Mister-Meeseeks · Oct 14, 2026 · Oct 13, 2026 · Oct 13, 2026
diff --git a/research/frontier1094-idle-star-ceiling/README.md b/research/frontier1094-idle-star-ceiling/README.md
@@ -0,0 +1,111 @@
+# Idle rank-48 repartitions on #1094: exact ceiling, PR1252 recount and a star-move lead
+
+**Scoped finding; no new selected κ and no admitted construction.** For one named
+residual subproblem of the selected #1094 packing, the idle rank-48 remainder, this
+package gives an exact price ceiling, an independent recount of PR1252's matching
+maximum, and the exact optimum of a broader move class with an explicit overlay. The
+overlay is structurally verified but has **not** been through the controller, invoice
+or strict assembly admission that PR1252 ran. The selected record, `upstream/` and
+every frozen input are unchanged.
+
+## Object
+
+The frozen #1094 replay (`research/community-frontier1094-audit/selected-replay.tar.gz`,
+`candidate/strict/global/hosted/IDLE-REMAINDER-FRAMES.json.gz`) has **4,866 idle
+rank-48 children** that are unhosted, have no pending offset and are absent from the
+T16 owner map. They lie in **1,792 (replica, port) groups** of sizes
+`{1: 45, 2: 703, 3: 761, 4: 283}`. The rest of the idle remainder (hosted rank-1/3/7
+children) and all local-word children are held fixed.
+
+PR1252 (Gonzalo-Mier, head `1b9db2fa`) moves 3 commuting axes inside 1,825 disjoint
+pairs, `48 + 48 → 45 + 51`. Its open questions ask for an independent reproduction of
+the matching maximum and for coordinated moves among more than two children.
+
+## Result
+
+Exact native prices from the pinned multicold `moment`/`base_two_moment`/`outer` engines
+(the price and outer recipe are PR1252's `math_support.price`, restated):
+
+| Profile | 48-children become | κ (native) | Gain over #1094 | Share of ceiling gain |
+|---|---|---|---:|---:|
+| #1094 selected | 4,866 × 48 | 0.001672660197076918 | — | 0% |
+| PR1252 pair class (independent recount: **1,825**) | 1,825×45, 1,216×48, 1,825×51 | 0.001672846574134982 | 1.864e-7 | 37.9% |
+| **Disjoint-star class optimum** | 796×42, 765×45, 948×48, 2,357×51 | **0.001672987659462603** | 3.275e-7 | 66.6% |
+| **Ceiling: any within-group repartition** | 283×39, 761×42, 703×45, 45×48, 3,074×51 | **0.001673151843213755** | 4.916e-7 | 100% |
+
+Conservative prices are 0.001672660194225008 (matches the selected record), 0.001672846571282754
+(matches PR1252), 0.001672987656610135 and 0.001673151840361007.
+
+1. **Ceiling.** In a group of `k` such children the mass is `48k`. No child can be
+   deleted, because `48k > 51(k−1)` for `k ≤ 16`, and splitting a child only raises the
+   subadditive moment. The halving guard caps ranks at 51 (`73² > 2·51²`). The main
+   moment `Σ r^(1−a)` is Schur-concave at a fixed child count, so the
+   majorization-maximal vector `(51^(k−1), 51−3k)` minimizes it. With the outer recipe
+   unchanged, **no within-group repartition of these children, under any commuting rule,
+   can exceed κ = 0.001673151843213755**. As a control, four deterministic random
+   repartitions are priced and all stay below the ceiling. This bounds PR1252's
+   "multi-child moves" route at about **+0.00000049** κ, which is 3.4% of PR1240's
+   fixed-word packing headroom (κ < 0.0016872026). Larger gains need hosted or
+   local-word mass, which PR1240 covers, or a different word.
+2. **PR1252 recount.** Under PR1252's rule, an independent per-group exhaustive
+   matching finds exactly **1,825** pairs. Under the same rule, this reproduces its κ
+   at both native and conservative precision.
+3. **Star class.** A centre child sends up to 3 axes to each of several leaves, and
+   leaves reach 51. Every moved axis satisfies the per-axis PR1252 rule: it commutes
+   with all original leaf axes, every stage entry/exit X and Y, the terminal Y
+   translation and the terminal translations of both centre and leaf. The class
+   contains the pair class. The exact per-group optimum comes from enumerating set
+   partitions × centres with bipartite b-matching of axes, and the choice is invariant
+   between `a = 0.0016756` and `a = 0.01`. It yields **2,357 moves from 1,561
+   centres** and κ = 0.001672987659462603. This is +1.41e-7 over PR1252 and two
+   thirds of the ceiling gain. The moves are in `STAR-MOVES.json.gz`.
+
+## Verification
+
+```sh
+python3 -B research/frontier1094-idle-star-ceiling/analyze.py --check   # ~5 s
+python3 -B research/frontier1094-idle-star-ceiling/verify.py            # ~2 s
+python3.12 scripts/research_package.py --package research/frontier1094-idle-star-ceiling
+```
+
+`analyze.py --check` recomputes everything from the committed replay archive and
+requires byte-identical `RESULT.json` and star payloads. Each price asserts acceptance
+at its exact endpoint, rejection at endpoint+1e-24, all 47 strict outer constraints,
+and rejection of the next 1e-18 κ grid point. `verify.py` does not import
+`analyze.py`. It re-derives every move's commuting conditions and checks the orthonormal
+resulting axis sets, unchanged per-group axis multisets (hence each group's Z4 phase
+polynomial), disjoint stars, the rank cap, the absence of hosted and owner children, and
+the profile. Seven mutation controls must be rejected: a non-commuting axis, an axis
+moved twice, a reused leaf, a rank-52 leaf, a foreign axis, a hosted leaf and a centre
+that is also a leaf. Its receipt is `STAR-AUDIT.json`.
+
+**Not checked:** the serialized controller/owner audit, the mixed-provider invoice and
+the strict global assembly with its negative controls, for the star overlay. The
+soundness of composing PR1252 transfers into stars is inherited from PR1252's pair
+argument and not proved here (see the open questions). The all-size compiler,
+common-label/mutable-source realization, hosting, precision/tape and analytic-transfer
+obligations of the selected record remain conditional.
+
+## Open next questions
+
+The structured list is in [TASK.json](TASK.json):
+
+- **Admission of the star overlay.** Generalize PR1252's `audit_allcold_idle_overlay.py`
+  from pairs to `(centre, leaf, moved_axes)` records, re-pin the SPEC and rerun its
+  controller and assembly controls. This is blocked on PR1252 being unmerged.
+- **Contract for multi-leaf moves.** Does a moved idle axis need to commute with
+  anything beyond the donor and receiver? The first step is to state the controller
+  contract.
+- **Stacking on stronger words.** Apply the same analysis to the idle remainders of the
+  open X54 / mixed-T37 candidates (PR1218/PR1261, κ ≈ 0.0016758), since the gain is
+  additive in the moment.
+
+## Credit
+
+Gonzalo-Mier's PR1252 provides the move, the commuting rule and the pricing recipe, and
+its open questions define this scope. The #1094 replay, T16 owner map and engines are
+those of the selected review and its retained supplier ancestry. PR1240 (Douglas Colkitt)
+provides the broader fixed-word packing ceiling. Coordination is in
+[Discussion #1265](https://github.com/CrocSwap/integer-mult-bounds/discussions/1265).
+This package was prepared by Guéorgui Maissouradze (GitHub: PatatePatatePatate) with
+substantial Claude (Anthropic) assistance. Apache-2.0.
diff --git a/research/frontier1094-idle-star-ceiling/RESULT.json b/research/frontier1094-idle-star-ceiling/RESULT.json
@@ -0,0 +1,174 @@
+{
+  "status": "PASS_IDLE48_REPARTITION_CEILING",
+  "replay_archive": {
+    "research/community-frontier1094-audit/selected-replay.tar.gz": "f1da80610bf33c58527f18ad63b4350d0b864e1d4486141ee17ac9bed4d529d6"
+  },
+  "inputs": {
+    "hosted/IDLE-REMAINDER-FRAMES.json.gz": "650c14a1de75015fdae5c02ea18f1d0fd2de35d249b5ff2cd77f3b1afe604985",
+    "base/work/complex-nine-native/DATA-PHASES.json": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa",
+    "audit/OWNERS-AND-PROFILE.json": "30bafff060afcfb3644e951214d66cfaffe58242881c652e28003958b42127cc",
+    "research/complex-t16-rare-pattern-exchanges/OWNERS.json.gz": "87902c528dfab7d503dcf8a784260e3ab6fb7424c418a1820f7261f25be7b6bc",
+    "research/community-frontier1094-audit/ASSEMBLY.json": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045"
+  },
+  "eligible_children": 4866,
+  "owner_slot_rank48_children": 0,
+  "groups": 1792,
+  "group_size_histogram": {
+    "1": 45,
+    "2": 703,
+    "3": 761,
+    "4": 283
+  },
+  "pair_class_maximum_pairs": 1825,
+  "star_vector_invariant_between_a": [
+    0.0016756,
+    0.01
+  ],
+  "star_vector_invariant": true,
+  "star_rank_histogram": {
+    "42": 796,
+    "45": 765,
+    "48": 948,
+    "51": 2357
+  },
+  "relaxation_rank_histogram": {
+    "39": 283,
+    "42": 761,
+    "45": 703,
+    "48": 45,
+    "51": 3074
+  },
+  "profiles": {
+    "baseline_1094": {
+      "1": 349604,
+      "2": 60192,
+      "3": 62585,
+      "4": 20448,
+      "5": 11808,
+      "7": 7,
+      "8": 1152,
+      "48": 4866
+    },
+    "pair_class_max (PR1252 rule, independent recount)": {
+      "1": 349604,
+      "2": 60192,
+      "3": 62585,
+      "4": 20448,
+      "5": 11808,
+      "7": 7,
+      "8": 1152,
+      "45": 1825,
+      "48": 1216,
+      "51": 1825
+    },
+    "star_class_optimum": {
+      "1": 349604,
+      "2": 60192,
+      "3": 62585,
+      "4": 20448,
+      "5": 11808,
+      "7": 7,
+      "8": 1152,
+      "42": 796,
+      "45": 765,
+      "48": 948,
+      "51": 2357
+    },
+    "relaxation_ceiling": {
+      "1": 349604,
+      "2": 60192,
+      "3": 62585,
+      "4": 20448,
+      "5": 11808,
+      "7": 7,
+      "8": 1152,
+      "39": 283,
+      "42": 761,
+      "45": 703,
+      "48": 45,
+      "51": 3074
+    }
+  },
+  "prices": {
+    "baseline_1094": {
+      "native": {
+        "endpoint": "67018507072324140917/40000000000000000000000",
+        "endpoint_decimal": "0.001675462676808103522925",
+        "kappa": "836330098538459/500000000000000000",
+        "kappa_decimal": "0.001672660197076918"
+      },
+      "conservative": {
+        "endpoint": "1675462673946628390003/1000000000000000000000000",
+        "endpoint_decimal": "0.001675462673946628390003",
+        "kappa": "104541262139063/62500000000000000",
+        "kappa_decimal": "0.001672660194225008"
+      }
+    },
+    "pair_class_max (PR1252 rule, independent recount)": {
+      "native": {
+        "endpoint": "104728104934992539263/62500000000000000000000",
+        "endpoint_decimal": "0.001675649678959880628208",
+        "kappa": "836423287067491/500000000000000000",
+        "kappa_decimal": "0.001672846574134982"
+      },
+      "conservative": {
+        "endpoint": "1675649676098086086493/1000000000000000000000000",
+        "endpoint_decimal": "0.001675649676098086086493",
+        "kappa": "836423285641377/500000000000000000",
+        "kappa_decimal": "0.001672846571282754"
+      }
+    },
+    "star_class_optimum": {
+      "native": {
+        "endpoint": "1675791237522814605259/1000000000000000000000000",
+        "endpoint_decimal": "0.001675791237522814605259",
+        "kappa": "1672987659462603/1000000000000000000",
+        "kappa_decimal": "0.001672987659462603"
+      },
+      "conservative": {
+        "endpoint": "167579123466077825869/100000000000000000000000",
+        "endpoint_decimal": "0.001675791234660778258690",
+        "kappa": "334597531322027/200000000000000000",
+        "kappa_decimal": "0.001672987656610135"
+      }
+    },
+    "relaxation_ceiling": {
+      "native": {
+        "endpoint": "837977986018757511763/500000000000000000000000",
+        "endpoint_decimal": "0.001675955972037515023526",
+        "kappa": "334630368642751/200000000000000000",
+        "kappa_decimal": "0.001673151843213755"
+      },
+      "conservative": {
+        "endpoint": "1675955969175197279329/1000000000000000000000000",
+        "endpoint_decimal": "0.001675955969175197279329",
+        "kappa": "1673151840361007/1000000000000000000",
+        "kappa_decimal": "0.001673151840361007"
+      }
+    }
+  },
+  "random_repartition_controls": [
+    {
+      "max_rank": 51,
+      "kappa": "836429700237559/500000000000000000",
+      "below_ceiling": true
+    },
+    {
+      "max_rank": 51,
+      "kappa": "836432225741851/500000000000000000",
+      "below_ceiling": true
+    },
+    {
+      "max_rank": 51,
+      "kappa": "1672865883743733/1000000000000000000",
+      "below_ceiling": true
+    },
+    {
+      "max_rank": 51,
+      "kappa": "418215852699067/250000000000000000",
+      "below_ceiling": true
+    }
+  ],
+  "star_moves": 2357,
+  "star_moves_payload_sha256": "d6d46d2acf47ce7e65434685d86a0542bed92b879579d81d5a115fa4c0f6f2dd"
+}
diff --git a/research/frontier1094-idle-star-ceiling/SOURCES.json b/research/frontier1094-idle-star-ceiling/SOURCES.json
@@ -0,0 +1,54 @@
+{
+  "schema_version": 1,
+  "base_commit": "dadfea6e8a78ff6ba901705f87532a71c6f5bbb8",
+  "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/community-frontier1094-audit/ASSEMBLY.json",
+      "usage": "input",
+      "git_blob": "5fd676df060f3b450782fdf5ff5f6beb63a5eeb0",
+      "sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045"
+    },
+    {
+      "path": "research/community-frontier1094-audit/selected-replay.tar.gz",
+      "usage": "input",
+      "git_blob": "c24daf067f9c543421d89ca2eeaa4eb78a57fb6f",
+      "sha256": "f1da80610bf33c58527f18ad63b4350d0b864e1d4486141ee17ac9bed4d529d6"
+    },
+    {
+      "path": "research/complex-k492-owner-repair/vendor/multicold/engines/base_two_moment.py",
+      "usage": "input",
+      "git_blob": "14a90fa0e788217856f070382c9e61bd7c603c61",
+      "sha256": "aee70c242a2f1d225528fabe2002e1564d1dc884797c07d28a831c7878ee9098"
+    },
+    {
+      "path": "research/complex-k492-owner-repair/vendor/multicold/engines/moment.py",
+      "usage": "input",
+      "git_blob": "7d697cb45203fe72d97037403d6da41f71f3b7fd",
+      "sha256": "81cbc864e106418fc7854322746f72180cb8110ff39f74450a3b8d55a82c1482"
+    },
+    {
+      "path": "research/complex-k492-owner-repair/vendor/multicold/engines/outer.py",
+      "usage": "input",
+      "git_blob": "43007488ad2c43187c77cc6798beff8b1cd76ad2",
+      "sha256": "c1ebb52163a7219504c728a5c2b098bf1e50bf20148a2879c5a74c3cadc7c03c"
+    },
+    {
+      "path": "research/complex-t16-rare-pattern-exchanges/OWNERS.json.gz",
+      "usage": "input",
+      "git_blob": "d3ba33d7e703a3229d9fb8a360f69d2f318bbb70",
+      "sha256": "87902c528dfab7d503dcf8a784260e3ab6fb7424c418a1820f7261f25be7b6bc"
+    }
+  ]
+}
diff --git a/research/frontier1094-idle-star-ceiling/STAR-AUDIT.json b/research/frontier1094-idle-star-ceiling/STAR-AUDIT.json
@@ -0,0 +1,42 @@
+{
+  "status": "PASS_INDEPENDENT_IDLE48_STAR_OVERLAY",
+  "moves": 2357,
+  "leaves": 2357,
+  "centres": 1561,
+  "profile": {
+    "1": 349604,
+    "2": 60192,
+    "3": 62585,
+    "4": 20448,
+    "5": 11808,
+    "7": 7,
+    "8": 1152,
+    "42": 796,
+    "45": 765,
+    "48": 948,
+    "51": 2357
+  },
+  "mutation_controls": {
+    "noncommuting_axis": "rejected",
+    "axis_moved_twice": "rejected",
+    "leaf_reused": "rejected",
+    "fourth_axis_rank52": "rejected",
+    "foreign_axis": "rejected",
+    "hosted_leaf": "rejected",
+    "centre_also_leaf": "rejected"
+  },
+  "checks": [
+    "per-axis PR1252 commuting rule",
+    "orthonormal resulting axis sets",
+    "group axis multisets unchanged",
+    "disjoint stars, leaves <= 51",
+    "no owner-map slots, no hosted/pending children",
+    "halving guard 73^2 > 2*51^2",
+    "profile equals RESULT.json star_class_optimum"
+  ],
+  "not_checked": [
+    "serialized controller/owner audit",
+    "mixed-provider invoice",
+    "strict global assembly and its negative controls"
+  ]
+}
diff --git a/research/frontier1094-idle-star-ceiling/STAR-MOVES.json.gz b/research/frontier1094-idle-star-ceiling/STAR-MOVES.json.gz