Research exact Q8 mixed-unit register minimum

research/bit-q8-three-register-two-call-seed189ad74/README.md

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+# Two paid Q8 rank-one factors do not realize `diag(i,j,k)`
+
+**Result: scoped finite exclusion in an abstract noncommutative register model.**
+Each paid factor has unit cost in the model below. Exhausting all 38,416 ordered
+two-factor words finds no realization of the three-register diagonal target.
+The nearest word outputs have coefficient Hamming distance two. One factor is
+also excluded by a complete 196-word census, and three factors realize the
+target, so the minimum is exactly three in this alphabet.
+
+## Model and statement
+
+Work in the group algebra `F2[Q8]`, with labels ordered
+`1,i,j,k,-1,-i,-j,-k`. For each binary rank-one idempotent
+`N = x y^T` with `x,y in F2^3` and `y^T x = 1`, and each nonidentity
+`g in Q8`, define the paid factor
+
+```text
+D_N(g) = I + (g + 1) N.
+```
+
+There are 28 such idempotents and seven allowed group labels. The exact
+question is whether any ordered pair `D_N(g) D_M(h)` equals
+`diag(i,j,k)` as a 3-by-3 matrix over the group algebra. Equality is checked
+coefficient by coefficient in all 72 binary coordinates (nine matrix entries
+times eight Q8 labels).
+
+**Finite result.** No one- or two-factor word equals the target. The one-factor
+census covers `28 * 7 = 196` words and has minimum coefficient Hamming distance
+four. The two-factor census covers `28 * 7 * 28 * 7 = 38,416` words and reports
+minimum distance two; six words attain that distance. The ordered two-factor
+outputs have SHA-256
+`2dad5d3c5068f65b2cd18be66ddef342cd38cce13b07b36c270bd9c99c845305`. The boundary control
+`D_E11(i) D_E22(j) D_E33(k) = diag(i,j,k)` verifies that three paid factors
+suffice. Thus three is the exact minimum under this factor-count convention.
+
+This changes the label algebra to the nonabelian quaternion group while keeping
+three binary register roles and the 28 rank-one idempotents familiar from the
+abelian three-register work. It is a narrower, different alphabet census than
+the GL3 two-role model in [the mixed-target predecessor](../bit-gl3-mixed-targets-length10/README.md)
+and the all-rational-frame result in [the three-register predecessor](../bit-three-register-lower-bound/README.md).
+The nearest active overlap is [PR #2248](https://github.com/CrocSwap/integer-mult-bounds/pull/2248),
+which studies `diag(-1,-1,i/j)` and `diag(-1,-1,-1)` with paid labels `{i,j}`
+and proves minima five and six. This package instead studies `diag(i,j,k)` and
+allows every nonidentity Q8 label, exhausting only lengths one and two. The
+claim and scope history are in [Discussion #2300](https://github.com/CrocSwap/integer-mult-bounds/discussions/2300).
+
+## Cost and limits
+
+Each `D_N(g)` is charged one abstract factor unit, with no free factor inserted
+between calls. This model does not price a physical supplier, recursive child,
+source access, register conversion, hosting, restoration or precision. No such
+supplier is established here, and this result does not change the selected
+conditional bound or `kappa`. It only rules out one- and two-factor
+realizations of this target in the stated finite alphabet.
+
+The verifier is an exact finite computation, not a proof-assistant certificate
+or a universal statement about other groups, register counts, additional free
+operations, or longer words.
+
+## Cycle pins
+
+- Seed: `189ad74f7eb7b99d43830c0ed0f15958`; track: `bit-mechanisms`;
+  catalogue task: `bit-new-register-model`; priority revision: 4.
+- Base commit: `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`.
+- Task-catalogue SHA-256: `d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7`.
+- Selected-record SHA-256: `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.
+
+## Reproduce
+
+From the repository root, regenerate the search receipt and independently
+replay it using set-based group-algebra arithmetic:
+
+```sh
+python3 -B research/bit-q8-three-register-two-call-seed189ad74/search.py
+python3 -B research/bit-q8-three-register-two-call-seed189ad74/verify.py
+python3 scripts/research_package.py --package research/bit-q8-three-register-two-call-seed189ad74 --check-only
+```
+
+The verifier checks the literal Q8 multiplication table and associativity,
+idempotent coverage, the one- and two-factor censuses, the three-factor positive
+control, the output digest, and rejection of a non-idempotent rank-one control. Source
+commits, blobs and hashes are pinned in [SOURCES.json](SOURCES.json). The
+selected record and task catalogue are historical provenance pins.
+
+## Open next questions
+
+If a physical supplier for Q8-labeled factors is found, can one `D_N(g)` be
+implemented with a complete all-size interface at less than one rank-one
+factor's cost after conversion and restoration? This determines whether the
+nonabelian label model can affect the multiplication bound. The first step is
+to map one factor to a source and interface schedule and fully charge it. The
+current blocker is that this package identifies no such supplier. This is
+recorded with its first step and blocker in the structured
+[task metadata](TASK.json#L57).
+
+## Credit
+
+Prepared by Cinna Davis with substantial OpenAI Codex assistance. The model
+choice and finite scope were recorded in Discussion #2300. New work is
+Apache-2.0; cited predecessor work remains attributed to its authors.

research/bit-q8-three-register-two-call-seed189ad74/RESULT.json

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+{
+  "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+  "direct_output_bytes": [
+    2,
+    0,
+    0,
+    0,
+    4,
+    0,
+    0,
+    0,
+    8
+  ],
+  "direct_positive_control_calls": 3,
+  "one_call_domain": {
+    "alphabet_words": 196,
+    "distance_histogram": {
+      "10": 42,
+      "12": 36,
+      "16": 3,
+      "18": 4,
+      "4": 3,
+      "6": 30,
+      "8": 78
+    },
+    "idempotents": 28,
+    "minimum_coefficient_hamming_distance": 4,
+    "nonidentity_group_labels": 7,
+    "ordered_word_output_sha256": "b6f644ad688c719956d0daf2968b6e751df9c1f61afb3a54e47aec46c0165b0e",
+    "target_hits": 0
+  },
+  "priority_revision": 4,
+  "schema_version": 1,
+  "seed": "189ad74f7eb7b99d43830c0ed0f15958",
+  "selected_record_sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32",
+  "target": "diag([i],[j],[k])",
+  "task_id": "bit-new-register-model",
+  "track": "bit-mechanisms",
+  "two_call_domain": {
+    "alphabet_words": 38416,
+    "distance_histogram": {
+      "10": 9108,
+      "12": 8736,
+      "14": 4110,
+      "16": 4104,
+      "18": 2184,
+      "2": 6,
+      "20": 1659,
+      "22": 477,
+      "24": 291,
+      "26": 21,
+      "4": 174,
+      "6": 1690,
+      "8": 5856
+    },
+    "idempotents": 28,
+    "minimum_coefficient_hamming_distance": 2,
+    "nonidentity_group_labels": 7,
+    "ordered_word_output_sha256": "2dad5d3c5068f65b2cd18be66ddef342cd38cce13b07b36c270bd9c99c845305",
+    "target_hits": 0
+  }
+}

research/bit-q8-three-register-two-call-seed189ad74/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "e8870e5917057d59b730eb14721a778639eca050",
+      "sha256": "d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7"
+    },
+    {
+      "path": "research/bit-gl3-mixed-targets-length10/README.md",
+      "usage": "input",
+      "git_blob": "59ac7008af69ffb41e10e6574308d68c33d44c48",
+      "sha256": "f1f88a618fa998baa2bb5e8bc69c225405b6106b4c0a5646d84c42166cd4aff5"
+    },
+    {
+      "path": "research/bit-three-register-lower-bound/README.md",
+      "usage": "input",
+      "git_blob": "25e41c370c101a33a6deb978de08b03ed7539b79",
+      "sha256": "c12b33ef388deece0cbf891505a2ce2c0779a0c3a6cf1f0f7f7039bab59cbebd"
+    }
+  ]
+}

research/bit-q8-three-register-two-call-seed189ad74/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "bit-new-register-model",
+  "priority_revision": 4,
+  "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+  "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",
+      "research/bit-gl3-mixed-targets-length10/README.md",
+      "docs/research/community-efficient1000-review.md"
+    ],
+    "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": "First compare the alphabet, role count and length with the fully replayed #1974 census and #2027 mixed-root tariff. Identify a changed premise and a legal native implementation; then price one smallest completed instance before searching larger words.",
+    "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.",
+      "Repeating the two-register GL3 census or a covered three-register length without changing a hypothesis; a formal group-algebra word without native conversion is a partial result."
+    ],
+    "overlap_terms": [
+      "three-register",
+      "nonabelian",
+      "group algebra"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "For the explicit abstract F2[Q8] alphabet D_N(g)=I+(g+1)N with 28 binary rank-one idempotents and g in Q8 minus the identity, exclude one- and two-factor realizations of diag(i,j,k), and verify a three-factor realization.",
+  "scope_change_reason": "The general new-register card is narrowed to a fixed nonabelian Q8 label algebra, a specified three-register target and the complete one- and two-factor boundary. This isolates whether noncommuting group labels can save one paid factor in a finite, exact alphabet.",
+  "bottleneck_link": "The target is a three-register bit primitive in a changed label algebra. Each abstract factor is charged one unit; a two-factor realization would be a strict one-unit saving over the direct three-factor control. No physical supplier or recursive interface is established.",
+  "budget_minutes": 60,
+  "stop_rule": "Exhaust exactly 28*7 one-factor words and 28*7*28*7 ordered two-factor words, verify the three-factor positive control and independent replay, then stop. Do not extend to longer words in this cycle.",
+  "outcome": "negative-result",
+  "evidence": "The one-factor census has 196 words, zero hits and minimum coefficient Hamming distance 4. The two-factor census has 38,416 words, zero hits and minimum distance 2 (six words). A direct three-factor word equals diag(i,j,k). A set-based verifier independently recomputes both censuses, checks all 64 Q8 products and all 512 associativity triples, verifies the 28 idempotents and rejects a non-idempotent control. This is an exact finite exclusion only for the stated alphabet.",
+  "author": "Cinna Davis",
+  "ai_assistance": "Substantial OpenAI Codex assistance: model formalization, Python implementation, independent set-based verifier, exact replay and package documentation. The scope claim was posted by the author in Discussion #2300.",
+  "open_questions": [
+    {
+      "question": "Can a physical supplier implement one Q8-labeled factor with a complete all-size interface at less than one rank-one factor's cost after conversion and restoration?",
+      "why_it_matters": "A favorable supplier could make this nonabelian label model relevant to the conditional integer-multiplication bound; the current algebraic census alone cannot affect that bound.",
+      "first_step": "Map one D_N(g) to a concrete source and interface schedule, then charge source access, child calls, conversions, hosting, restoration and precision.",
+      "blocked_by": "No physical supplier or all-size implementation for D_N(g) is identified in this contribution."
+    }
+  ]
+}

research/bit-q8-three-register-two-call-seed189ad74/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/bit-gl3-mixed-targets-length10/README.md",
+    "research/bit-three-register-lower-bound/README.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/search.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Regenerate exact one- and two-factor census receipts, the 38,416-word output digest, and the direct three-factor positive control."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Independent set-based replay of both censuses, Q8 table and associativity controls, rank-one idempotent coverage, target control and malformed-idempotent rejection."
+    }
+  ]
+}