Mister-Meeseeks · Oct 14, 2026 · Oct 14, 2026 · Oct 14, 2026
diff --git a/research/gl3-order7-dirty-register-seed39ae/CLAIM.md b/research/gl3-order7-dirty-register-seed39ae/CLAIM.md
@@ -0,0 +1,12 @@
+# Local claim: GL3 order-seven target pair with one dirty register
+
+- **Track / seed / task:** `bit-mechanisms`, `39ae4eb03387f147b3cd06da8e84b83e`, `bit-new-register-model`, catalogue scope key `7f5f4426ef796683c485`, refined scope key `06ec42af57445d26ed9d`, priority revision 3.
+- **Base / selected record:** `0b1cccf86e2f5132ed286a26ef0a681a495740ea`; selected-record SHA256 `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.
+- **Object / question:** `G=GL3(F2)`, all six coordinate-CNOT labels `Cij=I+Eij`, and `C=C01*C12*C20` (order seven, direct length three). In the formal algebra `F2[G]`, with exactly three scalar registers and free `GL3(F2)` scalar mixing, can paid factors `I+([g]+1)N`, `g` among all six labels and `N=uv^T`, `v^Tu=1`, realize `diag([C],[C],1)` in at most five calls?
+- **Changed assumption / budget:** the address algebra is the explicit nonabelian group algebra and a third arbitrary dirty role is allowed and restored. The direct witness costs six calls (three per target); improvement requires five. All six coordinate-CNOT labels and every inverse are priced as paid calls; no free address maps or clean storage are admitted.
+- **Method / bounded domain:** enumerate all 168 legal (address label, rank-one idempotent) factors. Build the complete identity-side ball through radius two and stream every formal target-side word of length at most three. This covers every word through five paid factors. Check the formal group-algebra endpoint, the six-call witness, all factor inverses and controls.
+- **Overlap:** the live public REST census paginated all 419 open PR records through 2026-10-14 08:22 UTC. #1567 covers two-register GL3 target pairs of direct cost at most nine using all six labels; this adds one restored dirty role. #1495 covers the same three-role/all-six-label shape for order-two swaps; #1496 covers two order-four GL3 targets at two roles; #1491 covers GL2 swap targets; #1720 uses an S4 order-four target. The latest complete cached Discussion snapshot had 371 records, fetched 08:16 UTC; the exact GL3 order-seven three-register query returned no match. Live Discussion refresh failed because the authenticated GitHub API rate limit was exceeded. See [OVERLAP.json](OVERLAP.json) for exact heads, coverage and limits.
+- **Finding:** the identity-side radius-two ball contains 22,051 states; all 4,770,025 target-side words were checked with zero intersections. No five-call word exists. The direct six-call construction passes all 504 regular-action basis rows. The exact minimum is six in this scoped model. Focused PR [#1741](https://github.com/CrocSwap/integer-mult-bounds/pull/1741) is open for review.
+- **Claim expiry:** 2026-10-14T10:11:00Z. Remote Discussion posting was unavailable under the API rate limit, so this file remains the local claim and finding; PR [#1741](https://github.com/CrocSwap/integer-mult-bounds/pull/1741) is the review handoff.
+
+No selected supplier, new `kappa`, or multiplication theorem is claimed.
diff --git a/research/gl3-order7-dirty-register-seed39ae/OVERLAP.json b/research/gl3-order7-dirty-register-seed39ae/OVERLAP.json
@@ -0,0 +1,71 @@
+{
+  "schema_version": 1,
+  "searched_at_utc": "2026-10-14T08:22:19Z",
+  "scope_query": "GL3 order-seven target pair exactly three registers full coordinate-CNOT alphabet",
+  "open_prs": {
+    "source": "GitHub public REST API /repos/CrocSwap/integer-mult-bounds/pulls?state=open&per_page=100",
+    "pages_checked": [
+      1,
+      2,
+      3,
+      4,
+      5
+    ],
+    "page_counts": [
+      100,
+      100,
+      100,
+      100,
+      19
+    ],
+    "complete_pagination": true,
+    "records": 419,
+    "exact_scope_match": false,
+    "relevant_neighbors": [
+      {
+        "number": 1567,
+        "title": "Scoped negative result: mixed GL3 targets have 27,308 exact minima; 624 natural aliases",
+        "head": "fd7f1a434da543e3fb4e1efbb6056dbdc3a292af",
+        "scope_difference": "Two scalar registers, full six-coordinate-CNOT alphabet, target pairs of direct cost at most nine. This package adds one arbitrary restored dirty scalar role for the same order-seven target pair."
+      },
+      {
+        "number": 1495,
+        "title": "Finite certificate: a third address coordinate does not cheapen two coordinate swaps (no new kappa)",
+        "head": "d5f8745962597a2e5d5bf2deeb7925705c37db8e",
+        "scope_difference": "Three scalar roles and all six labels, but two order-two swap targets. This package uses the order-seven target pair."
+      },
+      {
+        "number": 1496,
+        "title": "Finite certificate: two order-four GL3 targets need eight coordinate-CNOT calls (no new kappa)",
+        "head": "0b1420fcc47e2137c5f0a5b9aa347cd001fff9cd",
+        "scope_difference": "Two scalar roles and an order-four target with all six labels. This package adds one arbitrary dirty role and changes the target order."
+      },
+      {
+        "number": 1491,
+        "title": "Finite certificate: three coordinate swaps need nine paid calls on three registers (no new kappa)",
+        "head": "69638b36c691d3e1b8463526fd68c849e8d1aef9",
+        "scope_difference": "GL2 address group and order-two swap targets."
+      },
+      {
+        "number": 1720,
+        "title": "Scoped S4 result: two 4-cycle targets need six calls at r=3",
+        "head": "edb4a289a5f32c8209140144e384dcd3fa8cbfd1",
+        "scope_difference": "Adjacent-transposition S4 alphabet and order-four target."
+      }
+    ],
+    "refresh_through_utc": "2026-10-14T08:22:19Z"
+  },
+  "discussions": {
+    "source": "scripts/find_overlap.py cached snapshot",
+    "fetched_at_utc": "2026-10-14T08:16:34.312283Z",
+    "complete_pagination": true,
+    "records": 371,
+    "exact_scope_matches": 0,
+    "live_refresh": "Unavailable: authenticated GitHub API rate limit exceeded. The exact-scope search used the latest complete cached snapshot."
+  },
+  "limits": [
+    "Open PR records were searched by title and body; relevant bodies were inspected, but every code diff was not searched.",
+    "Discussion content was complete in the cached snapshot, not live-refreshed during this search.",
+    "No literature-priority or universal novelty claim is made."
+  ]
+}
diff --git a/research/gl3-order7-dirty-register-seed39ae/README.md b/research/gl3-order7-dirty-register-seed39ae/README.md
@@ -0,0 +1,90 @@
+# A dirty register does not shorten a GL3 order-seven target pair
+
+**Scoped exact minimum in a finite formal register model. No selected supplier or new `kappa` is claimed.**
+
+## Model and result
+
+Work over the formal group algebra `R = F2[GL3(F2)]`. The six paid address labels are all coordinate CNOTs
+
+```text
+C01 = I + E01,  C12 = I + E12,  C20 = I + E20,
+C10 = I + E10,  C21 = I + E21,  C02 = I + E02.
+```
+
+They generate all 168 elements of `GL3(F2)`. Let
+
+```text
+C = C01 C12 C20 = [[0,1,1], [1,1,1], [1,0,1]].
+```
+
+`C` has order seven and shortest coordinate-CNOT word length three. There are exactly three scalar registers. The endpoint is
+
+```text
+diag([C], [C], 1)
+```
+
+on arbitrary inputs, including an arbitrary dirty third register restored exactly. Scalar `GL3(F2)` mixing is free. Every paid call has the form
+
+```text
+F(g,N) = I + ([g] + 1) N,
+```
+
+where `g` is one of the six labels and `N=uv^T` is a rank-one idempotent over `F2` (`v^T u=1`). In characteristic two these calls are involutions. There are 28 allowed `N` and 168 paid factors total.
+
+The exact minimum is **six paid calls**. Three calls implement `C` on the first register, three on the second, and the third stays fixed. Exhaustive formal group-algebra search finds no five-call word, even with the free scalar mixing and dirty register.
+
+The free mixing normalization is exact: augmentation sends every address `[g]` to 1, so the scalar matrices in any successful boundary word multiply to the identity. Moving them through the paid calls conjugates each standard rank-one projector into some `uv^T` with `v^T u=1`. The census includes all 28 such projectors at each of the six paid labels. It includes every free scalar mixing pattern without charging or fixing a frame.
+
+## Complete finite evidence
+
+The verifier stores every coefficient in the 168-element group basis of `F2[GL3(F2)]`; it never accepts equality from the natural three-dimensional representation. A radius-two identity ball has unique-state shells `1, 168, 21,882`, for 22,051 states. On the target side, the verifier streams every raw word of length zero through three rather than storing duplicate states:
+
+| Target-side suffix length | Words checked |
+|---:|---:|
+| 0 | 1 |
+| 1 | 168 |
+| 2 | 28,224 |
+| 3 | 4,741,632 |
+
+None of the 4,770,025 target-side endpoints lies in the identity-side ball. Any word of length at most five splits into a prefix of length at most two and a suffix of length at most three. Every paid factor is its own inverse, so a target factorization would put `target * suffix^-1` in both sets. This excludes every word through length five.
+
+The six-call direct word is checked on all `3*168 = 504` basis rows in the faithful regular action. The verifier checks all 168 factor inverses, confirms the join logic with one-target three-call and identity positive controls, rejects the wrong CNOT order, and rejects omission of each direct call.
+
+## Scope and overlap
+
+This refines the `bit-new-register-model` card by changing the address algebra from rational abelian frames to the nonabelian formal algebra `F2[GL3(F2)]`, and by adding one arbitrary dirty scalar role to a two-target order-seven endpoint.
+
+The closest work is:
+
+- [PR #1567](https://github.com/CrocSwap/integer-mult-bounds/pull/1567) gives exact minima for all 27,308 two-register GL3 target pairs with the full six-label coordinate-CNOT alphabet and direct cost at most nine. This package asks whether a restored third role changes one such order-seven target pair.
+- [PR #1495](https://github.com/CrocSwap/integer-mult-bounds/pull/1495) adds a third role for two order-two swap targets using all six labels. The target order differs.
+- [PR #1496](https://github.com/CrocSwap/integer-mult-bounds/pull/1496) covers two order-four GL3 targets with two scalar registers and all six labels.
+- [PR #1491](https://github.com/CrocSwap/integer-mult-bounds/pull/1491) covers three GL2 coordinate swaps; [PR #1720](https://github.com/CrocSwap/integer-mult-bounds/pull/1720) uses an S4 adjacent-transposition alphabet and a four-cycle target.
+
+The live REST census covered all 419 open PR records through 2026-10-14 08:22 UTC. The latest complete cached Discussion snapshot had 371 records, fetched 08:16 UTC; the exact GL3 order-seven three-register query returned no match. [OVERLAP.json](OVERLAP.json) records the heads, coverage and API refresh limit. No literature-priority claim is made.
+
+This is a finite model result, not a multiplication supplier. It establishes no physical source access, selected-supplier occurrence, compiler or hosting cost, precision contract, recursive-moment change, new `kappa`, or full multiplication theorem. The selected record and its inherited assumptions are unchanged.
+
+## Reproduce
+
+From the repository root with Python 3.12 and the standard library:
+
+```sh
+python3.12 -I -B research/gl3-order7-dirty-register-seed39ae/verify.py
+python3.12 scripts/research_package.py --package research/gl3-order7-dirty-register-seed39ae --check-only
+python3.12 scripts/research_package.py --package research/gl3-order7-dirty-register-seed39ae
+```
+
+Expected direct result line:
+
+```text
+PASS_GL3_ORDER7_DIRTY_REGISTER registers=3 factors=168 prefix=22051 suffix_words=4770025 hits=0 minimum=6
+```
+
+## Open next questions
+
+Can a second dirty role change the minimum for this same order-seven target and all six labels? First derive the fully charged target and direct bill for exactly four registers, then use an optimistic bound to decide whether a radius search can affect that bill. The larger rank-one projector inventory and target-side ball are the main blockers.
+
+## Credit
+
+Prepared by Cinna Davis with substantial OpenAI Codex assistance in task refinement, algebra, exact search, verification and writing. No independent human review or proof-assistant formalization is claimed. New work is Apache-2.0. Predecessor work is attributed above and pinned in [SOURCES.json](SOURCES.json).
diff --git a/research/gl3-order7-dirty-register-seed39ae/RESULT.json b/research/gl3-order7-dirty-register-seed39ae/RESULT.json
@@ -0,0 +1,84 @@
+{
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "conclusion": "The exact minimum is six paid calls in this finite three-register full-coordinate-CNOT model; no scalar-mixing saving is found.",
+  "exact_search": {
+    "identity_side_ball_size": 22051,
+    "identity_side_sha256": "eeb9d169471466c9f1676d144d114883f4d5630d23d427ef15c1670a06e80cf4",
+    "identity_side_shells": [
+      1,
+      168,
+      21882
+    ],
+    "intersection_hits": 0,
+    "maximum_word_length": 5,
+    "split": [
+      2,
+      3
+    ],
+    "target_side_raw_word_counts": [
+      1,
+      168,
+      28224,
+      4741632
+    ],
+    "target_side_transitions": 4770024
+  },
+  "limitations": [
+    "This result covers only the full six-coordinate-CNOT alphabet, the stated order-seven target, and exactly three scalar registers.",
+    "No selected supplier, physical compilation, recursive moment change, new kappa, or multiplication theorem is established.",
+    "No human independent review or proof-assistant formalization is claimed."
+  ],
+  "model": {
+    "C_direct_word": [
+      "C01",
+      "C12",
+      "C20"
+    ],
+    "C_matrix_rows": [
+      [
+        0,
+        1,
+        1
+      ],
+      [
+        1,
+        1,
+        1
+      ],
+      [
+        1,
+        0,
+        1
+      ]
+    ],
+    "C_order": 7,
+    "address_group_order": 168,
+    "direct_paid_calls": 6,
+    "free_scalar_mixing": "GL3(F2)",
+    "paid_address_labels": [
+      "C01",
+      "C12",
+      "C20",
+      "C10",
+      "C21",
+      "C02"
+    ],
+    "paid_factors": 168,
+    "rank_one_idempotents": 28,
+    "scalar_registers": 3,
+    "target": "diag([C],[C],1)"
+  },
+  "negative_controls": {
+    "omitted_direct_calls_rejected": 6,
+    "wrong_CNOT_order_rejected": true
+  },
+  "positive_checks": {
+    "direct_regular_basis_columns": 504,
+    "factor_inverse_checks": 168,
+    "identity_join_hits": 1,
+    "one_target_join_hits": 1
+  },
+  "seed": "39ae4eb03387f147b3cd06da8e84b83e",
+  "selected_record_sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32",
+  "status": "PASS_GL3_ORDER7_DIRTY_REGISTER_FLOOR"
+}
diff --git a/research/gl3-order7-dirty-register-seed39ae/SOURCES.json b/research/gl3-order7-dirty-register-seed39ae/SOURCES.json
@@ -0,0 +1,60 @@
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+}