Exact length 6 for the semidihedral triple square

research/sd16-triple-square/README.md

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+# Semidihedral triple square: exact length 6
+
+**Scoped negative result; no new κ.** In the three-role semidihedral register
+model below, `diag([r^2], [r^2], [r^2])` has minimum paid length 6. The
+matching uniform identity is the doubled coordinate word. Reflections are in
+the paid alphabet and do not make a shorter word. The selected frontier is
+unchanged.
+
+This does not escape the scoped three-register obstruction. It prices one
+nonabelian alphabet outside that obstruction and shows that this target does
+not undercut its disjoint bill.
+
+## Scope
+
+Assignment: track `bit-mechanisms`, card `bit-new-register-model`, selector
+scope key `7084092dd726e7ba1f2f`, OS nonce `6f7a9ea6cd4e657dcf8c3a81f9840d4a`.
+Exact base `ef5f39403610d231a0c27e8a6b4f08c10209bcd6`. Selected-record SHA-256
+`10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`. Catalogue
+priority revision 2.
+
+**Changed hypothesis.** The predecessor in
+[research/bit-three-register-lower-bound](../bit-three-register-lower-bound/README.md)
+uses the additive abelian group `M_3(Q)` and forbids arbitrary free address
+maps between calls. This package replaces that address group by the
+semidihedral group of order 16,
+
+```text
+SD16 = <r, s | r^8 = s^2 = e, s r s = r^3>.
+```
+
+The relation is not the dihedral relation `s r s = r^{-1}`. Scope status is
+**refined**: the card allows one specified nonabelian family, and the family
+here is exactly this presentation, the charged labels `{r, s}`, and three
+output roles.
+
+**Roles and alphabet.** There are three roles and no scratch role. A paid call
+is
+
+```text
+D_N([g]) = I + ([g] + [e]) N,
+```
+
+where `g` is `r` or `s` and `N = x y^T` runs through the 28 binary rank-one
+idempotents on `F_2^3` (`y^T x = 1`). The alphabet has 56 calls. A final
+matrix equality in `M_3(F_2[SD16])` is an operator identity, so it holds for
+every dirty input state in `(F_2[SD16])^3`.
+
+**Target.** `diag([r^2], [r^2], [r^2])`.
+
+## Priced baselines
+
+| Bill | Length | What it says |
+|---|---:|---|
+| Disjoint diagonal word, two `r`-calls on each `E_ii` | 6 | Achieved by the identity below |
+| First-order abelianization, ignoring the `αβ` term | 2 | Necessary for the image to be `I`, and not sufficient |
+| Exact minimum in the 56-call alphabet | 6 | Shortest-word search |
+
+`SD16 / <r^2> ≅ C_2 × C_2`. Write `α = [x] + 1` and `β = [y] + 1` for the
+images of `r` and `s`. Then `α^2 = β^2 = 0`, while `αβ ≠ 0`. Every word maps
+to `I + α (∑ N_r) + β (∑ N_s) + αβ K`. The target maps to `I`, so both
+projector sums must vanish and so must `K`. Vanishing sums already force
+length at least 2. That optimistic floor is the priced relaxation. It does
+not meet the disjoint bill, so the exact search is what decides the target.
+
+## Uniform identity
+
+Each call `D_{E_ii}([g])` is diagonal, with `[g]` in position `i` and `[e]`
+elsewhere. Diagonal entries therefore multiply in call order. The word
+
+```text
+(E11, r), (E11, r), (E22, r), (E22, r), (E33, r), (E33, r)
+```
+
+has diagonal `[r][r] = [r^2]` in every role. The same reasoning gives two
+companion identities used as controls and upper bounds:
+
+- `(E11, r)` twice equals `diag([r^2], [e], [e])`.
+- `(E11, s), (E11, r), (E11, r), (E11, s)` equals `diag([r^6], [e], [e])`,
+  because `s r^2 s = r^6`. Two pure rotations do not isolate `[r^6]`.
+- Four rotations on one `E_ii` equal `[r^4]` there. Repeating either long
+  element on all three roles costs 12.
+
+The length-4 reflection word is the smallest witness that the semidihedral
+operation is not idle: `[r^6]` on one role has minimum length 4.
+
+## Exact minimum
+
+A shortest-path enumeration of the 56 calls produces 56, 2,051 and 68,899
+matrices at lengths 1, 2 and 3. The target is absent. Every length-2 and
+length-3 matrix in that frontier is invertible as an operator on
+`(F_2[SD16])^3`. Splitting a hypothetical shorter word into shortest halves
+gives no length-4 or length-5 factorization (0 pairs). The same split at
+length 6 finds 799 ordered pairs, in agreement with the explicit word.
+
+The search is a minimum-length argument. If a word of length `L ≤ 6` exists,
+some word of length `ℓ = min length` exists, and that shortest word splits
+into shortest halves of lengths `(2,2)`, `(3,2)` or `(3,3)`. Those halves lie
+in the enumerated frontiers. One matrix reachable by a two-call word is the
+identity itself: every repeated reflection square `(I + ([s] + [e]) N)^2`
+equals `I`. Exact length 2 therefore has 2,052 distinct matrices, one more
+than the shortest frontier.
+
+The same length-6 sieve shows that `diag([r^4], [r^4], [r^4])` and
+`diag([r^6], [r^6], [r^6])` have **no** word of length at most 6. The
+disjoint words give upper bound 12. Those two bounds are not claimed to be
+sharp.
+
+A positive sieve control keeps the empty counts meaningful:
+`diag([r], [r], [r])` has minimum length 3, and the length-5 split still
+finds 196 non-minimal factorizations of it.
+
+## What is not claimed
+
+- No new κ and no change to the selected record. The complex primitive binds.
+  This note does not alter either supplier. `make frontier1094-verify` was
+  not run.
+- No four-role or scratch-register statement. Three roles are the whole model.
+- No claim about dihedral groups, quaternion groups, or other presentations.
+  The dihedral order-16 reflection model (central involution, two roles) and
+  the D8 central-commutator models are different groups, alphabets or targets.
+- The length-6 absence for the two longer triples is not a minimum. Lengths
+  7 through 11 were not searched.
+- Finite `F_2`-column checks are not the certificate. The certificate is
+  equality of all nine group-algebra entries, recomputed by the diagonal
+  group word for the explicit identities and by the operator search for the
+  lower bound.
+- This is not a proof-assistant formalization and not an integer-multiplication
+  theorem.
+
+## Reproduce
+
+Standard library only, from the repository root. `PYTHONOPTIMIZE` is unset;
+the checks are explicit branches, not assertions.
+
+```sh
+python3 -B research/sd16-triple-square/verify.py
+python3 -B scripts/research_package.py --package research/sd16-triple-square --check-only
+```
+
+Expected final line of the verifier:
+
+```text
+PASS_SD16_TRIPLE_SQUARE length=6 frontier=1,56,2051,68899 exact2=2052 len4=0 len5=0 len6=799 r4_through6=0 r6_through6=0 generator_len5_hits=196 rejected=6 no_new_kappa
+```
+
+The run is about 20 seconds on CPython 3.12. The registered replay command is
+the same verifier. Session toolchain, not imported by this package: `c++`
+already selected `/usr/bin/g++`, `libboost-dev` 1.83.0.1ubuntu2 was present,
+and `research/e8-bit-p111-tuned/requirements.txt` was installed with
+`python3 -m pip install --user`. The positive control
+`python3 -B scripts/research_baseline.py` returned
+`PASS_RESEARCH_BASELINE_BINDINGS_ONLY`.
+
+Rejected fixtures: a dropped final call, an `r → s` substitution, a projector
+moved onto a role that already has its two calls, every trace-zero `x y^T`,
+the dihedral relation `s r s = r^7`, and the dirty state
+`([r^5], [s], [r^2] + [s])` on the dropped word. Accepted controls: the
+length-6 word, the single-role square, the reflection square back to `I`,
+the length-4 conjugation, the length-12 upper bounds, inverse roundtrip, and
+the 196 length-5 hits on the generator triple.
+
+## Open next questions
+
+1. **One restored scratch role.** Can `diag([r^2], [r^2], [r^2], [e])` be
+   realized in `M_4(F_2[SD16])` with the same labels and all 120 rank-one
+   idempotents in fewer than 6 calls? A scratch role is the usual way past a
+   three-role disjoint bill; the D8 central-pair precedent needed an extra
+   role for a different target. First step: shortest-path BFS to depth 3 on
+   the 240-call alphabet. Blocker: that depth-3 set was not built here, so
+   this package claims nothing about four roles.
+
+2. **Sharp length of the longer triples.** Is the minimum of
+   `diag([r^4], [r^4], [r^4])`, and likewise `[r^6]`, equal to 12 or to some
+   length in 7..11? A value below 12 would be the first triple-target saving
+   in this alphabet. The length-6 sieve already excludes 1..6, and the
+   disjoint words give 12. First step: meet in the middle at length 8 on the
+   depth-4 frontier. Blocker: that frontier has 2,198,276 matrices and was
+   not inverted here.
+
+## Credit and coordination
+
+The abelian three-register model and its nine-call theorem are the pinned
+predecessor, including eumemic's PR #288 scalar-idempotent model and the
+local package above. Nearby register-model work uses different groups or
+targets: D8 central commutators, the order-eight dihedral reflection alphabet
+on two roles with target `[r^4]`, odd-dihedral joint tariffs, and the Dic3
+generator alphabet. Those results are not inputs to this search.
+
+Overlap refresh at 2026-10-18T12:23:05Z covered 348 open PRs and 2,454
+discussion records with complete pagination. The exact terms `semidihedral`
+and `SD16` matched only this package. Closed PRs are visible to that search
+only through local notes and the ledger.
+
+Discussion writes for this account return FORBIDDEN / Resource not accessible
+by integration. No claim was posted, and the write was not retried.
+
+Prepared by an AI-assisted contributor. Substantial AI assistance was used to
+set up the multiplication table, implement the shortest-word search, and write
+these notes. New work is Apache-2.0.

research/sd16-triple-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"
+    },
+    {
+      "path": "research/bit-three-register-lower-bound/README.md",
+      "usage": "provenance",
+      "git_blob": "25e41c370c101a33a6deb978de08b03ed7539b79",
+      "sha256": "c12b33ef388deece0cbf891505a2ce2c0779a0c3a6cf1f0f7f7039bab59cbebd"
+    }
+  ]
+}

research/sd16-triple-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[SD16] on exactly three roles, with paid labels {r,s} and all 28 binary rank-one idempotents, the minimum word length of diag([r^2],[r^2],[r^2]) is 6.",
+  "scope_change_reason": "The card allows one specified nonabelian family. The family is the semidihedral group of order 16 with relation srs=r^3, charged generators {r,s}, three output roles and this one target. The abelian M3(Q) three-register model is not reused.",
+  "bottleneck_link": "The bit primitive does not bind, and this identity meets rather than undercuts the disjoint bill of 6, so selected kappa is unchanged. It prices a nonabelian alphabet outside the three-register obstruction: reflections do not reduce this triple-square target.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once the exact minimum for diag([r^2],[r^2],[r^2]) is known. Do not open a four-role search or a kappa claim. The r^4 and r^6 triple sieves stop at length 6.",
+  "outcome": "negative-result",
+  "evidence": "Selector nonce 6f7a9ea6cd4e657dcf8c3a81f9840d4a, scope key 7084092dd726e7ba1f2f. Verifier PASS_SD16_TRIPLE_SQUARE: shortest frontiers 56, 2051 and 68899; length-4 and length-5 factorizations 0; length-6 factorizations 799; explicit doubled coordinate word. Abelianization relaxation floor 2. 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 set up the semidihedral multiplication table, implement the shortest-word search, and write the package notes.",
+  "open_questions": [
+    {
+      "question": "Can diag([r^2],[r^2],[r^2],[e]) be realized in M4(F2[SD16]) with labels {r,s} and all 120 rank-one idempotents 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 240-call four-role alphabet and test the padded target.",
+      "blocked_by": "That four-role frontier was not built in this run."
+    },
+    {
+      "question": "What is the exact minimum length of diag([r^4],[r^4],[r^4]) and of diag([r^6],[r^6],[r^6]) in the same three-role alphabet?",
+      "why_it_matters": "A length below the disjoint bill of 12 would be a genuine triple-target saving. The length-6 sieve only excludes lengths 1 through 6.",
+      "first_step": "Meet in the middle at length 8 using the depth-4 frontier of 2198276 matrices.",
+      "blocked_by": "The depth-4 frontier was counted in a scratch run but not inverted or retained as a certificate."
+    }
+  ]
+}

research/sd16-triple-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": 120,
+      "checks": "Semidihedral relation, 56-call alphabet, length-6 triple-square identity, dirty-state and mutation controls, shortest frontiers, and length-4/5/6 factorizations"
+    }
+  ]
+}