Research: restricted three-register AGL2 target triple costs six calls

research/bit-affine-mixed-target-triple-seed818dbf/EXPECTED.json

@@ -0,0 +1,83 @@
+{
+  "controls": {
+    "all_generator_matrix_and_inverse_checks": 84,
+    "independent_dense_product_checks": 3,
+    "omitted_direct_call_checks": 6,
+    "wrong_label_check": true,
+    "wrong_target_check": true
+  },
+  "coordinate_labels": {
+    "a": 1,
+    "b": 5,
+    "h": 13,
+    "t_x": 7
+  },
+  "direct_cost": 6,
+  "direct_word": [
+    {
+      "label": "a",
+      "u": 1,
+      "v": 1
+    },
+    {
+      "label": "b",
+      "u": 2,
+      "v": 2
+    },
+    {
+      "label": "a",
+      "u": 4,
+      "v": 4
+    },
+    {
+      "label": "b",
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+      "v": 4
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+    {
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+    },
+    {
+      "label": "t_x",
+      "u": 4,
+      "v": 4
+    }
+  ],
+  "displacement_ranks": {
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+    "b": 1,
+    "h": 2
+  },
+  "exact_minimum_claim": "six in this 84-generator alphabet",
+  "group_order": 24,
+  "layer_counts": [
+    1,
+    84,
+    4697,
+    232720
+  ],
+  "layer_sha256_in_bfs_order": [
+    "044dde6d6a730099716e4e63b541be349bed3cb7645c2ec7681092fba7fbbe61",
+    "92062806bd0e0df4024b166d7a0cd0c762c6ad190a29a5e9b240fbe4fc215108",
+    "5064d7162057625a1f047261ce7c244f2cdfe2fbada42fcfe97c2445c63ab654",
+    "76e82cb9484516e0750f5a775a7d29d4f324ea1fd32b4c0215f9290da0570d2e"
+  ],
+  "left_join_probes": 4782,
+  "minimum_in_paid_alphabet_through_direct_cost": 6,
+  "model": "Mat_3(F2[AGL_2(F2)])",
+  "paid_generators": 84,
+  "paid_projectors": 28,
+  "radius": 3,
+  "rank_relaxation_lower_bound": 4,
+  "schema_version": 1,
+  "scope": "formal all-dirty register identities in the stated paid alphabet; no supplier or kappa claim",
+  "seed": "818dbf90f2581f79a4e4150cdd77e415",
+  "shortest_hit_through_five": null,
+  "target": [
+    "a",
+    "b",
+    "h=a*b*a*t_x"
+  ]
+}

research/bit-affine-mixed-target-triple-seed818dbf/OVERLAP.json

@@ -0,0 +1,62 @@
+{
+  "schema_version": 1,
+  "checked_at_utc": "2026-10-14T07:35Z",
+  "seed": "818dbf90f2581f79a4e4150cdd77e415",
+  "search_terms": [
+    "AGL2 mixed affine target",
+    "heterogeneous affine targets",
+    "three-register affine label",
+    "diag([a],[b],[h])",
+    "a*b*a*t_x"
+  ],
+  "methods": {
+    "repository_helper": "python3.12 scripts/find_overlap.py ... --refresh --json; discussion snapshot completed pagination",
+    "open_pull_requests": "gh api --paginate --method GET repos/CrocSwap/integer-mult-bounds/pulls -f state=open -f per_page=100, retaining titles and full bodies",
+    "discussion_cross_check": "Searched GitHub Discussions for repository-scoped AGL2, then fetched the current rendered Discussion #1317 page after helper refresh hit the API rate limit"
+  },
+  "coverage": {
+    "helper_discussion_snapshot_at": "2026-10-14T07:28:35Z",
+    "helper_discussion_pagination_complete": true,
+    "helper_discussion_records": 351,
+    "direct_open_pr_count": 403,
+    "direct_open_pr_search_matching_terms": [1711],
+    "direct_open_pr_relevant_nearby": [1696, 1711],
+    "repository_discussion_search": "repo:CrocSwap/integer-mult-bounds AGL2",
+    "repository_discussion_search_results": 1,
+    "repository_discussion_search_result": "Discussion #1317",
+    "direct_discussion_page_rechecked_at": "2026-10-14T07:35Z",
+    "direct_discussion_page_comment_count": 17
+  },
+  "nearest_work": [
+    {
+      "reference": "PR #1696",
+      "status": "open",
+      "distinction": "Exactly three registers and one each of a,b,aba in F2[GL2(F2)]; this package changes the address group and third target to h=aba*t_x in F2[AGL2(F2)]."
+    },
+    {
+      "reference": "Discussion #1317 claim and finding, adjacent two-target scope",
+      "status": "active claim with bounded result posted",
+      "distinction": "Two registers with diag([h],[h]) and four paid affine labels; this package has three registers, target diag([a],[b],[h]), and only a,b,t_x labels."
+    },
+    {
+      "reference": "PR #1711",
+      "status": "open",
+      "distinction": "Two-register endpoints diag([h],[h]) and diag([h],[h^-1]) with four paid labels a,b,t_x,t_y and 24 generators; this package has three registers, endpoint diag([a],[b],[h]), and the restricted three-label alphabet a,b,t_x with 84 generators."
+    },
+    {
+      "reference": "PR #1333",
+      "status": "merged predecessor",
+      "distinction": "General displacement-rank theorem for the complete rank-one affine-label alphabet; this package fixes a three-label alphabet and resolves a particular target gap by exact enumeration."
+    },
+    {
+      "reference": "PR #1414, #1491, #1541, #1567",
+      "status": "mixed merged/open predecessor set",
+      "distinction": "Repeated coordinate swaps or mixed GL3 targets; neither uses this affine target and label set."
+    }
+  ],
+  "warnings": [
+    "The helper's open-PR refresh failed because this gh version rejects --slurp; the direct paginated REST request succeeded.",
+    "The helper's Discussion refresh encountered GitHub API rate limiting; its latest fully paginated cache was cross-checked against the rendered live Discussion #1317 page."
+  ],
+  "result": "The open search found PR #1711 as nearby but not exact overlap; the complete Discussion snapshot and live thread showed no competing claim for this exact three-register target and three-label alphabet. Search results are overlap leads, not a novelty certificate."
+}

research/bit-affine-mixed-target-triple-seed818dbf/README.md

@@ -0,0 +1,158 @@
+# A mixed affine target triple needs six calls in a restricted label alphabet
+
+**Seed:** `818dbf90f2581f79a4e4150cdd77e415`
+
+**Track/task:** `bit-mechanisms` / `bit-new-register-model`
+
+**Priority revision:** 3
+
+**Base:** `0b1cccf86e2f5132ed286a26ef0a681a495740ea`
+
+**Selected-record SHA256:** `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`
+
+## Result
+
+In the exact three-register formal model below, the target
+
+\[
+  \operatorname{diag}([a],[b],[h]),\qquad h=a b a t_x,
+\]
+
+has paid-call cost **six**, even when arbitrary free binary changes of register
+basis are allowed. A complete radius-three search found no representation with
+five or fewer paid calls. The direct six-call word is
+
+```text
+D_(e0 e0^T)(a), D_(e1 e1^T)(b),
+D_(e2 e2^T)(a), D_(e2 e2^T)(b),
+D_(e2 e2^T)(a), D_(e2 e2^T)(t_x)
+```
+
+The target-specific homogeneous displacement-rank lower bound is four:
+`rho(a)=rho(b)=1`, while `h(x0,x1)=(x1,x0+1)` has `rho(h)=2`. Thus the
+experiment closes a two-call gap for this fixed label set. It is a scoped
+negative result about register mixing, not a new general affine theorem, a
+supplier construction, or a new conditional multiplication bound.
+
+## Exact model and complete charge
+
+Let `G = AGL_2(F2)` and `R = F2[G]` be the formal, noncommutative group
+algebra. Group elements act on all four points of `F2^2`; multiplication is
+composition with the rightmost map acting first. A register state is a
+`3 x 3` matrix over `R`, so equality means equality of every formal group
+algebra coefficient. This is the arbitrary-dirty-input contract: no register
+is assumed zero, copied clean, or discarded.
+
+The source labels are exactly
+
+```text
+a(x0,x1)   = (x0+x1,x1)
+b(x0,x1)   = (x0,x0+x1)
+t_x(x0,x1) = (x0+1,x1)
+```
+
+Each occurrence of
+
+\[
+D_N(g)=I+([g]-1)N,\qquad N=uv^T,\quad u,v\ne0,\quad v^Tu=1,
+\]
+
+costs one paid call. There are exactly 28 binary rank-one idempotents `N`
+and three labels, hence 84 paid generators. All three labels are involutions,
+so each paid generator is its own inverse.
+
+Arbitrary `B in GL_3(F2)` scalar mixing is free. It does not enlarge the
+normalized paid alphabet: move each `B` through the paid calls, conjugating
+each `N` to another rank-one idempotent; all 28 are already included. The
+augmentation `[g] -> 1` sends each paid call and the required target to the
+identity, so the total scalar map in a successful word must be `I`. This is
+the same normalization principle proved in the pinned
+[binary affine register bound](../binary-affine-register-bound/PROOF.md).
+
+For the optimistic charge screen, the direct bill is six calls. The
+homogeneous displacement rank gives the lower bound four because
+`[A-I | b]` has rank one for `a,b` and rank two for `h`. Every paid call has
+rank-one displacement in this representation. The full rank-one affine-label
+alphabet has a general theorem in the pinned predecessor; this experiment
+restricts the available paid address labels to `{a,b,t_x}` and asks whether
+dirty-register mixing can still reduce the direct bill to five. Extra address
+labels, precision, hosting, frame conversion, source access, and supplier
+interfaces are outside this finite model, so the answer does not translate to
+the selected `kappa`.
+
+## Exhaustive evidence
+
+`verify.py` stores each of the nine matrix entries as a 24-bit coefficient
+vector, one bit for each element of `AGL_2(F2)`. It performs a breadth-first
+search of the complete right-multiplication ball through radius three, then
+checks every canonical prefix through radius two against the radius-three
+ball. A word of length at most five must split into such a prefix and suffix.
+For prefix `U` and target `T`, the join checks whether `U^-1 T` lies in the
+right ball. The prefix gates are applied by left multiplication in forward
+word order, which yields `U^-1 T` because each gate is involutive.
+
+The completed layers contain `1, 84, 4,697, 232,720` distinct formal states.
+All `4,782` joins were checked and none hit the target. Since the explicit
+six-call word replays to `T`, the minimum in this stated paid alphabet is
+exactly six. The search does not quotient by a point-action representation;
+it retains all 24 formal basis coefficients.
+
+Controls also check all 84 generator matrices and involutions, compare three
+products with an independent dense group-algebra matrix multiplier, remove
+each direct call in turn, and reject a wrong affine label and wrong target.
+`EXPECTED.json` records deterministic breadth-first layer digests as well as
+the state counts and control totals.
+
+## Reproduce
+
+From the repository root, with Python 3.12 and no third-party dependencies:
+
+```sh
+python3.12 scripts/research_package.py \
+  --package research/bit-affine-mixed-target-triple-seed818dbf
+```
+
+Expected marker:
+
+```text
+PASS_BIT_AFFINE_MIXED_TARGET_TRIPLE group=24 generators=84 layers=1,84,4697,232720 probes=4782 minimum=6 hit=False
+PASS_REGISTERED_RESEARCH_PACKAGE research/bit-affine-mixed-target-triple-seed818dbf outcome=negative-result
+```
+
+Source commits, Git blobs, and SHA256 pins for the selected record, task
+catalogue, and predecessor results are in [SOURCES.json](SOURCES.json).
+The selected record was used only as the current frontier/provenance pin; it
+was not changed.
+
+## Overlap and attribution
+
+The final refresh covered all 403 open PRs and a repository-scoped `AGL2`
+Discussion search, which returned only #1317. The most recent complete
+paginated discussion snapshot had 351 records at 07:28 UTC; a later helper
+refresh hit GitHub's GraphQL rate limit, so I cross-checked the current
+rendered thread and repository search page. Open PR #1711 studies two-register
+endpoints `diag([h],[h])` and
+`diag([h],[h^-1])` with the four-label alphabet `{a,b,t_x,t_y}` (24
+generators). This result instead uses exactly three registers, target tuple
+`(a,b,h)`, and the restricted alphabet `{a,b,t_x}` (84 generators). The
+adjacent claim for two identical `h` targets is the same two-register,
+four-label scope as #1711. PR #1696 studies the related but linear target
+`(a,b,aba)` in `GL_2(F2)`. The two-coordinate-swap packages and mixed-`GL_3`
+target-pair work also change the target or address group. The exact claim and
+its expiry are recorded in Discussion
+[#1317](https://github.com/CrocSwap/integer-mult-bounds/discussions/1317#discussioncomment-18918291).
+
+This package builds on the affine rank argument in
+`research/binary-affine-register-bound/PROOF.md` and the coordinate CNOT
+formal-register model in `research/coordinate-cnot-register-bound/PROOF.md`.
+It does not claim either result as new. Prepared by Cinna Davis
+(@1337isnot1337), with substantial OpenAI Codex assistance in the scoped
+experiment design, verifier, exact-search implementation, and package draft.
+
+## Open next questions
+
+Can another three-label subset of the nine affine rank-one maps produce a
+dirty-register compression for a mixed target triple? A bounded next step is
+to enumerate those label subsets and select targets whose direct label-word
+bill exceeds the homogeneous-rank lower bound. This cycle deliberately fixed
+`{a,b,t_x}` and did not search other alphabets.

research/bit-affine-mixed-target-triple-seed818dbf/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "input",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
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+      "usage": "provenance",
+      "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1",
+      "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075"
+    },
+    {
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+      "git_blob": "b1152a39b559f7a058489ec74231488e4f8edd83",
+      "sha256": "36a33434222798434b0a17476ae5f9bc8a2a3891871c46e77cc50ce2e132206b"
+    },
+    {
+      "path": "research/binary-affine-register-bound/verify.py",
+      "usage": "input",
+      "git_blob": "41c7838c064d916456f94f2e73c171278997d5c9",
+      "sha256": "3d61e3e033208b3c06a0a4df26c6c53edeb7ccd3417ac0d86359df03692c4ec8"
+    },
+    {
+      "path": "research/coordinate-cnot-register-bound/PROOF.md",
+      "usage": "input",
+      "git_blob": "68fee5b5ccff5329e27300e40dcbd483640dd6b7",
+      "sha256": "bcee89f92dcce8253bd7eae6e924ddff598c900da81979c6171a412f03d765e1"
+    }
+  ]
+}