Construction: five plane queries for selected P111 bit decoder (no new kappa)

research/bit-p111-parseval-plane-decoder-seedceccc765/README.md

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+# Five plane queries for the selected P111 bit decoder
+
+**A finite bit-decoder construction and favorable coverage-law screen. No new supplier or `kappa` is claimed.**
+
+This package factors the exact 111-port decoder from the selected merged PR589 bit word into five rank-two queries. It uses one explicit 9-to-10 Parseval frame and keeps both source and target endpoint maps. The construction is at the decoder boundary; its plane-frame route, physical streaming schedule, hosting and precision invoice remain open.
+
+## Construction
+
+The frozen word has nine retained source channels. Write their target and source factors as matrices `P` and `Q`, so the scatter map is `P Q^T`. The ten columns below are vectors in `F2^9`:
+
+```text
+a1 = e1 + e2       a2 = e1 + e3
+a3 = e2 + e3       a4 = e1 + e2 + e3
+a5 = e4            a6 = e5
+a7 = e6             a8 = e7
+a9 = e8             a10 = e9
+```
+
+They satisfy the exact identity
+
+```text
+sum(j=1..10) aj aj^T = I9  over F2.
+```
+
+Transform both factor lists by these ten columns, then group columns `(1,2)`, `(3,4)`, `(5,6)`, `(7,8)`, `(9,10)`. Each pair gives one rank-two plane query. Therefore
+
+```text
+(P A)(Q A)^T = P (A A^T) Q^T = P Q^T.
+```
+
+The verifier reconstructs the nine original channels from the pinned word and checks the equality on all 661 formal input columns at once, including dirty targets and helpers. It also checks the expected `B0 = (1 + G) mod 2` map on all 111 by 111 source-target pairs. The five transformed source and target factor pairs each have exact F2 rank two.
+
+## Copy and ADD invoice
+
+The original decoder has nine rank-seven B14 pairing terms, for copy charge `63`. For each new plane, the rational pairing rank between the union of its target support and the union of its source support is exactly eight. The five plane queries therefore cost `40`, saving 23 copy-rank units (36.51%) at the algebraic decoder boundary.
+
+Forming the ten frame channels from the nine old channels takes five XOR additions on each side, ten ADDs total. The favorable local profile charges those ten as rank-one children, removes nine rank-seven copy children, and adds five rank-eight plane children:
+
+```text
+before: {1:1323, 2:1342, 3:79, 4:44, 5:102, 6:11, 7:19}, mass 5129
+after:  {1:1333, 2:1342, 3:79, 4:44, 5:102, 6:11, 7:10, 8:5}, mass 5116
+```
+
+This charges the binary basis change and every B14 copy rank in the proposed decoder. It does not charge physical frame transitions, temporary storage and cleanup, hosting, precision/tape, or composition with the rest of the bit word. Those costs can only reduce the favorable ceiling below.
+
+## Replacement coverage identity and budget
+
+For the pinned bit profile, `m=64`, `v=111`, `K=4`, and nine stages give `W=2 K v / 9 = 888/9 = 296/3` live families per local call. The selected line-port law is
+
+```text
+paid mass = (m - 1) W + 63 = 6279.
+```
+
+The proposed two-unpaid-coordinate plane law is
+
+```text
+paid mass = (m - 2) W + 40 = 18472/3.
+```
+
+The exact local profile leaves idle masses `1150` and `3124/3`, respectively. Applying the pinned ceiling routine with its selected nine-stage guard `R=42` gives:
+
+| Model | Certified `a` bracket | Idle mass |
+|---|---:|---:|
+| Existing line decoder | `[973733779/500000000000, 1947467559/10^12]` | `1150` |
+| Five-plane decoder, all other new interface costs set to zero | `[4260424781/500000000000, 8520849563/10^12]` | `3124/3` |
+
+At `a=1/99`, the favorable plane relaxation is already greater than one (`1.0047022529...`), so this construction does not reach that planning target. The result is a decoder-cost mechanism, not a replacement bit supplier. The selected result remains complex-bound; no overall `kappa` improvement follows from this screen.
+
+## Reproduction
+
+From the repository root:
+
+```sh
+python3 -B research/bit-p111-parseval-plane-decoder-seedceccc765/verify.py > research/bit-p111-parseval-plane-decoder-seedceccc765/RESULT.json
+python3 scripts/research_package.py --package research/bit-p111-parseval-plane-decoder-seedceccc765 --check-only
+python3.12 scripts/research_package.py --package research/bit-p111-parseval-plane-decoder-seedceccc765
+```
+
+The verifier uses exact F2 and rational arithmetic. [RESULT.json](RESULT.json) is its deterministic output. Four negative controls check a broken frame, an omitted plane, a switched source factor and a collapsed rank-two factor. Expected status is `PASS_BIT_P111_PARSEVAL_PLANE_DECODER`; the report includes the dirty-map SHA-256 and exact ceiling brackets. The registered replay uses Python 3.12; this local environment currently provides Python 3.14, so the command-level verifier and metadata check are the local evidence.
+
+## Provenance and scope
+
+- Cycle seed: `ceccc7658dd89c833863007fb63ad7b2`; track `recursive-interfaces`; task `coverage-law-change`; priority revision 3.
+- Base: `0b1cccf86e2f5132ed286a26ef0a681a495740ea`; selected-record SHA-256: `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.
+- Bit input: merged PR589's `research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz`, content SHA-256 `5642fd0e35a1acc11acff8d0c08350c1617e95f310d4d35ad63a651f9eeeba0b`; archive SHA-256 `e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f`.
+- Discussion claim: [#1587 comment 18924772](https://github.com/CrocSwap/integer-mult-bounds/discussions/1587#discussioncomment-18924772). The fresh 2026-10-14 overlap refresh covered 540 open PRs and 524 Discussions with complete pagination and returned no exact term matches. PR #1825 remains the selected complex-side paired-plane construction and lists this bit-side physical realization as open.
+- The work builds on PR589/PR555's bit word and PR1084's exact line-decoder pricing model. The plane endpoint question is from Discussion #1587; no code from PR #1825 is used.
+- Prepared by Cinna Davis with substantial OpenAI Codex assistance in construction, exact arithmetic, verification and writing. No independent human review is claimed. New work is Apache-2.0.
+
+## Open next questions
+
+- **Can these five coefficient planes be placed in a legal nested-frame schedule with streamed cleanup and whole-call hosting?** The exact decoder equality makes this the physical blocker. First derive source and target frame paths for each pair from the pinned 111-port geometry and test a two-plane dirty-register schedule. The current selected word records line-port routes only.
+- **Can a completed bit plane supplier compose with the complex paired-plane construction to improve the selected conditional bound?** The selected result is complex-bound, so this decoder alone cannot move `kappa`. First combine both all-cost invoices in one exact assembly after the route and precision obligations close. PR #1825 still leaves its all-cost complex route open.

research/bit-p111-parseval-plane-decoder-seedceccc765/RESULT.json

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+{"base_commit":"0b1cccf86e2f5132ed286a26ef0a681a495740ea","baseline_a_ceiling":["973733779/500000000000","1947467559/1000000000000"],"baseline_idle_mass":"1150","baseline_mass":"6279","basis_adds_charged":10,"controls":4,"copy_charge_line":63,"copy_charge_planes":40,"formal_dirty_input_columns":661,"full_dirty_decoder_sha256":"ae93f3411b3c799e482738a89db1e8a7f48bac8768030a649b2ffd63034e6b62","kappa_claim":false,"line_copy_costs":[7,7,7,7,7,7,7,7,7],"parseval_columns_hex":["0x3","0x5","0x6","0x7","0x8","0x10","0x20","0x40","0x80","0x100"],"plane_a_ceiling":["4260424781/500000000000","8520849563/1000000000000"],"plane_at_1_over_99_relaxation_lower":"116336572933712950230721466055437294176940130147802427912377566750618070383397/115792089237316195423570985008687907853269984665640564039457584007913129639936","plane_copy_costs":[8,8,8,8,8],"plane_factor_ranks":[[2,2],[2,2],[2,2],[2,2],[2,2]],"plane_target_source_support_sizes":[[89,85],[85,84],[86,88],[86,84],[90,87]],"planes_zero_based":[[0,1],[2,3],[4,5],[6,7],[8,9]],"ports":111,"replacement_idle_mass":"3124/3","replacement_mass":"18472/3","seed":"ceccc7658dd89c833863007fb63ad7b2","selected_record_sha256":"10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32","status":"PASS_BIT_P111_PARSEVAL_PLANE_DECODER","unpriced_interface_obligations":["physical plane-frame routes and transitions","streaming/register schedule and cleanup","owner hosting and whole-call placement","precision/tape and integration with the selected complex supplier"]}

research/bit-p111-parseval-plane-decoder-seedceccc765/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1",
+      "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075"
+    },
+    {
+      "path": "docs/research/community-batch1099-review.md",
+      "usage": "provenance",
+      "git_blob": "6cb6ec061a071a06a2d18cd4dc3e9a2b05d28099",
+      "sha256": "7e76d2c38e5fce30587835b7251df1430488ae4dd372611ef69a3f03f476b6cb"
+    },
+    {
+      "path": "research/bit-orthogonal-pair-decoders/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "be3243952be4a1a8f5901829c73ce425ab217d16",
+      "sha256": "d89b42d4aa2e6bf2c3ac454404fd1aeddbf2eb07f6ff1fc606057e3b2a928f78"
+    },
+    {
+      "path": "research/bit-orthogonal-pair-decoders/verify.py",
+      "usage": "provenance",
+      "git_blob": "1024748736844198a3ea92d0f35f651504f9ed04",
+      "sha256": "512df34b78791155d15467d1036a4fe217b007f40126abdcdc8a31c24ca94b1b"
+    },
+    {
+      "path": "research/community-frontier1094-audit/ASSEMBLY.json",
+      "usage": "input",
+      "git_blob": "5fd676df060f3b450782fdf5ff5f6beb63a5eeb0",
+      "sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045"
+    },
+    {
+      "path": "research/community-frontier1094-audit/SELECTED-SOURCE.json",
+      "usage": "provenance",
+      "git_blob": "6a75493790b4eff9c4a98051ef8b01d6aad0d5a8",
+      "sha256": "79b3de5db3dbcb9829789ba58e9e45758926031e2a7d80667a08429d369be53b"
+    },
+    {
+      "path": "research/complex-local-word-ceiling/ceiling.py",
+      "usage": "input",
+      "git_blob": "6c0f54527d879396f689523bfde19928a9b6157c",
+      "sha256": "86a1432870b3b21d8454106ec9162fddc0471472d9214c590a1420f2426cedfc"
+    },
+    {
+      "path": "research/e8-bit-p111-tuned/code/xcheck_p.py",
+      "usage": "provenance",
+      "git_blob": "184f5dcf94a486cc7db91ca4db5865916bf26009",
+      "sha256": "af3decc877e27c00001cf56f50c6c09566e8e101297ab34c7860aed96a06d80e"
+    },
+    {
+      "path": "research/e8-bit-p111-tuned/tune/EXPECTED.json",
+      "usage": "input",
+      "git_blob": "e293ddc557742637f9f679f083a1a282d87b1b13",
+      "sha256": "b9f0e5ca038abe837b0007c8fe006f366a6f02009244ac57705a8a200e2301f4"
+    },
+    {
+      "path": "research/e8-bit-p111-tuned/tune/README.md",
+      "usage": "provenance",
+      "git_blob": "5b1554369682c8344d4f0767111d9b2b35eecbec",
+      "sha256": "46c7917fb252949f38b7bcd88cf568efc282e3ea065b61d11e457e41673d0b2d"
+    },
+    {
+      "path": "research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz",
+      "usage": "input",
+      "git_blob": "85a4499bf9e4911a705d50567a2c2ef6cf23739a",
+      "sha256": "e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f"
+    }
+  ]
+}

research/bit-p111-parseval-plane-decoder-seedceccc765/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "coverage-law-change",
+  "priority_revision": 3,
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "original_card": {
+    "id": "coverage-law-change",
+    "question": "Can a template or call contract leave two or more unpaid coordinates per live family?",
+    "starting_points": [
+      "research/complex-local-word-ceiling/README.md",
+      "research/idle-axis-hosting/PROOF.md",
+      "certificates/selected-result.json"
+    ],
+    "methods": [
+      "State a replacement for the coverage identity (mass = 72W + 64N) and the contract change that realizes it, with its implementation cost.",
+      "Find an obstruction: show which data-endpoint obligation forces exactly one unpaid coordinate in the current template."
+    ],
+    "acceptance_gate": "An explicit replacement identity, verified on a small complete instance with dirty inputs, priced with ceiling.ceiling(unpaid=...).",
+    "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": "Explain where the single unpaid coordinate per family comes from in the hosted template; propose one change and price it. Ideal budget: two unpaid coordinates per family is 3.77x (levers.py).",
+    "bottleneck_link": "The deficit is one coordinate per family minus the centre charge; doubling unpaid coordinates per family scales it directly.",
+    "allowed_variations": [
+      "One template change or one endpoint-contract change.",
+      "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": [
+      "Counting a coordinate as unpaid while leaving it unadvanced in the data endpoint."
+    ],
+    "overlap_terms": [
+      "coverage identity",
+      "unpaid coordinate",
+      "deficit law",
+      "W - 64N"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Replace the pinned P111 bit decoder's nine rank-one factors by five rank-two plane factors using a fixed 9-to-10 Parseval frame; verify the complete dirty decoder boundary and price the candidate two-unpaid coverage law.",
+  "scope_change_reason": "The broad coverage-law card is refined to the bit-side plane-port obligation explicitly left open in Discussion #1587 and PR #1825. This isolates one exact P111 decoder, one frame and one grouping instead of claiming a general template or supplier change.",
+  "bottleneck_link": "The selected bit supplier has a nine-term copy charge of 63 and remains above the selected complex-bound kappa. A cheaper exact plane decoder could preserve bit-side headroom if the open complex plane-port work raises the other side.",
+  "budget_minutes": 60,
+  "stop_rule": "Use only the displayed ten-column Parseval frame and five fixed column pairs. Stop after exact factorization, rational copy ranks, all dirty decoder input columns, the four negative controls and the favorable replacement-law ceiling. Do not search alternative frames or infer a physical supplier or kappa.",
+  "outcome": "partial",
+  "evidence": "verify.py reports PASS_BIT_P111_PARSEVAL_PLANE_DECODER: exact 111-by-661 dirty decoder map, five rank-two plane factors, copy charge 63 to 40, ten basis ADDs charged, replacement mass 18472/3 with idle 3124/3, and favorable a ceiling 8520849563/10^12. Physical plane routes, streaming cleanup, hosting and precision remain open.",
+  "author": "Cinna Davis",
+  "ai_assistance": "Substantial OpenAI Codex assistance in construction, exact arithmetic, implementation, verification and writing. No independent human review is claimed.",
+  "open_questions": [
+    {
+      "question": "Can the five coefficient planes be realized by legal nested source and target frames in the P111 word, with streaming cleanup and owner hosting?",
+      "why_it_matters": "The exact matrix factorization and favorable ceiling do not establish that the five plane queries can replace the physical copy schedule.",
+      "first_step": "For each of the five explicit factor pairs, derive the required source/target frame from its 111-port supports and test a two-plane dirty-register schedule against the pinned route geometry.",
+      "blocked_by": "The selected P111 word records line-port geometry, not plane-port routes or a streamed 9-to-10 helper schedule."
+    },
+    {
+      "question": "Does this bit-side mechanism combine with the selected complex paired-plane decoder to raise the accepted kappa?",
+      "why_it_matters": "The selected result is complex-bound, so this bit-side headroom alone cannot improve its conditional bound.",
+      "first_step": "After both physical suppliers have complete route, hosting, restoration and precision invoices, substitute their changed laws into one exact joint assembly.",
+      "blocked_by": "PR #1825 leaves its all-cost complex route and the bit-side physical realization open."
+    }
+  ]
+}

research/bit-p111-parseval-plane-decoder-seedceccc765/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/community-frontier1094-audit/ASSEMBLY.json",
+    "research/e8-bit-p111-tuned/tune/EXPECTED.json",
+    "research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz",
+    "research/complex-local-word-ceiling/ceiling.py"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact 111-by-661 dirty decoder identity, Parseval frame identity, five rank-two planes, rational B14 copy ranks, four negative controls, replacement coverage mass and exact favorable ceiling brackets"
+    }
+  ]
+}