Close the 24 coordinate-kernel span-six pools by a 3+1 restriction

research/coordinate-kernel-span-six-closure/PROOF.md

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+# Closing the 24 coordinate-kernel pools at span six
+
+Prepared for Nicholas Stone with substantial OpenAI Codex assistance, 2026-10-25.
+New material: Apache-2.0. This is a written reduction using the explicitly cited
+predecessor lemma, with finite exact corroboration, not a proof-kernel theorem.
+
+## Statement
+
+In the four-scalar-register formal abelian BIT model, no eight-call word of
+rational address span six with noncoordinate projectors in a coordinate
+permutation of (0x0003,0x0022,0x0550) can have a diagonal monomial endpoint
+with three invertible rational 3-by-3 target addresses. The fourth endpoint
+address is arbitrary and all four input registers are arbitrary.
+
+A paid factor is D_N([A])=I+([A]-1)N over F2[G], where G is the additive
+group of rational 3-by-3 matrices, rank_Q(A)=1, and N=xy^t is a binary
+rank-one idempotent. Coordinate-projector calls may be interleaved arbitrarily.
+No arbitrary scalar basis change of a diagonal endpoint is made in this proof.
+
+## Inherited lemma
+
+David Govea's PR720, research/three-register-no-target-flag/README.md sections
+1--4, states: for a three-register word with torsion-free abelian exponents,
+relation lattice of rank at most two, and diagonal monomial endpoint, the
+three endpoint exponents are sums of pairwise disjoint subsets of the paid
+exponents. In particular their matrix ranks sum to at most the sum of the
+paid matrix ranks. No invertibility or target flag is required for this lemma.
+This is the low-corank lemma, not PR720's span-five exhaustive census.
+Its source and inherited PR468/PR288/PR203 attribution are recorded in SOURCES.json.
+The present package does not independently reprove that predecessor lemma.
+
+## 1. A literal coordinate block
+
+For packed bit 4i+j denoting entry (i,j), the representative is
+
+    X = [[1,1,0,0], [0,0,0,0], [0,0,0,0], [0,0,0,0]],
+    Y = [[0,1,0,0], [0,1,0,0], [0,0,0,0], [0,0,0,0]],
+    Z = [[0,0,0,0], [1,0,1,0], [1,0,1,0], [0,0,0,0]].
+
+Each has zero fourth row AND zero fourth column. Every coordinate projector
+also preserves the coordinate decomposition F2^4=F2^3 direct-sum F2.
+Thus EVERY allowed factor, including every interleaved coordinate factor,
+is block diagonal in this decomposition. E_33 acts only on the singleton;
+X,Y,Z,E_00,E_11,E_22 act only on the triple. Restricting a triple factor
+leaves a nonzero rank-one idempotent. No dirty input is discarded: the
+decomposition is an identity of full operators on arbitrary inputs.
+
+The 24 pools are coordinate permutations of this representative (PR5231).
+A coordinate permutation preserves diagonality and merely relocates the
+singleton. It need not preserve which coordinate is auxiliary, and the
+argument below makes no such assumption.
+
+## 2. Relation rank cannot grow on restriction
+
+Let J be the set of triple-block calls and K the singleton calls, with
+|J|+|K|=8. The full rational relation space is
+
+    L_Q = {c in Q^8 : sum_j c_j A_j=0},   dim L_Q=8-6=2.
+
+Every relation among (A_j)_{j in J}, extended by zeros on K, lies in L_Q.
+This is an injection, so the triple subword's relation rank is at most two.
+The generated address group is finitely generated and torsion free, so its
+integral relation lattice has this same rank. Relations involving both
+blocks do not invalidate this injection. No direct-sum assumption about
+their ADDRESS spans is needed.
+
+## 3. Rank budget
+
+The full diagonal endpoint restricts to a diagonal monomial endpoint on
+the triple. Apply the inherited disjoint-subset lemma to that subword:
+
+    sum_(i in triple) rank_Q(C_i) <= |J|.
+
+On the singleton each factor is multiplication by its monomial, so
+
+    C_singleton = sum_(j in K) A_j,
+    rank_Q(C_singleton) <= |K|.
+
+Adding gives sum_(i=0)^3 rank_Q(C_i) <= 8. Three invertible 3-by-3 targets
+already contribute nine, and the remaining rank is nonnegative. Contradiction.
+This covers a singleton data register as well as a singleton auxiliary.
+It allows arbitrary auxiliary diagonal address and arbitrary call order.
+
+Equivalently, the lemma supplies disjoint frame assignments on the triple;
+the singleton uses only the complementary indices K. Hence all four target
+addresses admit disjoint-subset assignments in this class at relation rank <=2.
+This stronger statement holds for any call count and any rational matrix size,
+provided the coordinate 3+1 decomposition and relation-rank condition hold.
+
+## 4. Scope and consequence
+
+This resolves the 24-pool handoff of PR5231 using coordinate isolation, not
+the kernel-support-union >=2 obstruction from that PR. The union is indeed
+one here; that old argument failing does not make the coordinate block coupled.
+
+Combining this result with the retained PR4119 classification, PR5075 cyclic-line
+closure, and PR5231 exclusion of the other 48 pools would exclude the entire
+eight-call span-six case for FOUR scalar registers. That combined statement
+additionally depends on those predecessor reductions; the new checker does
+not independently validate their universal proofs or re-enumerate the full
+four-register flag census. The local result proved here is the 24-pool closure.
+
+This does not settle span five or arbitrarily many auxiliary registers, and
+does not change a physical supplier or the selected multiplication exponent.
+For an upper-bound control, nine coordinate calls can produce three invertible
+targets with total address span six. They belong to the allowed alphabet but
+do not use all three exceptional projectors; no nine-call attainer with an
+exactly prescribed occurring exceptional triple is asserted.
+
+## Verification and credit
+
+verify.py checks all 24 coordinate permutations, the literal 3+1 block and
+restricted idempotents, and all positions of the three data targets. A separate
+dense-matrix formal Laurent engine checks block factorization for every word
+through length four in the seven-letter representative alphabet, and for
+nontrivial cancelling words. It checks a nine-call span-six positive control,
+rejects the two previously excluded 48-orbit representatives as coordinate-
+isolation inputs, and rejects an off-block bridge and a rank-two projector.
+These finite checks support the algebraic proof; they do not enumerate all
+eight-call rational-address words or replace the inherited lemma.
+
+Credit: woklerx/PR1846 (scalar orbit split), unpaidmoon/PR4119 (span-six
+classification), Nick Petalas/PR5231 (48-pool exclusion and the explicit
+24-pool question), David Govea/PR720 (arbitrary-target disjoint-subset lemma),
+Ali Romman/PR468, eumemic/PR288 and huxint/PR203 (underlying model and methods).
+JoyboyBrian/PR5075 supplies the separate cyclic-line closure. Only the
+coordinate-isolation application and its checks are new work here.
+
+Open next question: at four scalar registers, can frame span five permit eight
+calls? Its relation rank is three, so the lemma used here cannot decide the
+general coupled case. Start by retaining exact chronological endpoints while
+classifying three-dimensional off-diagonal pattern spaces. No search of that
+new problem was performed in this package.

research/coordinate-kernel-span-six-closure/README.md

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+# 剩余 24 类坐标核结构的八调用不可能性
+
+结论：在 PR5231 指定的四寄存器形式阿贝尔 BIT 模型中，剩余 24 类
+坐标核结构均不能在地址张成维数为 6 时，以 8 次秩一调用实现三个
+可逆的 3×3 有理地址矩阵目标。辅助输入任意，辅助输出允许任意对角单项式。
+
+核心是一个保留原坐标的 3+1 分块，而不是继续扩大搜索。
+代表结构的三个非坐标投影都在第四行和第四列为零，所有允许的坐标
+投影也保持这一分块。三寄存器子词的关系空间通过补零嵌入原关系空间，
+故关系秩至多为 2。由 PR720 第 1—4 节的互不相交子集引理，三块输出的
+地址秩之和不超过该子词调用数；孤立坐标的地址秩不超过其调用数。
+因此总秩 ≤8，与三个满秩目标所需的总秩 ≥9 矛盾。
+
+完整英文证明见 PROOF.md，依赖来源及当前主分支快照见 SOURCES.json
+。沿用的引理归 David Govea / PR720 及其 PR468、PR288、
+PR203 前驱工作；24/48 分类和题目来源归 woklerx、unpaidmoon 和
+Nick Petalas。新贡献仅为坐标隔离归约及相应检查。
+
+运行（仅 Python 标准库，无网络、无第三方依赖）：
+
+```sh
+python3 -I -B verify.py
+python3 -I -O -B verify.py
+```
+
+检查覆盖全部 24 个坐标置换池、96 种数据寄存器位置组合、七字母表中
+长度 1—4 的全部 2800 个形式词、逆元抵消、模型边界反例，以及一个
+地址维数为 6 的九调用正例。有限检查不是八调用全枚举；普遍结论由
+PROOF.md 的推导及明确引用的前驱引理支持。没有重新证明该引理、执行
+原有完整分类程序或进行证明助手核验。
+
+结果待维护者审阅；没有修改乘法指数 κ。
+跨度 5 的一般耦合结构与更多辅助寄存器的情况仍不在此结果内。
+
+Nicholas Stone，OpenAI Codex 提供实质性证明、代码及整理协助。
+新材料采用 Apache-2.0；前驱来源及其署名已在 PROOF.md 和 SOURCES.json 中注明。
+
+## Open next questions
+
+See TASK.json: frame-span five with one dirty auxiliary remains outside this proof. Its relation rank is three; begin with three-dimensional off-diagonal pattern spaces and exact chronological endpoints.

research/coordinate-kernel-span-six-closure/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "13be6ac685845c1ff671f5d26e7166165abd998e",
+  "files": [
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "fc3d4ba105e1f097d0d7d051a308281e86ab1446",
+      "sha256": "66974efdced4083c00025f97d95d83909a226cdf801edb9301c195c6641f6903"
+    },
+    {
+      "path": "research/flagless-kernel-obstruction/PROOF.md",
+      "usage": "input",
+      "git_blob": "78a9e1de94982b8158c50ec403e0d27d91fba59d",
+      "sha256": "9d45d578ef68bef05da4352d3c46449cffefe6917b42d760bd59fb4595930207"
+    },
+    {
+      "path": "research/auxiliary-span-six-flag/README.md",
+      "usage": "input",
+      "git_blob": "fcb3305be469805d6f6a15eb2c8ca6676af16730",
+      "sha256": "babb6e44c8ac7640a0cde329b7f6582e72adc2c9307de0d4b5bd2b2510489e1d"
+    },
+    {
+      "path": "research/three-register-no-target-flag/README.md",
+      "usage": "input",
+      "git_blob": "290bfb79db5f4177b8b03bc2027c3848671fac9e",
+      "sha256": "95d4483da9b19df051c7d13fe290eefb82da019ab4be8e66ad9216199ddda4eb"
+    }
+  ]
+}

research/coordinate-kernel-span-six-closure/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "bit-new-register-model",
+  "priority_revision": 7,
+  "base_commit": "13be6ac685845c1ff671f5d26e7166165abd998e",
+  "original_card": {
+    "id": "bit-new-register-model",
+    "question": "Can an explicitly larger or nonabelian register model escape the scoped three-register obstruction?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "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. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "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": "Exclude the remaining 24 coordinate-kernel pools for eight rank-one calls at frame span six in the four-register dirty diagonal-output contract.",
+  "scope_change_reason": "Explicit user-selected PR5231 handoff, resolved by coordinate 3+1 restriction and the inherited two-relation disjoint-subset lemma.",
+  "bottleneck_link": "Closes the remaining coordinate-kernel branch without a native supplier or kappa inference.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after the scoped proof, exact finite checks and publication; do not expand to span five.",
+  "outcome": "negative-result",
+  "evidence": "PROOF.md and verify.py",
+  "author": "Nicholas Stone / NicholasStone",
+  "ai_assistance": "Substantial OpenAI Codex assistance; no independent human or proof-kernel review.",
+  "open_questions": [
+    {
+      "question": "Can span five admit an eight-call word with one dirty auxiliary?",
+      "why_it_matters": "Its relation rank three remains outside this reduction.",
+      "first_step": "Classify three-dimensional pattern spaces while retaining exact chronological endpoints.",
+      "blocked_by": "The inherited disjoint-subset lemma only covers relation rank at most two."
+    }
+  ]
+}

research/coordinate-kernel-span-six-closure/VALIDATION.json

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+{
+  "python": "3.14.4 (main, Apr  7 2026, 13:13:20) [Clang 21.0.0 (clang-2100.0.123.102)]",
+  "platform": "macOS-26.5.2-arm64-arm-64bit-Mach-O",
+  "commands": [
+    {
+      "command": "python3 -I -B verify.py",
+      "exit_code": 0,
+      "result": {
+        "status": "PASS_COORDINATE_KERNEL_CLOSURE",
+        "pools": 24,
+        "data_placements": 96,
+        "formal_words": 2800,
+        "inverse_controls": 7,
+        "negative_controls": 4,
+        "positive_calls": 9,
+        "positive_address_span": 6,
+        "inherited_lemma_reproved": false
+      }
+    },
+    {
+      "command": "python3 -I -O -B verify.py",
+      "exit_code": 0,
+      "result": {
+        "status": "PASS_COORDINATE_KERNEL_CLOSURE",
+        "pools": 24,
+        "data_placements": 96,
+        "formal_words": 2800,
+        "inverse_controls": 7,
+        "negative_controls": 4,
+        "positive_calls": 9,
+        "positive_address_span": 6,
+        "inherited_lemma_reproved": false
+      }
+    }
+  ],
+  "review": "Single-agent written proof and exact finite checks. No independent human, agent, or proof-kernel review.",
+  "source_checks": "Three predecessor texts fetched at the pinned canonical commit. Full predecessor suites were not rerun.",
+  "publication_validation": {
+    "python": "3.14.4 (main, Apr  7 2026, 13:13:20) [Clang 21.0.0 (clang-2100.0.123.102)]",
+    "platform": "macOS-26.5.2-arm64-arm-64bit-Mach-O",
+    "source_pins": "API-fetched canonical commit; all five Git blob IDs recomputed and all declared live input SHA256 values checked.",
+    "metadata": "Upstream validate(check_pins=False) passed in a local Git fixture. Full canonical-history metadata replay and source-history preflight were not run locally; shell network DNS is unavailable.",
+    "replay": "Updated source-authenticating verify.py passed normal and optimized direct commands. Hosted Python 3.12 replay is pending.",
+    "overlap": "Bounded live PR title/body searches; no complete Discussion or arbitrary diff coverage; no exclusivity claim."
+  }
+}

research/coordinate-kernel-span-six-closure/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/flagless-kernel-obstruction/PROOF.md",
+    "research/auxiliary-span-six-flag/README.md",
+    "research/three-register-no-target-flag/README.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 30,
+      "checks": "All 24 pools, exact formal block restrictions, inverse and model-boundary controls, and nine-call span-six positive control; authenticate the declared predecessor inputs."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 30,
+      "checks": "All 24 pools, exact formal block restrictions, inverse and model-boundary controls, and nine-call span-six positive control; authenticate the declared predecessor inputs."
+    }
+  ]
+}