Mister-Meeseeks · Oct 16, 2026 · Oct 14, 2026 · Oct 14, 2026
diff --git a/research/affine-mixed-four-targets-seed5638/CERTIFICATE.json b/research/affine-mixed-four-targets-seed5638/CERTIFICATE.json
@@ -0,0 +1,111 @@
+{
+  "schema_version": 1,
+  "field": "F2",
+  "target": "diag(h,h^-1,tx,ty) in the homogeneous affine 3x3 representation",
+  "registers": 4,
+  "target_displacement_rank": 6,
+  "paid_labels": [
+    "a",
+    "b",
+    "tx",
+    "ty"
+  ],
+  "rank_one_idempotents": 120,
+  "factors": 480,
+  "q6_frontiers": [
+    {
+      "depth": 1,
+      "incoming_states": 1,
+      "strict_drop_edges": 112,
+      "unique_states": 112,
+      "states_sha256": "10df2803c837e379a55cb68c5c52b68befe13cb998db0f537ea99e69ff664939"
+    },
+    {
+      "depth": 2,
+      "incoming_states": 112,
+      "strict_drop_edges": 2816,
+      "unique_states": 960,
+      "states_sha256": "83d3033330efbec47b7f793a37366bc2414a267517979299f8edcfa65c04a528"
+    },
+    {
+      "depth": 3,
+      "incoming_states": 960,
+      "strict_drop_edges": 4928,
+      "unique_states": 656,
+      "states_sha256": "c4e849d46f4ca43aff8f8be6c02ba0f8450bea499f365976011c06e405cae63d"
+    },
+    {
+      "depth": 4,
+      "incoming_states": 656,
+      "strict_drop_edges": 520,
+      "unique_states": 32,
+      "states_sha256": "60e95c07370741151cdc96c1a97700f2c7ddb658ce7518ad8c464de1e38b8925"
+    },
+    {
+      "depth": 5,
+      "incoming_states": 32,
+      "strict_drop_edges": 0,
+      "unique_states": 0,
+      "states_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855"
+    }
+  ],
+  "q6_endpoint_found": false,
+  "controls": {
+    "direct_word_calls": 10,
+    "direct_word_endpoint": "target",
+    "drop_call_control": "rejected",
+    "wrong_label_control": "rejected",
+    "all_480_factors_involutive": true
+  },
+  "q7_trajectory": {
+    "program": "q7_trajectory_search.py",
+    "program_sha256": "dda3653279bf66b304c6ee0b9622ddae654eb875db2163e8802b96e4ac2b1362",
+    "output": "Q7-OUTPUT.txt",
+    "output_sha256": "8263e8941982eaa2840917f743f32db809644a3a2fcc481cea9f3c4f96dcb3d1",
+    "frontiers": [
+      {
+        "depth": 1,
+        "states": 352,
+        "neutral0": 112,
+        "neutral1": 240,
+        "goal": false
+      },
+      {
+        "depth": 2,
+        "states": 15368,
+        "neutral0": 960,
+        "neutral1": 14408,
+        "goal": false
+      },
+      {
+        "depth": 3,
+        "states": 58352,
+        "neutral0": 656,
+        "neutral1": 57696,
+        "goal": false
+      },
+      {
+        "depth": 4,
+        "states": 16772,
+        "neutral0": 32,
+        "neutral1": 16740,
+        "goal": false
+      },
+      {
+        "depth": 5,
+        "states": 164,
+        "neutral0": 0,
+        "neutral1": 164,
+        "goal": false
+      },
+      {
+        "depth": 6,
+        "states": 0,
+        "neutral0": 0,
+        "neutral1": 0,
+        "goal": false
+      }
+    ],
+    "endpoint_found": false
+  }
+}
diff --git a/research/affine-mixed-four-targets-seed5638/Q7-OUTPUT.txt b/research/affine-mixed-four-targets-seed5638/Q7-OUTPUT.txt
@@ -0,0 +1,8 @@
+G [(3, 2, 4), (1, 3, 4), (5, 2, 4), (1, 6, 4)] uv [((1, 0, 0), (0, 1, 0)), ((0, 1, 0), (1, 0, 0)), ((1, 0, 0), (0, 0, 1)), ((0, 1, 0), (0, 0, 1))]
+target [(2, 5, 4), (6, 1, 4), (5, 2, 4), (1, 6, 4)] rank 6 factors 480
+depth 1 states 352 neutral0 112 neutral1 240 goal False
+depth 2 states 15368 neutral0 960 neutral1 14408 goal False
+depth 3 states 58352 neutral0 656 neutral1 57696 goal False
+depth 4 states 16772 neutral0 32 neutral1 16740 goal False
+depth 5 states 164 neutral0 0 neutral1 164 goal False
+depth 6 states 0 neutral0 0 neutral1 0 goal False
diff --git a/research/affine-mixed-four-targets-seed5638/README.md b/research/affine-mixed-four-targets-seed5638/README.md
@@ -0,0 +1,130 @@
+# Four mixed affine targets in a four-register bit model
+
+This is a seed-pinned construction search in the noncommutative group algebra
+of AGL2(F2). It restricts the paid alphabet to four affine labels and tests
+whether a mixed four-target tuple can beat its direct ten calls. The closed
+PR #1333 theorem gives cost six when all rank-one affine labels are available;
+this experiment asks how much this fixed four-label subset costs.
+
+The paid labels are:
+
+* a(x0,x1)=(x0+x1,x1)
+* b(x0,x1)=(x0,x0+x1)
+* tx(x0,x1)=(x0+1,x1)
+* ty(x0,x1)=(x0,x1+1)
+
+The target is diag(h,h^-1,tx,ty), with h(x0,x1)=(x1,x0+1) and
+h^-1(x0,x1)=(x1+1,x0). A paid call is D_N(g)=I+([g]+[1])N for any rank-one
+idempotent N in M4(F2). Arbitrary dirty registers are allowed. Free GL4(F2)
+mixing is represented by conjugating the idempotents while moving scalar maps
+through the paid gates; augmentation forces the net scalar map to identity.
+
+## Cost screen
+
+The homogeneous 3-by-3 affine representation maps the four target blocks to
+displacement ranks 2, 2, 1, and 1, hence rank six total. Each paid generator
+has rank-one displacement and each register coefficient has rank one, so one
+call can alter the displacement rank by at most one. The direct construction
+uses ten calls, leaving at most four calls above this optimistic floor before
+any physical, recursive, helper, frame, copying, hosting, or restoration cost.
+This local finite model cannot itself change the selected kappa. The full
+rank-one affine alphabet in [PR #1333](https://github.com/CrocSwap/integer-mult-bounds/pull/1333)
+has exact minimum six for this target; the current four-label subset is
+strictly smaller and is the changed hypothesis tested here. The finished
+claim for that theorem is [Discussion #1317](https://github.com/CrocSwap/integer-mult-bounds/discussions/1317).
+
+## Exact q=6 result
+
+verify.py enumerates all 120 rank-one idempotents in M4(F2) and all four
+labels (480 factors). Any six-call identity from this target must lower the
+displacement rank at every step. The exhaustive lowering frontiers are
+recorded in CERTIFICATE.json; the search reaches no rank-one state after five
+drops, so no six-call word exists in this declared model.
+
+The certificate includes hashes of the complete unique-state frontiers. The
+verifier also checks the direct ten-call positive control, a deleted-call
+negative control, and a wrong-label negative control. All arithmetic is exact
+over F2.
+
+## Exact q=7 result
+
+A seven-call word must have exactly six rank drops and one neutral step:
+the target starts at displacement rank six, each factor changes rank by at
+most one, and seven steps must end at zero. q7_trajectory_search.py covers
+all such trajectories, allowing the neutral step at any position. The complete
+frontiers are in Q7-OUTPUT.txt:
+
+| Depth | Incoming states | Next states | Neutral unused | Neutral used |
+|---:|---:|---:|---:|---:|
+| 1 | 1 | 352 | 112 | 240 |
+| 2 | 352 | 15,368 | 960 | 14,408 |
+| 3 | 15,368 | 58,352 | 656 | 57,696 |
+| 4 | 58,352 | 16,772 | 32 | 16,740 |
+| 5 | 16,772 | 164 | 0 | 164 |
+| 6 | 164 | 0 | 0 | 0 |
+
+There are no length-six prefixes in the complete rank-compatible search,
+so no seven-call endpoint is possible. This is a necessary-condition
+obstruction in the homogeneous affine representation and therefore excludes
+an exact group-algebra word in the declared model. Combined with the rank-six
+floor and the direct ten-call word, the minimum with this four-label subset is
+between eight and ten calls. The exact minimum is unresolved. PR #1333 allows
+every rank-one affine displacement as a paid label and proves minimum six;
+therefore any six-call attaining word from that theorem must use a label
+outside this restricted set. No result here applies to the larger alphabet.
+
+Run the finite trajectory search with:
+
+    python3 -I -B research/affine-mixed-four-targets-seed5638/q7_trajectory_search.py
+
+The recorded run took about ten minutes. The q=6 verifier checks the pinned
+q=7 script/output hashes and frontier summary; rerunning this command checks
+the full q=7 enumeration from scratch.
+
+## Reproduction
+
+    python3 -I -B research/affine-mixed-four-targets-seed5638/verify.py
+
+Expected marker: PASS_AFFINE_MIXED_Q6_OBSTRUCTION. The output includes the
+certificate SHA-256.
+
+## Pins and overlap
+
+* Accepted selector seed: 5638c49abc39ff4644ef875d73a20996
+* Track/task: bit-mechanisms / bit-new-register-model, catalogue revision 3
+* Base commit: 0b1cccf86e2f5132ed286a26ef0a681a495740ea
+* Selected-result SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32
+* Source hashes: SOURCES.json
+* Active claim: https://github.com/CrocSwap/integer-mult-bounds/discussions/1727
+* Review PR: https://github.com/CrocSwap/integer-mult-bounds/pull/1745 (open)
+
+The final pre-submission overlap scan completed at 2026-10-14 08:28:13 UTC: it
+covered 419 open PRs and 375 Discussions with complete pagination and no warnings. Direct predecessor [PR #1333](https://github.com/CrocSwap/integer-mult-bounds/pull/1333)
+(closed; head 7f25b6d0e1d7e995d4d2840f231d98b1a1a9fec2) proves the exact
+six-call optimum for the full rank-one affine alphabet and names restricted
+coordinate CNOT/translation alphabets as a next question. This package fixes
+the four labels {a,b,tx,ty} and one four-register target tuple. Open PR #1711
+is a two-target affine minimum; open PR #1716 is a restricted three-register
+AGL2 triple. Discussion #1738 is a concurrent three-register UT3 central-pair
+claim with labels {x,y} and a full group-algebra radius search. Those have
+different target tuples, register counts, groups, and methods. Cycle 1's open
+[PR #1733](https://github.com/CrocSwap/integer-mult-bounds/pull/1733) concerns
+rational address relations rather than this binary-affine label restriction.
+Active Discussion #1727 is this task's separate four-register tuple with
+standalone translations; it was the original coordination claim for this run.
+
+The direct ten-call word and the interval 8..10 provide no kappa improvement.
+No physical supplier or recursive charge was established. No independent
+human review is claimed. Prepared with substantial OpenAI Codex assistance in
+framing, finite enumeration, verifier implementation, and writing.
+
+## Open next questions
+
+1. Can an exact eight- or nine-call word realize the same tuple? Any positive
+   candidate needs full formal group-algebra replay and full interface pricing.
+2. Can another four-register mixed affine tuple yield a real paid-call saving?
+   A distinct target would need its own optimistic rank screen and complete
+   helper, copying, hosting, recursive, frame, precision, and restoration bill.
+
+The q=6/q=7 obstruction above is the complete stopping point for this seed;
+no q=8..10 search was performed.
diff --git a/research/affine-mixed-four-targets-seed5638/SOURCES.json b/research/affine-mixed-four-targets-seed5638/SOURCES.json
@@ -0,0 +1,96 @@
+{
+  "schema_version": 1,
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "files": [
+    {
+      "path": "AGENTS.md",
+      "usage": "provenance",
+      "git_blob": "860762e04e963e1b163b8fa5267854c0040a9e67",
+      "sha256": "5f29c7cbf0eb28f4e87f0affa9cffdaef4e39a630a5878c8fed02b544c894d2e"
+    },
+    {
+      "path": "CONTRIBUTING.md",
+      "usage": "provenance",
+      "git_blob": "e51435543687f1bd88ab22248082dc689ce8b853",
+      "sha256": "df5b5646602f42cb5da5844f04ac925182ddcd0f627b9de1c96a5095bb7a277a"
+    },
+    {
+      "path": "autoresearch.md",
+      "usage": "provenance",
+      "git_blob": "8d5403e0f10f68a61d0cb9294fa2e83e872d1d82",
+      "sha256": "215b3222f6a943edc69422a28ac21ea1f397ddf54770f1d87ac4195385d73699"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "input",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1",
+      "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075"
+    },
+    {
+      "path": "docs/research/community-batch1099-review.md",
+      "usage": "input",
+      "git_blob": "6cb6ec061a071a06a2d18cd4dc3e9a2b05d28099",
+      "sha256": "7e76d2c38e5fce30587835b7251df1430488ae4dd372611ef69a3f03f476b6cb"
+    },
+    {
+      "path": "research/binary-affine-register-bound/PROOF.md",
+      "usage": "input",
+      "git_blob": "b1152a39b559f7a058489ec74231488e4f8edd83",
+      "sha256": "36a33434222798434b0a17476ae5f9bc8a2a3891871c46e77cc50ce2e132206b"
+    },
+    {
+      "path": "research/binary-affine-register-bound/README.md",
+      "usage": "input",
+      "git_blob": "7b68583becf8be6f3cf666b05b0943e7eb99270c",
+      "sha256": "7951868efdf35fdb2aacdd302ca825cbde0586a5fae6df23aee78e29f30c332a"
+    },
+    {
+      "path": "research/binary-affine-register-bound/SOURCES.json",
+      "usage": "input",
+      "git_blob": "1a6ff149a724392dfb829e06c7ad750aa999b590",
+      "sha256": "b3ecf3d47432eb6eb5271dcae1c9bd5cd97dfabb257006b3e59184ca038989d0"
+    },
+    {
+      "path": "research/binary-affine-register-bound/TASK.json",
+      "usage": "input",
+      "git_blob": "84dc9d73fdb009b91055b6451191d7dae86b7806",
+      "sha256": "75dbde42b53e785cc969b764d6477ea46eebf25a38946cf1ce56a2ee3c4e50e9"
+    },
+    {
+      "path": "research/bit-three-register-lower-bound/README.md",
+      "usage": "input",
+      "git_blob": "25e41c370c101a33a6deb978de08b03ed7539b79",
+      "sha256": "c12b33ef388deece0cbf891505a2ce2c0779a0c3a6cf1f0f7f7039bab59cbebd"
+    },
+    {
+      "path": "research/community-frontier1094-audit/ASSEMBLY.json",
+      "usage": "input",
+      "git_blob": "5fd676df060f3b450782fdf5ff5f6beb63a5eeb0",
+      "sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045"
+    },
+    {
+      "path": "research/community-frontier1094-audit/SELECTED-SOURCE.json",
+      "usage": "input",
+      "git_blob": "6a75493790b4eff9c4a98051ef8b01d6aad0d5a8",
+      "sha256": "79b3de5db3dbcb9829789ba58e9e45758926031e2a7d80667a08429d369be53b"
+    },
+    {
+      "path": "scripts/choose_research_task.py",
+      "usage": "provenance",
+      "git_blob": "695639622a283190948c4187bdbb41997e6e43d7",
+      "sha256": "f1af1e70858ab1d033ce20ee1968201bc592eb9e3fd05ab8ad78c842c0b028d1"
+    },
+    {
+      "path": "scripts/research_baseline.py",
+      "usage": "input",
+      "git_blob": "6b607b36eefd65718120ade3c8187681f11bf879",
+      "sha256": "59d9b04dc33e1bb3f81506d187243435de4902138a038b935e36ebaa5f37db67"
+    }
+  ]
+}