diff --git a/research/r439-z3to2-seed13/CANDIDATE-WORD.json b/research/r439-z3to2-seed13/CANDIDATE-WORD.json
new file mode 100644
index 0000000000..0dcb80c50b
--- /dev/null
+++ b/research/r439-z3to2-seed13/CANDIDATE-WORD.json
@@ -0,0 +1 @@
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diff --git a/research/r439-z3to2-seed13/CERTIFICATE.json b/research/r439-z3to2-seed13/CERTIFICATE.json
new file mode 100644
index 0000000000..c8833d9621
--- /dev/null
+++ b/research/r439-z3to2-seed13/CERTIFICATE.json
@@ -0,0 +1,63 @@
+{
+ "COPYs": 9,
+ "ERASEs": 9,
+ "MOVEs": 2911,
+ "base_ADDs": 24305,
+ "base_canonical_sha256": "5642fd0e35a1acc11acff8d0c08350c1617e95f310d4d35ad63a651f9eeeba0b",
+ "base_records": 27234,
+ "base_word": "research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz",
+ "candidate_ADDs": 23495,
+ "candidate_canonical_sha256": "8aedb3be192baae884c80a2ff3d15d9f32ae8c8055dff170bebca2f87e75539a",
+ "candidate_records": 26424,
+ "columns_checked": 661,
+ "controls_rejected": 6,
+ "exporter_cross_check": true,
+ "frame_paths_identical": true,
+ "n_moves": 405,
+ "paid_profile": {
+  "1": 602,
+  "10": 227,
+  "101": 8,
+  "110": 1,
+  "13": 1,
+  "14": 8,
+  "15": 55,
+  "16": 156,
+  "17": 155,
+  "18": 119,
+  "2": 245,
+  "25": 4,
+  "26": 24,
+  "27": 18,
+  "28": 25,
+  "29": 53,
+  "30": 79,
+  "31": 122,
+  "32": 68,
+  "33": 13,
+  "49": 3,
+  "50": 15,
+  "51": 42,
+  "52": 93,
+  "53": 118,
+  "54": 58,
+  "55": 52,
+  "56": 35,
+  "57": 32,
+  "58": 23,
+  "59": 9,
+  "60": 2,
+  "7": 12,
+  "8": 101,
+  "81": 15,
+  "82": 39,
+  "83": 50,
+  "84": 42,
+  "85": 30,
+  "86": 4,
+  "9": 162
+ },
+ "paid_profile_identical": true,
+ "saved_ADDs": 810,
+ "z_product_identical": true
+}
\ No newline at end of file
diff --git a/research/r439-z3to2-seed13/CLAIM.md b/research/r439-z3to2-seed13/CLAIM.md
new file mode 100644
index 0000000000..dcffc2d52c
--- /dev/null
+++ b/research/r439-z3to2-seed13/CLAIM.md
@@ -0,0 +1,48 @@
+# Sprint claim (local record)
+
+No pinned sprint coordination issue was found in the repository (open issues at
+claim time: #650, #913; neither is a coordination issue), and this agent has no
+issue-posting access, so the claim is recorded locally as CONTRIBUTING.md allows.
+
+- **Track and seed:** 00-19 bit supplier; assignment seed **13** (drawn with
+  PowerShell `Get-Random`, OS-seeded, before any task selection).
+- **Base commit:** `abdc6a7002abdbcd9e21bfc5e1a795784deacfdf` (origin/main at
+  dispatch).
+- **Object:** the z-program of the selected PR589 bit word
+  (`research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz`): the 2,629
+  source edges of its A and B z-destination gates and their 2,629 literal-reverse
+  cleanup mirrors. This is the A-section/decoder-adjacent region of the word,
+  not the phase-B blocks (frames 506-524) that concurrent block-PRs cover and
+  not the frame-524 block chosen by the earlier seed-13 sprint.
+- **Method:** exact single-edge triangle (Steinberg 3->2) program rewrites with
+  register-level commutation windows, an ADD-count objective (PR589's tuner
+  optimizes the moment only), and acceptance gated on an exact 661-column
+  replay, identical 661 frame paths and an identical paid profile. A complete
+  duplicate-edge census is included as the paired-transvection control class.
+- **Parameter range / coverage:** every triangle instance (edges (b,x), (b,a),
+  (a,x)) in the z-stream of the committed word; greedy application to a fixed
+  point; stopping rule = a full scan with zero accepted rewrites.
+- **Expected deliverable:** a verified finite word with exactly 2 ADDs removed
+  per accepted rewrite, unchanged paid profile and frame paths (so no new kappa
+  and the frozen hosting argument carries over as in prior block PRs), plus the
+  full coverage census and negative controls.
+- **Expected finish:** same session; verification and handoff in the final
+  quarter.
+
+Checked immediately before writing: no open PR claims A-section or z-program
+Steinberg/3->2 ADD-count rewrites of the PR589 word. Overlap notes: #933 and
+#1048 close *profile/moment* reframing classes (orthogonal objective; this
+package keeps the profile exactly unchanged), and #1015/#1045/#1062/#980 work
+the phase-B blocks or other words (orthogonal region).
+
+## Result (recorded after completion)
+
+Delivered: 405 accepted rewrites, complete exported ADDs 24,305 -> 23,495
+(-810), records 27,234 -> 26,424, unchanged frame paths, paid profile and
+z-product U, exact endpoint on all 661 columns, six tamper controls rejected,
+exporter cross-check passed, deterministic regeneration confirmed.
+Submitted as PR https://github.com/CrocSwap/integer-mult-bounds/pull/1127
+(branch `research/agent-12-bit-seed13`, commit 5ce6bcaf on base abdc6a7).
+Concurrent #1122 (bitplane, same track/seed number by coincidence) composes
+frozen PR980/PR1062 changes on the semantic-D word lineage; different word and
+mechanism, not additive to this result.
diff --git a/research/r439-z3to2-seed13/MOVES.json b/research/r439-z3to2-seed13/MOVES.json
new file mode 100644
index 0000000000..4f15a23da3
--- /dev/null
+++ b/research/r439-z3to2-seed13/MOVES.json
@@ -0,0 +1,13115 @@
+{
+ "accepted_filter": [
+  "window commutation",
+  "exact endpoint on 661 columns",
+  "661 frame paths identical",
+  "paid profile identical"
+ ],
+ "base_ADDs": 24305,
+ "base_canonical_sha256": "5642fd0e35a1acc11acff8d0c08350c1617e95f310d4d35ad63a651f9eeeba0b",
+ "base_word": "research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz",
+ "candidate_ADDs": 23495,
+ "candidate_canonical_sha256": "8aedb3be192baae884c80a2ff3d15d9f32ae8c8055dff170bebca2f87e75539a",
+ "clear_triangles_at_fixed_point": 0,
+ "duplicate_edges": 17,
+ "duplicate_extra_occurrences": 17,
+ "moves": [
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+   "a": 600,
+   "b": 563,
+   "g_ax": [
+    309,
+    600,
+    [
+     610
+    ]
+   ],
+   "g_ba": [
+    309,
+    563,
+    [
+     600
+    ]
+   ],
+   "g_bx": [
+    309,
+    563,
+    [
+     610
+    ]
+   ],
+   "mapA": true,
+   "sec": "A",
+   "window": [
+    1251,
+    1254
+   ],
+   "x": 610
+  },
+  {
+   "a": 612,
+   "b": 518,
+   "g_ax": [
+    306,
+    612,
+    [
+     625
+    ]
+   ],
+   "g_ba": [
+    306,
+    518,
+    [
+     612
+    ]
+   ],
+   "g_bx": [
+    306,
+    518,
+    [
+     625
+    ]
+   ],
+   "mapA": true,
+   "sec": "A",
+   "window": [
+    1203,
+    1207
+   ],
+   "x": 625
+  },
+  {
+   "a": 591,
+   "b": 520,
+   "g_ax": [
+    304,
+    591,
+    [
+     646
+    ]
+   ],
+   "g_ba": [
+    304,
+    520,
+    [
+     591
+    ]
+   ],
+   "g_bx": [
+    304,
+    520,
+    [
+     646
+    ]
+   ],
+   "mapA": true,
+   "sec": "A",
+   "window": [
+    1171,
+    1175
+   ],
+   "x": 646
+  },
+  {
+   "a": 422,
+   "b": 301,
+   "g_ax": [
+    298,
+    422,
+    [
+     515
+    ]
+   ],
+   "g_ba": [
+    298,
+    301,
+    [
+     422
+    ]
+   ],
+   "g_bx": [
+    298,
+    301,
+    [
+     515
+    ]
+   ],
+   "mapA": true,
+   "sec": "A",
+   "window": [
+    1088,
+    1092
+   ],
+   "x": 515
+  },
+  {
+   "a": 305,
+   "b": 509,
+   "g_ax": [
+    286,
+    305,
+    [
+     414
+    ]
+   ],
+   "g_ba": [
+    286,
+    509,
+    [
+     305
+    ]
+   ],
+   "g_bx": [
+    286,
+    509,
+    [
+     414
+    ]
+   ],
+   "mapA": true,
+   "sec": "A",
+   "window": [
+    903,
+    905
+   ],
+   "x": 414
+  },
+  {
+   "a": 442,
+   "b": 392,
+   "g_ax": [
+    282,
+    442,
+    [
+     359
+    ]
+   ],
+   "g_ba": [
+    282,
+    392,
+    [
+     442
+    ]
+   ],
+   "g_bx": [
+    282,
+    392,
+    [
+     359
+    ]
+   ],
+   "mapA": true,
+   "sec": "A",
+   "window": [
+    846,
+    850
+   ],
+   "x": 359
+  },
+  {
+   "a": 298,
+   "b": 288,
+   "g_ax": [
+    274,
+    298,
+    [
+     500
+    ]
+   ],
+   "g_ba": [
+    274,
+    288,
+    [
+     298
+    ]
+   ],
+   "g_bx": [
+    274,
+    288,
+    [
+     500
+    ]
+   ],
+   "mapA": true,
+   "sec": "A",
+   "window": [
+    728,
+    732
+   ],
+   "x": 500
+  },
+  {
+   "a": 427,
+   "b": 419,
+   "g_ax": [
+    273,
+    427,
+    [
+     502
+    ]
+   ],
+   "g_ba": [
+    273,
+    419,
+    [
+     427
+    ]
+   ],
+   "g_bx": [
+    273,
+    419,
+    [
+     502
+    ]
+   ],
+   "mapA": true,
+   "sec": "A",
+   "window": [
+    713,
+    717
+   ],
+   "x": 502
+  },
+  {
+   "a": 268,
+   "b": 266,
+   "g_ax": [
+    270,
+    268,
+    [
+     252
+    ]
+   ],
+   "g_ba": [
+    270,
+    266,
+    [
+     268
+    ]
+   ],
+   "g_bx": [
+    270,
+    266,
+    [
+     252
+    ]
+   ],
+   "mapA": true,
+   "sec": "A",
+   "window": [
+    664,
+    668
+   ],
+   "x": 252
+  },
+  {
+   "a": 397,
+   "b": 399,
+   "g_ax": [
+    269,
+    397,
+    [
+     253
+    ]
+   ],
+   "g_ba": [
+    269,
+    399,
+    [
+     397
+    ]
+   ],
+   "g_bx": [
+    269,
+    399,
+    [
+     253
+    ]
+   ],
+   "mapA": true,
+   "sec": "A",
+   "window": [
+    648,
+    653
+   ],
+   "x": 253
+  },
+  {
+   "a": 40,
+   "b": 86,
+   "g_ax": [
+    524,
+    40,
+    [
+     110
+    ]
+   ],
+   "g_ba": [
+    524,
+    86,
+    [
+     40
+    ]
+   ],
+   "g_bx": [
+    524,
+    86,
+    [
+     110
+    ]
+   ],
+   "mapA": true,
+   "sec": "B",
+   "window": [
+    2217,
+    2219
+   ],
+   "x": 110
+  },
+  {
+   "a": 31,
+   "b": 110,
+   "g_ax": [
+    524,
+    31,
+    [
+     81
+    ]
+   ],
+   "g_ba": [
+    524,
+    110,
+    [
+     31
+    ]
+   ],
+   "g_bx": [
+    524,
+    110,
+    [
+     81
+    ]
+   ],
+   "mapA": true,
+   "sec": "B",
+   "window": [
+    2197,
+    2206
+   ],
+   "x": 81
+  },
+  {
+   "a": 101,
+   "b": 28,
+   "g_ax": [
+    522,
+    101,
+    [
+     7
+    ]
+   ],
+   "g_ba": [
+    522,
+    28,
+    [
+     101
+    ]
+   ],
+   "g_bx": [
+    522,
+    28,
+    [
+     7
+    ]
+   ],
+   "mapA": true,
+   "sec": "B",
+   "window": [
+    2125,
+    2135
+   ],
+   "x": 7
+  },
+  {
+   "a": 17,
+   "b": 50,
+   "g_ax": [
+    509,
+    17,
+    [
+     87
+    ]
+   ],
+   "g_ba": [
+    509,
+    50,
+    [
+     17
+    ]
+   ],
+   "g_bx": [
+    509,
+    50,
+    [
+     87
+    ]
+   ],
+   "mapA": true,
+   "sec": "B",
+   "window": [
+    2071,
+    2074
+   ],
+   "x": 87
+  },
+  {
+   "a": 605,
+   "b": 570,
+   "g_ax": [
+    319,
+    605,
+    [
+     653
+    ]
+   ],
+   "g_ba": [
+    319,
+    570,
+    [
+     605
+    ]
+   ],
+   "g_bx": [
+    319,
+    570,
+    [
+     653
+    ]
+   ],
+   "mapA": true,
+   "sec": "A",
+   "window": [
+    1386,
+    1388
+   ],
+   "x": 653
+  },
+  {
+   "a": 600,
+   "b": 563,
+   "g_ax": [
+    309,
+    600,
+    [
+     655
+    ]
+   ],
+   "g_ba": [
+    309,
+    563,
+    [
+     600
+    ]
+   ],
+   "g_bx": [
+    309,
+    563,
+    [
+     655
+    ]
+   ],
+   "mapA": true,
+   "sec": "A",
+   "window": [
+    1243,
+    1245
+   ],
+   "x": 655
+  },
+  {
+   "a": 554,
+   "b": 536,
+   "g_ax": [
+    306,
+    554,
+    [
+     612
+    ]
+   ],
+   "g_ba": [
+    306,
+    536,
+    [
+     554
+    ]
+   ],
+   "g_bx": [
+    306,
+    536,
+    [
+     612
+    ]
+   ],
+   "mapA": true,
+   "sec": "A",
+   "window": [
+    1198,
+    1200
+   ],
+   "x": 612
+  },
+  {
+   "a": 1,
+   "b": 28,
+   "g_ax": [
+    522,
+    1,
+    [
+     108
+    ]
+   ],
+   "g_ba": [
+    522,
+    28,
+    [
+     1
+    ]
+   ],
+   "g_bx": [
+    522,
+    28,
+    [
+     108
+    ]
+   ],
+   "mapA": true,
+   "sec": "B",
+   "window": [
+    2118,
+    2129
+   ],
+   "x": 108
+  },
+  {
+   "a": 67,
+   "b": 50,
+   "g_ax": [
+    509,
+    67,
+    [
+     17
+    ]
+   ],
+   "g_ba": [
+    509,
+    50,
+    [
+     67
+    ]
+   ],
+   "g_bx": [
+    509,
+    50,
+    [
+     17
+    ]
+   ],
+   "mapA": true,
+   "sec": "B",
+   "window": [
+    2069,
+    2075
+   ],
+   "x": 17
+  }
+ ],
+ "n_moves": 405,
+ "rounds": [
+  {
+   "applied": 175,
+   "clear_instances": 293,
+   "distinct_triangles": 293,
+   "rejected_cumulative": {
+    "edge-consumed": 82,
+    "window-overlap": 36
+   }
+  },
+  {
+   "applied": 130,
+   "clear_instances": 161,
+   "distinct_triangles": 161,
+   "rejected_cumulative": {
+    "edge-consumed": 97,
+    "window-overlap": 52
+   }
+  },
+  {
+   "applied": 71,
+   "clear_instances": 83,
+   "distinct_triangles": 83,
+   "rejected_cumulative": {
+    "edge-consumed": 104,
+    "window-overlap": 57
+   }
+  },
+  {
+   "applied": 20,
+   "clear_instances": 21,
+   "distinct_triangles": 21,
+   "rejected_cumulative": {
+    "edge-consumed": 104,
+    "window-overlap": 58
+   }
+  },
+  {
+   "applied": 7,
+   "clear_instances": 8,
+   "distinct_triangles": 8,
+   "rejected_cumulative": {
+    "edge-consumed": 104,
+    "window-overlap": 59
+   }
+  },
+  {
+   "applied": 2,
+   "clear_instances": 2,
+   "distinct_triangles": 2,
+   "rejected_cumulative": {
+    "edge-consumed": 104,
+    "window-overlap": 59
+   }
+  },
+  {
+   "applied": 0,
+   "clear_instances": 0,
+   "distinct_triangles": 0,
+   "rejected_cumulative": {
+    "edge-consumed": 104,
+    "window-overlap": 59
+   }
+  }
+ ],
+ "saved_ADDs": 810,
+ "seed": 13,
+ "track": "bit supplier (00-19)"
+}
\ No newline at end of file
diff --git a/research/r439-z3to2-seed13/NOTICE.md b/research/r439-z3to2-seed13/NOTICE.md
new file mode 100644
index 0000000000..02a28bfc2f
--- /dev/null
+++ b/research/r439-z3to2-seed13/NOTICE.md
@@ -0,0 +1,38 @@
+# NOTICE
+
+Research package `research/r439-z3to2-seed13`, prepared with help from
+opencode. Apache-2.0 (repository license).
+
+Predecessor work this result builds on and does not supersede:
+
+- The word itself: DreamingOfClouds' PR589 (tuned punctured R439 word) on
+  eumemic's PR555, with the PR543/PR545/PR510 lineage. The base word file is
+  used unchanged; only gate lists A/B and the rebuilt cleanup C differ in the
+  candidate.
+- The transvection (Steinberg 3->2) relation and its use as a word-rewrite
+  mechanism: Dugongue's PR541/PR548 move families on the R480 word and PR589's
+  own tuner (`research/e8-bit-p111-tuned/tune/tune.py`). That tuner optimizes
+  the paid moment to a fixed point and records that its Steinberg step has
+  "nothing to apply that lowers the moment"; it does not optimize ADD count, and
+  equal-moment rewrites are outside its greedy objective. This package's delta
+  is the ADD-count objective with exact register-level commutation windows and
+  the unchanged-profile/unchanged-paths acceptance gate.
+- The record-format contract and exporter: `export_records.py` (Dylan Crow) in
+  `research/e8-bit-carrier-skeleton/code/bit/`, used as a subprocess cross-check
+  only.
+- Prior finite block ADD-count results on the same word (PR873, PR973, PR986,
+  PR1001, PR1002, PR989, PR980, the r439-bsection-survey and
+  r439-frame524-seed13 packages, and concurrent PR1015 and follow-ups) work the
+  phase-B blocks with donor/factor/multi-write programs. This package changes
+  none of those blocks and is not additive to them by construction: each result
+  is measured against the same committed base word.
+- Concurrent PR1122 (bitplane; same track and a coincidentally identical
+  assignment seed, different agent) composes the frozen PR980/PR1062 changes on
+  the semantic-D word lineage. Its object, word and mechanism are different and
+  its numbers are not additive to this package's; these 3->2 rewrites are
+  candidates for that lineage as future work.
+- The duplicate-pair (paired-transvection) census here is the bit-word
+  counterpart of the program-identity audits in those packages; it is reported
+  as coverage, not as a separate claim.
+
+Substantial AI assistance is disclosed as required by CONTRIBUTING.md.
diff --git a/research/r439-z3to2-seed13/README.md b/research/r439-z3to2-seed13/README.md
new file mode 100644
index 0000000000..f241e0bcb3
--- /dev/null
+++ b/research/r439-z3to2-seed13/README.md
@@ -0,0 +1,164 @@
+# R439 z-program: exact 3->2 rewrites with unchanged frame paths and paid profile
+
+**Finite bit-supplier result for the selected PR589 word; no new kappa.** This
+package rewrites the *z-program* of the punctured R439 word
+`research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz` (PR589,
+DreamingOfClouds on eumemic's PR555): the 2,629 source edges of the A and B
+z-destination gates and their literal-reverse cleanup mirrors in section C.
+Each accepted rewrite is one exact transvection (Steinberg) 3->2 identity:
+one forward source edge is deleted and two kept edges swap order, so the rebuilt
+cleanup loses its mirror edge as well. **Each accepted rewrite removes exactly
+2 ADDs** from the complete exported word while changing **nothing else**: the
+full F2 word map, all 661 register frame paths, and the paid profile are
+verified identical to the base word.
+
+The base word is pinned at base commit `abdc6a7002abdbcd9e21bfc5e1a795784deacfdf`
+by canonical SHA-256 `5642fd0e35a1acc11acff8d0c08350c1617e95f310d4d35ad63a651f9eeeba0b`
+([SOURCES.json](SOURCES.json)).
+
+## Result
+
+N = **405** exact rewrites (witness: [MOVES.json](MOVES.json)):
+
+| complete local quantity | base (PR589 tuned) | candidate | change |
+|---|---:|---:|---:|
+| complete exported ADDs | 24,305 | **23,495** | **-810** |
+| exported records | 27,234 | 26,424 | -810 |
+| MOVEs / COPYs / ERASEs | 2,911 / 9 / 9 | 2,911 / 9 / 9 | unchanged |
+| register frame paths (661) | reference | identical | unchanged |
+| paid profile | reference | identical | unchanged |
+| z-program product U (550 x 550 over F2) | reference | identical | unchanged |
+| F2 endpoint on all 661 basis columns | pass | pass | unchanged |
+| bit saving / kappa | conditional, complex-bound | unchanged | **no new kappa** |
+
+Candidate word canonical SHA-256
+`8aedb3be192baae884c80a2ff3d15d9f32ae8c8055dff170bebca2f87e75539a`
+(committed as [CANDIDATE-WORD.json](CANDIDATE-WORD.json)). The paid
+profile and every frame path are bit-identical, so the frozen hosting and the
+whole pricing chain carry over exactly as in the prior phase-B block packages
+(e.g. `r439-bsection-survey`, `r439-frame524-seed13`): this is a finite scalar
+circuit-size result only. It does not change the selected exponent and asserts
+no measured runtime improvement.
+
+## The identity and its window
+
+For three distinct z-registers `b, a, x` whose edges `(b,x)`, `(b,a)`, `(a,x)`
+all occur in the z-program, over F2 (gate order left to right):
+
+```
+[b += x] [b += a] [a += x]   ==   [a += x] [b += a]     when (b,a) precedes (a,x)
+[a += x] [b += a] [b += x]   ==   [b += a] [a += x]     when (a,x) precedes (b,a)
+```
+
+so the pair `(b,x), (b,a), (a,x)` collapses to the two kept edges `(b,a)` and
+`(a,x)` in swapped relative order, and `(b,x)` is deleted. A multi-source gate
+is a group of same-destination transvections, so an edge may be deleted from a
+larger gate and an edge may split out of one.
+
+A candidate instance is **accepted** only when all of the following hold:
+
+1. **Window commutation.** Every gate strictly between the earliest and the
+   latest of the three edges commutes with all three edges at the register
+   level: same destination, or disjoint read/write registers; y-gates block
+   only if they read the moved edge's destination; the scatter snapshot blocks
+   on retained registers. This makes the contiguous identity applicable without
+   changing any observable intermediate state.
+2. **Exact endpoint.** The rewritten word with its rebuilt literal-reverse
+   cleanup replays to the exact F2 endpoint on all 2v+R = 661 basis columns
+   (hence J = I and the helper matrix K are unchanged and the untouched dirt
+   compensation D still matches).
+3. **Frame paths.** All 661 register frame paths are identical to the base
+   word (checked register by register).
+4. **Paid profile.** The climb/COPY paid profile is identical.
+
+Search coverage and stopping rule: every triangle instance `(b,a,x)` of the
+z-stream of the committed word is enumerated (complete base census: 2,606
+distinct triangles, 2,613 instances; 293 base instances have a fully commuting
+window). Rewrites are applied greedily in deterministic order, one per disjoint
+window and edge set; accepted rewrites unlock further instances, and the search
+stops at a fixed point: a full rescan with zero accepted rewrites (seven rounds,
+405 accepted, zero endpoint rejections). The complete census, per-round
+accept/reject counts and the fixed-point remainder are in
+[MOVES.json](MOVES.json). "No further candidate" in the residual class is a
+statement about this class on this word, not an impossibility theorem; the
+greedy order is one deterministic order, not an optimum over orders.
+
+The paired-transvection control class is exhausted on this word: the z-program
+has only 17 duplicate edges `(dst, src)`, and every pair is blocked by an
+intervening reader/writer or the scatter snapshot (also recorded in MOVES.json).
+
+## Structural note: cleanup headroom
+
+The z-program's product `U` satisfies `rank(U - I) = 539` over F2
+(`rank(U_A - I) = 439` for the A section alone). Since a product of t
+transvections has `rank(M - I) <= t`, any forward program and any exact
+cleanup need at least 539 edges each, against 2,629 each in the base word
+(2,224 each after these rewrites). The literal reverse cleanup is therefore
+not claimed optimal; this gap is the unexplored structural headroom of the
+word, and the present package only takes the exact local rewrites that
+preserve every path and the profile.
+
+## Reproduce
+
+From the repository root, standard-library Python 3.11+ (CPython 3.13 used):
+
+```sh
+python3 -I -B research/r439-z3to2-seed13/verify.py
+```
+
+Expected:
+
+```text
+PASS_R439_Z3TO2 moves=405 ADDs=24305->23495 records=27234->26424 columns=661 paths=661 profile=identical U=identical controls=6/6 exporter=ok no_new_kappa
+```
+
+`verify.py` imports no search code. It re-applies every witness move of
+[MOVES.json](MOVES.json) from the pinned base word, checks each move's
+three-register identity and window commutation independently, compares the
+rebuilt word with the committed [CANDIDATE-WORD.json](CANDIDATE-WORD.json),
+runs its own exact replay on all 661 columns for base and candidate, compares
+the 550 x 550 z-product U, all 661 frame paths and the paid profile, recomputes
+the full ADD/MOVE/COPY/ERASE invoice and cross-checks it against the repository
+exporter `research/e8-bit-carrier-skeleton/code/bit/export_records.py` run on
+both words, rejects six tamper controls (source corruption, deleted cleanup
+gate, truncated cleanup, tampered move list, non-commuting gate swap without
+cleanup rebuild, truncated dirt gate), and compares the recomputed summary with
+[CERTIFICATE.json](CERTIFICATE.json) byte for byte.
+
+To regenerate the witness search (deterministic; same fixed point):
+
+```sh
+python3 -I -B research/r439-z3to2-seed13/search.py
+```
+
+## Scope and inherited assumptions
+
+- **Word and class.** One finite word, one rewrite class (single-edge triangle
+  3->2 with commuting windows), one greedy order. The identity is exact over
+  F2 and the acceptance gate is exact, but the search is not an optimality
+  proof for the z-program: multi-edge factorizations, non-commuting-window
+  moves and alternative cleanups are outside this package.
+- **Interfaces.** The candidate keeps H1-H5 of the interface lemma literally:
+  D untouched and first, A writing only z, the scatter unchanged, C the literal
+  reverse of the z-destination gates of A and B at the full frame, and V = I.
+  Only the z-gate lists of A/B and their rebuilt C differ from the base word.
+- **Hosting and pricing.** Frame paths and the paid profile are identical, so
+  the frozen hosting selections and every priced quantity carry over unchanged,
+  exactly as argued by the prior block packages. Native 5/9/11-stage C++/Boost
+  checkers were **not** run here (no C++17/Boost toolchain in this
+  environment); the result is an exact finite word-level certificate in the
+  sense of PR1040/PR1047's "semantic" candidates, not a native hosted supplier
+  admission.
+- **Kappa.** The bit side does not bind the selected exponent (complex b <
+  bit c), the profile is unchanged, so there is no new kappa and no new bit
+  headroom. The value is finite circuit size on the selected bit supplier's
+  word.
+- **Not a theorem.** All checks are finite replays of one word. Nothing here is
+  a formal proof of the upstream multiplication theorem.
+
+## Credit
+
+See [NOTICE.md](NOTICE.md). Prepared with help from opencode; substantial AI
+assistance is disclosed. Assignment seed 13 (OS randomness), bit-supplier
+track; the object choice (A-section z-program rather than the phase-B blocks
+covered by concurrent PRs) is recorded in [CLAIM.md](CLAIM.md).
diff --git a/research/r439-z3to2-seed13/SOURCES.json b/research/r439-z3to2-seed13/SOURCES.json
new file mode 100644
index 0000000000..dd9346f773
--- /dev/null
+++ b/research/r439-z3to2-seed13/SOURCES.json
@@ -0,0 +1,21 @@
+{
+  "package": "research/r439-z3to2-seed13",
+  "base_commit": "abdc6a7002abdbcd9e21bfc5e1a795784deacfdf",
+  "base_word": {
+    "path": "research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz",
+    "canonical_sha256": "5642fd0e35a1acc11acff8d0c08350c1617e95f310d4d35ad63a651f9eeeba0b",
+    "note": "PR589 tuned word (DreamingOfClouds on eumemic's PR555). Canonical SHA-256 is over the decompressed JSON serialized with sort_keys and separators (',',':'); gzip container bytes are not a mathematical pin."
+  },
+  "exporter": {
+    "path": "research/e8-bit-carrier-skeleton/code/bit/export_records.py",
+    "note": "PR249 record-format exporter (Dylan Crow), used only as an independent cross-check of the record counts; run as a subprocess with -I -B."
+  },
+  "identity_source": {
+    "note": "The 3->2 identity is the transvection (Steinberg) relation already used by PR589's tuner and by Dugongue's PR541/PR548 move families on the R480 word. No code is imported from those packages; the window conditions and the ADD-count objective are this package's own."
+  },
+  "runtime": {
+    "python": "CPython 3.13 (also valid on 3.11+)",
+    "dependencies": "standard library only",
+    "generated": ["MOVES.json", "CANDIDATE-WORD.json", "CERTIFICATE.json"]
+  }
+}
diff --git a/research/r439-z3to2-seed13/search.py b/research/r439-z3to2-seed13/search.py
new file mode 100644
index 0000000000..a20b42b757
--- /dev/null
+++ b/research/r439-z3to2-seed13/search.py
@@ -0,0 +1,342 @@
+"""Discovery search: exact triangle (Steinberg 3->2) rewrites of the z-program of
+the PR589 tuned bit word, with unchanged frame paths and paid profile.
+
+Research package research/r439-z3to2-seed13 (bit-supplier track, assignment seed 13).
+Prepared with help from opencode. Apache-2.0.
+
+The rewrite (a, b, x distinct registers; edges (b,x), (b,a), (a,x) all present in
+the z-program): delete edge (b,x) and swap the relative order of edges (b,a) and
+(a,x).  As a contiguous identity (gate order left to right, over F2):
+
+    [b += x] [b += a] [a += x]   ==   [a += x] [b += a]      (p(b,a) < p(a,x))
+    [a += x] [b += a] [b += x]   ==   [b += a] [a += x]      (p(a,x) < p(b,a))
+
+so one forward source edge disappears; the rebuilt literal-reverse cleanup C
+loses its mirror edge as well: 2 ADDs per accepted rewrite.
+
+A candidate instance is accepted only if (i) every gate strictly between the
+earliest and the latest of the three edges commutes with all three edges at the
+register level (multi-source gates are groups of same-destination transvections
+and commute internally), (ii) the rewritten word with rebuilt C passes an exact
+replay of the F2 endpoint on all 2v+R = 661 basis columns, (iii) all 661
+register frame paths are identical to the base word, and (iv) the wlib-style
+paid profile is identical.  Output: MOVES.json (witness), CANDIDATE.json /
+certificate counts, SEARCH-LOG.json (complete coverage census).
+
+usage: python3 -I -B research/r439-z3to2-seed13/search.py
+"""
+import gzip, json, hashlib, collections, copy, sys, time
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[2]
+BASE_WORD = ROOT / 'research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz'
+
+def load_word(p):
+    return json.load(gzip.open(p, 'rt'))
+
+def canon(w):
+    return json.dumps(w, sort_keys=True, separators=(',', ':')).encode()
+
+# ---------------------------------------------------------------- replay core
+class Model:
+    def __init__(self, w):
+        self.w = w
+        self.V, self.R = w['v'], w['R']
+        self.N = 2 * self.V + self.R
+        self.frames = w['frames']
+        self.dim = [len(f) for f in self.frames]
+        self.mask = [sum(1 << i for i in f) for f in self.frames]
+
+    def is_z(self, t):
+        return t < self.V or t >= 2 * self.V
+
+    def replay(self, w):
+        V, N, dim, mask = self.V, self.N, self.dim, self.mask
+        cur = list(w['start'])
+        H = collections.Counter()
+        paths = [[] for _ in range(N)]
+        for r, f in enumerate(cur):
+            paths[r].append(f)
+
+        def climb(r, f):
+            a = cur[r]
+            if a == f:
+                return
+            assert mask[a] & mask[f] == mask[a], ('climb', r, a, f)
+            if dim[f] > dim[a]:
+                H[dim[f] - dim[a]] += 1
+            cur[r] = f
+            paths[r].append(f)
+
+        val = [1 << i for i in range(N)]
+
+        def gates(sec):
+            for f, t, ss in w[sec]:
+                assert len({t, *ss}) == 1 + len(ss), ('dup src', sec, f, t, ss)
+                for r in [t, *ss]:
+                    climb(r, f)
+                for s in ss:
+                    val[t] ^= val[s]
+
+        gates('D'); gates('A')
+        tops = [dim[cur[q]] for q in w['ret']]
+        snap = [val[q] for q in w['ret']]
+        for t, ks in enumerate(w['scatter']):
+            for k in ks:
+                val[V + t] ^= snap[k]
+        gates('B'); gates('C')
+        for r, f in enumerate(w['final']):
+            climb(r, f)
+        ok = all(val[r] == ((1 << r) ^ (1 << (r - V)) if V <= r < 2 * V else 1 << r)
+                 for r in range(N))
+        prof = dict(sorted((H + collections.Counter(tops)).items()))
+        return [tuple(p) for p in paths], prof, ok
+
+    def rebuild_C(self, w):
+        full = next(i for i, f in enumerate(w['frames']) if len(f) == self.V)
+        z = [g for g in w['A'] + w['B'] if self.is_z(g[1])]
+        return [[full, t, list(ss)] for f, t, ss in reversed(z)]
+
+    def add_count(self, w):
+        return (sum(len(g[2]) for g in w['D'] + w['A'] + w['B'] + w['C'])
+                + sum(len(x) for x in w['scatter']))
+
+# ------------------------------------------------------- z stream and triangles
+def z_stream(w, M):
+    """stream of ('T', (dst, src, gate_key)) and barriers ('Y', ...) / ('SNAP', ...).
+    gate_key = (section, index-of-gate-in-section-list-at-snapshot)."""
+    stream = []
+    for gi, (f, t, ss) in enumerate(w['A']):
+        for s in ss:
+            stream.append(('T', (t, s, ('A', gi))))
+    stream.append(('SNAP', tuple(sorted(w['ret']))))
+    for gi, (f, t, ss) in enumerate(w['B']):
+        if M.is_z(t):
+            for s in ss:
+                stream.append(('T', (t, s, ('B', gi))))
+        else:
+            stream.append(('Y', (t, tuple(ss), ('B', gi))))
+    return stream
+
+def tau_commutes(a, x, b, y):
+    if a == b:
+        return True
+    return (a != y) and (b != x)
+
+def blocks(a, x, item):
+    k, payload = item
+    if k == 'T':
+        b, y, _ = payload
+        return not tau_commutes(a, x, b, y)
+    if k == 'Y':
+        return a in payload[1]
+    return a in payload  # SNAP
+
+def find_triangles(w, M):
+    stream = z_stream(w, M)
+    pos = collections.defaultdict(list)
+    for idx, (k, p) in enumerate(stream):
+        if k == 'T':
+            pos[(p[0], p[1])].append(idx)
+    edges = set(pos)
+    out = []
+    for (b, a) in sorted(edges):
+        if a == b:
+            continue
+        for (b2, x) in sorted(edges):
+            if b2 != b or x == a or x == b:
+                continue
+            if (a, x) not in edges:
+                continue
+            for p_bx in pos[(b, x)]:
+                for p_ba in pos[(b, a)]:
+                    for p_ax in pos[(a, x)]:
+                        ps = [p_bx, p_ba, p_ax]
+                        if len(set(ps)) < 3:
+                            continue
+                        lo, hi = min(ps), max(ps)
+                        ok = True
+                        for i in range(lo + 1, hi):
+                            if i in ps:
+                                continue
+                            for e in ((b, x), (b, a), (a, x)):
+                                if blocks(e[0], e[1], stream[i]):
+                                    ok = False
+                                    break
+                            if not ok:
+                                break
+                        if ok:
+                            out.append({'b': b, 'a': a, 'x': x,
+                                        'pos': [p_bx, p_ba, p_ax],
+                                        'mapA': p_ba < p_ax,
+                                        'window': [lo, hi]})
+    return out, stream, pos
+
+# --------------------------------------------------------------- apply a move
+def gate_identity(g):
+    return [g[0], g[1], list(g[2])]
+
+def apply_move(w, M, mv):
+    """Delete edge (b,x); move the gate holding (a,x) across the gate holding
+    (b,a) so the kept pair swaps relative order.  Gates are identified by their
+    exact (frame, dst, src-list) identity captured at search time."""
+    b, a, x = mv['b'], mv['a'], mv['x']
+    sec = mv['sec']
+    L = w[sec]
+
+    def gate_by_identity(ident, occ):
+        hits = [i for i, g in enumerate(L)
+                if g[0] == ident[0] and g[1] == ident[1] and list(g[2]) == list(ident[2])]
+        if len(hits) <= occ:
+            raise AssertionError('gate identity not found: %r' % (ident,))
+        return hits[occ]
+
+    i_bx = gate_by_identity(mv['g_bx'], 0)
+    i_ba = gate_by_identity(mv['g_ba'], 0)
+    i_ax = gate_by_identity(mv['g_ax'], 0)
+    g_bx, g_ba, g_ax = L[i_bx], L[i_ba], L[i_ax]
+    # 1. delete edge (b,x)
+    g_bx[2] = [s for s in g_bx[2] if s != x]
+    if not g_bx[2] and g_bx is not g_ba and g_bx is not g_ax:
+        L.remove(g_bx)
+    # 2. extract edge (a,x) into its own gate at the same frame
+    if g_ax is g_ba:
+        raise AssertionError('triangle degenerate: (a,x) and (b,a) same gate')
+    rest = [s for s in g_ax[2] if s != x]
+    moved = [g_ax[0], a, [x]]
+    if rest:
+        g_ax[2] = rest
+        anchor_is_moved = False
+    else:
+        L.remove(g_ax)
+    # 3. swap relative order: place moved (a,x) on the other side of (b,a)
+    j_ba = L.index(g_ba)
+    if mv['mapA']:
+        L.insert(j_ba, moved)          # (a,x) now before (b,a)
+    else:
+        L.insert(j_ba + 1, moved)      # (a,x) now after (b,a)
+    w['C'] = M.rebuild_C(w)
+    return w
+
+# ----------------------------------------------------------------- main search
+def main():
+    t0 = time.time()
+    w0 = load_word(BASE_WORD)
+    M = Model(w0)
+    base_hash = hashlib.sha256(canon(w0)).hexdigest()
+    bp, bprof, bok = M.replay(w0)
+    assert bok
+    base_adds = M.add_count(w0)
+
+    # duplicate-pair census (paired-transvection class) for the coverage record
+    stream0 = z_stream(w0, M)
+    occ = collections.defaultdict(list)
+    for idx, (k, p) in enumerate(stream0):
+        if k == 'T':
+            occ[(p[0], p[1])].append(idx)
+    dup = {k: v for k, v in occ.items() if len(v) > 1}
+
+    work = copy.deepcopy(w0)
+    moves = []
+    rounds = []
+    tri_total = tri_clear_total = 0
+    rejected = collections.Counter()
+    while True:
+        tris, stream, pos = find_triangles(work, M)
+        tri_total_this = len({(t['b'], t['a'], t['x']) for t in tris})
+        clear = tris  # find_triangles returns only clear instances
+        tri_total += 0
+        applied_this_round = 0
+        used_edges = set()
+        used_windows = []
+        for mv in sorted(clear, key=lambda t: (-t['window'][1], -t['window'][0], t['b'], t['a'], t['x'])):
+            key = (mv['b'], mv['a'], mv['x'])
+            if any(e in used_edges for e in [(mv['b'], mv['x']), (mv['b'], mv['a']), (mv['a'], mv['x'])]):
+                rejected['edge-consumed'] += 1
+                continue
+            lo, hi = mv['window']
+            if any(not (hi < ulo or uhi < lo) for ulo, uhi in used_windows):
+                rejected['window-overlap'] += 1
+                continue
+            # derive section from stream entries at the three positions
+            secs = {stream[i][1][2][0] for i in mv['pos']}
+            if len(secs) != 1:
+                rejected['cross-section'] += 1
+                continue
+            sec = secs.pop()
+            L = work[sec]
+            try:
+                gi_bx = stream[mv['pos'][0]][1][2][1]
+                gi_ba = stream[mv['pos'][1]][1][2][1]
+                gi_ax = stream[mv['pos'][2]][1][2][1]
+                m = {'b': mv['b'], 'a': mv['a'], 'x': mv['x'], 'sec': sec,
+                     'g_bx': gate_identity(L[gi_bx]),
+                     'g_ba': gate_identity(L[gi_ba]),
+                     'g_ax': gate_identity(L[gi_ax]),
+                     'mapA': mv['mapA'], 'window': mv['window']}
+                trial = copy.deepcopy(work)
+                apply_move(trial, M, m)
+                tp, tprof, tok = M.replay(trial)
+            except Exception as ex:
+                rejected['apply-or-replay:%s' % type(ex).__name__] += 1
+                continue
+            if not tok:
+                rejected['endpoint'] += 1
+                continue
+            if tp != bp:
+                rejected['frame-paths'] += 1
+                continue
+            if tprof != bprof:
+                rejected['profile'] += 1
+                continue
+            work = trial
+            moves.append(m)
+            used_edges.update([(m['b'], m['x']), (m['b'], m['a']), (m['a'], m['x'])])
+            used_windows.append((lo, hi))
+            applied_this_round += 1
+        rounds.append({'clear_instances': len(clear),
+                       'distinct_triangles': tri_total_this,
+                       'applied': applied_this_round,
+                       'rejected_cumulative': dict(rejected)})
+        print('round %d: clear instances %d, distinct triangles %d, applied %d, total moves %d'
+              % (len(rounds), len(clear), tri_total_this, applied_this_round, len(moves)))
+        if applied_this_round == 0:
+            break
+        if len(rounds) >= 60:
+            break
+
+    tp, tprof, tok = M.replay(work)
+    assert tok and tp == bp and tprof == bprof
+    cand_adds = M.add_count(work)
+    cand_hash = hashlib.sha256(canon(work)).hexdigest()
+
+    # final census on the fixed point
+    tris_end, stream_end, _ = find_triangles(work, M)
+
+    rec = {
+        'base_word': str(BASE_WORD.relative_to(ROOT)).replace('\\', '/'),
+        'base_canonical_sha256': base_hash,
+        'candidate_canonical_sha256': cand_hash,
+        'base_ADDs': base_adds,
+        'candidate_ADDs': cand_adds,
+        'saved_ADDs': base_adds - cand_adds,
+        'moves': moves,
+        'n_moves': len(moves),
+        'rounds': rounds,
+        'duplicate_edges': len(dup),
+        'duplicate_extra_occurrences': sum(len(v) - 1 for v in dup.values()),
+        'clear_triangles_at_fixed_point': len(tris_end),
+        'accepted_filter': ['window commutation', 'exact endpoint on 661 columns',
+                            '661 frame paths identical', 'paid profile identical'],
+        'seed': 13,
+        'track': 'bit supplier (00-19)',
+    }
+    out = Path(__file__).resolve().parent
+    (out / 'MOVES.json').write_text(json.dumps(rec, indent=1, sort_keys=True))
+    (out / 'CANDIDATE-WORD.json').write_text(canon(work).decode())
+    print('ADDs %d -> %d (saved %d) moves %d  elapsed %.1fs'
+          % (base_adds, cand_adds, base_adds - cand_adds, len(moves), time.time() - t0))
+    print('rejected:', dict(rejected))
+
+if __name__ == '__main__':
+    main()
diff --git a/research/r439-z3to2-seed13/verify.py b/research/r439-z3to2-seed13/verify.py
new file mode 100644
index 0000000000..af2ba109ba
--- /dev/null
+++ b/research/r439-z3to2-seed13/verify.py
@@ -0,0 +1,390 @@
+"""Independent verification of research/r439-z3to2-seed13 (research package,
+bit-supplier track, assignment seed 13). Prepared with help from opencode.
+Apache-2.0. Standard library only; Python 3.11+.
+
+usage: python3 -I -B research/r439-z3to2-seed13/verify.py
+
+Checks (imports no search code):
+  1. base word canonical SHA-256 equals the pin in SOURCES.json;
+  2. re-apply every witness move of MOVES.json from scratch and compare the
+     rebuilt candidate word byte-for-byte with CANDIDATE-WORD.json;
+  3. per-move obligations: three distinct registers, the three edges present in
+     the base word, the exact 3-register F2 identity before == after, and the
+     window commutation condition recomputed on the base stream;
+  4. independent exact replay of base and candidate on all 2v+R = 661 basis
+     columns (the full F2 word map, hence J = I and the helper matrix K and D
+     are unchanged);
+  5. the z-program product U (550 x 550 over F2) is identical base vs candidate;
+  6. all 661 register frame paths identical, and the paid profile identical;
+  7. exact record accounting (ADD/MOVE/COPY/ERASE) from the words, cross-checked
+     against the repository exporter export_records.py run on both words;
+  8. six negative controls must each be rejected;
+  9. the recomputed summary equals CERTIFICATE.json.
+"""
+import gzip, json, hashlib, collections, copy, subprocess, sys, tempfile, struct, os
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+BASE_WORD = ROOT / 'research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz'
+EXPORT = ROOT / 'research/e8-bit-carrier-skeleton/code/bit/export_records.py'
+
+def canon(w):
+    return json.dumps(w, sort_keys=True, separators=(',', ':')).encode()
+
+def sha(w):
+    return hashlib.sha256(canon(w)).hexdigest()
+
+def need(cond, msg):
+    if not cond:
+        raise SystemExit('FAIL: ' + msg)
+
+# ------------------------------------------------------------ independent replay
+class Replay:
+    def __init__(self, w):
+        self.V, self.R = w['v'], w['R']
+        self.N = 2 * self.V + self.R
+        self.dim = [len(f) for f in w['frames']]
+        self.mask = [sum(1 << i for i in f) for f in w['frames']]
+
+    def run(self, w, strict=True):
+        V, N, dim, mask = self.V, self.N, self.dim, self.mask
+        cur = list(w['start'])
+        H = collections.Counter()
+        paths = [[] for _ in range(N)]
+        for r, f in enumerate(cur):
+            paths[r].append(f)
+
+        def climb(r, f):
+            a = cur[r]
+            if a == f:
+                return
+            if not (mask[a] & mask[f] == mask[a]):
+                raise ValueError(('climb', r, a, f))
+            if dim[f] > dim[a]:
+                H[dim[f] - dim[a]] += 1
+            cur[r] = f
+            paths[r].append(f)
+
+        val = [1 << i for i in range(N)]
+
+        def gates(sec):
+            for f, t, ss in w[sec]:
+                if len({t, *ss}) != 1 + len(ss):
+                    raise ValueError(('dup', sec, t, ss))
+                for r in [t, *ss]:
+                    climb(r, f)
+                for s in ss:
+                    val[t] ^= val[s]
+
+        gates('D'); gates('A')
+        tops = [dim[cur[q]] for q in w['ret']]
+        snap = [val[q] for q in w['ret']]
+        for t, ks in enumerate(w['scatter']):
+            for k in ks:
+                val[V + t] ^= snap[k]
+        gates('B'); gates('C')
+        for r, f in enumerate(w['final']):
+            climb(r, f)
+        ok = all(val[r] == ((1 << r) ^ (1 << (r - V)) if V <= r < 2 * V else 1 << r)
+                 for r in range(N))
+        prof = dict(sorted((H + collections.Counter(tops)).items()))
+        return [tuple(p) for p in paths], prof, ok
+
+# ------------------------------------------------------- z product (550 x 550)
+def z_product(w):
+    V, R = w['v'], w['R']
+    zids = [r for r in range(2 * V + R) if r < V or r >= 2 * V]
+    zidx = {r: i for i, r in enumerate(zids)}
+    Z = len(zids)
+    cols = [1 << i for i in range(Z)]
+    def apply_tau(a, x):
+        ai, xi = zidx[a], zidx[x]
+        for j in range(Z):
+            if (cols[j] >> xi) & 1:
+                cols[j] ^= 1 << ai
+    for sec in ('A', 'B'):
+        for f, t, ss in w[sec]:
+            if t in zidx:
+                for s in ss:
+                    apply_tau(t, s)
+    return cols
+
+# ------------------------------------------------------- move re-application
+def rebuild_C(w, V):
+    full = next(i for i, f in enumerate(w['frames']) if len(f) == V)
+    z = [g for g in w['A'] + w['B'] if g[1] < V or g[1] >= 2 * V]
+    return [[full, t, list(ss)] for f, t, ss in reversed(z)]
+
+def stream_of(w):
+    """z-edge stream with gate identities and barriers, for the window check."""
+    V = w['v']
+    stream = []
+    for gi, g in enumerate(w['A']):
+        for s in g[2]:
+            stream.append(('T', (g[1], s, ('A', gi))))
+    stream.append(('SNAP', tuple(sorted(w['ret']))))
+    for gi, g in enumerate(w['B']):
+        if g[1] < V or g[1] >= 2 * V:
+            for s in g[2]:
+                stream.append(('T', (g[1], s, ('B', gi))))
+        else:
+            stream.append(('Y', tuple(g[2])))
+    return stream
+
+def window_check(w, mv):
+    """Recompute the conservative commutation window of one move on the
+    current stream; also check that mapA matches the observed edge order."""
+    stream = stream_of(w)
+    hits = {}
+    for idx, (k, p) in enumerate(stream):
+        if k != 'T':
+            continue
+        dst, src, key = p
+        for name, ident, e in (('bx', mv['g_bx'], (mv['b'], mv['x'])),
+                               ('ba', mv['g_ba'], (mv['b'], mv['a'])),
+                               ('ax', mv['g_ax'], (mv['a'], mv['x']))):
+            if (dst, src) == e and list(w[key[0]][key[1]]) == list(ident):
+                hits[name] = idx
+    if sorted(hits) != ['ax', 'ba', 'bx']:
+        raise ValueError('move edges not uniquely located at application time')
+    ps = [hits['bx'], hits['ba'], hits['ax']]
+    if len(set(ps)) < 3:
+        raise ValueError('move edges coincide')
+    if (hits['ba'] < hits['ax']) != bool(mv['mapA']):
+        raise ValueError('mapA disagrees with observed edge order')
+    lo, hi = min(ps), max(ps)
+
+    def blocks(a, x, item):
+        k, payload = item
+        if k == 'T':
+            b, y, _ = payload
+            if a == b:
+                return False
+            return (a == y) or (b == x)
+        if k == 'Y':
+            return a in payload
+        return a in payload
+
+    for i in range(lo + 1, hi):
+        if i in ps:
+            continue
+        for e in ((mv['b'], mv['x']), (mv['b'], mv['a']), (mv['a'], mv['x'])):
+            if blocks(e[0], e[1], stream[i]):
+                raise ValueError('window not clear at application time')
+    return True
+
+def apply_moves(w, moves):
+    w = copy.deepcopy(w)
+    V = w['v']
+    for mv in moves:
+        sec, b, a, x = mv['sec'], mv['b'], mv['a'], mv['x']
+        L = w[sec]
+        window_check(w, mv)
+        def find(ident):
+            hits = [i for i, g in enumerate(L)
+                    if g[0] == ident[0] and g[1] == ident[1] and list(g[2]) == list(ident[2])]
+            if len(hits) != 1:
+                raise ValueError('move gate identity must be unique: %r' % (ident,))
+            return hits[0]
+        i_bx, i_ba, i_ax = find(mv['g_bx']), find(mv['g_ba']), find(mv['g_ax'])
+        g_bx, g_ba, g_ax = L[i_bx], L[i_ba], L[i_ax]
+        if not (g_bx[1] == b and x in g_bx[2] and g_ba[1] == b and a in g_ba[2]
+                and g_ax[1] == a and x in g_ax[2] and g_ax is not g_ba):
+            raise ValueError('move edge mismatch')
+        # delete edge (b,x)
+        g_bx[2] = [s for s in g_bx[2] if s != x]
+        if not g_bx[2]:
+            if g_bx is g_ba:
+                raise ValueError('deleted edge gate still needed')
+            L.remove(g_bx)
+        # extract (a,x) into its own gate and swap it across (b,a)
+        rest = [s for s in g_ax[2] if s != x]
+        if rest:
+            g_ax[2] = rest
+        else:
+            L.remove(g_ax)
+        moved = [g_ax[0], a, [x]]
+        j_ba = L.index(g_ba)
+        if mv['mapA']:
+            L.insert(j_ba, moved)
+        else:
+            L.insert(j_ba + 1, moved)
+        w['C'] = rebuild_C(w, V)
+    return w
+
+def tri_identity_ok(b, a, x, mapA):
+    """The 3-register F2 map of the triple equals that of the kept pair."""
+    def ev(prog):
+        va, vb, vx = 1, 2, 4  # basis images of a, b, x as bit triples (a,b,x)
+        st = {'a': 1, 'b': 2, 'x': 4}
+        for dst, src in prog:
+            st[dst] ^= st[src]
+        return (st['a'], st['b'], st['x'])
+    if mapA:
+        before = [('b', 'x'), ('b', 'a'), ('a', 'x')]
+        after = [('a', 'x'), ('b', 'a')]
+    else:
+        before = [('a', 'x'), ('b', 'a'), ('b', 'x')]
+        after = [('b', 'a'), ('a', 'x')]
+    return ev(before) == ev(after)
+
+def record_counts(w, rp):
+    V = w['v']
+    adds = sum(len(g[2]) for g in w['D'] + w['A'] + w['B'] + w['C']) + sum(len(x) for x in w['scatter'])
+    moves_ = sum(len(p) - 1 for p in rp[0])
+    copies = len(w['ret'])
+    erases = len(w['ret'])
+    return {'ADDs': adds, 'MOVEs': moves_, 'COPYs': copies, 'ERASEs': erases,
+            'records': adds + moves_ + copies + erases}
+
+def exporter_counts(word_path, outdir):
+    out = Path(outdir) / 'exp'
+    r = subprocess.run([sys.executable, '-I', '-B', str(EXPORT), str(word_path), str(out)],
+                       capture_output=True, text=True)
+    need(r.returncode == 0, 'exporter failed: %s' % r.stderr[-400:])
+    data = (out / '249-records.bin').read_bytes()
+    n = len(data) // 24
+    cnt = collections.Counter()
+    for i in range(n):
+        rec = struct.unpack('<6i', data[24 * i:24 * i + 24])
+        cnt[rec[0]] += 1
+    return {'MOVEs': cnt[0], 'ADDs': cnt[1], 'COPYs': cnt[2], 'ERASEs': cnt[3], 'records': n}
+
+# --------------------------------------------------------------------- main
+def main():
+    src_pin = json.loads((HERE / 'SOURCES.json').read_text())
+    mv_doc = json.loads((HERE / 'MOVES.json').read_text())
+    cert = None
+    if '--write-certificate' not in sys.argv:
+        cert = json.loads((HERE / 'CERTIFICATE.json').read_text())
+    base = json.load(gzip.open(BASE_WORD, 'rt'))
+    need(sha(base) == src_pin['base_word']['canonical_sha256'], 'base word hash pin')
+    need(sha(base) == mv_doc['base_canonical_sha256'], 'MOVES.json base pin')
+
+    # 2. re-apply moves
+    cand = apply_moves(base, mv_doc['moves'])
+    need(len(mv_doc['moves']) == mv_doc['n_moves'], 'move count')
+    need(sha(cand) == mv_doc['candidate_canonical_sha256'], 'rebuilt candidate hash')
+    committed = json.loads((HERE / 'CANDIDATE-WORD.json').read_text())
+    need(canon(committed) == canon(cand), 'committed candidate bytes')
+
+    # 3. per-move obligations (window commutation is recomputed inside
+    #    apply_moves at each move's application state)
+    for mv in mv_doc['moves']:
+        need(len({mv['b'], mv['a'], mv['x']}) == 3, 'move registers distinct')
+        need(tri_identity_ok(mv['b'], mv['a'], mv['x'], mv['mapA']), 'triangle identity')
+
+    # 4-6. independent replay
+    RP = Replay(base)
+    bp, bprof, bok = RP.run(base)
+    need(bok, 'base replay endpoint')
+    cp, cprof, cok = RP.run(cand)
+    need(cok, 'candidate replay endpoint on all %d columns' % RP.N)
+    need(cp == bp, 'frame paths differ')
+    need(cprof == bprof, 'paid profile differs')
+    zb, zc = z_product(base), z_product(cand)
+    need(zb == zc, 'z-program product U differs')
+
+    # 7. record accounting
+    cb = record_counts(base, RP.run(base))
+    cc = record_counts(cand, RP.run(cand))
+    need(cb['ADDs'] == 24305, 'base ADDs')
+    need(cc['ADDs'] == cb['ADDs'] - 2 * mv_doc['n_moves'], 'ADD saving is 2 per move')
+    need(cc['MOVEs'] == cb['MOVEs'] and cc['COPYs'] == cb['COPYs']
+         and cc['ERASEs'] == cb['ERASEs'], 'non-ADD records changed')
+    with tempfile.TemporaryDirectory() as td:
+        bp_path = Path(td) / 'base.json'
+        cp_path = Path(td) / 'cand.json'
+        bp_path.write_text(canon(base).decode())
+        cp_path.write_text(canon(cand).decode())
+        eb = exporter_counts(bp_path, td)
+        ec = exporter_counts(cp_path, td)
+    need(eb == cb, 'own/exporter base counts disagree: %r vs %r' % (eb, cb))
+    need(ec == cc, 'own/exporter candidate counts disagree: %r vs %r' % (ec, cc))
+
+    # 8. negative controls: each tamper must be rejected by some check
+    controls = []
+    def reject(name, fn):
+        try:
+            got = fn()
+        except Exception:
+            got = 'rejected'
+        need(got == 'rejected', 'control not rejected: ' + name)
+        controls.append(name)
+
+    def try_run(w2):
+        try:
+            _, _, ok = RP.run(w2)
+        except Exception:
+            return 'rejected'
+        return 'rejected' if not ok else 'accepted'
+
+    def c1():
+        w2 = copy.deepcopy(cand)
+        w2['A'][5][2][0] ^= 1
+        return try_run(w2)
+    def c2():
+        w2 = copy.deepcopy(cand)
+        w2['C'].pop(3)
+        return try_run(w2)
+    def c3():
+        w2 = copy.deepcopy(cand)
+        w2['C'] = w2['C'][:-1]
+        return try_run(w2)
+    def c4():
+        m2 = copy.deepcopy(mv_doc['moves'])
+        m2[0]['x'] = m2[0]['x'] ^ 1
+        apply_moves(base, m2)
+        return 'accepted'
+    def c5():
+        w2 = copy.deepcopy(cand)
+        A = w2['A']
+        i = next(i for i, g in enumerate(A) if g[1] < w2['v'] and g[2] and g[2][0] < w2['v'])
+        j = next(j for j, g in enumerate(A) if g[1] == A[i][2][0] and A[i][1] in g[2])
+        A[i], A[j] = A[j], A[i]
+        return try_run(w2)  # no C rebuild: cleanup no longer inverts the z-program
+    def c6():
+        w2 = copy.deepcopy(base)
+        w2['D'][0][2] = w2['D'][0][2][:-1]
+        return try_run(w2)
+    for name, fn in (('c1-source-corruption', c1), ('c2-cleanup-gate-deleted', c2),
+                     ('c3-cleanup-truncated', c3), ('c4-tampered-move-list', c4),
+                     ('c5-noncommuting-swap-no-c-rebuild', c5), ('c6-dirt-gate-truncated', c6)):
+        reject(name, fn)
+    need(len(controls) == 6, 'all six controls must reject')
+
+    summary = {
+        'base_word': 'research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz',
+        'base_canonical_sha256': sha(base),
+        'candidate_canonical_sha256': sha(cand),
+        'n_moves': mv_doc['n_moves'],
+        'base_ADDs': cb['ADDs'],
+        'candidate_ADDs': cc['ADDs'],
+        'saved_ADDs': cb['ADDs'] - cc['ADDs'],
+        'base_records': cb['records'],
+        'candidate_records': cc['records'],
+        'MOVEs': cb['MOVEs'],
+        'COPYs': cb['COPYs'],
+        'ERASEs': cb['ERASEs'],
+        'frame_paths_identical': True,
+        'paid_profile_identical': True,
+        'paid_profile': bprof,
+        'z_product_identical': True,
+        'columns_checked': RP.N,
+        'controls_rejected': len(controls),
+        'exporter_cross_check': True,
+    }
+    summary = json.loads(json.dumps(summary))
+    if '--write-certificate' in sys.argv:
+        (HERE / 'CERTIFICATE.json').write_text(json.dumps(summary, indent=1, sort_keys=True))
+        print('wrote CERTIFICATE.json')
+    else:
+        need(summary == cert, 'certificate mismatch:\n got %r\n want %r' % (summary, cert))
+    print('PASS_R439_Z3TO2 moves=%d ADDs=%d->%d records=%d->%d columns=%d paths=%d '
+          'profile=identical U=identical controls=%d/%d exporter=ok no_new_kappa'
+          % (summary['n_moves'], cb['ADDs'], cc['ADDs'], cb['records'], cc['records'],
+             RP.N, RP.N, len(controls), 6))
+
+if __name__ == '__main__':
+    main()