Research exact local P104 I-G/2 tail correction

research/p104-tail-side-correction-seed749717/README.md

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+# P104 direct tail side correction
+
+This package gives an exact finite construction for the missing `I-G/2` map
+after the pinned P104 native `G/2` channel. It does not claim a new conditional
+bound or a globally admitted supplier.
+
+## Construction and cost
+
+The cut is immediately after the frozen 4,058-event channel prefix. At that
+point all 104 source registers `X_s` are full-frame and each target `Y_t` is in
+its final rank-eight hyperplane. Let `G[t,s]` be the exact root inner-product
+matrix on the 104 retained ports. For each target in order, the construction:
+
+1. Promotes `Y_t` from its rank-eight frame to the full nine-dimensional frame.
+2. Adds `delta[t,s] - G[t,s]/2` times `X_s` for every nonzero coefficient.
+3. Restores `Y_t` to its original rank-eight frame.
+
+Each row has 48 nonzero coefficients, so the correction uses 4,992 scalar
+additions and 208 rank-one native frame changes. Exact symbolic replay over all
+280 arbitrary input columns proves the final rows are `Y_t + X_t`; all other
+rows and all target frames are restored. Four negative controls reject a wrong
+sign, missing promotion, missing restoration, and missing matrix entry.
+
+The augmented local paid profile is
+`{1:1282, 2:58, 3:86, 4:104, 5:48, 6:24, 7:8, 8:8}` with rank mass 2,576.
+The exact favorable local-word ceiling is lower than the uncorrected channel's
+under the same pinned relaxation. See [RESULT.json](RESULT.json) for rational
+brackets and artifact hashes, [CORRECTION-ROWS.json](CORRECTION-ROWS.json) for
+the sparse matrix, [AUGMENTED-WITNESS.json.gz](AUGMENTED-WITNESS.json.gz) for
+the literal word, and [SIDE-TRANSPORTS.json.gz](SIDE-TRANSPORTS.json.gz) for
+the native recipes.
+
+This is a local tail construction. It does not assign global owners or hosts
+to all source reads, certify copied-information delivery, account for a global
+assembly, or establish an all-size supplier. Those obligations prevent any
+new kappa claim. The merged [#1940 chronological schedule](https://github.com/CrocSwap/integer-mult-bounds/pull/1940)
+attains the 32-chain target-order bound, but leaves exact-copy formation, dirty
+restoration, side corrections, hosting, and precision unresolved. This package
+does not integrate with that schedule.
+
+The frozen channel in `research/p104-native-channel` retains Rohan Arun's
+scalar/frame channel from [#1972](https://github.com/CrocSwap/integer-mult-bounds/pull/1972)
+and the native Clifford realization accepted in
+[#1991](https://github.com/CrocSwap/integer-mult-bounds/pull/1991). This
+package keeps that witness and its certificates unchanged. Source commit, Git
+blob, and SHA256 pins are listed in [SOURCES.json](SOURCES.json).
+
+The focused result is under review in [PR #2089](https://github.com/CrocSwap/integer-mult-bounds/pull/2089).
+Its canonical investigation thread is
+[Discussion #2092](https://github.com/CrocSwap/integer-mult-bounds/discussions/2092);
+the completed Cycle 16 claim is linked from the mechanism hub.
+
+## Reproduce
+
+From repository root, with Python 3.12:
+
+```sh
+python3 scripts/research_package.py --package research/p104-tail-side-correction-seed749717 --check-only
+python3.12 scripts/research_package.py --package research/p104-tail-side-correction-seed749717
+```
+
+The registered replay checks the pinned native positive control, rebuilds the
+row correction, verifies every native rank-one operator with the independent
+native interpreter, runs the exact rational endpoint induction and failure
+controls, and checks the deterministic artifact hashes.
+
+## Open next questions
+
+Can another sparse factorization or a different cut reduce the 208 paid
+rank-one frame changes while admitting a global owner/host schedule? The first
+step is to screen the changed inventory's optimistic paid profile before
+building a second word. The current global owner/host assignment is unresolved.
+
+## Credit
+
+Prepared by Cinna Davis with substantial OpenAI Codex assistance in verifier
+implementation, exact cost screening, and package preparation. The exact
+claims are those replayed by the included verifier. New package work is
+Apache-2.0; predecessor source notices remain in the pinned inputs.

research/p104-tail-side-correction-seed749717/RESULT.json

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+{
+  "artifact_sha256": {
+    "AUGMENTED-WITNESS.json.gz": "fa7487b0b0b4e8fb6bdb1bed1171498a4ed487a1efeb8bb93897e379200000a0",
+    "CORRECTION-ROWS.json": "59c145194ba070fda338619e890f25a4a8ae7f37facf3689c1c8c89f79e8bd36",
+    "SIDE-TRANSPORTS.json.gz": "21214b925da7d60953843a65b7a8e212cca49549adf9720f8258c4cf42cd3724"
+  },
+  "augmented_events": 9258,
+  "base_commit": "7c1e7811c730b17fe0eeef6c82ef011318b0465e",
+  "corrected_minus_native_ceiling_interval": [
+    "-32835661679/125000000000000",
+    "-26268529343/100000000000000"
+  ],
+  "correction_rows": 104,
+  "exact_moment_at_a_1_over_99": [
+    "117007905095648759337895851407138263526888781275233533712573438594931867332135/115792089237316195423570985008687907853269984665640564039457584007913129639936",
+    "117007905095648759337895851407138263526888781275233533712573438594931867332157/115792089237316195423570985008687907853269984665640564039457584007913129639936"
+  ],
+  "formal_columns": 280,
+  "native_channel_is_frozen": true,
+  "native_channel_local_ceiling_grid_bracket": [
+    "109658733077/31250000000000",
+    "701815891693/200000000000000"
+  ],
+  "native_channel_status": "PASS_CONCRETE_NATIVE_P104_LOW_RANK_CHANNEL",
+  "native_prefix_events": 4058,
+  "negative_controls": {
+    "missing_correction_entry": true,
+    "missing_promotion": true,
+    "missing_restoration": true,
+    "wrong_correction_sign": true
+  },
+  "nonzero_scalar_additions": 4992,
+  "optimistic_local_ceiling_grid_bracket": [
+    "3246394165033/1000000000000000",
+    "1623197082517/500000000000000"
+  ],
+  "paid_profile": {
+    "1": 1282,
+    "2": 58,
+    "3": 86,
+    "4": 104,
+    "5": 48,
+    "6": 24,
+    "7": 8,
+    "8": 8
+  },
+  "ports": 104,
+  "priority_revision": 4,
+  "rank_mass": 2576,
+  "rank_one_frame_changes": 208,
+  "remaining_obligation": "Prove a globally legal owner/host schedule for the appended source reads and rank-one transports, charge its copied information, and reassess complete supplier cost.",
+  "scope": "One finite rowwise I-G/2 correction appended at the final source-completed cut. Exact 280-column identity and native rank-one transports are verified. This does not supply global chronological helper hosting, all-size admission, precision accounting, or a new kappa.",
+  "seed": "749717109203f916f620456c85e26673",
+  "selected_record_sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32",
+  "status": "PASS_P104_EXACT_TAIL_SIDE_CORRECTION",
+  "track": "complex-mechanisms"
+}

research/p104-tail-side-correction-seed749717/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "7c1e7811c730b17fe0eeef6c82ef011318b0465e",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
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+      "usage": "provenance",
+      "git_blob": "e8870e5917057d59b730eb14721a778639eca050",
+      "sha256": "d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7"
+    },
+    {
+      "path": "docs/research/community-batch1099-review.md",
+      "usage": "input",
+      "git_blob": "6cb6ec061a071a06a2d18cd4dc3e9a2b05d28099",
+      "sha256": "7e76d2c38e5fce30587835b7251df1430488ae4dd372611ef69a3f03f476b6cb"
+    },
+    {
+      "path": "research/community-frontier1094-audit/ASSEMBLY.json",
+      "usage": "input",
+      "git_blob": "5fd676df060f3b450782fdf5ff5f6beb63a5eeb0",
+      "sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045"
+    },
+    {
+      "path": "research/community-frontier1094-audit/SELECTED-SOURCE.json",
+      "usage": "input",
+      "git_blob": "6a75493790b4eff9c4a98051ef8b01d6aad0d5a8",
+      "sha256": "79b3de5db3dbcb9829789ba58e9e45758926031e2a7d80667a08429d369be53b"
+    },
+    {
+      "path": "research/complex-local-word-ceiling/RESULT.json",
+      "usage": "input",
+      "git_blob": "77f55043a679dd722330f1e2499863e36376d9f9",
+      "sha256": "d91d6c46509875abf1f61de67c2609656d3dc685451299f83caa6f922493692d"
+    },
+    {
+      "path": "research/complex-local-word-ceiling/ceiling.py",
+      "usage": "input",
+      "git_blob": "6c0f54527d879396f689523bfde19928a9b6157c",
+      "sha256": "86a1432870b3b21d8454106ec9162fddc0471472d9214c590a1420f2426cedfc"
+    },
+    {
+      "path": "research/p104-native-channel/NATIVE-RECIPES.json.gz",
+      "usage": "input",
+      "git_blob": "ab129678c2df4f0d39b02b88457d02c37de0d9b6",
+      "sha256": "a1c4d83d3d5fe1338e99b59272e8cf5d43244c33888597da340b2aa7fef5b3e8"
+    },
+    {
+      "path": "research/p104-native-channel/SOURCES.json",
+      "usage": "input",
+      "git_blob": "901ad0fb42b62eefedecb467ad45c884f494c163",
+      "sha256": "2ad02321c659c4ebb75fae1a0265437506e7a1fb5ab5ff17444a8be5692fcf73"
+    },
+    {
+      "path": "research/p104-native-channel/check_native.py",
+      "usage": "input",
+      "git_blob": "f8e54afbf146cd6f250063bcb1b9dabf0cbc42ea",
+      "sha256": "e67b874a061a4a28007b9a0fabaf30c8db4a587b1e1c7dd9396c77ff1e3a9fe6"
+    },
+    {
+      "path": "research/p104-native-channel/compile.py",
+      "usage": "input",
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+      "sha256": "9d2fc2d81962ddb0b4dd7d6482d9918a87a6b05759690317dea79939ca5656a3"
+    },
+    {
+      "path": "research/p104-native-channel/inputs/LOWRANK-RESULT.json",
+      "usage": "input",
+      "git_blob": "cbd1c6bdc9e34b25c64b2636ccd6acd2392232d4",
+      "sha256": "7143ec7780f6f38a8c294df8009c3621936b281d05ae79946ce22f581b73d1b1"
+    },
+    {
+      "path": "research/p104-native-channel/inputs/LOWRANK-WITNESS.json",
+      "usage": "input",
+      "git_blob": "86beaf424e9f0f800ae3a62797d94ec9386b7706",
+      "sha256": "0f541bef7f19fdeffd67157684e9f03d785f42c89973632f508fc1a0452b580e"
+    },
+    {
+      "path": "research/p104-native-channel/inputs/P104-FIXTURE.json",
+      "usage": "input",
+      "git_blob": "2e16f99b295bb386fd13bd8fbd57b42ef54143a9",
+      "sha256": "516c06f6ec3cb84be588c546108de897650288576dd5fbf5fe44b9c5546ad8a4"
+    },
+    {
+      "path": "research/p104-native-channel/vendor/clifford_exact.py",
+      "usage": "input",
+      "git_blob": "166841a037f069249f6f7f6dedb89248812f59e2",
+      "sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962"
+    },
+    {
+      "path": "research/p104-native-channel/vendor/native_verify.py",
+      "usage": "input",
+      "git_blob": "2c209dfa9cfade3bc20ac5fc9d876b433a147f8c",
+      "sha256": "3049471905b47b385c88378a1a3be7a18b8858aadd035b8a6da2f4a03b470f38"
+    }
+  ]
+}

research/p104-tail-side-correction-seed749717/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "p104-side-correction",
+  "priority_revision": 4,
+  "base_commit": "7c1e7811c730b17fe0eeef6c82ef011318b0465e",
+  "upstream_checkpoint": {
+    "canonical_main_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+    "priority_revision": 4,
+    "selected_record_sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32",
+    "checked_at_utc": "2026-10-14T19:01:55.787429Z",
+    "changed_guidance": "Current main adds mandatory upstream refresh checkpoints and asks loops to maintain a canonical investigation thread with public artifact status. Task catalogue, selected result and scientific inputs stayed unchanged from the pinned experiment base.",
+    "overlap_snapshot": {
+      "open_prs": 352,
+      "discussion_records": 736,
+      "complete_pagination": true,
+      "warnings": [],
+      "narrow_exact_queries": [
+        "P104 I-G/2 direct tail correction",
+        "rank-one target restoration"
+      ],
+      "narrow_exact_matches": 0,
+      "broader_task_search": "Found relevant but nonoverlapping predecessors #1940, #1991 and #1972; #1940 leaves side correction and copy/hosting unresolved."
+    },
+    "relevant_predecessors": [
+      {
+        "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1940",
+        "head": "a9b7dfe2809092a9a4c90784d12bb60061e995e8",
+        "status": "closed",
+        "distinction": "Attains the 32-chain target-order bound with 416 reads, but leaves exact-copy formation, dirty restoration, side corrections, hosting and precision unresolved. This package supplies only a local side-correction word and does not integrate that schedule."
+      },
+      {
+        "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1991",
+        "head": "6e222517e182e98499239bc9a4009f540e1c5520",
+        "status": "merged",
+        "distinction": "Supplies the exact native Clifford realization of the P104 G/2 channel; the missing I-G/2 correction remains outside its scope."
+      },
+      {
+        "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1972",
+        "head": "fd82d172f46933fd253c2c6de4bd5eaff5429cfd",
+        "status": "open-draft",
+        "distinction": "Original G/2 scalar/frame channel predecessor; no side correction or complete supplier is claimed."
+      }
+    ]
+  },
+  "original_card": {
+    "id": "p104-side-correction",
+    "question": "Can the native P104 low-rank channel be completed with a legal, fully charged I-G/2 side correction?",
+    "starting_points": [
+      "research/p104-native-channel/README.md",
+      "research/p104-native-channel/PROOF.md",
+      "docs/research/community-efficient1000-review.md"
+    ],
+    "methods": [
+      "Construct one side-correction word at an explicit chronological cut with the needed frames still available.",
+      "Derive a lower bound on the side correction and hosting costs that decides whether this channel can beat the selected budget."
+    ],
+    "acceptance_gate": "An exact arbitrary-input identity, source/target frame availability and restoration at the chosen cut, plus the complete channel+correction+idle/hosting bill or an explicit favorable bound. The native channel alone is not a supplier.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read the P104 open next questions and its fixed local histogram; inspect why appending I-G/2 after source completion is not justified. #1991 supplies the checked native channel, not the missing correction.",
+      "scope": "Source-bound orientation only. Read the named package and review before relying on its local evidence; this command is not its replay or supplier admission."
+    },
+    "first_experiment": "Choose one exact side-correction cut and bound its allowed cost from a favorable completed moment at a declared target. Stop or hand off if the necessary source/target access cannot be justified or even the optimistic total misses the target.",
+    "bottleneck_link": "Completing an already realized changed channel tests a concrete construction opportunity, instead of refining the incumbent packing. A positive local result still needs global and all-size admission.",
+    "allowed_variations": [
+      "One correction factorization, one chronological cut, or one explicit hosting obstruction.",
+      "A changed correction schedule with an updated full bill and native frame contract."
+    ],
+    "scope_change_triggers": [
+      "Recompiling the already certified low-rank channel without closing a new obligation.",
+      "Treating a pending ancestor, local histogram or unpriced side correction as an admitted global supplier."
+    ],
+    "overlap_terms": [
+      "P104",
+      "I-G/2",
+      "side correction",
+      "native channel",
+      "1972",
+      "1991"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "At the final source-completed cut of the pinned 4,058-event P104 G/2 channel, construct the direct rowwise I-G/2 correction, restore each target frame, and measure its exact local paid profile.",
+  "scope_change_reason": "The catalogue permits one correction factorization at an explicit chronological cut. This run fixes the cut after all 104 X registers are full-frame and all 104 Y registers are in their final rank-eight hyperplanes, then tests the direct sparse row factorization only.",
+  "bottleneck_link": "The pinned channel stops at Y + (G/2)X. This construction supplies the exact missing matrix I-G/2 with native rank-one target promotion and restoration; its additional 208 paid rank-one changes are included in the local profile. Global hosting and copied-information admission remain open.",
+  "budget_minutes": 60,
+  "stop_rule": "Use only the final cut after the pinned 4,058-event prefix and the literal correction coefficient delta[t,s]-G[t,s]/2. Stop after exact replay of all 104 rows, 4,992 nonzero additions, 208 rank-one transports, and four failure controls; do not search alternative factorizations or cuts in this cycle.",
+  "outcome": "partial",
+  "publication": {
+    "submitted_at_utc": "2026-10-14T19:10:37Z",
+    "pr_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2089",
+    "pr_status_at_handoff": "open",
+    "canonical_discussion_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/2092",
+    "completed_hub_claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1167#discussioncomment-18932282"
+  },
+  "evidence": "Registered package replay passes. The 280-column rational induction proves the appended operator changes every Y_t row from y_t+(G/2)x to y_t+x and leaves every other row unchanged. Each of 208 frame changes has an independently interpreted exact native rank-one recipe. The charged local profile is {1:1282,2:58,3:86,4:104,5:48,6:24,7:8,8:8}, rank mass 2576. The exact optimistic local ceiling worsens relative to the uncorrected channel by the interval recorded in RESULT.json. No global owner/host or copied-information schedule is certified, so this is not an admitted supplier or a new kappa. The nearest completed schedule #1940 does not provide these correction or copy obligations, and this package does not claim to resolve its global integration gap.",
+  "author": "Cinna Davis",
+  "ai_assistance": "OpenAI Codex provided substantial assistance with verifier implementation, exact cost screening, and package preparation. The result is limited to the explicit rational identity and native recipes checked by the reproducible package.",
+  "open_questions": [
+    {
+      "question": "Can a different sparse factorization or interleaved correction schedule reduce the number or rank of native frame changes while retaining a globally legal owner/host schedule?",
+      "why_it_matters": "The direct tail word adds 208 rank-one paid frame changes and worsens its same-model local ceiling; further copied-information and hosting costs can only affect admission after a complete schedule is proved.",
+      "first_step": "Factor I-G/2 by a mechanism with a changed helper inventory or cut, then compute its optimistic fully charged local profile before constructing the word.",
+      "blocked_by": "No global owner/host assignment for the current 104-by-104 source reads has been established."
+    }
+  ]
+}