Conditional κ = 0.001730189061233850760095: idle rank-48 star exchanges on #3148's T740 fill (+0.0238%)

research/rank7-idle-star-exchanges-t740/AUDIT.json

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+{
+ "fill_rebound": true,
+ "gain_over_pr3148": "411566555253748209/1000000000000000000000000",
+ "kappa": "17301890612338507600949999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000",
+ "moves": 125706,
+ "profile": {
+  "1": 16376940,
+  "2": 4821840,
+  "3": 1567431,
+  "4": 679320,
+  "42": 45543,
+  "45": 34620,
+  "48": 76700,
+  "5": 147075,
+  "51": 125706,
+  "8": 53280
+ },
+ "rejected_controls": {
+  "axis_moved_twice": "axis moved twice",
+  "centre_is_leaf": "centre is also a leaf",
+  "foreign_axis": "foreign axis",
+  "hosted_child": "centre/leaf must be idle children of the group",
+  "next_root_grid": "moment < 1 at lower endpoint",
+  "non_commuting_axis": "axis does not commute with leaf axes",
+  "reused_leaf": "leaf reused",
+  "two_axes_only": "exactly three distinct axes",
+  "wrong_profile": "declared profile"
+ },
+ "status": "PASS_IDLE48_STAR_EXCHANGE_AUDIT"
+}

research/rank7-idle-star-exchanges-t740/CLAIM.json

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+{
+ "kappa": "17301890612338507600949999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000",
+ "kappa_decimal": "0.0017301890612338507600949999999999999999999998999999999999999999999999",
+ "root": [
+  "17301890612338507600957/10000000000000000000000000",
+  "8650945306169253800479/5000000000000000000000000"
+ ],
+ "baseline": "PR3148 (open; Jeremy Diviney) head f2927a4b8",
+ "baseline_kappa": "17297774946785970118859999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000",
+ "gain": "411566555253748209/1000000000000000000000000",
+ "profile": {
+  "1": 16376940,
+  "2": 4821840,
+  "3": 1567431,
+  "4": 679320,
+  "5": 147075,
+  "8": 53280,
+  "42": 45543,
+  "45": 34620,
+  "48": 76700,
+  "51": 125706
+ },
+ "T": 740,
+ "changed_hypothesis": "idle capacity moved between completely idle capacity-48 children of one (replica, port) by PR1252-rule star exchanges",
+ "status": "author-checked conditional candidate; complete T74/T740 native admission and outer release with the overlay not rerun",
+ "credits": "PR1252 Gonzalo-Mier; frontier1094-idle-star-ceiling Gueorgui Maissouradze; PR3148 Jeremy Diviney; PR3133/PR3042 dcharb78; PR2979 Marius-Paul Dumitrean; OpenCode (Space bunny free); Anthropic Claude"
+}

research/rank7-idle-star-exchanges-t740/README.md

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+# Idle rank-48 star exchanges on PR3148's T740 fill: conditional κ = 0.001730189061233850760095 − 10⁻⁴⁶ − 10⁻⁷⁰
+
+**Author-checked conditional candidate; not maintainer-reviewed.** The selected record
+on `main` remains #1094 (κ = 0.001672660197076918). This package stacks on the open,
+unreviewed PR3148 (Jeremy Diviney: PR3133's words at their T740 packing cap) and does not
+change its words, helpers, fill rows, order or dispatch.
+
+```text
+                       complex root bracket (1e-25)                         κ headline
+PR3148 (baseline)      [0.0017297774946785970118869, ...8870]               0.001729777494678597011886 − 1e-46 − 1e-70
+this package           [0.0017301890612338507600957, ...0958]               0.001730189061233850760095 − 1e-46 − 1e-70
+gain                                                                         +4.11566555253748209e-7  (+0.0238%)
+```
+
+## Changed hypothesis
+
+PR3133's fixed-capacity relaxation cap (RELAX-CAP) and PR3148's cap-attaining fill both
+charge **one idle child per bin, of rank `capacity − used`**. They never move idle
+capacity *between* bins. In PR3148's fill, **282,569** capacity-48 bins are completely
+idle: no helper row touches them. They lie in 82,880 (replica, port) groups of sizes
+`{1: 414, 2: 19103, 3: 9503, 4: 53860}`.
+
+Gonzalo-Mier's PR1252 showed on #1094 that two such children can exchange three axes,
+`48 + 48 → 45 + 51`. The move is legal when every moved axis commutes with all of the
+receiver's axes, every stage entry/exit X and Y translation, the terminal Y translation
+and both children's terminal translations. The group's total axis set, and hence its Z4
+phase polynomial, is unchanged. `research/frontier1094-idle-star-ceiling` (Guéorgui
+Maissouradze) extended this to **stars**: one centre sends three axes to each of several
+leaves.
+
+**The data needed for this rule is on main.** The rank-seven T74/T740 lineage uses the
+same exact data-phase recipe as #1094 (`DATA-PHASES.json`, SHA-256
+`c5916ff3…e8251fa`; GLOBAL-TEMPLATE `exact_data_phase_recipe_sha256`). Its per-port idle
+children carry the orthogonal axis sets. For all 4,866 #1094 idle rank-48 children, these
+axes equal the hosted `IDLE-REMAINDER-FRAMES` that PR1252 used, so legality is a
+port-local property. It does not depend on the replica or on the local word.
+
+## Result
+
+The exhaustive per-group star optimum covers every role assignment of the idle children
+(untouched, centre, or leaf of one centre). Each candidate is checked by an exact
+disjoint 3-axis matching:
+
+| Group shape after exchange | Groups |
+|---|---:|
+| `[51,51,48,42]` | 37,080 |
+| `[48,48]` (no movable axes) | 19,083 |
+| `[51,51,45,45]` | 16,780 |
+| `[51,51,42]` | 8,463 |
+| `[51,48,45]` | 1,040 |
+| `[48]` (singleton) | 414 |
+| `[51,45]` | 20 |
+
+The search makes 125,706 three-axis moves and changes 205,869 children. The idle bill
+`{48: 282569}` becomes `{42: 45543, 45: 34620, 48: 76700, 51: 125706}`. The rank mass
+`65088·740` is unchanged. The maximum child rank is 51, and the finite halving guard still
+holds: `73² = 5329 > 2·51² = 5202`.
+
+The complete paid profile is
+`{1:16376940, 2:4821840, 3:1567431, 4:679320, 5:147075, 8:53280, 42:45543, 45:34620, 48:76700, 51:125706}`.
+
+The complex side still binds: the bit supplier is #589 at 0.001733427… > 0.00173019. The
+within-group ceiling (any repartition of each group, `51^(k−1), 51−3k`) has root
+0.00173069001404…. The star class captures 45% of that headroom.
+
+## Reproduce
+
+```sh
+# baseline input: PR3148's own deterministic materializer (head f2927a4b)
+python3 -B research/coupled-carrier-cap740/materialize.py --output /tmp/t740   # on PR3148's branch
+python3 -B research/rank7-idle-star-exchanges-t740/build.py --fill /tmp/t740/COMPOSED-FILL.json.gz
+python3 -B research/rank7-idle-star-exchanges-t740/verify.py --fill /tmp/t740/COMPOSED-FILL.json.gz
+```
+
+`build.py` authenticates DATA-PHASES and the decoded PR3148 fill payload
+(`8f4d6965…`). It then derives `IDLE-48-BINS.json.gz` and `STAR-MOVES.json.gz`, and
+writes `RESULT.json` with an exact rational root bracket. That bracket reproduces
+PR3148's published one exactly for the baseline profile. Building takes about 15 s.
+
+`verify.py` does not import `build.py`. It works from the committed files and does not
+need the fill; the optional `--fill` flag rebinds the idle bins. It re-derives:
+
+- every moved axis's commuting rule;
+- disjoint stars, single-use axes and single-centre leaves;
+- orthonormal, independent final axis sets;
+- unchanged per-group axis multisets;
+- the rank cap and halving guard;
+- the complete profile and rank mass;
+- the root bracket, using its own directed-rounding interval arithmetic (10⁻⁷²);
+- the headline κ, gated above PR3148 and PR3143.
+
+All nine mutation controls are rejected: non-commuting axis, axis moved twice, reused
+leaf, foreign axis, centre that is also a leaf, hosted child, two-axis move, wrong profile
+and the next root grid point. The receipt is `AUDIT.json`; the run takes about 3.5 min.
+
+## Scope and remaining obligations
+
+What is established: a literal, port-local axis overlay on PR3148's untouched idle
+children. Every move satisfies PR1252's commuting rule, and the exact complete-profile
+root with the lineage's headline convention is computed.
+
+What is **not** run here:
+
+- PR3148/PR3133's complete native admission, controller/owner audit and modern outer
+  release with this overlay. That audit, `audit_mixed.py` in the T74 release chain, must
+  be generalized to read idle axis overlays, as PR1252 did for the #1094 async auditor.
+- Re-pinning the outer finite check `halving_integer_inequality = [5329, 4608]` to
+  `[5329, 5202]`.
+
+Soundness of the star class is inherited from PR1252's pair argument (unchanged group
+axis set and phase polynomial, commuting moved axes), as in the star-ceiling package. All
+inherited all-size compiler, realization, hosting, precision/tape, recovery and
+analytic-transfer obligations of the lineage remain. PR3148 itself is an unreviewed
+author claim; if it is not admitted, the same overlay applies to PR3133's T74 fill.
+
+## Open next questions
+
+- **Does the star overlay pass the T74 controller/owner audit and outer release?** Only then
+  is this κ an admitted finite candidate. First step: generalize the T74 `audit_mixed.py` to
+  read the idle-axis overlay and rerun PR3148's release. Blocked by: the release chain has no
+  overlay reader.
+- **Partially hosted remainders as donors** (PR3190) push the complex root past the bit side.
+
+## Credit
+
+- Gonzalo-Mier (PR1252): the idle exchange move, its commuting rule and its pricing
+  recipe.
+- Guéorgui Maissouradze (PatatePatatePatate): the star class and the per-group optimum
+  (`frontier1094-idle-star-ceiling`).
+- Jeremy Diviney (PR3148): the T740 cap-attaining fill.
+- dcharb78 (PR3133, PR3042): the words.
+- Marius-Paul Dumitrean (PR2979).
+- Jeremy Kothe (PR3015).
+- Nick Petalas (PR3024 lift).
+- The credited rank-seven, T37-envelope and data-phase lineage.
+
+Prepared by Rohan Arun. This was done with help from **OpenCode (Space bunny free)**,
+which identified the idle-exchange lever and proposed porting it to the rank-seven line,
+and with substantial Anthropic Claude assistance. Apache-2.0.

research/rank7-idle-star-exchanges-t740/RESULT.json

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+{
+ "status": "PASS_IDLE48_STAR_EXCHANGES_T740",
+ "T": 740,
+ "idle48": 282569,
+ "groups": 82880,
+ "group_sizes": {
+  "1": 414,
+  "2": 19103,
+  "3": 9503,
+  "4": 53860
+ },
+ "movable_axis_counts": {
+  "0": 448,
+  "3": 280,
+  "15": 616
+ },
+ "moves": 125706,
+ "changed_children": 205869,
+ "shapes": [
+  [
+   [
+    51,
+    51,
+    48,
+    42
+   ],
+   37080
+  ],
+  [
+   [
+    48,
+    48
+   ],
+   19083
+  ],
+  [
+   [
+    51,
+    51,
+    45,
+    45
+   ],
+   16780
+  ],
+  [
+   [
+    51,
+    51,
+    42
+   ],
+   8463
+  ],
+  [
+   [
+    51,
+    48,
+    45
+   ],
+   1040
+  ],
+  [
+   [
+    48
+   ],
+   414
+  ],
+  [
+   [
+    51,
+    45
+   ],
+   20
+  ]
+ ],
+ "baseline_profile": {
+  "1": 16376940,
+  "2": 4821840,
+  "3": 1567431,
+  "4": 679320,
+  "5": 147075,
+  "8": 53280,
+  "48": 282569
+ },
+ "profile": {
+  "1": 16376940,
+  "2": 4821840,
+  "3": 1567431,
+  "4": 679320,
+  "5": 147075,
+  "8": 53280,
+  "42": 45543,
+  "45": 34620,
+  "48": 76700,
+  "51": 125706
+ },
+ "baseline_root": [
+  "17297774946785970118869/10000000000000000000000000",
+  "1729777494678597011887/1000000000000000000000000"
+ ],
+ "root": [
+  "17301890612338507600957/10000000000000000000000000",
+  "8650945306169253800479/5000000000000000000000000"
+ ],
+ "root_decimal": [
+  "17301890612338507600957e-25",
+  "17301890612338507600958e-25"
+ ],
+ "kappa": "17301890612338507600949999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000",
+ "pr3148_kappa": "17297774946785970118859999999999999999999998999999999999999999999999/10000000000000000000000000000000000000000000000000000000000000000000000",
+ "gain_over_pr3148": "411566555253748209/1000000000000000000000000",
+ "within_group_ceiling_root_lower": "17306900140466462208079/10000000000000000000000000",
+ "max_child_rank": 51,
+ "halving": [
+  5329,
+  5202
+ ]
+}

research/rank7-idle-star-exchanges-t740/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "997e6a9ca7b4f95d50e366f61412fd640d72a88a",
+  "files": [
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "research/adaptive-kernel-demotion/inputs/DATA-PHASES.json",
+      "usage": "input",
+      "git_blob": "d9a4f84211182ca200c081ea061d724c8a2cfb70",
+      "sha256": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa"
+    },
+    {
+      "path": "research/frontier1094-idle-star-ceiling/README.md",
+      "usage": "provenance",
+      "git_blob": "dc62e4569948a9e10bb1c72731a9ddfdc81fe0cd",
+      "sha256": "2f640127f48be627f10a8db1d224d8a75219772525b3c0d5d3251dc8dc57b121"
+    }
+  ]
+}