Conditional κ = 0.001673628681907313 by composing a cycle with PR3366

research/t16-post-unlock-cycle/PROOF.md

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+# One cycle after the neutral unlock
+
+Reconstruct the complete PR3366 array from authenticated canonical data and
+the retained delta; compare every canonical row to the original payload digest.
+Rotate placement fields for rows43909,0,60, preserving each logical stage,
+replica, invocation time, helper and width. Compact offsets in affected bins.
+The full independent owner auditor checks complete inventory, actual source
+windows, physical family identities/uniqueness and all capacity constraints.
+
+The old idle residuals are2,1,3 and the new residuals0,3,3. Therefore the
+complete paid profile changes by `H3-H2-H1`, conserving rank mass.
+For every saving `0<a<1`, strict concavity yields
+`3^(1-a) < 2^(1-a)+1`, proving a strict decrease in the complex moment.
+This does not by itself price the magnitude or admit the physical construction.
+
+The unchanged eleven-stage strict replay regenerates the controller and charts,
+then binds the full paid invoice to the fresh source files. Every typed cost
+field is preserved; only three authenticated source identifiers differ.
+Both exact moment engines price both suppliers and conservative surcharge
+variants. Both bootstrap paths and all47 strict outer inequalities per path
+are reconstructed. RESULT.json records the resulting exact conditional κ.
+
+The predecessor's rank-two residual is essential to this witness. The direct
+cycle obstruction on the older PR3359 map is not an obstruction on PR3366.
+The fixed-word relaxation remains
+`κ < 1690054109089/1001690054109089`; this is not an attainable optimum.
+All inherited semantic and all-size premises listed in README remain open.

research/t16-post-unlock-cycle/README.md

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+# Compose a further cycle with PR3366's neutral unlock
+
+Conditional finite-composition **κ = 0.001673628681907313**, improving the
+latest [PR3366](https://github.com/CrocSwap/integer-mult-bounds/pull/3366) by
+`261943251/250000000000000000 = 1.047773004e-9`.
+The complex side still binds. This is a packing witness, not an optimum or
+complete multiplication theorem.
+
+Starting from PR3366's exact owner map, rotate rows `(43909,0,60)` forward.
+The three bins' idle ranks change from `(2,1,3)` to `(0,3,3)`.
+The complete paid delta is `-H1-H2+H3`. PR3366's five-helper construction and
+all seven earlier cycles are retained. The selected word, helper widths,
+144-call order, suppliers and complete physical invoice stay fixed.
+
+The profile is
+`{1:349709,2:60192,3:62327,4:20448,5:11808,7:2,8:1152,16:2,48:4880}`.
+Rank mass remains1,041,408 and deficit5,120. There are69,120 complete helper
+roles and10,005 actual charts. Prefix397 is paid at512; row coefficient16,469.
+See [PROOF.md](PROOF.md) for the strict concavity argument and scope.
+
+## Reproduction
+
+Standard-library CPython3.12, from the repository root:
+
+```sh
+python3.12 -Werror -B scripts/research_package.py --package research/t16-post-unlock-cycle
+python3.12 -Werror -B research/t16-post-unlock-cycle/verify.py --work /absolute/new/output
+```
+
+The verifier authenticates43 canonical inputs, reconstructs PR3366 from
+canonical PR2169 data plus a236-row patch, applies the exact cycle, audits
+the full owner map and regenerates OWNERS/PLAN/SPEC. It runs unchanged
+11-stage strict admission, both exact supplier prices, both seven-step
+bootstraps and all47 strict outer inequalities per tariff path. The entire
+typed physical invoice is compared, allowing only three freshly bound source
+identifiers to differ. Private search or solver tools are not needed.
+
+Exact predecessor: PR3366 head
+`3f59a00257b3e29ad54e349f826d5ffdc59f7cfb`.
+Its construction, original delta, source export, result, README and proof are
+retained under `predecessor/`. Its unchanged admission wrapper is `replay.py`.
+The canonical predecessor payload digest is
+`96f0dc4e3b8f08cdfd385992e8675f033ed83bcd61bb14f8f6b9671b8c125fe3`.
+Canonical source base is `6f100d4bb5083f8c35d96fd95a539811c62f245c`.
+
+## Relation to parallel work
+
+PR3366 superseded our previous PR3359. Reusing its new map reveals a direct
+cycle that was unavailable on3359. PR3369's direct-cycle obstruction concerns
+that earlier map, so this result does not contradict its scope. A private
+mixed-call solver independently found a37-role rearrangement with the same
+paid gain; this contribution publishes the simpler three-role witness.
+The alternative is preserved privately for future composition.
+
+## Scope and credit
+
+The selected all-size/compiler, common-label/mutable-source, whole-call
+hosting, precision/tape, exact recovery, analytic and bit H1–H5 premises
+remain. Native local ancestry is reused by source binding. Fresh full cold
+native rebuilding, formal-kernel verification and independent human review
+are not claimed. Selected-result.json is unchanged.
+
+Credit Nick Petalas for PR3366/3354/3338, Pierre/hexapode for PR3359,
+Gonzalo Mier for PR3308, Philippe Van Velzen for PR2169, George Lydakis,
+Rohan Arun and Jeremy Kothe for the packing lineage, Josie Elliston for the
+independent owner auditor, Hao Yan for the T16 controller and the retained
+compiler/native/bit/moment authors. Original notices remain. New work by
+Pierre (hexapode), with substantial OpenAI Codex assistance from four
+concurrent agents. Apache-2.0.
+
+## Open next questions
+
+Can a mixed-call exchange move several helpers through one bin to pool more
+residuals? Preserve this full map, nominate explicit moves and replay the
+complete invoice. No global packing or multi-exchange optimum is claimed.

research/t16-post-unlock-cycle/RESULT.json

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+{
+  "status": "PASS_T16_POST_UNLOCK_CYCLE",
+  "cycle": [
+    43909,
+    0,
+    60
+  ],
+  "owner_canonical_sha256": "402dc5357441bc8fb4d36fbb5595dcad64d117f24d58cc25ce4b25d9239f70b8",
+  "profile": {
+    "1": 349709,
+    "2": 60192,
+    "3": 62327,
+    "4": 20448,
+    "5": 11808,
+    "7": 2,
+    "8": 1152,
+    "16": 2,
+    "48": 4880
+  },
+  "idle": {
+    "1": 2381,
+    "3": 119,
+    "7": 2,
+    "16": 2,
+    "48": 4880
+  },
+  "kappa": "1673628681907313/1000000000000000000",
+  "complex_lower": "1676434410620263951161/1000000000000000000000000",
+  "bit_lower": "1733427039589844234749/1000000000000000000000000",
+  "conservative": {
+    "endpoint": "41910860193947972751/25000000000000000000000",
+    "endpoint_decimal": "0.001676434407757918910040",
+    "kappa": "1673628679054541/1000000000000000000",
+    "kappa_decimal": "0.001673628679054541"
+  },
+  "gain_vs_pr3366": "261943251/250000000000000000",
+  "gain_vs_selected": "193696966079/200000000000000000",
+  "roles": 69120,
+  "rank_mass": 1041408,
+  "deficit": 5120,
+  "actual_charts": 10005,
+  "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": "1676457204938259661245488867857/1690054109089000000000000000000",
+  "scope": "Conditional selected-word finite composition; all inherited selected assumptions remain."
+}

research/t16-post-unlock-cycle/SOURCE-EXPORT.json

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+{
+  "source_pr": 3366,
+  "source_commit": "3f59a00257b3e29ad54e349f826d5ffdc59f7cfb",
+  "source_path": "research/t16-neutral-unlock/verify.py",
+  "source_git_blob": "76d586b40c034decec976dc76349feb8a1fb2848",
+  "source_sha256": "91f08f8a69796d3488d6d97fffc63fd6910e770f515321f07c0d784e0e09226c",
+  "predecessor_canonical_sha256": "96f0dc4e3b8f08cdfd385992e8675f033ed83bcd61bb14f8f6b9671b8c125fe3",
+  "patch_rows": 236,
+  "retained_files": {
+    "predecessor/PROOF.md": "3c80d1dc9514b288a109a82d1192f7151deaa51360c18bdbdd419a3708cbf6a4",
+    "predecessor/README.md": "b7767510af201d4a22f5164f402648c22a1c84c914bbf8881d439a84fc31d3cd",
+    "predecessor/RESULT.json": "f6afaa5731bcc7d5df06fa9a335b1687efe1419c94db84cb80a0ac0570435e95",
+    "predecessor/SOURCE-EXPORT.json": "a382314320249e54a4641f427c42f22b672aa6673429ea5c65552d5b033ae41a",
+    "predecessor/predecessor.patch.json": "c23ef350c07cb0a6aa3f1ab0f4633dda229c4ab14f4684665fd4cfbf7ffa11a3",
+    "predecessor/verify.py": "91f08f8a69796d3488d6d97fffc63fd6910e770f515321f07c0d784e0e09226c",
+    "predecessor.patch.json": "eb74c3db039cca850e3263823a63b90e328e70331b75368519afcbb6f81c93cc",
+    "replay.py": "a7d79c93c8e45614dc6855bbcef3cd687cae5f48051bb06389d1cdb71d6969a2"
+  },
+  "replay_source": "Byte-identical PR3366 admission wrapper. Complete predecessor map reconstructed from authenticated original code, original delta, neutral pair and triple; canonical payload matches original RESULT."
+}