Scoped negative: third dirty register does not shorten D8 central pair

research/d8-third-register-seed89de/CERTIFICATE.json

@@ -0,0 +1,43 @@
+{
+  "status": "PASS_D8_THIRD_REGISTER",
+  "seed": "89dea4bc3a42ef85f785169cd2b6977e",
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "selected_record_sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32",
+  "group": "UT3(F2)",
+  "target": "diag([xyxy],[xyxy],[1])",
+  "registers": 3,
+  "rank_one_idempotents": 28,
+  "generators": 56,
+  "identity_layers": [
+    1,
+    56,
+    1750,
+    45424
+  ],
+  "target_layers": [
+    1,
+    56,
+    1750,
+    45424
+  ],
+  "identity_states_radius3": 47231,
+  "target_states_radius3": 47231,
+  "identity_ball_sha256": "2d66bf31796711244a41efc678d433da22b9ee110855ec00d21c34c332f54de5",
+  "target_ball_sha256": "1e9e3dac8b229db93c6ee82ef19d165ce7a9ceb475b2b1a59815b48f136b5e92",
+  "radius3_intersection_states": 0,
+  "target_radius4_edges_checked": 2543744,
+  "radius4_intersection_edges": 0,
+  "maximum_paid_word_searched": 7,
+  "direct_upper_bound_calls": 8,
+  "exact_minimum_in_model": 8,
+  "free_gl3_mixing": "normalized by augmentation into the complete 56-generator alphabet",
+  "controls": {
+    "full_group_associativity": true,
+    "all_generators_involutive": true,
+    "direct_word_exact": true,
+    "all_eight_single_omissions_rejected": true,
+    "compact_transition_matches_full_algebra": true,
+    "changed_third_target_rejected": true
+  },
+  "new_kappa": false
+}

research/d8-third-register-seed89de/CLAIM.md

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+CLAIM
+
+**Track / assignment:** `bit-mechanisms`; accepted OS seed `89dea4bc3a42ef85f785169cd2b6977e`; task `bit-new-register-model`, refined and changed to `D8-UT3(F2)-r3-central-pair`; priority revision 3; catalogue SHA-256 `6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075`. Exact base `0b1cccf86e2f5132ed286a26ef0a681a495740ea`; selected-record SHA-256 `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.
+
+**Question / object:** Let `G=UT3(F2)` with `x=(1,0,0)`, `y=(0,1,0)`, and central commutator `z=xyxy=(0,0,1)`. On three arbitrary dirty scalar registers, can a word of at most seven paid factors `I+([g]+1)N`, `g in {x,y}`, `N=uv^T` any rank-one idempotent in `M3(F2)`, realize `diag([z],[z],[1])` exactly in `F2[G]`? This asks whether one restored dirty register can share work across the two duplicate central targets.
+
+**Changed hypothesis and full optimistic charge:** The predecessor fixes two registers and proves minimum eight; this scope adds exactly one arbitrary dirty register, which must return unchanged. Every paid factor is charged one unit, including all restoration. The direct two-register construction costs eight, so a useful candidate must use at most seven. Arbitrary interleaved `GL3(F2)` scalar mixing is allowed at zero price only as the explicit favorable normalization by augmentation; no COPY, clean state, frame operation, or additional role is available in this formal model. Any physical hosting/recursive cost for the added register is outside this scalar construction and would still be needed for supplier adoption. No immediate kappa claim is possible from this abstraction.
+
+**Method / stopping rule:** Enumerate all 28 rank-one idempotents and both labels (56 paid generators). Use exact packed coefficient matrices over the full eight-element group algebra. Exhaust words of length at most seven by a radius-3 identity ball and radius-4 target-side enumeration, checking every expanded state against the exact opposite ball. A meeting yields an exact candidate and its paid word; no meeting excludes only this stated alphabet/role model through seven calls. Stop after all depth-four expansions and exact direct-word/control replays.
+
+**Predecessors and overlap:** The exact two-register predecessor is `research/bit-dihedral-commutator/README.md` and its pinned verifier; it establishes the eight-call baseline and explicitly leaves a restored third dirty register open. The refreshed overlap search at 2026-10-14 08:12 UTC had complete pagination (415 open PRs, 328 discussion records, no warnings) and found no exact D8/UT3 third-register claim. Nearest structurally related work is PR #1720 (three-register S4 4-cycle targets) and Discussion #1317 (coordinate-CNOT labels), which use different groups/targets; active #1700 and #1727 use four-register rational/AGL2 objects. This scope changes the group object and target from those results.
+
+**Expected evidence:** A source-pinned package with the complete radius counts and hashes, a direct eight-call positive control, exact independent endpoint multiplication, and altered-target/omitted-call negative controls. Outcome is either a seven-call formal construction or a bounded no-seven-call result; neither would establish physical supplier admission or a multiplication bound.
+
+Claim expiry: 2026-10-14 10:15 UTC.