Research: conditional κ = 0.001673620586419375 via a three-helper T16 cycle

research/t16-three-helper-cycle/PROOF.md

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+# A three-helper cycle on the selected T16 word
+
+## Literal construction
+
+The starting map is PR3338 at
+`dba99522cef0028eaeda1da408a23ea9eec37df2`, not a newly selected frontier.
+`predecessor.patch.json` reconstructs its complete owner payload from the
+canonical PR2169 map. The 169 changed rows have checked preimages. The SHA256
+of the entire canonical JSON result is checked against the exported PR3338
+payload digest, not just the changed rows. `SOURCE-EXPORT.json` records the
+original source blob, container digest, representation and retained-file hashes.
+No unmerged Git object is needed by replay.
+
+Write a bin as owner replica, idle child, port. The three original placements are:
+
+| Row | Call time | Helper | Width | Bin | Capacity | Free rank |
+|---|---:|---:|---:|---|---:|---:|
+| 1811 | 3 | 371 | 5 | 0,0,63 | 16 | 2 |
+| 2977 | 6 | 97 | 4 | 2,4,109 | 32 | 1 |
+| 531 | 1 | 51 | 4 | 9,2,103 | 16 | 0 |
+
+Move row 1811 into row 2977's bin, row 2977 into row 531's bin, and
+row 531 into row 1811's bin. Each role retains its stage, source replica,
+call time, helper and width. Copy all five physical placement fields together.
+Compact only those three bins, in stable old-offset order. The free ranks,
+in the table's bin order, become 3, 0, 0. Exactly three rows relocate and
+four other rows change offsets. No role is split or reordered in its call.
+
+The independent PR1091 owner auditor checks the complete 69,120-row map,
+including actual event-derived windows, active-pair exclusion, one physical
+family per helper per call, disjoint prefix slices and exact whole profile.
+The original map passes the same audit. The unchanged PR1087 strict pipeline
+then rebuilds the dispatch and physical charts and checks its own full contract.
+The cycle is a witness, not a classification of all possible cycles or an optimum.
+
+## Strict gain
+
+For every `0 < p < 1`, strict concavity with the continuous value at zero gives
+
+```text
+(3/73)^p + 0 + 0 < (2/73)^p + (1/73)^p + 0.
+```
+
+Every other recursive child remains unchanged. Thus the complete complex
+moment strictly decreases whenever the inherited physical charges stay fixed.
+That last condition is checked by comparing the entire newly emitted selected
+invoice with the selected invoice, excluding only its three rebound source
+identifiers. All other keys, including future keys, must agree.
+
+The resulting complete profile is
+`{1:349718,2:60198,3:62320,4:20448,5:11808,7:2,8:1152,16:2,48:4880}`.
+Rank mass remains 1,041,408 and deficit remains 5,120. The local word, helper
+inventory, 144-call order and selected supplier contracts are unchanged.
+
+## Full finite composition
+
+Two exact moment engines check the native and conservative complex endpoints.
+The retained bit endpoint remains
+`1733427039589844234749/1000000000000000000000000`.
+The native complex endpoint is
+`419106571991632694507/250000000000000000000000`.
+The complex side binds. The seven-step bootstrap and strict outer assembly give
+
+```text
+kappa = 2677792938271/1600000000000000
+      = 0.001673620586419375.
+```
+
+This exceeds PR3338 by exactly `1047762837/10^18` and the selected headline
+by exactly `960389342457/10^18`. The conservative path gives
+`0.001673620583566572`. Actual chart count remains 10,008, global prefix
+397 is still paid at 512, and physical row coefficient remains 16,469.
+The full strict pipeline retains controls 69, 12, 6 and 15; the separately
+implemented owner auditor adds eleven rejection controls. Ten package
+behavioral groups run normally and with Python assertions disabled.
+
+The fixed-word relaxation computed by `ceiling.ceiling()` gives
+`kappa < 1690054109089/1001690054109089`. The exact fraction attained by
+this witness is recorded in `RESULT.json`. This ceiling does not prove
+attainability or global packing optimality.
+
+## Scope
+
+This is one conditional finite composition under the selected all-size/compiler,
+common-label/mutable-source realization, whole-call hosting, precision/tape,
+exact recovery, analytic transfer and bit H1-H5 premises. The selected native
+ancestry is reused by source binding, not rebuilt cold. The unchanged strict
+pipeline does rebuild the changed whole-owner controller, charts, finite invoice
+and both supplier compositions. No unconditional multiplication theorem,
+formal-kernel verification, independent human review or selected-frontier
+update is claimed. Larger pending claims may use different unadmitted sources.

research/t16-three-helper-cycle/README.md

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+# A three-helper T16 cycle improves the complete paid moment
+
+One literal three-bin cycle on PR3338's owner map gives conditional finite
+`κ = 0.001673620586419375`. The exact gain is `1047762837/10^18` over
+PR3338 and `960389342457/10^18` over the selected headline. The complex
+side still binds. This is a packing improvement on the unchanged selected
+word and physical contract. The selected manifest is unchanged.
+
+Rows 1811, 2977 and 531 move cyclically between bins of capacities 16, 32
+and 16, across calls 3, 6 and 1. Idle ranks `2,1,0` become `3,0,0`.
+Strict concavity proves the moment decreases. Three helpers relocate and
+four other offsets change. [PROOF.md](PROOF.md) gives the exact construction,
+full paid ledger, predecessor credit and inherited obligations.
+
+The complete profile is
+`{1:349718,2:60198,3:62320,4:20448,5:11808,7:2,8:1152,16:2,48:4880}`.
+The complete unchanged eleven-stage PR1087 admission passes on all 69,120
+roles and 144 calls. Both exact moment engines, both seven-step bootstrap
+paths and all 47 strict outer inequalities on each tariff path pass.
+The whole typed JSON invoice agrees with selected1094 except its three newly bound
+source identifiers. Actual chart count is 10,008, prefix 397 is paid at
+512, and physical row coefficient is 16,469. Controls remain 69/12/6/15;
+the independent owner auditor adds eleven. Ten new behavioral groups
+pass normally and with assertions disabled.
+
+The unchanged fixed-word relaxation gives
+`κ < 1690054109089/1001690054109089`.
+This witness attains approximately 99.1949954% of that cap. There is no
+global-best, exhaustive three-cycle or packing-optimality claim.
+
+## Reproduce
+
+From the repository root with standard-library Python 3.12:
+
+```sh
+python3.12 -Werror -B scripts/research_package.py --package research/t16-three-helper-cycle
+```
+
+To retain every generated artifact, use a fresh external directory:
+
+```sh
+python3.12 -Werror -B research/t16-three-helper-cycle/verify.py --work /absolute/fresh/output
+python3.12 -Werror -B research/t16-three-helper-cycle/test_verify.py
+python3.12 -O -Werror -B research/t16-three-helper-cycle/test_verify.py
+```
+
+`uv run --no-project --python 3.12 python` can replace `python3.12`.
+Set `TMPDIR` to the session output directory when running the generic hook
+to keep its disposable outputs there. The verifier reconstructs the owner
+map, PLAN and SPEC itself; no solver, network, foreign commit object or
+pre-existing replay output is needed. Its generated compressed container
+may differ between zlib versions. The complete canonical owner payload,
+all profile fields and every invoice field remain checked.
+
+`SOURCE-EXPORT.json` authenticates the complete PR3338 owner payload,
+reconstructed from canonical PR2169 data plus a 169-row patch. It also
+pins the byte-identical PR3338 admission wrapper retained as `replay.py`.
+`SOURCES.json` pins all 43 canonical live inputs. `RESULT.json` is compared
+with freshly computed mathematical outputs; evidence receipts record the
+actual executed strict stages. The source baseline is canonical main
+`6f100d4bb5083f8c35d96fd95a539811c62f245c`, priority revision 6.
+
+## Limits
+
+This retains the selected all-size/compiler, common-label/mutable-source,
+whole-call hosting, precision/tape, exact recovery, analytic and bit H1-H5
+assumptions. Native local ancestry is source-bound reuse, not a fresh cold
+native rebuild. The changed global controller, physical charts, finite
+invoice and both supplier compositions are rebuilt. There is no full
+multiplication theorem, formal-kernel or independent human review claim.
+Larger pending claims are not treated as admitted suppliers.
+
+## Credit
+
+Nick Petalas's PR3338 supplies the starting two-role witness and its open
+three-bin question. Gonzalo Mier's PR3308 supplies the preceding map;
+Philippe Van Velzen's PR2169 supplies the canonical two-stage-cycle map
+and replay lineage. The vacancy-chain and neutral-reshuffle lineage includes
+George Lydakis, Rohan Arun and Jeremy Kothe. Josie Elliston supplies the
+separately implemented PR1091 owner auditor. Hao Yan supplies the T16
+controller, and the retained compiler/native/bit/moment sources keep their
+original notices. This package adds one explicit mixed-call three-helper
+cycle, its proof, complete source-bound reproduction and failure controls.
+New material is Apache-2.0.
+
+## Open next questions
+
+Can further disjoint three-bin cycles improve this exact map? Recompute
+capacities and actual windows after this cycle, nominate one next cycle,
+then rerun complete admission and invoice comparison. The present result
+does not classify all cycles or establish that separate nominations compose.
+
+The publication refresh on 2026-10-17 retained the same canonical main.
+PR3338 live head `f2e33dd2c07caf5dac2ee42505852a081d8fa7ca` changes
+supporting receipt serialization transport; its OWNERS, PLAN, SPEC, replay
+and swap source blobs equal the frozen scientific pin used here.

research/t16-three-helper-cycle/RESULT.json

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+{
+  "status": "PASS_T16_THREE_HELPER_CYCLE",
+  "cycle": [
+    1811,
+    2977,
+    531
+  ],
+  "owner_canonical_sha256": "e6428d9661bc4f0f2744347927b164f63ea6f153522f2aec1656b3ed24533b5c",
+  "profile": {
+    "1": 349718,
+    "2": 60198,
+    "3": 62320,
+    "4": 20448,
+    "5": 11808,
+    "7": 2,
+    "8": 1152,
+    "16": 2,
+    "48": 4880
+  },
+  "idle": {
+    "1": 2390,
+    "2": 6,
+    "3": 112,
+    "7": 2,
+    "16": 2,
+    "48": 4880
+  },
+  "kappa": "2677792938271/1600000000000000",
+  "complex_lower": "419106571991632694507/250000000000000000000000",
+  "bit_lower": "1733427039589844234749/1000000000000000000000000",
+  "conservative": {
+    "endpoint": "41910657127603871281/25000000000000000000000",
+    "endpoint_decimal": "0.001676426285104154851240",
+    "kappa": "418405145891643/250000000000000000",
+    "kappa_decimal": "0.001673620583566572"
+  },
+  "gain_vs_pr3338": "1047762837/1000000000000000000",
+  "gain_vs_selected": "960389342457/1000000000000000000",
+  "roles": 69120,
+  "rank_mass": 1041408,
+  "deficit": 5120,
+  "actual_charts": 10008,
+  "owner_controls": 11,
+  "strict_controls": {
+    "admission": 69,
+    "bit_supplier": 12,
+    "independent_t16": 6,
+    "controller_mutations": 15
+  },
+  "complete_invoice_unchanged": true,
+  "prefix_bits": 397,
+  "row_coefficient": 16469,
+  "packing_kappa_cap": "1690054109089/1001690054109089",
+  "fraction_of_kappa_cap": "2682318553229614410477045119/2704086574542400000000000000",
+  "scope": "Conditional selected-word finite composition; all inherited selected assumptions remain."
+}

research/t16-three-helper-cycle/SOURCE-EXPORT.json

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+{
+  "source_pr": 3338,
+  "source_commit": "dba99522cef0028eaeda1da408a23ea9eec37df2",
+  "source_path": "research/t16-two-role-moment-swap/data/OWNERS.json.gz",
+  "source_git_blob": "87ab85d5a8267c0593e48d12828eacbc8d8165a0",
+  "source_sha256": "abb13ccd2b1f408ec6d7b759b7acd6a40a5000890ab12ae3cdf6aa973d8b8b29",
+  "predecessor_canonical_sha256": "074799f046ee0e51a5e48411ec9fb30b60a9083a625846bbddc9b04a00e8ea74",
+  "patch_rows": 169,
+  "retained_files": {
+    "predecessor.patch.json": "a71b1c1761f670ef50722b0e28e40c90d7a38b288793889a531eb3644e70c989",
+    "replay.py": "a7d79c93c8e45614dc6855bbcef3cd687cae5f48051bb06389d1cdb71d6969a2"
+  },
+  "replay_source": "Byte-identical PR3338 replay.py; attributed PR1087/PR2169/PR3308 admission wrapper. Map is a complete canonical-payload export reconstructed from canonical data plus the retained patch; compressed container bytes are not required."
+}