research: screen rank-2 block helper inventory

research/complex-rank2-block-helper-screen/README.md

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+# Rank-2 block-helper inventory screen
+
+**Exploratory exact planning bound; no new supplier or selected kappa.**
+
+The pinned task/scope, source hashes and replay contract are in
+[TASK.json](TASK.json), [SOURCES.json](SOURCES.json) and [VERIFY.json](VERIFY.json).
+
+## Question and contract
+
+Can the selected #1094 complex local word replace blocks of eight rank-1
+children implementing one 2-by-2 matrix product with one recursive rank-2
+helper, and what is the best optimistic moment if every eligible block is
+replaced?
+
+The proposed helper contract is the ordinary map
+`(A,B) -> AB` for two 2-by-2 matrices. Its schoolbook implementation uses
+eight scalar products. If eight rank-1 children in the selected local word
+form one complete, disjoint instance of that map, they can be represented as
+one rank-2 recursive call. For `t` such blocks per call, the changed local
+profile is
+
+```text
+n1 = 2412 - 8t, n2 = 418 + t,
+n3 = 432, n4 = 142, n5 = 82, n8 = 8,   0 <= t <= 301.
+```
+
+For `A=[[a,b],[c,d]]` and `B=[[e,f],[g,h]]`, the eight terms are
+`ae,bg,af,bh,ce,dg,cf,dh`; the four outputs are respectively
+`ae+bg`, `af+bh`, `ce+dg`, and `cf+dh`. Thus this is one coupled four-output
+matrix product, not a direct sum of independent scalar products.
+
+Each block lowers the paid rank mass by six; the optimistic screen returns
+those six units to the rank-51 idle remainder. It charges no block extraction,
+input/output framing, addition, source restoration, hosting, precision, or
+other consumer cost. The arithmetic therefore prices a proposed interface; it
+does not show that its eight-term blocks occur in the selected word.
+
+## Exact screen
+
+The prescribed baseline example passes. The parent model has `m=73`,
+`W=896/9` live families per call, nine stages, centre charge 64, and the
+halving-guard idle rank cap `R=51`. The source-bound selected local profile is
+`{1:2412, 2:418, 3:432, 4:142, 5:82, 8:8}` with local mass 5,586 and idle
+mass 1,646.
+
+The best profile in this one-rewrite class uses all 301 complete blocks:
+
+```text
+{1:4, 2:719, 3:432, 4:142, 5:82, 8:8}
+local mass = 3,780; idle mass = 3,452; total = 7,232.
+```
+
+Using `ceiling.ceiling()` with exact rational intervals at grid `10^-12`, its
+optimistic root is
+`a in [0.002632122649, 0.002632122650]`, hence the complex-side planning cap
+is `kappa <= 0.002625212768...`. The endpoint signs are certified by the
+parent interval engine. Even this maximum inventory does not reach the
+planning target `kappa=1/99`.
+
+The exact first point where this optimistic complex ceiling clears the
+unchanged selected bit root is `t=22`:
+
+```text
+{1:2236, 2:440, 3:432, 4:142, 5:82, 8:8}
+local mass = 5,454; idle mass = 1,778.
+```
+
+At `a=bit_lower`, the previous inventory (`t=21`) has certified moment
+strictly above one, and `t=22` has certified moment strictly below one. The
+verifier prints the exact rational intervals for both signs. This is only a
+zero-overhead threshold for source matching; it does not price the proposed
+block extraction or its consumers.
+
+More importantly, with the selected bit supplier unchanged, the overall cap
+is still the bit-side value `b/(1+b)`, where the selected assembly has
+`b=1733427039589844234749/1000000000000000000000000`; this is below a 5%
+increase over the selected `kappa=0.001672660197076918`. This helper change
+alone cannot meet the 5% system target. It could matter only as part of work
+that also raises the bit-side bound.
+
+For each `a>0`, replacing a block strictly lowers the relaxed moment because,
+after factoring out `73^(-(1-a))`, its decrease is
+
+```text
+8 - 2^(1-a) - 6*51^(-a) > 0.
+```
+
+The expression is zero at `a=0` and has positive derivative for `a>=0`.
+Thus `t=301` is the best ceiling in this restricted rewrite class, not a
+sampled point.
+
+## Reproduction
+
+From the repository root (Python standard library only):
+
+```sh
+python3 -B research/complex-local-word-ceiling/verify.py
+python3 -I -B research/complex-rank2-block-helper-screen/verify.py --check
+```
+
+The first command prints `PASS_COMPLEX_LOCAL_WORD_CEILING`. The second checks
+the selected-source pins, the full per-call rank/idle bill, the exact root
+bracket and both endpoint signs, the strict improvement over the original
+ceiling, the unchanged bit cap, the 5% stop test, and out-of-range controls.
+Neither command replays the complete selected construction.
+Use `python3 -I -B research/complex-rank2-block-helper-screen/verify.py --write`
+to regenerate the exact rational receipt `RESULT.json`.
+
+## Scope, source and credit
+
+Base: `20ecf2891d80b5acf18f980b2164fb44ea7faa90` (the session's pinned
+canonical main). The
+selected record is `certificates/selected-result.json`, kappa
+`836330098538459/500000000000000000`; its reviewed source is
+`docs/research/community-batch1099-review.md`, its assembly is
+`research/community-frontier1094-audit/ASSEMBLY.json`, and its source manifest
+is `research/community-frontier1094-audit/SELECTED-SOURCE.json`. The finite
+coverage identity is attributed to PR #1012. The ceiling lemma and exact
+interval engine are Douglas Colkitt's
+`research/complex-local-word-ceiling/` contribution, prepared with Claude
+(Anthropic) assistance.
+
+This screen does not find any complete eight-term block in the selected
+producer, prove compatibility with its frames, or charge the physical
+realization of a block. It does not admit a supplier, change kappa or the
+selected frontier, or establish the full multiplication theorem. No
+independent human review is claimed. Prepared with OpenAI Codex assistance.
+
+## Relation to nearby work
+
+This candidate changes one *complete* 2-by-2 matrix-product subword. It is
+different from PR #1960's direct sums of consecutive independent rank-2/rank-3
+children, from the rank-one common-product/output-anchor claims in #1167, and
+from PR #2224's six-coordinate `Q[S3]` convolution factorization. Those are
+close block or bilinear mechanisms, so this package makes no novelty claim
+beyond the exact object and moment screen stated here. It also differs from
+PR #3874's combined ideal-lever grid: it changes the local parent subword and
+releases the resulting six units of rank mass per block into idle capacity.
+
+## Open next questions
+
+Inspect the exact selected producer behind #1094 and determine whether even
+one disjoint eight-term subword has the complete 2-by-2 input/output map above.
+If so, price its frame, addition, restoration, hosting, and bit-side effects
+before seeking more blocks. Otherwise stop this contract. Any continuation
+must address the unchanged bit cap as well as the complex moment.

research/complex-rank2-block-helper-screen/RESULT.json

@@ -0,0 +1,65 @@
+{
+  "base": "20ecf2891d80b5acf18f980b2164fb44ea7faa90",
+  "baseline_a_bracket_1e-12": [
+    "1690054109/1000000000000",
+    "169005411/100000000000"
+  ],
+  "bit_b": "1733427039589844234749/1000000000000000000000000",
+  "bit_crossing_idle_mass": "1778",
+  "bit_crossing_local_mass": "5454",
+  "bit_crossing_profile_per_call": {
+    "1": 2236,
+    "2": 440,
+    "3": 432,
+    "4": 142,
+    "5": 82,
+    "8": 8
+  },
+  "bit_kappa_cap_upper": "1733427039589844234749/1001733427039589844234749",
+  "blocks_per_call_max": 301,
+  "candidate_a_bracket_1e-12": [
+    "2632122649/1000000000000",
+    "52642453/20000000000"
+  ],
+  "candidate_moment_at_a_hi": [
+    "115792089237429756355399434788156137141182667572608663281727707418166298314853/115792089237316195423570985008687907853269984665640564039457584007913129639936",
+    "57896044618714878177699717394078068570591333786304331640863853709083149157435/57896044618658097711785492504343953926634992332820282019728792003956564819968"
+  ],
+  "candidate_moment_at_a_lo": [
+    "3618502788662927505738195365636877584726214511141390350157743396969395891349/3618502788666131106986593281521497120414687020801267626233049500247285301248",
+    "7237005577325855011476390731273755169452429022282780700315486793938791782699/7237005577332262213973186563042994240829374041602535252466099000494570602496"
+  ],
+  "candidate_moment_at_baseline_hi": [
+    "28897204584848427825436451363931202947323668317080618356325526673841573167429/28948022309329048855892746252171976963317496166410141009864396001978282409984",
+    "28897204584848427825436451363931202947323668317080618356325526673841573167433/28948022309329048855892746252171976963317496166410141009864396001978282409984"
+  ],
+  "candidate_profile_per_call": {
+    "1": 4,
+    "2": 719,
+    "3": 432,
+    "4": 142,
+    "5": 82,
+    "8": 8
+  },
+  "complex_kappa_cap_upper": "52642453/20052642453",
+  "contract": "8 rank-1 scalar products -> 1 recursive rank-2 2x2 block helper",
+  "five_percent_system_target": "17562932069307639/10000000000000000000",
+  "idle_mass": "3452",
+  "local_mass": "3780",
+  "min_blocks_per_call_to_clear_bit_root": 22,
+  "moment_at_bit_root_first_crossing": [
+    "115791433861598405212192388976899848948633134343156221001587985922425666556379/115792089237316195423570985008687907853269984665640564039457584007913129639936",
+    "57895716930799202606096194488449924474316567171578110500793992961212833278207/57896044618658097711785492504343953926634992332820282019728792003956564819968"
+  ],
+  "moment_at_bit_root_previous_blocks": [
+    "115792126583934581881974895553767938252482618389908562191656991839663548837927/115792089237316195423570985008687907853269984665640564039457584007913129639936",
+    "57896063291967290940987447776883969126241309194954281095828495919831774418981/57896044618658097711785492504343953926634992332820282019728792003956564819968"
+  ],
+  "overall_cap_with_bit_fixed": "1733427039589844234749/1001733427039589844234749",
+  "planning_target_1_over_99": "1/99",
+  "scope": "optimistic profile only; source matching and extra physical charges are unproved",
+  "selected_kappa": "836330098538459/500000000000000000",
+  "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129",
+  "status": "PASS_RANK2_BLOCK_HELPER_SCREEN",
+  "total_coverage_mass": "7232"
+}

research/complex-rank2-block-helper-screen/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "files": [
+    {
+      "path": "certificates/checkpoints/frontier1042.json",
+      "usage": "input",
+      "git_blob": "f451ced6626a1f03aba6915ce23a136b01ebe51d",
+      "sha256": "207d1f49f113c585024b9d3c17eadbb8b1be17c1321e3d190aabe2f18ce1c5b7"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "input",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    },
+    {
+      "path": "docs/research/community-batch1099-review.md",
+      "usage": "input",
+      "git_blob": "6cb6ec061a071a06a2d18cd4dc3e9a2b05d28099",
+      "sha256": "7e76d2c38e5fce30587835b7251df1430488ae4dd372611ef69a3f03f476b6cb"
+    },
+    {
+      "path": "research/community-frontier1042-audit/ASSEMBLY.json",
+      "usage": "input",
+      "git_blob": "4e308dc23da97373d66e5d8d4a68b0661caea8e9",
+      "sha256": "093f35445812ea96fe1bfe6a3af7d769cb7f274e08f4fb30f71a17ed1161e54e"
+    },
+    {
+      "path": "research/community-frontier1094-audit/ASSEMBLY.json",
+      "usage": "input",
+      "git_blob": "5fd676df060f3b450782fdf5ff5f6beb63a5eeb0",
+      "sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045"
+    },
+    {
+      "path": "research/community-frontier1094-audit/INDEPENDENT1094.json",
+      "usage": "input",
+      "git_blob": "009a385e26858058a18188364078d162aa6f9519",
+      "sha256": "301fb8a0e4ab7e3c383ff974c54c08178283b2a3453c5282757f692a3c7a4bfd"
+    },
+    {
+      "path": "research/community-frontier1094-audit/SELECTED-SOURCE.json",
+      "usage": "input",
+      "git_blob": "6a75493790b4eff9c4a98051ef8b01d6aad0d5a8",
+      "sha256": "79b3de5db3dbcb9829789ba58e9e45758926031e2a7d80667a08429d369be53b"
+    },
+    {
+      "path": "research/complex-local-word-ceiling/README.md",
+      "usage": "input",
+      "git_blob": "f5f742f8196da29d2c3f4dffb3c27685ca9401f6",
+      "sha256": "b60be039cb0e03e064a27407d0bce9e58908d793da5ee405c876de5aebfa5f81"
+    },
+    {
+      "path": "research/complex-local-word-ceiling/ceiling.py",
+      "usage": "input",
+      "git_blob": "6c0f54527d879396f689523bfde19928a9b6157c",
+      "sha256": "86a1432870b3b21d8454106ec9162fddc0471472d9214c590a1420f2426cedfc"
+    },
+    {
+      "path": "research/complex-local-word-ceiling/verify.py",
+      "usage": "input",
+      "git_blob": "069f09de381a52ac91ec3edb372bd4536a051d91",
+      "sha256": "06b59955c69a2e7f49aa031d2ebaac2f39dbe107f4087122704f2a37ce9806d5"
+    }
+  ]
+}

research/complex-rank2-block-helper-screen/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-new-helper-inventory",
+  "priority_revision": 6,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "original_card": {
+    "id": "complex-new-helper-inventory",
+    "question": "Can a new local cost interface change the deficit-to-recursive-cost ratio beyond current packing?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/complex-local-word-ceiling/README.md",
+      "research/heterogeneous-owner-assembly/README.md",
+      "certificates/selected-result.json",
+      "research/selected-k8-unrestricted-ends/README.md"
+    ],
+    "methods": [
+      "Propose a changed helper inventory and derive an optimistic exact moment.",
+      "Design a small completed instance and independently verify its endpoint and rank bill."
+    ],
+    "acceptance_gate": "A new physically realizable contract outside the frozen catalogue whose per-call profile, screened with ceiling.ceiling(), beats the selected word's packing ceiling; a lower cap does not establish identical circuitry or order attainable optima, and is insufficient to pass the stronger-budget target under this model. 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",
+        "research/complex-local-word-ceiling/verify.py"
+      ],
+      "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING",
+      "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.",
+      "scope": "Positive control and planning budget only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Before any synthesis, write the candidate per-call local profile (and centre charge / unpaid coordinates if changed) and screen it with ceiling.ceiling(); reject it if even the ideal ceiling does not beat the selected word's by your stated margin.",
+    "bottleneck_link": "Packing of the selected word is capped at +0.87% (ceiling model); only a lower local excess spend or a changed coverage law raises the cap.",
+    "allowed_variations": [
+      "Narrow to one changed boundary or helper type with explicit counts.",
+      "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."
+    ],
+    "scope_change_triggers": [
+      "Mixing only providers already inside the pinned catalogue.",
+      "Only changing helper terminal subspaces and monotone frames within the fixed scalar word, entrances and cleanup contract already covered by #1995/#2013.",
+      "Continuing a partial cost improvement while leaving the same unpriced consumer, restoration or central-readout obligation unresolved; refresh the family expansion decision first. Local product-retiming evidence has a narrow scope and is not a ban on changed graphs or contracts."
+    ],
+    "overlap_terms": [
+      "helper inventory",
+      "heterogeneous-owner",
+      "mutable leaver"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Screen the new complete 2x2 matrix-product helper contract: replace eight rank-1 children by one rank-2 recursive helper; derive the best exact optimistic moment under the selected 73/896/9/64 coverage model and stop at the unchanged bit cap.",
+  "scope_change_reason": "Narrow the card to one coupled four-output M(2) bilinear subword, distinct from PR3874 ideal lever-grid enumeration, PR1960 independent direct-sum blocks, and earlier common-product output-anchor work. The verifier gives an optimistic profile screen; selected-source occurrence and physical charges remain open.",
+  "bottleneck_link": "The selected complex word ceiling is limited by paid rank-1 mass under PR1012 coverage. This M(2) subword would replace eight rank-1 children with one rank-2 child, releasing six rank-mass units per complete block to ideal rank-51 idle capacity.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after the exact maximum t=301 inventory and bit-root threshold are certified. The maximum complex cap remains below 1/99, and an unchanged bit supplier keeps the overall cap below the 5% system target. Reopen only with a source-mapped complete block and a changed bit-side route.",
+  "outcome": "partial",
+  "evidence": "PASS_RANK2_BLOCK_HELPER_SCREEN: exact root bracket [2632122649/1000000000000, 52642453/20000000000] at t=301; first bit-root crossing t=22; source occurrence and physical charges unproved.",
+  "author": "OpenAI Codex, homelab-06",
+  "ai_assistance": "Research and verifier prepared with GPT-6 Luna xhigh via OpenAI Codex; independent human review is unclaimed.",
+  "open_questions": [
+    {
+      "question": "Does the source-bound #1094 local word contain at least 22 disjoint complete schoolbook M(2) subwords per call, with the required frames and restoration?",
+      "why_it_matters": "The exact zero-cost screen needs 22 substitutions per call merely to move the complex root above the selected bit root; physical charges could erase that headroom, and the unchanged bit supplier still blocks a 5% system gain.",
+      "first_step": "Trace the selected producer source and annotate each rank-1 child input/output pair; search for the exact eight-term 2x2 map in its two orientations.",
+      "blocked_by": "The selected assembly/profile does not encode ordered source expressions and frame/consumer mappings for these candidate groups in this package."
+    }
+  ]
+}

research/complex-rank2-block-helper-screen/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "certificates/checkpoints/frontier1042.json",
+    "certificates/selected-result.json",
+    "docs/research/community-batch1099-review.md",
+    "research/community-frontier1042-audit/ASSEMBLY.json",
+    "research/community-frontier1094-audit/ASSEMBLY.json",
+    "research/community-frontier1094-audit/INDEPENDENT1094.json",
+    "research/community-frontier1094-audit/SELECTED-SOURCE.json",
+    "research/complex-local-word-ceiling/README.md",
+    "research/complex-local-word-ceiling/ceiling.py",
+    "research/complex-local-word-ceiling/verify.py"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py",
+        "--check"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Source pins, selected per-call profile, coverage mass, exact moment bracket endpoint signs, minimal t=22 crossing of the selected bit root, 5% and 1/99 stop tests, and invalid block-count control."
+    }
+  ]
+}