Certify resonant partial-channel rank 100 at width four (no new κ)

research/resonant-channel-rank/README.md

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+# Exact width-four rank of the resonant partial target channel
+
+The frozen coefficient in JoyboyBrian's [PR #4797](https://github.com/CrocSwap/integer-mult-bounds/pull/4797)
+has
+
+\[
+\operatorname{rank}_{\mathbb C}(A_4)=100,
+\qquad A_4:\mathbb C^{4096}\longrightarrow\mathbb C^{4096}.
+\]
+
+This is a **finite certificate** for one width. It answers the first new-width
+case of that package's second open question. The predecessor establishes a pure
+partial channel `A_f K_f x`, gives its norm at every width, and determines rank
+at widths one and two. Those results are cited inputs. This package does not
+replay its decoder or audit its construction. Since `K_4` is invertible, the
+delivered partial channel also has rank 100.
+
+There is no general even-width rank formula here. No complete target, paid
+saving, new kappa, supplier admission or multiplication theorem is claimed.
+
+## Frozen object and source binding
+
+The source is `research/resonant-owner-extraction/RESULT.json` at exact
+predecessor head `121910f1890494d6134cd48c422facbe116db36d`.
+[SOURCE-RESULT.json](SOURCE-RESULT.json) retains those exact bytes. Its SHA256 is
+`a0fe212792f1bffca701a01185b89596ff03bb7fccf5b71accf63a92ca07fd4d`.
+The pending predecessor commit need not be present in a replay checkout.
+
+[TARGET.json](TARGET.json) exports its 256 actual monomial terms, with real
+coefficients `+1/16` or `-1/16`. Its canonical JSON digest is
+`445b3e5f8edb6f345981c02c1feb91f2ce473763be74ca7d949421e1ca64d03d`,
+exactly the predecessor receipt's `target_coefficient_sha256`. Canonical JSON
+here means `json.dumps(records, sort_keys=True).encode()` with the default
+Python separators. This digest authenticates the exported coefficient without
+rechecking the earlier elimination or norm computation. The source URL and
+byte hash are also recorded in [SOURCES.json](SOURCES.json).
+
+For a record with permutation `p` and phase exponents `s`, the base monomial is
+
+\[
+g e_j=i^{s(j)}e_{p(j)},\qquad 0\le j<8.
+\]
+
+For a column `j=j_0+8j_1+8^2j_2+8^3j_3`, its contribution to `A_4 e_j` is
+
+\[
+c_g\,i^{s(j_0)+s(j_1)+s(j_2)+s(j_3)}
+e_{p(j_0)+8p(j_1)+8^2p(j_2)+8^3p(j_3)}.
+\]
+
+The rational coefficient `c_g` occurs once. In particular, evaluation is
+`sum_g c_g g^{tensor 4}`, not the fourth tensor power of the width-one sum.
+All central phases are included in the displayed phase sum. The extra affine
+and phase-polynomial fields in TARGET.json are retained for source-digest
+equality; the rank calculation uses the literal permutation and phase arrays.
+
+## Exact certificate
+
+The verifier enumerates all 4,096 input columns and all 256 terms per column.
+It accumulates real and imaginary integer numerators before dividing by 16.
+For this width the resulting matrix is real integral. It has 6,440 nonzero
+entries, 496 nonzero rows and 446 nonzero columns. Every other entry is zero
+by that complete enumeration, not by a support estimate.
+
+Two exact calculations establish the rank.
+
+1. Rational sparse row reduction processes every nonzero row. It creates 100
+   normalized pivot rows and makes 1,808 reductions by existing pivots. Every
+   original row is in their span, so the rank is at most 100.
+2. The selected original row and column labels in [RESULT.json](RESULT.json)
+   define a 100 by 100 minor. Separate integer Bareiss elimination gives its
+   determinant as `-16777216 = -2^24`. Thus the rank is at least 100.
+
+The computation uses exact integers and fractions throughout. A rational
+matrix has the same rank over the rationals and the complex numbers, because
+its rank is the maximum order of a nonzero minor. The resulting 100-dimensional
+image leaves a 3,996-dimensional kernel. No conclusion about other widths is
+inferred from this value.
+
+RESULT.json is generated by the verifier. It records both label lists, the
+minor determinant and the SHA256 of the full canonical sparse matrix, together
+with the source digests and enumerated sizes. Default replay compares the
+generated object to the frozen result and prints a receipt binding every
+package file by SHA256. It does not write files.
+
+Six controls reject an invalid base permutation, an invalid phase, an invalid
+coefficient, a duplicate minor row, the wrong determinant orientation, and a
+changed coefficient with otherwise valid syntax. The duplicate minor has
+determinant zero; swapping two rows negates the determinant. These controls
+exercise source identity and the independent lower-bound calculation.
+
+## Reproduction
+
+Run from the repository root, using Python 3.12 and the standard library:
+
+```sh
+~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/resonant-channel-rank/verify.py
+~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/resonant-channel-rank/verify.py
+~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/resonant-channel-rank
+~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/resonant-channel-rank --check-source-history
+```
+
+Expected marker is `PASS_RESONANT_CHANNEL_RANK`, with `rank: 100` and
+`dimension: 4096`. Both registered checks have a 60-second limit. To inspect
+the generated scientific result without changing the frozen file, add
+`--emit-result` to the direct verifier command.
+
+The historical card's first working command was attempted. After its omitted
+historical input was materialized, it failed with `KeyError: 'complex_profile'`
+against the current selected certificate schema. The current
+`scripts/research_selected_baseline.py` orientation passed. Neither control
+is evidence for the rank result. The package is explicitly a side question
+under `complex-rank-shape-decoder`, not an improved paid profile.
+
+The verifier has no recursion, subprocesses or dynamic code execution. All
+loops range over fixed width four, 4,096 columns, 256 records, finite matrix
+supports or a minor of order at most 100. Input files are capped at 512 KiB.
+No function exceeds 60 lines. Python's exact arithmetic allocates during this
+bounded offline computation, a Power of Ten memory-allocation deviation.
+The dimension caps and registered timeouts bound this use; no persistent or
+shared state is written. Warning-strict normal and optimized Python checks
+are registered. Explicit checks remain active under optimization.
+
+## Limits and remaining obligations
+
+The source channel identity is inherited from the exact head cited above.
+This certificate determines its coefficient's rank at width four. It does
+not independently establish the five-call preparation, free-alphabet
+implementation or norm theorem. There is no physical construction campaign
+in this package.
+
+Original `x,y`, the fresh-owner complement and retained intermediates remain
+as in the predecessor. Complete target extraction, physical preparation,
+storage/routing, terminal frames, cleanup, dirty restoration and recursive
+supplier costs remain unknown here. Unknown costs are not zero. Independent
+human and proof-kernel review, Lean, Docker and the selected scientific cold
+replay are unrun.
+
+## Open next questions
+
+The structured questions are in [TASK.json](TASK.json).
+
+1. Find an analytic image/kernel factorization of the frozen word sum that
+   determines rank at general even widths. Start with the tensor evaluation
+   above and use rank 100 as a boundary value. The norm formula alone cannot
+   supply the missing factorization.
+2. Decide whether a complete charged consumer can use this image. First
+   specify independent inputs, retained complements and every added call.
+   Missing coordinates, physical routing and restoration still block a
+   supplier-level conclusion.
+
+## Credit
+
+This contribution was produced by Nick Petalas. The frozen partial channel,
+its coefficient and the explicit rank question are due to JoyboyBrian,
+[PR #4797](https://github.com/CrocSwap/integer-mult-bounds/pull/4797).
+Its retained lineage credits Yiyang for the phase-aligned repair and staggered
+continuation in #4710, lee-phys for the original two-image prefix in #4688,
+and Jacob Sussman for the small-unit model and free alphabet in #4628.
+The fixed-original-owner obstruction in #4751 is unchanged. The new result
+here is the exact width-four rank certificate. Apache-2.0.

research/resonant-channel-rank/RESULT.json

@@ -0,0 +1,227 @@
+{
+  "claim_type": "finite certificate",
+  "dimension": 4096,
+  "exact_row_reductions": 1808,
+  "matrix_sha256": "9d5487d45601277fa0fc6851153690a1c6cab858c82332f200a029744b9b2139",
+  "minor_columns": [
+    9,
+    18,
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+    715,
+    854,
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+    903,
+    594,
+    603
+  ],
+  "minor_determinant": "-16777216",
+  "minor_rows": [
+    9,
+    18,
+    27,
+    54,
+    65,
+    72,
+    83,
+    90,
+    101,
+    108,
+    119,
+    126,
+    130,
+    139,
+    144,
+    153,
+    166,
+    175,
+    180,
+    189,
+    195,
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+  ],
+  "nonzero_columns": 446,
+  "nonzero_entries": 6440,
+  "nonzero_rows": 496,
+  "rank": 100,
+  "rejection_controls": [
+    "invalid_permutation",
+    "invalid_phase",
+    "invalid_coefficient",
+    "duplicated_minor_row",
+    "wrong_minor_orientation",
+    "changed_valid_coefficient_digest"
+  ],
+  "source_receipt_sha256": "a0fe212792f1bffca701a01185b89596ff03bb7fccf5b71accf63a92ca07fd4d",
+  "status": "PASS_RESONANT_CHANNEL_RANK",
+  "target_coefficient_sha256": "445b3e5f8edb6f345981c02c1feb91f2ce473763be74ca7d949421e1ca64d03d",
+  "width": 4
+}