Research: Rounded global Lambda

research/current-rounded-lambda/ARITHMETIC.json

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+{
+  "accepted": 1708,
+  "additional_endpoint_cases": 1536,
+  "exhaustive_width_1_2_cases": 544,
+  "oracle": "direct four signed integers; no table recurrence imported",
+  "stage_rows": {
+    "-1": 5275,
+    "1": 5275
+  }
+}

research/current-rounded-lambda/COMPOSITION.json

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+{
+  "cases": [
+    {
+      "accepted": true,
+      "actual_source_nine_axis": false,
+      "endpoint_sha256": "3953d77479ecd5ff46517c4312203154cc4d59a906b9aa1d5ce32200d323a1f0",
+      "error": null,
+      "f": 1,
+      "materialized_private_cells": 45478,
+      "ordinary_runtime_calls": 0,
+      "owned_ports": 1837,
+      "physical_actions": 5033209,
+      "q": 1,
+      "rank": 1,
+      "records": 16,
+      "sign": 1,
+      "width": 3
+    },
+    {
+      "accepted": false,
+      "actual_source_nine_axis": false,
+      "endpoint_sha256": "bdb5c260c467b45d7ae412fa679baa720d5eea7c1240095322b30edbfb37b358",
+      "error": "Lambda signed range or canonical format",
+      "f": 1,
+      "materialized_private_cells": 45478,
+      "ordinary_runtime_calls": 0,
+      "owned_ports": 1837,
+      "physical_actions": 5027459,
+      "q": 1,
+      "rank": 1,
+      "records": 16,
+      "sign": 1,
+      "width": 3
+    },
+    {
+      "accepted": true,
+      "actual_source_nine_axis": false,
+      "endpoint_sha256": "3953d77479ecd5ff46517c4312203154cc4d59a906b9aa1d5ce32200d323a1f0",
+      "error": null,
+      "f": 1,
+      "materialized_private_cells": 45478,
+      "ordinary_runtime_calls": 0,
+      "owned_ports": 1837,
+      "physical_actions": 5033209,
+      "q": 1,
+      "rank": 1,
+      "records": 16,
+      "sign": 1,
+      "width": 3
+    },
+    {
+      "accepted": true,
+      "actual_source_nine_axis": false,
+      "endpoint_sha256": "933f5416771fd022da257c253f8343a8b88001de28261285435fc19ed60904db",
+      "error": null,
+      "f": 1,
+      "materialized_private_cells": 45478,
+      "ordinary_runtime_calls": 0,
+      "owned_ports": 1837,
+      "physical_actions": 5033209,
+      "q": 1,
+      "rank": 1,
+      "records": 16,
+      "sign": -1,
+      "width": 3
+    }
+  ],
+  "late_failure_public_and_context_restored": true,
+  "positive_and_negative_signs": true,
+  "whole_source_nine_axis_executed": false
+}

research/current-rounded-lambda/CONTROLS.json

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+{
+  "controls": [
+    "global_vs_tensor_Lambda",
+    "rounding_not_sign_equivariant",
+    "late_public_write_protection"
+  ],
+  "detected_table_mutants": [
+    "tie-down",
+    "omit-range"
+  ],
+  "typed_binding_controls": [
+    "boolean_table_sign",
+    "boolean_provider_sign",
+    "boolean_site",
+    "boolean_port"
+  ]
+}

research/current-rounded-lambda/DESCRIPTOR.json

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+{
+  "cases": [
+    {
+      "actions": 37639,
+      "f": 1,
+      "old_remainder": 0,
+      "parent_bits": 1115
+    },
+    {
+      "actions": 38253,
+      "f": 1,
+      "old_remainder": 8,
+      "parent_bits": 1137
+    },
+    {
+      "actions": 47831,
+      "f": 1,
+      "old_remainder": 152,
+      "parent_bits": 1441
+    },
+    {
+      "actions": 67656,
+      "f": 2,
+      "old_remainder": 0,
+      "parent_bits": 2045
+    },
+    {
+      "actions": 68270,
+      "f": 2,
+      "old_remainder": 8,
+      "parent_bits": 2067
+    },
+    {
+      "actions": 77686,
+      "f": 2,
+      "old_remainder": 152,
+      "parent_bits": 2365
+    },
+    {
+      "actions": 216919,
+      "f": 7,
+      "old_remainder": 0,
+      "parent_bits": 6699
+    },
+    {
+      "actions": 217533,
+      "f": 7,
+      "old_remainder": 8,
+      "parent_bits": 6721
+    },
+    {
+      "actions": 226885,
+      "f": 7,
+      "old_remainder": 152,
+      "parent_bits": 7017
+    },
+    {
+      "actions": 928271,
+      "f": 31,
+      "old_remainder": 0,
+      "parent_bits": 28911
+    },
+    {
+      "actions": 928885,
+      "f": 31,
+      "old_remainder": 8,
+      "parent_bits": 28933
+    },
+    {
+      "actions": 938237,
+      "f": 31,
+      "old_remainder": 152,
+      "parent_bits": 29229
+    }
+  ],
+  "changed_fields": [
+    "remainder",
+    "selected_axes"
+  ],
+  "old_source_frame_export_assumed": false,
+  "private_ports": 39
+}

research/current-rounded-lambda/OVERLAP.md

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+# Bounded overlap and source boundary
+
+The precision package at56bc9383594a3efc303148e00bb746851abd1ed5 Sections2/5
+explicitly leaves one-rounded Lambda with exact Q² production open. The
+#6936 retained physical audit already establishes the total-Q convention
+and source phase equations; this package credits and binds them to a new
+literal monomial/header/Gaussian join, rather than claiming new character
+algebra. #6937 already discusses total rounded division abstractly.
+
+The generic monomial7d1ff237... and serial2ec95e06... providers already
+supply exact address/phase transport and genuine one-column execution.
+This package does not re-prove their internal algorithms or infer membership
+in the selected#6030 source catalogue. Its identity9/offset-q/phase0 recipe
+and new outer header domain are derived directly from proposed#6936.
+
+Targeted canonical reads at7c49fe32... included:
+* research/reflection-rounding-radius/README.md, blob14123795584429567e395216d493041175b94408:
+  optimal equivariant rounding radii for nonorthogonal integer reflections;
+  different map/domain and no physical Lambda join.
+* research/d10-reflection-sharing-seeda3def8/README.md, blob658429a7889af7a7e145be353dfe70ab28d217f8:
+  finite group-algebra paid-call lower bound, not numeric arithmetic.
+* research/format-limits-corrected/scripts/c5_lambda.py, blob4eb6c0bd57a58ac4e13d6a7d520a74deef2b70e5:
+  exact classical phase-frame butterfly/kernel identities. These are prior
+  algebra, not this current physical provider and lifecycle construction.
+
+The native worker's contemporaneous whole-MOVE rounding task is separate;
+it does not identify the outer source Q² or implement this global Lambda.
+The lead's refresh retained canonical7c49fe32, source#6936 head d360b118...
+(timeout-only delta), and#6937 head a98df73.... No source-semantic delta is
+assumed from a PR title. This is a bounded review, not exhaustive novelty.

research/current-rounded-lambda/PR-DESCRIPTION.md

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+## Summary
+
+The source precision proof leaves a completed one-rounded global LAMBDA
+primitive open. This package supplies a literal Gaussian table and a
+source-specific exact Q² monomial/header join, retaining all routing,
+owned-copy, source, head-return and cleanup costs. **No new kappa, complete
+caller admission or physical tariff-cap claim.**
+
+For the actual source direction q and positive column multiplicity f,
+Q_f² is simultaneous address translation by q in every central column,
+with phase zero. The new fixed recipe is identity9/offset-q/phase0.
+It is executed through the unchanged serial monomial provider, with a
+physically encoded private view changing exactly remainder to 0 and
+selected_axes to 9f. All other descriptor fields and the full row are retained.
+
+Global Lambda_s(Q_f)=((1+s*i)I+(1-s*i)Q_f²)/2 stays outside tensor powers.
+Two odd factor characters explicitly distinguish it from a product of
+one-column Lambda kernels. The two Gaussian numerators are formed at W+2
+with cancellation intact, then rounded once per real/imaginary component.
+STAGE writes only an owned translated copy; a separate literal COMMIT
+writes the public field after every range check passes.
+
+## Source and exact result
+
+Canonical base: `7c49fe32c55de02c2c873655c804b16e4c3fef0a`.
+Source: DreamingOfClouds' CrocSwap/integer-mult-bounds#6936 at tested head
+`03626137d0bd57f09b0bf9597376183ce6fa9683`; current head `d360b118...`
+changes only the verification timeout. The QUERY-only caller is public
+`617903497387a98f7ca929ad0b6928dd32abffec`, descended from
+CrocSwap/integer-mult-bounds#6937 at `a98df73...`.
+
+The caller has 14,480 expanded LAMBDA occurrences at 104 actual directions.
+Each sign table has 5,275 rows. The rank-nine provider has 1,965 fixed owned
+roles, including all retained private inventory, one new full-field bank
+and two numerator banks. No new recursive paid request or productive helper
+credit is introduced. Every unchanged dependency has exact commit/path/
+Git-blob/SHA-256 authentication.
+
+For valid current geometry, canonical common-grid data and admitted signed
+range, the output defect from Lambda applied to the actual input is at most
+eta_X=2^((X-1)/2)Delta on each containing complete X-fibre. The fibre includes
+all 9f translated axes. Named late range rejection preserves the public value,
+retires the partial copy/work and restores public/context heads explicitly.
+No arbitrary malformed/native-interruption rollback is claimed.
+
+## Full cost and remaining interfaces
+
+The full serial bill remains
+
+    O_9(f*((g+1)*(Vraw+M1*A1^6)+V+(E+1)^4+P)),
+
+where g=popcount(q). The new Gaussian tape/control bill is bounded by
+4096(S+1), with additional positive copy, header, retirement, setup and old-
+backing charges. Historical capacities of every role remain allocated.
+Only explicit padding absorptions give O_9(fV). This is not converted into
+the old 1000-unit LAMBDA tariff or a supplier raw cap.
+
+The new outer token/header contract is explicit. The old compiler has not
+been shown to emit it. The selected #6030 catalogue has 22,803 events/3,453
+MOVEs; proposed #6936 has 71,144/6,323. This package derives its new Q² recipe
+from #6936 and does not inherit selected-catalogue membership. Actual caller
+width/radius substitution, full recursive stock and the full source invoice
+remain open. All five foundational framework assumptions remain global
+premises, and the selected certificate is unchanged.
+
+## Reproduction and verification
+
+```bash
+python3 -I -B research/current-rounded-lambda/verify.py --suite all
+python3 -I -O -B research/current-rounded-lambda/verify.py --suite all
+```
+
+Expected: five PASS_CURRENT_ROUNDED_LAMBDA_* markers. Registered suites have
+120-second caps; standard library only, with no runtime Git/network/solver.
+
+Author checks cover 544 exhaustive four-input cases, 1,536 wider endpoints,
+12 literal 23-field views, both-sign complete reduced-rank composition,
+late rejection/retry, two arithmetic mutants, four strict typed-source
+controls, and 53,248 source-character/compiled-flip equations. Separate lead
+AI checks cover 48 wider cases through 129-bit components, a read-only-public
+STAGE trap, dirty displaced work/context, retained backing after long-to-short
+reuse, an independent rational oracle, and 1,020 global character cases.
+Unchanged independent runner/receipt and reviewed hashes are supplied.
+
+The complete actual nine-axis numerical case is explicitly unexecuted.
+Its minimum 4,096 records was screened above the 120-second attempt cap from
+the measured 16-record reduced run before extra flips. This is a feasibility
+screen, not a formal wall-time lower bound. Reduced physical evidence is
+not described as a whole source replay; the universal join uses the retained
+provider proof and the source-specific recipe/header proof.
+
+## Credit
+
+Prepared for **@CubeJerry**, with substantial **OpenAI assistance**. Credit
+DreamingOfClouds' CrocSwap/integer-mult-bounds#6936/#6937 source lineage;
+the retained monomial/phase work including CrocSwap/integer-mult-bounds#3087
+and CrocSwap/integer-mult-bounds#4172; CubeJerry's lifecycle, scalar and
+serial-provider packages; and the source precision package at
+`56bc9383594a3efc303148e00bb746851abd1ed5`. Classical Gaussian/rounding
+identities are not claimed new. This contribution is the literal physical
+source join, signed/defect proof and complete stated bill. Separate AI review
+is not human or proof-kernel acceptance.