Exact lengths for three modular-group generator triples

research/m16-generator-triple/README.md

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+# Modular generator triples meet the disjoint bill
+
+**Finite register-model certificate; no new κ.** The selected result, both
+primitive suppliers, and `upstream/` are unchanged. The complex primitive
+binds. These identities meet their disjoint bills, so register mixing does
+not undercut them.
+
+In the group algebra \(\mathbb F_2[M_{16}]\),
+
+\[
+M_{16}=\langle a,b\mid a^8=e,\ b^2=e,\ bab=a^5\rangle,
+\]
+
+with exactly three roles and paid calls
+
+\[
+D_N([g])=I+([g]+[e])N,
+\qquad g\in\{a,b\},
+\]
+
+the minimum word lengths are
+
+| Target | Disjoint bill | Minimum |
+|---|---:|---:|
+| \(\operatorname{diag}([a^2],[a^2],[a^2])\) | 6 | 6 |
+| \(\operatorname{diag}([a^3],[a^3],[a^3])\) | 9 | 9 |
+| \(\operatorname{diag}([a^5],[a^5],[a^5])\) | 9 | 9 |
+
+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. Elements
+are encoded by \(a^k=k\) and \(a^kb=8+k\).
+
+The changed hypothesis is the address group. The abelian three-register model
+uses arbitrary rational rank-one frames and concludes that nine calls are
+necessary for three invertible lower-triangular targets. This alphabet uses
+only the two modular generators and charges every call. There is no free
+scalar mixing outside the 28 idempotents and no fourth role. The group is not
+the generalized quaternion or semidihedral group of order 16: its center is
+\(\{e,a^2,a^4,a^6\}\) and its derived subgroup is \(\{e,a^4\}\).
+
+## Priced baseline
+
+Generator lengths, from the right Cayley graph on \(\{a,b\}\), computed
+before the matrix search:
+
+| Element | Shortest generator word | Disjoint three-role bill |
+|---|---|---:|
+| \(a^2\) | \(aa\) | 6 |
+| \(a^3\) | \(aaa\) | 9 |
+| \(a^4\) | \(aaaa\) | 12 |
+| \(a^5\) | \(bab\) | 9 |
+| \(a^6\) | \(baba=(ab)^2\) | 12 |
+
+The degree-1 relaxation does not decide these bills. The augmentation powers
+of \(\mathbb F_2[M_{16}]\) have dimensions \(15,13,11,9,7,5,3,1,0\). The
+quotient \(\Delta/\Delta^2\) has basis \(\alpha=[a]+1\) and \(\beta=[b]+1\).
+Odd powers of \(a\) are congruent to \(\alpha\); \(a^2\), \(a^4\) and \(a^6\)
+lie in \(\Delta^2\); \(b\) is congruent to \(\beta\).
+
+Every paid deviation \(S=([g]+1)N\) has entries in \(\Delta\). For every pair
+of paid calls, the cross term \(S_iS_j\) has entries in \(\Delta^2\). By
+induction on those pairs, a product of calls is
+
+\[
+I+\alpha\sum_{a\text{-calls}}N_j+\beta\sum_{b\text{-calls}}N_j
+\]
+
+modulo \(M_3(\Delta^2)\). Both odd triples are congruent to \(I+\alpha I\).
+Each admitted projector has trace 1 and \(I\) has trace 1, so every word for
+either odd triple has an odd number of \(a\)-calls and an even number of
+\(b\)-calls. Even lengths are impossible. The same congruence gives only an
+even-length floor for \(a^2\), \(a^4\) and \(a^6\).
+
+## Result
+
+Lengths 1 through 4 were enumerated in \(M_3(\mathbb F_2[M_{16}])\). The exact
+frontiers have sizes 56, 2052, 68955 and 2199152. Meet-in-the-middle on these
+frontiers finds no word of length at most 7 for either odd triple. The
+coordinate words
+
+- \((E_{00},a)\) three times, then the same for \(E_{11}\) and \(E_{22}\),
+- \((E_{00},b),(E_{00},a),(E_{00},b)\), then the same for \(E_{11}\) and \(E_{22}\),
+
+equal the two odd targets and have length 9. Both minima are therefore 9.
+
+The same frontiers find \(\operatorname{diag}([a^2]^3)\) at length 6 and not
+earlier, realized by two \(a\)-calls on each coordinate projector. The
+disjoint bill of 6 is sharp.
+
+Separately, \(\operatorname{diag}([a^4]^3)\) and \(\operatorname{diag}([a^6]^3)\)
+have no word of length at most 7. The symbol lemma already excludes the odd
+lengths. The coordinate words above have length 12. Both exact minima
+therefore lie in \(\{8,10,12\}\) and are not claimed to be sharp.
+
+## Reproduce
+
+From the repository root, with Python 3.12 and `PYTHONOPTIMIZE` unset:
+
+```sh
+python3 -B research/m16-generator-triple/verify.py
+python3 -B scripts/research_package.py --package research/m16-generator-triple
+```
+
+Expected final verifier line:
+
+```text
+PASS_M16_GENERATOR_TRIPLE a2=6 a3=9 a5=9 a4_through7=0 a6_through7=0 a4_upper=12 a6_upper=12 idempotents=28 calls=56 frontier2=2052 frontier3=68955 frontier4=2199152 center=4 derived=2 controls=7 no_new_kappa
+```
+
+The script uses the standard library only. It checks the presentation,
+associativity, the order-4 center, the derived subgroup, the augmentation
+dimensions, the 28-idempotent scan of all 512 matrices, call inverses,
+associativity of the paid calls and of their inverse-label calls, the
+\(\Delta^2\) cross-term identity for every pair, and the frontiers. Seven
+controls are required: the generator triples \(\operatorname{diag}([a]^3)\)
+and \(\operatorname{diag}([b]^3)\) at length 3; the two-role square
+\(\operatorname{diag}([a^2],[a^2],[e])\) at length 4, one role of \(a^4\) at
+length 4 and one role of \(a^5\) at length 3; a dropped call failing on the
+dirty state \(([a^5]+[b],[a^2],[a^4]+[a^3b])\); an \(a\)-for-\(b\) substitution
+failing; the dihedral relation \(bab=a^7\) making the \(a^5\) witness fail;
+and one explicit seven-call word whose matrix is first reached at length 7.
+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
+`python3 -m pip install --user` (numpy 2.5.4, sympy 1.14.0, scipy 1.18.1).
+`PYTHONOPTIMIZE` was unset. This package imports none of those libraries.
+Direct verifier runtime on this machine was about 60 seconds.
+
+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 words. Odd lengths for the \(a^4\) and \(a^6\) triples are excluded by
+the symbol argument, not by a separate search beyond the length-7 sieve.
+
+## Open next questions
+
+1. Is the minimum of \(\operatorname{diag}([a^4]^3)\) equal to 8, 10, or 12?
+   The same question applies to \(\operatorname{diag}([a^6]^3)\). A value
+   below 12 would be a real mixing saving in this alphabet. First step:
+   store one length-4 word for each of the 2,199,152 depth-4 matrices and
+   test the two targets by inverse meet-in-the-middle. This package builds
+   that set for the length-7 sieve and does not store the words.
+2. Can one restored scratch role realize
+   \(\operatorname{diag}([a^5],[a^5],[a^5],[e])\) in fewer than 9 calls?
+   A scratch role is the usual route past a three-role disjoint bill. First
+   step: shortest-path search 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,
+section 1.2. That theorem concerns rational rank-one frames, not this
+generator alphabet. Quaternion and semidihedral order-16 certificates use
+different relations and have center of order 2; their frontiers and targets
+are not reused. 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
+`a1330c55bf86b0ea1c2eb0ac57772471`, card `bit-new-register-model`, scope key
+`7084092dd726e7ba1f2f`, priority revision 2. Catalogue SHA-256
+`89b5928bc92c81e2647b6d702a7ad85a6795b8f44e2eb267b2ff3145d8f2b2ce`.
+Selected-record SHA-256
+`10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.

research/m16-generator-triple/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"
+    },
+    {
+      "path": "research/bit-three-register-lower-bound/README.md",
+      "usage": "provenance",
+      "git_blob": "25e41c370c101a33a6deb978de08b03ed7539b79",
+      "sha256": "c12b33ef388deece0cbf891505a2ce2c0779a0c3a6cf1f0f7f7039bab59cbebd"
+    }
+  ]
+}

research/m16-generator-triple/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[M16] on exactly three roles, with paid labels {a,b} and all 28 binary rank-one idempotents, the minimum word lengths of diag([a^2]^3), diag([a^3]^3) and diag([a^5]^3) are 6, 9 and 9.",
+  "scope_change_reason": "The card allows one specified nonabelian family. The family is the modular group of order 16 with relations a^8=e, b^2=e and bab=a^5, charged generators {a,b}, three output roles, and these three diagonal targets. The abelian M3(Q) three-register model is not reused.",
+  "bottleneck_link": "The complex primitive binds. These identities meet rather than undercut the disjoint generator bills, so the bit primitive and the selected kappa do not move. A finite-address sample would not have been enough; the certificate is equality in M_3(F2[M16]).",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once the three named triples are priced through length 7, the symbol parity lemma is checked on every paid pair, and the length-6 and length-9 coordinate witnesses are checked. Do not build a four-role alphabet or enumerate lengths 8, 10 and 12.",
+  "outcome": "completed",
+  "evidence": "PASS_M16_GENERATOR_TRIPLE a2=6 a3=9 a5=9 a4_through7=0 a6_through7=0 a4_upper=12 a6_upper=12 idempotents=28 calls=56 frontier2=2052 frontier3=68955 frontier4=2199152 center=4 derived=2 controls=7 no_new_kappa",
+  "author": "AI-assisted contributor",
+  "ai_assistance": "Substantial AI assistance was used to build the modular-group multiplication table, search the paid words, and write the package notes.",
+  "open_questions": [
+    {
+      "question": "Is the minimum of diag([a^4]^3) equal to 8, 10, or 12, and is the minimum of diag([a^6]^3) the same set?",
+      "why_it_matters": "A value below the disjoint bill of 12 would be a mixing saving inside this charged generator alphabet.",
+      "first_step": "Store one length-4 word for each of the 2199152 depth-4 matrices and test both targets by inverse meet-in-the-middle.",
+      "blocked_by": "The length-7 sieve builds the depth-4 set and does not store words. Lengths 8, 10 and 12 were not enumerated. Odd lengths are already excluded by the degree-1 symbol."
+    },
+    {
+      "question": "Can one restored scratch role realize diag([a^5],[a^5],[a^5],[e]) in fewer than 9 calls?",
+      "why_it_matters": "A scratch role is the usual route past a three-role disjoint bill that mixing among the three output roles did not beat.",
+      "first_step": "Shortest-path search to depth 4 on four roles, using both generators and every rank-one idempotent in M_4(F2).",
+      "blocked_by": "That four-role alphabet was not built."
+    }
+  ]
+}

research/m16-generator-triple/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": 300,
+      "checks": "M16 presentation, order-4 center and derived subgroup; all 28 rank-one idempotents; augmentation dimensions and degree-1 classes; Delta^2 cross terms for every paid pair; exhaustive absence through length 7 for the a^3 and a^5 triples and the sharp length-9 witnesses; a^2 minimum 6; a^4 and a^6 absence through length 7 with length-12 upper words; seven positive and negative controls."
+    }
+  ]
+}