Certify losing-first two-edit obstruction at displacement two on staged544

[Clear1nsight]

Change and mathematical scope

Adds only research/staged544-losing-first-d2/: an exact source-bound obstruction on both literal PR3734 providers. A legal first ADD retiming crosses exactly two scalar neighbours and has a nonzero strictly losing raw moment; the second ADD may move any distance in its allowed scalar corridor with any legal common frame. Other frames and all entrances, cuts, native maps, helper identities/exits and caller interfaces stay fixed.

Per provider: 797 legal losing first subspace records (607 A, 180 B, 10 restore), 9,149,777 interacting second cut/dimension prices and 48 combined signed-profile types. Every combined type is exactly non-improving for all real 0<a<1. The first loss is included. Both providers therefore supply 18,299,554 explicit second prices; these are relaxed price cells, not distinct physical circuits.

All first subspaces are enumerated, not one representative per dimension. The maximum quotient dimension is three. Same-ADD and disjoint-register second moves use the credited PR3942 reductions and a freshly regenerated PR3886 single-move premise. All first records pass strict boundaries and the retained frame auditor. No new supplier, kappa, selected-frontier change or global two-edit barrier is claimed.

Validation and complete ledger

python3 -B scripts/research_package.py --package research/staged544-losing-first-d2
python3 -B scripts/check_research_commit.py --package research/staged544-losing-first-d2 --check-source-history

Both complete scientific generations pass on CPython 3.12.3 / Linux x86_64, standard library only. Every one of the 1,594 first scalar blocks passes an exact rational identity on independent dirty symbols. Complete free-operator replay additionally covers both baselines, 18 edited first words and 18 final second-edit words: 632 live inputs plus 91 arbitrary dirty buffers, both orientations, reverse pull-back and cleanup boundaries. Whole-word sampling is explicit; universal first-class coverage comes from the exact local identities and fixed-boundary composition argument.

Independent controls enumerate all 1,594 first records by direct event scans and 3,130 interval/dimension cases by quotient-basis subset XOR, reconstruct all 96 provider sign rows, exercise 24 three-gate SCALE conjugations and reject source/scalar/frame/cleanup/barrier mutations. An actual production-path omission of the first loss changes the retained price digest. The existing Jeremy Kothe #2957 rollback/recovery remains a sensitivity control, not a new gain. VALIDATION.json retains development control repairs and final exact execution coverage. Both complete provider commands printed PASS and matched the frozen receipts inside the official registered invocation. The enclosing invocation then ended with status 143 before the final control/overall PASS marker; its cause is not established. The assertion-disabled controls were rerun separately with python3 -I -O -B and passed in 27.949 seconds. The interrupted log and exact resumed coverage are preserved in VALIDATION.json; no uninterrupted overall registry PASS or hosted success is claimed. Post-creation committed-byte/source-history checking is reported below.

LEDGER.json retains the complete local bill: baseline 3,519 children, rank mass 5,134, 408 helper roles and 91 dirty buffers; all intermediate profiles use 3,519–3,521 children and unchanged rank mass. All combined histograms and primitive/native inventories are retained. A conservative finite bound on final retimed scalar numerator/denominator size is 11 bits each; it is not a uniform precision/tape tariff. Fresh native/phase, physical hosting/allocation, uniform stock/precision/tape, all-size/compiler, outer/supplier, analytic and BIT obligations remain unknown or inherited, never zero.

Task, overlap and source pins

Explicit user-assigned continuation of #3941/#3942 under scalar-to-moment; changed scope recorded in TASK.json, with canonical base 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority 6. Current AGENTS.md, autoresearch.md and CONTRIBUTING.md were read. Entry overlap fetched all 649 open PR bodies over seven REST pages; the pre-submission refresh fetched all 656 open PR bodies and reran the current helper. New PR3949 concerns staged entries and changed widths, a distinct contract. Signed-in retiming handoff coverage is partial. Public claim. No exhaustive novelty claim.

PR3941 covers neutral changed-frame first moves through displacement two and adjacent frame-preserving ADD swaps. PR3942 covers first displacement at most one, including losing starts. This contribution adds the strictly losing displacement-exactly-two class. Counts overlap historical domains only as explained in their source contracts; they are not added as independent circuit counts. PR3945 changes terminal/native interfaces and is outside this frozen-input comparison.

Frozen provider/source pins: dataisfire #3734 fa041037a3374eaa08feeca0b26a5d6893194147; #3886 scientific 359fe0fe77d29544972fd22ee674085ed5a3b4d0; #3942 scientific 1416f8cd6bfb2f2eb9aa5cc22f0388a85fdbac91; #3941 scientific 001ebcdb3f86a7f9d5f5897f9cf5a84e74dd1822. Exact byte/hash/blob snapshots make scientific replay Git-free. No frozen predecessor is edited.

Credit

Credit dataisfire, #3886/#3941/#3942, Jeremy Kothe #2957/#2979 and the staged-entry/retiming chain, Dugongue #3863 and the retained ranked-pair/MathProof predecessors. The first-subspace enumeration, same-gate/disjoint reductions and sign criterion are explicitly credited reuse/adaptation. Sagar Thakrar's auditor and roger-sato's adapter are unchanged, with Arham Tahir's frame-contract attribution; they are not new independent interpreters. New displacement-two census, controls and packaging by Chris R / Clear1nsight with substantial OpenAI Codex assistance. Apache-2.0 and retained notices. No independent human or formal-kernel acceptance claimed.

Open next questions

README.md/TASK.json preserve the remaining boundaries. Neutral frame-preserving first moves across two neighbours are excluded here and not exhausted by the earlier domains; their new insertion gaps need a separate reduction/census. Farther losing first moves and simultaneous frame rewrites without a legal intermediate also remain open. Neither this result nor the union of earlier certificates proves a global or complete radius-two obstruction. Stop this session at the declared result and hand off those questions with exact dirty/physical bills.

Exact publication checks

Scientific head: c8285dbaffdc254f3124c7e8975d4dc66031ec62. GitHub stores exactly the locally verified full canonical-based tree: one commit, 27 added files, no deletions or other changes. check_research_commit.py --check-source-history passes on this exact GitHub-authored commit in the actual full-history worktree: all 27 working/committed byte streams match, and declared Git source pin 20ecf2891d80b5acf18d980b2164fb44ea7faa90 is an ancestor. External pending source snapshots are separately authenticated by exact bytes/SHA256/blob IDs, not claimed as Git ancestors. Hosted CI remains separate from author verification.