 2,201 of rank 1, 1,363 consecutive rank-1 pairs on 509 fields.

• S1: no pair is idle; some event touches the field at the intermediate frame.
• S2: in all 1,261 pairs whose intermediate events are plain ADDs, some partner field never occupies the target frame at any time, so the interaction cannot follow a merged climb.
• S3: the 12 pairs whose partners all visit the start frame (cold helpers 241/240 … 417/416) are each blocked before the merged climb by exactly one main-field event on the partner at the intermediate frame; that main never shares a frame with the helper later.
• Frame-level relocation test: 0 relocatable pairs. Controls: deleting the single blocking main event makes exactly one pair relocatable (positive); a corrupted climb rank is rejected (negative).
• Price order at t* = 1 − 0.0030233: one fusion gains 2 − 2^t* = 0.0041868 Λ per call; any repair adding a paid rank-1 job costs 1; a descent ≥ 1.99022 (Fragmentation-loss ceiling: exact Λ ship threshold for the #4610 family, with Lean (no κ change) #5226 Theorem D); the bar needs 0.0209332 per call.

Evidence beyond the certificate (recorded in RESULT.json, scripts in tools/, not replayed by CI): an exact rational row engine (the convention of #5377's native.py) on the backward word finds, for the 2,111 contributions lost by deleting the intermediate events, 12 repayable by free same-frame ADDs (all one-sided) and 0 fully repayable pairs; a greedy downstream-propagation search over 2,522 pair-variants finds 0 successes. The 102 pairs that warp or scale the field between the climbs are counted and left outside the claim. This is a scoped obstruction on the frozen word, not a universal barrier: a fusion needs new frames or native maps.

Task and overlap

Base 1ec1119972c9aefe42525172d07b332fcff81cb4 (main at the time of the work; the branch merges cleanly onto current main, which has since promoted #5398). Task card scalar-to-moment, priority revision 7, scope refined: the card asks whether a changed frame itinerary can remove a paid transition; #5226/#5243 made the instance on the current word explicit (one fusion per native call → 0.0030220, five → 0.0030233) and asked for a physical pair. Nearest work: #5226 (framework and prices), #5243 (compiler target, pair wanted), #5301/#5324 (no-gain over helper/contact charts and chart heights, different family), #5377/#5383/#5397 (helper-input relations, outside this claim; #5397's traversal corroborates that only the eight known relations exist). No PR or discussion supplies or excludes a fusion pair; a fresh check of open PRs through #5412 found none.

Validation

python3 -I -W error -B research/fusion-pair-scalar-rewiring/verify.py
python3 -I -O -W error -B research/fusion-pair-scalar-rewiring/verify.py
python3 -B scripts/research_package.py --package research/fusion-pair-scalar-rewiring
python3 -B scripts/check_research_commit.py --package research/fusion-pair-scalar-rewiring --ref HEAD

Marker PASS_NO_FUSION_PAIR_SCALAR_REWIRING; registered normal and optimised replays, metadata/pins and commit-byte checks pass locally on CPython 3.12 (Windows). The verifier binds the word by SHA256, replays the nested-climb contract, and compares RESULT.json byte for byte. Not run: forward row-level searches (the forward program is the transpose of the backward one and shows the same S1–S3 obstruction), Lean, native/phase/compiler replays, hosted CI. Finite exact certificate plus computational search evidence; not a theorem about other words.

Credit

Cao Yifan (intqwq, #4610) for the word; DreamingOfClouds (#5226) for the fragmentation framework and exchange values; Clear1nsight (#5243) for the one-fusion target; Ali Romman (#5377) for the exact row convention; Cinna Davis, zxv, Jacob Alan Sussman and the retained native/controller lineage. Prepared by Mark (Matarener) with substantial Claude Code (Claude) assistance. Apache-2.0; author-checked, not independent acceptance.

Open next questions

TASK.json: (1) a native word in the (112, 8) family whose intermediate meeting frames are nested in the next climb target, so fusions become legal (first step: feed the exported climb paths and partner frames to helper_line_milp.py; blocked by re-admission of a changed word); (2) whether the 102 warp pairs admit a fusion if the warp is re-bound to the merged frames (blocked by the fixed frame pairs of the 72 native maps).

🤖 Generated with Claude Code