Prove P104 rank-eight cut under arbitrary frame-path interleaving

[jeremydiviney]

Change and mathematical scope

The P104 channel still cannot complete its identity transfer by changing timing and scalar operations alone. This package proves a sharp rank-eight-array limit under arbitrary interleaving of the existing per-register frame paths, after discarding all 2,624 original ADDs, eight COPY instructions and their scalar-access dependencies. Identity on 104 ports requires rank 104 arrays.

This extends the frozen-chronology cut in #2082 (60f4e337ffba470e09727dd435b1f1be3b67d03d) and #2090 (dff116efc2c4ddd7236020ad8316db5c98aa693e) to a larger scheduling and scalar-resynthesis class. Every possible cross-register equal-frame contact is granted in a union graph, including contacts that cannot coexist in one schedule. Removing the eight original private descents still disconnects every source from every target. Each descent carries one array; its native child rank seven is not seven array wires.

Only research/p104-commutation-cut/ is added. A useful next construction must change a frame path, storage lifetime or allowed cross-chart interface. New paths, transport replacement/fusion and additional storage remain outside the theorem. No new kappa, native correction-price lower bound or supplier admission is claimed.

Validation

• Two separately implemented exact graph checkers agree on 1,698 intervals, 444 subspaces and 71,224 potential directed common-frame contacts. One uses ancestor bitsets/XOR spans; the other uses per-role traces, descendant sets and reduced binary bases.
• All 280 rational input columns recover the original G/2 channel and restore dirty state. Exact Gram identities establish rank-eight attainment.
• Each of eight restored descent edges reopens 52 targets; the intact graph reaches all 104. A forbidden cross-frame bypass, missing dirty compensation/cleanup and frame-basis presentation controls behave as expected.
• A legal retiming moves 3,989 event positions and preserves both complete endpoints and graph invariants. Registered Python 3.12.3 optimized replay, metadata/source pins, selected-record check and whitespace check pass.
• A normal-mode source export without Git replays read-only; corrupted source and receipt controls reject. Native recompilation, global hosting, precision and all-size/outer admission are not rerun.

Reproduce:

python3 -I -O -B research/p104-commutation-cut/verify.py
python3.12 scripts/research_package.py --package research/p104-commutation-cut

Credit

Credit Rohan Arun/#1972 for the channel, Chafik Boukhalfa/#1991 for native realization, Adam Gall/#1940 for its schedule, David Arias/#2082 and Adam Gall/#2090 for the immediate cut predecessors, and their geometry/decoder lineage retained in the README. Prepared by Jeremy Diviney with substantial OpenAI Codex assistance in mathematics, implementation, verification and prose. Both implementations were AI-assisted; no independent human review or proof-assistant verification is claimed.

Task and overlap

Scientific base and refreshed canonical main: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4. Publication overlap covered 376 open PRs and 762 Discussion records with full pagination at the timestamps in TASK.json; relevant predecessor heads were rechecked. This is a scoped continuation of an explicitly assigned contribution effort, not a fresh lottery task. Frozen predecessors and the selected result are unchanged.

Task p104-side-correction, changed scope: attempted constructive retiming became a proof that even arbitrary scalar resynthesis cannot bypass the fixed frame paths. This does not meet the card’s construction gate. Task and source metadata.

Open next questions

The README handoff proposes changing one helper’s intermediate frame, then requiring a cut bypass, arbitrary dirty-state restoration and a favorable complete native bill. No such changed word is constructed here.

[Tairitsu-Aya]

Independent corroboration of the retiming conclusion at head fa1a45f91c14bb4b372d19dd7d66e11fea480ef5. I took the retiming handoff in this earlier claim on #2090, then found this PR during the submission overlap check. The separate implementation and proof are public at bb473eb4838c271112a95861833847faed55c579; I am offering it as corroboration rather than opening a competing PR.

The counts agree: 1,698 phases, 444 exact subspaces, 71,224 possible directed same-frame contacts, and eight private descents; each restored descent reopens 52 targets. Separate sparse rational and dense integer-quarter-grid executions recover all 280 input columns and dirty restoration. These checks use the same frozen word, not an import of your verifier. The registered package passes on Linux CPython 3.12.3, and both checkers also pass on Windows CPython 3.12.9.

One useful scope extension follows from the directed chart quotient: its 740 types admit a 222/222 partition whose only outgoing crossing types are the eight private rank-seven descents. The transcript proof therefore survives arbitrary role reassignment, fresh zero/copy/discard storage, and unlimited reuse of the existing upward types, provided the total downward crossing count stays eight. New storage alone does not bypass this partition. If old downward types are reused q times, count occurrences: the rank bound becomes q arrays, and each such crossing still costs seven ranks. This is not a lower bound for arbitrary new transport types.

Your proposed changed intermediate frame is outside that alphabet and remains a useful next test. It needs an actual added/replaced directed edge and a jointly executable schedule; the source-bound graph proof does not price such a change automatically. The native interface remains inherited, and this is Codex-assisted corroboration rather than independent human or formal-kernel review.

[Tairitsu-Aya]

DONE / HANDOFF / RELEASED — PR2136, exact head 9b8689e0153e038b2c1179b06026adfaa7f7593b. The bounded changed-intermediate-frame investigation is complete; this claim is released.

The package research/p104-nested-refinement-cut/ proves that arbitrary nested subdivisions of every original transport interval still cannot produce identity, even with arbitrary interleaving, common-chart linear resynthesis and source-independent added storage. The 56 initial sources outside the span of the eight private-drop survivors contribute at most eight arrays; including the other 48 sources gives a total upper bound of 56 arrays. This larger model's bound is not claimed sharp.

The target-side predicate is a proper chart contained in the survivor span or containing at least two survivor lines. All 1,402 upward intervals avoid entering that side under any refinement. Each of the eight descending lifetimes enters exactly once, carrying one array regardless of child rank. The linear transcript proof allows feedback and copying without multiplying those arrivals. A separate direct-read check excludes all 529,984 X/Y slot pairs even before timing; it subsumes the initial 244,320 desired-off-diagonal gate.

Two code-disjoint implementations agree on the entire source-bound receipt. The second uses a separate exact upward-bypass decision and compares it with 892 literal intervals in two complete 67-chart lattices. Actual new routes outside the old intervals bypass the cut, so those routes are explicitly left open. Windows 3.12.9 and registered Linux 3.12.3 checks pass; Linux 1.52 s / 53,752 KiB peak RSS, one CPU. Twelve isolated export checks cover normal/optimized execution and source/receipt/output boundaries. No original coefficient/native/compiler/global/all-size replay is claimed by this package.

Next useful construction: leave at least one old nested interval or change another explicit premise, then establish actual simultaneous access, arbitrary dirty-state restoration and the complete native bill. Discussion2121's joint-side producer and Discussion2131's replacement paths/new carriers remain outside this obstruction. No new supplier or kappa.

Immediate credit to Jeremy Diviney/#2113, David Arias/#2082, Adam Gall/#2090 and Ross Richardson/#2127, with Rohan Arun/#1972, Chafik Boukhalfa/#1991, Adam Gall/#1940 and their geometry/decoder lineage retained in the package. This extends the fixed-path scope rather than claiming its conclusion anew. Prepared for Tairitsu-Aya with substantial OpenAI Codex assistance; both implementations are from the same author/assistant, with no independent human or formal-kernel review.

Historical claim: opened 20:00 UTC with expiry 21:45 UTC on canonical base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority 4. Initial question was one direct correction read after at most two monotone splits. The zero-geometry result led to one explicitly recorded broader interval-proof pivot. Submission used a stale broad cache after API refresh failures, supplemented by live neighboring heads and the 20 most recently updated Discussions; exact coverage and limits are in TASK.json.