Independently audit native equality-cut signs and caller contracts

research/native-equality-independent-audit/README.md

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+# Independent review of the native equality-cut tariff
+
+The signed Clifford tariff and caller implications in **PR #5588 PROOF.md
+§§3–5 hold under their stated hypotheses**. No mathematical defect was found
+in this scope. [PROOF.md](PROOF.md) independently derives the intersection
+by a trace calculation and proves the complete-word, cached-history and
+approximation statements. This takes the
+[explicit independent-review request](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19183713).
+
+For `a_k = 2^(n(f-k))`, a Fourier-rank-r Clifford call has crossing rank
+`a_k (1-2^(-kr))` across the cut equating k disjoint pairs of columns.
+Rectangular rank subadditivity then bounds any chronological linear word,
+including changed paid inputs, feedback, COPY, CLEAR and independent dirty
+helpers. A cached boundary contributes **one joint crossing rank of its
+complete availability map J**. The exact rank bound permits invertible
+ordinary endpoint codes; the stated unit error floor needs unitary,
+cut-preserving endpoint frames.
+
+This validates one proposed supplier-screening theorem. It does not produce
+a supplier, extract J from P104, or discharge any selected implementation
+premise. No κ change or native admission is claimed.
+
+## What the independent checks add
+
+The checkers import no implementation from PR #5588. They use standard-library
+rational and Gaussian-integer arithmetic and independently check:
+
+- All **366 signed ordered commuting Pauli image frames** for one/two axes,
+  covering 18 image subgroups. The 54 literal projector checks verify
+  symmetry, idempotence, trace and rational rank. These are image-frame
+  enumerations, not an enumeration of all Clifford unitaries. Separate
+  two-pair checks give crossing ranks 3 and 6 at widths 4 and 5.
+- A complex Clifford with `C Z C* = -Y`. Its same-factor tensor has a
+  Bell-minus intersection; conjugating the second factor changes it to
+  Bell-plus without changing its rank. This catches a sign/orientation
+  distinction that rank-only tests miss.
+- One **18-operation complete word**, with four paid overwrites, feedback
+  into the third query, an arbitrary dirty field, and a zero temporary.
+  All 19 prefixes obey the cut bound; the complete endpoint is
+  `(2x + Ty, Tx + y, d, 0)`. Omitting dirty subtraction, the final CLEAR,
+  or the first crossing charge is detected. Five two-bank paid updates
+  and five self updates also cover zero, negative and fractional coefficients,
+  including singular self updates.
+- `J = [I; T; T]`: joint crossing rank 1, while summing separate cache ranks
+  gives 2. Selecting a cached response realizes T with no additional call,
+  demonstrating why omitting J gives a false obstruction.
+- Invertible ordinary codes preserve rank. Scaling a unitary target by 1/8
+  preserves positive crossing rank but lowers its zero-program operator
+  error to 1/8. A non-columnwise unitary monomial endpoint instead gives
+  cut Gram `I/2`, invalidating the partial-isometry step. Both distinguish
+  restrictions already present in the source contract.
+
+[RESULT.json](RESULT.json) is a deterministic receipt. Finite controls support
+the written all-size argument; they do not replace it. This is an agent-assisted
+independent derivation and implementation, not human or proof-kernel acceptance.
+
+## Source and scope
+
+Reviewed source: [Ali Romman's PR #5588](https://github.com/CrocSwap/integer-mult-bounds/pull/5588),
+head `97db3fb45cd93a350e300781d75440d14e6a1f73`.
+The unchanged proof and README are pinned in [SOURCES.json](SOURCES.json)
+at canonical main `df0137efd35e11b4445734bf612bdd4c7ebc330c`.
+The selected baseline at this checkpoint is P104 PR #5599,
+κ=`62009/20000000`; its five inherited premises remain.
+
+Excluded: source §6's profile algorithm, §7's semisimple output-packing
+question, extraction of an actual retained-observation ledger, full supplier
+replay, compiler/precision/storage/routing costs, and the complete rounded
+multiplication theorem. The source verifier and selected cold replay were not
+run. The common factor assumes equal field capacities. Free maps must preserve
+the cut and its complement; arbitrary inter-column maps are not included.
+
+The picker ran once with seed `81093d593dbc22d5b1e5cb3cc7d46405`.
+[TASK.json](TASK.json) retains the `uniform-transfer-audit` card and records
+the handoff-first scope change from its historical #4100 target to this explicit
+current native-lift review. The first overlap snapshot covered 7,676 local
+notes, 370 open PRs and 4,998 Discussion records with complete reported
+pagination and no warnings. Exact-thread and targeted all-status searches
+found no competing audit; PR #5617's stronger stage cuts are distinct.
+Searches do not cover unpublished work or every historical code file.
+The canonical claim is in
+[Discussion #5582](https://github.com/CrocSwap/integer-mult-bounds/discussions/5582#discussioncomment-19188980).
+
+## Reproduce
+
+Python 3.12, standard library only; the registered replay is bounded at 120 s.
+
+```sh
+python3.12 -B scripts/research_package.py --package research/native-equality-independent-audit --check-only
+python3.12 -B scripts/research_package.py --package research/native-equality-independent-audit
+python3.12 -I -O -B research/native-equality-independent-audit/verify.py
+```
+
+[VERIFICATION.md](VERIFICATION.md) records executed commands and transport
+checks. Direct replay needs only the package and its three declared input
+files; Git and network are unnecessary. It reads exact pinned source bytes
+and compares the recomputed receipt. Normal execution never regenerates it.
+
+## Open next questions
+
+At one unclaimed physical caller boundary, extract the complete actual J,
+including row versions and independent dirty inputs, and certify its joint
+cut rank. The blocker is a source-bound availability ledger, not this rank
+inequality. For a concrete caller with inter-column operations or unequal
+field sizes, first specify typed projectors and charge every crossing map;
+the present normalized tariff cannot be imported unchanged. See the
+structured relevance, first steps and blockers in [TASK.json](TASK.json).
+
+## Credit
+
+Ali Romman retains credit for the equality-cut invariant, Clifford tariff,
+complete-word theorem and explicit boundary/approximation limitations in
+PR #5588. Its fused-block and dirty-host predecessors, an664 and reviewers,
+Adam Gall, Ross Richardson, Chris R/Clear1nsight, and the P104/native/helper/
+compiler lineage retain their credit. Pauli/stabilizer theory, trace
+orthogonality and product-rank inequalities are classical.
+
+New derivation, code and review prepared for Hiroki KUMAZAKI, with substantial
+OpenAI Codex assistance. Delegated agents separately scoped overlap,
+reviewed the proof and signs, and implemented chronological/cache controls.
+No predecessor code is copied or imported. Apache-2.0.

research/native-equality-independent-audit/RESULT.json

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+{
+  "authenticated_inputs": 3,
+  "contracts": {
+    "cached_history": {
+      "additional_paid_calls": 0,
+      "available_rows": [
+        "x",
+        "T*x",
+        "T*x"
+      ],
+      "exact_target_cut_rank": 1,
+      "joint_cut_rank": 1,
+      "rejected_control": "omit initial availability map J",
+      "sum_individual_cut_ranks": 2
+    },
+    "changed_alphabet": {
+      "crossing_rank": 1,
+      "inter_column_permutation": [
+        0,
+        1,
+        3,
+        2
+      ],
+      "rejected_control": "grant inter-column CNOT as a zero-rank free map",
+      "repeated_one_column_positive_controls": 2
+    },
+    "chronological": {
+      "endpoint": [
+        "2*x + T*y",
+        "T*x + y",
+        "d",
+        "0"
+      ],
+      "endpoint_cut_rank": 2,
+      "feedback_paid_input_index": 3,
+      "field_dimension": 4,
+      "missing_first_charge": {
+        "mutant_prefix_budget": 0,
+        "observed_prefix_rank": 1
+      },
+      "operations": 18,
+      "original_and_dirty_input_columns": 12,
+      "owned_zero_registers": 1,
+      "paid_overwrites": 4,
+      "prefix_bounds": [
+        0,
+        0,
+        0,
+        1,
+        1,
+        1,
+        1,
+        2,
+        2,
+        2,
+        2,
+        2,
+        3,
+        3,
+        3,
+        3,
+        4,
+        4,
+        4
+      ],
+      "prefix_cut_ranks": [
+        0,
+        0,
+        0,
+        1,
+        1,
+        1,
+        1,
+        2,
+        2,
+        1,
+        1,
+        1,
+        2,
+        2,
+        2,
+        2,
+        3,
+        3,
+        2
+      ],
+      "rejected_controls": [
+        "omit dirty-response subtraction",
+        "omit final CLEAR",
+        "omit first paid crossing charge"
+      ]
+    },
+    "endpoint_codes": {
+      "coded_target_cut_rank": 2,
+      "left_and_right_code_ranks": [
+        8,
+        8
+      ],
+      "rejected_control": "retain unit error floor after nonunitary scaling",
+      "scaled_target_cut_rank": 1,
+      "zero_program_operator_error": "1/8",
+      "zero_program_paid_calls": 0
+    },
+    "paid_updates": {
+      "cases": [
+        {
+          "coefficient": "0",
+          "self_update_cut_rank": 0,
+          "self_update_full_rank": 4,
+          "two_bank_cut_rank": 0
+        },
+        {
+          "coefficient": "1",
+          "self_update_cut_rank": 1,
+          "self_update_full_rank": 2,
+          "two_bank_cut_rank": 1
+        },
+        {
+          "coefficient": "-1",
+          "self_update_cut_rank": 1,
+          "self_update_full_rank": 2,
+          "two_bank_cut_rank": 1
+        },
+        {
+          "coefficient": "1/2",
+          "self_update_cut_rank": 1,
+          "self_update_full_rank": 4,
+          "two_bank_cut_rank": 1
+        },
+        {
+          "coefficient": "-3/2",
+          "self_update_cut_rank": 1,
+          "self_update_full_rank": 4,
+          "two_bank_cut_rank": 1
+        }
+      ],
+      "self_update_cases": 5,
+      "two_bank_cases": 5,
+      "two_bank_independent_input_columns_per_case": 8
+    }
+  },
+  "new_kappa_or_supplier_admission": false,
+  "reviewed_source_head": "97db3fb45cd93a350e300781d75440d14e6a1f73",
+  "scope": "PR5588 PROOF.md sections3\u20135; written all-size proof plus finite exact controls",
+  "source_verifier_imported": false,
+  "stabilizers": {
+    "controls": {
+      "k2_f4_cut_rank": 3,
+      "k2_f5_cut_rank": 6,
+      "mismatched_factor_control": 1,
+      "rejected_frame_controls": 3
+    },
+    "n1": {
+      "fourier_rank_counts": {
+        "0": 2,
+        "1": 4
+      },
+      "frames_with_imaginary_Pauli_entries": 2,
+      "image_subgroups": 3,
+      "literal_projectors": 9,
+      "signed_ordered_frames": 6
+    },
+    "n2": {
+      "fourier_rank_counts": {
+        "0": 24,
+        "1": 144,
+        "2": 192
+      },
+      "frames_with_imaginary_Pauli_entries": 216,
+      "image_subgroups": 15,
+      "literal_projectors": 45,
+      "signed_ordered_frames": 360
+    },
+    "signed_matrix_controls": {
+      "conjugated_factor_intersection": "Bell plus",
+      "exact_Gaussian_integer_controls": 3,
+      "noncolumnwise_unitary_endpoint_cut_gram": "I/2",
+      "same_factor_intersection": "Bell minus"
+    }
+  },
+  "status": "PASS_NATIVE_EQUALITY_INDEPENDENT_AUDIT"
+}

research/native-equality-independent-audit/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+  "files": [
+    {
+      "path": "LICENSE",
+      "usage": "input",
+      "git_blob": "261eeb9e9f8b2b4b0d119366dda99c6fd7d35c64",
+      "sha256": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "581f8d87008354eb5b869f7ed9de1317ad837ce8",
+      "sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/native-equality-cut/PROOF.md",
+      "usage": "input",
+      "git_blob": "bd4964f17868e4f016d47db887bae6906e8e9f2b",
+      "sha256": "8f0043ff2084b3ff503cd357a186bda4ffa3cd4f1e50df4d7d9b63d23064e38d"
+    },
+    {
+      "path": "research/native-equality-cut/README.md",
+      "usage": "input",
+      "git_blob": "83fcc474f5b79a7d1118090d4687d55974708d5a",
+      "sha256": "2fab505ce913ef8edf474a355b3e7d8c40552e37f9e3a9f03e380b4d4332f9c2"
+    }
+  ]
+}