Conditional κ = 0.00166202617947539: U8 with two rank-seven helper ends

research/complex-rank7-end-pair/NOTICE.md

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+# Attribution and scope
+
+New contribution by Alexander Schwartzman (`AlexS-scale`), prepared with
+substantial OpenAI Codex assistance, including delegated research and review.
+The new code and research artifacts use the repository's Apache-2.0 license.
+No independent human peer review or full formal proof is claimed.
+
+This contribution changes two helper ends in the retained PR869 U8 construction
+and supplies a new physical owner assignment. It builds on Jeremy Kothe's
+(`jeremykothe2`) PR869 triangle-hybrid deliveries, credited to NewBArch, and
+PR799 X4 assembly with its credited JointXF read exchanges,
+Chafik Boukhalfa's (`chafreaky`) PR741 frame coalescences and PR685 ownership
+work, icekylinx's Gauge44 construction, and the retained PR635 cold-end and
+residual-wrapper machinery. BlockInvert's entrance promotions, G44Pack's
+packing work, and the other predecessor layers retain their original credit.
+The new end recipe uses the existing cold-end principle; it does not claim
+that principle as new.
+
+The source packages are fetched at exact commits, with their complete source
+manifests, notices, and licenses preserved. PR777 by Tekalign (`tekuuu`)
+provides the unchanged generic global runner, spec freezing, and independent
+owner/moment checker. `code/prepare_candidate.py` adapts its data preparation
+to fresh native widths and supplied owners. The mathematical admission
+predicates come from the pinned PR799 package and its retained predecessors.
+
+The bit supplier is the cited PR589 record, including its PR555/eumemic
+ancestry and DreamingOfClouds' transport. Douglas Colkitt's repository,
+integration, and retained analytic framework are also prerequisites. This
+notice supplements the original notices; it does not replace them.
+
+The comparison snapshot is Jonathan Capalbo's (`jon314159`) PR942 selected
+result, following Chafik Boukhalfa's PR890 and the PR858/869/870 lineage.
+Its fixed-width all-replica bound does not cover the new width inventory.
+
+The investigation and submission branch start at
+`66849e1475c930fd7732b1a0067ffee7e1e95f60`. The final development and comparison
+snapshot is `407d52ff15b2217b6e22bce44728b4c3907b6515`, which selects PR942.
+`SOURCES.json` records both bases and all consumed source commits. Only the
+new research package is submitted from the earlier branch base. No source
+from PR784 is used.

research/complex-rank7-end-pair/PROOF.md

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+# The end pair and its complete cost
+
+All indices below refer to PR869's pinned U8 raw word. The recipe records both
+the old and new value at every changed position and checks the complete input
+and output content hashes.
+
+For both helper registers 588 and 589, replace the old dimension-eight end
+with the subspace
+
+```
+U = span_GF(2) {256, 128, 96, 17, 9, 5, 3}.
+```
+
+The integers encode vectors in GF(2)^9. The seven generators are independent,
+so U has 128 points. In the exact scalar prefix, inverse row 588 is
+`e588 - e412`, and inverse row 589 is `e589 - e414`. Each of the two helper
+dirt columns has only its diagonal entry. For each helper, the required
+end-of-B frames from its inverse row span precisely U. Thus U contains the
+complete inverse-row and dirt-column cones required by the retained residual
+wrapper contract.
+
+The new end requires eight B-frame changes, at zero-based positions
+1018, 1019, 1050, 1051, 1130, 1131, 1162, and 1163. The two early-cleanup
+frames at entries 148 and 149 also become U. No scalar gate changes. The
+independent point audit checks containment for every adjacent frame, every
+source and target boundary, the center interface, all paid jumps, and the
+unchanged scalar program. The unchanged native auditors independently derive
+the scalar prefix and check both end cones and both complete orientations.
+
+There are eight paid changes from rank 3 to rank 2, ten from rank 1 to rank 2,
+and four from rank 2 to rank 1. Therefore the local profile change is exactly
+
+```
+ΔH_local = {1:-6, 2:+14, 3:-8}.
+```
+
+Both helpers have empty entrances. Each end reduction releases one coordinate,
+so the helper width becomes 3776 per call. The width inventory is
+`{4:80, 5:32, 7:56, 8:120, 9:216}`. Keeping the predecessor owner assignment
+would increase the complete cost. The new assignment is an essential part of
+the construction.
+
+The frozen flow lists width patterns and call counts for the 144 existing
+physical windows, each with 112 ports. For a pattern used N times, each width
+slot receives N items and each call supplies at most N items. Split the width
+counts between those slots and add dummy slots to obtain an N-regular
+bipartite multigraph. Repeated perfect matchings give N colors. Each color is
+one physical bin with the stated width pattern and at most one item from any
+call. Dummy slots do not create helper roles.
+
+`build_owners.py` performs this integer construction deterministically. It
+then checks the literal assignment separately: all 36,288 call/helper pairs
+occur exactly once; families are distinct within each call; every role uses
+a valid window; and the coordinate intervals are disjoint and fit the stated
+capacity. The retained global auditor independently reconstructs the physical
+windows and checks the final assignment against the native helper widths.
+The complete unused-space profile is exactly `{48:2400}`.
+
+Across all 72 calls, the global profile change from the U8 construction is
+
+```
+ΔH_global = {1:-433, 2:+1008, 3:-577, 44:-1, 48:+4}.
+```
+
+The rank mass does not change:
+`-433 + 2*1008 - 3*577 - 44 + 4*48 = 0`.
+The resulting mass is 520704, retaining the deficit 2560 and the same physical
+geometry. The certified global prefix bound is 257 bits, within the retained
+512-bit allowance. The physical row coefficient is 16467, below 20161.
+Admission checks both retained rational interval engines and the complete
+precision and tape invoice.
+
+The native complex recurrence is accepted at
+`b = 6659172436825204659 / 4000000000000000000000`
+and rejected at `b + 10^-24`. The complete bit profile retains its bad-residue
+surcharge. Both moment engines and all 47 outer inequalities certify
+
+```
+κ = 166202617947539 / 100000000000000000.
+```
+
+This exceeds the selected PR942 value by exactly
+`65631761443 / 1000000000000000000`. PR942's all-replica bound fixes the
+previous U8 profile and width inventory. The new helper ends change both,
+so that bound does not exclude this construction.
+
+The scalar program and its finite envelopes retain their source predicates.
+The new frames, wrapper ranks, local profile, owner map, global profile,
+moments, and outer witness are freshly admitted. This establishes a finite
+conditional improvement under the predecessor interfaces, with the scope
+stated in `README.md`.

research/complex-rank7-end-pair/README.md

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+# U8 with two rank-seven helper ends: conditional κ = 0.00166202617947539
+
+This construction gives **κ = 0.00166202617947539**, above the pinned selected
+PR942 result, `0.001661960547713947`. The exact increase is
+`65631761443 / 1000000000000000000`, approximately **0.00394906%** of that κ.
+This compares asymptotic exponent savings, not measured runtime.
+
+The construction combines PR869's U8 word with two smaller helper ends and
+an explicit owner assignment. Helpers 588 and 589 finish in dimension seven
+instead of eight. Eight B frames and two early-cleanup frames change with
+those ends. The owner assignment combines the released space into rank-48
+residuals. All raw scalar gates and their order retain their U8 values. The
+ports, center interface, physical geometry, and 72-call schedule retain their
+PR799 values.
+
+| Quantity | U8 before this change | This construction |
+| --- | ---: | ---: |
+| Local rank-one children | 2548 | 2542 |
+| Local rank-two children | 338 | 352 |
+| Local rank-three children | 446 | 438 |
+| Local rank-four / five / eight children | 162 / 72 / 8 | 162 / 72 / 8 |
+| Helper width per call | 3778 | 3776 |
+| Global idle profile | `{1:1, 3:1, 44:1, 48:2396}` | `{48:2400}` |
+
+The complete global profile is
+`{1:183024, 2:25344, 3:31536, 4:11664, 5:5184, 8:576, 48:2400}`.
+Its rank mass is 520704, with deficit 2560. The certified complex endpoint is
+`0.001664793109206301164750`; the retained PR589 bit endpoint is
+`0.001733427039589844234749`. The complex supplier remains the limiting one.
+The conservative calculation separately gives `κ = 0.001662026176592891`.
+
+The 72-call construction has a certified global prefix bound of 257 bits and
+retains the 512-bit allowance. Its physical row coefficient is 16467, below
+the retained limit 20161. The comparison result uses 333 calls and a 1024-bit
+allowance. Its all-replica bound fixes the previous local profile and helper
+widths; this construction changes both.
+
+From the repository root, use Python 3.11 or newer and Git:
+
+```sh
+python3 -B research/complex-rank7-end-pair/reproduce.py --work /tmp/rank7-end-pair
+```
+
+Use a new output directory. The command fetches the four exact source commits
+in `SOURCES.json`. It rebuilds the PR741 ancestry, PR799 word, and PR869 U8
+word; applies the twelve changed annotations; rebuilds the owner assignment;
+and runs complete native and global admission. No optimizer or Python package
+installation is required. If a local repository contains the pinned commits,
+add `--git-repository /absolute/path/to/repository` to use its objects without
+fetching. Source files are extracted into the output directory in either case.
+
+Expected final output:
+
+```text
+PASS_COLD_RANK7_END_PAIR_COMPLETE_ADMISSION 0.00166202617947539
+```
+
+The required checks include all 38 U8 construction controls, every 728-input
+column in both orientations, all 2,950 phase edges, exact residual wrappers,
+25 native controls, and the 89 applicable inherited cold-end controls.
+Dedicated controls cover the new dimension-seven ends and their frame changes.
+The independent point audit uses explicit subsets of the 512-point frame
+space. Owner checks cover all 36,288 roles, complete availability windows,
+distinct families per call, capacity, and disjoint compact coordinate prefixes.
+Discovery is followed by frozen strict admission, 69 admission controls,
+12 bit-supplier controls, seven independent owner controls, and both exact
+moment engines with all 47 outer inequalities. `data/EXPECTED.json` pins
+the resulting word, programs, phase recipes, owner map, profile, and κ.
+
+`evidence/VALIDATION.json` records the final cold run's environment, coverage,
+and input hashes. `evidence/ASSEMBLY.json.gz` preserves the unchanged
+assembler's complete output. Its historical descriptions and comparison
+labels refer to its predecessors. The profile, moment endpoints, κ, and
+outer inequalities are the computed results for this candidate.
+
+`PROOF.md` explains the end frames, packing certificate, and complete cost
+change. `SEARCH.json` records the finite discovery and subsequent composition
+with U8. Search optimality is not needed to verify the construction.
+`data/SELECTED942.json.gz` preserves the selected-result snapshot used only
+for the numerical comparison; its source commit and hash are in `SOURCES.json`.
+
+This is a finite conditional witness for
+`T(n) = O(n (log n)^(1-kappa))`. The inherited all-size compiler,
+common-label and mutable-source realization, whole-call hosting, precision,
+tape, exact-recovery, analytic-transfer, and bit H1–H5 assumptions remain.
+The command reprices the pinned PR589 bit supplier; it does not repeat its
+bit-circuit construction. Finite replay does not prove the full multiplication
+theorem. The selected certificate is outside this contribution's scope.
+
+The contribution is independent of PR784. See `NOTICE.md` for predecessor
+credit, licensing, and the disclosure of AI assistance.

research/complex-rank7-end-pair/SEARCH.json

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+{
+  "scope": "Discovery of one finite construction; no universal or global optimality claim.",
+  "research_seed": 143265410407226848,
+  "winning_frame_search": {
+    "source_pr": 799,
+    "source_commit": "80e6463157cfc80b85af09e7e2f14748fe2dac60",
+    "domain": "20 groups of existing dimension-eight helper ends with dimension-seven lower spaces under fixed scalar code and interfaces. Exact subspace refitting of B and early-cleanup frames.",
+    "completed_groups": 20,
+    "selected_group": 17,
+    "selected_helpers": [
+      588,
+      589
+    ],
+    "solver_seed": 73,
+    "stopping_rule": "Select a legal word with positive complete packing headroom, obtain a literal owner map, and require unchanged full native and global admission.",
+    "minimalization": "Retain only the eight B frames, two tail frames, and two ends necessary for the verified profile.",
+    "screen_only": "Local cost and aggregate packing screens are not admitted kappa witnesses."
+  },
+  "packing_search": {
+    "date": "2026-10-13",
+    "status": "FROZEN_FEASIBLE_INTEGER_FLOW_NOT_A_GENERAL_OPTIMALITY_CLAIM",
+    "source_pr": 799,
+    "source_commit": "80e6463157cfc80b85af09e7e2f14748fe2dac60",
+    "owner_lineage": "chafreaky PR685, reused by PR741 and adapted by G44Pack for PR799",
+    "command": "pattern_flow.py --plan source-799/research/gauge44-frames-x/data/PLAN.json --owners source-799/research/gauge44-frames-x/data/OWNERS.json.gz --word exact-search/end-deepen-17-0.word.json.gz --out order-search/packing/end17-free4 --free-legacy --distance 4 --seconds 60 --workers 2",
+    "random_seed": 73,
+    "distance": 4,
+    "free_legacy": true,
+    "workers": 2,
+    "time_limit_seconds": 60,
+    "objective": "feasibility; all nonempty physical bins are full",
+    "solver": "OR-Tools CP-SAT",
+    "solver_version": "9.15.6755",
+    "python_version": "3.12.12",
+    "solver_receipt": {
+      "status": "OPTIMAL",
+      "pattern_variables": 503,
+      "flow_variables": 25699,
+      "seconds": 58.33655075000206,
+      "input_word_sha256": "9f63de243f867e2c52212b18d2253059a3bc965db4a981ecf18b73d83969006c",
+      "fixed_partial_bins": []
+    },
+    "search_code_sha256": "88e905223d3691c00bef5e6873d6709273ab2bdf7b2142797448ebab6a390730",
+    "search_word_content_sha256": "9f63de243f867e2c52212b18d2253059a3bc965db4a981ecf18b73d83969006c",
+    "rebinding": "The first minimal end-pair word removed unrelated frame rechoices while preserving the search profile and helper widths. Later U8 composition changes the local paid profile but retains the entire helper-width vector, frozen flow and literal owners.",
+    "original_flow_content_sha256": "7ff3e4c2bbe9e0beb2592888de7db962fd34971a025be841c8f1711b5139f8f1",
+    "owner_content_sha256": "02bde66a90b8ae9b0c4d8d563ca2218ad1de851fa6dec3d1abb5d3a30ca3dd2a",
+    "idle_profile": {
+      "48": 2400
+    },
+    "rank_mass": 520704,
+    "deficit": 2560,
+    "search_moment_delta_at_baseline_b": "-0.119035908243901811918210669949724217035275333404321692154",
+    "required_admission": "unchanged native local and strict global replay plus independent exact repricing",
+    "end_pair_only_minimal_word_content_sha256": "796a40505ed9dc38d64c5b7856a0908e70bd9f22cd869edf01ebe66ac02ab15a",
+    "search_global_profile": {
+      "1": 183024,
+      "2": 25344,
+      "3": 30384,
+      "4": 13968,
+      "5": 4032,
+      "8": 576,
+      "48": 2400
+    }
+  },
+  "earlier_searches": [
+    {
+      "domain": "Both pinned words, fixed scalar order and fixed entrance/end interfaces; full induced GF(2)^9 subspace intervals in 140 components.",
+      "result": "No lower profile found. Integer-rounded moment objectives do not prove a universal real-objective optimum."
+    },
+    {
+      "domain": "Legal local commuting reorders and exact adjacent shear rewrites with correction gates.",
+      "result": "No improved candidate passed the cost screen; these searches are not impossibility results."
+    },
+    {
+      "domain": "Six E4-to-E5 entrance promotions, independently verified natively, plus uniform and small mixed owner inventories.",
+      "result": "Local savings did not overcome global packing cost. The new end dimensions change the width inventory and escape that restriction."
+    }
+  ],
+  "final_raw_word_sha256": "096ed4abee4bb02c0d6f03bb8191510e57f6b1fcd34e8644f4fd336254e46411",
+  "final_owner_sha256": "02bde66a90b8ae9b0c4d8d563ca2218ad1de851fa6dec3d1abb5d3a30ca3dd2a",
+  "admission_status": "PASS_STRICTLY_IMPROVED_KAPPA_UNCHANGED_ADMISSION",
+  "kappa": "166202617947539/100000000000000000",
+  "composition": {
+    "date": "2026-10-13",
+    "source_pr": 869,
+    "source_commit": "73e86f69bf9a391e3c334999f7bbe3825dc42759",
+    "method": "Apply the same end subspace and eight B frames to the pinned U8 word. Its eight new early tails move the two affected tail entries from 140/141 to 148/149. All 504 helper widths otherwise match the prior end-pair candidate.",
+    "new_search_required": false,
+    "independent_point_audit": "Exact twelve-annotation difference and local delta {1:-6,2:+14,3:-8}; U8 scalar gates unchanged.",
+    "owner_reuse": "The full per-helper width vector is identical, so the frozen integer flow and literal owner payload remain valid. All roles, windows, charts, and controls are rechecked.",
+    "new_global_profile": {
+      "1": 183024,
+      "2": 25344,
+      "3": 31536,
+      "4": 11664,
+      "5": 5184,
+      "8": 576,
+      "48": 2400
+    },
+    "selected_comparison_pr": 942,
+    "selected_comparison_kappa": "1661960547713947/1000000000000000000",
+    "scope": "PR942 all-replica bound fixes the previous U8 profile and helper widths; this construction changes both."
+  },
+  "original_end_pair_only_kappa": "1661712108827881/1000000000000000000"
+}

research/complex-rank7-end-pair/SOURCES.json

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+{
+  "schema": "complex-rank7-end-pair-sources/1",
+  "base_commit": "407d52ff15b2217b6e22bce44728b4c3907b6515",
+  "sources": [
+    {
+      "pr": 741,
+      "repository": "https://github.com/CrocSwap/integer-mult-bounds.git",
+      "commit": "e516b1463606840a15f80aac1995d64ce0092e89",
+      "path": "research/complex-sw4-frame8",
+      "manifest_sha256": "b9b39c206fcd2613382b0dcbda23a371e8781d1bedaf8e2f8e7df1e79dc1fa89"
+    },
+    {
+      "pr": 799,
+      "repository": "https://github.com/CrocSwap/integer-mult-bounds.git",
+      "commit": "80e6463157cfc80b85af09e7e2f14748fe2dac60",
+      "path": "research/gauge44-frames-x",
+      "manifest_sha256": "e1b7998c8f7cb24f8674d250d0510f2b5b351b2ed0b35f17a17e6bacf3d9824b"
+    },
+    {
+      "pr": 777,
+      "repository": "https://github.com/CrocSwap/integer-mult-bounds.git",
+      "commit": "eca5c317d8176401a1f71a5ea1e81384c2eab1e0",
+      "path": "research/pr699-frame-retiming",
+      "manifest_sha256": "e0102586186663b2a667768565f61db184638b370f68f0ea58dd673e523173ec"
+    },
+    {
+      "pr": 869,
+      "repository": "https://github.com/CrocSwap/integer-mult-bounds.git",
+      "commit": "73e86f69bf9a391e3c334999f7bbe3825dc42759",
+      "path": "research/gauge44-barch",
+      "manifest_sha256": "ffcafdcb4693211fdb95d1b7b3f5bfc54f203aa0cd2cd9dbdcdad32f42104e17"
+    }
+  ],
+  "research_start_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "submission_branch_base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "comparison_snapshot": {
+    "pr": 942,
+    "artifact": "data/SELECTED942.json.gz",
+    "repository": "https://github.com/CrocSwap/integer-mult-bounds.git",
+    "commit": "407d52ff15b2217b6e22bce44728b4c3907b6515",
+    "path": "certificates/selected-result.json",
+    "source_content_sha256": "1d35283cb0ce3f8db1f285683933517f9e87475d295e2b6316399ad9928c81c3",
+    "audited_source_commit": "1960cfa919a0945d0ca4cb14ea96237000037903",
+    "scope": "Immutable selected-result snapshot for numerical comparison; no T37 construction or enlarged prefix allowance is used."
+  }
+}