diff --git a/research/round6-pr71/.gitignore b/research/round6-pr71/.gitignore
new file mode 100644
index 0000000000..1efebcb3ce
--- /dev/null
+++ b/research/round6-pr71/.gitignore
@@ -0,0 +1 @@
+baseline/
diff --git a/research/round9-balanced-columns/README.md b/research/round9-balanced-columns/README.md
new file mode 100644
index 0000000000..6bcba0f6b4
--- /dev/null
+++ b/research/round9-balanced-columns/README.md
@@ -0,0 +1,58 @@
+# Balanced columns with envelope ordering and carried-signal exchanges
+
+The selected conditional witness has κ = **51915547765237/10^18**
+(**5.1915547765237e-5**) and bit saving **3244890195589/62500000000000000**.
+It combines balanced coarse columns with anchored split frames, envelope
+ordering and carried-signal exchanges. Physical width is **133,585,058**,
+with auxiliary role counts **27,128 / 35,598**.
+
+Compared with PR #74 at `5830ccfca8466aff613767222558e29b57fa8225`, the
+conditional κ improves by approximately **0.163125609609%**. Compared with
+PR #76 it improves by **0.3418559067%**. These are comparisons of pinned
+conditional claims; no global optimality or unconditional theorem is claimed.
+
+[Proof and attribution](selected-both/PROOF.md) ·
+[Exact certificate](selected-both/arithmetic.json) ·
+[Source and word manifest](selected-both/MANIFEST.json) ·
+[Completed finite validation](package-validation-receipt.json) ·
+[Exact comparison](selected-both/comparison.json)
+
+Passed checks include dense scalar/data-output identities, full compiler
+input/dirty bases, independent serialized-word replay in both orientations,
+exact frame/event reconstruction, regenerated fixed-basis profiles with zero
+CRT disagreements, independent rational moments, all 47 strict assembly
+constraints and seven margins, and adjacent-grid rejection. Fresh portable
+deterministic compilation and broad repository checks remain pending.
+
+The package validation receipt independently reproduces all saved-word, profile
+and arithmetic checks against the current manifest. Its mathematical results
+match the original receipt exactly. The original validation receipt binds
+the original freeze manifest archived
+in `provenance/frozen-manifest.json`. Runtime sources, words, profiles and
+arithmetic are byte-identical to that freeze; only proof/comparison metadata
+has been updated for this publication. The current manifest records those
+two documentation changes. No CI configuration or dependencies are changed.
+
+From the repository root, restore the exact inherited source archive:
+
+```sh
+git fetch https://github.com/CrocSwap/integer-mult-bounds.git 1bef94fd40a746452548c84a4a8f8834670a3113
+mkdir -p research/round6-pr71/baseline
+git archive 1bef94fd40a746452548c84a4a8f8834670a3113 | tar -x -C research/round6-pr71/baseline
+python3 research/round9-balanced-columns/dense_audit.py
+python3 research/round9-balanced-columns/validate_selected.py --bundle research/round9-balanced-columns/selected-both --output /tmp/balanced-columns-validation.json
+```
+
+The proof gives sequential compiler regeneration commands. Use assert-enabled
+Python and C++17; `CXX` can select a compiler and SDK. All inherited all-size
+compiler, fixed-tape, analytic, routing, prime-selection, precision and recovery
+hypotheses remain conditional. Finite checks do not establish practical speedup.
+
+Credit eumemic's PR #69 balanced coarse columns; huxint's PR #77 anchored
+adapter; Thomas DiFiore's PR #74 envelope ordering; Chafik Boukhalfa's PR #71
+anchored splits and next-use scoring; Alejandro Zarzuelo Urdiales's PR #70
+exchanges; Rohan Arun's PR #65/#67 schedules, costs and arithmetic; Dominik
+Scholz's PR #68/#76 controls and composition; Avi Eisenberg's PR #62,
+eumemic's PR #57 and the complete inherited contributor chain. This composition
+was prepared by Dominik Scholz with substantial OpenAI GPT-6 Astra assistance.
+All original notices and Apache-2.0 terms remain preserved.
diff --git a/research/round9-balanced-columns/SOURCE.json b/research/round9-balanced-columns/SOURCE.json
new file mode 100644
index 0000000000..5e26581927
--- /dev/null
+++ b/research/round9-balanced-columns/SOURCE.json
@@ -0,0 +1,14 @@
+{
+  "pr77_commit": "83298467291d3bd4b53f76faaef60dbf585be171",
+  "pr76_base_manifest_sha256": "52c1a59a4372376f64df937a2f19268e2ebd553574386fcdda08b01db9a5c345",
+  "base_pair_graph_sha256": "3d47e89d2649c90800a276bdd0480131def50e0bdf92c122a19494b55bf17420",
+  "base_pair_graph_matches_pr77_bytes": true,
+  "inherited_files": {
+    "inherited/SOURCE.json": "813564d76569fd54a5aa1645da15f1a33b12de8a0bcc53056e69087394b0cfcd",
+    "inherited/graph.py": "349c74a3684cc26c14f4b7001efb086825a4398461709ba13ade2cd10e336d23",
+    "inherited/NOTICE": "3bfe7a693e10fbe710e7fadedb7caeb21e272ce5899f48d043f636e2b06562b0",
+    "inherited/README.md": "e9d7ae955d27fb1832744583f1d251263e9fc499bae2d664b31f28fc83dc2592",
+    "inherited/vendor/pr69-compiler.py": "bd8169bf9f5df185a8aa4aec22234b4fd483000a1cecc78b71bf32e479af1499"
+  },
+  "scope": "Only PR69/77 balanced coarse-column graph mechanism composed with PR76 node ordering + carry exchanges; no PR77 cost-basis compiler changes."
+}
diff --git a/research/round9-balanced-columns/cache-stale-failure.log b/research/round9-balanced-columns/cache-stale-failure.log
new file mode 100644
index 0000000000..6667276f30
--- /dev/null
+++ b/research/round9-balanced-columns/cache-stale-failure.log
@@ -0,0 +1,25 @@
+Traceback (most recent call last):
+  File "/Users/dominik/integer-mult-bounds-incremental/research/round9-balanced-columns/run.py", line 102, in <module>
+    if __name__=='__main__':main()
+                            ~~~~^^
+  File "/Users/dominik/integer-mult-bounds-incremental/research/round9-balanced-columns/run.py", line 93, in main
+    try:result,word=compiler.compile_(a.h,matching=True,reclaim=True,dirty=True)
+                    ~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+  File "/Users/dominik/integer-mult-bounds-incremental/research/round6-pr71/baseline/scripts/experiments/split_pair_engine.py", line 120, in compile_
+    t0=time.time();c,blocks,uses,value_uses,owner,signal,order,contains=build(h);v=len(c.inputs)
+                                                                        ~~~~~^^^
+  File "/Users/dominik/integer-mult-bounds-incremental/research/round6-pr71/baseline/references/frame-compiler/pr65/research/reordered-rank-pair/verify.py", line 34, in build
+    parts=list(original(h))
+               ~~~~~~~~^^^
+  File "/Users/dominik/integer-mult-bounds-incremental/research/round6-pr71/baseline/scripts/experiments/split_pair_engine.py", line 39, in build
+    c=graph(h);groups={};owner={};signal={};blocks=[]
+  File "/Users/dominik/integer-mult-bounds-incremental/research/round9-balanced-columns/run.py", line 33, in <lambda>
+    compiler.graph=lambda h: schedule_variant.reorder(coarse_graph.graph(h,coarse='columns'));compiler.build=producer.ordered_build(compiler.build,a.h)
+                             ~~~~~~~~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+  File "/Users/dominik/integer-mult-bounds-incremental/research/round9-balanced-columns/schedule_variant.py", line 34, in reorder
+    c.verify()
+    ~~~~~~~~^^
+  File "/Users/dominik/integer-mult-bounds-incremental/research/round6-pr71/baseline/references/frame-compiler/pr48/scripts/partial_swap/shared.py", line 111, in verify
+    assert not A&B
+           ^^^^^^^
+AssertionError
diff --git a/research/round9-balanced-columns/coarse_graph.py b/research/round9-balanced-columns/coarse_graph.py
new file mode 100644
index 0000000000..03ac7d8723
--- /dev/null
+++ b/research/round9-balanced-columns/coarse_graph.py
@@ -0,0 +1,68 @@
+"""Controlled combinations of the PR62, PR69 and PR71 scalar graphs.
+
+PR62 graph: Avi Eisenberg with Claude assistance. PR59 split operation:
+Rohan Garg. PR69 balanced coarse columns: eumemic with Codex assistance.
+PR71 anchored split vector: Chafik Boukhalfa with Codex assistance.
+This configurable composition/search adapter: huxint with Codex assistance.
+Apache-2.0 and all inherited notices apply. No global search claim.
+"""
+import sys
+if sys.flags.optimize:
+    raise ValueError('Assertions must remain enabled')
+from hashlib import sha256
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parent.parent/'round6-pr71/baseline/references/frame-compiler/pr65'
+BASE = ROOT/'research/pair-assembly/pair_graph.py'
+BEFORE = '''            coarse = {(i, j): self.total([e(a, b) for a in groups[i] for b in groups[j]])
+                      for i, j in combinations(range(ng), 2)}'''
+AFTER_COLUMNS = '''            def coarse_sum(i, j):
+                columns = [self.total([e(a,b) for a in groups[i]]) for b in groups[j]]
+                return self.total(columns)
+            coarse = {(i,j):coarse_sum(i,j) for i,j in combinations(range(ng),2)}'''
+AFTER_ROWS = '''            def coarse_sum(i, j):
+                rows = [self.total([e(a,b) for b in groups[j]]) for a in groups[i]]
+                return self.total(rows)
+            coarse = {(i,j):coarse_sum(i,j) for i,j in combinations(range(ng),2)}'''
+
+
+def graph(h, groups=(1, 1, 2), anchor='first', coarse='none'):
+    if h not in (23, 25) or any(type(x) is not int or not 0 <= x <= 3 for x in groups):
+        raise ValueError('Unsupported finite graph configuration')
+    source = BASE.read_text()
+    if coarse != 'none':
+        assert source.count(BEFORE) == 1
+        source = source.replace(BEFORE, {'columns': AFTER_COLUMNS, 'rows': AFTER_ROWS}[coarse])
+    namespace = {'__file__': str(BASE), '__name__': 'next_use_graph_composition'}
+    exec(compile(source, str(BASE)+':next_use_composition', 'exec'), namespace)
+    original_class = namespace['circuit_class']
+
+    def circuit_class(dimension):
+        class SplitPair(original_class(dimension)):
+            def grouping(self, points):
+                partition = super().grouping(points)
+                choice = groups[self._level] if self._level < len(groups) else 0
+                if len(points) > 3 and choice:
+                    index = {1: 0, 2: (len(partition)-1)//2, 3: len(partition)-1}[choice]
+                    if len(partition[index]) == 2:
+                        partition[index:index+1] = [[x] for x in partition[index]]
+                assert all(len(g) in (1, 2) for g in partition)
+                assert len(partition) < len(points)
+                assert sorted(x for g in partition for x in g) == sorted(points)
+                return partition
+        return SplitPair
+
+    namespace['circuit_class'] = circuit_class
+    original_points = namespace['alternating_points']
+    if anchor != 'none':
+        def points(dimension, common):
+            old = original_points(dimension, common)
+            pairs = [(a, a+1) for a in range(0, dimension-1, 2) if common not in (a, a+1)]
+            chosen = {'first': pairs[0], 'second': pairs[min(1, len(pairs)-1)], 'last': pairs[-1]}[anchor]
+            result = list(chosen)+[x for x in old if x not in chosen]
+            assert sorted(result) == [x for x in range(dimension) if x != common]
+            return result
+        namespace['alternating_points'] = points
+    result = namespace['graph'](h)
+    result.verify()
+    return result
diff --git a/research/round9-balanced-columns/dense-audit.json b/research/round9-balanced-columns/dense-audit.json
new file mode 100644
index 0000000000..c40a04a61d
--- /dev/null
+++ b/research/round9-balanced-columns/dense-audit.json
@@ -0,0 +1,28 @@
+{
+  "scope": "Finite dense scalar/data-output identity audit",
+  "axes": {
+    "23": {
+      "h": 23,
+      "inputs": 1771,
+      "additions": 67460,
+      "outputs": 5336,
+      "all_dense_additions_disjoint": true,
+      "all_dense_partial_outputs_exact": true,
+      "original_common_point_envelopes_exact": true
+    },
+    "25": {
+      "h": 25,
+      "inputs": 2300,
+      "additions": 94993,
+      "outputs": 6925,
+      "all_dense_additions_disjoint": true,
+      "all_dense_partial_outputs_exact": true,
+      "original_common_point_envelopes_exact": true
+    }
+  },
+  "sources": {
+    "dense_audit.py": "b282e743d9d8c48cbc97eb2c2597a6c52894d33ea1967dd1ff4c269b0e0a72e3",
+    "coarse_graph.py": "b5bbabb8e5bf69769af4b6cd67163954d990dc489e3c9488ed84bd09f0a767c8",
+    "schedule_variant.py": "3ce3b4c0886868e3fcd2b75c54b61180d103447cfdb9d3ece1de47dd6d46eecc"
+  }
+}
diff --git a/research/round9-balanced-columns/dense_audit.py b/research/round9-balanced-columns/dense_audit.py
new file mode 100644
index 0000000000..979cfe41cc
--- /dev/null
+++ b/research/round9-balanced-columns/dense_audit.py
@@ -0,0 +1,53 @@
+"""Independent dense supports for both balanced-column scalar graphs.
+
+Dominik Scholz with substantial OpenAI GPT-6 Astra assistance. Apache-2.0.
+Does not use the graph's memoized/provenance-based support routine.
+"""
+from hashlib import sha256
+from pathlib import Path
+import json
+import run
+
+
+def audit(h):
+    c = run.schedule_variant.reorder(run.coarse_graph.graph(h, coarse='columns'))
+    support = [0] * len(c.args)
+    point_rows = [0] * h
+    for i, triple in enumerate(c.inputs):
+        support[i + 1] = 1 << i
+        for point in triple:
+            point_rows[point] |= 1 << i
+    additions = 0
+    for node in sorted(c.active):
+        if not c.args[node]:
+            assert node <= len(c.inputs)
+            continue
+        a, b = c.args[node]
+        assert a < node and b < node
+        assert support[a] and support[b]
+        assert not support[a] & support[b]
+        support[node] = support[a] | support[b]
+        assert c.core[node] == c.core[a] & c.core[b]
+        assert c.union[node] == c.union[a] | c.union[b]
+        additions += 1
+    for (common, target), node in c.outputs.items():
+        expected = point_rows[common]
+        for point in target:
+            if point != common:
+                expected &= ~point_rows[point]
+        assert support[node] == expected
+    return dict(h=h, inputs=len(c.inputs), additions=additions,
+                outputs=len(c.outputs), all_dense_additions_disjoint=True,
+                all_dense_partial_outputs_exact=True,
+                original_common_point_envelopes_exact=True)
+
+
+if __name__ == '__main__':
+    root = Path(__file__).resolve().parent
+    result = {'scope': 'Finite dense scalar/data-output identity audit',
+              'axes': {str(h): audit(h) for h in (23, 25)},
+              'sources': {name: sha256((root/name).read_bytes()).hexdigest()
+                          for name in ('dense_audit.py', 'coarse_graph.py',
+                                       'schedule_variant.py')}}
+    (root/'dense-audit.json').write_text(json.dumps(result, indent=2) + '\n')
+    print(json.dumps(result, indent=2))
diff --git a/research/round9-balanced-columns/inherited/NOTICE b/research/round9-balanced-columns/inherited/NOTICE
new file mode 100644
index 0000000000..be4f46cf39
--- /dev/null
+++ b/research/round9-balanced-columns/inherited/NOTICE
@@ -0,0 +1,28 @@
+Balanced coarse sums on anchored split frames
+Prepared for huxint with substantial OpenAI Codex assistance, October 9, 2026.
+Distributed under the repository's Apache License 2.0.
+
+The selected finite network composes previously attributed constructions.
+Preserve all repository notices and these source-specific attributions:
+
+eumemic: PR57 joint reversible frame compiler and independent physical replay;
+PR69 balanced coarse column sums, with OpenAI Codex assistance.
+Avi Eisenberg / ikeboy: PR62 interval strips and core-aware pair assembly,
+with Anthropic Claude assistance.
+Rohan Garg / rohangar1: PR59 split-pair operation, with OpenAI Codex assistance.
+Rohan Arun: PR65 schedule/refined arithmetic and PR67 profile-cost compiler,
+with OpenAI Codex assistance.
+Dominik Scholz: PR63 ranked composition and PR68 compatible pending controls,
+with OpenAI GPT-6 Astra / Codex assistance.
+Chafik Boukhalfa: PR60 ranked reclamation and PR71 anchored split graph and
+next-use scoring, with OpenAI Codex assistance.
+
+The original code comments and preserved vendor snapshots retain their own
+attributions. The inherited framework also credits icekylinx, RaD / hipotures,
+James Chang, Zhihao Chen / jacklightChen, Aurel Prosz / Paureel, Swapnil Jain,
+Douglas Colkitt, OpenAI, David Harvey, Joris van der Hoeven, and other
+contributors identified in the root NOTICE and inherited source files.
+
+The new package's finite checks are conditional evidence. They are not a
+complete theorem formalization, an independent human referee report, or a
+practical multiplication benchmark.
diff --git a/research/round9-balanced-columns/inherited/README.md b/research/round9-balanced-columns/inherited/README.md
new file mode 100644
index 0000000000..bd922144c1
--- /dev/null
+++ b/research/round9-balanced-columns/inherited/README.md
@@ -0,0 +1,90 @@
+# Balanced coarse sums on anchored split frames
+
+The selected finite construction gives the conditional integer-multiplication
+bound
+
+\[
+T(n)=O\bigl(n(\log n)^{1-\kappa}\bigr),\qquad
+\kappa=\frac{12886963972497}{250000000000000000}
+=5.1547855889988\times10^{-5}.
+\]
+
+This is **0.2590891819% above PR71** at commit
+`1bef94fd40a746452548c84a4a8f8834670a3113`. It is a comparison of conditional
+asymptotic exponent bounds, with no global-optimality or measured-runtime claim.
+
+The construction combines eumemic's PR69 balanced coarse columns with Chafik
+Boukhalfa's PR71 anchored `[1,1,2]` split graph. It uses the inherited joint
+reversible frame compiler and compatible pending controls, and orders both
+eligible live controls and retired dependency candidates by their actual
+profile cost. The latter changes the temporary dependency basis, not just the
+choice among the first basis's dependent rows. Pending controls are priced
+against their known next use. Every selected clearing expression is emitted
+as literal XORs, and all frame moves and final cleanup remain charged.
+
+| Quantity | Pinned PR71 | This construction |
+|---|---:|---:|
+| Roles at h=23 | 27,455 | 27,338 |
+| Roles at h=25 | 36,015 | 35,939 |
+| Physical width W | 135,075,665 | 134,671,969 |
+| Complete rank mass | 77,666,660,475 | 77,434,535,275 |
+| Rank deficit | 1,846,900 | 1,846,900 |
+
+The strict bit-saving witness is `51550513208569/10^18`. The full paid child
+list of PR71 has a rigorous moment lower bound above one at this value. The
+new list has a rigorous upper bound below one. Its next `10^-18` bit grid
+point is rejected, as is the next kappa grid point in the unchanged balanced
+assembly. All 47 strict inequalities and seven margins are checked exactly.
+
+## Reproduction
+
+From the repository root, with Python 3.11+ and a C++17 compiler:
+
+```sh
+make -C research/balanced-split-reclamation verify
+make -j1 verify
+```
+
+The first command regenerates both selected words byte-for-byte, independently
+replays every input/target/dirty basis vector in both orientations, derives
+all physical transitions from literal XOR incidences, recomputes every
+fixed-basis profile with the inherited bounded-minor/CRT checks, rebuilds
+the complete global profile and exact arithmetic, and runs twelve failure-mode
+tests. A second logarithm/exponential evaluation uses a different range
+reduction and Taylor degree. This is a second arithmetic implementation,
+not independent expert review of the theorem.
+
+`selected/` contains the words, actual profiles, receipts and exact certificate.
+`SOURCE.json` freezes consumed sources and deliverables; verification never
+rewrites this manifest. `pin.py` is an explicit maintenance operation.
+`generate.py --output <directory>` produces fresh artifacts without modifying
+the selected witness. The proof is in [paper.tex](paper.tex) and [paper.pdf](paper.pdf).
+
+The selected graph uses columns rather than a three-addition coarse chain,
+split vector `[1,1,2]`, and the lowest intact pair first in each common-point
+ordering. The h23 region order is `cover-core`; h25 uses `reverse-node`.
+The oracle's integer entropy score is only a deterministic search heuristic.
+The final complete physical profile, not the score, determines acceptance.
+
+## Proof boundary and attribution
+
+The original-envelope interpretation, all-size residual/frame compiler,
+fixed finite-alphabet tape layout, semantic and analytic multiplication
+reduction, routing, prime setup and exact recovery remain inherited
+assumptions. The complex branch is unchanged. Exact finite checks do not
+constitute a complete formal proof of the multiplication theorem.
+
+This composition, cost-ordered dependency basis experiment, independent
+reproduction and package were prepared for **huxint with OpenAI Codex
+assistance**. Credit eumemic for PR57's reversible region compiler and PR69's
+coarse-column identity; Avi Eisenberg for PR62's interval/core-aware graph;
+Rohan Garg for PR59's split operation; Rohan Arun for PR65's schedules and
+PR67's cost oracle and numerical refinement; Dominik Scholz for PR63's
+composition and PR68's compatible pending controls; and Chafik Boukhalfa for
+PR60's ranked reclamation and PR71's anchored split/next-use composition.
+All prior contributors retain their original notices. See [NOTICE](NOTICE).
+
+The preserved contributor files in `vendor/` are unmodified snapshots, used
+for provenance and the complete PR71 comparison. The inherited repository
+files are retained unchanged. No endorsement or completed expert review by
+those contributors is implied.
diff --git a/research/round9-balanced-columns/inherited/SOURCE.json b/research/round9-balanced-columns/inherited/SOURCE.json
new file mode 100644
index 0000000000..cc93786f91
--- /dev/null
+++ b/research/round9-balanced-columns/inherited/SOURCE.json
@@ -0,0 +1,64 @@
+{
+  "files": {
+    ".github/workflows/balanced-split.yml": "9f45519e842d7412c2c68d349252b57872a1a73e25447ca2ac63c625eb3849d3",
+    "references/frame-compiler/pr48/SOURCE.json": "6ccc673f88d3650d75ed600b082fe2fc2866dd88a172d182a38c1d0cd27d24b3",
+    "references/frame-compiler/pr48/research/copied-fixed/balanced_assembly.py": "4d27575dcad2b45a281a34054d311409ecd954f373c0c3a62c29bcc843f06a86",
+    "references/frame-compiler/pr48/research/copied-fixed/certificate.json": "33b3373c04e18a7f4cee33ba68da6ed71934acc3e0179cca6197696be4befa22",
+    "references/frame-compiler/pr48/research/copied-fixed/changed_graph.py": "375a7ca6cac6d153c1880fd5ea5b4525e5dca9d4ee3018d24d679c219b59c5b3",
+    "references/frame-compiler/pr48/research/copied-fixed/profiles.cpp": "b9fbd71c746609948b60c441ba4601e88436956fe636d1810f270ce4ce6a63a6",
+    "references/frame-compiler/pr48/research/copied-fixed/verify.py": "44932182ce6c4d39776650311262ef0ac1ea9b5da7220bc0f728e2789881f84a",
+    "references/frame-compiler/pr48/scripts/exclusion_circuit.py": "b253f9d7318becda1c1f3b24e2287ed44a9c27fcea00cf9d531afb950d0457a0",
+    "references/frame-compiler/pr48/scripts/partial_swap/binary_io.hpp": "7e143eb75bb342ac8f0c82a60005e7c35c9ec827de479bf6fb773151f3af5fa1",
+    "references/frame-compiler/pr48/scripts/partial_swap/paired.py": "ddcecace3a8e04d690914cbd64873ab8dde8c0c3477a255767b0ed6f639bb0c7",
+    "references/frame-compiler/pr48/scripts/partial_swap/shared.py": "81e9af92cdf38243a540cdcff2062778e449467848d444da37a9553c12ca4c34",
+    "research/balanced-split-reclamation/Makefile": "2417f68960e98c9fc5cba82511236c555a3dc8c93650da2b61a7011573d0e015",
+    "research/balanced-split-reclamation/NOTICE": "3bfe7a693e10fbe710e7fadedb7caeb21e272ce5899f48d043f636e2b06562b0",
+    "research/balanced-split-reclamation/README.md": "e9d7ae955d27fb1832744583f1d251263e9fc499bae2d664b31f28fc83dc2592",
+    "research/balanced-split-reclamation/arithmetic.py": "574873dd24ed83592abe198340af94918d484095c0c077c8886628200439eae6",
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+    "research/balanced-split-reclamation/paper.pdf": "960361b250fd3033cb1b5a7c5f8ce738432a525bc4795f9bf503c28d8fcd4c42",
+    "research/balanced-split-reclamation/paper.tex": "875411ec7f0578c521645fb07ede2071a1b0e1b0d98c90e27c1d41abbc434538",
+    "research/balanced-split-reclamation/pin.py": "16734634d7856d6001a4fe8f5247bb1da47bd6852210d51c881b14afde6babc7",
+    "research/balanced-split-reclamation/profile_oracle.cpp": "d1f6b650de9fb5842b412595369271d9eeb6f54045c76c0fb5cad29854955c35",
+    "research/balanced-split-reclamation/selected/audit.json": "10040d0a140cf6185881801687985134d3b950400db73192be5e995cde4e81c7",
+    "research/balanced-split-reclamation/selected/certificate.json": "683bc658daf2fd4da8083c2853932ebe641607df84dcf5bb8701713a318df72f",
+    "research/balanced-split-reclamation/selected/compiled-23.json": "7ca13046c2a84f86d97eb056d609c8d635bff2670d2f4e622f9444e4e3f1593c",
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+    "research/balanced-split-reclamation/vendor/pr69-compiler.py": "bd8169bf9f5df185a8aa4aec22234b4fd483000a1cecc78b71bf32e479af1499",
+    "research/balanced-split-reclamation/vendor/pr71-certificate.json": "0fcb53fa374273efab6370953c966e8fa065dbfdba43691843a4788cb2cda42e",
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+    "research/balanced-split-reclamation/verify.py": "a205f2f73cd07cd6af94a4cf11f2c669cdd18b44fcd810df0ac6193eaff5a613",
+    "research/pair-assembly/frame/frame-compiler.json": "86eeb10989a61e571938178b91dfe1b81ba39ec97cc93124c00cecd91dc6ef13",
+    "research/pair-assembly/frame/frame_verify.py": "fafe25d57166a747eeb847444a2e02dd7eed19038b93a2c495014d147d04a8fb",
+    "research/pair-assembly/pair_graph.py": "3d47e89d2649c90800a276bdd0480131def50e0bdf92c122a19494b55bf17420",
+    "scripts/exclusion_circuit.py": "b253f9d7318becda1c1f3b24e2287ed44a9c27fcea00cf9d531afb950d0457a0",
+    "scripts/experiments/binary_frame_math.py": "1f96e8aafc8ae89f8eeec5d03800131d93d58e01bf8a7fca8feb03702d1740d0",
+    "scripts/experiments/binary_frame_profile_prepare.py": "8a50381cab00b33e62b3a057d199ba5e19647929135c95b0fd63fc8a8ef7c9b3",
+    "scripts/experiments/binary_frame_profiles.cpp": "8e2e391d3ee57cbb1372c68e4aa031e1b861321536eabd642ada54e12e678669",
+    "scripts/experiments/binary_frame_replay.py": "392e78e62d7685c99a5888a7da743ea6c3dcad631af4a743c91e0c09348442bc",
+    "scripts/partial_swap/paired.py": "ddcecace3a8e04d690914cbd64873ab8dde8c0c3477a255767b0ed6f639bb0c7",
+    "scripts/partial_swap/shared.py": "81e9af92cdf38243a540cdcff2062778e449467848d444da37a9553c12ca4c34"
+  },
+  "scope": "Explicit source and artifact freeze; no automatic refresh in verification",
+  "upstream": {
+    "pr67": "b3745601e947a94316bf25c2c6263d93c06e3364",
+    "pr68": "8778fdb52e834a2a14371590d80ea6e0c734a866",
+    "pr69": "91aa1f17e6e3fc063241686a41a34ddd0dc24c50",
+    "pr71": "1bef94fd40a746452548c84a4a8f8834670a3113"
+  }
+}
diff --git a/research/round9-balanced-columns/inherited/graph.py b/research/round9-balanced-columns/inherited/graph.py
new file mode 100644
index 0000000000..ba0382efaa
--- /dev/null
+++ b/research/round9-balanced-columns/inherited/graph.py
@@ -0,0 +1,68 @@
+"""Controlled combinations of the PR62, PR69 and PR71 scalar graphs.
+
+PR62 graph: Avi Eisenberg with Claude assistance. PR59 split operation:
+Rohan Garg. PR69 balanced coarse columns: eumemic with Codex assistance.
+PR71 anchored split vector: Chafik Boukhalfa with Codex assistance.
+This configurable composition/search adapter: huxint with Codex assistance.
+Apache-2.0 and all inherited notices apply. No global search claim.
+"""
+import sys
+if sys.flags.optimize:
+    raise ValueError('Assertions must remain enabled')
+from hashlib import sha256
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[2]
+BASE = ROOT/'research/pair-assembly/pair_graph.py'
+BEFORE = '''            coarse = {(i, j): self.total([e(a, b) for a in groups[i] for b in groups[j]])
+                      for i, j in combinations(range(ng), 2)}'''
+AFTER_COLUMNS = '''            def coarse_sum(i, j):
+                columns = [self.total([e(a,b) for a in groups[i]]) for b in groups[j]]
+                return self.total(columns)
+            coarse = {(i,j):coarse_sum(i,j) for i,j in combinations(range(ng),2)}'''
+AFTER_ROWS = '''            def coarse_sum(i, j):
+                rows = [self.total([e(a,b) for b in groups[j]]) for a in groups[i]]
+                return self.total(rows)
+            coarse = {(i,j):coarse_sum(i,j) for i,j in combinations(range(ng),2)}'''
+
+
+def graph(h, groups=(1, 1, 2), anchor='first', coarse='none'):
+    if h not in (23, 25) or any(type(x) is not int or not 0 <= x <= 3 for x in groups):
+        raise ValueError('Unsupported finite graph configuration')
+    source = BASE.read_text()
+    if coarse != 'none':
+        assert source.count(BEFORE) == 1
+        source = source.replace(BEFORE, {'columns': AFTER_COLUMNS, 'rows': AFTER_ROWS}[coarse])
+    namespace = {'__file__': str(BASE), '__name__': 'next_use_graph_composition'}
+    exec(compile(source, str(BASE)+':next_use_composition', 'exec'), namespace)
+    original_class = namespace['circuit_class']
+
+    def circuit_class(dimension):
+        class SplitPair(original_class(dimension)):
+            def grouping(self, points):
+                partition = super().grouping(points)
+                choice = groups[self._level] if self._level < len(groups) else 0
+                if len(points) > 3 and choice:
+                    index = {1: 0, 2: (len(partition)-1)//2, 3: len(partition)-1}[choice]
+                    if len(partition[index]) == 2:
+                        partition[index:index+1] = [[x] for x in partition[index]]
+                assert all(len(g) in (1, 2) for g in partition)
+                assert len(partition) < len(points)
+                assert sorted(x for g in partition for x in g) == sorted(points)
+                return partition
+        return SplitPair
+
+    namespace['circuit_class'] = circuit_class
+    original_points = namespace['alternating_points']
+    if anchor != 'none':
+        def points(dimension, common):
+            old = original_points(dimension, common)
+            pairs = [(a, a+1) for a in range(0, dimension-1, 2) if common not in (a, a+1)]
+            chosen = {'first': pairs[0], 'second': pairs[min(1, len(pairs)-1)], 'last': pairs[-1]}[anchor]
+            result = list(chosen)+[x for x in old if x not in chosen]
+            assert sorted(result) == [x for x in range(dimension) if x != common]
+            return result
+        namespace['alternating_points'] = points
+    result = namespace['graph'](h)
+    result.verify()
+    return result
diff --git a/research/round9-balanced-columns/inherited/vendor/pr69-compiler.py b/research/round9-balanced-columns/inherited/vendor/pr69-compiler.py
new file mode 100644
index 0000000000..271b44d795
--- /dev/null
+++ b/research/round9-balanced-columns/inherited/vendor/pr69-compiler.py
@@ -0,0 +1,93 @@
+#!/usr/bin/env python3
+"""Balance each four-edge coarse sum in PR62 into two column pair-star sums.
+
+Instead of a three-addition chain, use (e(a,b)+e(a2,b))+(e(a,b2)+e(a2,b2)).
+The existing interning shares pair-star sums with the strip computations.
+Only the coarse-edge construction changes. The full graph is checked and
+compiled, so all remaining copies and clearing operations remain charged.
+
+Base graph: Avi Eisenberg / ikeboy, with Anthropic Claude assistance (#62).
+Compiler: eumemic with OpenAI Codex assistance (#57). Rank priority: Chafik
+Boukhalfa with OpenAI Codex assistance (#60). Their unchanged composition
+is also published by Dominik Scholz with OpenAI Codex assistance (#63).
+This coarse-sum experiment was prepared with OpenAI Codex assistance.
+All inherited licenses and notices remain.
+"""
+import sys
+sys.dont_write_bytecode = True
+if sys.flags.optimize:
+    raise ValueError('Assertions must remain enabled')
+import argparse
+import gzip
+from hashlib import sha256
+import inspect
+import json
+from pathlib import Path
+
+import binary_frame_compiler as compiler
+import ranked_interval_compiler as predecessor
+
+ROOT = Path(__file__).resolve().parents[2]
+PIN = ROOT / 'references/frame-compiler/balanced-coarse'
+BEFORE = '''            coarse = {(i, j): self.total([e(a, b) for a in groups[i] for b in groups[j]])
+                      for i, j in combinations(range(ng), 2)}'''
+AFTER = '''            def coarse_sum(i, j):
+                columns = [self.total([e(a,b) for a in groups[i]]) for b in groups[j]]
+                return self.total(columns)
+            coarse = {(i,j):coarse_sum(i,j) for i,j in combinations(range(ng),2)}'''
+
+
+def check_sources():
+    predecessor.check_sources()
+    manifest = json.loads((PIN / 'SOURCE.json').read_text())
+    for name, digest in manifest['files'].items():
+        assert sha256((PIN / name).read_bytes()).hexdigest() == digest, name
+    dependency = PIN / manifest['shared_dependency_manifest']
+    assert sha256(dependency.read_bytes()).hexdigest() == manifest['shared_dependency_manifest_sha256']
+    return manifest
+
+
+def graph(h):
+    assert h in (23, 25)
+    check_sources()
+    path = predecessor.PIN / 'research/pair-assembly/pair_graph.py'
+    source = path.read_text()
+    assert source.count(BEFORE) == 1
+    namespace = {'__file__': str(path), '__name__': 'balanced_coarse_pr62'}
+    exec(compile(source.replace(BEFORE, AFTER), str(path) + ':balanced_coarse', 'exec'), namespace)
+    result = namespace['graph'](h)
+    result.verify()
+    return result
+
+
+def compile_axis(h):
+    circuit = graph(h)
+    source = inspect.getsource(compiler.compile_)
+    before = 'for s in sorted(retired):'
+    after = "for s in sorted(retired, key=lambda s: (-blocks[frames[s]]['rank'], s)):"
+    assert source.count(before) == 1
+    namespace = dict(vars(compiler))
+    exec(compile(source.replace(before, after), __file__ + ':ranked_compile', 'exec'), namespace)
+    # build() is the original function and resolves graph in compiler's globals.
+    previous = compiler.graph
+    compiler.graph = lambda _: circuit
+    try:
+        result, word = namespace['compile_'](h, matching=True, reclaim=True, dirty=True)
+    finally:
+        compiler.graph = previous
+    result['scalar'] = circuit.verify()
+    result['baseline_pr62_stacked_roles'] = {23: 27918, 25: 36586}[h]
+    return result, word
+
+
+if __name__ == '__main__':
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--h', type=int, choices=(23, 25), required=True)
+    parser.add_argument('--output', type=Path, required=True)
+    parser.add_argument('--word', type=Path, required=True)
+    args = parser.parse_args()
+    result, word = compile_axis(args.h)
+    raw = (json.dumps(word, separators=(',', ':')) + '\n').encode()
+    args.word.write_bytes(gzip.compress(raw, mtime=0))
+    args.output.write_text(json.dumps(result, indent=2) + '\n')
+    print(json.dumps(result), flush=True)
diff --git a/research/round9-balanced-columns/package-validation-receipt.json b/research/round9-balanced-columns/package-validation-receipt.json
new file mode 100644
index 0000000000..0b83d4a9fb
--- /dev/null
+++ b/research/round9-balanced-columns/package-validation-receipt.json
@@ -0,0 +1,160 @@
+{
+  "status": "PASS finite source binding, both serialized words, all CRT profiles, rational moment and assembly",
+  "manifest_sha256": "861f48772c29ce35eace43fe277ce4b06a598bf4338b9f545d349a2577889485",
+  "axes": {
+    "23": {
+      "replay": {
+        "status": "independent serialized word, exact frames, full input and dirty basis both orientations PASS",
+        "h": 23,
+        "roles": 27128,
+        "output_roles": 5336,
+        "elementary_xors": 167539,
+        "word_length": 694950,
+        "full_basis_vectors": 30670,
+        "rank_mass": 624450,
+        "histogram": {
+          "1": 58230,
+          "0": 185200,
+          "2": 81090,
+          "4": 14624,
+          "3": 10015,
+          "5": 6757,
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+          "12": 691,
+          "14": 1893,
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+          "11": 97,
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+          "17": 543,
+          "18": 3197,
+          "19": 6482,
+          "15": 267,
+          "22": 23,
+          "7": 278,
+          "21": 429
+        }
+      },
+      "transitions": {
+        "h": 23,
+        "v": 1771,
+        "R": 27128,
+        "frames": 59803,
+        "distinct_transitions": 151935,
+        "singles": 15939,
+        "rank_mass": 624450,
+        "rank_histogram_with_side_growth_split_into_singletons": {
+          "1": 68856,
+          "2": 75777,
+          "3": 10015,
+          "4": 14624,
+          "5": 6757,
+          "6": 2127,
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+          "14": 1893,
+          "15": 267,
+          "16": 430,
+          "17": 543,
+          "18": 3197,
+          "19": 6482,
+          "21": 429,
+          "22": 23
+        },
+        "word_sha256": "7e452d6cec73d2433c9ec996fac2bfaf97317294ee515f86cef95c3ad9f154d7",
+        "copied_centers": 23,
+        "transition_events_equal_independent_xor_word_reconstruction": true
+      },
+      "actual_profiles_equal": true
+    },
+    "25": {
+      "replay": {
+        "status": "independent serialized word, exact frames, full input and dirty basis both orientations PASS",
+        "h": 25,
+        "roles": 35598,
+        "output_roles": 6925,
+        "elementary_xors": 235195,
+        "word_length": 972980,
+        "full_basis_vectors": 40198,
+        "rank_mass": 890550,
+        "histogram": {
+          "1": 76952,
+          "0": 260118,
+          "2": 118303,
+          "3": 13346,
+          "4": 18638,
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+          "15": 125,
+          "17": 583,
+          "20": 4217,
+          "21": 8479,
+          "24": 25,
+          "23": 462
+        }
+      },
+      "transitions": {
+        "h": 25,
+        "v": 2300,
+        "R": 35598,
+        "frames": 84845,
+        "distinct_transitions": 212411,
+        "singles": 20700,
+        "rank_mass": 890550,
+        "rank_histogram_with_side_growth_split_into_singletons": {
+          "1": 90752,
+          "2": 111403,
+          "3": 13346,
+          "4": 18638,
+          "5": 8916,
+          "6": 3400,
+          "7": 576,
+          "8": 1196,
+          "9": 553,
+          "10": 854,
+          "11": 21,
+          "12": 776,
+          "13": 162,
+          "14": 756,
+          "15": 125,
+          "16": 2229,
+          "17": 583,
+          "18": 572,
+          "19": 879,
+          "20": 4217,
+          "21": 8479,
+          "23": 462,
+          "24": 25
+        },
+        "word_sha256": "fc6f429363b3c8208fd9bff10ec20386de280a3e1ea4f109ca5e0834d2c60fc4",
+        "copied_centers": 25,
+        "transition_events_equal_independent_xor_word_reconstruction": true
+      },
+      "actual_profiles_equal": true
+    }
+  },
+  "kappa": "51915547765237/1000000000000000000",
+  "bit_saving": "3244890195589/62500000000000000",
+  "strict_constraints": 47,
+  "margins": 7,
+  "independent_moment_and_adjacent_grid": true,
+  "scope": "Finite conditional witness; inherited all-size and transfer assumptions remain."
+}
diff --git a/research/round9-balanced-columns/provenance/frozen-manifest.json b/research/round9-balanced-columns/provenance/frozen-manifest.json
new file mode 100644
index 0000000000..c269419787
--- /dev/null
+++ b/research/round9-balanced-columns/provenance/frozen-manifest.json
@@ -0,0 +1,193 @@
+{
+  "baseline_files": {
+    "certificates/split-pair-compiler.json": "6561fab8609e8aa3a291a8197cd8b834b0881ee0b9a5b991926c2c93c4d5f884",
+    "certificates/split-pair-kappa.json": "0fcb53fa374273efab6370953c966e8fa065dbfdba43691843a4788cb2cda42e",
+    "certificates/split-pair-profiles-23.json": "867302152b62abf28a6e1b119580b99eaffa237082b1d295000a3ee2637d5db4",
+    "certificates/split-pair-profiles-25.json": "1ecc63972ae4df2ed7af7278d5f65790dd4aca3032cc17304f39ac063cd2c067",
+    "certificates/split-pair-repository-validation.json": "a4700ef48968e0bc6afadf8048d82bb4f4cfc9f21db9a8bf355bd9c2745da53e",
+    "certificates/split-pair-transitions-23.json": "6ca709c1e101f7c9a828d9ceb8a7f2991a2d954b84c9ae8c616deabfc1e321c8",
+    "certificates/split-pair-transitions-25.json": "4364441b6a7c444bb2c84ab5b973c81e4069d753be7f1cf5a3c836096a739f1e",
+    "certificates/split-pair-validation.json": "9b64dd6382fc592e2cda6bb54be27ba8f705ede24c50fd2f9b4336ce2e402c7e",
+    "certificates/split-pair-word-23.json.gz": "ca504cb3912358a9560d4916eaeea2e415c24991db0bc7d397e50ea53d9cb24b",
+    "certificates/split-pair-word-25.json.gz": "54f612f688d6ba3b1fffdd83aa6053035006fee90cb4b3899fc7880f35c38b4f",
+    "references/frame-compiler/pr48/LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
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+    "scripts/experiments/shared_core.py": "d3c59b61229956cb54ae1ef9145ed213abdf1ae743929de145a77c43656f01a8",
+    "scripts/experiments/signed_sparse_core.py": "21da8e1de5ab6ac8479183326c2000313d24b47cc5be2f46b4fe6ac3cd59779f",
+    "scripts/experiments/single_intersection.py": "08ddae46e5621a5b4c525b504e67ae655abe9b49461d894feaa7abc89a51847d",
+    "scripts/experiments/single_intersection_witnesses.json": "c2308a3d21991f5828b0519b191b470f48b2fbbcaa5af0f7aed3986cc7534185",
+    "scripts/experiments/skip_frame_README.txt": "c92a53e608edb95a266ad27f4543c27668451a009c72fd5860619e6f0a355852",
+    "scripts/experiments/skip_frame_compiler.py": "8ffd079fd2e4f055b5befefecc5f33e686de9898219e9252f5d3913df8b195d8",
+    "scripts/experiments/skip_frame_compose.py": "d48e688ebb9a490f4eaf50d9ab0a4d4e3e76b6588b55d213df14c703c27a7f66",
+    "scripts/experiments/split_pair_arithmetic.py": "fb982a0ac8758b2e308dc8bd145f21548c28f4f2ee97d28e77fa849b9560de4a",
+    "scripts/experiments/split_pair_compiler.py": "3de0ececd6ab108d8780834c3f1a1a84ce7cae49dca4019882bea7f63b588661",
+    "scripts/experiments/split_pair_compose.py": "9c6337096d5517344cc5029bd80a1379fa407047662ab54c469db7b11e9daa34",
+    "scripts/experiments/split_pair_engine.py": "58f13f82ce1bb7f2b5549e17628bc6034e5e05cb24db2acb9a91c24820872f70",
+    "scripts/experiments/split_pair_graph.py": "9d4ee7179f880f9576e81c7ad3a5290fe61380a7531d559ef3cf0939bb30089c",
+    "scripts/experiments/subset_core_family.py": "373a45a8fb4294cdf72d510d54e16d21535227b085aefb008aa092b3ad26e4de",
+    "scripts/experiments/ternary_side.py": "93379b6e70cf58bb55432d9f2d1c0e0305c20a075bfe3cc219deabd927afd375",
+    "scripts/experiments/verify_joint_dual.py": "7a31bba1e9c0ded32ae285df2ca9bc91ef20c330f6789fdba361876f3c7a1985",
+    "scripts/experiments/verify_skip_frame.py": "fc42dc3d59239ee3908d0e3e95402eb53fd6815a100521911fde1b541070c292",
+    "scripts/experiments/verify_split_pair.py": "5cc7031569b17b560f68d4238191a32e5d7f66e7fdcbaabd0ca87020c89f89c8"
+  },
+  "bit_saving": "3244890195589/62500000000000000",
+  "files": {
+    "SOURCE.json": "20273d507b6074ce91fd7467e2d23101bf35f8d60a2d81684c58f611d91f9108",
+    "cache-stale-failure.log": "edb8b32a81ab277326e3deb19d34119982c10e998ca0f1ae9cf22b1fa4afea21",
+    "coarse_graph.py": "b5bbabb8e5bf69769af4b6cd67163954d990dc489e3c9488ed84bd09f0a767c8",
+    "dense-audit.json": "055a31465e8c42c27b8c80115cb9a2edef5b32b52b61ce3c4dac9f718c86f954",
+    "dense_audit.py": "b282e743d9d8c48cbc97eb2c2597a6c52894d33ea1967dd1ff4c269b0e0a72e3",
+    "inherited/NOTICE": "3bfe7a693e10fbe710e7fadedb7caeb21e272ce5899f48d043f636e2b06562b0",
+    "inherited/README.md": "e9d7ae955d27fb1832744583f1d251263e9fc499bae2d664b31f28fc83dc2592",
+    "inherited/SOURCE.json": "813564d76569fd54a5aa1645da15f1a33b12de8a0bcc53056e69087394b0cfcd",
+    "inherited/graph.py": "349c74a3684cc26c14f4b7001efb086825a4398461709ba13ade2cd10e336d23",
+    "inherited/vendor/pr69-compiler.py": "bd8169bf9f5df185a8aa4aec22234b4fd483000a1cecc78b71bf32e479af1499",
+    "run.py": "3b8f3a355799882fd1431e3ced934e34149c77e36dce09f62e214fc6156e4c0a",
+    "schedule_variant.py": "3ce3b4c0886868e3fcd2b75c54b61180d103447cfdb9d3ece1de47dd6d46eecc",
+    "screen_width.py": "9087cac4d745d3525814323462efa7927db820e86bcc9e93b81a9fd7dcf6d372",
+    "selected-both/PROOF.md": "5343fcb8dca9ab48c86a8c0a6735b2afbb07fca75f6d0e62b3d3d640f7f190d0",
+    "selected-both/arithmetic.json": "7ed59a2d4b546f3375b4f95e9005831ca9741bb56092bc49a61bc7b2e770eebb",
+    "selected-both/comparison.json": "c69207a10663d760232439ae82f18785bb14172c5422ab7092aee93d69f3e41b",
+    "selected-both/profile-23.json": "af5194d09494e76a9c02cfb4b102dab07f20ad7758acaf2867158af0cf6a4731",
+    "selected-both/profile-25.json": "4c915181d485f4d4fcb64b6f9d0a73ccd87c28cea3dd45fc5297e41d15c41ae7",
+    "selected-both/result-23.json": "7ef1ed08ffcd9782419dabb2cb1f40a6541ffc479a0cfaaedb07314704a7864c",
+    "selected-both/result-25.json": "561326c6f0f188736c73f05a999b773a65b78b568c2df15fa0c7092cd14b7225",
+    "selected-both/transitions-23.bin.json": "8905306796e045dd91d711f3696e05785c0db07cd0f90cfd93690ab438ff239b",
+    "selected-both/transitions-25.bin.json": "28a702c6d487e1e878c307f40191c1386acbb58139ed9573bb139a27aee4ad83",
+    "selected-both/word-23.json.gz": "7c33d64688332fce70c5e451fd00e86e58d2d44bdc648ff7139756f6a6c774fe",
+    "selected-both/word-25.json.gz": "eec0b667f3c2ba199fe291c11f19d6d0112576dc34a113372a9867ca5509f4ba",
+    "validate_selected.py": "45f8d580df7c6897b8c8f96d552ca0dfa3c78bbeed837c8c4f2ead4e21e4036b",
+    "width_bridge.py": "6486e6b9508357e33a3e8ee6740bf4ae46e306dfdd8f270eb2dd235270e77dfd"
+  },
+  "kappa": "51915547765237/1000000000000000000",
+  "public_PR76": "51738676045141/1000000000000000000",
+  "publication_eligible": false,
+  "status": "Internal subthreshold composition; compiler dirty bases/dense scalar audit/actual profiles/exact47/7 PASS. Separate independent receipt follows.",
+  "threshold_difference": "-16364332025941/200000000000000000000"
+}
diff --git a/research/round9-balanced-columns/run.py b/research/round9-balanced-columns/run.py
new file mode 100644
index 0000000000..9d292dfb1b
--- /dev/null
+++ b/research/round9-balanced-columns/run.py
@@ -0,0 +1,102 @@
+#!/usr/bin/env python3
+"""PR70 bounded carried-signal exchanges composed with pinned PR71.
+
+PR70 exchanges: Alejandro; PR71 split/next-use compiler: Chafik Boukhalfa.
+PR67 profile oracle/cost: Rohan Arun; PR68 live controls: Dominik Scholz.
+Composition and integer-entropy pricing prepared with substantial OpenAI
+Codex assistance. Discovery heuristic only; exact profiles accept candidates.
+"""
+from pathlib import Path
+from collections import Counter
+from hashlib import sha256
+import argparse,gzip,json,os,shlex,struct,subprocess,sys,time
+if sys.flags.optimize:raise ValueError('Assertions must remain enabled')
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent
+BASE=HERE.parent/'round6-pr71/baseline'
+sys.path.insert(0,str(BASE/'scripts/experiments'))
+import split_pair_engine as compiler
+import split_pair_graph as producer
+import schedule_variant
+import coarse_graph
+from binary_frame_math import logs
+
+
+def main():
+ p=argparse.ArgumentParser();p.add_argument('--h',type=int,choices=(23,25),required=True);p.add_argument('--work-dir',type=Path,required=True);p.add_argument('--passes',type=int,default=3);a=p.parse_args()
+ a.work_dir.mkdir(parents=True,exist_ok=True)
+ source=Path(compiler.__file__)
+ expected=json.loads((BASE/'research/split-pair/engine-provenance.json').read_text())['portable_engine_sha256']
+ assert sha256(source.read_bytes()).hexdigest()==expected
+ executable=a.work_dir/'oracle'
+ subprocess.run([*shlex.split(os.environ.get('CXX','c++')),'-O3','-std=c++17','-I',str(BASE/'references/frame-compiler/pr48/scripts/partial_swap'),str(BASE/'references/frame-compiler/pr67/research/slot-cost-rank-pair/profile_oracle.cpp'),'-o',str(executable)],check=True)
+ compiler.graph=lambda h: schedule_variant.reorder(coarse_graph.graph(h,coarse='columns'));compiler.build=producer.ordered_build(compiler.build,a.h)
+ original_match=compiler.match
+ def exchange_match(blocks,uses,enabled):
+  result=original_match(blocks,uses,enabled)
+  edges,chosen,right,stats=result
+  initial_cardinality=len(chosen)
+  if not a.passes:return result
+  owner={x:g for g,b in enumerate(blocks) for x in b['nodes']}
+  path=a.work_dir/'matching-oracle.bin'
+  with path.open('wb')as stream:
+   h=a.h;v=h*(h-1)*(h-2)//6
+   stream.write(struct.pack('<6I2Q',h,v,0,len(blocks)+2,0,0,h*(h-1),h*(h-1)))
+   stream.write(struct.pack('<2QI',0,0,0));stream.write(struct.pack('<2QI',0,0,h))
+   for b in blocks:stream.write(struct.pack('<2QI',*b['frame'],b['rank']))
+  oracle=subprocess.Popen([str(executable),str(path)],stdin=subprocess.PIPE,stdout=subprocess.PIPE,text=True,bufsize=1)
+  weights=[0]+[int(t*sum(logs(t))*10**30//2)for t in range(1,h+1)];cache={}
+  def entropy(x,y):
+   if (x,y)not in cache:
+    oracle.stdin.write(f'{x} {y}\n');oracle.stdin.flush();line=oracle.stdout.readline();assert line
+    values=list(map(int,line.split()));assert len(values)==h+1
+    cache[x,y]=sum(n*w for n,w in zip(values,weights))
+   return cache[x,y]
+  def cost(edge):
+   g,u,i=edge;p=owner[blocks[g]['inputs'][i]];t=uses[u][1]
+   return -entropy(g+2,t+2)+entropy(g+2,1)+entropy(0,p+2)+entropy(p+2,t+2)
+  try:
+   prices=[]
+   for j,e in enumerate(edges):
+    prices.append(cost(e))
+    if j%20000==0:print('edge prices',j,len(edges),'pairs',len(cache),flush=True)
+  finally:
+   oracle.stdin.close();assert oracle.wait(timeout=10)==0
+  priority=sorted(range(len(edges)),key=lambda e:(prices[e],edges[e]));initial=sum(prices[e]for e in chosen)
+  def can(e,drop=None):
+   g,u,i=edges[e]
+   rows=blocks[g]['outbasis']+[1<<edges[f][2]for f in blocks[g]['selected']if f!=drop]
+   return compiler.independent(compiler.basis(rows),1<<i)
+  for phase in range(a.passes):
+   changes=0
+   for e in priority:
+    if e in chosen:continue
+    g,u,i=edges[e];f=right.get(u)
+    if f is not None:
+     if prices[e]>=prices[f]:continue
+     if not can(e,f if edges[f][0]==g else None):continue
+    else:
+     f=next((z for z in sorted(blocks[g]['selected'],key=lambda z:(-prices[z],z)) if prices[e]<prices[z]and can(e,z)),None)
+     if f is None:continue
+    fg,fu,fi=edges[f];chosen.remove(f);blocks[fg]['selected'].remove(f);del right[fu]
+    chosen.add(e);blocks[g]['selected'].add(e);assert u not in right;right[u]=e;changes+=1
+   stats['carried_signal_exchanges']+=changes
+   print('exchanges',phase,changes,flush=True)
+   if not changes:break
+  assert len(chosen)==len(right)==initial_cardinality
+  for g,b in enumerate(blocks):assert len(compiler.basis(b['outbasis']+[1<<edges[e][2]for e in b['selected']]))==len(b['outbasis'])+len(b['selected'])
+  stats['exchange_entropy_gain_1e30']=initial-sum(prices[e]for e in chosen)
+  stats['exchange_profile_pairs']=len(cache)
+  (a.work_dir/f'matching-{a.h}.json').write_text(json.dumps(dict(edges=edges,chosen=sorted(chosen),stats=dict(stats)),separators=(',',':'))+'\n')
+  return edges,chosen,right,stats
+ compiler.match=exchange_match;compiler.oracles=[];compiler.ORACLE_EXE=str(executable);compiler.ORACLE_INPUT=str(a.work_dir/'compile-oracle.bin');compiler.PENDING_COST=True
+ try:result,word=compiler.compile_(a.h,matching=True,reclaim=True,dirty=True)
+ finally:
+  for process in compiler.oracles:process.stdin.close();assert process.wait(timeout=10)==0
+ result.pop('seconds');result['scalar']=compiler.graph(a.h).verify();result['graph_configuration']=dict(producer.configuration(a.h),scalar_node_order='core-reverse',coarse='columns');result['source_sha256']={p.name:sha256(p.read_bytes()).hexdigest()for p in(HERE/'run.py',HERE/'schedule_variant.py',HERE/'coarse_graph.py')}
+ raw=(json.dumps(word,separators=(',',':'))+'\n').encode()
+ (a.work_dir/f'word-{a.h}.json.gz').write_bytes(gzip.compress(raw,mtime=0))
+ (a.work_dir/f'result-{a.h}.json').write_text(json.dumps(result,indent=2)+'\n')
+ print(json.dumps(result),flush=True)
+
+if __name__=='__main__':main()
diff --git a/research/round9-balanced-columns/schedule_variant.py b/research/round9-balanced-columns/schedule_variant.py
new file mode 100644
index 0000000000..8e9a0e06e7
--- /dev/null
+++ b/research/round9-balanced-columns/schedule_variant.py
@@ -0,0 +1,37 @@
+"""PR74 scalar-envelope order composed with PR71 and PR70 carry exchanges.
+
+The reorder function is preserved from Thomas DiFiore's PR74 producer,
+prepared with OpenAI Codex assistance. PR71 is Chafik Boukhalfa's split
+construction; the copied carried-signal runner credits Alejandro's PR70.
+Composition experiment: Dominik Scholz, substantial OpenAI GPT-6 Astra
+assistance. Apache-2.0 and all inherited notices apply.
+"""
+
+
+def reorder(c):
+    """Preserve source IDs and same-envelope order; verify scalar identities."""
+    groups = {}
+    for node in sorted(c.active):
+        if c.args[node]:
+            groups.setdefault((c.core[node], c.union[node]), []).append(node)
+
+    def key(item):
+        (core, union), nodes = item
+        return union.bit_count() - core.bit_count(), -core, union, min(nodes)
+
+    variables = len(c.inputs)
+    ids = list(range(variables + 1)) + [
+        x for _, nodes in sorted(groups.items(), key=key) for x in nodes
+    ]
+    mapping = {x: i for i, x in enumerate(ids)}
+    args, core, union, provenance = c.args, c.core, c.union, c.provenance
+    c.args = [tuple(mapping[y] for y in args[x]) if args[x] else None for x in ids]
+    c.core = [core[x] for x in ids]
+    c.union = [union[x] for x in ids]
+    c.provenance = [provenance[x] for x in ids]
+    c.active = {mapping[x] for x in c.active}
+    c.outputs = {k: mapping[v] for k, v in c.outputs.items()}
+    # PR77 verifies before returning, so node-index memo entries must be reset.
+    c.support_in.cache_clear()
+    c.verify()
+    return c
diff --git a/research/round9-balanced-columns/screen_width.py b/research/round9-balanced-columns/screen_width.py
new file mode 100644
index 0000000000..2d11a3c8ac
--- /dev/null
+++ b/research/round9-balanced-columns/screen_width.py
@@ -0,0 +1,59 @@
+#!/usr/bin/env python3
+"""Actual paid transition profiles and PR71 exact conditional assembly."""
+from pathlib import Path
+from fractions import Fraction as Q
+from collections import Counter
+from math import comb
+from hashlib import sha256
+import argparse,gzip,json,math,os,shlex,subprocess,sys
+if sys.flags.optimize:raise ValueError('Assertions must remain enabled')
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent;BASE=HERE.parent/'round6-pr71/baseline'
+sys.path.insert(0,str(BASE/'scripts/experiments'))
+from binary_frame_profile_prepare import prepare
+from binary_frame_replay import replay
+import binary_frame_math as arithmetic
+from split_pair_arithmetic import refine,audit
+from width_bridge import bridge_for_width
+
+
+def main():
+ p=argparse.ArgumentParser();p.add_argument('--h',type=int,choices=(23,25),required=True);p.add_argument('--work-dir',type=Path,required=True);p.add_argument('--other-profile',type=Path);p.add_argument('--independent-replay',action='store_true');a=p.parse_args()
+ word=a.work_dir/f'word-{a.h}.json.gz';transitions=a.work_dir/f'transitions-{a.h}.bin'
+ prepared=prepare(word,transitions)
+ if a.independent_replay:
+  receipt=replay(word);(a.work_dir/f'replay-{a.h}.json').write_text(json.dumps(receipt,indent=2)+'\n')
+ exe=a.work_dir/'profiles'
+ subprocess.run([*shlex.split(os.environ.get('CXX','c++')),'-O3','-std=c++17','-I',str(BASE/'references/frame-compiler/pr48/scripts/partial_swap'),str(BASE/'scripts/experiments/binary_frame_profiles.cpp'),'-o',str(exe)],check=True)
+ subprocess.run([str(exe),str(transitions)],check=True)
+ f=json.loads(Path(str(transitions)+'.profiles.json').read_text());(a.work_dir/f'profile-{a.h}.json').write_text(json.dumps(f,indent=2)+'\n')
+ axes=[f if h==a.h else json.loads((a.other_profile or BASE/f'certificates/split-pair-profiles-{h}.json').read_text())for h in(23,25)]
+ m=575;N=comb(23,3)*comb(25,3);W=2*N+sum(N//v['v']*v['R']for v in axes);L=sum(N//v['v']*v['loss']for v in axes)
+ parts={'data':Counter({1:18*N,21:2*N,17:2*N,481:2*N}),'paid_endpoint_copy':Counter({1:N})}
+ for v in axes:
+  h=v['h'];rep=N//v['v'];bank=rep*v['R']
+  assert v['crt_disagreements']==0 and v['field_prime']==2**61-1
+  assert v['v']==comb(h,3)and v['loss']==h*(h-1)
+  assert v['rank_sum']==sum(t*n for t,n in enumerate(v['blocks']))==h*v['R']+v['loss']
+  assert v['blocks'][0]==v['blocks'][h]==0
+  parts[f'internal_{h}']=Counter({t:n*rep for t,n in enumerate(v['blocks'])if t and n})
+  parts[f'exterior_{h}']=Counter({h:bank,m-2*h:bank})
+  parts[f'data_growth_{h}']=Counter({1:2*N,h-2:2*N})
+ rows=sum(parts.values(),Counter());mass=sum(t*n for t,n in rows.items());assert mass==m*W-N+L
+ low,high=0.,0.001
+ for _ in range(60):
+  mid=(low+high)/2
+  if sum(n*(t/m)**(1-mid)/W for t,n in rows.items())<1:low=mid
+  else:high=mid
+ den=10**18;k=math.floor(low*den)
+ while refine.exact_moment(m,W,rows,Q(k,den))['upper']>=1:k-=1
+ while refine.exact_moment(m,W,rows,Q(k+1,den))['upper']<1:k+=1
+ bit=Q(k,den);accepted=refine.exact_moment(m,W,rows,bit);excluded=refine.exact_moment(m,W,rows,bit+Q(1,den));assert accepted['upper']<1<excluded['lower']
+ bridge=json.loads((BASE/'references/frame-compiler/pr48/research/copied-fixed/certificate.json').read_text())['finite_bridge'];bridge=bridge_for_width(bridge,W)
+ assembly=refine.assemble(bridge,bit,Q(1,10**12),den)
+ profile=dict(m=m,N=N,W=W,L=L,total_rank=mass,deficit=m*W-mass,child_multiplicities=dict(sorted(rows.items())),parts=parts,axes=axes)
+ result=dict(scope='Finite conditional screen; inherited all-size and transfer hypotheses remain.',kappa=assembly['kappa'],bit_saving=bit,profile=profile,moment=accepted,excluded_next_bit=excluded,assembly=assembly,independent_serialized_replay=a.independent_replay,profile_reconstruction=prepared,comparison_pr71=assembly['kappa']-Q(51414646104039,10**18))
+ (a.work_dir/'arithmetic.json').write_text(json.dumps(arithmetic.js(result),indent=2)+'\n')
+ print(json.dumps(dict(kappa=str(assembly['kappa']),bit_saving=str(bit),kappa_float=float(assembly['kappa']),gain_over_pr71=str(result['comparison_pr71']),roles={v['h']:v['R']for v in axes},W=W,all_47_constraints_and_7_margins=True,independent_serialized_replay=a.independent_replay),indent=2),flush=True)
+
+if __name__=='__main__':main()
diff --git a/research/round9-balanced-columns/selected-both/MANIFEST.json b/research/round9-balanced-columns/selected-both/MANIFEST.json
new file mode 100644
index 0000000000..504348d485
--- /dev/null
+++ b/research/round9-balanced-columns/selected-both/MANIFEST.json
@@ -0,0 +1,196 @@
+{
+  "baseline_files": {
+    "certificates/split-pair-compiler.json": "6561fab8609e8aa3a291a8197cd8b834b0881ee0b9a5b991926c2c93c4d5f884",
+    "certificates/split-pair-kappa.json": "0fcb53fa374273efab6370953c966e8fa065dbfdba43691843a4788cb2cda42e",
+    "certificates/split-pair-profiles-23.json": "867302152b62abf28a6e1b119580b99eaffa237082b1d295000a3ee2637d5db4",
+    "certificates/split-pair-profiles-25.json": "1ecc63972ae4df2ed7af7278d5f65790dd4aca3032cc17304f39ac063cd2c067",
+    "certificates/split-pair-repository-validation.json": "a4700ef48968e0bc6afadf8048d82bb4f4cfc9f21db9a8bf355bd9c2745da53e",
+    "certificates/split-pair-transitions-23.json": "6ca709c1e101f7c9a828d9ceb8a7f2991a2d954b84c9ae8c616deabfc1e321c8",
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+  },
+  "bit_saving": "3244890195589/62500000000000000",
+  "documentation_changes_since_freeze": [
+    "selected-both/PROOF.md",
+    "selected-both/comparison.json"
+  ],
+  "files": {
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+    "cache-stale-failure.log": "edb8b32a81ab277326e3deb19d34119982c10e998ca0f1ae9cf22b1fa4afea21",
+    "coarse_graph.py": "b5bbabb8e5bf69769af4b6cd67163954d990dc489e3c9488ed84bd09f0a767c8",
+    "dense-audit.json": "055a31465e8c42c27b8c80115cb9a2edef5b32b52b61ce3c4dac9f718c86f954",
+    "dense_audit.py": "b282e743d9d8c48cbc97eb2c2597a6c52894d33ea1967dd1ff4c269b0e0a72e3",
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+    "run.py": "3b8f3a355799882fd1431e3ced934e34149c77e36dce09f62e214fc6156e4c0a",
+    "schedule_variant.py": "3ce3b4c0886868e3fcd2b75c54b61180d103447cfdb9d3ece1de47dd6d46eecc",
+    "screen_width.py": "9087cac4d745d3525814323462efa7927db820e86bcc9e93b81a9fd7dcf6d372",
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+    "selected-both/arithmetic.json": "7ed59a2d4b546f3375b4f95e9005831ca9741bb56092bc49a61bc7b2e770eebb",
+    "selected-both/comparison.json": "162a69efcac46cf132a3f05fc87202c82d1f4664970bc8b7d90c41311baac781",
+    "selected-both/profile-23.json": "af5194d09494e76a9c02cfb4b102dab07f20ad7758acaf2867158af0cf6a4731",
+    "selected-both/profile-25.json": "4c915181d485f4d4fcb64b6f9d0a73ccd87c28cea3dd45fc5297e41d15c41ae7",
+    "selected-both/result-23.json": "7ef1ed08ffcd9782419dabb2cb1f40a6541ffc479a0cfaaedb07314704a7864c",
+    "selected-both/result-25.json": "561326c6f0f188736c73f05a999b773a65b78b568c2df15fa0c7092cd14b7225",
+    "selected-both/transitions-23.bin.json": "8905306796e045dd91d711f3696e05785c0db07cd0f90cfd93690ab438ff239b",
+    "selected-both/transitions-25.bin.json": "28a702c6d487e1e878c307f40191c1386acbb58139ed9573bb139a27aee4ad83",
+    "selected-both/word-23.json.gz": "7c33d64688332fce70c5e451fd00e86e58d2d44bdc648ff7139756f6a6c774fe",
+    "selected-both/word-25.json.gz": "eec0b667f3c2ba199fe291c11f19d6d0112576dc34a113372a9867ca5509f4ba",
+    "validate_selected.py": "45f8d580df7c6897b8c8f96d552ca0dfa3c78bbeed837c8c4f2ead4e21e4036b",
+    "width_bridge.py": "6486e6b9508357e33a3e8ee6740bf4ae46e306dfdd8f270eb2dd235270e77dfd"
+  },
+  "kappa": "51915547765237/1000000000000000000",
+  "original_freeze_manifest_sha256": "c6e947aef048434c57a2570d1c97026283c84b61b974ad5495996f5fffc5129d",
+  "public_PR76": "51738676045141/1000000000000000000",
+  "status": "Finite compiler, dense scalar audit, independent serialized replay, actual profiles and exact arithmetic PASS; deterministic regeneration and broad checks pending"
+}
diff --git a/research/round9-balanced-columns/selected-both/PROOF.md b/research/round9-balanced-columns/selected-both/PROOF.md
new file mode 100644
index 0000000000..fa0b88a385
--- /dev/null
+++ b/research/round9-balanced-columns/selected-both/PROOF.md
@@ -0,0 +1,101 @@
+# Balanced columns composed with envelope ordering and carry exchanges
+
+The selected finite witness has
+
+    kappa = 51915547765237 / 10^18
+    bit saving = 3244890195589 / 62500000000000000
+    roles = 27128 / 35598, W = 133585058
+    total recursive rank = 76809561450, deficit = 1846900.
+
+Against PR #74 at `5830ccfca8466aff613767222558e29b57fa8225`, whose
+reported conditional exponent is `10366199626713/200000000000000000`,
+the exact gain is `10568703959/125000000000000000`, or approximately 0.163125609609%.
+Against PR #76 the gain is `5527241253/31250000000000000`, or
+0.3418559067%. These are comparisons of specific conditional claims, not
+a proof of global optimality.
+
+## Exact graph change
+
+The PR69/77 balanced-column identity replaces each coarse total over two
+groups A and B by
+
+    sum_{b in B} (sum_{a in A} e(a,b)).
+
+Every summand still appears once, and each addition has disjoint support.
+Support interning may reuse the column sums in interval strips. The exact
+adapter is imported from huxint's PR77 at
+`83298467291d3bd4b53f76faaef60dbf585be171`; its underlying PR62 pair graph
+is byte-identical to the graph preserved in our PR71/PR65 closure.
+The `[1,1,2]` split vector and lowest-intact-pair anchor stay fixed.
+
+The graph then uses PR74 envelope-node renumbering, PR65's cover-core h23
+and reverse-node h25 schedules, and the three-pass PR70 carry-exchange
+composition frozen in PR76. The PR77 cost-ordered clearing compiler is
+not imported. The PR71 engine, frame-containment rules, next-use score,
+dirty wrapper and every literal charge remain inherited and unmodified.
+Every schedule dependency and eligible carrier is checked forward; every
+carry exchange preserves cardinality, unique uses and basis independence.
+
+PR77's graph adapter verifies before returning, populating a node-ID
+support cache. Renumbering invalidates those cache keys. The local reorder
+therefore explicitly clears `support_in` memoization after applying the
+node bijection and before verifying again. The initial stale-cache failure
+is preserved in `cache-stale-failure.log`. No assertion is disabled.
+
+`dense_audit.py` separately evaluates all active scalar additions over
+dense global input bitsets without using the memoized/provenance support
+routine. All additions are disjoint, every partial output equals its
+specified global support, and the common-point envelopes are exact for
+both axes. This preserves the original scalar/data-output interface and
+I+J geometry; it does not assume a cheaper data layer.
+
+## Physical and arithmetic evidence
+
+Both compiler words pass complete input/dirty basis execution in both
+orientations. Actual fixed-I+J profiles have zero CRT disagreements. The
+full paid child list includes all promotions, clearing controls, endpoint
+copies, data growth and the inherited complex branch. The exact moment
+accepts the stated bit saving and rejects the next denominator-10^18 grid
+point. All 47 strict assembly inequalities and seven margins pass, with
+the original h=1/10^12 backoff. Width-derived bridge fields are recomputed
+by the preserved round7 helper: bit wire logarithm 27, product-row
+coefficient 843, degree 2000 and degree gap 7007/25.
+
+The manifest freezes the selected artifacts and exact source closure.
+Independent serialized replay, profile regeneration and the independent
+rational moment check are recorded in the completed separate validation receipt. The receipt binds the original
+freeze manifest preserved under `provenance`; only proof/comparison metadata
+changed in this package, with every runtime and physical artifact unchanged. Broad repository checks
+and portable deterministic regeneration are not claimed here.
+
+All inherited all-size compiler, fixed finite-alphabet tape, analytic and
+semantic reduction, routing, precision, prime setup and exact recovery
+obligations remain conditional. Arithmetic cutoffs are not practical
+runtime thresholds or a proof of the complete multiplication theorem.
+
+## Reproduction and credit
+
+From this directory with the exact PR71 archive at
+`../round6-pr71/baseline`, execute sequentially:
+
+```sh
+python3 run.py --h 23 --work-dir /tmp/columns23
+python3 run.py --h 25 --work-dir /tmp/columns25
+python3 screen_width.py --h 23 --work-dir /tmp/columns23
+python3 screen_width.py --h 25 --work-dir /tmp/columns25 --other-profile /tmp/columns23/profile-23.json
+python3 dense_audit.py
+python3 validate_selected.py --bundle selected-both --output validation-receipt.json
+```
+
+Use assert-enabled Python and C++17. This machine needs
+`CXX='c++ -isysroot /Library/Developer/CommandLineTools/SDKs/MacOSX15.2.sdk'`.
+
+Credit eumemic PR69 for balanced coarse columns; huxint PR77 for the
+anchored balanced-column adapter; Thomas DiFiore PR74 for envelope ordering;
+Chafik Boukhalfa PR71 for anchored splits and next-use scoring; Alejandro
+Zarzuelo Urdiales PR70 for exchanges; Rohan Arun PR65/67 for schedules,
+profile costs and arithmetic; Dominik Scholz PR68/76 for pending controls
+and the prior composition; Avi Eisenberg PR62, eumemic PR57 and the full
+retained author chain. This further composition was prepared for Dominik
+Scholz with substantial OpenAI GPT-6 Astra assistance. Apache-2.0 and all
+inherited notices remain. No CI or repository integration changes are included.
diff --git a/research/round9-balanced-columns/selected-both/arithmetic.json b/research/round9-balanced-columns/selected-both/arithmetic.json
new file mode 100644
index 0000000000..6b5f852da0
--- /dev/null
+++ b/research/round9-balanced-columns/selected-both/arithmetic.json
@@ -0,0 +1,945 @@
+{
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diff --git a/research/round9-balanced-columns/selected-both/comparison.json b/research/round9-balanced-columns/selected-both/comparison.json
new file mode 100644
index 0000000000..748f9f9a80
--- /dev/null
+++ b/research/round9-balanced-columns/selected-both/comparison.json
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+{
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new file mode 100644
index 0000000000..869fbc9e8b
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+++ b/research/round9-balanced-columns/selected-both/profile-23.json
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diff --git a/research/round9-balanced-columns/selected-both/profile-25.json b/research/round9-balanced-columns/selected-both/profile-25.json
new file mode 100644
index 0000000000..b98279deab
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diff --git a/research/round9-balanced-columns/selected-both/result-23.json b/research/round9-balanced-columns/selected-both/result-23.json
new file mode 100644
index 0000000000..0de212965f
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diff --git a/research/round9-balanced-columns/selected-both/result-25.json b/research/round9-balanced-columns/selected-both/result-25.json
new file mode 100644
index 0000000000..540287acb9
--- /dev/null
+++ b/research/round9-balanced-columns/selected-both/result-25.json
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diff --git a/research/round9-balanced-columns/selected-both/transitions-23.bin.json b/research/round9-balanced-columns/selected-both/transitions-23.bin.json
new file mode 100644
index 0000000000..fbc07d359c
--- /dev/null
+++ b/research/round9-balanced-columns/selected-both/transitions-23.bin.json
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diff --git a/research/round9-balanced-columns/selected-both/transitions-25.bin.json b/research/round9-balanced-columns/selected-both/transitions-25.bin.json
new file mode 100644
index 0000000000..dac6567ef8
--- /dev/null
+++ b/research/round9-balanced-columns/selected-both/transitions-25.bin.json
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+    "24": 25
+  },
+  "word_sha256": "fc6f429363b3c8208fd9bff10ec20386de280a3e1ea4f109ca5e0834d2c60fc4",
+  "copied_centers": 25,
+  "transition_events_equal_independent_xor_word_reconstruction": true
+}
diff --git a/research/round9-balanced-columns/selected-both/word-23.json.gz b/research/round9-balanced-columns/selected-both/word-23.json.gz
new file mode 100644
index 0000000000..8d077760e3
Binary files /dev/null and b/research/round9-balanced-columns/selected-both/word-23.json.gz differ
diff --git a/research/round9-balanced-columns/selected-both/word-25.json.gz b/research/round9-balanced-columns/selected-both/word-25.json.gz
new file mode 100644
index 0000000000..a353dfdbd8
Binary files /dev/null and b/research/round9-balanced-columns/selected-both/word-25.json.gz differ
diff --git a/research/round9-balanced-columns/validate_selected.py b/research/round9-balanced-columns/validate_selected.py
new file mode 100644
index 0000000000..9f79237c40
--- /dev/null
+++ b/research/round9-balanced-columns/validate_selected.py
@@ -0,0 +1,49 @@
+#!/usr/bin/env python3
+"""Independent finite validation of a frozen compiler-exchange witness."""
+from pathlib import Path
+from hashlib import sha256
+from fractions import Fraction as Q
+from collections import Counter
+from math import comb
+import argparse,json,os,shlex,subprocess,sys,tempfile
+if sys.flags.optimize:raise ValueError('Assertions must remain enabled')
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent;BASE=HERE.parent/'round6-pr71/baseline'
+sys.path.insert(0,str(BASE/'scripts/experiments'))
+from binary_frame_replay import replay
+from binary_frame_profile_prepare import prepare
+from split_pair_arithmetic import refine,audit
+from width_bridge import bridge_for_width
+
+
+def main():
+ p=argparse.ArgumentParser();p.add_argument('--bundle',type=Path,required=True);p.add_argument('--output',type=Path,required=True);a=p.parse_args()
+ selected=a.bundle.resolve();manifest=json.loads((selected/'MANIFEST.json').read_text())
+ for name,digest in manifest['files'].items():assert sha256((HERE/name).read_bytes()).hexdigest()==digest,name
+ for name,digest in manifest['baseline_files'].items():assert sha256((BASE/name).read_bytes()).hexdigest()==digest,name
+ receipts={}
+ with tempfile.TemporaryDirectory(prefix='exchange-validation-')as directory:
+  work=Path(directory);exe=work/'profiles'
+  subprocess.run([*shlex.split(os.environ.get('CXX','c++')),'-O3','-std=c++17','-I',str(BASE/'references/frame-compiler/pr48/scripts/partial_swap'),str(BASE/'scripts/experiments/binary_frame_profiles.cpp'),'-o',str(exe)],check=True)
+  for h in(23,25):
+   word=selected/f'word-{h}.json.gz';r=replay(word);transitions=work/f'word{h}.bin';prepared=prepare(word,transitions)
+   subprocess.run([str(exe),str(transitions)],check=True,capture_output=True,text=True)
+   actual=json.loads(Path(str(transitions)+'.profiles.json').read_text());expected=json.loads((selected/f'profile-{h}.json').read_text());assert actual==expected
+   receipts[h]=dict(replay=r,transitions=prepared,actual_profiles_equal=True)
+ a0=json.loads((selected/'arithmetic.json').read_text());profile=a0['profile'];rows={int(t):n for t,n in profile['child_multiplicities'].items()};bit=Q(a0['bit_saving']);kappa=Q(a0['kappa'])
+ actual_axes=[json.loads((selected/f'profile-{h}.json').read_text())for h in(23,25)]
+ assert profile['axes']==actual_axes
+ N=comb(23,3)*comb(25,3);width=2*N;rebuilt=Counter({1:19*N,21:2*N,17:2*N,481:2*N});loss=0
+ for v in actual_axes:
+  h=v['h'];rep=N//v['v'];bank=rep*v['R'];width+=bank;loss+=rep*v['loss']
+  rebuilt.update({t:rep*n for t,n in enumerate(v['blocks'])if t and n})
+  rebuilt.update({h:bank,575-2*h:bank});rebuilt.update({1:2*N,h-2:2*N})
+ assert profile['m']==575 and profile['W']==width and rebuilt==rows
+ assert sum(t*n for t,n in rebuilt.items())==575*width-N+loss==profile['total_rank']
+ accepted=refine.exact_moment(profile['m'],profile['W'],rows,bit);rejected=refine.exact_moment(profile['m'],profile['W'],rows,bit+Q(1,10**18));assert accepted['upper']<1<rejected['lower']
+ lower,upper=audit.independent_moment(profile,bit,a0['moment']['terms']);rl,ru=audit.independent_moment(profile,bit+Q(1,10**18),a0['moment']['terms']);assert upper<1<rl
+ bridge=json.loads((BASE/'references/frame-compiler/pr48/research/copied-fixed/certificate.json').read_text())['finite_bridge'];bridge=bridge_for_width(bridge,profile['W']);assembly=refine.assemble(bridge,bit,Q(1,10**12),10**18);assert assembly['kappa']==kappa
+ result=dict(status='PASS finite source binding, both serialized words, all CRT profiles, rational moment and assembly',manifest_sha256=sha256((selected/'MANIFEST.json').read_bytes()).hexdigest(),axes=receipts,kappa=str(kappa),bit_saving=str(bit),strict_constraints=47,margins=7,independent_moment_and_adjacent_grid=True,scope='Finite conditional witness; inherited all-size and transfer assumptions remain.')
+ a.output.write_text(json.dumps(result,indent=2)+'\n');print(json.dumps({k:v for k,v in result.items()if k!='axes'},indent=2))
+
+if __name__=='__main__':main()
diff --git a/research/round9-balanced-columns/validation-receipt.json b/research/round9-balanced-columns/validation-receipt.json
new file mode 100644
index 0000000000..c3bbaa6cb5
--- /dev/null
+++ b/research/round9-balanced-columns/validation-receipt.json
@@ -0,0 +1,160 @@
+{
+  "status": "PASS finite source binding, both serialized words, all CRT profiles, rational moment and assembly",
+  "manifest_sha256": "c6e947aef048434c57a2570d1c97026283c84b61b974ad5495996f5fffc5129d",
+  "axes": {
+    "23": {
+      "replay": {
+        "status": "independent serialized word, exact frames, full input and dirty basis both orientations PASS",
+        "h": 23,
+        "roles": 27128,
+        "output_roles": 5336,
+        "elementary_xors": 167539,
+        "word_length": 694950,
+        "full_basis_vectors": 30670,
+        "rank_mass": 624450,
+        "histogram": {
+          "1": 58230,
+          "0": 185200,
+          "2": 81090,
+          "4": 14624,
+          "3": 10015,
+          "5": 6757,
+          "6": 2127,
+          "8": 1105,
+          "10": 599,
+          "9": 364,
+          "12": 691,
+          "14": 1893,
+          "13": 208,
+          "11": 97,
+          "16": 430,
+          "17": 543,
+          "18": 3197,
+          "19": 6482,
+          "15": 267,
+          "22": 23,
+          "7": 278,
+          "21": 429
+        }
+      },
+      "transitions": {
+        "h": 23,
+        "v": 1771,
+        "R": 27128,
+        "frames": 59803,
+        "distinct_transitions": 151935,
+        "singles": 15939,
+        "rank_mass": 624450,
+        "rank_histogram_with_side_growth_split_into_singletons": {
+          "1": 68856,
+          "2": 75777,
+          "3": 10015,
+          "4": 14624,
+          "5": 6757,
+          "6": 2127,
+          "7": 278,
+          "8": 1105,
+          "9": 364,
+          "10": 599,
+          "11": 97,
+          "12": 691,
+          "13": 208,
+          "14": 1893,
+          "15": 267,
+          "16": 430,
+          "17": 543,
+          "18": 3197,
+          "19": 6482,
+          "21": 429,
+          "22": 23
+        },
+        "word_sha256": "7e452d6cec73d2433c9ec996fac2bfaf97317294ee515f86cef95c3ad9f154d7",
+        "copied_centers": 23,
+        "transition_events_equal_independent_xor_word_reconstruction": true
+      },
+      "actual_profiles_equal": true
+    },
+    "25": {
+      "replay": {
+        "status": "independent serialized word, exact frames, full input and dirty basis both orientations PASS",
+        "h": 25,
+        "roles": 35598,
+        "output_roles": 6925,
+        "elementary_xors": 235195,
+        "word_length": 972980,
+        "full_basis_vectors": 40198,
+        "rank_mass": 890550,
+        "histogram": {
+          "1": 76952,
+          "0": 260118,
+          "2": 118303,
+          "3": 13346,
+          "4": 18638,
+          "5": 8916,
+          "6": 3400,
+          "8": 1196,
+          "7": 576,
+          "9": 553,
+          "10": 854,
+          "12": 776,
+          "14": 756,
+          "13": 162,
+          "16": 2229,
+          "18": 572,
+          "19": 879,
+          "11": 21,
+          "15": 125,
+          "17": 583,
+          "20": 4217,
+          "21": 8479,
+          "24": 25,
+          "23": 462
+        }
+      },
+      "transitions": {
+        "h": 25,
+        "v": 2300,
+        "R": 35598,
+        "frames": 84845,
+        "distinct_transitions": 212411,
+        "singles": 20700,
+        "rank_mass": 890550,
+        "rank_histogram_with_side_growth_split_into_singletons": {
+          "1": 90752,
+          "2": 111403,
+          "3": 13346,
+          "4": 18638,
+          "5": 8916,
+          "6": 3400,
+          "7": 576,
+          "8": 1196,
+          "9": 553,
+          "10": 854,
+          "11": 21,
+          "12": 776,
+          "13": 162,
+          "14": 756,
+          "15": 125,
+          "16": 2229,
+          "17": 583,
+          "18": 572,
+          "19": 879,
+          "20": 4217,
+          "21": 8479,
+          "23": 462,
+          "24": 25
+        },
+        "word_sha256": "fc6f429363b3c8208fd9bff10ec20386de280a3e1ea4f109ca5e0834d2c60fc4",
+        "copied_centers": 25,
+        "transition_events_equal_independent_xor_word_reconstruction": true
+      },
+      "actual_profiles_equal": true
+    }
+  },
+  "kappa": "51915547765237/1000000000000000000",
+  "bit_saving": "3244890195589/62500000000000000",
+  "strict_constraints": 47,
+  "margins": 7,
+  "independent_moment_and_adjacent_grid": true,
+  "scope": "Finite conditional witness; inherited all-size and transfer assumptions remain."
+}
diff --git a/research/round9-balanced-columns/width_bridge.py b/research/round9-balanced-columns/width_bridge.py
new file mode 100644
index 0000000000..4f9c12957b
--- /dev/null
+++ b/research/round9-balanced-columns/width_bridge.py
@@ -0,0 +1,22 @@
+"""Refresh derived role-logarithm/product-row constants after a width change.
+
+Formulas are inherited from PR48/PR65 balanced_assembly.validate_bridge.
+Dominik Scholz with substantial OpenAI GPT-6 Astra assistance. Apache-2.0.
+"""
+from copy import deepcopy
+from fractions import Fraction as Q
+
+
+def bridge_for_width(inherited, width):
+    assert type(width) is int and width > 0
+    bridge = deepcopy(inherited)
+    bridge['bit']['W'] = width
+    bridge['bit']['wire_bits'] = width.bit_length()
+    coefficient = sum(
+        bridge[axis]['halving_degree'] * bridge[axis]['wire_bits']
+        for axis in ('bit', 'complex')
+    )
+    bridge['rows']['coefficient'] = coefficient
+    bridge['rows']['degree_gap'] = Q(bridge['rows']['degree']) - Q(51, 25) * coefficient
+    assert bridge['rows']['degree_gap'] > 0
+    return bridge