Selected packing: slack-7 three-bin cycles all fail windows

research/selected-slack7-cycle-obstruction/README.md

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+# Slack-7 three-bin cycles on the selected owner map
+
+**Negative result. No new κ.** On the selected PR1094 owner map, every directed
+three-role cycle on three distinct occupied bins that touches a slack-7 bin and
+strictly majorizes the slack triple is illegal for the selected source-event
+windows. The complex primitive still binds. The selected record is unchanged.
+
+Selector nonce `c6124726e8f026f5858c9552e1483c86`, track
+`incumbent-optimization`, card `selected-packing` (scope key
+`ab5d8f0f0d19bbab8625`), priority revision 2. Base
+`ef5f39403610d231a0c27e8a6b4f08c10209bcd6`. Scope is **refined** to this one
+residual. The baseline binding check returned
+`PASS_RESEARCH_BASELINE_BINDINGS_ONLY`.
+
+## Cycle and sign
+
+A directed cycle of roles A, B, C in three distinct occupied bins sends A into
+B's bin, B into C's bin, and C into A's bin. One role leaves each bin and one
+role arrives, so the slack update is
+
+```text
+slack' = slack + departing width - arriving width
+```
+
+and every new slack must be nonnegative. Prefix compaction is the only offset
+freedom used: the test keeps total width inside capacity. It does not rebuild
+charts.
+
+The selected idle slacks on occupied bins are only `{0,1,3,7}`. Exactly ten
+roles sit in the seven slack-7 bins, in classes `(7,4)`, `(7,8)` and `(7,9)`.
+Every directed cycle that touches a slack-7 bin is written once, starting at
+its lowest-index slack-7 role. There are 13 occupied `(slack, width)` classes
+and `3 * 13 * 13 = 507` anchored class patterns:
+
+| sign | patterns |
+|---|---:|
+| uniformly improving | 77 |
+| uniformly worsening | 96 |
+| histogram-neutral | 39 |
+| capacity-infeasible | 275 |
+| not majorization-ordered | 20 |
+
+For an equal-sum triple, strict majorization is the usual strict partial-sum
+order. For every real `0<a<1`, `t -> t^(1-a)` is strictly concave on
+`[0, infinity)`, so a strictly majorizing new triple has a strictly smaller
+idle power sum. Neutral patterns do not change the multiset. Incomparable
+patterns are not claimed to have one sign on the whole interval.
+
+## What was counted
+
+Among the 77 improving patterns there are **32,981,939** capacity-feasible
+directed cycles and **0** whose three landings all lie in the destination
+window lists of `PLAN.json`. Those lists are the selected plan's strict
+source-event intervals. The same count includes **0** same-call cycles.
+Recomputing the two published rank-8 patterns gives the recorded sizes:
+
+| classes `(slack, width)` | old slacks | new slacks | capacity | window-legal |
+|---|---|---|---:|---:|
+| `(7,8), (0,8), (1,7)` | `(7,0,1)` | `(8,0,0)` | 17,567 | 0 |
+| `(7,8), (1,7), (0,7)` | `(7,1,0)` | `(8,0,0)` | 4,319 | 0 |
+
+The other 75 improving patterns, including every pattern that uses a slack-3
+bin, are window-empty as well. The 20 incomparable patterns contribute
+**325,142** further capacity-feasible cycles, also with **0** window-legal
+landings. That is an exclusion, not a moment-sign theorem.
+
+The identity check accepts all 69,120 selected roles inside their own windows.
+A repair control opens the two blocked legs of one `(7,8),(0,8),(1,7)` triple
+whose middle time is unique among landings that already accept the slack-7
+role. The census then reports exactly one window hit, so an empty real count
+is not a blind counter. No repaired triple is part of the mathematical claim.
+
+## What this does not show
+
+Active-pair exclusion, per-call family uniqueness, chart reconstruction,
+controller emission, native replay, invoice, moment repricing and the outer
+assembly were not run. The window lists were not regenerated from the parent
+execution schedule; the census uses the committed `PLAN.json` lists. Windows already reject every candidate, so those later
+checks cannot create a legal cycle inside this domain. Cycles that never touch
+a slack-7 bin, cycles on fewer than three bins, and sequential moves are
+outside the count. No supplier adoption or selected-frontier change is claimed.
+Discussion writes for this account were rejected with FORBIDDEN, so no claim
+was posted; that limitation is recorded here and was not retried.
+
+## Reproduce
+
+Python 3.12, standard library, from the repository root. `PYTHONOPTIMIZE` was
+unset for the first command. The C++ toolchain was aligned for the session and
+is not used by this census.
+
+```sh
+python3 -B research/selected-slack7-cycle-obstruction/verify.py
+python3 -I -O -B research/selected-slack7-cycle-obstruction/verify.py
+python3 -B scripts/research_package.py --package research/selected-slack7-cycle-obstruction
+```
+
+Expected primary status:
+
+```text
+PASS_SELECTED_SLACK7_CYCLE_OBSTRUCTION improving_patterns=77 capacity=32981939 window=0 same_call=0 known=17567+4319 incomparable_patterns=20 incomparable_capacity=325142 incomparable_window=0 synthetic_window_hits=1 roles=69120 no_new_kappa
+```
+
+## Open next questions
+
+Does any directed three-bin cycle that avoids every slack-7 bin strictly
+majorize its slack triple and land inside the selected source-event windows?
+A yes answer would be an idle-histogram witness on the binding complex side.
+The smallest next join anchors the same majorization test at the 754 slack-3
+roles and drops triples that also touch slack 7. The blocker is the larger
+class products, together with three-component transitions that are not
+majorization-ordered.
+
+## Credit
+
+The owner map is Rohan Arun's PR1094, retaining Jeremy Kothe's packing lineage
+and Hao Yan's window geometry. The rank-8 counts 17,567 and 4,319 were
+published by Aaron Wang / navinor in PR2145; this package recomputes them and
+extends the census to every slack-7-touching majorizing pattern. The same-call
+three-role census in PR1873 is a different slice; the zero same-call count
+here is only inside the slack-7 anchor. Josie Elliston's PR1091 owner audit
+remains the geometry predecessor and was not rerun. Apache-2.0.
+
+Prepared with substantial AI assistance for the census design, exact checks,
+implementation and write-up. No independent human review or formal-kernel
+verification is claimed.

research/selected-slack7-cycle-obstruction/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "ef5f39403610d231a0c27e8a6b4f08c10209bcd6",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "c05fafaa25ba95b2d5af86b1800427198e3d20eb",
+      "sha256": "89b5928bc92c81e2647b6d702a7ad85a6795b8f44e2eb267b2ff3145d8f2b2ce"
+    },
+    {
+      "path": "research/complex-t16-rare-pattern-exchanges/data/OWNERS.json.gz",
+      "usage": "input",
+      "git_blob": "d3ba33d7e703a3229d9fb8a360f69d2f318bbb70",
+      "sha256": "87902c528dfab7d503dcf8a784260e3ab6fb7424c418a1820f7261f25be7b6bc"
+    },
+    {
+      "path": "research/complex-t16-rare-pattern-exchanges/data/PLAN.json",
+      "usage": "input",
+      "git_blob": "f1bec76e8e9b7a1f09c1e05cb6cce973b6e59f1b",
+      "sha256": "7bd786ba8e66408f02aa9ef446eaf20c1bfe981ea2e4dd1e5a929f0aa7767bc0"
+    }
+  ]
+}

research/selected-slack7-cycle-obstruction/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "selected-packing",
+  "priority_revision": 2,
+  "base_commit": "ef5f39403610d231a0c27e8a6b4f08c10209bcd6",
+  "original_card": {
+    "id": "selected-packing",
+    "question": "Is there a genuinely unsearched, explicitly bounded improvement to the selected packing?",
+    "starting_points": [
+      "certificates/selected-result.json",
+      "research/heterogeneous-owner-assembly/README.md"
+    ],
+    "methods": [
+      "Search a named unclaimed canonical candidate domain with a reproducible partition.",
+      "Prove a stronger scoped relaxation or exact optimum for one named residual subproblem."
+    ],
+    "acceptance_gate": "Check live PR overlap first, price both suppliers and distinguish a search best from an optimum.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Locate the selected packing source in the review and check live PRs for the same residual problem.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Define an unclaimed canonical domain and a favorable bound before starting a new search.",
+    "bottleneck_link": "An actual paid-profile gain can improve the frontier up to the other supplier cap.",
+    "allowed_variations": [
+      "One bounded residual packing instance or exact relaxation."
+    ],
+    "scope_change_triggers": [
+      "Rerunning an existing search with only a different seed as a separate mathematical contribution."
+    ],
+    "overlap_terms": [
+      "T16",
+      "owner exchange",
+      "packing"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "On the selected PR1094 owner map, every directed three-role cycle on three distinct occupied bins that touches a slack-7 bin and strictly majorizes the slack triple fails the selected source-event window list.",
+  "scope_change_reason": "The card allows one bounded residual of the selected packing. The full T16 search and the single rank transition {0,1,7}->{0,0,8} are already published. This census is the remaining slack-7-touching majorization class, including that transition as a recomputed control.",
+  "bottleneck_link": "A legal idle-histogram change is what could raise the binding complex primitive. This residual has 32,981,939 capacity-feasible uniform improvements and zero window-legal ones, so it contributes no paid-profile or kappa gain.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after the complete slack-7-anchored three-bin census, its incomparable companion, and the synthetic window control. Do not open a slack-3 anchor search in this session.",
+  "outcome": "negative-result",
+  "evidence": "PASS_SELECTED_SLACK7_CYCLE_OBSTRUCTION improving_patterns=77 capacity=32981939 window=0 same_call=0 known=17567+4319 incomparable_patterns=20 incomparable_capacity=325142 incomparable_window=0 synthetic_window_hits=1 roles=69120 no_new_kappa",
+  "author": "AI-assisted research session",
+  "ai_assistance": "Substantial AI assistance was used for the census design, exact arithmetic, implementation, checks, and write-up.",
+  "open_questions": [
+    {
+      "question": "Does any directed three-bin cycle that avoids every slack-7 bin strictly majorize its slack triple and land inside the selected source-event windows?",
+      "why_it_matters": "The complex primitive still binds. A window-legal majorizing cycle outside this anchor would be a concrete idle-histogram improvement to price.",
+      "first_step": "Repeat this majorization join with the anchor restricted to the 754 roles in slack-3 bins, and reject any triple that also touches slack 7 so the domains stay disjoint.",
+      "blocked_by": "Slack-3 class products are much larger than the slack-7 join, and three-component transitions that are not majorization-ordered still need a separate exact sign rule."
+    }
+  ]
+}

research/selected-slack7-cycle-obstruction/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/complex-t16-rare-pattern-exchanges/data/OWNERS.json.gz",
+    "research/complex-t16-rare-pattern-exchanges/data/PLAN.json"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Recompute the slack-7 three-bin census, the known 17567+4319 rank-8 counts, the incomparable companion, and the one-hit synthetic window repair."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Repeat the census with assertions stripped and isolated imports; the explicit require checks must still pass."
+    }
+  ]
+}