Exact length 6 for two quaternion triple squares

research/q16-central-square/README.md

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+# Quaternion triple squares need six calls
+
+**Finite register-model certificate; no new κ.** The selected result, both
+primitive suppliers, and `upstream/` are unchanged. The complex primitive
+binds, and these identities meet their disjoint bills.
+
+In the group algebra \(\mathbb F_2[Q_{16}]\),
+
+\[
+Q_{16}=\langle x,y\mid x^8=e,\ y^2=x^4,\ yxy^{-1}=x^{-1}\rangle,
+\]
+
+with exactly three roles and paid calls
+
+\[
+D_N([g])=I+([g]+[e])N,
+\qquad g\in\{x,y\},
+\]
+
+the minimum word lengths of \(\operatorname{diag}([x^2],[x^2],[x^2])\) and
+\(\operatorname{diag}([x^4],[x^4],[x^4])\) are both 6. Here \(N\) runs over
+all 28 rank-one idempotents in \(M_3(\mathbb F_2)\). Matrix equality is an
+operator identity on arbitrary dirty states. The element \(x^4\) is the unique
+order-2 element and is central; it is also \(y^2\).
+
+The changed hypothesis is the address group. The abelian three-register model
+uses arbitrary rational rank-one frames. This alphabet uses only the two
+quaternion generators and charges every call. There is no free scalar mixing
+between calls and no fourth role.
+
+## Priced baseline
+
+Generator lengths, computed before the matrix search:
+
+| Target element | Shortest generator word | Disjoint three-role bill |
+|---|---|---:|
+| \(x^2\) | \(xx\) | 6 |
+| \(x^4\) | \(yy\) | 6 |
+| \(x^6\) | \(xxyy\) | 12 |
+
+The degree-1 relaxation does not decide these bills. The augmentation powers
+of \(\mathbb F_2[Q_{16}]\) have dimensions \(15,13,11,9,7,5,3,1,0\). The
+quotient \(\Delta/\Delta^2\) is spanned by \([x]+1\) and \([y]+1\). A product
+of calls has degree-1 symbol equal to the sum of its projectors on each
+generator: every call augments to \(I\), and the cross term \((M-I)\alpha N\)
+has entries in \(\Delta^2\). Both triple squares, and the \(x^6\) triple, lie
+in \(I+M_3(\Delta^2)\). Vanishing of the symbol forces each projector sum to
+be zero. Every admitted projector has trace 1, so both label counts are even.
+The relaxation therefore gives only the even-length floor 2.
+
+## Result
+
+Lengths 1 through 5 were enumerated in \(M_3(\mathbb F_2[Q_{16}])\). The exact
+frontiers have sizes 56, 2352 and 92374. None contains either triple square.
+The doubled coordinate words
+
+- \((E_{00},x)\) twice, then the same for \(E_{11}\) and \(E_{22}\),
+- \((E_{00},y)\) twice, then the same for \(E_{11}\) and \(E_{22}\),
+
+equal the two targets. Both minima are therefore 6. A released two-role note
+already observed that two copies of \(x^4\) cost 4 by the direct word. The
+three-role search is the new statement: every admissible mixer still leaves
+the disjoint bill of 6 unchanged.
+
+There are 799 length-3 prefix matrices with a length-3 continuation to the
+central target. All 799 prefixes are pure \(y\)-words, and 792 of them use a
+projector outside \(\{E_{00},E_{11},E_{22}\}\). Off-diagonal mixing occurs at
+length 6 and still does not produce a shorter word. The same prefix-matrix
+count 799 appears for the \(x^2\) triple. An earlier semidihedral package
+reported the same count for a different presentation and different frontiers
+(2051 and 68899). The count is not treated as a new mechanism.
+
+Separately, \(\operatorname{diag}([x^6],[x^6],[x^6])\) has no word of length
+at most 6, and the coordinate word \(xxyy\) on each role has length 12. The
+parity lemma above excludes the odd lengths 7, 9 and 11, which were not
+enumerated. The exact minimum is therefore in \(\{8,10,12\}\) and is not
+claimed to be sharp.
+
+## Reproduce
+
+From the repository root, with Python 3.12 and `PYTHONOPTIMIZE` unset:
+
+```sh
+python3 -B research/q16-central-square/verify.py
+python3 -B scripts/research_package.py --package research/q16-central-square
+```
+
+Expected final verifier line:
+
+```text
+PASS_Q16_CENTRAL_SQUARE x2=6 x4=6 x6_through6=0 x6_upper=12 idempotents=28 calls=56 frontier2=2352 frontier3=92374 len4=0 len6_x2=799 len6_x4=799 len6_x6=0 central_nondiagonal_prefixes=792 group_lengths=2,2,4 controls=7 no_new_kappa
+```
+
+The script uses the standard library only. It checks the presentation,
+associativity, the unique order-2 element, the augmentation dimensions, the
+28-idempotent scan of all \(512\) matrices, call inverses, and the frontiers.
+Seven controls are required: \(\operatorname{diag}([x]^3)\) and
+\(\operatorname{diag}([y]^3)\) at length 3, the two-role square
+\(\operatorname{diag}([x^2],[x^2],[e])\) at length 4, one role of \(x^4\) at
+length 2, a dropped call failing on the dirty state
+\(([x^5],[y],[x^2]+[y])\), an \(x\)-for-\(y\) substitution failing, and the
+false relation \(y^2=e\) making the central witness fail. The rank-2 matrix
+\(E_{00}+E_{11}\) is not admitted.
+
+Recorded environment: CPython 3.12.3, `g++` 13.3.0 with `/usr/bin/c++` pointing
+at `/usr/bin/g++`, `libboost-dev` 1.83.0.1ubuntu2, and
+`research/e8-bit-p111-tuned/requirements.txt` already satisfied by a
+`python3 -m pip install --user` run. This package imports none of those
+libraries.
+
+Not run: `make frontier1094-verify`, the three-register C++ replay, a
+four-role search, and lengths 8, 10 and 12 except for the explicit length-12
+upper word. Odd lengths 7, 9 and 11 are excluded by the parity argument, not
+by enumeration.
+
+## Open next questions
+
+1. Is the minimum of \(\operatorname{diag}([x^6]^3)\) equal to 8, 10, or 12?
+   A value below the disjoint bill of 12 would be a real mixing saving in
+   this alphabet. First step: meet in the middle at length 8 on the depth-4
+   frontier. That frontier was not built; depth 3 already has 92374 matrices.
+2. Can one restored scratch role realize
+   \(\operatorname{diag}([x^4],[x^4],[x^4],[e])\) in fewer than 6 calls?
+   A scratch role is the usual route past a three-role disjoint bill. First
+   step: shortest-path search to depth 3 on the four-role alphabet of both
+   generators and every rank-one idempotent in \(M_4(\mathbb F_2)\). That
+   alphabet was not built.
+
+## Credit
+
+Predecessor model: the scalar-idempotent calls in
+`research/bit-three-register-lower-bound`, from eumemic's pinned PR #288.
+Discussion comment
+[18926868](https://github.com/CrocSwap/integer-mult-bounds/discussions/1167#discussioncomment-18926868)
+records the same presentation and releases a two-role schedule: \(x^4=y^2\),
+so two copies cost 4 by the direct word, not an assumed 8. That note does
+not determine the three-role minimum. The \(Q_8\) central-pair claims are the
+order-8 group. The semidihedral triple-square certificate uses
+\(srs=r^3\) and \(s^2=e\); its frontiers are 2051 and 68899, not these.
+Apache-2.0.
+
+Substantial AI assistance was used to build the multiplication table, search
+the paid words, and write these notes.
+
+Discussion writes for this account return FORBIDDEN / Resource not accessible
+by integration. No claim was posted, and the write was not retried.
+
+Base commit `ef5f39403610d231a0c27e8a6b4f08c10209bcd6`. Selector nonce
+`bfe60b39ce1400e38d330c639d8e904a`, card `bit-new-register-model`, scope key
+`7084092dd726e7ba1f2f`, priority revision 2.

research/q16-central-square/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "ef5f39403610d231a0c27e8a6b4f08c10209bcd6",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "c05fafaa25ba95b2d5af86b1800427198e3d20eb",
+      "sha256": "89b5928bc92c81e2647b6d702a7ad85a6795b8f44e2eb267b2ff3145d8f2b2ce"
+    }
+  ]
+}

research/q16-central-square/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "bit-new-register-model",
+  "priority_revision": 2,
+  "base_commit": "ef5f39403610d231a0c27e8a6b4f08c10209bcd6",
+  "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": "refined",
+  "actual_scope": "In F2[Q16] on exactly three roles, with paid labels {x,y} and all 28 binary rank-one idempotents, the minimum word lengths of diag([x^2]^3) and diag([x^4]^3) are both 6.",
+  "scope_change_reason": "The card allows one specified nonabelian family. The family is the generalized quaternion group of order 16 with relations x^8=e, y^2=x^4 and yxy^{-1}=x^{-1}, charged generators {x,y}, three output roles, and these two triple-square targets. The abelian M3(Q) three-register model is not reused.",
+  "bottleneck_link": "The complex primitive binds, and both identities meet rather than undercut the disjoint bill of 6, so selected kappa is unchanged. The degree-1 relaxation only forces even length and does not decide the targets. This prices a nonabelian generator alphabet outside the three-register obstruction.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once both triple squares have an exact minimum. Do not open a four-role search or a length-8 search for diag([x^6]^3), and do not claim a new kappa.",
+  "outcome": "negative-result",
+  "evidence": "Selector nonce bfe60b39ce1400e38d330c639d8e904a, scope key 7084092dd726e7ba1f2f. Verifier PASS_Q16_CENTRAL_SQUARE: frontiers 2352 and 92374; both minima 6; 799 length-6 prefix matrices for each square, 792 of the central prefixes nondiagonal; x^6 absent through length 6 with upper bound 12. No new kappa. Discussion writes are forbidden for this account and were not retried.",
+  "author": "AI-assisted contributor",
+  "ai_assistance": "Substantial AI assistance was used to build the quaternion multiplication table, search the paid words, and write the package notes.",
+  "open_questions": [
+    {
+      "question": "Is the minimum length of diag([x^6],[x^6],[x^6]) equal to 8, 10, or 12 in this same three-role alphabet?",
+      "why_it_matters": "A length below the disjoint bill of 12 would be a genuine mixing saving. The parity lemma and the length-6 sieve only leave those three candidates.",
+      "first_step": "Meet in the middle at length 8 using the depth-4 frontier obtained by extending the 92374 length-3 matrices by one call.",
+      "blocked_by": "The depth-4 frontier was not built or stored in this run."
+    },
+    {
+      "question": "Can diag([x^4],[x^4],[x^4],[e]) be realized with labels {x,y} and all rank-one idempotents in M4(F2) in fewer than 6 calls?",
+      "why_it_matters": "A restored scratch role is the usual route past a three-role disjoint bill. This package only rules out savings with exactly three roles.",
+      "first_step": "Run a shortest-path search to depth 3 on the four-role generator alphabet.",
+      "blocked_by": "That four-role alphabet and frontier were not built in this run."
+    }
+  ]
+}

research/q16-central-square/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 180,
+      "checks": "Q16 presentation and unique central involution; all 28 rank-one idempotents; augmentation dimensions; exhaustive absence of words of length at most 5 for both triple squares; length-6 coordinate witnesses; x^6 sieve through length 6 and the length-12 word; seven positive and negative controls."
+    }
+  ]
+}