Mister-Meeseeks · Oct 14, 2026 · Oct 14, 2026
diff --git a/research/bit-affine-generator-sharing/CERTIFICATE.json b/research/bit-affine-generator-sharing/CERTIFICATE.json
@@ -0,0 +1,120 @@
+{
+  "conclusion": "both fixed two-register endpoints require 8 calls",
+  "controls": {
+    "all_24_formal_group_elements_enumerated": true,
+    "all_24_paid_factors_included": true,
+    "all_6_rank_one_idempotents_enumerated": true,
+    "changed_affine_target_mutation_rejected": true,
+    "direct_formal_endpoint_passes": true,
+    "inverse_target_pair_verified": true,
+    "omitted_call_mutations_rejected": true,
+    "two_endpoints_share_identical_complete_state_balls": true
+  },
+  "direct_controls": {
+    "all_8_single_call_omissions_rejected": true,
+    "changed_target_and_target_order_rejected": true,
+    "inverse_pair_word": [
+      "a:N=1000",
+      "b:N=1000",
+      "a:N=1000",
+      "v:N=1000",
+      "a:N=0001",
+      "b:N=0001",
+      "a:N=0001",
+      "u:N=0001"
+    ],
+    "same_target_word": [
+      "a:N=1000",
+      "b:N=1000",
+      "a:N=1000",
+      "v:N=1000",
+      "a:N=0001",
+      "b:N=0001",
+      "a:N=0001",
+      "v:N=0001"
+    ]
+  },
+  "first_endpoint": {
+    "direct_calls_total": 8,
+    "direct_word_per_target": "abav",
+    "formal_endpoint": "diag([h],[h])",
+    "map_per_target": "h(x0,x1)=(x1,x0+1)",
+    "minimum": 8,
+    "rank_floor": 4,
+    "search": {
+      "endpoint_reachable_within_7": false,
+      "left_radius": 3,
+      "left_state_sha256": "a0dade172de4088b86500e54ae8cc2389b447719fe7f250c2503151262633793",
+      "left_states": 6457,
+      "left_states_examined": 6457,
+      "minimum_word_length_found": null,
+      "right_radius": 4,
+      "right_state_sha256": "7ff93fe51d26c756210384dc31e97472f405853e00dc90413448f7f5a3c22bbe",
+      "right_states": 90859
+    },
+    "short_witness": null
+  },
+  "model": {
+    "address_group": "AGL_2(F_2)",
+    "calls_per_label_per_idempotent": 1,
+    "factor": "D_N(g)=I+([g]+[1])N",
+    "formal_ring": "F_2[AGL_2(F_2)]",
+    "free_scalar_mixing": "GL_2(F_2), normalized into the six paid idempotents by augmentation",
+    "paid_factor_count": 24,
+    "paid_labels": {
+      "a": {
+        "matrix_row_major": "1011",
+        "translation": "00"
+      },
+      "b": {
+        "matrix_row_major": "1101",
+        "translation": "00"
+      },
+      "u": {
+        "matrix_row_major": "1001",
+        "translation": "01"
+      },
+      "v": {
+        "matrix_row_major": "1001",
+        "translation": "10"
+      }
+    },
+    "rank_one_idempotents": [
+      "0001",
+      "0011",
+      "0101",
+      "1000",
+      "1010",
+      "1100"
+    ],
+    "registers": 2,
+    "workspace": "none"
+  },
+  "pivot_endpoint": {
+    "direct_calls_total": 8,
+    "direct_words": [
+      "abav",
+      "abau"
+    ],
+    "formal_endpoint": "diag([h],[h_inverse])",
+    "maps": [
+      "h(x0,x1)=(x1,x0+1)",
+      "h_inverse(x0,x1)=(x1+1,x0)"
+    ],
+    "minimum": 8,
+    "rank_floor": 4,
+    "search": {
+      "endpoint_reachable_within_7": false,
+      "left_radius": 3,
+      "left_state_sha256": "a0dade172de4088b86500e54ae8cc2389b447719fe7f250c2503151262633793",
+      "left_states": 6457,
+      "left_states_examined": 6457,
+      "minimum_word_length_found": null,
+      "right_radius": 4,
+      "right_state_sha256": "7ff93fe51d26c756210384dc31e97472f405853e00dc90413448f7f5a3c22bbe",
+      "right_states": 90859
+    },
+    "short_witness": null
+  },
+  "schema_version": 1
+}
diff --git a/research/bit-affine-generator-sharing/README.md b/research/bit-affine-generator-sharing/README.md
@@ -0,0 +1,125 @@
+# Generator-priced affine target sharing
+
+**Scoped negative result; no new \(\kappa\).** In the formal
+\(\mathbb F_2[\mathrm{AGL}_2(\mathbb F_2)]\) model with two target registers,
+four paid address generators, and all six rank-one scalar idempotents, both
+\[
+\operatorname{diag}([h],[h])
+\quad\text{and}\quad
+\operatorname{diag}([h],[h^{-1}]),
+\qquad h(x_0,x_1)=(x_1,x_0+1),
+\]
+have exact minimum cost **8 calls**. The direct words use four calls per
+target. A complete formal radius-seven search finds no saving, even though the
+general homogeneous displacement-rank floor is only four calls.
+
+This tests a new cost rule: only the two coordinate shears and two coordinate
+translations have unit price. PR #1333 prices every rank-one affine address
+map at one call and proves the general displacement-rank floor. Its stated
+generator-priced follow-up leaves room for a sharing gain; these two exact
+endpoints do not realize it. The result is limited to two registers and the
+two displayed target profiles.
+
+## Exact model and complete cost
+
+Work in the noncommutative group algebra
+\(\mathbb F_2[G]\), \(G=\mathrm{AGL}_2(\mathbb F_2)\). The paid labels are
+\[
+a(x_0,x_1)=(x_0+x_1,x_1),\quad
+b(x_0,x_1)=(x_0,x_1+x_0),\quad
+u(x_0,x_1)=(x_0+1,x_1),\quad
+v(x_0,x_1)=(x_0,x_1+1).
+\]
+Each costs one. A call chooses any of the six rank-one idempotents
+\(N\in M_2(\mathbb F_2)\) and applies
+\[
+D_N(g)=I+([g]+[1])N.
+\]
+Free scalar \(GL_2(\mathbb F_2)\) mixing is retained as the inherited
+optimistic relaxation. Moving it through the paid factors conjugates each
+idempotent; augmentation \([g]\mapsto1\) forces the net scalar map to be the
+identity, so enumerating all six idempotents covers interleaved free mixing.
+Address labels outside \(\{a,b,u,v\}\) are forbidden.
+
+The target \(h\) has a shortest address word *abav*, length four, and
+\(\operatorname{rank}_{\mathbb F_2}[A_h-I\mid b_h]=2\). Its inverse
+\(h^{-1}(x_0,x_1)=(x_1+1,x_0)\) has shortest word *abau* and the same rank.
+Thus each endpoint has direct cost 8 and the general displacement-rank floor
+4. The mechanism had at most four calls of optimistic headroom. There is no
+helper register or copied information; both arbitrary-input target registers
+are represented in the exact endpoint. Each address call is charged once.
+Physical/compiler realization, recursive supplier costs, precision, hosting,
+and admission into the selected bit construction are outside this local model.
+
+The selected conditional record remains **complex-binding** according to
+scripts/research_baseline.py. Consequently even a local bit saving would not
+by itself change its \(\kappa\); this result claims no supplier-wide saving.
+
+## Exact search and controls
+
+Each paid factor is a formal involution because every allowed address label
+satisfies \(g^2=1\) and every \(N\) satisfies \(N^2=N\). For a word of at most
+seven calls, split its factor sequence after at most three calls. The verifier
+enumerates exact states in the radius-three and radius-four balls. If \(P\) is
+the prefix and \(T=QP\) the endpoint, then the required suffix is
+\(Q=TP^{-1}\). Every state is the full \(2\times2\) matrix of 24 formal
+coefficients; no finite address representation is used for acceptance.
+
+The two independent implementations agree on 6,457 radius-three states and
+90,859 radius-four states, with their canonical state digests in
+CERTIFICATE.json. The first uses packed row-major \(AGL_2\) matrices and a
+rank-one update. The second enumerates all 24 permutations of the four address
+points and multiplies the two formal coefficients of each factor directly.
+Neither finds a word of length at most seven for either endpoint. The direct
+eight-call words replay coefficient by coefficient, so the minima are exactly
+eight in both stated cases.
+
+The replay also verifies all 24 paid factors, all 24 formal group elements,
+all six idempotents, factor involution, both direct words, rejection of every
+single-call deletion from each direct word, and a changed-target/order
+mutation. These checks establish the finite algebraic result only.
+
+## Reproduce
+
+From the repository root:
+
+    python3 -I -B research/bit-affine-generator-sharing/verify.py
+    python3 -I -B research/bit-affine-generator-sharing/independent.py
+    python3 scripts/research_package.py --package research/bit-affine-generator-sharing
+
+The first two commands should print PASS_BIT_AFFINE_GENERATOR_SHARING and
+PASS_BIT_AFFINE_GENERATOR_SHARING_INDEPENDENT. TASK.json records seed, base,
+changed scope, exact cost model, stopping rule, and next question. SOURCES.json
+pins the retained AGL predecessor review/proof. No predecessor implementation
+is imported.
+
+## Overlap and attribution
+
+Gareth Davidson (@bitplane), PR #1333, supplies the broader AGL group model,
+the displacement-rank floor, and the generator-priced next question. Related
+coordinate-CNOT-only linear targets are recorded in TASK.json. Two nearby
+Discussion #1317 claims use three registers: comment 18918042 studies
+\(\operatorname{diag}([a],[b],[aba])\) with only the two CNOT labels, while
+comment 18918291 studies \(\operatorname{diag}([a],[b],[h])\) with labels
+\(a,b,t_x\). The latter shares the map \(h\), but changes both the target tuple
+and the address alphabet. This package prices \(a,b,t_x,t_y\) and gives exact
+two-register minima for \(\operatorname{diag}([h],[h])\) and
+\(\operatorname{diag}([h],[h^{-1}])\). Open PR #1696 submits a scoped exact
+five-call result for the three-register CNOT-only target
+\(\operatorname{diag}([a],[b],[aba])\); this package adds the two affine
+two-target profiles under the four-label alphabet. A separate current claim,
+Discussion #1700, studies rational rank-one frames with two independent
+address relations; its group and cost premise differ. No selected-result or
+physical-supplier change is claimed.
+
+Prepared by Cinna Davis with substantial OpenAI Codex assistance. There has
+been no independent human review. New work is Apache-2.0; predecessor files
+remain unchanged.
+
+## Open next questions
+
+Could a third target register or another explicitly priced affine target family
+permit sharing? First choose one three-target endpoint, compute its direct
+generator lengths and displacement-rank floor, and stop if the fully charged
+optimistic gap is zero. This package does not cover larger target tuples,
+workspace, or a physical/compiler admission.
diff --git a/research/bit-affine-generator-sharing/SOURCES.json b/research/bit-affine-generator-sharing/SOURCES.json
@@ -0,0 +1,30 @@
+{
+  "schema_version": 1,
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1",
+      "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075"
+    },
+    {
+      "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"
+    }
+  ]
+}
diff --git a/research/bit-affine-generator-sharing/TASK.json b/research/bit-affine-generator-sharing/TASK.json
@@ -0,0 +1,101 @@
+{
+  "schema_version": 1,
+  "task_id": "bit-new-register-model",
+  "priority_revision": 3,
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "session_seed": "f024daf023fbb77c64b7b0501cb0db7e",
+  "track": "bit-mechanisms",
+  "catalog_sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075",
+  "selected_record_sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32",
+  "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1317#discussioncomment-18918046",
+  "finding_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1317#discussioncomment-18918154",
+  "claim_expires_utc": "2026-10-14T09:20:00Z",
+  "original_card": {
+    "id": "bit-new-register-model",
+    "question": "Can an explicitly larger or nonabelian register model escape the scoped three-register obstruction?",
+    "starting_points": [
+      "research/bit-three-register-lower-bound/README.md",
+      "certificates/selected-result.json"
+    ],
+    "methods": [
+      "Construct the smallest exact identity in a newly specified legal operation alphabet.",
+      "Derive a target-specific relaxation bound for that same explicit alphabet."
+    ],
+    "acceptance_gate": "Name the changed hypothesis, all counted roles and a uniform identity; finite column tests alone do not certify it.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read Model section 1 of the three-register bound and identify the exact excluded free operations.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Write a complete operation alphabet outside one named premise and price a baseline before searching identities.",
+    "bottleneck_link": "A small-width completed saving changes the bit primitive; a finite-address identity alone is insufficient.",
+    "allowed_variations": [
+      "Choose a fixed larger register count or a specified nonabelian operation family."
+    ],
+    "scope_change_triggers": [
+      "Remaining inside the already-excluded three-register abelian model."
+    ],
+    "overlap_terms": [
+      "three-register",
+      "nonabelian",
+      "group algebra"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Exact minimum-call results for two fixed two-register endpoints in formal F2[AGL_2(F2)], where only the two coordinate shears and two coordinate translations are unit-price address labels: diag([h],[h]) and diag([h],[h_inverse]), with h(x0,x1)=(x1,x0+1).",
+  "scope_change_reason": "PR1333 already proves the all-register displacement-rank floor when every rank-one affine address map costs one call. This task instead charges only four elementary affine generators and asks whether their longer direct words permit register sharing. The helper-contract suggestion overlaps the R439 helper-star lineage and PR1194/PR1306. The first identical-target search found no saving, so one bounded pivot changed the endpoint to an inverse affine pair to test inter-target cancellation.",
+  "bottleneck_link": "A real saving in a paid bit-call alphabet could lower a bit-circuit profile. The selected conditional record remains complex-binding, and this abstract result supplies no physical bit supplier or kappa change.",
+  "budget_minutes": 60,
+  "stop_rule": "Complete the exact radius-seven 3+4 join for the identical and inverse-pair endpoints, compare an independent permutation-group coefficient engine, and stop without enlarging the target/register domain.",
+  "outcome": "negative-result",
+  "evidence": "CERTIFICATE.json records the two radius-seven outcomes and canonical state hashes; verify.py and independent.py agree on both formal searches, the 6457/90859 state-ball counts, and the direct eight-call controls.",
+  "author": "Cinna Davis",
+  "ai_assistance": "Substantial OpenAI Codex assistance in scoping, implementation, verification and writing; no independent human review claimed.",
+  "open_questions": [
+    {
+      "question": "Can a third target register or a different affine target family enable sharing in this four-generator alphabet?",
+      "why_it_matters": "The two fixed two-register profiles have no saving, but the exact search does not bound larger target tuples or workspace.",
+      "first_step": "Choose one explicit three-target AGL2 endpoint, compute every direct generator length and displacement-rank floor, then test whether the optimistic budget leaves room for a strict saving.",
+      "blocked_by": "This package does not enumerate larger register dimensions and does not give a physical/compiler admission for the affine address alphabet."
+    }
+  ],
+  "predecessors": [
+    "PR1333: broad AGL displacement-rank theorem and generator-priced follow-up question",
+    "PR1414, PR1491, PR1495, PR1496, PR1567: related coordinate-CNOT linear target scopes, not this translation-bearing AGL2 alphabet"
+  ],
+  "predecessor_heads": {
+    "PR1333": "7f25b6d0e1d7e995d4d2840f231d98b1a1a9fec2",
+    "PR1414": "9e1640094f649ba6db9742796825a8d6c238989b",
+    "PR1496": "0b1420fcc47e2137c5f0a5b9aa347cd001fff9cd"
+  },
+  "concurrent_scope": "Discussion 1317 comments 18918042 and 18918291 claim three-register heterogeneous endpoints. Comment 18918291 uses the same affine target h once, but only labels a,b,t_x and the endpoint diag([a],[b],[h]); this package prices a,b,t_x,t_y and gives exact minima for the two-register endpoints diag([h],[h]) and diag([h],[h_inverse]). Discussion 1700 claims a four-register rational two-relation model, a different address group and cost premise.",
+  "overlap_snapshot": {
+    "observed_utc": "2026-10-14T07:30:58Z",
+    "pull_requests": {
+      "complete_pagination": true,
+      "open_count": 398,
+      "relevant_heads": {
+        "PR1696": "4dadd2745da777f166c842c19a497791b0ec755c",
+        "PR1690": "dfd15686309d26eb98740942c4e84cddeafae7d8",
+        "PR1680": "a281330e84e64ad357e35b338796c96144a345ce",
+        "PR1662": "01cc02b164ecc4f9d84df3f56a1fe65b88daf390",
+        "PR1496": "0b1420fcc47e2137c5f0a5b9aa347cd001fff9cd",
+        "PR1705": "ad6b7d641b3e8576a95add7981fa75b6a8278fbc"
+      },
+      "closest": "PR1696 proves an exact five-call minimum for diag([a],[b],[aba]) on three registers over GL2(F2) with only two CNOT labels. It is a related but different endpoint, register count, and alphabet.",
+      "other_nearby": "PR1705 studies four repeated linear coordinate-SWAP targets at four registers; it has no translation-bearing affine endpoint or generator-priced four-label alphabet."
+    },
+    "discussions": {
+      "complete_top_level_pagination": true,
+      "top_level_count": 118,
+      "relevant_thread": "Discussion 1317, all 11 comments inspected",
+      "other_active_model": "Discussion 1700 is a four-register rational two-relation scope."
+    },
+    "limits": "The repository overlap helper could not refresh PR data because installed gh rejects --slurp. Direct complete REST pagination was used for all 398 open PR bodies; the current Discussion 1317 thread was fully read at updatedAt 2026-10-14T07:24:28Z, including the neighboring three-register affine claim. A subsequent all-discussions GraphQL refresh hit the GitHub GraphQL rate limit. This is an overlap check, not a novelty certificate."
+  }
+}