diff --git a/research/coordinated-gauge-async/EXPECTED.json b/research/coordinated-gauge-async/EXPECTED.json
new file mode 100644
index 0000000000..4da2016e90
--- /dev/null
+++ b/research/coordinated-gauge-async/EXPECTED.json
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+{
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diff --git a/research/coordinated-gauge-async/LICENSE b/research/coordinated-gauge-async/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/coordinated-gauge-async/LICENSE
@@ -0,0 +1,201 @@
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diff --git a/research/coordinated-gauge-async/MANIFEST.json b/research/coordinated-gauge-async/MANIFEST.json
new file mode 100644
index 0000000000..93f00c7718
--- /dev/null
+++ b/research/coordinated-gauge-async/MANIFEST.json
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+  "vendor/complex/e8-complex-gauge-kernel/code/audit_gk.py": "cb79c6b788333a031e14c7f134823f5f6147d7f1dcc6e5e3b1420743c32f8ebc",
+  "vendor/complex/e8-complex-gauge-kernel/code/emit_gk.py": "4e415bff1d9f154496ed38e6889929b600aa0f3d583a375e8ca257ad8f51fa23",
+  "vendor/complex/e8-complex-gauge-kernel/code/gk_common.py": "e68d190fd436b2f3955decb590adb53344d5d4ea0a19d62d86df1e4b55ecd198",
+  "vendor/complex/e8-complex-gauge-kernel/code/gk_word.py": "f027a9273bcf4e0971258c160b7d3b3c196366b32a529cff6e6a0e666b03826b",
+  "vendor/complex/e8-complex-gauge-kernel/code/integrate_gk.py": "2e807893180a3716b36a81f1a7d933bd55f2299489b6f236ba5c644d87739209",
+  "vendor/complex/e8-complex-gauge-kernel/code/onehost_gk.py": "416b318c1b6b8c7b72e24713b1b18d8772ec368e042a321b4418c63a74711590",
+  "vendor/complex/e8-complex-gauge-kernel/code/replay_gk.py": "2422b44fc2fb4742bb575572337158a15fb6831630f8746a1c7ee3133e0ee2e3",
+  "vendor/complex/e8-complex-gauge-kernel/data/selections/gauge80-kernel42.json": "436e4acadee227dbfcb66386c9ccaa38e68054e8ed4870bb174eeb41363375bb",
+  "vendor/complex/split-centres-global/code/complex_phases_full.py": "0f2795aaffafe0c64d77f3748ef2109a3c42c58da867e4cf269a0a8cb6109466",
+  "vendor/complex/split-centres-global/vendor/clifford_exact.py": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962",
+  "verify.py": "c553dbf909d703e5afb0333d7da43481f5df18a71e932c23986d9bfd8b953e2e",
+  "verify.sh": "2b1b51f757d8ddf81941c8c96007a2256336856c2e017bcf6c3a37576c82bc8b"
+}
diff --git a/research/coordinated-gauge-async/NOTICE.md b/research/coordinated-gauge-async/NOTICE.md
new file mode 100644
index 0000000000..d95d36e8af
--- /dev/null
+++ b/research/coordinated-gauge-async/NOTICE.md
@@ -0,0 +1,21 @@
+# Scoped attribution
+
+Prepared with OpenAI Codex assistance. Apache-2.0; see LICENSE.
+Original authorship notices in included vendor code are retained.
+
+PR484 supplies the gauge80-kernel42 complex word and wrappers. PR496 supplies
+whole-owner pooling and invoice/assembly ancestry. PR493 supplies the unchanged
+cited bit operator; its relevant attribution excerpt is in references/PR493-NOTICE.md.
+PR495 supplies body-frame retiming. PR501 and PR502 transfer it onto the gauge
+word; the included build_retimed.py comes from PR502. PR473's single-helper
+rank4-to-rank2 proposal directly inspired the coordinated four-helper change.
+
+Earlier retained ancestry includes PR433/425/370 (geometry and scalar/phase
+compiler), PR458 (whole-owner spectator wrappers), PR361/315 (exact moments and
+outer arithmetic), and PR434/417/435/440/401/408-416 (kernel and gauge methods).
+
+The new local contribution is coordinated shared target-level collapse, composed
+with our prior asynchronous complete-invocation schedule. OpenAI math source
+inspection informed scheduling and response-matroid research; no fresh Lean
+certification is claimed. Detailed provenance and hashes are in SOURCES.json
+and SOURCE-FILES.json. This notice includes only relevant project attribution.
diff --git a/research/coordinated-gauge-async/PORTABILITY.md b/research/coordinated-gauge-async/PORTABILITY.md
new file mode 100644
index 0000000000..8ac01d2eee
--- /dev/null
+++ b/research/coordinated-gauge-async/PORTABILITY.md
@@ -0,0 +1,27 @@
+# Portable serialization and safety checks
+
+Every child runs through code/run_portable.py. The wrapper preserves its script
+arguments, import directory and working directory, and forces newly generated
+Path.write_text output to UTF-8/LF. It changes no mathematical operation.
+A fixture checked sibling imports, argv, cwd and exact LF output bytes. A separate
+optimized-Python test confirmed the verifier refuses to run with assertions off.
+
+This fixes a concrete issue: adapters_gk.py writes RESIDUAL-ADAPTERS.json as text;
+integrate_gk.py embeds its raw SHA in the final template. Default Windows CRLF
+and Linux LF produce different hashes despite identical parsed mathematics.
+Pinned source/input bytes remain unchanged; `.gitattributes` uses `** -text`.
+
+Generated gzip headers can differ across Python/zlib versions. Compressed hashes
+are fresh same-run provenance, not fixed cross-platform mathematical identities.
+The final template identity is the hash of decompressed canonical JSON. No claim
+is made that Python3.14 or Linux was executed in this local Windows3.11 run.
+
+The reconstructed word was compared against the independently nominated word.
+Every program field matched. Differences were only an optional redundant R=560
+field and two unused floating estimates in report. The constructor checks the
+redundant dimension, removes only those metadata fields from its semantic hash,
+and binds every remaining field. No numerical proof uses the removed estimates.
+See research-notes/RECONSTRUCTION-METADATA-DIFF.json.
+
+The final manifest binds all source and evidence bytes. The output directory must
+be new and outside the package. No cleanup, network or publication is performed.
diff --git a/research/coordinated-gauge-async/PROOF.md b/research/coordinated-gauge-async/PROOF.md
new file mode 100644
index 0000000000..407e40bed0
--- /dev/null
+++ b/research/coordinated-gauge-async/PROOF.md
@@ -0,0 +1,78 @@
+# Sufficient composition of retiming, coordinated gauge collapse and asynchronous hosting
+
+## Local reconstruction
+
+Begin with the pinned PR484/496 gauge80-kernel42 complex supplier. Transfer only
+frame labels of the identical scalar body from PR495 through the PR502 constructor.
+The transfer changes224 A labels and504 B labels; any changed scalar coefficient
+or register is rejected. The new gauge selection replaces helper entrances
+540,541,542,543 by the rank2 plane spanned by binary vectors408 and102.
+
+The plane lies inside each original rank4 entrance. Reconstruct every helper
+chain, compensation target level and centre descent from this joint selection.
+Targets81,82,84,87 all fall from rank4 to rank2. The42 response-kernel pivots and
+the underlying scalar operator are retained. The exact local paid-profile delta
+is -16 rank2 +4 rank4 +4 rank6. Four width5 residuals become width7, giving
+18 width7,104 width8 and438 width9 residuals. The source program is independently
+checked on every arbitrary-dirty input column in both orientations. Exact
+Gaussian phase checks and emitted paid spectator wrappers are additional gates,
+not consequences assumed from the scalar check.
+
+The target level is a shared maximum. A single gauge withdrawal leaves other
+contributors at rank4. Only withdrawing the complete group releases the target
+and its centre-descending cost. Thus the joint profile is reconstructed; four
+separately priced gains cannot be summed. research-notes/RESEARCH-NOTE.md gives
+the exact comparison with the four individual moves.
+
+## Global sufficient scheduling theorem
+
+Assume the retained complete normalized invocation contract: the invocation
+implements its declared active data shear on arbitrary dirty inputs, each helper
+uses one distinct physical family, paid wrappers restore borrowed spectators,
+private workspace is cleared, and only the declared residual transform persists
+on the assigned pending block. The all-size realization of this contract remains
+an explicit inherited condition.
+
+Take eight replicas of the nine-stage controller, with disjoint active data
+banks. Sort their complete calls by (7*stage+6*replica,stage,replica), preserving
+each replica's entire source-event order. Every invocation, including its cover
+sweep and wrappers, finishes before the next. Between consecutive touches of an
+owner family, allocate disjoint blocks of that family's pending child to helpers.
+Pay the exact remaining block before the owner's next touch.
+
+Induct over the emitted events. A complete invocation has the retained active
+shear endpoint; arbitrary dirt in a borrowed owner is permitted by the contract.
+Its spectator coordinates are restored, and its persistent residual has a
+disjoint footprint in the pending child. These residuals commute with each other
+and multiply with the explicitly paid remainder to the original pending child.
+At every bridge or next active touch, the owner therefore has exactly its original
+controller state. Each replica retains its original data/bridge/terminal order.
+After all replicas finish, apply the common full-stock correction once. The global
+operator is unchanged. No helper is split across physical owners, and no scalar
+gate is commuted through an unfinished cover invocation.
+
+The frozen witness gives every one of40320 helper uses. The independent audit
+reconstructs actual windows from local calls and bridge touches, checks all800
+families in every call are distinct, reconstructs actual73-column orthogonal
+charts and paid adapter frames, and verifies disjoint residuals plus remainder
+exactly cover each pending child. A floating optimizer is unnecessary in replay.
+
+## Complete cost and sufficient analytic transfer
+
+The global histogram is171720 rank1,17856 rank2,41568 rank3,6336 rank4,
+23280 rank5,6912 rank6,1752 rank7,1224 rank8 and432 rank48 children.
+Ambient73, live7680, physical8258 and deficit2496 are unchanged. Maximum rank48
+satisfies73^2>2*48^2, so the retained depth2 invoice applies after recounting.
+
+The prefix bound counts72 complete invocations and72 individual bridge sweeps,
+plus local coefficient envelope20, input sup-to-L2 conversion37, and margin4:
+205 bits suffice and512 are charged. The row invoice pays another2*512.
+The exact assembly uses the inherited bad-cover allowance, two independent
+rational moment engines, six bootstrap steps and all47 strict outer inequalities.
+RESULT.json and evidence/EXACT-ASSEMBLY.json record the attained exponent.
+
+This establishes a sufficient conditional finite construction under the retained
+all-size compiler, mutable-basis/common-label, Lemma A realization, restored-row,
+precision/tape and analytic-transfer contracts and the pinned PR493 bit supplier.
+It is not a proof of those inherited contracts, a fresh Lean certification, or
+an unconditional faster integer-multiplication theorem.
diff --git a/research/coordinated-gauge-async/README.md b/research/coordinated-gauge-async/README.md
new file mode 100644
index 0000000000..78d6b92bf9
--- /dev/null
+++ b/research/coordinated-gauge-async/README.md
@@ -0,0 +1,72 @@
+# Coordinated gauge collapse with asynchronous whole-owner hosting
+
+Conditional kappa = **0.001372554837726243**, exactly `1372554837726243/1000000000000000000`.
+This is **2.161253% above the pinned PR501/502 whole-owner bound** and
+1.093987% above our preceding asynchronous construction, under the same retained
+supplier and analytic contracts. It does not exceed PR499's larger disputed-interface
+claim. No unconditional multiplication theorem or fresh Lean certification is claimed.
+
+Complex saving b = `1374441333828159774039/1000000000000000000000000`; unchanged cited PR493 bit saving
+c = `391360506745414227149/250000000000000000000000`. The complex supplier binds.
+
+## What is new
+
+Four rank4 entrance gauges are replaced together by one existing rank2 plane.
+Their shared target levels then drop together, saving costs that remain when any
+one gauge is changed alone. This is an actual reconstructed group change,
+not a sum of single-gauge estimates. It composes with PR495/501/502 body retiming
+and our asynchronous complete-invocation scheduling, now at offset6.
+
+- Helpers540..543 use the plane spanned by408 and102.
+- Targets81,82,84,87 fall from rank4 to rank2.
+- Local profile delta: -16 rank2 +4 rank4 +4 rank6.
+- Residual inventory:18 width7,104 width8,438 width9.
+- All72 calls use this same provider; no heterogeneous dispatch is required.
+- The exact global profile is `{1:171720,2:17856,3:41568,4:6336,5:23280,6:6912,7:1752,8:1224,48:432}`.
+
+Ambient73, live7680, physical8258 and deficit2496 remain unchanged. There are
+40320 explicitly assigned helper uses,2972 actual orthogonal73-column charts,
+9360 paid idle remainders and32113 cross-replica helper uses. Each helper stays
+whole on one physical owner and all800 families in every call are distinct.
+
+## Reproduce offline
+
+Python3.11 or later, standard library only. From this directory:
+
+```
+python -B verify.py --work /absolute/path/to/a-new-output-directory
+```
+
+Use an available drive path on Windows. The output must be outside this package
+and must not already exist. No optimizer, compiler, Lean installation, network or
+shell extension is required. Assertions must stay enabled. `bash verify.sh` is
+an optional convenience. The run takes several minutes; per-stage logs and
+STEPS.json are written in the chosen output directory.
+
+The verifier reconstructs the retimed word and coordinated gauges, checks all
+scalar columns and exact Gaussian phases, emits actual paid wrappers and the
+asynchronous physical program, independently checks ownership and chronology,
+recounts the prefix/invoice, and runs two exact rational moment engines and all
+47 strict outer constraints. It rejects19 controller corruptions and4 outer
+controls, including the next kappa grid point. The unchanged PR493 bit operator
+is a cited dependency; its published histogram is repriced, not its whole C++
+operator freshly reconstructed.
+
+Generated text uses UTF-8/LF via the documented process-scoped run_portable.py
+launcher. This prevents Windows CRLF from changing embedded adapter hashes.
+Frozen source bytes are covered by MANIFEST.json and `.gitattributes`.
+Two stale discovery floats and a redundant R field are excluded only from the
+constructor's semantic comparison after their exact irrelevance was checked;
+all program fields remain bound. See PORTABILITY.md.
+
+## Scope and research archive
+
+The inherited all-size compiler, mutable-basis/common-label, normalized Lemma A
+realization, restored-row, precision/tape and analytic-transfer contracts remain
+explicit conditions. Finite checks do not prove these contracts.
+
+PROOF.md states the sufficient composition. RESEARCH.md distinguishes achieved
+mechanisms from additional sufficient identities and bounded failed searches.
+SOURCES.json and SOURCE-FILES.json give credits and pins. Evidence records the
+fresh replay. The public snapshot was frozen through PR504 before packaging.
+No publishing or fork modification is performed by this package.
diff --git a/research/coordinated-gauge-async/RESEARCH.md b/research/coordinated-gauge-async/RESEARCH.md
new file mode 100644
index 0000000000..f9c38f1cba
--- /dev/null
+++ b/research/coordinated-gauge-async/RESEARCH.md
@@ -0,0 +1,57 @@
+# Research record and future routes
+
+The achieved numerical contribution is the coordinated four-gauge target-level
+collapse combined with public retiming and asynchronous whole-owner packing.
+See research-notes/RESEARCH-NOTE.md for the non-additive local identity and
+research-notes/SYNTHESIS-PROOF.md for its whole-profile selection.
+
+## Mathematical connections that were used
+
+1. Shared-maximum optimization: a complete group of gauge withdrawals releases
+   a common target level. Marginal, one-at-a-time search misses this saving.
+2. Order-preserving complete-block scheduling: replicas may progress at different
+   stages while each physical owner's actual pending interval is respected.
+   OpenAI WordScheduling and earlier controller work inspired this route.
+3. Response-matroid cancellation: exact donor relations cancel dirty-response
+   pivots. The retained42-pivot kernel is preserved while gauges and inventory
+   are reconstructed together. More aggressive local matroid variants were
+   checked but did not improve the complete price.
+4. Bipartite matching decomposes aggregate inventory nominations into literal
+   per-invocation owners. The frozen witness, not a solver status, establishes
+   that every helper fits and active data families remain distinct.
+
+These are combinations and applications of established ideas; no claim is made
+that matroids, matching, scheduling or shared maxima are newly invented theories.
+
+## Additional sufficient identities, not counted in kappa
+
+research-notes/PENDING-FACTOR-TRANSPORT.md proves the exact bridge intertwiner
+identity Q_y A=A Q_x and a sufficient translated-bridge compensation rule.
+This is the operator-transport equation familiar from tensor-isomorphism work.
+A safe specialization postpones complete terminal suffixes until all local
+calls finish. It is mathematically usable but yielded no extra price gain for
+this fixed inventory. No unconstructed bridge or unknown gadget enters the claim.
+
+## Bounded searches preserved
+
+- 382 proper single-gauge subframes:114 frame-valid cases,11 distinct inventory
+ types, no individually profitable full-profile candidate.
+- 20 coordinated choices:15 legal words; four rank4-to-rank2 replacements win.
+-Six combinations with additional response-matroid and gauge changes: no further
+ gain; all72 calls select the same winning provider.
+-K3/K5 controllers and heterogeneous portfolios: no better tested construction.
+-A stock-only numerical relaxation ignoring timing attained the same retimed
+ baseline profile as offset6. This is restricted solver evidence, not a theorem
+ of global optimality.
+-The native clean-row alias screen found56 same-value candidates and zero
+ compatible fixed-frame merges. It used a floating discovery screen and is not
+ an exact universal obstruction. Source and result are retained.
+
+Every discovery witness remains in the local round archive. This upload includes
+the winning portable reconstruction, concise screen receipts, the positive
+transport proof, and prior response-matroid reconstruction companions. The older
+research document is explicitly historical at references/PRIOR-RESEARCH.md;
+its old exponent is not this package's result.
+
+The OpenAI source revision and inspected paths are recorded in SOURCES.json and
+the prior research document. No Lean module was newly kernel-certified here.
diff --git a/research/coordinated-gauge-async/SNAPSHOT.md b/research/coordinated-gauge-async/SNAPSHOT.md
new file mode 100644
index 0000000000..2437b77ea1
--- /dev/null
+++ b/research/coordinated-gauge-async/SNAPSHOT.md
@@ -0,0 +1,12 @@
+# Frozen public comparison
+
+Snapshot: 81 recently created/updated PR conversation pages, through PR504.
+No public refresh is performed while this package is completed.
+
+- PR501/502: comparable retimed whole-owner construction, kappa0.001343518018117309.
+- PR473: draft stage-specific single-helper gauge withdrawal; inspiration for our joint change.
+- PR503: stronger bit-only draft; complex remains binding, so we retain pinned PR493.
+- PR504: draft read-then-climb alternative below the retained route; studied as a lead.
+- PR499: higher0.001452948511251494 claim inherits the disputed PR494 split-residual interface. We neither adopt nor claim to exceed it.
+
+Source heads are pinned in SOURCES.json. Reading a PR is not independent validation. The verifier reconstructs our changed complex derivative and reprices the cited bit histogram; it does not rerun every unchanged upstream artifact.
diff --git a/research/coordinated-gauge-async/SOURCE-FILES.json b/research/coordinated-gauge-async/SOURCE-FILES.json
new file mode 100644
index 0000000000..147dc3147a
--- /dev/null
+++ b/research/coordinated-gauge-async/SOURCE-FILES.json
@@ -0,0 +1,462 @@
+[
+  {
+    "path": ".gitattributes",
+    "sha256": "4ce4b6e6b8435f8c3a8387387732964a2ebf0183e8d77b8247ae8061b033e462",
+    "origin": "Unchanged preceding asynchronous whole-owner package; see SOURCES.json for upstream ancestry"
+  },
+  {
+    "path": "code/assemble.py",
+    "sha256": "9db68cad2738d76052b25c8bdf6485edd6f14025bf47107d2425b10a2bdf69e0",
+    "origin": "Unchanged preceding asynchronous whole-owner package; see SOURCES.json for upstream ancestry"
+  },
+  {
+    "path": "code/audit_async.py",
+    "sha256": "bbc2bd87ab2aa8f7711da15d410bffeb96b0176da0fbfb80c6625cd4c61e06ca",
+    "origin": "Current coordinated gauge and asynchronous synthesis round; source inspection and exact replay scope in README"
+  },
+  {
+    "path": "code/audit_async_invoice.py",
+    "sha256": "bcfdc0b4aae2257327b3a643d7dfc0c6de85f3f392b686aa76df64b0c519f907",
+    "origin": "Current coordinated gauge and asynchronous synthesis round; source inspection and exact replay scope in README"
+  },
+  {
+    "path": "code/build_rank2_collapse.py",
+    "sha256": "96695ffd96c81a5e06049cb61f91f5c69e1aa9a4223ada63986a2198a82d2847",
+    "origin": "Current coordinated gauge and asynchronous synthesis round; source inspection and exact replay scope in README"
+  },
+  {
+    "path": "code/build_retimed.py",
+    "sha256": "5177c0f1200d30fb532e58964d67d3cb08f7b9854ea97107f776f1883e36de97",
+    "origin": "Unchanged PR502 transfer constructor at pinned4d30b8322184598e65e90a92389e17e1baedd9eb"
+  },
+  {
+    "path": "code/emit_async.py",
+    "sha256": "a0ac854f279a538e289e74e04ce6b7d216215eb0ab1ea582d4ac98122bbf1ef9",
+    "origin": "Current coordinated gauge and asynchronous synthesis round; source inspection and exact replay scope in README"
+  },
+  {
+    "path": "code/run_portable.py",
+    "sha256": "01030376f8a3afb86c559bc5036d1524165fc3b4119bcca12cc7c6e1990a7548",
+    "origin": "Current coordinated gauge and asynchronous synthesis round; source inspection and exact replay scope in README"
+  },
+  {
+    "path": "evidence/ASYNC-AUDIT.json",
+    "sha256": "1aa61bed3ff2819cf5c1634d90df720cfda7a9cd32e570d752c3a6698a4405c1",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "evidence/EXACT-ASSEMBLY.json",
+    "sha256": "5455ed1f38c252a1e888a29357aa544ed449d2d4485bd485b6881aa48f4b5964",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "evidence/GAUGE-CONSTRUCTION.json",
+    "sha256": "10200234894ea7097c0f40b83b92c6a37185b7ed6a13e693cefdd658903fd980",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "evidence/GLOBAL-PREFIX.json",
+    "sha256": "8cf6fdbffc834d82703c7b1403cc4f670eec58362d965baec994f2a465aceae5",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "evidence/LOCAL-INDEPENDENT-REPLAY.json",
+    "sha256": "58cd590cf9690155163e86fc4bb60df004cdb02e80c9c0d3608aff7b277c7b2e",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "evidence/LOCAL-PHASE-REVIEW.json",
+    "sha256": "0ad76fc07e2931504b5f12277c6342107cce31f6be9cc29c3e474f0f86dc8c29",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "evidence/LOCAL-SCALAR-FRAME-AUDIT.json",
+    "sha256": "ff003df1bab281cfdc82acb5ed8fcc3d24cb60f65a579cf84a716bbfb711e251",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "evidence/OWNERS-AND-PROFILE.json",
+    "sha256": "90b024ed70f38ef14f03646d91a4104aab578fdbdd358505664c2fbcf54961a6",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "evidence/PACKAGE-REVIEW.json",
+    "sha256": "c1885a8fd7c0ccbab88273b5d45e7f5c627a83c5562bd38d455bd83938503eea",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "evidence/PACKAGE-REVIEW.md",
+    "sha256": "ac1979274d00d4dc436e4bcfc61ef61b1e0e0a6ab7ad2e3c1be22e6dce3e057f",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "evidence/PREFIX-AND-INVOICE.json",
+    "sha256": "b0c2f45bc2aebf61d1b071f8805996c51383cd338c6d25a6fbc06f85268ce23b",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "evidence/RESULT.json",
+    "sha256": "9be31086d73fb8b0dfa400173fad661b3b62c0407968a957b885e317d73a58e9",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "evidence/RETIMING-TRANSFER.json",
+    "sha256": "5b7140fd41f52d3ad8f0a73a53acebd0034b2704f2a7e78770657ddede26998e",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "evidence/RUN-STEPS.json",
+    "sha256": "f49c933305b32c1774a375fd74b9edf398d4efce75d1947600cc0138c15965b0",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "EXPECTED.json",
+    "sha256": "e077e7260eb4316d1fb94389f191a82f66399c0e3b0188e1581ca51037252fc7",
+    "origin": "Current synthesis verification receipt or publication documentation"
+  },
+  {
+    "path": "inputs/ASYNC-6.json",
+    "sha256": "bfa1ac46493e39c23007c3c2a4efbdc719de1f53b37ad3a4d9a3ecc112153ebe",
+    "origin": "Current coordinated gauge and asynchronous synthesis round; source inspection and exact replay scope in README"
+  },
+  {
+    "path": "inputs/ASYNC-OWNERS.json.gz",
+    "sha256": "27dde4588521328868af5cbd28216869d3fefb03c1ca8248e2e12cacd30c3c29",
+    "origin": "Current coordinated gauge and asynchronous synthesis round; source inspection and exact replay scope in README"
+  },
+  {
+    "path": "inputs/BASE-WORD.json.gz",
+    "sha256": "da4c5007092fd9456403179ef2a4391aef9fc5310e3eec5f2ab6c49b24147387",
+    "origin": "PR433 base before gauge80-kernel42; preserved cached input"
+  },
+  {
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+    "origin": "Current coordinated gauge and asynchronous synthesis round; source inspection and exact replay scope in README"
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+    "origin": "PR496 pinned vendor of PR484; unchanged"
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+    "origin": "PR496 pinned vendor of PR484; unchanged"
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+  },
+  {
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+  {
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+  },
+  {
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+    "origin": "PR496 pinned vendor of PR484; unchanged"
+  },
+  {
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+    "origin": "PR496 pinned vendor of PR484; unchanged"
+  },
+  {
+    "path": "vendor/complex/split-centres-global/code/complex_phases_full.py",
+    "sha256": "0f2795aaffafe0c64d77f3748ef2109a3c42c58da867e4cf269a0a8cb6109466",
+    "origin": "PR496 pinned vendor of PR484; unchanged"
+  },
+  {
+    "path": "vendor/complex/split-centres-global/vendor/clifford_exact.py",
+    "sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962",
+    "origin": "PR496 pinned vendor of PR484; unchanged"
+  },
+  {
+    "path": "verify.py",
+    "sha256": "c553dbf909d703e5afb0333d7da43481f5df18a71e932c23986d9bfd8b953e2e",
+    "origin": "Current coordinated gauge and asynchronous synthesis round; source inspection and exact replay scope in README"
+  },
+  {
+    "path": "verify.sh",
+    "sha256": "2b1b51f757d8ddf81941c8c96007a2256336856c2e017bcf6c3a37576c82bc8b",
+    "origin": "Unchanged preceding asynchronous whole-owner package; see SOURCES.json for upstream ancestry"
+  }
+]
diff --git a/research/coordinated-gauge-async/SOURCES.json b/research/coordinated-gauge-async/SOURCES.json
new file mode 100644
index 0000000000..5ea4ce58d7
--- /dev/null
+++ b/research/coordinated-gauge-async/SOURCES.json
@@ -0,0 +1,65 @@
+{
+  "snapshot": "81 recent/updated conversation pages downloaded; snapshot frozen through PR504 before packaging",
+  "direct": [
+    {
+      "pr": 496,
+      "head": "00cf70efd6a8637531deafef698157ca83e249ee",
+      "role": "whole-owner parent, invoice and assembly ancestry"
+    },
+    {
+      "pr": 484,
+      "head": "035845ce95f31d0a2dcfb42b8c8e13411271fa5f",
+      "role": "PR496 pinned gauge80-kernel42 parent, not newer moving head"
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+      "head": "4d30b8322184598e65e90a92389e17e1baedd9eb",
+      "role": "retiming transfer constructor and PR495 schedule fixture"
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+      "head": "1dc056496fdbbc74073ed26d948a2dc235dbf356",
+      "role": "parallel retiming integration; credited, no duplicated gain"
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+      "head": "628b454d3df00367d2d4f78de7d4eb9cbcbd47ca",
+      "role": "single-helper gauge withdrawal inspiration"
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+  "openai_math": {
+    "repository": "https://github.com/openai/math",
+    "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
+    "status": "source inspection and mathematical adaptation; no new Lean audit"
+  },
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+    497,
+    458,
+    442,
+    434,
+    433,
+    425,
+    417,
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+  ],
+  "excluded": {
+    "494": "under-review split-residual interface not used",
+    "491": "withdrawn GL sampling argument not used",
+    "499": "larger claim inherits disputed split-residual interface; not used"
+  }
+}
diff --git a/research/coordinated-gauge-async/code/assemble.py b/research/coordinated-gauge-async/code/assemble.py
new file mode 100644
index 0000000000..de2a375d13
--- /dev/null
+++ b/research/coordinated-gauge-async/code/assemble.py
@@ -0,0 +1,42 @@
+"""Two exact moment engines and strict outer composition of a new PR484 rehosting
+with the unchanged published PR493 bit supplier. Prepared by OpenAI Codex; arithmetic
+engines are pinned inherited PR361/Mercator and independent base-two code.
+Usage: assemble_candidate.py AUDIT_OUT PARENT_VARIANT HOSTED BIT_RESULT
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+import json,gzip,sys,hashlib,importlib.util
+O,V,H,B=map(Path,sys.argv[1:5]);sha=lambda p:hashlib.sha256(p.read_bytes()).hexdigest()
+def load(n):
+ p=V/'work/pr361/code/engines'/(n+'.py');s=importlib.util.spec_from_file_location('candidate_'+n,p);m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m
+M,Other,Outer=[load(n)for n in ['moment','base_two_moment','outer']]
+owner=json.loads((O/'OWNERS-AND-PROFILE.json').read_text());inv=json.loads((O/'PREFIX-AND-INVOICE.json').read_text());host=json.loads((H/'HOSTED.json').read_text());bit=json.loads(B.read_text())["eleven"];P={int(r):n for r,n in owner['paid_profile'].items()};m,W=73,owner['live'];assert m*W-sum(r*n for r,n in P.items())==2496
+rc=M.certify(P,m,W,True);b=Q(rc['lower']);bp=Q(rc['upper']);assert bp-b==Q(1,10**24)
+fallback=32*m*m*sum(P.values());receipts=[]
+for z,accept in[(b,True),(bp,False)]:
+ ml,mh=M.moment(P,m,W,z,True);a,c=Other.moment(m,W,list(P.items()),z);d,e=Other.moment(m,W,[(1,fallback)],z);a+=d/10**16;c+=e/10**16
+ assert (mh<1 and c<1)if accept else(ml>1 and a>1)
+ receipts.append(dict(endpoint=str(z),mercator=[str(ml-1),str(mh-1)],base_two=[str(a-1),str(c-1)],accept=accept))
+# Reprice the cited unchanged bit histogram, clearly separate from bit operator verification.
+bitP={int(r):n for r,n in bit['profile'].items()};bitc=Q(bit['bit_lower']);bitnext=bitc+Q(1,10**24)
+for zz,accepted in [(bitc,True),(bitnext,False)]:
+ bl,bh=M.moment(bitP,78,9600,zz,True)
+ ol,oh=Other.moment(78,9600,list(bitP.items()),zz)
+ fl,fh=Other.moment(78,9600,[(1,32*78*78*sum(bitP.values()))],zz)
+ assert (bh<1 and oh+fh/10**16<1) if accepted else (bl>1 and ol+fl/10**16>1)
+c=Q(bit['bit_lower']);eta=beta=Q(1,10**18);grid=Q(1,10**18);s=min(c,(1-beta)*b)-Q(1,10**24)
+assert 0<s<c and s<(1-beta)*b
+coeff=int(bit['bit']['charged_three_level_coefficient_bits']);row=max(20161,int(bit['bit']['row_coefficient']),inv['finite']['physical_row_coefficient']);chain=[Q(384599,10**10)];cuts=[]
+def ceil(x):return -(-x.numerator//x.denominator)
+for i in range(6):
+ old=chain[-1];a=(1-s)*s+s*old;gap=min(s-a,1-a-s,1-a-s*(1-old),1-s);assert old<a<s<1-a and gap>0
+ cut=max(ceil(36/gap**2),ceil(2*(4+coeff)/gap));assert cut*gap**2>=36 and cut*gap>=2*(4+coeff);cuts.append(dict(gap=str(gap),cutoff_log2=str(cut)));chain.append(a)
+q=chain[-1]*(1-2*eta);margin=(1-eta)*q/(1+q);ticks=margin/grid;kap=grid*((ticks.numerator-1)//ticks.denominator)
+bridge={'semantic':{'C0':{'base':2,'exponent':210000},'strict_literal_gap':1},'rows':{'degree_gap':Q(10**6)-Q(51*row,25)}}
+a=Outer.assembly(chain[-1],b,bridge,kap,eta=eta,beta=beta);assert len(a['strict_constraints'])==47 and all(x>0 for x in a['strict_constraints'].values());controls=[]
+for name,k,e,be in [('next_grid',kap+grid,eta,beta),('zero_eta',kap,Q(0),beta),('zero_beta',kap,eta,Q(0)),('unit_beta',kap,eta,Q(1))]:
+ try:Outer.assembly(chain[-1],b,bridge,k,eta=e,beta=be)
+ except AssertionError:controls.append(name)
+ else:raise AssertionError(name)
+res=dict(status='PASS_ASYNC_COMPLEX_AND_CITED_BIT_EXACT_ASSEMBLY',complex_lower=str(b),complex_decimal=rc['lower'],next_b_rejected=str(bp),exact_moment_receipts=receipts,bit_lower=str(c),bit_result_sha256=sha(B),bit_template_sha256=bit['emitted_program_sha256'],bit_prefix_bits=bit['bit']['charged_prefix_bits'],bit_charged_three_level_coefficient_bits=coeff,external_row_allowance=row,kappa=str(kap),kappa_decimal=f'{float(kap):.18f}',binding_component='bit'if c<(1-beta)*b else'complex',strict_outer_constraints={k:str(v)for k,v in a['strict_constraints'].items()},outer_controls=controls,bootstrap=cuts,complex_template_sha256=host['template_sha256'],owner_audit_sha256=sha(O/'OWNERS-AND-PROFILE.json'),invoice_sha256=sha(O/'PREFIX-AND-INVOICE.json'),bit_scope='PR493 published profile and operator reused as a pinned dependency; this verifier reprices its published profile but does not reconstruct that unchanged bit operator.',scope='Conditional finite witness. Inherited compiler, precision/tape and analytic-transfer contracts remain assumed; changed-host operator admission recorded separately.')
+(O/'EXACT-ASSEMBLY.json').write_text(json.dumps(res,indent=2)+'\n');print(res['status'],'b',res['complex_decimal'],'kappa',res['kappa_decimal'],flush=True)
diff --git a/research/coordinated-gauge-async/code/audit_async.py b/research/coordinated-gauge-async/code/audit_async.py
new file mode 100644
index 0000000000..edb8e4873f
--- /dev/null
+++ b/research/coordinated-gauge-async/code/audit_async.py
@@ -0,0 +1,149 @@
+"""OpenAI Codex independent serialized ASYNCHRONOUS controller/spectator audit.
+Imports no emitter, optimizer, expander, edge-coloring, or source hosting code.
+CLI: VERIFIED_VARIANT POOLED_HOSTED OUT.json
+The local provider is read from VERIFIED_VARIANT, never from a baseline width histogram.
+"""
+import sys,json,gzip,hashlib,copy
+from pathlib import Path
+from collections import Counter,defaultdict
+BASE,OUT,REPORT=map(lambda p:Path(p).resolve(),sys.argv[1:4]);read=lambda p:json.loads(gzip.decompress(p.read_bytes())if p.suffix=='.gz'else p.read_bytes())
+parent=read(BASE/'split-centres/GLOBAL-TEMPLATE.json.gz');canon=read(BASE/'split-centres/CANONICAL-WORD.json.gz');checks=read(BASE/'split-centres/LOCAL-CHECKS.json');ph=read(BASE/'work/complex-nine-native/DATA-PHASES.json');ct=read(OUT/'GLOBAL-TEMPLATE.json.gz');summary=read(OUT/'HOSTED.json');V=canon['v'];R=canon['R'];N=2*V+R;CENTRES=len(canon['centers']);assert V==120 and canon['h']==9; width=[9-len(canon['frames'][canon['start'][2*V+h]])for h in range(R)];prank={e['child']:e['rank']for e in parent['idle_children']};M=73;T=8;DATA=7680;PHYSICAL=DATA+CENTRES+R+2;CALLS=72;HOSTED=CALLS*R
+
+def orthobasis(port):
+ q=canon['ports'][port];z=next(i for i in range(9)if(q>>i)&1==0);a=q^(1<<z);order=list(range(9));order[z],order[0]=order[0],order[z]
+ b=[(1<<i)^(a if a>>i&1 else 0)for i in order]+[1<<i for i in range(9,73)]
+ assert len(b)==73 and all((x&y).bit_count()%2==int(i==j)for i,x in enumerate(b)for j,y in enumerate(b));return b
+bases=[orthobasis(p)for p in range(120)]
+# Bind the selected provider's actual normalization recipes, including newly
+# selected entrance frames; a baseline recipe registry cannot silently survive.
+required_artifacts={'split-centres/CANONICAL-WORD.json.gz','split-centres/LOCAL-PROGRAMS.json.gz','split-centres/LOCAL-CHECKS.json','split-centres/RESIDUAL-ADAPTERS.json','work/complex-nine-native/DATA-PHASES.json'}
+assert required_artifacts <= set(summary['source_artifacts'])
+adapter_path=BASE/'split-centres/RESIDUAL-ADAPTERS.json'
+assert hashlib.sha256(adapter_path.read_bytes()).hexdigest()==parent['residual_adapter_sha256']
+adapters=read(adapter_path)['recipes']
+for h,w in enumerate(width):
+ if w<9:
+  f=canon['start'][240+h];a=adapters[str(f)]
+  assert a['frame']==f and a['vectors']==canon['frames'][f] and a['rank']==w and a['entrance_rank']==9-w
+
+def audit(t):
+ assert(t['live'],t['physical'])==(DATA,PHYSICAL)
+ assert t['local_programs']==parent['local_programs']and t['recipe']==parent['recipe']and 'each local invocation completes before the next' in t['loop_order'].lower()
+ assert len(t['calls'])==72 and t['hosting']['max_helpers_per_physical_family_per_call']==1 and t['hosting']['banks']==[0]*9
+ for key in('exact_data_phase_recipe_sha256','exact_local_clifford_recipe_sha256','residual_adapter_sha256','adapter_policy'):assert t[key]==parent[key]
+ # Independently project the literal schedule onto every data replica.
+ src=parent['execution_schedule'];assert t['async_schedule']['source_schedule']==src
+ assert t['async_schedule']['source_call_order']==[[c['stage'],c['replica']]for c in parent['calls']]
+ order=[(c['stage'],c['replica'])for c in t['calls']]
+ assert [list(x)for x in order]==t['async_schedule']['order'] and len(set(order))==72
+ all_local=[];perrep={r:[]for r in range(8)};globals_=[]
+ for e in t['execution_schedule']:
+  j=e['source_event'];old=copy.deepcopy(src[j])
+  if 'replica'in e:
+   r=e['replica'];assert 0<=r<8 and j<30;perrep[r].append(j)
+   expected=old|dict(source_event=j,replica=r,replicas=[r],time=e['time'])
+   assert e==expected
+   assert 0<=e['time']<72 and order[e['time']][1]==r
+   if e['kind']=='local_cover_sweep':
+    assert order[e['time']]==(e['stage'],r);all_local.append((e['stage'],r,e['time']))
+  else:
+   globals_.append(j);expected=old|dict(source_event=j)
+   if e['kind']=='bank_padding_cover_sweep':expected['families']=[7680,7680]
+   if e['kind']=='translate_every_live_family':expected.update(families=[0,7680],charged_calls_per_cover=7680)
+   assert e==expected
+ assert all(v==list(range(30))for v in perrep.values()) and globals_==[30,31]
+ assert all_local==[(st,r,i)for i,(st,r)in enumerate(order)]
+ assert all('replica'in e for e in t['execution_schedule'][:-2])
+ assert [e['source_event']for e in t['execution_schedule'][-2:]]==[30,31]
+ # Local cover calls are indivisible; a block then runs the source events
+ # up to that replica's next local invocation.
+ block=-1
+ for e in t['execution_schedule'][:-2]:
+  if e['kind']=='local_cover_sweep':block+=1
+  assert e['time']==block
+ assert t['bank_padding']==dict(parent['bank_padding'],families=[])
+ assert t['cover_final_correction']==dict(parent['cover_final_correction'],families=[0,7680],charged_calls_per_cover=7680)
+ assert t['private_namespace']=={'centers':[DATA,DATA+CENTRES],'reserved_tableau':[DATA+CENTRES,DATA+CENTRES+R],'coefficient':[DATA+CENTRES+R,PHYSICAL]}
+ # Actual physical touch intervals include every individual bridge.
+ pairs={};sides={};windows={};last={};callpos={};stagepair={e['stage']:e['pair']for e in parent['recipe']if e['kind']=='local'}
+ for pos,e in enumerate(t['execution_schedule']):
+  if 'replica'not in e:continue
+  rep=e['replica']
+  if e['kind']=='local_cover_sweep':
+   pair=stagepair[e['stage']];callpos[e['stage'],rep]=pos;last[rep,pair]=pos
+  elif e['kind']=='bridge':last[rep,e['src']]=last[rep,e['dst']]=pos
+  elif e['kind']=='idle_cover_sweep':
+   for side,i in enumerate(e['child_indices']):
+    if i not in prank:continue
+    pairs[i]=e['pair'];sides[i]=side;windows[rep,i]=(last.get((rep,e['pair']),-1),pos)
+ used=defaultdict(set);roles={};paid=Counter();sigmas={};digest=hashlib.sha256();entering=0;cross=0;touches=defaultdict(list)
+ for call in t['calls']:
+  old=next(z for z in parent['calls']if(z['stage'],z['replica'])==(call['stage'],call['replica']))
+  assert {k:v for k,v in call.items()if k not in('owners','time')}=={k:v for k,v in old.items()if k!='owners'}
+  s,rep,pair=call['stage'],call['replica'],call['pair'];assert call['time']==t['async_schedule']['order'].index([s,rep]);own=call['owners'];assert len(own)==N and len({f for f,rt in own})==N and all(0<=f<DATA for f,rt in own)
+  assert own[:240]==old['owners'][:240]
+  for h,(family,route)in enumerate(own[240:]):
+   w=width[h];typ,i,off=route[:3];ownerrep,sub=divmod(family,960);fampair,sub=divmod(sub,240);side,port=divmod(sub,120)
+   assert windows[ownerrep,i][0]<callpos[s,rep]<windows[ownerrep,i][1] and fampair==pairs[i] and side==sides[i] and 0<=off and off+w<=prank[i]
+   B=bases[port];pend=[int(x)for x in ph['recipes'][port]['idle_children'][i]['orthogonal_axes']];assert len(pend)==prank[i]
+   occ=[B.index(v)for v in pend[off:off+w]];sp=[j for j in range(M)if j not in occ][:9-w]
+   assert route==([3,i,off]if w==9 else[4,i,off,w,sp]);assert len(set(occ+sp))==9
+   order=occ+sp+[j for j in range(M)if j not in occ and j not in sp];assert sorted(order)==list(range(M));sigma=[B[j]for j in order]
+   assert sigma[:w]==pend[off:off+w]and sigma[w:9]==[B[j]for j in sp]
+   sh=hashlib.sha256(json.dumps(sigma,separators=(',',':')).encode()).hexdigest();sigmas[sh]=sigma;digest.update(json.dumps([s,rep,h,family,sh],separators=(',',':')).encode()+b'\n')
+   key=(i,ownerrep,port);cells=set(range(off,off+w));assert not used[key]&cells;used[key]|=cells
+   role=(s,rep,h);assert role not in roles;roles[role]=dict(stage=s,replica=rep,helper=h,family=family,owner_replica=ownerrep,child=i,copy=ownerrep*120+port,port=port,offset=off,width=w)
+   touches[family].append((s,rep,h));entering+=w<9;cross+=rep!=ownerrep
+  for r,n in checks[call['orientation']]['paid_profile'].items():paid[int(r)]+=n
+ assert set(roles)=={(s,rep,h)for s in range(9)for rep in range(8)for h in range(R)}
+ # The frozen pooled metadata must bind the actual serialized owner maps.
+ assert len(t['hosting']['pooled_slots'])==HOSTED
+ for z in t['hosting']['pooled_slots']:
+  key=(z['stage'],z['replica'],z['helper']);assert key in roles
+  assert {k:z[k]for k in roles[key]}==roles[key]
+ assert len({(z['stage'],z['replica'],z['helper'])for z in t['hosting']['pooled_slots']})==HOSTED
+ rem={};wanted=[]
+ for old in parent['idle_children']:
+  i,port=old['child'],old['port']
+  for rep in range(T):
+   u=used[i,rep,port];assert u==set(range(len(u)));r=old['rank']-len(u)
+   if r:
+    f=rep*960+(2*pairs[i]+sides[i])*120+port;wanted.append(dict(child=i,port=port,replica=rep,replicas=1,family=f,rank=r,hosted=len(u),pending_offset=len(u)));paid[r]+=1
+ assert t['idle_children']==wanted
+ mass=sum(r*n for r,n in paid.items());assert73=73*7680-mass;assert assert73==2496 and 73**2>2*max(paid)**2
+ return dict(profile=dict(sorted(paid.items())),mass=mass,hosted=HOSTED,entering=entering,cross_replica=cross,sigma_digest=digest.hexdigest(),sigmas=sigmas,idle_remainders=len(wanted),busy_family_checks=len(roles),invocations=72)
+
+r=audit(ct);raw=gzip.decompress((OUT/'GLOBAL-TEMPLATE.json.gz').read_bytes());assert hashlib.sha256(raw).hexdigest()==summary['template_sha256'];assert r['profile']=={int(k):v for k,v in summary['global_profile'].items()};assert r['sigmas']==read(OUT/'ACTUAL-SIGMAS.json.gz');assert r['sigma_digest']==summary['sigma_stream_sha256']
+assert all(hashlib.sha256((BASE/name).read_bytes()).hexdigest()==value for name,value in summary['source_artifacts'].items())
+controls=[]
+def reject(name,mut):
+ q=copy.deepcopy(ct);mut(q)
+ try:audit(q)
+ except(AssertionError,KeyError,IndexError,ValueError,TypeError):controls.append(name)
+ else:raise AssertionError('ACCEPTED '+name)
+reject('wrong call time',lambda q:q['calls'][0].__setitem__('time',71))
+enter=lambda q:next(o for c in q['calls']for o in c['owners'][240:]if o[1][0]==4)
+reject('alias two whole owners',lambda q:q['calls'][0]['owners'].__setitem__(241,q['calls'][0]['owners'][240]))
+reject('helper on active data family',lambda q:q['calls'][0]['owners'][240].__setitem__(0,q['calls'][0]['owners'][0][0]))
+reject('owner in private namespace',lambda q:q['calls'][0]['owners'][240].__setitem__(0,7680))
+reject('unbound change of owner replica',lambda q:q['calls'][0]['owners'][240].__setitem__(0,(q['calls'][0]['owners'][240][0]+960)%7680))
+reject('omit complete invocation',lambda q:q['calls'].pop())
+reject('reorder invocation replicas',lambda q:q['calls'].__setitem__(0,q['calls'][1]))
+reject('interleave local invocations',lambda q:q.__setitem__('loop_order','interleaved local instructions'))
+reject('change local instruction stream',lambda q:q['local_programs']['forward'].pop())
+reject('wrong residual width',lambda q:enter(q)[1].__setitem__(3,enter(q)[1][3]-1))
+reject('wrong spectator coordinate',lambda q:enter(q)[1][4].__setitem__(0,72))
+reject('omit spectator route',lambda q:enter(q).__setitem__(1,enter(q)[1][:3]))
+reject('overlap residual footprint',lambda q:q['calls'][0]['owners'][240][1].__setitem__(2,100))
+reject('omit idle remainder',lambda q:q['idle_children'].pop())
+reject('underpay idle remainder',lambda q:q['idle_children'][0].__setitem__('rank',0))
+reject('omit bridge touch',lambda q:q['execution_schedule'].pop(next(i for i,e in enumerate(q['execution_schedule'])if e['kind']=='bridge')))
+reject('remove one-owner rule',lambda q:q['hosting'].pop('max_helpers_per_physical_family_per_call'))
+reject('duplicate frozen role',lambda q:q['hosting']['pooled_slots'].__setitem__(1,q['hosting']['pooled_slots'][0]))
+reject('understate physical reserve',lambda q:q.__setitem__('physical',PHYSICAL-1))
+report=dict(status='PASS_INDEPENDENT_ASYNC_WHOLE_OWNER_CONTROLLER_PHASES_AND_SPECTATORS',template_sha256=summary['template_sha256'],complex_template_sha256=summary['template_sha256'],complex_live=7680,complex_profile=r['profile'],complex_hosted_residuals=HOSTED,physical=PHYSICAL,rank_mass=r['mass'],deficit=2496,entering_helper_roles=r['entering'],cross_replica_helper_roles=r['cross_replica'],one_owner_per_complete_invocation=True,helper_busy_family_window_checks=r['busy_family_checks'],complete_invocations=r['invocations'],actual_orthogonal_73column_maps=len(r['sigmas']),actual_sigma_stream_sha256=r['sigma_digest'],all_actual_residual_and_spectator_charts=True,idle_remainders=r['idle_remainders'],mutations_rejected=controls,widths=dict(sorted(Counter(width).items())),scope='Independent exact asynchronous chronology, per-replica phase projection, owner/chart/paid-profile audit. Uses unchanged source local program and normalized residual-wrapper identities; all-size compilation and complete coefficient invoice are separate inherited obligations.')
+REPORT.write_text(json.dumps(report,indent=2)+'\n');print(json.dumps(report,indent=2))
+
+owners=dict(status='PASS_INDEPENDENT_ASYNC_OWNERS_AND_PROFILE',template_sha256=summary['template_sha256'],paid_profile=r['profile'],live=7680,physical=PHYSICAL,rank_mass=r['mass'],deficit=2496,hosted_residuals=HOSTED,banked_residuals=0,all_800_owners_per_call_distinct=True,actual_helper_owner_charts=HOSTED,actual_73column_sigmas=len(r['sigmas']),cross_replica_helper_roles=r['cross_replica'],explicit_idle_remainder_children=r['idle_remainders'],source_idle_children_charged=sum(x['replicas']for x in parent['idle_children']),source_artifacts=summary['source_artifacts'],controller_audit_sha256=hashlib.sha256(REPORT.read_bytes()).hexdigest())
+(REPORT.parent/'OWNERS-AND-PROFILE.json').write_text(json.dumps(owners,indent=2)+'\n')
+
diff --git a/research/coordinated-gauge-async/code/audit_async_invoice.py b/research/coordinated-gauge-async/code/audit_async_invoice.py
new file mode 100644
index 0000000000..c0cfbcad5e
--- /dev/null
+++ b/research/coordinated-gauge-async/code/audit_async_invoice.py
@@ -0,0 +1,100 @@
+"""Async PR484 scalar/prefix lowering and finite overcharge recount. OpenAI Codex.
+Reads emitted programs, canonical word and raw routes, importing no supplier code.
+Inherited all-size compiler admissibility is a condition, not inferred here.
+"""
+from pathlib import Path
+from fractions import Fraction as F
+from collections import Counter
+import gzip,json,math,hashlib
+import sys
+O=Path(sys.argv[1]);V=Path(sys.argv[2]);HOSTED=Path(sys.argv[3])
+def read(p):return json.loads(gzip.decompress(p.read_bytes())if p.suffix=='.gz'else p.read_bytes())
+w=read(V/'split-centres/CANONICAL-WORD.json.gz');p=read(V/'split-centres/LOCAL-PROGRAMS.json.gz');old=read(V/'split-centres/GLOBAL-TEMPLATE.json.gz');new=read(HOSTED/'GLOBAL-TEMPLATE.json.gz');integ=read(V/'split-centres/INTEGRATION.json');checks=read(V/'split-centres/LOCAL-CHECKS.json');prior=read(V/'split-centres/INDEPENDENT-REPLAY.json');owners=read(O/'OWNERS-AND-PROFILE.json')
+VV=w['v'];R=w['R'];N=2*VV+R;M=73;L=9;T=8;K=4
+assert VV==120 and w['h']==9
+PAYLOAD=512
+control=read(O/'ASYNC-AUDIT.json')
+raw=gzip.decompress((HOSTED/'GLOBAL-TEMPLATE.json.gz').read_bytes());th=hashlib.sha256(raw).hexdigest()
+assert th==control['complex_template_sha256']==owners['template_sha256']
+assert control['one_owner_per_complete_invocation'] and control['complete_invocations']==72
+assert new['local_programs']==old['local_programs'] and new['recipe']==old['recipe']
+assert Counter(c['orientation']for c in new['calls'])==Counter(c['orientation']for c in old['calls'])
+local={}
+for ori,evs in p.items():
+ rows={('L',i):{i:F(1)}for i in range(N)};env={('L',i):F(1)for i in range(N)};peak=F(1);real=macros=0;paid=Counter();workops=0
+ for e in evs:
+  if e[0]=='climb':paid[e[5]]+=1
+  elif e[0]=='center':paid[e[4]]+=1
+  elif e[0]=='scalar':
+   macros+=1;x=e[1];op=x[0];a=tuple(x[1]);count=0
+   if op=='clear':rows[a]={};env[a]=F(0);count=1
+   elif op=='copy':rows[a]=dict(rows[tuple(x[2])]);env[a]=env[tuple(x[2])];count=1
+   elif op=='scale':
+    q=F(x[2],x[3]);num,den=abs(q.numerator),q.denominator
+    assert num and num&(num-1)==0 and den&(den-1)==0
+    rows[a]={j:q*v for j,v in rows[a].items()};env[a]*=abs(q)
+    count=abs(num.bit_length()-den.bit_length())+int(q<0)
+   elif op=='add':
+    b=tuple(x[2]);q=F(x[3],x[4]);num,den=abs(q.numerator),q.denominator
+    assert a!=b and num>0 and den&(den-1)==0
+    # Lower actual coefficient to cleared Horner work. Every intermediate
+    # magnitude envelope is explicitly included, before denominator halvings.
+    acc=0;count=1
+    for i,ch in enumerate(bin(num)[2:]):
+     if i:acc*=2;count+=1;peak=max(peak,acc*env[b])
+     if ch=='1':acc+=1;count+=1;peak=max(peak,acc*env[b])
+    v=F(acc)
+    for i in range(den.bit_length()-1):v/=2;count+=1;peak=max(peak,v*env[b])
+    assert v==abs(q);count+=2 # signed accumulation and private work clear
+    env[a]=env.get(a,F(0))+abs(q)*env[b];rr=rows.setdefault(a,{})
+    for j,z in list(rows[b].items()):
+     value=rr.get(j,F(0))+q*z
+     if value:rr[j]=value
+     else:rr.pop(j,None)
+    workops+=count
+   else:raise AssertionError(op)
+   real+=2*count;peak=max(peak,env[a])
+ assert all(not row for reg,row in rows.items()if reg[0]=='C')
+ for i in range(N):
+  expected={i:F(1)}
+  if ori=='forward'and 120<=i<240:expected[i-120]=F(1)
+  if ori=='backward'and i<120:expected[i+120]=F(-1)
+  assert rows['L',i]==expected
+ assert peak==F(prior['local'][ori]['prefix_l1_upper']) and peak<2**192
+ assert real==checks[ori]['real_component_primitive_bound']and macros==checks[ori]['scalar_macros']
+ assert dict(paid)=={int(r):z for r,z in checks[ori]['paid_profile'].items()}
+ local[ori]=dict(events=len(evs),macros=macros,real_component_primitives=real,prefix_l1=str(peak),prefix_bits=peak.numerator.bit_length()-peak.denominator.bit_length()+1,coefficient_work_primitive_count=workops,all_exact_columns=N,all_560_dirty_helper_columns_restored=True,private_centres_and_coefficient_work_cleared=True)
+centres=len(w['centers']);widths=[9-len(w['frames'][f])for f in w['start'][240:]];enter=sum(x<9 for x in widths)
+children=sum(checks['forward']['paid_profile'].values());macros=max(x['macros']for x in local.values());compiler=64*(M+1)**3
+adapters=16*(macros+children+N+centres)+64*M
+primitive_bound=max(x['real_component_primitives']for x in local.values())+adapters*compiler
+bridges=sum(e['kind']=='bridge'for e in old['recipe']);bridgeadd=2*VV*T*bridges;assert sum(e['kind']=='bridge'for e in new['execution_schedule'])==T*bridges
+shifts=K*VV*T;moves=2*K*VV*T
+oldW=old['live'];oldP=old['physical'];oldNB=(oldW-2*VV*K*T)//L
+scalar=sum(local[c['orientation']]['real_component_primitives']for c in old['calls'])
+nonlocal_items=4*VV*T*L+2*VV*T*L+2*L*T*R+sum(e['replicas']for e in old['idle_children'])+L*oldNB+bridgeadd+shifts+moves+oldW+oldP+2*L*T*enter*144
+outside=8*nonlocal_items+64*M
+pervertex=scalar+bridgeadd+shifts+moves+oldW+(72*adapters+outside)*compiler+64*oldP
+invoice=integ['finite_invoice'];assert (primitive_bound,adapters,pervertex,compiler)==(invoice['local_primitive_bound'],invoice['local_adapters'],invoice['pervertex'],invoice['compiler'])
+# Independent hosted stock rebase, retaining the larger unhosted operation
+# overcharge; only the stock and mass in the common router/literal change.
+W=new['live'];P=new['physical'];mass=owners['rank_mass'];profile=owners['paid_profile']
+assert M*W-mass==2496 and P==W+R+centres+2
+rebase=pervertex-oldP*65+P*65-oldW+W;used=max(rebase,pervertex)
+r=M//2;group=2**(r*r)
+for i in range(1,r+1):group*=2**(2*i)-1
+stock=group*P;smass=group*mass;logical=group*used+64;router=compiler*(logical+1)*(stock+1)**2
+G,E,C0=2**30000,2**100000,2**210000;B=smass+E;literal=2*G*stock**2+8*smass+4*stock+4+32*M;row=2*stock.bit_length()+9909+252+2*PAYLOAD;maxchild=max(map(int,profile))
+ineq=dict(local_primitive_bound=primitive_bound<2**48,logical=logical<2**6050,router=router<G,literal=literal<E,mass=B<2**100001,precision=32*M*B*B<C0,tape=2*B+18<C0,row=row<20161,halving=M*M>2*maxchild*maxchild)
+assert all(ineq.values())
+localbits=max(x['prefix_bits']for x in local.values());input_l2_bits=(M+1)//2
+bridge_sweeps=sum(e['kind']=='bridge'for e in new['execution_schedule'])
+required_prefix=localbits+input_l2_bits+len(new['calls'])+bridge_sweeps+4
+assert localbits==20 and bridge_sweeps==72 and required_prefix==205 and required_prefix<PAYLOAD
+assert owners['explicit_idle_remainder_children']<=owners['source_idle_children_charged']
+assert new['physical']<=old['physical'] and new['live']<=old['live'] and not new['bank_padding']['families']
+out=dict(status='PASS_INDEPENDENT_ASYNC_SCALAR_PREFIX_AND_FINITE_INVOICE',template_sha256=th,global_prefix_bits=required_prefix,charged_prefix_bits=PAYLOAD,complete_invocation_sweeps=72,bridge_sweeps=bridge_sweeps,row_prefix_extra=2*PAYLOAD,local=local,entering_helpers=enter,entrance_rank=sum(9-x for x in widths),local_adapters=adapters,local_primitive_bound=primitive_bound,wrapper_elementary_factors_charged=2*72*enter*144,compiler=compiler,unhosted_pervertex=pervertex,hosted_rebased_pervertex=rebase,conservative_pervertex_used=used,finite=dict(physical_stock_bits=stock.bit_length(),router_bits=router.bit_length(),literal_bits=literal.bit_length(),physical_row_coefficient=row,retained_row_coefficient=20161,required_C0_bits=(32*M*B*B).bit_length(),halving_integer_inequality=[M*M,2*maxchild*maxchild],G_exponent=30000,E_exponent=100000,C0_exponent=210000),strict_checks=ineq,scope='Exact finite recount conditional on inherited all-size compiler/precision/tape and analytic-transfer contracts. Signed accumulation uses the supplier scalar primitive convention. Async prefix counts all72individual bridges and72complete invocations;512bits are charged, and an additional2*512 enters the two-level physical row bound.')
+(O/'PREFIX-AND-INVOICE.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2),flush=True)
+
+prefix=dict(status='PASS_ASYNC_GLOBAL_PREFIX',template_sha256=th,global_prefix_bits=required_prefix,retained_payload_bits=PAYLOAD,complete_invocation_cover_sweeps=len(new['calls']),bridge_factor_two_sweeps=bridge_sweeps,local_scalar_coefficient_work_prefix_bits=localbits,stream_input_sup_to_l2_bits=input_l2_bits,primitive_arithmetic_margin_bits=4,controller_audit_sha256=hashlib.sha256((O/'ASYNC-AUDIT.json').read_bytes()).hexdigest(),proof='Each complete normalized invocation has unitary helper endpoints and a data shear with at most two unitary summands. Distinct full orthogonal owner charts act by right-translation bijections on the retained cover group. Thus the maximum full-stream L2 norm grows by at most2per complete local cover sweep and individual bridge. All72local sweeps and72individual bridge sweeps are counted. Unit-modulus phases, idle children, terminal translations and normalized spectator endpoints are unitary. The independently replayed scalar/Horner envelope costs20bits, initial sup-to-L2 conversion37bits, primitive margin4bits: total205<512. All-size child/compiler and precision/tape interfaces remain inherited.')
+(O/'GLOBAL-PREFIX.json').write_text(json.dumps(prefix,indent=2)+'\n')
diff --git a/research/coordinated-gauge-async/code/build_rank2_collapse.py b/research/coordinated-gauge-async/code/build_rank2_collapse.py
new file mode 100644
index 0000000000..bdc52df699
--- /dev/null
+++ b/research/coordinated-gauge-async/code/build_rank2_collapse.py
@@ -0,0 +1,35 @@
+"""Reconstruct the coordinated rank-two gauge collapse after PR502 body retiming.
+Apache-2.0. Prepared with OpenAI Codex assistance.
+Usage: python -B build_rank2_collapse.py GK_CODE RETIMED_BASE BASELINE_SELECTION OUTPUT
+The body retiming must be independently source-bound by its retained constructor.
+"""
+from pathlib import Path
+import json,gzip,sys,hashlib,copy
+from collections import Counter
+if not __debug__:raise RuntimeError('Assertions must remain enabled')
+code,base_path,sel_path,out=map(Path,sys.argv[1:]);assert not out.exists();sys.dont_write_bytecode=True;sys.path.insert(0,str(code));import gk_word as g
+base=json.loads(gzip.decompress(base_path.read_bytes()));s=json.loads(sel_path.read_bytes());before=g.build(base,s);old=copy.deepcopy(s)
+for h in ('540','541','542','543'):
+ assert len(s['gauge'][h])==4
+ assert g.span([408,102])<=g.span(s['gauge'][h])
+ s['gauge'][h]=[408,102]
+word=g.build(base,s);H0=g.profile(before);H=g.profile(word);delta={r:H[r]-H0[r]for r in set(H)|set(H0)if H[r]!=H0[r]}
+assert delta=={2:-16,4:4,6:4},delta
+changed={int(t):(len(E),len(word['target_levels'].get(t,[])))for t,E in before['target_levels'].items()if E!=word['target_levels'].get(t,[])}
+assert changed=={81:(4,2),82:(4,2),84:(4,2),87:(4,2)},changed
+assert Counter(g.widths_of(word))=={7:18,8:104,9:438}
+assert old['kernel']==s['kernel']
+for field in ('B','cut','ports','centers','center_frames','precondition'):assert word[field]==before[field],field
+# Two inherited discovery floats differ in the independently reconstructed base;
+# optional top-level R duplicates report.R. Every other field remains hash-bound.
+normalized=copy.deepcopy(word)
+assert normalized.get('R', normalized['report']['R']) == normalized['report']['R'] == 560
+normalized.pop('R',None)
+for key in ('complex_coarse_float','gain_percent_float'):
+    normalized['report'].pop(key,None)
+semantic_sha=hashlib.sha256(json.dumps(normalized,sort_keys=True,separators=(',',':')).encode()).hexdigest()
+assert semantic_sha=='879c0e6ade3f1c84a515609a848dc917bddd1c1499e0cc37b2862809dfb8655b'
+raw=g.dump(word)
+out.mkdir(parents=True);(out/'FULL-WORD.json.gz').write_bytes(raw);(out/'SELECTION.json').write_text(json.dumps(s,indent=2)+'\n')
+receipt=dict(status='PASS_EXACT_COORDINATED_GAUGE_RECONSTRUCTION',source_hashes={'retimed_base':hashlib.sha256(base_path.read_bytes()).hexdigest(),'baseline_selection':hashlib.sha256(sel_path.read_bytes()).hexdigest(),'gk_word.py':hashlib.sha256((code/'gk_word.py').read_bytes()).hexdigest(),'gk_common.py':hashlib.sha256((code/'gk_common.py').read_bytes()).hexdigest()},word_sha256=hashlib.sha256(raw).hexdigest(),normalized_word_sha256=semantic_sha,normalization_ignored_metadata=['optional redundant R','report.complex_coarse_float','report.gain_percent_float'],changed_target_dimensions=changed,local_profile=dict(H),delta=delta,widths=dict(Counter(g.widths_of(word))),scope='Exact local construction and frame nesting. Scalar, Gaussian phase, physical ownership, prefix and outer assembly independently required.')
+(out/'CONSTRUCTION.json').write_text(json.dumps(receipt,indent=2)+'\n');print(receipt['status'])
diff --git a/research/coordinated-gauge-async/code/build_retimed.py b/research/coordinated-gauge-async/code/build_retimed.py
new file mode 100644
index 0000000000..8ed4e8f141
--- /dev/null
+++ b/research/coordinated-gauge-async/code/build_retimed.py
@@ -0,0 +1,90 @@
+"""Transfer PR495's main-word frame schedule onto PR484's frozen gauge selection.
+
+Prepared by OpenAI Codex. Apache-2.0.
+Usage: build_retimed.py COMPLEX484_PACKAGE BASE_WORD RETIMED495_WORD OUT
+The caller must independently replay the resulting physical programs. This
+builder certifies the structural transfer, not the inherited all-size contracts.
+"""
+from pathlib import Path
+import copy
+import gzip
+import hashlib
+import json
+import sys
+
+if not __debug__:
+    raise RuntimeError('assertions must remain enabled')
+assert len(sys.argv) == 5, __doc__
+package, base_path, schedule_path, out = map(lambda s: Path(s).resolve(), sys.argv[1:])
+assert not out.exists()
+sys.path.insert(0, str(package / 'code'))
+import gk_word as G
+
+
+def read(path):
+    raw = path.read_bytes()
+    return json.loads(gzip.decompress(raw) if path.suffix == '.gz' else raw)
+
+
+def transfer(base, candidate):
+    word = copy.deepcopy(base)
+    changed, prefixes = {}, {}
+    for part in ('A', 'B'):
+        assert len(candidate[part]) >= len(base[part])
+        tail = candidate[part][-len(base[part]):]
+        # Discard the separate PR495 entrance kernel. Only the frame field of
+        # the identical original scalar word is permitted to cross this gate.
+        assert [e[:-1] for e in tail] == [e[:-1] for e in base[part]], part
+        changed[part] = sum(x[-1] != y[-1] for x, y in zip(base[part], tail))
+        prefixes[part] = len(candidate[part]) - len(base[part])
+        word[part] = copy.deepcopy(tail)
+    word['cleanup'] = [G.inverse(e, G.FULL) for e in reversed(word['A'] + word['B'])
+                       if not G.V <= e[1] < 2 * G.V]
+    return word, changed, prefixes
+
+
+base, candidate = read(base_path), read(schedule_path)
+assert len(base['A']) == 2736 and len(base['B']) == 840
+selection_path = package / 'data/selections/gauge80-kernel42.json'
+selection = read(selection_path)
+word, changed, discarded = transfer(base, candidate)
+assert changed == {'A': 224, 'B': 504}
+assert discarded == {'A': 229, 'B': 0}
+original_gauge = G.build(base, selection)
+retimed_gauge = G.build(word, selection)
+preserved = ['centers', 'center_frames', 'target_levels', 'root_lines', 'ports',
+             'cut', 'precondition', 'start_extra', 'scatter']
+assert all(original_gauge[k] == retimed_gauge[k] for k in preserved)
+assert len(retimed_gauge['start_extra']) == 122
+assert sum(map(len, retimed_gauge['start_extra'].values())) == 148
+
+# Check that the transfer gate cannot smuggle a changed scalar coefficient or
+# register through the otherwise identical main-word tail.
+controls = []
+for part, field in [('A', 1), ('B', 3)]:
+    bad = copy.deepcopy(candidate)
+    event = bad[part][-len(base[part])]
+    event[field] += 1
+    try:
+        transfer(base, bad)
+    except AssertionError:
+        controls.append([part, field])
+    else:
+        raise AssertionError('changed scalar field accepted')
+assert len(controls) == 2
+
+out.mkdir(parents=True)
+for name, value in [('RETIMED-BASE.json.gz', word), ('FULL-WORD.json.gz', retimed_gauge)]:
+    (out / name).write_bytes(G.dump(value))
+receipt = dict(
+    status='PASS_STRUCTURAL_MAIN_WORD_RETIMING_TRANSFER',
+    source_files={str(p): hashlib.sha256(p.read_bytes()).hexdigest()
+                  for p in [base_path, schedule_path, selection_path]},
+    changed_frames=changed, discarded_prefix_events=discarded,
+    preserved_gauge_fields=preserved, entering_helpers=122, entrance_rank=148,
+    local_profile=retimed_gauge['report']['local_profile'],
+    transfer_controls_rejected=controls,
+    outputs={p.name: hashlib.sha256(p.read_bytes()).hexdigest() for p in out.iterdir()},
+    scope='Structural transfer; scalar, frame, phase, whole-program, hosting and finite assembly checks must run independently.')
+(out / 'TRANSFER.json').write_text(json.dumps(receipt, indent=2) + '\n')
+print(receipt['status'], changed, flush=True)
diff --git a/research/coordinated-gauge-async/code/emit_async.py b/research/coordinated-gauge-async/code/emit_async.py
new file mode 100644
index 0000000000..8e9895ac21
--- /dev/null
+++ b/research/coordinated-gauge-async/code/emit_async.py
@@ -0,0 +1,89 @@
+"""OpenAI Codex whole-owner cross-replica derivative of pinned PR484.
+CLI: SOURCE484_PACKAGE VERIFIED_VARIANT ASYNC-OWNERS.json.gz ASYNC-PLAN.json OUT
+No optimizer or unpublished word builder runs. The frozen map explicitly names
+one whole helper per family per complete invocation, plus borrowed spectators.
+"""
+import sys,json,gzip,hashlib,copy
+from pathlib import Path
+from collections import Counter,defaultdict
+SOURCE,BASE,MAP,PLAN,OUT=map(lambda p:Path(p).resolve(),sys.argv[1:6]);assert not OUT.exists();OUT.mkdir(parents=True)
+sys.path.insert(0,str(SOURCE/'code'));import onehost_gk as OH
+read=lambda p:json.loads(gzip.decompress(p.read_bytes())if p.suffix=='.gz'else p.read_bytes())
+plan=read(PLAN);call_order=[tuple(x)for x in plan['order']];assert len(call_order)==len(set(call_order))==72;at={x:i for i,x in enumerate(call_order)}
+parent=read(BASE/'split-centres/GLOBAL-TEMPLATE.json.gz');t=copy.deepcopy(parent);canon=read(BASE/'split-centres/CANONICAL-WORD.json.gz');checks=read(BASE/'split-centres/LOCAL-CHECKS.json');ph=read(BASE/'work/complex-nine-native/DATA-PHASES.json');roles=read(MAP);DATA=7680;T=8;V=120;M=73;R=560
+assert(len(roles),len(canon['start']),len(ph['recipes']))==(40320,800,120)
+widths=[9-len(canon['frames'][canon['start'][240+h]])for h in range(R)];assert widths==plan['baseline_widths'], 'source-bound selected provider widths'
+rank={z['child']:z['rank']for z in parent['idle_children']};side={};pairs={}
+for e in parent['execution_schedule']:
+ if e['kind']=='idle_cover_sweep':
+  for sd,ch in enumerate(e['child_indices']):side[ch]=sd;pairs[ch]=e['pair']
+bycall=defaultdict(dict);used=defaultdict(set);sigmas={};digest=hashlib.sha256();entering=0
+for z in roles:
+ s,rep,h=z['stage'],z['replica'],z['helper'];i,cp,off,w,f=z['child'],z['copy'],z['offset'],z['width'],z['family'];orp,port=divmod(cp,V)
+ assert 0<=s<9 and 0<=rep<T and 0<=h<R and w==widths[h] and 0<=orp<T
+ assert f==orp*960+(2*pairs[i]+side[i])*120+port
+ assert(h not in bycall[s,rep])and off>=0 and off+w<=rank[i]
+ assert z['time']==at[s,rep] and z['time'] in plan['windows'][f'{orp},{i}'][1]
+ B=OH.idle_basis(canon['ports'][port]);pending=list(map(int,ph['recipes'][port]['idle_children'][i]['orthogonal_axes']));assert len(pending)==rank[i]
+ occ=[B.index(v)for v in pending[off:off+w]];sp=OH.spectators(canon['ports'][port],pending,off,w)if w<9 else[]
+ order=occ+sp+[j for j in range(M)if j not in occ and j not in sp];assert len(order)==len(set(order))==M
+ sigma=[B[j]for j in order];sigmahash=hashlib.sha256(json.dumps(sigma,separators=(',',':')).encode()).hexdigest();sigmas[sigmahash]=sigma
+ route=[4,i,off,w,sp]if w<9 else[3,i,off];bycall[s,rep][h]=[f,route];entering+=w<9
+ cells=set(range(off,off+w));assert not used[i,orp,port]&cells;used[i,orp,port]|=cells
+ digest.update(json.dumps([s,rep,h,f,sigmahash],separators=(',',':')).encode()+b'\n')
+for c in t['calls']:
+ k=(c['stage'],c['replica']);assert set(bycall[k])==set(range(R))
+ c['owners']=c['owners'][:240]+[bycall[k][h]for h in range(R)];assert len({f for f,r in c['owners']})==800 and all(f<DATA for f,r in c['owners'])
+newidle=[];paid=Counter()
+for c in t['calls']:
+ for r,n in checks[c['orientation']]['paid_profile'].items():paid[int(r)]+=n
+for old in parent['idle_children']:
+ i,port=old['child'],old['port']
+ for rep in range(T):
+  u=used[i,rep,port];assert u==set(range(len(u)));left=old['rank']-len(u)
+  if left:
+   f=rep*960+(2*pairs[i]+side[i])*120+port;newidle.append(dict(child=i,port=port,replica=rep,replicas=1,family=f,rank=left,hosted=len(u),pending_offset=len(u)));paid[left]+=1
+assert M*DATA-sum(r*n for r,n in paid.items())==2496
+C=len(canon['centers']);physical=DATA+C+R+2
+assert parent['physical']-parent['live']==C+R+2
+# Local circuits, exact phase recipes, normalized wrappers and data routes stay
+# byte-identical; only full-family helper placement and idle remainders change.
+t.update(version='complex484-asynchronous-replica-whole-owner-v1',live=DATA,physical=physical,idle_children=newidle,private_namespace={'centers':[DATA,DATA+C],'reserved_tableau':[DATA+C,DATA+C+R],'coefficient':[physical-2,physical]})
+t['hosting']=dict(rule='one-helper-per-physical-family-per-complete-invocation',max_helpers_per_physical_family_per_call=1,scope='asynchronously interleaved data replicas; each whole helper occurs once per(stage,invocation replica)',pooled_slots=roles,banks=[0]*9,owner_map_sha256=hashlib.sha256(MAP.read_bytes()).hexdigest(),idle_remainder_semantics='Each idle_children item acts once, on its explicit replica/family, at pending_offset with the stated rank; replicas=1.')
+t['route_semantics']['helper']='Whole physical family may belong to another replica; actual orthogonal port basis followed by residual/spectator coordinate permutation. Entering helpers have paid wrappers around the complete invocation; at most one helper and no active data wire uses that family in that invocation. Route[4,child,offset,width,spectators], width9 route[3,child,offset].'
+t['bank_padding']['families']=[];t['cover_final_correction']['families']=[0,DATA];t['cover_final_correction']['charged_calls_per_cover']=DATA
+for e in t['execution_schedule']:
+ if e['kind']=='bank_padding_cover_sweep':e['families']=[DATA,DATA]
+ elif e['kind']=='translate_every_live_family':e.update(t['cover_final_correction'])
+# Expand the retained controller into one-replica blocks; each block consists
+# of its full local cover sweep and every following replica-local event before
+# its next local sweep. Final whole-stock correction remains after all replicas.
+blocks={};globals_=[];st=None
+for ei,e in enumerate(parent['execution_schedule']):
+ if e['kind']in('bank_padding_cover_sweep','translate_every_live_family'):
+  globals_.append((ei,e));continue
+ if e['kind']=='local_cover_sweep':st=e['stage'];blocks[st]=[]
+ assert st is not None
+ blocks[st].append((ei,e))
+new_schedule=[]
+for tm,(s,rep)in enumerate(call_order):
+ for ei,e in blocks[s]:
+  ne=copy.deepcopy(e);ne.update(replica=rep,replicas=[rep],source_event=ei,time=tm)
+  new_schedule.append(ne)
+for ei,e in globals_:
+ ne=copy.deepcopy(e);ne['source_event']=ei
+ if e['kind']=='bank_padding_cover_sweep':ne['families']=[DATA,DATA]
+ else:ne.update(t['cover_final_correction'])
+ new_schedule.append(ne)
+t['execution_schedule']=new_schedule
+cmap={(c['stage'],c['replica']):c for c in t['calls']}
+t['calls']=[dict(cmap[s,rep],time=tm)for tm,(s,rep)in enumerate(call_order)]
+t['loop_order']='Follow the explicit asynchronous execution_schedule. Each local invocation completes before the next, including its entire cover sweep and normalization wrappers. Per-replica source event order is preserved; each idle remainder executes before that replica next touches its family.'
+t['async_schedule']=dict(order=[list(x)for x in call_order],source_plan_sha256=hashlib.sha256(PLAN.read_bytes()).hexdigest(),source_schedule=parent['execution_schedule'],source_call_order=[[c['stage'],c['replica']]for c in parent['calls']],scope='Only independent replica controllers are interleaved. Owner windows are measured on the owner replica timeline; no within-cover or within-invocation interleaving.')
+assert dict(sorted(paid.items()))=={int(k):v for k,v in plan['profile'].items()}
+assert M*M>2*max(paid)**2
+
+assert t['local_programs']==parent['local_programs']
+raw=json.dumps(t,sort_keys=True,separators=(',',':')).encode();(OUT/'GLOBAL-TEMPLATE.json.gz').write_bytes(gzip.compress(raw,mtime=0))
+res=dict(status='EMITTED_ASYNCHRONOUS_WHOLE_OWNER_COMPLEX_TEMPLATE',template_sha256=hashlib.sha256(raw).hexdigest(),parent_template_sha256=hashlib.sha256(gzip.decompress((BASE/'split-centres/GLOBAL-TEMPLATE.json.gz').read_bytes())).hexdigest(),live=DATA,physical=physical,global_profile=dict(sorted(paid.items())),rank_mass=sum(r*n for r,n in paid.items()),deficit=2496,maxchild=max(paid),hosted_residuals=len(roles),banks_per_stage=[0]*9,entering_helper_roles=entering,unique_actual_73column_sigmas=len(sigmas),sigma_stream_sha256=digest.hexdigest(),cross_replica_helper_roles=sum(z['replica']!=z['owner_replica']for z in roles),asynchronous_order_sha256=hashlib.sha256(json.dumps(call_order,separators=(',',':')).encode()).hexdigest(),source_artifacts={p.relative_to(BASE).as_posix():hashlib.sha256(p.read_bytes()).hexdigest()for p in [BASE/'split-centres/CANONICAL-WORD.json.gz',BASE/'split-centres/LOCAL-PROGRAMS.json.gz',BASE/'split-centres/LOCAL-CHECKS.json',BASE/'split-centres/RESIDUAL-ADAPTERS.json',BASE/'work/complex-nine-native/DATA-PHASES.json']},scope='New emitted complete-invocation placement; independent controller/spectator/normalization and complete finite invoice admission required.')
+(OUT/'HOSTED.json').write_text(json.dumps(res,indent=2)+'\n');(OUT/'ACTUAL-SIGMAS.json.gz').write_bytes(gzip.compress(json.dumps(sigmas,separators=(',',':')).encode(),mtime=0));print(json.dumps(res,indent=2))
diff --git a/research/coordinated-gauge-async/code/run_portable.py b/research/coordinated-gauge-async/code/run_portable.py
new file mode 100644
index 0000000000..dcfe7128e1
--- /dev/null
+++ b/research/coordinated-gauge-async/code/run_portable.py
@@ -0,0 +1,15 @@
+"""Run one retained verifier with UTF-8/LF text outputs on every platform.
+This changes output serialization only; no mathematical operation is replaced.
+"""
+from pathlib import Path
+import runpy,sys
+if not __debug__:
+    raise RuntimeError('Assertions must remain enabled')
+original_write_text=Path.write_text
+def write_text_lf(self,data,encoding=None,errors=None,newline=None):
+    return original_write_text(self,data,encoding=encoding or 'utf-8',errors=errors,newline='\n')
+Path.write_text=write_text_lf
+script=Path(sys.argv[1]).resolve()
+sys.argv=sys.argv[1:]
+sys.path.insert(0,str(script.parent))
+runpy.run_path(str(script),run_name='__main__')
diff --git a/research/coordinated-gauge-async/evidence/ASYNC-AUDIT.json b/research/coordinated-gauge-async/evidence/ASYNC-AUDIT.json
new file mode 100644
index 0000000000..faf0e72851
--- /dev/null
+++ b/research/coordinated-gauge-async/evidence/ASYNC-AUDIT.json
@@ -0,0 +1,57 @@
+{
+  "status": "PASS_INDEPENDENT_ASYNC_WHOLE_OWNER_CONTROLLER_PHASES_AND_SPECTATORS",
+  "template_sha256": "3a3e6f87997f15815d11b1ee389d0783482e0eca087fff8ebd29164eecf974f0",
+  "complex_template_sha256": "3a3e6f87997f15815d11b1ee389d0783482e0eca087fff8ebd29164eecf974f0",
+  "complex_live": 7680,
+  "complex_profile": {
+    "1": 171720,
+    "2": 17856,
+    "3": 41568,
+    "4": 6336,
+    "5": 23280,
+    "6": 6912,
+    "7": 1752,
+    "8": 1224,
+    "48": 432
+  },
+  "complex_hosted_residuals": 40320,
+  "physical": 8258,
+  "rank_mass": 558144,
+  "deficit": 2496,
+  "entering_helper_roles": 8784,
+  "cross_replica_helper_roles": 32113,
+  "one_owner_per_complete_invocation": true,
+  "helper_busy_family_window_checks": 40320,
+  "complete_invocations": 72,
+  "actual_orthogonal_73column_maps": 2972,
+  "actual_sigma_stream_sha256": "0a3efb2551fc0ed53035e7224a38a5bbaa0b1f9cb532c37de3bf9113d6d007fb",
+  "all_actual_residual_and_spectator_charts": true,
+  "idle_remainders": 9360,
+  "mutations_rejected": [
+    "wrong call time",
+    "alias two whole owners",
+    "helper on active data family",
+    "owner in private namespace",
+    "unbound change of owner replica",
+    "omit complete invocation",
+    "reorder invocation replicas",
+    "interleave local invocations",
+    "change local instruction stream",
+    "wrong residual width",
+    "wrong spectator coordinate",
+    "omit spectator route",
+    "overlap residual footprint",
+    "omit idle remainder",
+    "underpay idle remainder",
+    "omit bridge touch",
+    "remove one-owner rule",
+    "duplicate frozen role",
+    "understate physical reserve"
+  ],
+  "widths": {
+    "7": 18,
+    "8": 104,
+    "9": 438
+  },
+  "scope": "Independent exact asynchronous chronology, per-replica phase projection, owner/chart/paid-profile audit. Uses unchanged source local program and normalized residual-wrapper identities; all-size compilation and complete coefficient invoice are separate inherited obligations."
+}
diff --git a/research/coordinated-gauge-async/evidence/EXACT-ASSEMBLY.json b/research/coordinated-gauge-async/evidence/EXACT-ASSEMBLY.json
new file mode 100644
index 0000000000..44c1f6d142
--- /dev/null
+++ b/research/coordinated-gauge-async/evidence/EXACT-ASSEMBLY.json
@@ -0,0 +1,127 @@
+{
+  "status": "PASS_ASYNC_COMPLEX_AND_CITED_BIT_EXACT_ASSEMBLY",
+  "complex_lower": "1374441333828159774039/1000000000000000000000000",
+  "complex_decimal": "0.001374441333828159774039",
+  "next_b_rejected": "34361033345703994351/25000000000000000000000",
+  "exact_moment_receipts": [
+    {
+      "endpoint": "1374441333828159774039/1000000000000000000000000",
+      "mercator": [
+        "-28291815341633545409236243687551323657234293/13479973333575319897333507543509815336818572211270286240551805124608",
+        "-7072953835408386352309060921887830914308573/3369993333393829974333376885877453834204643052817571560137951281152"
+      ],
+      "base_two": [
+        "-10494022411949002506793/5000000000000000000000000000000000000000000000",
+        "-20988034823898005013471/10000000000000000000000000000000000000000000000"
+      ],
+      "accept": true
+    },
+    {
+      "endpoint": "34361033345703994351/25000000000000000000000",
+      "mercator": [
+        "30953950173653324095534537518255255473208761/26959946667150639794667015087019630673637144422540572481103610249216",
+        "30953950173653324095534537518255255473208763/26959946667150639794667015087019630673637144422540572481103610249216"
+      ],
+      "base_two": [
+        "11481455176457850684391/10000000000000000000000000000000000000000000000",
+        "5740732588228925342253/5000000000000000000000000000000000000000000000"
+      ],
+      "accept": false
+    }
+  ],
+  "bit_lower": "391360506745414227149/250000000000000000000000",
+  "bit_result_sha256": "3494ba83280cc86548fe0cc6695c50d23e6b6ffca85aeb1a33337c1196e280cb",
+  "bit_template_sha256": "e37b52e4861c39154b5891460fb45df8651c92cd68961078f8f06596f057c822",
+  "bit_prefix_bits": 2176,
+  "bit_charged_three_level_coefficient_bits": 6464,
+  "external_row_allowance": 20161,
+  "kappa": "1372554837726243/1000000000000000000",
+  "kappa_decimal": "0.001372554837726243",
+  "binding_component": "complex",
+  "strict_outer_constraints": {
+    "bit_positive": "1374441333828159763657025210474097063071084428770185212744198659759863764456562241372079108639576639945415444179598377964782511788681826833133887787942198363260943218133543024094534487189715815236378411554889827340225709330011818664519322661189190430126225974452649731410285505180363525104679/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "complex_above_bit": "10381974789525902936928915571229814787255801340240136235543437758627920891360423360054584555820401622035217488211318173166866112212057801636739056781866456975905465512810284184763621588445110172659774290669988181335480677338810809569873774025547350268589714494819636474895321/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "complex_below_one_over32": "29875558666171840225961/1000000000000000000000000",
+    "beta_positive": "1/1000000000000000000",
+    "beta_below_one": "999999999999999999/1000000000000000000",
+    "leaf_saving_above_bit": "9007533455697743162889915571229814787255801340240136235543437758627920891360423360054584555820401622035217488211318173166866112212057801636739056781866456975905465512810284184763621588445110172659774290669988181335480677338810809569873774025547350268589714494819636474895321/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "q_positive": "687220666914079880454071271408888767878517003910995543301014901109746669484082460926175789863226078600628613450222549036975811714742535451784432105289272348496583821124573148786324025461314883523654718587729098433734443110116081992033952000582776550543790326037134435578916778137532031142053994819636474895321/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "q_below_internal": "1374441333828159763657025210474097063071084428770185212744198659759863764456562241372079108639576639945415444179598377964782511788681826833133887787942198363260943218133543024094534487189715815236378411554889827340225709330011818664519322661189190430126225974452649731410285505180363525104679/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "q_below_leaf": "5878208061677031345101982996089004456698985098890253330515917539073824210136773921399371386549777450963024188285257464548215567894710727651503416178875426851213675974538685116476345281412270901566265556889883918007966047999417223449456209673962865564421083221862467968857946005180363525104679/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "c_positive": "687220666914080005454071271408888767878517003910995543301014901109746669484082460926175789863226078600628613450222549036975811714742535451784432105289272348496583821124573148786324025461314883523654718587729098433734443110116081992033952000582776550543790326037134435578916778137532031142053994819636474895321/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "c_below_one": "499312779333085919994545928728591111232121482996089004456698985098890253330515917539073824210136773921399371386549777450963024188285257464548215567894710727651503416178875426851213675974538685116476345281412270901566265556889883918007966047999417223449456209673962865564421083221862467968857946005180363525104679/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "q_below_reservations": "1/4000000000000000000",
+    "lambda_above_tau": "1374441333828159763657025210474097063071084428770185212744198659759863764456562241372079108639576639945415444179598377964782511788681826833133887787942198363260943218133543024094534487189715815236378411554889827340225709330011818664519322661189190430126225974452649731410285505180363525104679/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "lambda_above_sigma": "11756416123354062700585940781703911850326885769010321448287636418387784655816985601426663664459978261980632932390916551131648624000739628469872944569808655339166408730943827208858156075634825987896152702224878008675706386668822628234393096686736540698715940469272286206305606505180363525104679/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "lambda_above_internal": "1374441333828159763657025210474097063071084428770185212744198659759863764456562241372079108639576639945415444179598377964782511788681826833133887787942198363260943218133543024094534487189715815236378411554889827340225709330011818664519322661189190430126225974452649731410285505180363525104679/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "lambda_prime_above_lambda": "1374441333828159763657025210474097063071084428770185212744198659759863764456562241372079108639576639945415444179598377964782511788681826833133887787942198363260943218133543024094534487189715815236378411554889827340225709330011818664519322661189190430126225974452649731410285505180363525104679/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "compact_leaf": "5878208061677031345101982996089004456698985098890253330515917539073824210136773921399371386549777450963024188285257464548215567894710727651503416178875426851213675974538685116476345281412270901566265556889883918007966047999417223449456209673962865564421083221862467968857946005180363525104679/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "compact_reservations": "1/4000000000000000000",
+    "lambda_prime_below_one": "687220666914079880454071271408888767878517003910995543301014901109746669484082460926175789863226078600628613450222549036975811714742535451784432105289272348496583821124573148786324025461314883523654718587729098433734443110116081992033952000582776550543790326037134435578916778137532031142053994819636474895321/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "epsilon_positive": "499999999999999999500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000/500687220666914079880454071271408888767878517003910995543301014901109746669484082460926175789863226078600628613450222549036975811714742535451784432105289272348496583821124573148786324025461314883523654718587729098433734443110116081992033952000582776550543790326037134435578916778137532031142053994819636474895321",
+    "epsilon_below_one": "687220666914080380454071271408888767878517003910995543301014901109746669484082460926175789863226078600628613450222549036975811714742535451784432105289272348496583821124573148786324025461314883523654718587729098433734443110116081992033952000582776550543790326037134435578916778137532031142053994819636474895321/500687220666914079880454071271408888767878517003910995543301014901109746669484082460926175789863226078600628613450222549036975811714742535451784432105289272348496583821124573148786324025461314883523654718587729098433734443110116081992033952000582776550543790326037134435578916778137532031142053994819636474895321",
+    "guard_width": "687220666914080380454071271408888767878517003910995543301014901109746669484082460926175789863226078600628613450222549036975811714742535451784432105289272348496583821124573148786324025461314883523654718587729098433734443110116081992033952000582776550543790326037134435578916778137532031142053994819636474895321/500687220666914079880454071271408888767878517003910995543301014901109746669484082460926175789863226078600628613450222549036975811714742535451784432105289272348496583821124573148786324025461314883523654718587729098433734443110116081992033952000582776550543790326037134435578916778137532031142053994819636474895321",
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+    "dimension_above_kappa": "499312779333085919624597876700605464383834491565652444817325085415914119181576128336573262785421040524793234526108367137455474899183098668444328859534258476834628419067180150319821068036902109734781016414407129669043307820522501898244862725169979233569823901708217287350171074706257048735301708603480402436294441000840344920390997/500687220666914079880454071271408888767878517003910995543301014901109746669484082460926175789863226078600628613450222549036975811714742535451784432105289272348496583821124573148786324025461314883523654718587729098433734443110116081992033952000582776550543790326037134435578916778137532031142053994819636474895321000000000000000000"
+  },
+  "outer_controls": [
+    "next_grid",
+    "zero_eta",
+    "zero_beta",
+    "unit_beta"
+  ],
+  "bootstrap": [
+    {
+      "gap": "1836228103880433291398542476426906988370579360282332180021168519310327538373521/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "10676999947950"
+    },
+    {
+      "gap": "2523787804310175454715447176821293207202255617805608896940928268471124779853446531573884911704955880558012747059178681/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "5651930671462759834"
+    },
+    {
+      "gap": "3468798276055320231711557144387303338852481417280110204303882152370570951680955412219527932797033402979496802110748859462363286292429397843767148782113937441/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "2991881658776002015587982"
+    },
+    {
+      "gap": "4767659729322295512729219692797036118516320702189050904092239698198332543513371413525281293638818450690796506702925782485555918821974109708538946374605577514727212271755985462544578671108858105801/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "1583769579007161960210219534901"
+    },
+    {
+      "gap": "6552868597608539028708223079706561719383711679234392833003118738466329798169127808734065554528412321480369680984826991010559681020889366842170965547699104625094834440477019355054499762265316488242190657484446163719485700795530284899761/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "838377437834454763996195214956705598"
+    },
+    {
+      "gap": "9006533455697743162889915571229814787255801340240136235543437758627920891360423360054584555820401622035217488211318173166866112212057801636739056781866456975905465512810284184763621588445110172659774290669988181335480677338810809569873774025547350268589714494819636474895321/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "443799866840785294166734569567404475997436"
+    }
+  ],
+  "complex_template_sha256": "3a3e6f87997f15815d11b1ee389d0783482e0eca087fff8ebd29164eecf974f0",
+  "owner_audit_sha256": "90b024ed70f38ef14f03646d91a4104aab578fdbdd358505664c2fbcf54961a6",
+  "invoice_sha256": "b0c2f45bc2aebf61d1b071f8805996c51383cd338c6d25a6fbc06f85268ce23b",
+  "bit_scope": "PR493 published profile and operator reused as a pinned dependency; this verifier reprices its published profile but does not reconstruct that unchanged bit operator.",
+  "scope": "Conditional finite witness. Inherited compiler, precision/tape and analytic-transfer contracts remain assumed; changed-host operator admission recorded separately."
+}
diff --git a/research/coordinated-gauge-async/evidence/GAUGE-CONSTRUCTION.json b/research/coordinated-gauge-async/evidence/GAUGE-CONSTRUCTION.json
new file mode 100644
index 0000000000..20f664875f
--- /dev/null
+++ b/research/coordinated-gauge-async/evidence/GAUGE-CONSTRUCTION.json
@@ -0,0 +1,55 @@
+{
+  "status": "PASS_EXACT_COORDINATED_GAUGE_RECONSTRUCTION",
+  "source_hashes": {
+    "retimed_base": "77c5a22608644e4634cbabeb3f53d40f50f4f5632405e8f0381c3fc4c4b845f4",
+    "baseline_selection": "436e4acadee227dbfcb66386c9ccaa38e68054e8ed4870bb174eeb41363375bb",
+    "gk_word.py": "f027a9273bcf4e0971258c160b7d3b3c196366b32a529cff6e6a0e666b03826b",
+    "gk_common.py": "e68d190fd436b2f3955decb590adb53344d5d4ea0a19d62d86df1e4b55ecd198"
+  },
+  "word_sha256": "91c7de4fef095352b0995a56eebbd2821d268b0da0632cccd808e2cd921abca0",
+  "normalized_word_sha256": "879c0e6ade3f1c84a515609a848dc917bddd1c1499e0cc37b2862809dfb8655b",
+  "normalization_ignored_metadata": [
+    "optional redundant R",
+    "report.complex_coarse_float",
+    "report.gain_percent_float"
+  ],
+  "changed_target_dimensions": {
+    "81": [
+      4,
+      2
+    ],
+    "82": [
+      4,
+      2
+    ],
+    "84": [
+      4,
+      2
+    ],
+    "87": [
+      4,
+      2
+    ]
+  },
+  "local_profile": {
+    "1": 2385,
+    "2": 248,
+    "3": 530,
+    "4": 88,
+    "6": 96,
+    "5": 270,
+    "8": 17,
+    "7": 1
+  },
+  "delta": {
+    "2": -16,
+    "4": 4,
+    "6": 4
+  },
+  "widths": {
+    "9": 438,
+    "8": 104,
+    "7": 18
+  },
+  "scope": "Exact local construction and frame nesting. Scalar, Gaussian phase, physical ownership, prefix and outer assembly independently required."
+}
diff --git a/research/coordinated-gauge-async/evidence/GLOBAL-PREFIX.json b/research/coordinated-gauge-async/evidence/GLOBAL-PREFIX.json
new file mode 100644
index 0000000000..fee6202532
--- /dev/null
+++ b/research/coordinated-gauge-async/evidence/GLOBAL-PREFIX.json
@@ -0,0 +1,13 @@
+{
+  "status": "PASS_ASYNC_GLOBAL_PREFIX",
+  "template_sha256": "3a3e6f87997f15815d11b1ee389d0783482e0eca087fff8ebd29164eecf974f0",
+  "global_prefix_bits": 205,
+  "retained_payload_bits": 512,
+  "complete_invocation_cover_sweeps": 72,
+  "bridge_factor_two_sweeps": 72,
+  "local_scalar_coefficient_work_prefix_bits": 20,
+  "stream_input_sup_to_l2_bits": 37,
+  "primitive_arithmetic_margin_bits": 4,
+  "controller_audit_sha256": "1aa61bed3ff2819cf5c1634d90df720cfda7a9cd32e570d752c3a6698a4405c1",
+  "proof": "Each complete normalized invocation has unitary helper endpoints and a data shear with at most two unitary summands. Distinct full orthogonal owner charts act by right-translation bijections on the retained cover group. Thus the maximum full-stream L2 norm grows by at most2per complete local cover sweep and individual bridge. All72local sweeps and72individual bridge sweeps are counted. Unit-modulus phases, idle children, terminal translations and normalized spectator endpoints are unitary. The independently replayed scalar/Horner envelope costs20bits, initial sup-to-L2 conversion37bits, primitive margin4bits: total205<512. All-size child/compiler and precision/tape interfaces remain inherited."
+}
diff --git a/research/coordinated-gauge-async/evidence/LOCAL-INDEPENDENT-REPLAY.json b/research/coordinated-gauge-async/evidence/LOCAL-INDEPENDENT-REPLAY.json
new file mode 100644
index 0000000000..821b3c231f
--- /dev/null
+++ b/research/coordinated-gauge-async/evidence/LOCAL-INDEPENDENT-REPLAY.json
@@ -0,0 +1,71 @@
+{
+  "status": "PASS_INDEPENDENT_NEW_STRUCTURAL_IR_REPLAY",
+  "template_sha256": "e61382a16a757dfe122c7718c79b1c4e84014d267dc8701805c55decb4f6042e",
+  "local": {
+    "forward": {
+      "all_exact_columns": 800,
+      "prefix_l1_upper": "32209",
+      "prefix_bits_upper": 15,
+      "charged_payload_bits": 192,
+      "private_work_erased": true
+    },
+    "backward": {
+      "all_exact_columns": 800,
+      "prefix_l1_upper": "1039266",
+      "prefix_bits_upper": 20,
+      "charged_payload_bits": 192,
+      "private_work_erased": true
+    }
+  },
+  "mutation_controls_rejected": [
+    "wrong_scale",
+    "missing_center",
+    "wrong_common_frame",
+    "wrong_child_rank",
+    "missing_pre_wrapper",
+    "missing_post_wrapper",
+    "wrapper_with_wrong_frame",
+    "wrapper_in_wrong_phase",
+    "wrapper_on_a_data_register",
+    "wrapper_on_a_target_register",
+    "wrapper_on_a_centre_register",
+    "centre_step_not_nested",
+    "centre_descent_step_skipped",
+    "missing_padding",
+    "aliased_owner",
+    "missing_cover_phase",
+    "two_entering_helpers_on_one_bank_family",
+    "bank_spectator_is_a_residual_cell",
+    "bank_spectator_shifted",
+    "bank_spectator_record_dropped",
+    "bank_spectator_count_wrong",
+    "second_owner_on_an_active_data_family",
+    "helper_owner_on_an_active_data_family",
+    "active_wire_on_a_bank_family",
+    "owner_in_the_private_namespace_centres",
+    "owner_in_the_private_namespace_reserved_tableau",
+    "owner_in_the_private_namespace_coefficient",
+    "active_owner_on_the_family_of_another_replica"
+  ],
+  "all_eight_global_scalar_columns": true,
+  "bank_owners_and_padding_exact": true,
+  "paid_profile": {
+    "1": 175662,
+    "2": 17856,
+    "3": 38160,
+    "4": 6336,
+    "5": 19440,
+    "6": 6912,
+    "7": 72,
+    "8": 6984,
+    "9": 936,
+    "16": 3840,
+    "17": 162,
+    "32": 3840,
+    "48": 3840
+  },
+  "two_rational_moment_engines": true,
+  "complex_b": "592163915290447/500000000000000000",
+  "next_b_grid_rejected": true,
+  "scope": "Independent finite replay; retains the written generic scalar-interface lemma and inherited all-size/analytic assumptions."
+}
diff --git a/research/coordinated-gauge-async/evidence/LOCAL-PHASE-REVIEW.json b/research/coordinated-gauge-async/evidence/LOCAL-PHASE-REVIEW.json
new file mode 100644
index 0000000000..d62428c5c6
--- /dev/null
+++ b/research/coordinated-gauge-async/evidence/LOCAL-PHASE-REVIEW.json
@@ -0,0 +1,21 @@
+{
+  "status": "PASS_NEW_WORD_EXACT_LOCAL_PHASE_RECIPES",
+  "canonical_word_sha256": "e88cd2b8a34a012542ab126a12518ab16888435ec42c4065ee63c7a85b5ec9f2",
+  "frames": 971,
+  "directed_edges": 2950,
+  "ranks": {
+    "1": 1784,
+    "3": 530,
+    "8": 24,
+    "6": 102,
+    "2": 248,
+    "7": 2,
+    "5": 172,
+    "4": 88
+  },
+  "endpoint_adapters": 240,
+  "all_child_equalities_include_global_phase": true,
+  "recipe_sha256": "3084c0b2d1bbb47b8deed98d96727e63ead03c434ce720557f491177a1756017",
+  "seconds": 200.67199999999139,
+  "scope": "Every local frame and center transition. Expanded scalar-interface integration and global emitted program are separate obligations."
+}
diff --git a/research/coordinated-gauge-async/evidence/LOCAL-SCALAR-FRAME-AUDIT.json b/research/coordinated-gauge-async/evidence/LOCAL-SCALAR-FRAME-AUDIT.json
new file mode 100644
index 0000000000..a81d5932ee
--- /dev/null
+++ b/research/coordinated-gauge-async/evidence/LOCAL-SCALAR-FRAME-AUDIT.json
@@ -0,0 +1,67 @@
+{
+  "status": "PASS_EXACT_SCALAR_AND_FRAME_RESEARCH_CANDIDATE_NOT_YET_ADMITTED",
+  "full_word_sha256": "91c7de4fef095352b0995a56eebbd2821d268b0da0632cccd808e2cd921abca0",
+  "canonical_word_sha256": "e88cd2b8a34a012542ab126a12518ab16888435ec42c4065ee63c7a85b5ec9f2",
+  "live_columns_each_orientation": 800,
+  "arbitrary_helper_columns": 560,
+  "all_live_source_and_helper_rows_restored": true,
+  "all_target_rows_correct": true,
+  "negative_controls_rejected": [
+    "missing_dirty_compensation",
+    "missing_Y_inverse_basis",
+    "missing_source_cleanup",
+    "wrong_dual_event_order"
+  ],
+  "old_response_additions": 12082,
+  "complete_forward_scalar_operations": 19660,
+  "coefficient_histogram": {
+    "-1/2": 1896,
+    "1/2": 1552,
+    "-1": 2636,
+    "9/8": 46,
+    "1/8": 1450,
+    "-1/8": 1276,
+    "1/4": 3260,
+    "-1/4": 3084,
+    "3/8": 402,
+    "-3/8": 476,
+    "-5/8": 250,
+    "-7/8": 106,
+    "5/8": 174,
+    "3/4": 24,
+    "7/8": 46,
+    "-11/8": 38,
+    "1": 2502,
+    "2": 104,
+    "-4": 1,
+    "-2": 308,
+    "4": 29
+  },
+  "frames": 971,
+  "nested_directed_edges": 2926,
+  "common_frame_incidences": 7976,
+  "orthogonal_transport_vector": 43,
+  "transported_ports_exactly_match_fixed_routes": true,
+  "local_profile": {
+    "1": 2385,
+    "2": 248,
+    "3": 530,
+    "4": 88,
+    "5": 270,
+    "6": 96,
+    "7": 1,
+    "8": 17
+  },
+  "prospective_complex_b": "592163915290447/500000000000000000",
+  "prospective_b_float": 0.001184327830580894,
+  "prospective_gain_percent": 18.505237681363564,
+  "two_rational_moment_engines": true,
+  "next_b_grid_rejected": true,
+  "remaining": [
+    "new general scalar-interface proof integration",
+    "exact phase and emitted local/global program",
+    "finite invoice and outer assembly"
+  ],
+  "scope": "Substantial new scalar-network candidate. All parameter values remain prospective until the expanded scalar interface and complete phase-aware supplier are admitted.",
+  "seconds": 2.1560000000172295
+}
diff --git a/research/coordinated-gauge-async/evidence/OWNERS-AND-PROFILE.json b/research/coordinated-gauge-async/evidence/OWNERS-AND-PROFILE.json
new file mode 100644
index 0000000000..bde2721272
--- /dev/null
+++ b/research/coordinated-gauge-async/evidence/OWNERS-AND-PROFILE.json
@@ -0,0 +1,35 @@
+{
+  "status": "PASS_INDEPENDENT_ASYNC_OWNERS_AND_PROFILE",
+  "template_sha256": "3a3e6f87997f15815d11b1ee389d0783482e0eca087fff8ebd29164eecf974f0",
+  "paid_profile": {
+    "1": 171720,
+    "2": 17856,
+    "3": 41568,
+    "4": 6336,
+    "5": 23280,
+    "6": 6912,
+    "7": 1752,
+    "8": 1224,
+    "48": 432
+  },
+  "live": 7680,
+  "physical": 8258,
+  "rank_mass": 558144,
+  "deficit": 2496,
+  "hosted_residuals": 40320,
+  "banked_residuals": 0,
+  "all_800_owners_per_call_distinct": true,
+  "actual_helper_owner_charts": 40320,
+  "actual_73column_sigmas": 2972,
+  "cross_replica_helper_roles": 32113,
+  "explicit_idle_remainder_children": 9360,
+  "source_idle_children_charged": 17280,
+  "source_artifacts": {
+    "split-centres/CANONICAL-WORD.json.gz": "e88cd2b8a34a012542ab126a12518ab16888435ec42c4065ee63c7a85b5ec9f2",
+    "split-centres/LOCAL-PROGRAMS.json.gz": "7f59bfbfc174adac07ae70afe290d6a41e9d8aa4e2550edb45e0004788c99d98",
+    "split-centres/LOCAL-CHECKS.json": "66c2f1da830b0b4f6da8af8c2cd81affa0f057947a4cdf304d59ffbc0c417a9c",
+    "split-centres/RESIDUAL-ADAPTERS.json": "c46c93e3c87a9a561f2d7f617b72665b8aa59eb531167151e5b9592bd1233084",
+    "work/complex-nine-native/DATA-PHASES.json": "f621f6d4296cb215f334830356c5f79fe10d6d04d15b3ccda76ba0885ab67d0a"
+  },
+  "controller_audit_sha256": "1aa61bed3ff2819cf5c1634d90df720cfda7a9cd32e570d752c3a6698a4405c1"
+}
diff --git a/research/coordinated-gauge-async/evidence/PACKAGE-REVIEW.json b/research/coordinated-gauge-async/evidence/PACKAGE-REVIEW.json
new file mode 100644
index 0000000000..3e90491c1e
--- /dev/null
+++ b/research/coordinated-gauge-async/evidence/PACKAGE-REVIEW.json
@@ -0,0 +1,59 @@
+{
+  "status": "PASS_REVIEW_OF_COMPLETE_PORTABLE_REPLAY_AND_COMPOSITION",
+  "reviewed_sha256": {
+    "verify.py": "c553dbf909d703e5afb0333d7da43481f5df18a71e932c23986d9bfd8b953e2e",
+    "code/assemble.py": "9db68cad2738d76052b25c8bdf6485edd6f14025bf47107d2425b10a2bdf69e0",
+    "code/build_retimed.py": "5177c0f1200d30fb532e58964d67d3cb08f7b9854ea97107f776f1883e36de97",
+    "code/build_rank2_collapse.py": "96695ffd96c81a5e06049cb61f91f5c69e1aa9a4223ada63986a2198a82d2847",
+    "code/audit_async.py": "bbc2bd87ab2aa8f7711da15d410bffeb96b0176da0fbfb80c6625cd4c61e06ca",
+    "code/audit_async_invoice.py": "bcfdc0b4aae2257327b3a643d7dfc0c6de85f3f392b686aa76df64b0c519f907",
+    "code/run_portable.py": "01030376f8a3afb86c559bc5036d1524165fc3b4119bcca12cc7c6e1990a7548",
+    ".gitattributes": "4ce4b6e6b8435f8c3a8387387732964a2ebf0183e8d77b8247ae8061b033e462"
+  },
+  "new_runtime_result": "Parent single portable replay completed all12 stages successfully; reviewer inspected receipts and independently compared serialized operators. No duplicate heavy replay launched.",
+  "checks": [
+    "Retiming transfer permits only frame changes on identical scalar event tails.",
+    "Coordinated four-gauge collapse explicitly preserves kernel selection and checks target dimensions, residual widths and paid-profile delta.",
+    "Actual source frame/phase/normalization constructors precede independent physical owner audit.",
+    "Every whole owner remains unique within each complete invocation, including240active data owners.",
+    "Actual event chronology and source-phase projection govern availability; sigma and spectator maps are reconstructed from actual pending axes.",
+    "All emitted children and all remainder children counted before exact moment pricing.",
+    "Prefix overcharge counts72complete invocations and72individual bridges and charges512bits.",
+    "Outer assembly uses two independent rational moment engines, a pinned cited bit profile,47strict inequalities and adjacent-grid rejection.",
+    "No optional discovery optimizer is required by the standalone verifier."
+  ],
+  "scope": "Conditional finite whole-owner construction; inherited all-size compiler, common-label, normalized Lemma A realization, restored-row, precision/tape and analytic-transfer contracts remain dependencies. PR493 bit operator is cited, not freshly reconstructed. No unconditional multiplication bound or Lean certification.",
+  "noncontributing_research": [
+    "The sufficient deferred-terminal-suffix theorem is valid but stock-capacity screen gives no additional gain here.",
+    "General paired-factor phase transport is a sufficient construction rule, not an achieved merged-bin witness."
+  ],
+  "kappa": "1372554837726243/1000000000000000000",
+  "kappa_decimal": "0.001372554837726243",
+  "complex_lower": "1374441333828159774039/1000000000000000000000000",
+  "bit_lower": "391360506745414227149/250000000000000000000000",
+  "template_sha256": "3a3e6f87997f15815d11b1ee389d0783482e0eca087fff8ebd29164eecf974f0",
+  "strict_constraints": 47,
+  "whole_owner_roles": 40320,
+  "orthogonal_73column_maps": 2972,
+  "explicit_remainders": 9360,
+  "rejected_mutations": 19,
+  "global_prefix_bits": 205,
+  "charged_prefix_bits": 512,
+  "portable_serialization_review": {
+    "only_parsed_template_change": "residual_adapter_sha256",
+    "local_event_streams_identical": true,
+    "normalization_recipes_identical": true,
+    "local_checks_identical": true,
+    "text_byte_changes_exactly_CRLF_to_LF": true,
+    "actual_Linux_execution_claimed": false,
+    "pinned_input_bytes_retained": true
+  },
+  "outstanding_finite_replay_obligations": [],
+  "receipt_sha256": {
+    "RESULT.json": "e79dec872757e3dc49648e362e0997b23bfcd88b9f18779ac71259aaacedd952",
+    "audit/ASYNC-AUDIT.json": "1aa61bed3ff2819cf5c1634d90df720cfda7a9cd32e570d752c3a6698a4405c1",
+    "audit/OWNERS-AND-PROFILE.json": "90b024ed70f38ef14f03646d91a4104aab578fdbdd358505664c2fbcf54961a6",
+    "audit/PREFIX-AND-INVOICE.json": "b0c2f45bc2aebf61d1b071f8805996c51383cd338c6d25a6fbc06f85268ce23b",
+    "audit/EXACT-ASSEMBLY.json": "5455ed1f38c252a1e888a29357aa544ed449d2d4485bd485b6881aa48f4b5964"
+  }
+}
diff --git a/research/coordinated-gauge-async/evidence/PACKAGE-REVIEW.md b/research/coordinated-gauge-async/evidence/PACKAGE-REVIEW.md
new file mode 100644
index 0000000000..60594f7d1b
--- /dev/null
+++ b/research/coordinated-gauge-async/evidence/PACKAGE-REVIEW.md
@@ -0,0 +1,45 @@
+# Independent package review
+
+The complete portable replay passed all 12 stages. Its exact conditional kappa is `1372554837726243/1000000000000000000` = **0.001372554837726243**, with 47 strictly positive assembly inequalities. No finite replay item remains outstanding in this review. The reviewer inspected the receipts and independently compared emitted programs; it did not launch a redundant heavy replay.
+
+## Exact verified scope
+
+Conditional finite whole-owner construction; inherited all-size compiler, common-label, normalized Lemma A realization, restored-row, precision/tape and analytic-transfer contracts remain dependencies. PR493 bit operator is cited, not freshly reconstructed. No unconditional multiplication bound or Lean certification.
+
+The independent controller audit covers 40,320 whole-helper assignments, 2,972 actual orthogonal 73-column maps, 9,360 explicit idle remainders, and 19 rejected malformed constructions. The local scalar audit uses all 800 input columns in both orientations. The prefix invoice requires 205 bits and charges 512, with physical row coefficient 16,467 below the retained 20,161 allowance. The final uncompressed template SHA-256 is `3a3e6f87997f15815d11b1ee389d0783482e0eca087fff8ebd29164eecf974f0`.
+
+## Construction and composition review
+
+- Retiming transfer permits only frame changes on identical scalar event tails.
+- Coordinated four-gauge collapse explicitly preserves kernel selection and checks target dimensions, residual widths and paid-profile delta.
+- Actual source frame/phase/normalization constructors precede independent physical owner audit.
+- Every whole owner remains unique within each complete invocation, including240active data owners.
+- Actual event chronology and source-phase projection govern availability; sigma and spectator maps are reconstructed from actual pending axes.
+- All emitted children and all remainder children counted before exact moment pricing.
+- Prefix overcharge counts72complete invocations and72individual bridges and charges512bits.
+- Outer assembly uses two independent rational moment engines, a pinned cited bit profile,47strict inequalities and adjacent-grid rejection.
+- No optional discovery optimizer is required by the standalone verifier.
+
+## Serialization correction independently checked
+
+The first successful replay and the portable replay have exactly identical parsed local instruction streams, normalization recipes and local checks. Their parsed complete global templates differ only in the raw residual-adapter SHA-256 field. The changed text bytes are exactly CRLF versus LF. The new launcher changes only `Path.write_text` serialization, retaining original scripts, arithmetic, arguments and import paths. Both completed replays give the same exact kappa. This is not a claim that a Linux process was run.
+
+Pinned inherited source/input bytes remain unchanged under `** -text`; new generated verifier outputs use UTF-8/LF. Final upload manifest and batch preflight are parent packaging work, separate from the mathematical audit reviewed here.
+
+## Research not counted as a numerical contribution
+
+- The sufficient deferred-terminal-suffix theorem is valid but stock-capacity screen gives no additional gain here.
+- General paired-factor phase transport is a sufficient construction rule, not an achieved merged-bin witness.
+
+## Reviewed executable source hashes
+
+- `verify.py`: `c553dbf909d703e5afb0333d7da43481f5df18a71e932c23986d9bfd8b953e2e`
+- `code/assemble.py`: `9db68cad2738d76052b25c8bdf6485edd6f14025bf47107d2425b10a2bdf69e0`
+- `code/build_retimed.py`: `5177c0f1200d30fb532e58964d67d3cb08f7b9854ea97107f776f1883e36de97`
+- `code/build_rank2_collapse.py`: `96695ffd96c81a5e06049cb61f91f5c69e1aa9a4223ada63986a2198a82d2847`
+- `code/audit_async.py`: `bbc2bd87ab2aa8f7711da15d410bffeb96b0176da0fbfb80c6625cd4c61e06ca`
+- `code/audit_async_invoice.py`: `bcfdc0b4aae2257327b3a643d7dfc0c6de85f3f392b686aa76df64b0c519f907`
+- `code/run_portable.py`: `01030376f8a3afb86c559bc5036d1524165fc3b4119bcca12cc7c6e1990a7548`
+- `.gitattributes`: `4ce4b6e6b8435f8c3a8387387732964a2ebf0183e8d77b8247ae8061b033e462`
+
+Comparison is against compatible pinned whole-owner constructions. Higher disputed split-owner claims are neither silently accepted nor disproved here. The numerical contribution is coordinated rank-two gauge collapse combined with public body retiming and the literal asynchronous owner allocation.
diff --git a/research/coordinated-gauge-async/evidence/PREFIX-AND-INVOICE.json b/research/coordinated-gauge-async/evidence/PREFIX-AND-INVOICE.json
new file mode 100644
index 0000000000..0e4cc36e5a
--- /dev/null
+++ b/research/coordinated-gauge-async/evidence/PREFIX-AND-INVOICE.json
@@ -0,0 +1,69 @@
+{
+  "status": "PASS_INDEPENDENT_ASYNC_SCALAR_PREFIX_AND_FINITE_INVOICE",
+  "template_sha256": "3a3e6f87997f15815d11b1ee389d0783482e0eca087fff8ebd29164eecf974f0",
+  "global_prefix_bits": 205,
+  "charged_prefix_bits": 512,
+  "complete_invocation_sweeps": 72,
+  "bridge_sweeps": 72,
+  "row_prefix_extra": 1024,
+  "local": {
+    "backward": {
+      "events": 25074,
+      "macros": 19708,
+      "real_component_primitives": 221516,
+      "prefix_l1": "1039266",
+      "prefix_bits": 20,
+      "coefficient_work_primitive_count": 109970,
+      "all_exact_columns": 800,
+      "all_560_dirty_helper_columns_restored": true,
+      "private_centres_and_coefficient_work_cleared": true
+    },
+    "forward": {
+      "events": 25074,
+      "macros": 19708,
+      "real_component_primitives": 221420,
+      "prefix_l1": "32209",
+      "prefix_bits": 15,
+      "coefficient_work_primitive_count": 109906,
+      "all_exact_columns": 800,
+      "all_560_dirty_helper_columns_restored": true,
+      "private_centres_and_coefficient_work_cleared": true
+    }
+  },
+  "entering_helpers": 122,
+  "entrance_rank": 140,
+  "local_adapters": 391216,
+  "local_primitive_bound": 10145927414092,
+  "wrapper_elementary_factors_charged": 2529792,
+  "compiler": 25934336,
+  "unhosted_pervertex": 1298986174975248,
+  "hosted_rebased_pervertex": 1298986174642608,
+  "conservative_pervertex_used": 1298986174975248,
+  "finite": {
+    "physical_stock_bits": 2641,
+    "router_bits": 7984,
+    "literal_bits": 35282,
+    "physical_row_coefficient": 16467,
+    "retained_row_coefficient": 20161,
+    "required_C0_bits": 200012,
+    "halving_integer_inequality": [
+      5329,
+      4608
+    ],
+    "G_exponent": 30000,
+    "E_exponent": 100000,
+    "C0_exponent": 210000
+  },
+  "strict_checks": {
+    "local_primitive_bound": true,
+    "logical": true,
+    "router": true,
+    "literal": true,
+    "mass": true,
+    "precision": true,
+    "tape": true,
+    "row": true,
+    "halving": true
+  },
+  "scope": "Exact finite recount conditional on inherited all-size compiler/precision/tape and analytic-transfer contracts. Signed accumulation uses the supplier scalar primitive convention. Async prefix counts all72individual bridges and72complete invocations;512bits are charged, and an additional2*512 enters the two-level physical row bound."
+}
diff --git a/research/coordinated-gauge-async/evidence/RESULT.json b/research/coordinated-gauge-async/evidence/RESULT.json
new file mode 100644
index 0000000000..c312de588c
--- /dev/null
+++ b/research/coordinated-gauge-async/evidence/RESULT.json
@@ -0,0 +1,73 @@
+{
+  "status": "PASS_CONDITIONAL_ASYNCHRONOUS_COMPLEX_DERIVATIVE",
+  "kappa": "1372554837726243/1000000000000000000",
+  "kappa_decimal": "0.001372554837726243",
+  "complex_lower": "1374441333828159774039/1000000000000000000000000",
+  "bit_lower": "391360506745414227149/250000000000000000000000",
+  "template_sha256": "3a3e6f87997f15815d11b1ee389d0783482e0eca087fff8ebd29164eecf974f0",
+  "strict_constraints": 47,
+  "scope": "New complex program independently replayed. The unchanged PR493 bit operator and inherited all-size compiler, common-label, Lemma A realization, restored-row, precision/tape and analytic-transfer contracts remain dependencies. No unconditional multiplication bound or fresh Lean certification is claimed.",
+  "stages": [
+    {
+      "stage": "retiming",
+      "exit": 0,
+      "seconds": 1.360000000015134
+    },
+    {
+      "stage": "construct",
+      "exit": 0,
+      "seconds": 1.3589999999967404
+    },
+    {
+      "stage": "scalar",
+      "exit": 0,
+      "seconds": 2.2809999999881256
+    },
+    {
+      "stage": "emit-local",
+      "exit": 0,
+      "seconds": 1.3440000000118744
+    },
+    {
+      "stage": "phases",
+      "exit": 0,
+      "seconds": 200.84399999998277
+    },
+    {
+      "stage": "adapters",
+      "exit": 0,
+      "seconds": 1.4690000000118744
+    },
+    {
+      "stage": "parent-integration",
+      "exit": 0,
+      "seconds": 1.3280000000086147
+    },
+    {
+      "stage": "parent-replay",
+      "exit": 0,
+      "seconds": 9.625
+    },
+    {
+      "stage": "emit-async",
+      "exit": 0,
+      "seconds": 5.48399999999674
+    },
+    {
+      "stage": "independent-async",
+      "exit": 0,
+      "seconds": 26.90600000001723
+    },
+    {
+      "stage": "invoice",
+      "exit": 0,
+      "seconds": 2.25
+    },
+    {
+      "stage": "two-moments-assembly",
+      "exit": 0,
+      "seconds": 0.7030000000086147
+    }
+  ],
+  "package_manifest_sha256": null
+}
diff --git a/research/coordinated-gauge-async/evidence/RETIMING-TRANSFER.json b/research/coordinated-gauge-async/evidence/RETIMING-TRANSFER.json
new file mode 100644
index 0000000000..9ea674075c
--- /dev/null
+++ b/research/coordinated-gauge-async/evidence/RETIMING-TRANSFER.json
@@ -0,0 +1,54 @@
+{
+  "status": "PASS_STRUCTURAL_MAIN_WORD_RETIMING_TRANSFER",
+  "source_files": {
+    "F:\\proof\\kappa-synthesis-20261011\\portable-replay-3\\base\\BASE-WORD.json.gz": "da4c5007092fd9456403179ef2a4391aef9fc5310e3eec5f2ab6c49b24147387",
+    "F:\\proof\\kappa-synthesis-20261011\\package\\research\\coordinated-gauge-async\\inputs\\PR495-FULL-WORD.json.gz": "66655cd858afb515dfd4cad43c4fef01f9185af670e144750e9977588ebd4ecf",
+    "F:\\proof\\kappa-synthesis-20261011\\package\\research\\coordinated-gauge-async\\vendor\\complex\\e8-complex-gauge-kernel\\data\\selections\\gauge80-kernel42.json": "436e4acadee227dbfcb66386c9ccaa38e68054e8ed4870bb174eeb41363375bb"
+  },
+  "changed_frames": {
+    "A": 224,
+    "B": 504
+  },
+  "discarded_prefix_events": {
+    "A": 229,
+    "B": 0
+  },
+  "preserved_gauge_fields": [
+    "centers",
+    "center_frames",
+    "target_levels",
+    "root_lines",
+    "ports",
+    "cut",
+    "precondition",
+    "start_extra",
+    "scatter"
+  ],
+  "entering_helpers": 122,
+  "entrance_rank": 148,
+  "local_profile": {
+    "1": 2385,
+    "2": 264,
+    "3": 530,
+    "4": 84,
+    "5": 270,
+    "6": 92,
+    "7": 1,
+    "8": 17
+  },
+  "transfer_controls_rejected": [
+    [
+      "A",
+      1
+    ],
+    [
+      "B",
+      3
+    ]
+  ],
+  "outputs": {
+    "FULL-WORD.json.gz": "c2ae016c463be382567afa47ee0718f83572d5372474237e2f74fb58a13fd264",
+    "RETIMED-BASE.json.gz": "77c5a22608644e4634cbabeb3f53d40f50f4f5632405e8f0381c3fc4c4b845f4"
+  },
+  "scope": "Structural transfer; scalar, frame, phase, whole-program, hosting and finite assembly checks must run independently."
+}
diff --git a/research/coordinated-gauge-async/evidence/RUN-STEPS.json b/research/coordinated-gauge-async/evidence/RUN-STEPS.json
new file mode 100644
index 0000000000..6e9a714f35
--- /dev/null
+++ b/research/coordinated-gauge-async/evidence/RUN-STEPS.json
@@ -0,0 +1,62 @@
+[
+  {
+    "stage": "retiming",
+    "exit": 0,
+    "seconds": 1.360000000015134
+  },
+  {
+    "stage": "construct",
+    "exit": 0,
+    "seconds": 1.3589999999967404
+  },
+  {
+    "stage": "scalar",
+    "exit": 0,
+    "seconds": 2.2809999999881256
+  },
+  {
+    "stage": "emit-local",
+    "exit": 0,
+    "seconds": 1.3440000000118744
+  },
+  {
+    "stage": "phases",
+    "exit": 0,
+    "seconds": 200.84399999998277
+  },
+  {
+    "stage": "adapters",
+    "exit": 0,
+    "seconds": 1.4690000000118744
+  },
+  {
+    "stage": "parent-integration",
+    "exit": 0,
+    "seconds": 1.3280000000086147
+  },
+  {
+    "stage": "parent-replay",
+    "exit": 0,
+    "seconds": 9.625
+  },
+  {
+    "stage": "emit-async",
+    "exit": 0,
+    "seconds": 5.48399999999674
+  },
+  {
+    "stage": "independent-async",
+    "exit": 0,
+    "seconds": 26.90600000001723
+  },
+  {
+    "stage": "invoice",
+    "exit": 0,
+    "seconds": 2.25
+  },
+  {
+    "stage": "two-moments-assembly",
+    "exit": 0,
+    "seconds": 0.7030000000086147
+  }
+]
diff --git a/research/coordinated-gauge-async/inputs/ASYNC-6.json b/research/coordinated-gauge-async/inputs/ASYNC-6.json
new file mode 100644
index 0000000000..48c0434ac2
--- /dev/null
+++ b/research/coordinated-gauge-async/inputs/ASYNC-6.json
@@ -0,0 +1,11626 @@
+{
+  "status": "UNIFORM_SELECTED_EXACT_INTEGER_ASYNC_NOMINATION",
+  "mode": "offset6",
+  "K": 4,
+  "T": 8,
+  "M": 73,
+  "L": 9,
+  "live": 7680,
+  "deficit": 2496,
+  "profile": {
+    "1": 171720,
+    "2": 17856,
+    "3": 41568,
+    "4": 6336,
+    "5": 23280,
+    "6": 6912,
+    "7": 1752,
+    "8": 1224,
+    "48": 432
+  },
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diff --git a/research/coordinated-gauge-async/inputs/PR493-EXPECTED.json b/research/coordinated-gauge-async/inputs/PR493-EXPECTED.json
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+{
+ "note": "Exact values that verify.sh must reproduce. The bit c is on the 10^-24 grid of PR #472's assembly (largest grid point admitted by both PR #315 moment engines with the inherited 10^-16 bad-branch allowance; the next grid point is rejected); kappa values are on the 10^-18 grid of the six-level, 47-inequality outer assembly.",
+ "word": {
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+    "kappa_decimal": "0.001325031497415498",
+    "binding_component": "complex",
+    "complex_lower": "663394767670133/500000000000000000",
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+    "binding_component": "complex",
+    "complex_lower": "1305231506267591/1000000000000000000",
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+   "kappa": "646049779721139/500000000000000000",
+   "published": "research/e8-r715-whole-hosted/EXPECTED.json at 259c684: bit_lower 323442810173099014031/250000000000000000000000, kappa 646049779721139/500000000000000000"
+  },
+  "pr445": {
+   "bit_lower": "316240894019344628547/250000000000000000000000",
+   "published": "research/r740-hosted-eleven README at 40ab91d: 1264963576077378514188/10^24"
+  },
+  "pr451": {
+   "bit_lower": "158442042752174644233/125000000000000000000000",
+   "published": "PR #451 body at f3a3815: c = 0.001267536342017397153864"
+  }
+ }
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new file mode 100644
index 0000000000..6f128ce2d2
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+      ],
+      "terminal_Y_translation": 127
+    }
+  ],
+  "cover_correction": {
+    "formula": "delta(g)_i = (weight(row_i(g))-1)/2 mod 2; apply X_delta(g) to EVERY final live stream in cover class g.",
+    "reason": "P_g C_all P_g^-1 = C_all X_delta(g) exactly. Orthogonality cancels all quadratic cross terms; both sides fix the uniform vector.",
+    "mandatory": true,
+    "complete_small_cover_checks": [
+      {
+        "n": 1,
+        "orthogonal_matrices": 1,
+        "nonzero_full_kernel_corrections": 0
+      },
+      {
+        "n": 2,
+        "orthogonal_matrices": 2,
+        "nonzero_full_kernel_corrections": 0
+      },
+      {
+        "n": 3,
+        "orthogonal_matrices": 6,
+        "nonzero_full_kernel_corrections": 0
+      },
+      {
+        "n": 4,
+        "orthogonal_matrices": 48,
+        "nonzero_full_kernel_corrections": 24
+      },
+      {
+        "n": 5,
+        "orthogonal_matrices": 720,
+        "nonzero_full_kernel_corrections": 600
+      }
+    ]
+  },
+  "global_phase_argument": "Every C_q and every linear address permutation fixes the uniform vector. Equality of the exact Z4 Fourier multiplier therefore fixes the scalar phase as well.",
+  "scope": "All compressed data boundary operators, idle children, terminal X_q and all-live cover correction. Internal arbitrary-subspace canonical labels are checked separately."
+}
\ No newline at end of file
diff --git a/research/coordinated-gauge-async/inputs/work/complex-nine-native/GLOBAL-TEMPLATE.json.gz b/research/coordinated-gauge-async/inputs/work/complex-nine-native/GLOBAL-TEMPLATE.json.gz
new file mode 100644
index 0000000000..975ce5c0df
Binary files /dev/null and b/research/coordinated-gauge-async/inputs/work/complex-nine-native/GLOBAL-TEMPLATE.json.gz differ
diff --git a/research/coordinated-gauge-async/inputs/work/higher-stage-e8/verified/gcert1-e8-retimed.json.gz b/research/coordinated-gauge-async/inputs/work/higher-stage-e8/verified/gcert1-e8-retimed.json.gz
new file mode 100644
index 0000000000..ccbdc0596c
Binary files /dev/null and b/research/coordinated-gauge-async/inputs/work/higher-stage-e8/verified/gcert1-e8-retimed.json.gz differ
diff --git a/research/coordinated-gauge-async/inputs/work/odd-stages/clifford_exact.py b/research/coordinated-gauge-async/inputs/work/odd-stages/clifford_exact.py
new file mode 100644
index 0000000000..166841a037
--- /dev/null
+++ b/research/coordinated-gauge-async/inputs/work/odd-stages/clifford_exact.py
@@ -0,0 +1,149 @@
+"""Small exact Gaussian-dyadic Clifford compiler, independent of gx internals.
+Paulis use i^p X^x Z^z. State vectors use integer Gaussian numerators / 2^d.
+The tableau is supplemented by a literal circuit, so global phases are kept.
+"""
+from dataclasses import dataclass
+
+def parity(x):return x.bit_count()&1
+def rank(rows):
+ p={}
+ for x in rows:
+  while x:
+   b=x.bit_length()-1
+   if b not in p:p[b]=x;break
+   x^=p[b]
+ return len(p)
+def transpose(rows,n):return tuple(sum(((rows[j]>>i)&1)<<j for j in range(n)) for i in range(n))
+def mv(rows,x):return sum(parity(r&x)<<i for i,r in enumerate(rows))
+def inv(rows,n):
+ a=[r|(1<<(n+i)) for i,r in enumerate(rows)]
+ for j in range(n):
+  k=next(i for i in range(j,n) if a[i]>>j&1);a[j],a[k]=a[k],a[j]
+  for i in range(n):
+   if i!=j and a[i]>>j&1:a[i]^=a[j]
+ assert [r&((1<<n)-1) for r in a]==[1<<i for i in range(n)]
+ return tuple(r>>n for r in a)
+def basis_extend(cols,n):
+ out=[]
+ for x in list(cols)+[1<<i for i in range(n)]:
+  if rank(out+[x])>len(out):out.append(x)
+ assert len(out)==n
+ return tuple(out)
+def nullspace(rows,n):
+ a=list(rows);p=[];k=0
+ for j in range(n):
+  t=next((i for i in range(k,len(a)) if a[i]>>j&1),None)
+  if t is None:continue
+  a[t],a[k]=a[k],a[t]
+  for i in range(len(a)):
+   if i!=k and a[i]>>j&1:a[i]^=a[k]
+  p.append(j);k+=1
+ return tuple((1<<j)|sum(((a[i]>>j)&1)<<v for i,v in enumerate(p)) for j in range(n) if j not in p)
+def pmul(a,b):
+ x,z,p=a;y,w,q=b
+ return x^y,z^w,(p+q+2*parity(z&y))%4
+def identity(n):return tuple([(1<<i,0,0) for i in range(n)]+[(0,1<<i,0) for i in range(n)])
+def apply(T,p):
+ n=len(T)//2;x,z,ph=p;out=(0,0,ph)
+ for i in range(n):
+  if x>>i&1:out=pmul(out,T[i])
+ for i in range(n):
+  if z>>i&1:out=pmul(out,T[n+i])
+ return out
+def compose(A,B):return tuple(apply(A,p) for p in B)
+def inverse(T):
+ n=len(T)//2;cols=[x|(z<<n) for x,z,p in T];rows=transpose(cols,2*n);cols_inv=transpose(inv(rows,2*n),2*n);out=[]
+ for j,w in enumerate(cols_inv):
+  x=w&((1<<n)-1);z=w>>n;u,v,p=apply(T,(x,z,0));assert (u,v)==identity(n)[j][:2];out.append((x,z,(-p)%4))
+ return tuple(out)
+def qvalue(Q,alpha,x,n):
+ return (sum(alpha[i] for i in range(n) if x>>i&1)+2*sum(parity((Q[i]>>(i+1))&(x>>(i+1))) for i in range(n) if x>>i&1))%4
+def mul_i(a,p):
+ x,y=a;p%=4
+ return [(x,y),(-y,x),(-x,-y),(y,-x)][p]
+@dataclass
+class Op:
+ n:int
+ tab:tuple
+ circuit:list
+ def then(self,b):
+  assert self.n==b.n
+  return Op(self.n,compose(b.tab,self.tab),self.circuit+b.circuit)
+ def adjoint(self):
+  circ=[]
+  for e in reversed(self.circuit):
+   if e[0]=='P':circ.append(('P',inv(e[1],self.n)))
+   elif e[0]=='D':circ.append(('D',e[1],tuple((-x)%4 for x in e[2])))
+   elif e[0]=='C':circ.append(('C',e[1],-e[2]))
+   elif e[0]=='phase':circ.append(('phase',(-e[1])%4))
+   else:circ.append(e)
+  return Op(self.n,inverse(self.tab),circ)
+ def zero_state(self,basis=0):
+  N=1<<self.n;vec=[(0,0)]*N;vec[basis]=(1,0);den=0
+  for e in self.circuit:
+   if e[0]=='P':
+    v=[(0,0)]*N
+    for x,a in enumerate(vec):v[mv(e[1],x)]=a
+    vec=v
+   elif e[0]=='X':vec=[vec[x^e[1]] for x in range(N)]
+   elif e[0]=='D':vec=[mul_i(a,qvalue(e[1],e[2],x,self.n)) for x,a in enumerate(vec)]
+   elif e[0]=='phase':vec=[mul_i(a,e[1]) for a in vec]
+   elif e[0]=='C':
+    q,sgn=e[1:];v=[]
+    for x,(a,b) in enumerate(vec):
+     c,d=vec[x^q]
+     # C=(1+i)I/2+(1-i)X/2; inverse conjugates both coefficients.
+     v.append((a-sgn*b+c+sgn*d,b+sgn*a+d-sgn*c))
+    vec=v;den+=1
+   else:raise AssertionError(e)
+   while den and all((a&1)==(b&1)==0 for a,b in vec):vec=[(a//2,b//2) for a,b in vec];den-=1
+  return vec,den
+def ident(n):return Op(n,identity(n),[])
+def perm(n,G):
+ cols=transpose(G,n);gi=inv(G,n)
+ return Op(n,tuple([(a,0,0) for a in cols]+[(0,a,0) for a in gi]),[('P',tuple(G))])
+def diag(n,Q,alpha=None):
+ Q=tuple(Q);assert transpose(Q,n)==Q
+ alpha=tuple(alpha) if alpha is not None else tuple((Q[i]>>i)&1 for i in range(n))
+ assert all(alpha[i]%2==(Q[i]>>i)&1 for i in range(n))
+ return Op(n,tuple([(1<<i,Q[i],alpha[i]) for i in range(n)]+[(0,1<<i,0) for i in range(n)]),[('D',Q,alpha)])
+def cgate(n,q,sgn=1):
+ return Op(n,tuple([(1<<i,0,0) for i in range(n)]+[(q if q>>i&1 else 0,1<<i,(3*sgn)%4 if q>>i&1 else 0) for i in range(n)]),[('C',q,sgn)])
+def xgate(n,b):return Op(n,tuple([(1<<i,0,0) for i in range(n)]+[(0,1<<i,2*((b>>i)&1)) for i in range(n)]),[('X',b)])
+def fourier(n,axes=None):
+ out=ident(n)
+ for i in range(n) if axes is None else axes:out=out.then(cgate(n,1<<i))
+ return out
+def hadamard(n,axes):
+ axes=list(axes);Q=tuple(1<<i if i in axes else 0 for i in range(n));S=diag(n,Q)
+ return S.then(fourier(n,axes)).then(S)
+def canonical_frame(n,U):
+ r=len(U)
+ if r==0:return ident(n)
+ if r==n:return fourier(n)
+ G=transpose(basis_extend(U,n),n);gi=inv(G,n);Q=tuple((1<<i)^sum(parity(gi_i&gi_j)<<j for j,gi_j in enumerate(transpose(gi,n))) for i,gi_i in enumerate(transpose(gi,n)))
+ K=perm(n,G).then(diag(n,Q))
+ return K.adjoint().then(fourier(n,range(r))).then(K)
+def monomial(op):
+ n=op.n;T=op.tab;assert all(x==0 and p%2==0 for x,z,p in T[n:]);G=transpose([x for x,z,p in T[:n]],n);b=mv(G,sum((p//2)<<i for i,(x,z,p) in enumerate(T[n:])))
+ gt=transpose(G,n);cols=[mv(gt,z) for x,z,p in T[:n]];Q=transpose(cols,n);alpha=[(p-2*parity(b&z))%4 for x,z,p in T[:n]]
+ out=diag(n,Q,alpha).then(perm(n,G)).then(xgate(n,b));assert out.tab==T
+ vec,den=op.zero_state();assert den==0 and all(a==(0,0) for i,a in enumerate(vec) if i!=b);phase=[(1,0),(0,1),(-1,0),(0,-1)].index(vec[b]);out=out.then(Op(n,identity(n),[('phase',phase)]));assert out.zero_state()==op.zero_state()
+ return out,dict(matrix_rows=G,offset=b,quadratic_rows=Q,linear_Z4=alpha,phase=phase)
+def lower_child(op):
+ n=op.n;Bcols=[x for x,z,p in op.tab[n:]];r=rank(Bcols)
+ if not r:
+  mon,rec=monomial(op);return dict(rank=0,left=rec,right=None),mon
+ G=transpose(basis_extend(Bcols,n),n);A=inv(G,n);PA=perm(n,A);part=op.then(PA);Bcols=[x for x,z,p in part.tab[n:]];Dcols=[z for x,z,p in part.tab[n:]]
+ picks=[];selected=[]
+ for j,x in enumerate(Bcols):
+  if rank(selected+[x])>len(selected):picks.append(j);selected.append(x)
+ assert len(picks)==r and all(x>>r==0 for x in Bcols)
+ Br=transpose(selected,r);Bi=inv(Br,r);Dr=transpose([Dcols[j]&((1<<r)-1) for j in picks],r)
+ Qsmall=tuple(sum(parity(row&col)<<j for j,col in enumerate(transpose(Bi,r))) for row in Dr);Q=Qsmall+(0,)*(n-r);DQ=diag(n,Q)
+ V=part.then(DQ).then(hadamard(n,range(r)));assert all(x==0 for x,z,p in V.tab[n:]);Vmon,Vrec=monomial(V)
+ S=diag(n,tuple(1<<i if i<r else 0 for i in range(n)))
+ left=S.adjoint().then(DQ.adjoint()).then(PA.adjoint());right=Vmon.then(S.adjoint()).then(xgate(n,(1<<r)-1))
+ lmon,lrec=monomial(left);rmon,rrec=monomial(right);compiled=rmon.then(fourier(n,range(r))).then(lmon)
+ assert compiled.tab==op.tab and compiled.zero_state()==op.zero_state()
+ return dict(rank=r,left=lrec,right=rrec),compiled
diff --git a/research/coordinated-gauge-async/inputs/work/pr361/code/engines/base_two_moment.py b/research/coordinated-gauge-async/inputs/work/pr361/code/engines/base_two_moment.py
new file mode 100644
index 0000000000..14a90fa0e7
--- /dev/null
+++ b/research/coordinated-gauge-async/inputs/work/pr361/code/engines/base_two_moment.py
@@ -0,0 +1,63 @@
+#!/usr/bin/env python3
+"""Maintainer arithmetic cross-check of the pinned PR39 certificate.
+
+Uses no producer/checker imports. Counts are inputs, not proved by this script.
+80-term rational logarithms and 12-term exponentials independently enclose
+the two moments. The accompanying written review addresses their realization.
+"""
+import argparse
+from fractions import Fraction as Q
+from hashlib import sha256
+import json
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[1]
+GRID = 10**30
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def log_unit(x):
+    require(1 <= x <= 2, "logarithm range")
+    z = (x-1)/(x+1)
+    lo = 2*sum((z**(2*j+1)/(2*j+1) for j in range(80)), Q(0))
+    return lo, lo+2*z**161/(161*(1-z*z))
+
+
+def log_bounds(x):
+    k = 0
+    while x > 2:
+        x /= 2
+        k += 1
+    lo, hi = log_unit(x)
+    l2, u2 = log_unit(Q(2))
+    lo, hi = lo+k*l2, hi+k*u2
+    # Outward rational rounding keeps later certificates compact.
+    return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID)
+
+
+def exp_bounds(x):
+    require(0 <= x < 1, "exponential range")
+    term = total = Q(1)
+    for j in range(1, 13):
+        term *= x/j
+        total += term
+    tail = term*x/13/(1-x/14)
+    return total, total+tail
+
+
+def moment(m, W, entries, saving):
+    lo = hi = Q(0)
+    for t, count in entries:
+        t, count = int(t), int(count)
+        require(0 < t < m and count > 0, "improper child")
+        l, u = log_bounds(Q(m, t))
+        lower = exp_bounds(saving*l)[0]
+        upper = exp_bounds(saving*u)[1]
+        lo += Q(t*count, m*W)*lower
+        hi += Q(t*count, m*W)*upper
+    return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID)
+
diff --git a/research/coordinated-gauge-async/inputs/work/pr361/code/engines/moment.py b/research/coordinated-gauge-async/inputs/work/pr361/code/engines/moment.py
new file mode 100644
index 0000000000..7d697cb452
--- /dev/null
+++ b/research/coordinated-gauge-async/inputs/work/pr361/code/engines/moment.py
@@ -0,0 +1,113 @@
+"""Independent prospective ordinary five-stage cost, never supplier admission.
+
+Consumes a raw one-invocation ledger made from the actual source/internal/target
+paths.  No packed profile is used as an input.  Missing physical/finite/compiler
+obligations must remain explicit in the accompanying report.
+"""
+from collections import Counter
+from decimal import Decimal, localcontext, ROUND_FLOOR
+from fractions import Fraction as F
+from functools import lru_cache
+from math import factorial
+import hashlib, json, sys
+from pathlib import Path
+
+DYAD = 1 << 224
+
+def floor(x): return F(x.numerator*DYAD//x.denominator, DYAD)
+def ceil(x): return -floor(-x)
+
+@lru_cache(None)
+def logarithm(x):
+    """Mercator bounds after binary reduction; ln(2) via two Mercator sums.
+
+    For 1 <= y < 2, log(y) is an alternating series in z=y-1.
+    We instead reduce to 1 <= y < 3/2, and bound log(3/2) and
+    log(2) by the same alternating series at z=1/2 and z=1/3,
+    using log(2)=log(3/2)+log(4/3). All sums are exact rationals.
+    """
+    assert x >= 1
+    k = 0
+    while x >= 2: x /= 2; k += 1
+    j = int(x >= F(3, 2))
+    if j: x /= F(3, 2)
+    def mercator(y):
+        z=y-1
+        assert 0 <= z <= F(1,2)
+        # An even partial sum is below log; the next term bounds its tail.
+        t=z; s=F(0)
+        for n in range(1, 225):
+            s += (t if n & 1 else -t)/n
+            t *= z
+        return s, s+t/225
+    a,b=mercator(x); c,d=mercator(F(3,2)); e,f=mercator(F(4,3))
+    return floor(a+(k+j)*c+k*e), ceil(b+(k+j)*d+k*f)
+
+def exponential(l,u):
+    assert 0 <= l <= u < F(1,10)
+    def poly(x):
+        s=t=F(1)
+        for j in range(1,25): t=t*x/j; s+=t
+        return s,t
+    lo,_=poly(l);hi,last=poly(u)
+    return floor(lo),ceil(hi+last*u/25/(1-u/26))
+
+def moment(H,m,W,a,bad=False):
+    lo=hi=F(0)
+    for r,n in H.items():
+        assert 0 < r < m and n > 0
+        l,u=logarithm(F(m,r)); e,f=exponential(a*l,a*u)
+        p=F(r*n,m*W);lo+=p*e;hi+=p*f
+    if bad:
+        # Existing envelope shape, only conditionally portable to the new cover.
+        l,u=logarithm(F(m));e,f=exponential(a*l,a*u)
+        p=F(32*m*sum(H.values()),10**16*W)
+        lo+=p*e;hi+=p*f
+    return floor(lo),ceil(hi)
+
+def decimal(x):
+    with localcontext() as c:
+        c.prec=70
+        return str(Decimal(x.numerator)/Decimal(x.denominator))
+
+def certify(H,m,W,bad=False):
+    with localcontext() as c:
+        c.prec=70
+        terms=[(Decimal(r*n)/Decimal(m*W),(Decimal(m)/Decimal(r)).ln()) for r,n in H.items()]
+        fallback=Decimal(32*m*sum(H.values()))/Decimal(10**16*W) if bad else Decimal(0)
+        lm=Decimal(m).ln();lo=Decimal(0);hi=Decimal('.003')
+        for _ in range(210):
+            a=(lo+hi)/2
+            if sum(p*(a*l).exp() for p,l in terms)+fallback*(a*lm).exp()<1:lo=a
+            else:hi=a
+        q=int(((lo+hi)/2*10**24).to_integral_value(rounding=ROUND_FLOOR))
+    lo=F(q,10**24);hi=lo+F(1,10**24)
+    lv=moment(H,m,W,lo,bad);hv=moment(H,m,W,hi,bad)
+    assert lv[1]<1<hv[0],(lo,hi,lv,hv)
+    k=F(694782719690783,10**18);required=k/(1-k)
+    at=moment(H,m,W,required,bad)
+    return dict(lower=decimal(lo),upper=decimal(hi),lower_endpoint_residual_upper=decimal(lv[1]-1),upper_endpoint_residual_lower=decimal(hv[0]-1),strict_rational_signs=True,moment_minus_one_at_current_k_required_rate=[decimal(x-1) for x in at],nominal_k_ceiling_interval=[decimal(lo/(1+lo)),decimal(hi/(1+hi))])
+
+def price(core,gauges,R,v,h):
+    core=Counter({int(k):n for k,n in core.items() if n})
+    gauges=Counter({int(k):n for k,n in gauges.items() if n})
+    m=5*h;W=4*v+R;ell=h*(h-2)  # p = 10 port: was the p = 12 literal 24*22
+    assert sum(r*n for r,n in core.items())==h*R+2*v*(h-1)+ell-sum(r*n for r,n in gauges.items())
+    idle=Counter({r:2*v for r in (2*h-2,h-1,2*h+2,4)})
+    ans={}
+    for name,bundled in [('separate_rank_a_corrections',False),('bundled_rank_5a_corrections',True)]:
+        H=Counter({r:5*n for r,n in core.items()});H.update(idle)
+        for a,n in gauges.items():H[5*a if bundled else a]+=n if bundled else 5*n
+        mass=sum(r*n for r,n in H.items());assert m*W-mass==4*v-5*ell
+        ans[name]=dict(histogram=dict(sorted(H.items())),calls=sum(H.values()),rank_mass=mass,deficit=m*W-mass,maxchild=max(H),root=certify(H,m,W),root_with_inherited_bad_envelope=certify(H,m,W,True))
+    return dict(status='PROSPECTIVE_COST_ONLY_NOT_PHYSICAL_OR_ASSEMBLY_ADMISSION',m=m,W=W,R=R,v=v,h=h,copy_calls_per_invocation=h,copy_rank=h-2,ell=ell,raw_core=dict(sorted(core.items())),auxiliary_entrances=dict(sorted(gauges.items())),idle_histogram=dict(sorted(idle.items())),profiles=ans,interval_method='Independent exact rational alternating Mercator log bounds (224 terms), 24-degree exponential with geometric tail, 224-bit outward dyadic rounding; Decimal only proposes endpoints.',bad_envelope_scope='10^-16 and 32*m^2 rank-one fallback calls per paid child are inherited assumptions, not a new m = 5h admission.')
+
+if __name__=='__main__':
+    path=Path(sys.argv[1]);raw=path.read_bytes();data=json.loads(raw)
+    core=data['one_stage_helper_histogram_including_copies']
+    gauges=data['auxiliary_entrance_rank_histogram']
+    result=price(core,gauges,data['physical_R'],data['v'],data['h'])
+    result['input_ledger_sha256']=hashlib.sha256(raw).hexdigest()
+    result['input_head']=data['head']
+    result['one_invocation_parts']={key:data[key] for key in ('physical_internal_excluding_center_copies','paid_center_copy_histogram','physical_source_histogram','physical_target_histogram')}
+    print(json.dumps(result,indent=2))
diff --git a/research/coordinated-gauge-async/inputs/work/pr361/code/engines/outer.py b/research/coordinated-gauge-async/inputs/work/pr361/code/engines/outer.py
new file mode 100644
index 0000000000..43007488ad
--- /dev/null
+++ b/research/coordinated-gauge-async/inputs/work/pr361/code/engines/outer.py
@@ -0,0 +1,58 @@
+#!/usr/bin/env python3
+"""Exact semantic assembly with the balanced prefix, for the paired-cube network.
+
+Prepared by eumemic with Anthropic Claude assistance. Apache-2.0. The 47 strict constraints, the seven margins and
+the finite bridge are unchanged from icekylinx's scripts/structured_bulk_assembly.py (PR144, adapted from Zhihao
+Chen's PR23). The prefix is the RaD project's (hipotures) balanced positional transform, reviewed in
+references/semantic-bulk/rad20/reports/review-balanced-transform.md: the prefix margin becomes 1-eps instead of
+1-eps(1+c), and eps(1+c)<1 remains a strict geometric condition (K_geometry) whose gap need not exceed kappa.
+The parameters c=q+eta/4 and eps=(1-eta)/(1+q) are those of the reviewed balanced assembly in
+research/copied-fixed/balanced_assembly.py, with its backoff h equal to eta. Nothing else differs from
+structured_bulk_assembly.assembly.
+"""
+from fractions import Fraction as Q
+
+
+def assembly(bit_saving,complex_saving,bridge,kappa,eta=Q(1,10**8),beta=Q(1,4)):
+    a,b=bit_saving,complex_saving
+    tau,sigma=1-a,1-b
+    q=a*(1-2*eta); lp=1-q; lam=(tau+lp)/2
+    c=q+eta/4; eps=(1-eta)/(1+q)
+    minimum=eps*q; r=(minimum+1-eps)/2; delta=eta/8
+    internal=tau+(1-beta)*max(sigma-tau,Q(0));leaf=sigma+beta*(1-sigma)
+    margins=dict(balanced_prefix=1-eps,coordinate_movement=a,
+        compact_phase_layer=minimum,bulk_exposure=a,
+        Gaussian_arithmetic=min(1-eps-delta,r-delta),
+        scalar_work=1-eps-delta,dimension=eps)
+    slacks=dict(bit_positive=a,complex_above_bit=b-a,
+        complex_below_one_over32=Q(1,32)-b,beta_positive=beta,
+        beta_below_one=1-beta,leaf_saving_above_bit=(1-beta)*b-a,
+        q_positive=q,q_below_internal=1-internal-q,q_below_leaf=1-leaf-q,
+        c_positive=c,c_below_one=1-c,q_below_reservations=c-q,
+        lambda_above_tau=lam-tau,lambda_above_sigma=lam-sigma,
+        lambda_above_internal=lam-internal,lambda_prime_above_lambda=lp-lam,
+        compact_leaf=lp-leaf,compact_reservations=lp-(1-c),
+        lambda_prime_below_one=q,epsilon_positive=eps,epsilon_below_one=1-eps,
+        guard_width=1-eps,K_geometry=1-eps*(1+c),K_dominates_log=eps*c,
+        record_suffix=1-eps,phase_local=1-eps-delta,phase_boundary=r-delta,
+        gamma_sublinear=1-eps-r,cell_above_band=eps-(1-r)/2,
+        prime_interval_packing=1-eps,alpha_positive=r,alpha_below_one=1-r,
+        alpha_below_one_fourth=Q(1,4)-r,delta_positive=delta,
+        delta_below_one_eighth=Q(1,8)-delta,short_record_fallback=eps-a,
+        small_field_exposure=1-eps-minimum,
+        artificial_boundary=8-eps+r-delta-minimum,
+        literal_scalar_guard=Q(bridge['semantic']['strict_literal_gap']),
+        row_product_gap=bridge['rows']['degree_gap'])
+    slacks.update({name+'_above_kappa':val-kappa for name,val in margins.items()})
+    assert len(slacks)==47 and len(margins)==7
+    assert all(v>0 for v in slacks.values()),{k:str(v) for k,v in slacks.items() if v<=0}
+    assert min(margins.values())==minimum
+    assert margins['balanced_prefix']-minimum==eta and 1-eps-r==eta/2
+    assert 1-eps*(1+c)==eta-eps*eta/4
+    return dict(parameters=dict(a_bit=a,a_complex=b,tau=tau,sigma=sigma,
+            eta=eta,beta=beta,q=q,c=c,epsilon=eps,lambda_=lam,
+            lambda_prime=lp,alpha_squared_power=r,delta=delta,
+            C0=bridge['semantic']['C0'],C1=1,kappa=kappa),
+        strict_constraints=slacks,margins=margins,minimum_margin=minimum,
+        absorption_gap=minimum-kappa,
+        recurrence=dict(internal=internal,leaf=leaf,reservations=1-c))
diff --git a/research/coordinated-gauge-async/inputs/work/pr370-source/research/e8-bit/data/bit/bitE8-r865.json.gz b/research/coordinated-gauge-async/inputs/work/pr370-source/research/e8-bit/data/bit/bitE8-r865.json.gz
new file mode 100644
index 0000000000..8bec1751f7
Binary files /dev/null and b/research/coordinated-gauge-async/inputs/work/pr370-source/research/e8-bit/data/bit/bitE8-r865.json.gz differ
diff --git a/research/coordinated-gauge-async/references/APPENDIX-SOURCES.json b/research/coordinated-gauge-async/references/APPENDIX-SOURCES.json
new file mode 100644
index 0000000000..485655d40f
--- /dev/null
+++ b/research/coordinated-gauge-async/references/APPENDIX-SOURCES.json
@@ -0,0 +1,112 @@
+[
+  {
+    "repository": "https://github.com/openai/math",
+    "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
+    "relative_path": "lean/OAI/Combinatorics/MatroidCounting/CommonBases.lean",
+    "working_sha256": "0f2d97183b830e8900a73c5637a6c243b03eb817cf514487b0db83db94434484",
+    "matches": [
+      {
+        "name": "greedyOracle_correct",
+        "lines": [
+          10965
+        ]
+      },
+      {
+        "name": "greedyOracle_rank",
+        "lines": [
+          11000
+        ]
+      },
+      {
+        "name": "pair_rank_parallel",
+        "lines": [
+          549
+        ]
+      },
+      {
+        "name": "pair_rank_nonparallel",
+        "lines": [
+          556
+        ]
+      }
+    ],
+    "status": "Direct source inspection; no fresh Lean kernel audit in this round"
+  },
+  {
+    "repository": "https://github.com/openai/math",
+    "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
+    "relative_path": "lean/OAI/Topology/ArtinGroups/Cancellation/BlockPivot.lean",
+    "working_sha256": "d3a8dbe1f5b68b8ebde555a1f85be1f655c20d6195a857135347aa88c1560c39",
+    "matches": [
+      {
+        "name": "homotopy_square",
+        "lines": [
+          108
+        ]
+      },
+      {
+        "name": "homotopy_retract",
+        "lines": [
+          118
+        ]
+      },
+      {
+        "name": "remainder_retained",
+        "lines": [
+          198
+        ]
+      }
+    ],
+    "status": "Direct source inspection; no fresh Lean kernel audit in this round"
+  },
+  {
+    "repository": "https://github.com/openai/math",
+    "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
+    "relative_path": "lean/OAI/NumberTheory/TwoPoint/Walks/ColumnRankRelations.lean",
+    "working_sha256": "535603fcb618d881bb0dc829fed17790f697c7dd90fa06153b0efe8605fc1f46",
+    "matches": [
+      {
+        "name": "columnDifference_disjoint_minor",
+        "lines": [
+          94
+        ]
+      }
+    ],
+    "status": "Direct source inspection; no fresh Lean kernel audit in this round"
+  },
+  {
+    "repository": "https://github.com/openai/math",
+    "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
+    "relative_path": "lean/OAI/Computability/FourierCircuit/TensorTools.lean",
+    "working_sha256": "40f36b086973e91bdbb9b94184e866375793e45ad57da5dd5c60b66d83542cf8",
+    "status": "Direct source inspection, not fresh Lean certification in this round"
+  },
+  {
+    "repository": "https://github.com/openai/math",
+    "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
+    "relative_path": "lean/OAI/Computability/FourierCircuit/SparseShear.lean",
+    "working_sha256": "1a711340e5b75cf5b94909659e97becf1158bf4414770392d09e14de0549af8c",
+    "status": "Direct source inspection, not fresh Lean certification in this round"
+  },
+  {
+    "repository": "https://github.com/openai/math",
+    "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
+    "relative_path": "lean/OAI/Computability/FourierCircuit/PivotJoin.lean",
+    "working_sha256": "06a9a4bdbdc3903dc021a2a8a402e5b67ee25b84a898339efbb7b747339ab389",
+    "status": "Direct source inspection, not fresh Lean certification in this round"
+  },
+  {
+    "repository": "https://github.com/openai/math",
+    "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
+    "relative_path": "lean/OAI/Computability/FourierCircuit/MonomialFeedback.lean",
+    "working_sha256": "534b24551437a70259fd11916ec9bfd114f4353d225d965be401e3ab28defaa0",
+    "status": "Direct source inspection, not fresh Lean certification in this round"
+  },
+  {
+    "repository": "https://github.com/openai/math",
+    "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
+    "relative_path": "lean/OAI/Computability/FourierCircuit/WordScheduling.lean",
+    "working_sha256": "0f1b0c89091f6acc5310e95c07102792749a38b0e35a760b7feae25682ef2713",
+    "status": "Direct source inspection, not fresh Lean certification in this round"
+  }
+]
diff --git a/research/coordinated-gauge-async/references/PR493-NOTICE.md b/research/coordinated-gauge-async/references/PR493-NOTICE.md
new file mode 100644
index 0000000000..d9e0c531bc
--- /dev/null
+++ b/research/coordinated-gauge-async/references/PR493-NOTICE.md
@@ -0,0 +1,11 @@
+# Attribution for the included PR493 material
+
+Apache-2.0. Prepared by eumemic with substantial Anthropic Claude assistance.
+The only copied PR493 payload in this package is its published EXPECTED.json.
+Its R489 bit operator is cited, not vendored or freshly reconstructed here.
+
+The original complete notice is available with the pinned source:
+https://github.com/CrocSwap/integer-mult-bounds/blob/c3a9a436b84fe44fdc223a5784d84736d1c2d3c7/research/e8-bit-carrier-skeleton/NOTICE.md
+
+This attribution excerpt omits sections for code and historical receipt files
+that are not included in this package. It does not modify any upstream file.
diff --git a/research/coordinated-gauge-async/references/PR496-PROOF.md b/research/coordinated-gauge-async/references/PR496-PROOF.md
new file mode 100644
index 0000000000..aa8230f063
--- /dev/null
+++ b/research/coordinated-gauge-async/references/PR496-PROOF.md
@@ -0,0 +1,105 @@
+# Whole-owner pooling of the complex supplier
+
+This construction is conditional on the all-size compiler, common-label,
+restored-row, precision/tape and analytic-transfer contracts retained by the
+pinned PR493 bit and PR484 complex packages. It also retains PR484's explicit
+Lemma A realization contract for paid residual wrappers with borrowed spectators.
+Its Lemma G schedule/conjugation identities are checked on the serialized local
+programs, with independent pre-operation integer bounds. Finite verification
+alone does not establish the inherited all-size contracts.
+
+## Placement
+
+Keep PR484's gauge80-kernel42 local word, its exact Gaussian phases, both
+orientations, and all entry/exit wrappers. There are 560 helper roles in each of
+72 complete local invocations, indexed by nine stages and eight replicas. A
+helper of entrance rank e has one residual of width 9-e. No residual is split.
+
+Pool the idle coordinates of all eight physical replicas. A frozen allocation
+places every whole residual in a single pending idle child, and a frozen
+bipartite edge coloring assigns that slot to its invocation replica. Left
+vertices are physical families; right vertices are actual helpers. Each helper
+has degree eight and every physical family has degree at most eight at the
+stage. The eight matching colors give each invocation exactly one of every
+helper and at most one helper on any family. The verifier reads every actual
+role, rather than trusting existence of a coloring or the optimizer status.
+
+Every host is an idle pair at that stage. Thus no host is an active data wire,
+even if its physical replica differs from its invocation replica. The literal
+loop order is retained: recipe phase, replica, cover class, complete local
+invocation. Entry wrapper, local word and exit wrapper all finish before the
+next replica or cover class. This is necessary when spectators are borrowed
+from coordinates that another invocation later uses.
+
+## Actual maps and restoration
+
+For each physical port q, take the pinned orthogonal 73-column basis B_q. A
+helper's first w columns are precisely its pending residual coordinates. Its
+next 9-w columns are the canonical unused spectator coordinates. Append every
+remaining B_q column in increasing basis-index order. The resulting full
+matrix is an orthogonal coordinate permutation of B_q, with inverse its
+transpose. Both reflected local programs use this same whole-family chart.
+
+The fresh PR484 local check establishes the normalized helper action
+L_E^-1 S_E R_E^-1 = C_[w] tensor I on all 512 local address columns, including
+spectators, with all global phases. Its emitted scalar word restores arbitrary
+dirty helper inputs and its private work. Unique physical owners within a
+complete invocation let the scalar/common-label identity be transported through
+these full charts without aliasing. Because that invocation restores the
+borrowed work before the next, reusing the physical family in a later replica
+is allowed under the same whole-invocation realization contract.
+
+Across all calls, occupied cells of each pending idle child are disjoint. The
+explicit remainder child uses that family's actual replica, starts at the
+recorded pending offset, and executes once. The old `replicas=8` shorthand is
+not applied to these heterogeneous remainders. The independent controller audit
+checks these cells, execution windows, full charts, source-program hashes,
+private-family rebase and cover correction. Deliberately corrupted programs
+must fail the controller audit.
+
+## Paid profile and finite charge
+
+All 40,320 whole residuals fit on the 7,680 existing data families. There are no
+bank families. The physical stock is 8,258, including unchanged private centres,
+reserved tableau and two coefficient registers. The complete paid profile is
+
+```
+rank   1       2      3     4      5     6    7     8
+count 148440  66816  7472  19728  25664  144  1800  4120
+```
+
+Its rank mass is 558,144, and 73*7680 - 558144 = 2496. Both exact moment engines
+include the inherited bad-cover fallback. The complete invoice counts unchanged
+local primitives, wrappers and selectors, and rebases stock, private families,
+idle remainders, routing and final correction to the actual emitted program.
+No new factor is hidden in a cached coefficient. For the global prefix, use
+maximum stream L2 norm over every physical family and cover class. Each
+completed helper endpoint is unitary; each completed data shear or bridge has
+at most two unitary summands. For each actual orthogonal chart, right
+multiplication permutes the cover classes, so an active stream is visited once
+per complete call sweep. The bound uses all 72 ordered call sweeps and nine
+bridges, without assuming replica independence. The local scalar/Horner envelope
+costs 20 bits; conversion from the initial supremum norm to L2 on 2^73 entries
+costs 37 bits; four bits cover primitive arithmetic. Thus 142 bits suffice,
+within the retained 192-bit allowance. The verifier checks the actual matrices,
+complete-call endpoints, scalar envelope and controller premises used here. Each remainder is at most
+rank eight, so the degree-two halving inequality is 73^2 > 2*8^2.
+
+The bit side is reconstructed from PR493 with real Boost. Its physical prefix,
+charged three-level coefficient and row coefficient are read from the fresh
+receipt. Six bootstrap levels and all 47 strict outer inequalities use the
+fresh complex exponent, common row allowance, and C0=2^210000. The next grid
+point and zero-eta, zero-beta and unit-beta controls must all fail.
+
+## Provenance and limits
+
+The selection is a finite nomination, not an optimality theorem. Search used
+floating-point MILP objectives and matching only to propose the frozen map;
+certification uses the actual scalar words, complete emitted role map, exact
+integer/rational checks and fresh source replays. The earlier unpooled padding
+experiment is retained as provenance, not a separately published record.
+
+This does not use the cross-family split of PR448/480/494. Those submissions
+carry a distinct disputed operator interface. This package makes a conditional
+whole-owner claim only and does not claim the largest number among all draft
+submissions under different assumptions.
diff --git a/research/coordinated-gauge-async/references/PRIOR-RESEARCH.md b/research/coordinated-gauge-async/references/PRIOR-RESEARCH.md
new file mode 100644
index 0000000000..b258199878
--- /dev/null
+++ b/research/coordinated-gauge-async/references/PRIOR-RESEARCH.md
@@ -0,0 +1,62 @@
+# Research appendix: corpus reuse, the achieved mechanism, and preserved candidates
+
+The delivered conditional bound is **kappa = 0.001357701757628940**. Its new contribution is asynchronous scheduling of complete replica invocations. The local gauge/kernel word is unchanged from PR484/496; the stronger bit supplier comes from PR493. The package's final receipts, physical audit and exact assembly are the evidence for the numerical claim. The inherited complete-invocation, compiler and analytic contracts remain explicit conditions; this is not a new Lean certification of integer multiplication.
+
+## The mechanism that improved the bound
+
+Earlier packing advanced all eight replicas in lockstep. The new schedule preserves each replica's nine-stage controller order while allowing replicas to occupy different stages. A helper may then use an idle block whenever its entire invocation lies between consecutive touches on the **owner's** timeline. The selected order sorts calls by `(stage + 3*replica/7, stage, replica)`.
+
+Correctness uses completed invocation endpoints. Each invocation restores arbitrary dirty workspace and borrowed spectators, and leaves only its specified residual on its assigned pending block. Different residual blocks of an idle child are disjoint; their product with the paid remainder is the original child. Completing that product before the owner's next touch preserves each replica's original controller. No scalar gate is commuted through an unfinished invocation, and no residual is split between owners.
+
+A flow nomination is converted to literal owners using bipartite edge coloring. There are 72 calls and 40,320 helper occurrences, with 560 distinct helper families per call. Ambient dimension 73, live stock7680 and deficit 2496 stay fixed. The changed paid profile is:
+
+    1:148240, 2:66816, 3:7032, 4:19728, 5:26109,
+    6:144, 7:2209, 8:1224, 16:1, 48:408.
+
+The complete schedule needs its own physical chart checks and prefix invoice. Its gain does not follow from the floating packing objective alone.
+
+## What was actually reused from the OpenAI corpus
+
+The inspected snapshot is `adc7f1241b42e322a6451854ab7e4b4c146bf78a` of [openai/math](https://github.com/openai/math). `APPENDIX-SOURCES.json` records repository-relative source paths and hashes. Direct source inspection and Git consistency checks are distinct from compiling these modules in this round; no claim is made that every paper in the collection is formalized.
+
+- [FourierCircuit/WordScheduling.lean](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/FourierCircuit/WordScheduling.lean): `scheduledStep_product` and `scheduledLayer_product` preserve products under order-preserving schedules on fixed disjoint supports. `Word.matrix_substitute` and `Word.weight_substitute` preserve semantics and account for substituted calls. These supplied the structural template for scheduling complete normalized invocations. The new owner-relative interval argument is stated and checked separately; the fixed-support theorem is not silently applied to aliased unfinished programs.
+- [MatroidCounting/CommonBases.lean](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/MatroidCounting/CommonBases.lean): `greedyOracle_correct`, `greedyOracle_rank`, and the parallel/nonparallel pair-rank lemmas motivated response-matroid basis and circuit-component selection. The common-integer-polymatroid preprint, problem114, also suggested moving capacity across fixed stage partitions. The actual packing family is not asserted to be a matroid, and no counting FPRAS is used as a packing certificate.
+- `FourierCircuit/TensorTools.lean`, `RowCompression.sparse_row_compression`, and `SparseShear.lean`, `cancellation_identity`, provide explicit row-basis changes and cancellation of a shared exterior block. Independent dirty replica coordinates cannot be deleted merely by compressing a displayed response matrix. No additional gain was obtained from this route.
+- `FourierCircuit/PivotJoin.lean`, `PivotJoin.of_join`, and `MonomialFeedback.lean`, `Dense2x2.price`, suggested alternative local compositions. The latter's abstract normalized price bound is not a literal bounded-coefficient routine in this model. The exact-Fourier documentation also limits its theorem to a subsequence; it does not supply the missing all-length bit-complexity implementation.
+- `Topology/ArtinGroups/Cancellation/BlockPivot.lean`, `homotopy_square`, `homotopy_retract`, `remainder_retained`, and `NumberTheory/TwoPoint/Walks/ColumnRankRelations.lean`, `columnDifference_disjoint_minor`, were inspected for simultaneous elimination and donor/pivot separation. They did not produce another priced gain in this round.
+
+## Three checked local constructions, not added to the achieved gain
+
+For a common entrance frame, choose a basis of a response-matroid component as donors. Each remaining column is an exact rational donor combination. Prepending its cancelling donor shears makes that pivot's dirty response zero; reverse cleanup restores the helper state. All required frame nesting and source spans must still hold.
+
+The following concrete words passed exact rational reconstruction, all 800 arbitrary-dirt columns in both forward and inverse-transpose orientations, and every individual pivot-correction omission control:
+
+| Candidate | Local profile change from PR484/496 | Residual counts (width:count) | Omission controls |
+|---|---|---|---:|
+| COMPONENT | -10 rank 1, -14 rank 2, +14 rank 3 | 5:4, 7:14, 8:100, 9:442 | 38 |
+| JOINT-47 | -14 rank 1, -2 rank 2, -4 rank 3, +4 rank 4, +4 rank 5 | 5:4, 7:12, 8:102, 9:442 | 48 |
+| JOINT-51 | -14 rank 1, -4 rank 2, +8 rank 3, -2 rank 4, +2 rank 5 | 5:4, 7:14, 8:100, 9:442 | 40 |
+
+These improve selected local objectives but lose scarce narrow residual inventory. Their tested lockstep allocations, including stage-selective portfolios and paid banks, did not improve the full bound. None is counted in the delivered kappa. They remain useful sufficient constructions for future schedules.
+
+The companion base, three selections and `REBUILD.json` permit source-bound reconstruction through `rebuild_candidate.py`; `audit_component_scalar.py` checks each reconstructed word independently. `LOCAL-CANDIDATE-AUDITS.json` preserves the three exact scalar receipts. The constructor reuses the retained, hash-pinned `gk_word.py` and `gk_common.py` with their attribution.
+
+## Other bounded lessons
+
+All14,851 subspaces contained in first-touch frames were screened. Under the fixed80 gauge exclusions, only 14 classes had positive response nullity, all dimension 1. Allowing excluded gauge helpers exposed63 additional components, still dimension 1. A catalogue of 646 components from deletion minors up to depth two did not beat the inherited42-pivot width budget. These are finite-search results, not global impossibility theorems.
+
+The proposed gauged-donor ordering is algebraically sufficient when original dirt compensation precedes the kernel update. Adding individual donors while retaining the current palette failed exact target/centre nesting in all 86 tested cases; geometric admissibility alone was insufficient. Whole-palette replacements remain a separate route.
+
+Native frame-retiming searches on the full joint gauge word and its cleanup extension found no better profile. A direct lift of PR493's bit skeleton to the complex side also failed a specific structural test: all 14,400 port pairs obey complemented adjacency. An even-parity-frame repair and the identity `G Hadamard G + A_orth - J = 3 I` are constructive leads, but require extra paid copies and a different decoder; they are not achieved improvements.
+
+The productive change was therefore at the scheduling level: reuse the already checked local engine, remove replica lockstep, and reconstruct the complete physical ownership and prefix bill.
+
+
+Portable companion files are under `research-notes/prior-kernel/`. From the package root, rebuild a local candidate with:
+
+```
+python -B research-notes/prior-kernel/rebuild_candidate.py vendor/complex/e8-complex-gauge-kernel/code research-notes/prior-kernel JOINT-47 /new/output-word.json.gz
+python -B research-notes/prior-kernel/audit_component_scalar.py /new/output-word.json.gz research-notes/prior-kernel/JOINT-47-SELECTION.json /new/scalar-receipt.json
+```
+
+These separate local identities are not added to the achieved kappa.
diff --git a/research/coordinated-gauge-async/references/PRIOR-SCHEDULING-LEMMA.md b/research/coordinated-gauge-async/references/PRIOR-SCHEDULING-LEMMA.md
new file mode 100644
index 0000000000..9328fb1ee5
--- /dev/null
+++ b/research/coordinated-gauge-async/references/PRIOR-SCHEDULING-LEMMA.md
@@ -0,0 +1,25 @@
+# Sufficient asynchronous whole-owner scheduling lemma
+
+This is a mathematical composition proof, not Lean certification. It applies to the complete-call interfaces already retained by PR496. It does not permit splitting a residual across physical owners or stopping before dirty workspace restoration.
+
+Let physical families be mutually independent vector spaces. A local invocation is an explicitly implemented finite word whose COMPLETE endpoint factors as its intended data operation on one replica and one prescribed residual operator on each distinct helper owner; all private workspace and borrowed spectator coordinates are restored. A pending idle child on a family factors into commuting coordinate residuals on disjoint coordinate blocks and its complementary remainder.
+
+A finite schedule is sufficient if:
+
+1. For each data replica it preserves the order of all of that replica's data operations, including bridges and recipe phases.
+2. Every helper use occurs inside the pending child's interval on the OWNER replica's timeline, and that owner is not an active data family in the invocation.
+3. Pending coordinate blocks allocated to different uses are disjoint; all blocks and the one complementary remainder cover the original pending child exactly.
+4. Every complete invocation has distinct physical helper owners and finishes its entry wrappers, scalar body, cleanup, exit wrappers and full cover sweep before the next invocation. It restores spectators even when later calls will use those coordinates actively.
+5. All allocated residuals and the remainder finish before the owner replica's next operation that depends on completion of that pending child.
+
+Then the schedule has exactly the same complete endpoint as the original controller. Proof: replace each completed local invocation by its established block-diagonal endpoint. This is valid on arbitrary dirty input, hence remains valid after any preceding invocation. On each family, data operations occur in their original order. Between two consecutive such operations, the only newly ordered factors are residuals on disjoint coordinate blocks; they commute, and their product with the remainder is the original pending child. Regroup these factors inside each family interval. Their product is unchanged, so the overall controller endpoint is unchanged. No operation is commuted through an unfinished scalar body, and no independent-dirt claim is inferred merely from a logical alias. The full-cover right-translation bijection and actual owner maps remain premises of each complete invocation.
+
+Replica lockstep is therefore unnecessary for this sufficient composition argument. The construction still needs an explicit schedule, owner injection, full maps, interval checks, and paid profile.
+
+## Prefix bill
+
+The old synchronized prefix bound cannot be copied without checking its batching premise. If nine bridge rounds become as many as seventy-two ordered bridge sweeps, a safe logarithmic envelope should count those sweeps individually, alongside the seventy-two complete local sweeps, exact local transient norm, and conversion from entry sup norm to full-family L2. Enlarging the finite coefficient allowance is allowed if the outer assembly pays it. Alternatively an independent argument can retain a tighter norm bound by tracking disjoint active replicas. This is an invoice issue, not an extra recursive child count.
+
+## Structural source
+
+OpenAI FourierCircuit/WordScheduling.lean gives fixed-support scheduling and substitution identities. The proof above uses the same support-sensitive mechanism, with completed normalized invocations as its indivisible units. SparseShear's whole-call-index constructions reinforce why physical supports, not just abstract row dimensions, must remain explicit.
diff --git a/research/coordinated-gauge-async/research-notes/ALIAS-SCREEN.json b/research/coordinated-gauge-async/research-notes/ALIAS-SCREEN.json
new file mode 100644
index 0000000000..dcb3ccdeef
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/ALIAS-SCREEN.json
@@ -0,0 +1,338 @@
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index 0000000000..c8d28e6104
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diff --git a/research/coordinated-gauge-async/research-notes/COORDINATED-RANK2-SCALAR.json b/research/coordinated-gauge-async/research-notes/COORDINATED-RANK2-SCALAR.json
new file mode 100644
index 0000000000..accf1ef508
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/COORDINATED-RANK2-SCALAR.json
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+    "212": "REJECTED_OMITTED_KERNEL_CORRECTION",
+    "213": "REJECTED_OMITTED_KERNEL_CORRECTION"
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+}
diff --git a/research/coordinated-gauge-async/research-notes/COORDINATED-RANK2-WIDTHS.json b/research/coordinated-gauge-async/research-notes/COORDINATED-RANK2-WIDTHS.json
new file mode 100644
index 0000000000..b862562801
--- /dev/null
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+}
\ No newline at end of file
diff --git a/research/coordinated-gauge-async/research-notes/GAUGE-SUBFRAME-SCREEN.json b/research/coordinated-gauge-async/research-notes/GAUGE-SUBFRAME-SCREEN.json
new file mode 100644
index 0000000000..bae24d0702
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/GAUGE-SUBFRAME-SCREEN.json
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diff --git a/research/coordinated-gauge-async/research-notes/PAIRED-PARITY-SCREEN.json b/research/coordinated-gauge-async/research-notes/PAIRED-PARITY-SCREEN.json
new file mode 100644
index 0000000000..280fbb2180
--- /dev/null
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diff --git a/research/coordinated-gauge-async/research-notes/PENDING-FACTOR-TRANSPORT.md b/research/coordinated-gauge-async/research-notes/PENDING-FACTOR-TRANSPORT.md
new file mode 100644
index 0000000000..d9d895af34
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/PENDING-FACTOR-TRANSPORT.md
@@ -0,0 +1,29 @@
+# Constructive pending-factor transport and deferred terminal suffixes
+
+## Sufficient retiming identity
+
+Let X_v be translation by v on a finite F2 address space, and C_v=((1+i)I+(1-i)X_v)/2. Direct expansion gives C_v C_w=C_w C_v and C_v^2=X_v. Let a scalar array shear be S_A(x,y)=(x,y+A x). Then diag(Q_x,Q_y) commutes with S_A exactly when Q_y A=A Q_x. In particular identical products Q of C_v commute through both sides of a shear whose A is a polynomial in translations.
+
+This applies to the actual completed complex local endpoint: its blocks are products F C_q^-1 and translations (the source defines F as the product of C_coordinate, NOT the Hadamard matrix). It does not apply blindly to intermediate canonical-frame gates. The complete invocation must finish, all private registers clear, and all borrowed spectators restore before invoking the identity.
+
+A more flexible sufficient identity allows monomial bridge changes. If a retiming changes cumulative exponents on bridge endpoints by d_x(v),d_y(v), and all differences d_y(v)-d_x(v) are even, the conjugated bridge coefficient differs by the translation product of X_v raised to half those differences. Emit and charge that translation in the bridge; the endpoint identity is then exact on arbitrary dirty input arrays. Equal parities are checked for each actual vector v, not for unrelated local coordinate indices. This criterion is sufficient for the specifically serialized translated bridge. It is not a license to change a bridge without emitting its compensation.
+
+## A usable specialization requiring no new bridge at all
+
+Take an asynchronous controller preserving each replica's original local and bridge order. When a replica completes its last local invocation, postpone its entire remaining suffix: late idle-child completions, its terminal bridge chain, its terminal translations and terminal data operation. Finish all replicas' local invocations first, then execute these suffixes. Leave the common global correction last.
+
+Every postponed event touches only its replica's data families (plus unchanged cleared private workspace). Other replicas may borrow one of those families only through a complete normalized helper invocation. Such an invocation applies its declared residual to a subset of that owner's still-pending idle axes, restores all other coordinates including spectators, and clears all private registers. Distinct residual footprints on the owner commute, and their product with the explicit remainder is exactly the original pending idle child. Therefore regrouping all helper endpoints before the deferred idle completion gives exactly the old per-replica endpoint. The terminal bridges see fully completed original data states. The construction is valid for arbitrary initial owner dirt.
+
+Concrete K4 consequence: late children 12..17 can remain available after their owner's stage8 local invocation, until the common end of all local calls. Their release points remain after the owner's stage2, stage4 or stage6 negative bridge, respectively. Whole owner uniqueness within every invocation, distinct active data families, literal spectator maps, disjoint pending footprints and explicit remainder billing are unchanged. No residual is split across physical families. No pending child's rank is reduced on the strength of this lemma alone.
+
+## Invoice and audit requirements
+
+The local scalar words, number of complete local sweeps and number of bridges are unchanged. Only chronology and owner allocation change; all owner charts must still be emitted and charged. The existing max-L2 prefix overcharge counts complete local sweeps and individual bridges, so its conservative bit bound remains valid independent of this reordered chronology. Each replica's projection must still equal its original source event sequence; all tail sweeps must precede that replica's first terminal bridge. Recompute the profile from actual assignments and actual unfilled remainders before reporting any saving.
+
+## Exact pilot, scope and provenance
+
+PAIRED-PARITY-SCREEN.json enumerates possible single-use, paired-X/Y factor retimings for the literal 73-axis K4 controller. The seven actual-vector incidence classes reproduce the original early and narrow bridge bounds. The meaningful extra freedom is precisely the terminal suffix timing above. This restricted enumeration does not exclude redesigns of the local word, multiple uses with cancellations, or different controller recipes.
+
+Source definitions: PR496 finite_serial_pool.py gives C_v; its vendored split-centres-global/vendor/clifford_exact.py defines fourier as a product of C_v. The vendored e8-complex-gauge-kernel PROOF.md states both completed data endpoints. The source DATA-PHASES.json supplies actual pending vectors and all phase translations. The algebra is the same intertwiner equation used to transport operators in tensor-isomorphism reductions; OpenAI FourierCircuit/SparseShear and WordScheduling supply related complete-block cancellation and support-safe substitution structures, not an imported proof of this new schedule.
+
+No new kappa is claimed in this note. The hosting lane must price and emit an actual deferred-suffix allocation, and the independent audit must validate it before composition.
diff --git a/research/coordinated-gauge-async/research-notes/PORTABILITY-TEST.json b/research/coordinated-gauge-async/research-notes/PORTABILITY-TEST.json
new file mode 100644
index 0000000000..31d1e32b96
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/PORTABILITY-TEST.json
@@ -0,0 +1,5 @@
+{
+  "LF_argv_cwd_sibling_imports": "PASS",
+  "optimized_python_rejected": "PASS",
+  "environment": "3.11.9 (tags/v3.11.9:de54cf5, Apr  2 2024, 10:12:12) [MSC v.1938 64 bit (AMD64)]"
+}
diff --git a/research/coordinated-gauge-async/research-notes/RECONSTRUCTION-METADATA-DIFF.json b/research/coordinated-gauge-async/research-notes/RECONSTRUCTION-METADATA-DIFF.json
new file mode 100644
index 0000000000..c3c4f06777
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/RECONSTRUCTION-METADATA-DIFF.json
@@ -0,0 +1,22 @@
+{
+  "actual_difference": [
+    [
+      "root.R",
+      "missing",
+      false,
+      true
+    ],
+    [
+      "root.report.complex_coarse_float",
+      0.001183986549206271,
+      0.0011839865492062917
+    ],
+    [
+      "root.report.gain_percent_float",
+      18.471088665042544,
+      18.471088665044633
+    ]
+  ],
+  "normalized_equal": true,
+  "normalized_sha256": "879c0e6ade3f1c84a515609a848dc917bddd1c1499e0cc37b2862809dfb8655b"
+}
\ No newline at end of file
diff --git a/research/coordinated-gauge-async/research-notes/RESEARCH-NOTE.md b/research/coordinated-gauge-async/research-notes/RESEARCH-NOTE.md
new file mode 100644
index 0000000000..5432b13548
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/RESEARCH-NOTE.md
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+# Shared target levels make gauge choices non-additive
+
+A helper entrance gauge moves that helper's dirty-response compensation to a chosen frame. The target must reach the largest compensation frame used by any helper feeding it, and each centre must descend through the resulting target levels. Consequently, changing one helper's entrance and changing a coordinated group have different paid effects.
+
+Start from the PR484/496 gauge80-kernel42 scalar word with the body retiming transferred from PR495 through PR501/502. Helpers540,541,542,543 have rank4 entrances. Replace all four by the already-present rank2 frame [408,102], leaving all kernel classes and the scalar body fixed. The four target levels at81,82,84,87 then drop from rank4 to rank2. Every first-touch inclusion, helper frame chain, target compensation chain and centre descent is checked by the exact constructor.
+
+Changing only one of these helpers, as suggested by PR473's stage-specific helper540 experiment, gives the local delta -rank2 +rank4. Four separate such changes would therefore suggest -4 rank2 +4 rank4. The actual joint change gives **-16 rank2 +4 rank4 +4 rank6**: the additional -12 rank2 +4 rank6 comes from the shared target-level collapse. This extra contribution is missed by summing individual gauge scores.
+
+The four helper residuals grow from width5 to width7. The complete local inventory becomes18 width7,104 width8 and438 width9 residuals. They must all remain whole on their assigned physical owners. The asynchronous packing is recomputed with that inventory; the benefit is not obtained by adding a local saving to a published exponent.
+
+The selected word has passed an independent exact rational audit on all800 arbitrary-dirty input columns, both forward and inverse-transpose, and42 omitted-kernel-correction controls. A portable constructor reproduces the selected word, checks its exact local profile delta, and binds the complete word after normalizing only two stale discovery floats and an optional redundant R field. It does not by itself certify physical aliasing, Gaussian normalization, finite precision or the outer exponent; those belong to the final global replay and its inherited conditional contracts.
+
+The bounded search retained three retimed response-matroid variants, all382 proper single-gauge subframes,20 coordinated choices and six further combinations. Only the coordinated rank2 collapse improved the full tested asynchronous inventory. These are local search results, not an optimality theorem. The exact selected construction, unsuccessful witnesses and rejected frame chains are preserved for future work.
diff --git a/research/coordinated-gauge-async/research-notes/RETIMED-VARIANTS.json b/research/coordinated-gauge-async/research-notes/RETIMED-VARIANTS.json
new file mode 100644
index 0000000000..ac302db5a0
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/RETIMED-VARIANTS.json
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+}
diff --git a/research/coordinated-gauge-async/research-notes/SIX-COMBOS.json b/research/coordinated-gauge-async/research-notes/SIX-COMBOS.json
new file mode 100644
index 0000000000..b255f70cd8
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/SIX-COMBOS.json
@@ -0,0 +1,180 @@
+{
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+        "8": -2,
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+    },
+    {
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diff --git a/research/coordinated-gauge-async/research-notes/SYNTHESIS-PROOF.md b/research/coordinated-gauge-async/research-notes/SYNTHESIS-PROOF.md
new file mode 100644
index 0000000000..8c18dcc69a
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/SYNTHESIS-PROOF.md
@@ -0,0 +1,38 @@
+# Synthesis result: retimed local word, coordinated gauges, and saturated asynchronous packing
+
+This round produced a concrete uniform provider selected by the COMPLETE paid profile. It combines the public retiming from PR501/502 with one new coordinated local change: all four higher-rank gauges540..543 are reduced together from rank4 to the existing rank2 frame[408,102]. Their shared target-frame effects are reconstructed jointly; the profile is not obtained by adding individual-gauge estimates. The selected supplier is then placed in our whole-owner asynchronous offset6 controller.
+
+## Sufficient finite construction
+
+Use the kernel lane's exact reconstructed word and its independently checked800-column forward/inverse-transpose scalar identity. Its local paid profile is
+
+    {1:2385,2:248,3:530,4:88,5:270,6:96,7:1,8:17},
+
+and its ordered residual widths agree literally with COORDINATED-RANK2-WIDTHS.json:18 width7,104 width8,438 width9. There are560 helpers. Whole owner normalization and spectator restoration are inherited from the reconstructed admitted local interface; no helper is divided between physical owners.
+
+Run eight independent copies of the nine-stage controller in the order sorted by(stage+6*replica/7,stage,replica), completing each whole cover invocation and its immediate source-event block before the next. The complete-invocation asynchronous theorem from the prior round applies unchanged. Each owner is idle on its own replica timeline. Every original pending child is tiled by disjoint helper residual blocks and an explicitly paid remainder.
+
+WINNING-PLAN.json fixes the order and patterns. WINNING-OWNERS.json.gz gives every one of40,320 helper occurrences, including its physical family, owner replica, child, port, offset and width. Constructive integer assignment followed by regular bipartite perfect-match decomposition yields560 distinct helper families per call; no active data family aliases a helper. Every ordered width independently matches the kernel source receipt. The actual map SHA-256 is27dde4588521328868af5cbd28216869d3fefb03c1ca8248e2e12cacd30c3c29.
+
+The exact global profile is
+
+    {1:171720,2:17856,3:41568,4:6336,5:23280,
+     6:6912,7:1752,8:1224,48:432}.
+
+Ambient dimension73, live families7680 and deficit2496 remain fixed. The nominal moment root is0.0013744413363825925. This number is a discovery price, not the final certified kappa. The parent performs exact moment/outer assembly and an independent worker audits the emitted physical template and enlarged prefix invoice. The retained all-size compiler, precision and analytic contracts remain explicit.
+
+## What combines successfully
+
+The newly retimed public word with ordinary offset6 packing gives nominal b=0.0013733114437630143. The coordinated rank2 change improves this to0.0013744413363825925. The previous delivered offset3 construction had b=0.0013595476203535626. These are composed profiles, not added gains.
+
+Individual gauge deletions and earlier response-matroid component variants did not improve the full bill. Coordinating four gauges changes shared target levels and succeeds where the individual proposals did not. This is the original local contribution in this round; retiming is attributed to its public source, and asynchronous whole-owner scheduling is retained from our previous construction.
+
+## Bounded routes and honest scope
+
+- K3 asynchronous all-hosted placement was infeasible in the tested finite model. K5 offset4 gave b about0.00135477 before the new retiming, worse than K4. No general impossibility theorem is claimed.
+- All three earlier exact component/joint variants, all11 single-gauge profile types,15 legal coordinated proposals and six additional combinations were gated by complete recurrence profiles. The sole winning coordinated provider was chosen uniformly in every call; a heterogeneous-provider compiler is unnecessary.
+- Delaying terminal suffixes until all local invocations finish is a valid sufficient schedule transformation, independently established by the complex lane, but gave no numerical gain over offset6 here. It remains a future route, not a claimed contribution to this kappa.
+- A compressed stock-only numerical MILP, with all timing restrictions removed, found the same profile as offset6 for the fixed retimed baseline. This is useful discovery evidence to stop pure scheduling sweeps; it is not a formal universal cap and is not the research deliverable.
+- The complex lane's exact per-axis transport study found that naive pending-bin mergers across bridges require more than a monomial compensation under unchanged local endpoints. No unproved merger or split-residual admission is used.
+
+All earlier drafts and failed screens are preserved. The replay consumes frozen exact input maps, not a floating optimizer. No Lean formalization, deletion or publication was performed.
diff --git a/research/coordinated-gauge-async/research-notes/WINNING-OWNERS.json.receipt.json b/research/coordinated-gauge-async/research-notes/WINNING-OWNERS.json.receipt.json
new file mode 100644
index 0000000000..0d34819398
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/WINNING-OWNERS.json.receipt.json
@@ -0,0 +1,10 @@
+{
+  "status": "PASS_CONSTRUCTIVE_ASYNC_WHOLE_OWNER_MATCHING",
+  "source_sha256": "bfa1ac46493e39c23007c3c2a4efbdc719de1f53b37ad3a4d9a3ecc112153ebe",
+  "roles_sha256": "27dde4588521328868af5cbd28216869d3fefb03c1ca8248e2e12cacd30c3c29",
+  "roles": 40320,
+  "distinct_calls": 72,
+  "per_call_helpers": 560,
+  "physical_data_families": 7680,
+  "window_groups": 144
+}
diff --git a/research/coordinated-gauge-async/research-notes/prior-kernel/APPENDIX-MANIFEST.json b/research/coordinated-gauge-async/research-notes/prior-kernel/APPENDIX-MANIFEST.json
new file mode 100644
index 0000000000..0d911fff2f
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/prior-kernel/APPENDIX-MANIFEST.json
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+  "JOINT-47-SELECTION.json": "89ca715918bfbb04fad2eaee8a95a40324859cd0955512eb13077502e7d950e3",
+  "JOINT-51-SELECTION.json": "7094bb647cc9951a5385660a9ec5f012a427fd118baacaf3d5e9f28cb436acc7",
+  "rebuild_candidate.py": "f8063d539f772fb63712bed787a1c472da28c9baf38b88e3ae14806632d4afb7",
+  "audit_component_scalar.py": "feab9aa6dbcd8f9fffa6dbdb981d15ffc850402ae91cf45f946ff069cac54037"
+}
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index 0000000000..8f51ec043c
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diff --git a/research/coordinated-gauge-async/research-notes/prior-kernel/JOINT-51-SELECTION.json b/research/coordinated-gauge-async/research-notes/prior-kernel/JOINT-51-SELECTION.json
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diff --git a/research/coordinated-gauge-async/research-notes/prior-kernel/LOCAL-CANDIDATE-AUDITS.json b/research/coordinated-gauge-async/research-notes/prior-kernel/LOCAL-CANDIDATE-AUDITS.json
new file mode 100644
index 0000000000..bfef2c4569
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/prior-kernel/LOCAL-CANDIDATE-AUDITS.json
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+  },
+  "JOINT-47": {
+    "status": "PASS_EXACT_RATIONAL_FORWARD_AND_INVERSE_TRANSPOSE",
+    "word_sha256": "f42a6761ec00798d75669b77006260bcaa75606dd652f802174c9e2b503dc98a",
+    "selection_sha256": "89ca715918bfbb04fad2eaee8a95a40324859cd0955512eb13077502e7d950e3",
+    "columns_each_orientation": 800,
+    "compensations": 12108,
+    "program_operations": 19658,
+    "kernel_pivots": 48,
+    "controls": {
+      "390": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "391": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "418": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "419": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "381": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "385": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "441": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "445": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "498": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "499": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "500": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "501": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "156": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "157": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "358": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "359": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "386": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "387": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "301": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "357": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "413": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "436": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "437": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "439": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "443": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "496": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "497": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "502": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "503": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "242": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "243": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "244": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "245": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "268": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "269": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "216": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "217": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "214": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "215": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "299": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "212": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "213": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "327": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "409": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "470": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "471": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "494": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "495": "REJECTED_OMITTED_KERNEL_CORRECTION"
+    },
+    "scope": "Exact local scalar identity and preserved frame-zero compensation; physical owner routing, complete invoice and outer kappa separate."
+  },
+  "JOINT-51": {
+    "status": "PASS_EXACT_RATIONAL_FORWARD_AND_INVERSE_TRANSPOSE",
+    "word_sha256": "b880f20ece93a26eaa649e3b895c76b224b5052775ad8dea9f2239825d36ce10",
+    "selection_sha256": "7094bb647cc9951a5385660a9ec5f012a427fd118baacaf3d5e9f28cb436acc7",
+    "columns_each_orientation": 800,
+    "compensations": 12102,
+    "program_operations": 19648,
+    "kernel_pivots": 40,
+    "controls": {
+      "390": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "391": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "418": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "419": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "156": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "157": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "358": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "359": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "386": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "387": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "301": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "357": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "413": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "436": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "437": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "439": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "443": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "496": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "497": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "502": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "503": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "242": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "243": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "244": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "245": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "268": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "269": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "216": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "217": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "214": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "215": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "299": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "212": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "213": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "385": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "411": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "440": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "441": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "466": "REJECTED_OMITTED_KERNEL_CORRECTION",
+      "467": "REJECTED_OMITTED_KERNEL_CORRECTION"
+    },
+    "scope": "Exact local scalar identity and preserved frame-zero compensation; physical owner routing, complete invoice and outer kappa separate."
+  }
+}
diff --git a/research/coordinated-gauge-async/research-notes/prior-kernel/REBUILD.json b/research/coordinated-gauge-async/research-notes/prior-kernel/REBUILD.json
new file mode 100644
index 0000000000..a20e7b3889
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/prior-kernel/REBUILD.json
@@ -0,0 +1,26 @@
+{
+  "base_sha256": "8b91151043fd61afe379314e4fb1e647cf9cd3cfe98462920d3faba5b1f268f2",
+  "base_parent_kernel79_sha256": "2e13b2dd1e1b0dfa341dbd57b91ee1758ad041cd5ce2e734bf2f9fd0bac155cb",
+  "base_derivation": "Remove the229 entering gauge-prefix gates from parent kernel79.A, empty start_extra and target_levels, rebuild cleanup as inverse of remaining A+B non-target gates. These exact base bytes are retained.",
+  "gk_sources": {
+    "gk_word.py": "f027a9273bcf4e0971258c160b7d3b3c196366b32a529cff6e6a0e666b03826b",
+    "gk_common.py": "e68d190fd436b2f3955decb590adb53344d5d4ea0a19d62d86df1e4b55ecd198"
+  },
+  "candidates": {
+    "COMPONENT": {
+      "selection": "COMPONENT-SELECTION.json",
+      "selection_sha256": "b682f7cff6de4f40bdadfc64a9ecb9b03a8d77dbf29be3748fa9d742dc894071",
+      "word_sha256": "8cd4987c01252cfa0adcc21b083d84aac89e7b25966299d7f8c26446043627de"
+    },
+    "JOINT-47": {
+      "selection": "JOINT-47-SELECTION.json",
+      "selection_sha256": "89ca715918bfbb04fad2eaee8a95a40324859cd0955512eb13077502e7d950e3",
+      "word_sha256": "f42a6761ec00798d75669b77006260bcaa75606dd652f802174c9e2b503dc98a"
+    },
+    "JOINT-51": {
+      "selection": "JOINT-51-SELECTION.json",
+      "selection_sha256": "7094bb647cc9951a5385660a9ec5f012a427fd118baacaf3d5e9f28cb436acc7",
+      "word_sha256": "b880f20ece93a26eaa649e3b895c76b224b5052775ad8dea9f2239825d36ce10"
+    }
+  }
+}
diff --git a/research/coordinated-gauge-async/research-notes/prior-kernel/RESEARCH-APPENDIX.md b/research/coordinated-gauge-async/research-notes/prior-kernel/RESEARCH-APPENDIX.md
new file mode 100644
index 0000000000..6748d26366
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/prior-kernel/RESEARCH-APPENDIX.md
@@ -0,0 +1,52 @@
+# Research appendix: corpus reuse, the achieved mechanism, and preserved candidates
+
+The delivered conditional bound is **kappa = 0.001357701757628940**. Its new contribution is asynchronous scheduling of complete replica invocations. The local gauge/kernel word is unchanged from PR484/496; the stronger bit supplier comes from PR493. The package's final receipts, physical audit and exact assembly are the evidence for the numerical claim. The inherited complete-invocation, compiler and analytic contracts remain explicit conditions; this is not a new Lean certification of integer multiplication.
+
+## The mechanism that improved the bound
+
+Earlier packing advanced all eight replicas in lockstep. The new schedule preserves each replica's nine-stage controller order while allowing replicas to occupy different stages. A helper may then use an idle block whenever its entire invocation lies between consecutive touches on the **owner's** timeline. The selected order sorts calls by `(stage + 3*replica/7, stage, replica)`.
+
+Correctness uses completed invocation endpoints. Each invocation restores arbitrary dirty workspace and borrowed spectators, and leaves only its specified residual on its assigned pending block. Different residual blocks of an idle child are disjoint; their product with the paid remainder is the original child. Completing that product before the owner's next touch preserves each replica's original controller. No scalar gate is commuted through an unfinished invocation, and no residual is split between owners.
+
+A flow nomination is converted to literal owners using bipartite edge coloring. There are 72 calls and 40,320 helper occurrences, with 560 distinct helper families per call. Ambient dimension73, live stock7680 and deficit2496 stay fixed. The changed paid profile is:
+
+    1:148240, 2:66816, 3:7032, 4:19728, 5:26109,
+    6:144, 7:2209, 8:1224, 16:1, 48:408.
+
+The complete schedule needs its own physical chart checks and prefix invoice. Its gain does not follow from the floating packing objective alone.
+
+## What was actually reused from the OpenAI corpus
+
+The inspected snapshot is `adc7f1241b42e322a6451854ab7e4b4c146bf78a` of [openai/math](https://github.com/openai/math). `APPENDIX-SOURCES.json` records repository-relative source paths and hashes. Direct source inspection and Git consistency checks are distinct from compiling these modules in this round; no claim is made that every paper in the collection is formalized.
+
+- [FourierCircuit/WordScheduling.lean](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/FourierCircuit/WordScheduling.lean): `scheduledStep_product` and `scheduledLayer_product` preserve products under order-preserving schedules on fixed disjoint supports. `Word.matrix_substitute` and `Word.weight_substitute` preserve semantics and account for substituted calls. These supplied the structural template for scheduling complete normalized invocations. The new owner-relative interval argument is stated and checked separately; the fixed-support theorem is not silently applied to aliased unfinished programs.
+- [MatroidCounting/CommonBases.lean](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/MatroidCounting/CommonBases.lean): `greedyOracle_correct`, `greedyOracle_rank`, and the parallel/nonparallel pair-rank lemmas motivated response-matroid basis and circuit-component selection. The common-integer-polymatroid preprint, problem114, also suggested moving capacity across fixed stage partitions. The actual packing family is not asserted to be a matroid, and no counting FPRAS is used as a packing certificate.
+- `FourierCircuit/TensorTools.lean`, `RowCompression.sparse_row_compression`, and `SparseShear.lean`, `cancellation_identity`, provide explicit row-basis changes and cancellation of a shared exterior block. Independent dirty replica coordinates cannot be deleted merely by compressing a displayed response matrix. No additional gain was obtained from this route.
+- `FourierCircuit/PivotJoin.lean`, `PivotJoin.of_join`, and `MonomialFeedback.lean`, `Dense2x2.price`, suggested alternative local compositions. The latter's abstract normalized price bound is not a literal bounded-coefficient routine in this model. The exact-Fourier documentation also limits its theorem to a subsequence; it does not supply the missing all-length bit-complexity implementation.
+- `Topology/ArtinGroups/Cancellation/BlockPivot.lean`, `homotopy_square`, `homotopy_retract`, `remainder_retained`, and `NumberTheory/TwoPoint/Walks/ColumnRankRelations.lean`, `columnDifference_disjoint_minor`, were inspected for simultaneous elimination and donor/pivot separation. They did not produce another priced gain in this round.
+
+## Three checked local constructions, not added to the achieved gain
+
+For a common entrance frame, choose a basis of a response-matroid component as donors. Each remaining column is an exact rational donor combination. Prepending its cancelling donor shears makes that pivot's dirty response zero; reverse cleanup restores the helper state. All required frame nesting and source spans must still hold.
+
+The following concrete words passed exact rational reconstruction, all800 arbitrary-dirt columns in both forward and inverse-transpose orientations, and every individual pivot-correction omission control:
+
+| Candidate | Local profile change from PR484/496 | Residual counts (width:count) | Omission controls |
+|---|---|---|---:|
+| COMPONENT | -10 rank1, -14 rank2, +14 rank3 | 5:4, 7:14, 8:100, 9:442 | 38 |
+| JOINT-47 | -14 rank1, -2 rank2, -4 rank3, +4 rank4, +4 rank5 | 5:4, 7:12, 8:102, 9:442 | 48 |
+| JOINT-51 | -14 rank1, -4 rank2, +8 rank3, -2 rank4, +2 rank5 | 5:4, 7:14, 8:100, 9:442 | 40 |
+
+These improve selected local objectives but lose scarce narrow residual inventory. Their tested lockstep allocations, including stage-selective portfolios and paid banks, did not improve the full bound. None is counted in the delivered kappa. They remain useful sufficient constructions for future schedules.
+
+The companion base, three selections and `REBUILD.json` permit source-bound reconstruction through `rebuild_candidate.py`; `audit_component_scalar.py` checks each reconstructed word independently. `LOCAL-CANDIDATE-AUDITS.json` preserves the three exact scalar receipts. The constructor reuses the retained, hash-pinned `gk_word.py` and `gk_common.py` with their attribution.
+
+## Other bounded lessons
+
+All14,851 subspaces contained in first-touch frames were screened. Under the fixed80 gauge exclusions, only14 classes had positive response nullity, all dimension1. Allowing excluded gauge helpers exposed63 additional components, still dimension1. A catalogue of646 components from deletion minors up to depth two did not beat the inherited42-pivot width budget. These are finite-search results, not global impossibility theorems.
+
+The proposed gauged-donor ordering is algebraically sufficient when original dirt compensation precedes the kernel update. Adding individual donors while retaining the current palette failed exact target/centre nesting in all86 tested cases; geometric admissibility alone was insufficient. Whole-palette replacements remain a separate route.
+
+Native frame-retiming searches on the full joint gauge word and its cleanup extension found no better profile. A direct lift of PR493's bit skeleton to the complex side also failed a specific structural test: all14,400 port pairs obey complemented adjacency. An even-parity-frame repair and the identity `G Hadamard G + A_orth - J = 3 I` are constructive leads, but require extra paid copies and a different decoder; they are not achieved improvements.
+
+The productive change was therefore at the scheduling level: reuse the already checked local engine, remove replica lockstep, and reconstruct the complete physical ownership and prefix bill.
diff --git a/research/coordinated-gauge-async/research-notes/prior-kernel/audit_component_scalar.py b/research/coordinated-gauge-async/research-notes/prior-kernel/audit_component_scalar.py
new file mode 100644
index 0000000000..df3f46411c
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/prior-kernel/audit_component_scalar.py
@@ -0,0 +1,55 @@
+"""Independent exact scalar witness for a selected complex kernel word.
+Uses immutable correct-word compensation for omission controls; no repaired
+compensation is silently supplied to a mutant. No physical-hosting claim.
+"""
+import sys,json,gzip,hashlib,copy
+from pathlib import Path
+from fractions import Fraction as Q
+from collections import Counter
+p=Path(__file__).resolve().parent
+path=Path(sys.argv[1]);selection=Path(sys.argv[2]);out=Path(sys.argv[3]);w=json.loads(gzip.decompress(path.read_bytes()));sel=json.loads(selection.read_text());n=240+w['report']['R']
+def ident():return[{r:Q(1)}for r in range(n)]
+def apply(e,rows):
+ if e[0]=='scale':
+  _,t,a,b,*_=e;q=Q(a,b);rows[t]={j:q*x for j,x in rows[t].items()};return
+ _,t,s,a,b,*_=e;q=Q(a,b)
+ for j,x in list(rows[s].items()):
+  z=rows[t].get(j,Q(0))+q*x
+  if z:rows[t][j]=z
+  else:rows[t].pop(j,None)
+def inv(e,dual=False):
+ e=copy.deepcopy(e)
+ if e[0]=='add':
+  e[3]*=-1
+  if dual:e[1],e[2]=e[2],e[1]
+ else:e[2],e[3]=e[3],e[2]
+ return e
+def replay(es):
+ rows=ident()
+ for e in es:apply(e,rows)
+ return rows
+main=w['A']+w['cut']+w['B'];mx=replay(main);dirty=[{}for _ in range(n)]
+for r in range(120,240):dirty[r]={j:x for j,x in mx[r].items()if j>=240}
+for e in w['precondition']:apply(e,dirty)
+comp=[]
+for r in range(120,240):
+ for j,x in sorted(dirty[r].items()):
+  assert not w.get('start_extra',{}).get(str(j)),('nonzero-frame compensation',j)
+  comp.append(['add',r,j,-x.numerator,x.denominator,[]])
+cleanup=[inv(e)for e in reversed(w['A']+w['B'])if not 120<=e[1]<240]
+assert[e[:-1]for e in cleanup]==[e[:-1]for e in w['cleanup']]
+program=w['precondition']+comp+main+cleanup;expected=ident()
+for t in range(120):expected[120+t][t]=Q(1)
+assert replay(program)==expected
+back=ident()
+for t in range(120):back[t][120+t]=Q(-1)
+assert replay([inv(e,True)for e in program])==back
+controls={}
+for cl in sel['kernel']:
+ for h in cl['pivots']:
+  aa=[e for e in w['A']if not(e[0]=='add'and e[2]==240+h and e[1]>=240 and e[-1]==cl['E'])]
+  assert len(aa)<len(w['A']);cc=[inv(e)for e in reversed(aa+w['B'])if not 120<=e[1]<240]
+  assert replay(w['precondition']+comp+aa+w['cut']+w['B']+cc)!=expected
+  controls[str(h)]='REJECTED_OMITTED_KERNEL_CORRECTION'
+r=dict(status='PASS_EXACT_RATIONAL_FORWARD_AND_INVERSE_TRANSPOSE',word_sha256=hashlib.sha256(path.read_bytes()).hexdigest(),selection_sha256=hashlib.sha256(selection.read_bytes()).hexdigest(),columns_each_orientation=n,compensations=len(comp),program_operations=len(program),kernel_pivots=len(controls),controls=controls,scope='Exact local scalar identity and preserved frame-zero compensation; physical owner routing, complete invoice and outer kappa separate.')
+out.write_text(json.dumps(r,indent=2)+'\n');print(r['status'],n,len(controls))
diff --git a/research/coordinated-gauge-async/research-notes/prior-kernel/rebuild_candidate.py b/research/coordinated-gauge-async/research-notes/prior-kernel/rebuild_candidate.py
new file mode 100644
index 0000000000..fc15fc255e
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/prior-kernel/rebuild_candidate.py
@@ -0,0 +1,13 @@
+"""Rebuild one checked local candidate from a pinned pre-gauge base and selection.
+python rebuild_candidate.py GK_CODE_DIR LANE_DATA CANDIDATE OUTPUT_WORD
+CANDIDATE is COMPONENT, JOINT-47 or JOINT-51. Uses retained Apache-2.0 upstream
+code with original attribution. Physical hosting and pricing are separate.
+"""
+import sys,json,gzip,hashlib
+from pathlib import Path
+code,data,name,out=Path(sys.argv[1]),Path(sys.argv[2]),sys.argv[3],Path(sys.argv[4])
+assert not out.exists();r=json.loads((data/'REBUILD.json').read_bytes());sha=lambda p:hashlib.sha256(p.read_bytes()).hexdigest()
+for f,h in r['gk_sources'].items():assert sha(code/f)==h,(f,'source hash')
+assert sha(data/'BASE-WORD.json.gz')==r['base_sha256'];c=r['candidates'][name];assert sha(data/c['selection'])==c['selection_sha256']
+sys.dont_write_bytecode=True;sys.path.insert(0,str(code.resolve()));import gk_word as g
+word=g.build(json.loads(gzip.decompress((data/'BASE-WORD.json.gz').read_bytes())),json.loads((data/c['selection']).read_bytes()));raw=g.dump(word);assert hashlib.sha256(raw).hexdigest()==c['word_sha256'];out.write_bytes(raw);print('PASS_SOURCE_BOUND_RECONSTRUCTION',name,c['word_sha256'])
diff --git a/research/coordinated-gauge-async/research-notes/screen-complex-alias.cpp b/research/coordinated-gauge-async/research-notes/screen-complex-alias.cpp
new file mode 100644
index 0000000000..a7fc7294a3
--- /dev/null
+++ b/research/coordinated-gauge-async/research-notes/screen-complex-alias.cpp
@@ -0,0 +1,21 @@
+#include <bits/stdc++.h>
+#include "json.hpp"
+using namespace std;using J=nlohmann::json;
+using Frame=bitset<512>;
+Frame space(J a){Frame f;f[0]=1;for(auto z:a){int v=z;Frame g=f;for(int i=0;i<512;i++)if(f[i])g[i^v]=1;f=g;}return f;}
+bool subset(Frame a,Frame b){return (a&~b).none();}
+int main(int argc,char**argv){ifstream in(argv[1]);J w;in>>w;vector<J> ev;for(auto e:w["A"])ev.push_back(e);for(auto e:w["cut"])ev.push_back(e);for(auto e:w["B"])ev.push_back(e);
+const int N=800,V=120;vector<array<double,120>> rows(N);vector<unsigned long long> hash(N);for(int i=0;i<V;i++)rows[i][i]=1;
+auto rehash=[&](int r){unsigned long long h=1469598103934665603ull;for(double v:rows[r]){if(v==0)v=0;unsigned long long z;memcpy(&z,&v,8);h=(h^z)*1099511628211ull;}hash[r]=h;};for(int i=0;i<N;i++)rehash(i);
+vector<bitset<800>> cand(N);vector<int> reads(N);for(int r=240;r<N;r++){cand[r].set();for(int t=120;t<240;t++)cand[r][t]=0;cand[r][r]=0;}
+for(auto e:ev){string op=e[0];int r=e[1];if(op=="add"){int s=e[2];if(s>=240){reads[s]++;for(int t=0;t<N;t++)if(cand[s][t]&&(hash[t]!=hash[s]||rows[t]!=rows[s]))cand[s][t]=0;}double q=double(e[3].get<int>())/e[4].get<int>();for(int i=0;i<V;i++)rows[r][i]+=q*rows[s][i];}
+else {double q=double(e[2].get<int>())/e[3].get<int>();for(double&x:rows[r])x*=q;}rehash(r);}
+ J out=J::array();set<int> centres;for(auto x:w["centers"])centres.insert(x.get<int>());
+vector<Frame> frames;for(auto e:ev)frames.push_back(space(e.back()));
+for(int r=240;r<N;r++)if(reads[r]&&!centres.count(r))for(int t=0;t<N;t++)if(cand[r][t]){
+ Frame f;f[0]=1;if(t<120)f=space(J::array({w["ports"][t]}));bool ok=true;int crossings=0;
+ for(int k=0;k<(int)ev.size();k++){auto e=ev[k];int dst=e[1];if(dst==r)continue;string op=e[0];int src=op=="add"?e[2].get<int>():-1;if(dst==t||src==t||src==r){if(!subset(f,frames[k])){ok=false;break;}f=frames[k];crossings++;if(src==r&&dst==t){ok=false;break;}}}
+ out.push_back({{"drop",r},{"keep",t},{"frame_chain_legal",ok},{"touches",crossings}});
+}
+ofstream o(argv[2]);o<<out.dump(2)<<"\n";cout<<"same-value candidates "<<out.size()<<", compatible "<<count_if(out.begin(),out.end(),[](auto x){return x["frame_chain_legal"].template get<bool>();})<<"\n";
+}
diff --git a/research/coordinated-gauge-async/vendor/audit_scalar_invoice.py b/research/coordinated-gauge-async/vendor/audit_scalar_invoice.py
new file mode 100644
index 0000000000..b03fdf510f
--- /dev/null
+++ b/research/coordinated-gauge-async/vendor/audit_scalar_invoice.py
@@ -0,0 +1,82 @@
+"""Fresh PR484 scalar/prefix lowering and finite overcharge recount. OpenAI Codex.
+Reads emitted programs, canonical word and raw routes, importing no supplier code.
+Inherited all-size compiler admissibility is a condition, not inferred here.
+"""
+from pathlib import Path
+from fractions import Fraction as F
+from collections import Counter
+import gzip,json,math,hashlib
+import sys
+O=Path(sys.argv[1]);V=Path(sys.argv[2]);HOSTED=Path(sys.argv[3])
+def read(p):return json.loads(gzip.decompress(p.read_bytes())if p.suffix=='.gz'else p.read_bytes())
+w=read(V/'split-centres/CANONICAL-WORD.json.gz');p=read(V/'split-centres/LOCAL-PROGRAMS.json.gz');old=read(V/'split-centres/GLOBAL-TEMPLATE.json.gz');new=read(HOSTED/'GLOBAL-TEMPLATE.json.gz');integ=read(V/'split-centres/INTEGRATION.json');checks=read(V/'split-centres/LOCAL-CHECKS.json');prior=read(V/'split-centres/INDEPENDENT-REPLAY.json');owners=read(O/'OWNERS-AND-PROFILE.json')
+N=800;M=73;R=560;L=9;T=8;VV=120;K=4
+local={}
+for ori,evs in p.items():
+ rows={('L',i):{i:F(1)}for i in range(N)};env={('L',i):F(1)for i in range(N)};peak=F(1);real=macros=0;paid=Counter();workops=0
+ for e in evs:
+  if e[0]=='climb':paid[e[5]]+=1
+  elif e[0]=='center':paid[e[4]]+=1
+  elif e[0]=='scalar':
+   macros+=1;x=e[1];op=x[0];a=tuple(x[1]);count=0
+   if op=='clear':rows[a]={};env[a]=F(0);count=1
+   elif op=='copy':rows[a]=dict(rows[tuple(x[2])]);env[a]=env[tuple(x[2])];count=1
+   elif op=='scale':
+    q=F(x[2],x[3]);num,den=abs(q.numerator),q.denominator
+    assert num and num&(num-1)==0 and den&(den-1)==0
+    rows[a]={j:q*v for j,v in rows[a].items()};env[a]*=abs(q)
+    count=abs(num.bit_length()-den.bit_length())+int(q<0)
+   elif op=='add':
+    b=tuple(x[2]);q=F(x[3],x[4]);num,den=abs(q.numerator),q.denominator
+    assert a!=b and num>0 and den&(den-1)==0
+    # Lower actual coefficient to cleared Horner work. Every intermediate
+    # magnitude envelope is explicitly included, before denominator halvings.
+    acc=0;count=1
+    for i,ch in enumerate(bin(num)[2:]):
+     if i:acc*=2;count+=1;peak=max(peak,acc*env[b])
+     if ch=='1':acc+=1;count+=1;peak=max(peak,acc*env[b])
+    v=F(acc)
+    for i in range(den.bit_length()-1):v/=2;count+=1;peak=max(peak,v*env[b])
+    assert v==abs(q);count+=2 # signed accumulation and private work clear
+    env[a]=env.get(a,F(0))+abs(q)*env[b];rr=rows.setdefault(a,{})
+    for j,z in list(rows[b].items()):
+     value=rr.get(j,F(0))+q*z
+     if value:rr[j]=value
+     else:rr.pop(j,None)
+    workops+=count
+   else:raise AssertionError(op)
+   real+=2*count;peak=max(peak,env[a])
+ assert all(not row for reg,row in rows.items()if reg[0]=='C')
+ for i in range(N):
+  expected={i:F(1)}
+  if ori=='forward'and 120<=i<240:expected[i-120]=F(1)
+  if ori=='backward'and i<120:expected[i+120]=F(-1)
+  assert rows['L',i]==expected
+ assert peak==F(prior['local'][ori]['prefix_l1_upper']) and peak<2**192
+ assert real==checks[ori]['real_component_primitive_bound']and macros==checks[ori]['scalar_macros']
+ assert dict(paid)=={int(r):z for r,z in checks[ori]['paid_profile'].items()}
+ local[ori]=dict(events=len(evs),macros=macros,real_component_primitives=real,prefix_l1=str(peak),prefix_bits=peak.numerator.bit_length()-peak.denominator.bit_length()+1,coefficient_work_primitive_count=workops,all_exact_columns=N,all_560_dirty_helper_columns_restored=True,private_centres_and_coefficient_work_cleared=True)
+centres=len(w['centers']);widths=[9-len(w['frames'][f])for f in w['start'][240:]];enter=sum(x<9 for x in widths)
+children=sum(checks['forward']['paid_profile'].values());macros=max(x['macros']for x in local.values());compiler=64*(M+1)**3
+adapters=16*(macros+children+N+centres)+64*M
+primitive_bound=max(x['real_component_primitives']for x in local.values())+adapters*compiler
+bridges=sum(e['kind']=='bridge'for e in old['recipe']);bridgeadd=2*VV*T*bridges;shifts=K*VV*T;moves=2*K*VV*T
+oldW=old['live'];oldP=old['physical'];oldNB=(oldW-2*VV*K*T)//L
+scalar=sum(local[c['orientation']]['real_component_primitives']for c in old['calls'])
+nonlocal_items=4*VV*T*L+2*VV*T*L+2*L*T*R+sum(e['replicas']for e in old['idle_children'])+L*oldNB+bridgeadd+shifts+moves+oldW+oldP+2*L*T*enter*144
+outside=8*nonlocal_items+64*M
+pervertex=scalar+bridgeadd+shifts+moves+oldW+(72*adapters+outside)*compiler+64*oldP
+invoice=integ['finite_invoice'];assert (primitive_bound,adapters,pervertex,compiler)==(invoice['local_primitive_bound'],invoice['local_adapters'],invoice['pervertex'],invoice['compiler'])
+# Independent hosted stock rebase, retaining the larger unhosted operation
+# overcharge; only the stock and mass in the common router/literal change.
+W=new['live'];P=new['physical'];mass=owners['rank_mass'];profile=owners['paid_profile']
+assert M*W-mass==2496 and P==W+R+centres+2
+rebase=pervertex-oldP*65+P*65-oldW+W;used=max(rebase,pervertex)
+r=M//2;group=2**(r*r)
+for i in range(1,r+1):group*=2**(2*i)-1
+stock=group*P;smass=group*mass;logical=group*used+64;router=compiler*(logical+1)*(stock+1)**2
+G,E,C0=2**30000,2**100000,2**210000;B=smass+E;literal=2*G*stock**2+8*smass+4*stock+4+32*M;row=2*stock.bit_length()+9909+252;maxchild=max(map(int,profile))
+ineq=dict(local_primitive_bound=primitive_bound<2**48,logical=logical<2**6050,router=router<G,literal=literal<E,mass=B<2**100001,precision=32*M*B*B<C0,tape=2*B+18<C0,row=row<20161,halving=M*M>2*maxchild*maxchild)
+assert all(ineq.values())
+out=dict(status='PASS_INDEPENDENT_SCALAR_PREFIX_AND_HOSTED_FINITE_INVOICE',local=local,entering_helpers=enter,entrance_rank=sum(9-x for x in widths),local_adapters=adapters,local_primitive_bound=primitive_bound,wrapper_elementary_factors_charged=2*72*enter*144,compiler=compiler,unhosted_pervertex=pervertex,hosted_rebased_pervertex=rebase,conservative_pervertex_used=used,finite=dict(physical_stock_bits=stock.bit_length(),router_bits=router.bit_length(),literal_bits=literal.bit_length(),physical_row_coefficient=row,retained_row_coefficient=20161,required_C0_bits=(32*M*B*B).bit_length(),halving_integer_inequality=[M*M,2*maxchild*maxchild],G_exponent=30000,E_exponent=100000,C0_exponent=210000),strict_checks=ineq,scope='Exact finite recount conditional on inherited all-size compiler/precision/tape and analytic-transfer contracts. Signed accumulation uses the supplier scalar primitive convention.')
+(O/'PREFIX-AND-INVOICE.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2),flush=True)
diff --git a/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/adapters_gk.py b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/adapters_gk.py
new file mode 100644
index 0000000000..252ca9393d
--- /dev/null
+++ b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/adapters_gk.py
@@ -0,0 +1,59 @@
+"""Paid residual wrappers for entering helpers (Lemma A): generation and independent decoding.  Apache-2.0.
+Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.  The wrapper construction follows PR #458
+(chafreaky): a helper that starts at frame E and ends at FULL has the physical residual S_E = T_FULL T_E^-1 (apply T_E^-1 first);
+S_E = L_E (C_[w] x I) R_E with R_E applied first; the wrappers  R_E^-1  before the whole local invocation and  L_E^-1  after its
+full-frame cleanup turn the residual into the coordinate factor C_[w] x I and restore the other 9 - w coordinates identically.
+usage: adapters_gk.py CANONICAL-WORD.json.gz RESIDUAL-ADAPTERS.json OUT-LEMMA-A.json
+Writes the recipes (generation uses clifford_exact.lower_child) and then DECODES the serialized monomials with the primitive
+constructors of clifford_exact (not lower_child) and checks, for every entrance frame, that  L_E^-1 S_E R_E^-1  has exactly the
+tableau and zero-state vector (global phase included) of  fourier(9, range(w)); mutated recipes (wrong phase, swapped sides, wrong
+translation, wrong width) must be rejected."""
+import copy, gzip, json, sys
+from pathlib import Path
+R = Path(__file__).resolve().parent.parent.parent
+sys.path.insert(0, str(R / 'split-centres-global/vendor'))
+from clifford_exact import canonical_frame, lower_child, fourier, diag, perm, xgate, Op, identity
+n = 9; FULL = tuple(1 << i for i in range(n))
+def monomial_op(rec):
+    """decode a serialized monomial (matrix rows, quadratic rows, Z4 linear part, translation, phase)"""
+    out = diag(n, tuple(rec['quadratic_rows']), tuple(rec['linear_Z4'])).then(perm(n, tuple(rec['matrix_rows']))).then(xgate(n, rec['offset']))
+    return out.then(Op(n, identity(n), [('phase', rec['phase'])]))
+def S_of(U):
+    return canonical_frame(n, tuple(U)).adjoint().then(canonical_frame(n, FULL))
+def wrapped_ok(U, rec):
+    """exactly: L^-1 S R^-1 = C_[w] x I  (apply order: R^-1, S, L^-1)"""
+    w = n - len(U); S = S_of(U); Rm = monomial_op(rec['right']); Lm = monomial_op(rec['left'])
+    net = Rm.adjoint().then(S).then(Lm.adjoint()); target = fourier(n, range(w))
+    return rec['rank'] == w and net.tab == target.tab and net.zero_state() == target.zero_state()
+def main():
+    c = json.load(gzip.open(sys.argv[1])); frames = c['frames']; v = c['v']
+    starts = sorted({c['start'][r] for r in range(2 * v, len(c['start'])) if c['start'][r] != 0})
+    recipes = {}
+    for f in starts:
+        U = frames[f]; rec, compiled = lower_child(S_of(U))
+        recipes[str(f)] = dict(frame=f, vectors=list(U), entrance_rank=len(U), rank=rec['rank'], left=rec['left'], right=rec['right'])
+    Path(sys.argv[2]).write_text(json.dumps(dict(status='RESIDUAL_ADAPTER_RECIPES', recipes=recipes), sort_keys=True, separators=(',', ':')) + '\n')
+    # ---- independent decode of the serialized file
+    data = json.loads(Path(sys.argv[2]).read_text())['recipes']
+    dot = lambda a, b: bin(a & b).count('1') & 1
+    detail = []
+    for f in starts:
+        rec = data[str(f)]; U = frames[f]
+        assert wrapped_ok(U, rec), ('wrapper product is not C_[w] x I', f)
+        detail.append(dict(frame=f, entrance_rank=len(U), residual_width=n - len(U), vectors=list(U), all_even_weight=all(bin(x).count('1') % 2 == 0 for x in U),
+                           totally_isotropic=all(dot(a, b) == 0 for a in U for b in U)))
+    controls = []
+    def reject(name, fn):
+        f = starts[0]; rec = copy.deepcopy(data[str(f)]); fn(rec)
+        if wrapped_ok(frames[f], rec): raise AssertionError('mutated wrapper accepted: ' + name)
+        controls.append(name)
+    reject('wrong global phase', lambda r: r['right'].__setitem__('phase', (r['right']['phase'] + 1) % 4))
+    reject('wrong translation', lambda r: r['left'].__setitem__('offset', r['left']['offset'] ^ 1))
+    reject('wrong quadratic form', lambda r: r['right'].__setitem__('quadratic_rows', [x ^ 2 if i == 0 else x for i, x in enumerate(r['right']['quadratic_rows'])]) if False else r['right']['linear_Z4'].__setitem__(0, (r['right']['linear_Z4'][0] + 2) % 4))
+    reject('sides swapped', lambda r: (r.__setitem__('left', r['right']), r.__setitem__('right', r['left'])))
+    reject('wrong width', lambda r: r.__setitem__('rank', r['rank'] + 1))
+    res = dict(status='PASS_RESIDUAL_WRAPPERS_ARE_EXACT', frames=len(detail), wrappers_reject=controls, all_even_weight=all(d['all_even_weight'] for d in detail),
+               all_totally_isotropic=all(d['totally_isotropic'] for d in detail), detail=detail)
+    Path(sys.argv[3]).write_text(json.dumps(res, indent=1) + '\n')
+    print('Lemma A (exact): L^-1 S_E R^-1 = C_[w] x I for', len(detail), 'entrance frames, global phase included;', len(controls), 'wrapper mutations rejected; all-even-weight:', res['all_even_weight'])
+if __name__ == '__main__': main()
diff --git a/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/audit_gk.py b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/audit_gk.py
new file mode 100644
index 0000000000..e80ca6b8b3
--- /dev/null
+++ b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/audit_gk.py
@@ -0,0 +1,189 @@
+# GK: modified copy of PR #433's complex_audit.py (icekylinx with Codex; PR #425) by DreamingOfClouds with Claude Opus 5.5
+# (Anthropic), via Claude Code.  Every change is marked "GK:"; with a word that has no start_extra / target_levels the
+# behaviour and the written files are PR #433's.
+# Modified from PR425 by icekylinx with Codex assistance.
+"""Read-back audit, independent of the Hamming producer and its row state."""
+import copy,gzip,hashlib,importlib.util,json,math,time
+from collections import Counter
+from fractions import Fraction as Q
+from pathlib import Path
+
+OUT=Path('split-centres');p=OUT/'FULL-WORD.json.gz'
+word=json.load(gzip.open(p));v=120;n=240+word['report']['R'];start=time.monotonic()
+def apply(e,rows):
+    if e[0]=='scale':
+        _,t,a,b,*_=e;q=Q(a,b);assert q
+        rows[t]={j:q*x for j,x in rows[t].items()}
+    else:
+        _,t,s,a,b,*_=e;q=Q(a,b);assert t!=s
+        for j,x in list(rows[s].items()):
+            z=rows[t].get(j,Q(0))+q*x
+            if z:rows[t][j]=z
+            else:rows[t].pop(j,None)
+def ident():return [{r:Q(1)} for r in range(n)]
+def inv(e):
+    e=copy.deepcopy(e)
+    if e[0]=='add':e[3]*=-1
+    else:e[2],e[3]=e[3],e[2]
+    return e
+def dual(e):
+    e=inv(e)
+    if e[0]=='add':e[1],e[2]=e[2],e[1]
+    return e
+
+main=word['A']+word['cut']+word['B']
+assert all(not 120<=e[2]<240 for e in word['A']+word['B'] if e[0]=='add' and not 120<=e[1]<240)
+matrix=ident()
+for e in main:apply(e,matrix)
+print('full main columns',n,'replayed',flush=True)
+# Compute -A_Y^-1 B_S using only rows of the explicit main operator.
+dirty=[{} for r in range(n)]
+for r in range(v,2*v):dirty[r]={j:x for j,x in matrix[r].items() if j>=2*v}
+for e in word['precondition']:apply(e,dirty)
+comp=[]
+for r in range(v,2*v):
+    for j,q in sorted(dirty[r].items()):comp.append(['add',r,j,-q.numerator,q.denominator,[]])
+
+cleanup=[inv(e) for e in reversed(word['A']+word['B']) if not v<=e[1]<2*v]
+assert [e[:-1] for e in cleanup]==[e[:-1] for e in word['cleanup']]
+program=word['precondition']+comp+main+cleanup
+actual=ident()
+for e in program:apply(e,actual)
+expected=ident()
+for t in range(v):expected[v+t][t]=Q(1)
+assert actual==expected
+print('PASS exact arbitrary dirt and restored sources/helpers',flush=True)
+back=ident()
+for e in program:apply(dual(e),back)
+expected_back=ident()
+for t in range(v):expected_back[t][v+t]=Q(-1)
+assert back==expected_back
+print('PASS exact inverse-transpose on every column',flush=True)
+# Independent sparse vector controls avoid repeating a dense full replay.
+def vector(es,x):
+    x=list(map(Q,x))
+    for e in es:
+        if e[0]=='scale':x[e[1]]*=Q(e[2],e[3])
+        else:x[e[1]]+=Q(e[3],e[4])*x[e[2]]
+    return x
+sample=[((i*137+13)%101)-50 for i in range(n)]
+want=vector(program,sample)
+controls={
+    'missing_dirty_compensation':word['precondition']+main+cleanup,
+    'missing_Y_inverse_basis':comp+main+cleanup,
+    'missing_source_cleanup':word['precondition']+comp+main,
+    'wrong_dual_event_order':[dual(e) for e in reversed(program)],
+}
+for name,es in controls.items():
+    expected_sample=vector([dual(e) for e in program],sample) if name=='wrong_dual_event_order' else want
+    assert vector(es,sample)!=expected_sample,name
+
+# Orthogonal transvection and a port permutation bring the new geometry onto
+# the already fixed 120-port routing family. Every incidence is transformed.
+old=json.load(gzip.open('work/higher-stage-e8/verified/gcert1-e8-retimed.json.gz'))
+def tau(x):return x^(43 if (x&43).bit_count()%2 else 0)
+assert 43 .bit_count()%2==0
+assert set(map(tau,word['ports']))==set(old['ports'])
+mapping={i:old['ports'].index(tau(q)) for i,q in enumerate(word['ports'])}
+mapping.update({v+i:v+mapping[i] for i in range(v)})
+mapping.update({r:r for r in range(2*v,n)})
+def basis(vs):
+    b={}
+    for x in vs:
+        for k in sorted(b,reverse=True):
+            if x>>k&1:x^=b[k]
+        if x:
+            k=x.bit_length()-1
+            for j in b:
+                if b[j]>>k&1:b[j]^=x
+            b[k]=x
+    return tuple(b[k] for k in sorted(b,reverse=True))
+def perp(x):return basis([z for z in range(1,512) if not (x&z).bit_count()%2])
+full=tuple(1<<i for i in range(8,-1,-1));frames=[(),full];fi={f:i for i,f in enumerate(frames)}
+def fid(f):
+    f=basis(f)
+    if f not in fi:fi[f]=len(frames);frames.append(f)
+    return fi[f]
+def canonical(e):
+    e=copy.deepcopy(e);e[1]=mapping[e[1]]
+    if e[0]=='add':e[2]=mapping[e[2]]
+    if e[-1] is not None:e[-1]=fid([tau(x) for x in e[-1]])
+    return e
+can={key:[canonical(e) for e in word[key]] for key in ('A','B','cut','cleanup','precondition')}
+can['compensation']=[canonical(e) for e in comp]
+can['start']=[fid([q]) for q in old['ports']]+[0]*(v+word['report']['R'])
+for _r,_E in word.get('start_extra',{}).items():
+    assert 2*v<=int(_r)<2*v+word['report']['R'],('an entrance (residual wrapper, spectators, bank footprint) is defined for helper registers only',_r)   # GK
+    can['start'][int(_r)]=fid([tau(x) for x in _E])   # GK: helper entrances
+can['final']=[1]*v+[fid(perp(q)) for q in old['ports']]+[1]*word['report']['R']
+can['ports']=old['ports'];can['centers']=[mapping[r] for r in word['centers']]
+can['center_frames']=[fid([tau(x) for x in f]) for f in word['center_frames']]
+tlev={}   # GK: nonzero-frame compensation raises targets before the centre scatter
+for _t,_E in word.get('target_levels',{}).items():tlev[mapping[v+int(_t)]-v]=fid([tau(x) for x in _E])
+if tlev:can['target_levels']=tlev
+can['h']=9;can['v']=v;can['R']=word['report']['R'];can['frames']=frames
+cur=list(can['start']);paid=Counter();edges=set();incidences=0
+def climb(r,f):
+    old=cur[r]
+    assert basis(frames[old]+frames[f])==frames[f]
+    if old!=f:
+        delta=len(frames[f])-len(frames[old]);assert delta>0
+        edges.add((old,f));paid[delta]+=1;cur[r]=f
+for part in ('A','B'):
+    if part=='B':
+        assert all(cur[r]==f for r,f in zip(can['centers'],can['center_frames']))
+        assert all(cur[r]==tlev.get(r-v,0) for r in range(v,2*v))   # GK: targets sit at their level
+        for r_c,f in zip(can['centers'],can['center_frames']):
+            # GK: a centre copy descends cf -> served levels (descending chain) -> lowest served level; at the defaults the
+            # only served level is 0 and this is PR #433's single child of rank |cf|
+            served={tlev.get(e[1]-v,0) for e in can['cut'] if e[2]==r_c}
+            o=f
+            for Lv in sorted(served,key=lambda x:-len(frames[x])):
+                assert basis(frames[Lv]+frames[o])==tuple(frames[o]),('centre level outside the previous frame',r_c)
+                if Lv!=o:
+                    paid[len(frames[o])-len(frames[Lv])]+=1
+                    if not(o==f and Lv==0):edges.add((o,Lv));edges.add((Lv,o))   # (cf,0),(0,cf) are added by the phase script
+                    o=Lv
+    for e in can[part]:
+        rs=[e[1]]+([e[2]] if e[0]=='add' else [])
+        for r in rs:climb(r,e[-1]);incidences+=1
+for r,f in enumerate(can['final']):climb(r,f)
+assert dict(paid)=={int(r):x for r,x in word['report']['local_profile'].items()}
+# GK: entrances lower the helper climb by their rank; centre descents may stop above frame zero
+if not word.get('start_extra') and not word.get('target_levels'):assert sum(r*x for r,x in paid.items())==9*can['R']+240*8+72
+else:assert sum(r*x for r,x in paid.items())==sum(int(r)*x for r,x in word['report']['local_profile'].items())
+can['local_profile']=dict(sorted(paid.items()));can['edges']=sorted(edges)
+cp=OUT/'CANONICAL-WORD.json.gz';cp.write_bytes(gzip.compress(json.dumps(can,sort_keys=True,separators=(',',':')).encode(),mtime=0))
+
+def engine(name):
+    p=Path('work/pr361/code/engines')/(name+'.py');s=importlib.util.spec_from_file_location('hamming_'+name,p)
+    m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m
+M,M2=engine('moment'),engine('base_two_moment');m,W=73,word['report'].get('live',12720)   # GK: unhosted stock of the word
+HG={int(r):z for r,z in word['report']['global_profile'].items()}
+root=M.certify(HG,m,W,True);b=Q(int(Q(root['lower'])*10**18),10**18);grid=Q(1,10**18)
+assert M.moment(HG,m,W,b,True)[1]<1<M.moment(HG,m,W,b+grid,True)[0]
+fallback=32*m*m*sum(HG.values())
+def second(a):
+    lo,hi=M2.moment(m,W,list(HG.items()),a)
+    l2,h2=M2.moment(m,W,[(1,fallback)],a)
+    return lo+l2/10**16,hi+h2/10**16
+assert second(b)[1]<1<second(b+grid)[0]
+coeff=Counter(str(Q(e[3],e[4]) if e[0]=='add' else Q(e[2],e[3])) for e in program)
+report={'status':'PASS_EXACT_SCALAR_AND_FRAME_RESEARCH_CANDIDATE_NOT_YET_ADMITTED',
+        'full_word_sha256':hashlib.sha256(p.read_bytes()).hexdigest(),
+        'canonical_word_sha256':hashlib.sha256(cp.read_bytes()).hexdigest(),
+        'live_columns_each_orientation':n,'arbitrary_helper_columns':can['R'],
+        'all_live_source_and_helper_rows_restored':True,'all_target_rows_correct':True,
+        'negative_controls_rejected':list(controls),'old_response_additions':len(comp),
+        'complete_forward_scalar_operations':len(program),'coefficient_histogram':dict(coeff),
+        'frames':len(frames),'nested_directed_edges':len(edges),'common_frame_incidences':incidences,
+        'orthogonal_transport_vector':43,'transported_ports_exactly_match_fixed_routes':True,
+        'local_profile':dict(sorted(paid.items())),
+        'prospective_complex_b':str(b),'prospective_b_float':float(b),
+        'prospective_gain_percent':float(100*(b/Q(999388595603943,10**18)-1)),
+        'two_rational_moment_engines':True,'next_b_grid_rejected':True,
+        'remaining':['new general scalar-interface proof integration','exact phase and emitted local/global program','finite invoice and outer assembly'],
+        'scope':'Substantial new scalar-network candidate. All parameter values remain prospective until the expanded scalar interface and complete phase-aware supplier are admitted.',
+        'seconds':time.monotonic()-start}
+(OUT/'SCALAR-FRAME-AUDIT.json').write_text(json.dumps(report,indent=2)+'\n')
+print('PASS prospective b',float(b),'gain',report['prospective_gain_percent'],'percent; seconds',report['seconds'],flush=True)
diff --git a/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/emit_gk.py b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/emit_gk.py
new file mode 100644
index 0000000000..0dd1853bb5
--- /dev/null
+++ b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/emit_gk.py
@@ -0,0 +1,156 @@
+# GK: modified copy of PR #433's complex_emit.py (icekylinx with Codex assistance; PR #425) by DreamingOfClouds with Claude Opus 5.5
+# (Anthropic), via Claude Code.  Every change is marked "GK:".  On a word without start_extra / target_levels the behaviour
+# and every written file equal PR #433's (regression anchor in verify.py).
+import sys as _sys, os as _os; _sys.path.insert(0, _os.path.dirname(_os.path.abspath(__file__)))
+# Adapted from PR425 by icekylinx with Codex assistance.
+"""Emit both local programs, integrate the fixed nine-stage geometry, and bill.
+
+The expanded scalar interface is justified in PROOF.txt. This script does not
+claim a Lean proof or discharge the inherited all-size/analytic interfaces.
+"""
+import copy,gzip,hashlib,importlib.util,json,math
+from collections import Counter
+from fractions import Fraction as Q
+from pathlib import Path
+
+OUT=Path('split-centres');cp=OUT/'CANONICAL-WORD.json.gz'
+c=json.load(gzip.open(cp))
+frames=c['frames'];v,R,h=c['v'],c['R'],c['h'];n=2*v+R
+def digest(x):return hashlib.sha256(json.dumps(x,sort_keys=True,separators=(',',':')).encode()).hexdigest()
+def basis(vs):
+    b={}
+    for x in vs:
+        for p in sorted(b,reverse=True):
+            if x>>p&1:x^=b[p]
+        if x:
+            p=x.bit_length()-1
+            for q in b:
+                if b[q]>>p&1:b[q]^=x
+            b[p]=x
+    return tuple(b[p] for p in sorted(b,reverse=True))
+def dual(e):
+    e=list(e)
+    if e[0]=='add':e[1],e[2]=e[2],e[1];e[3]*=-1
+    else:e[2],e[3]=e[3],e[2]
+    return e
+def scalar(e,back):
+    e=dual(e) if back else list(e)
+    if e[0]=='add':return ['add',['L',e[1]],['L',e[2]],e[3],e[4]]
+    return ['scale',['L',e[1]],e[2],e[3]]
+def primitive_bound(e):
+    if e[0] in ('clear','copy'):return 1
+    q=Q(e[3],e[4]) if e[0]=='add' else Q(e[2],e[3])
+    a,b=abs(q.numerator),q.denominator
+    assert b&(b-1)==0
+    if e[0]=='scale':
+        assert a&(a-1)==0
+        return abs(a.bit_length()-b.bit_length())+int(q<0)
+    # Exact binary Horner multiplication on a cleared private stream, repeated
+    # halving, signed accumulation into the target, then clear the private stream.
+    return 3+(a.bit_length()-1)+a.bit_count()+(b.bit_length()-1)
+
+def emit(back):
+    out=[];cur=c['start'].copy()
+    def event(e):out.append(e)
+    def scal(e,f):event(['scalar',e,f])
+    event(['phase','zero_frame_precondition'])
+    for e in c['precondition']+c['compensation']:
+        regs=[e[1]]+([e[2]] if e[0]=='add' else [])
+        assert all(c['start'][r]==0 for r in regs)
+        scal(scalar(e,back),0)
+    event(['phase','entry_adapters'])
+    # GK: a helper that starts at a nonzero entrance frame has NO entry adapter; its paid wrapper R_E^-1 (Lemma A) precedes every gate
+    # that touches it and L_E^-1 follows the full-frame cleanup.  Words without entrances emit PR #433's events.
+    entering=[r for r,f in enumerate(c['start']) if r>=2*v and f!=0]
+    for r,f in enumerate(c['start']):
+        if r in entering:event(['residual_adapter','pre',r,f])
+        else:event(['adapter','old_to_new',r,f])
+    def climb(r,f,part):
+        old=cur[r]
+        if old==f:return
+        assert basis(frames[old]+frames[f])==tuple(frames[f])
+        rank=len(frames[f])-len(frames[old]);assert rank>0
+        event(['climb',part,r,old,f,rank]);cur[r]=f
+    for part in ('A','centers','B','final'):
+        event(['phase',part])
+        if part in ('A','B'):
+            for e in c[part]:
+                regs=[e[1]]+([e[2]] if e[0]=='add' else [])
+                for r in regs:climb(r,e[-1],part)
+                scal(scalar(e,back),e[-1])
+        elif part=='centers':
+            # GK: targets sit at their nonzero-frame-compensation level (0 at the defaults); a centre copy descends
+            # cf -> served levels (descending chain) -> lowest served level and scatters into each target at its level
+            lv=lambda r:c.get('target_levels',{}).get(str(r-v),0)
+            assert all(cur[r]==lv(r) for r in range(v,2*v))
+            for k,r in enumerate(c['centers']):
+                cf=c['center_frames'][k];assert cur[r]==cf
+                terms=[e for e in c['cut'] if e[2]==r]
+                path=[cf]+sorted({lv(e[1]) for e in terms}-{cf},key=lambda x:-len(frames[x]))
+                C=['C',k]
+                if not back:
+                    scal(['clear',C],cf);scal(['copy',C,['L',r]],cf)
+                    for a in range(len(path)):
+                        if a>0:event(['center',k,path[a-1],path[a],len(frames[path[a-1]])-len(frames[path[a]])])
+                        for e in terms:
+                            if lv(e[1])==path[a]:scal(['add',['L',e[1]],C,e[3],e[4]],path[a])
+                    scal(['clear',C],path[-1])
+                else:
+                    scal(['clear',C],path[-1])
+                    for a in range(len(path)-1,-1,-1):
+                        for e in terms:
+                            if lv(e[1])==path[a]:scal(['add',C,['L',e[1]],e[3],e[4]],path[a])
+                        if a>0:event(['center',k,path[a],path[a-1],len(frames[path[a-1]])-len(frames[path[a]])])
+                    scal(['add',['L',r],C,-1,1],cf);scal(['clear',C],cf)
+        else:
+            for r,f in enumerate(c['final']):climb(r,f,part)
+    event(['phase','exit_adapters'])
+    for r,f in enumerate(c['final']):event(['adapter','new_to_old',r,f])
+    event(['phase','full_frame_cleanup'])
+    for e in c['cleanup']:
+        regs=[e[1]]+([e[2]] if e[0]=='add' else [])
+        assert all(c['final'][r]==1 for r in regs)
+        scal(scalar(e,back),1)
+    for r in entering:event(['residual_adapter','post',r,c['start'][r]])   # GK: L_E^-1 after the full-frame cleanup
+    event(['phase','done'])
+    return out
+
+def check_local(events,back):
+    # Literal scalar replay of the emitted stream, including center lifetimes.
+    state={('L',r):{r:Q(1)} for r in range(n)};live=[];paid=Counter();prims=0;macros=0
+    for e in events:
+        if e[0]=='climb':assert not live;paid[e[5]]+=1
+        elif e[0]=='center':assert live==[e[1]];paid[e[4]]+=1
+        elif e[0]=='scalar':
+            op=e[1];tag=op[0];t=tuple(op[1]);macros+=1;prims+=2*primitive_bound(op)
+            if tag=='clear':
+                state[t]={}
+                if t[0]=='C' and t[1] in live:live.remove(t[1])
+            elif tag=='copy':state[t]=dict(state.get(tuple(op[2]),{}));live.append(t[1])
+            elif tag=='scale':state[t]={j:Q(op[2],op[3])*x for j,x in state[t].items()}
+            else:
+                s=tuple(op[2]);q=Q(op[3],op[4]);row=state.setdefault(t,{})
+                if t[0]=='C' and t[1] not in live:live.append(t[1])
+                for j,x in list(state.get(s,{}).items()):
+                    z=row.get(j,Q(0))+q*x
+                    if z:row[j]=z
+                    else:row.pop(j,None)
+    assert not live and all(not row for r,row in state.items() if r[0]=='C')
+    for r in range(n):
+        expected={r:Q(1)}
+        if back and r<v:expected[v+r]=Q(-1)
+        elif not back and v<=r<2*v:expected[r-v]=Q(1)
+        assert state['L',r]==expected,('emitted column mismatch',back,r)
+    assert dict(paid)=={int(r):z for r,z in c['local_profile'].items()}
+    return {'sha256':digest(events),'events':len(events),'scalar_macros':macros,
+            'real_component_primitive_bound':prims,'paid_profile':dict(sorted(paid.items())),
+            'all_columns':n,'private_centers_cleared':True,'private_coefficient_stream_cleared_after_each_macro':True}
+
+programs={};checks={}
+for name,back in [('forward',False),('backward',True)]:
+    programs[name]=emit(back);checks[name]=check_local(programs[name],back)
+    print('PASS emitted',name,checks[name]['events'],'events',flush=True)
+
+
+(OUT/'LOCAL-PROGRAMS.json.gz').write_bytes(gzip.compress(json.dumps(programs,sort_keys=True,separators=(',',':')).encode(),mtime=0))
+(OUT/'LOCAL-CHECKS.json').write_text(json.dumps(checks,indent=2)+'\n')
diff --git a/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/gk_common.py b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/gk_common.py
new file mode 100644
index 0000000000..0e5fde920d
--- /dev/null
+++ b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/gk_common.py
@@ -0,0 +1,94 @@
+"""Shared helpers for the complex shared-donor-kernel / entrance-gauge word.  Apache-2.0.
+Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.
+
+A "gk word" is PR #433's FULL-WORD.json.gz with optional extension keys (absent in #433's word, so every default is
+#433's behaviour):
+  start_extra   {register: entrance frame (list of F2^9 vectors)}  helpers that start at a nonzero frame (width 9 - dim)
+  target_levels {target: frame}                                    frame a target reaches through nonzero-frame compensation
+Entrance gauges are ordinary gates at the head of A (target += c * helper at the entrance frame); kernel updates are
+ordinary gates at the head of A (donor += q * pivot at the entrance frame).
+"""
+import math
+from collections import Counter
+
+V, K, T, L, M = 120, 4, 8, 9, 73
+DATA = 2 * K * V * T          # 7680
+FULL = tuple(1 << i for i in range(8, -1, -1))
+
+
+def basis(vs):
+    b = {}
+    for x in vs:
+        for p in sorted(b, reverse=True):
+            if x >> p & 1: x ^= b[p]
+        if x:
+            p = x.bit_length() - 1
+            for q in b:
+                if b[q] >> p & 1: b[q] ^= x
+            b[p] = x
+    return tuple(b[p] for p in sorted(b, reverse=True))
+
+
+def widths_of(word):
+    """residual width of every helper: 9 - dim(entrance frame); 9 when the word has no extension"""
+    R = word['report']['R'] if 'R' not in word else word['R']
+    w = [L] * R
+    for r, E in word.get('start_extra', {}).items():
+        assert 2 * V <= int(r) < 2 * V + R
+        w[int(r) - 2 * V] = L - len(basis(E))
+    return w
+
+
+def split_pad(p):
+    """a pad of rank p is paid as nested commuting children of rank <= 51 (so that 73^2 > 2*child^2); rank 1 stays 1"""
+    if p == 0: return []
+    n = -(-p // 51); return [p // n + (1 if i < p % n else 0) for i in range(n)]
+
+
+def tile(items):
+    """Bank tiling of unhosted residuals.  items: list of (rep, helper, width); a bank has M = 73 cells.
+    * width-9 residuals (no wrapper acts on them): first fit in the given order, packed from offset 0, eight per bank, pad rank 1
+      (PR #433/#442's tiling and routes [2, slot]);
+    * ENTERING residuals (width w < 9, paid wrappers, e = 9 - w spectators): banks of their own in which every entering helper is the ONLY
+      owner of its bank family in its invocation (the one-helper rule of PR #458 extended to banks): bank j holds the j-th entering
+      helper of each replica, residuals packed from offset 0 in replica order (at most one helper per replica, hence per invocation).
+      The spectators of an entering helper are the first e cells of the bank beyond every residual cell of the bank (the pad, which no call
+      uses for a residual, is paid as before: rank 73 - sum of widths; the spectators are borrowed inside it and not deducted).
+    Returns (assign, banks, pads): assign[(rep, helper)] = (bank, offset, width, pure, spectators) with pure = the bank holds only width-9
+    residuals and spectators = () for width 9; pads = Counter of pad ranks (split into children of rank <= 51)."""
+    assign = {}; pads = Counter(); fill = []
+    for z, (rep, h, w) in enumerate(items):             # width 9: first fit, as PR #433 (all widths 9 reproduces it exactly)
+        if w != L: continue
+        for b, f in enumerate(fill):
+            if f + w <= M: break
+        else:
+            b = len(fill); fill.append(0)
+        assign[(rep, h)] = (b, fill[b], w, True, ()); fill[b] += w
+    nwide = len(fill); byrep = {}
+    for rep, h, w in items:
+        if w < L: byrep.setdefault(rep, []).append((h, w))
+    cells = [0] * max((len(v) for v in byrep.values()), default=0); emax = [0] * len(cells); place = {}
+    for rep in sorted(byrep):                           # the j-th entering helper of every replica goes to entering bank j
+        for j, (h, w) in enumerate(byrep[rep]):
+            place[(rep, h)] = (nwide + j, cells[j], w); cells[j] += w; emax[j] = max(emax[j], L - w)
+    for j in range(len(cells)): assert cells[j] + emax[j] <= M, ('entering bank without room for its spectators', j)
+    for (rep, h), (b, off, w) in place.items():
+        assign[(rep, h)] = (b, off, w, False, tuple(range(cells[b - nwide], cells[b - nwide] + L - w)))
+    fill += cells
+    for f in fill:
+        for part in split_pad(M - f): pads[part] += 1
+    return assign, len(fill), pads
+
+
+def route(off, w, pure, sp=()):
+    """owner route of a banked helper residual: PR #433's [2, slot] for width 9 (pure width-9 bank); an entering helper (w < 9) is
+    [2, offset, width, [spectator cells]] with e = 9 - w spectators"""
+    if w == L: assert pure and not sp; return [2, off // L]
+    assert not pure and len(sp) == L - w; return [2, off, w, list(sp)]
+
+
+def unhosted_stock(widths):
+    """live stock and padding histogram of the unhosted (parent) template, per stage x nine"""
+    R = len(widths)
+    _, nb, pads = tile([(rep, h, widths[h]) for rep in range(T) for h in range(R)])
+    return DATA + L * nb, Counter({r: z * L for r, z in pads.items()})
diff --git a/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/gk_word.py b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/gk_word.py
new file mode 100644
index 0000000000..15782ba625
--- /dev/null
+++ b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/gk_word.py
@@ -0,0 +1,194 @@
+"""Apply a frozen selection (shared-donor kernel + entrance gauges) to PR #433's complex word.  Apache-2.0.
+Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.
+usage: gk_word.py BASE_FULL-WORD.json.gz SELECTION.json OUT_FULL-WORD.json.gz
+An empty selection ({}) copies the input byte for byte (regression anchor).
+
+selection = {"kernel": [{"E": [x], "donors": [h..], "pivots": [h..]}..], "gauge": {"h": [x..]}}   (h = helper index 0..R-1)
+Kernel (PR #417/#435/#434/#440): a pivot starts at E; every donor receives  donor += -q * pivot  at frame E before the main
+word, where col(pivot) = sum q_d col(d) are the exact dirty-response columns; the pivot's frame-zero compensation vanishes.
+Gauge (#401/#408-#416): the compensation  target += -C[t,h] * helper  of helper h is executed at frame E_h as the first gates
+of the main word; h starts at E_h and its targets climb to the level L_t = largest E over their entering helpers.
+The exact verification is the arbitrary-dirt replay of audit_gk.py; this module also checks the structural conditions."""
+import copy, gzip, hashlib, json, sys
+from collections import Counter, defaultdict
+from fractions import Fraction as Q
+from pathlib import Path
+sys.path.insert(0, str(Path(__file__).resolve().parent))
+from gk_common import *
+
+
+def apply_op(e, rows):
+    if e[0] == 'scale':
+        _, t, a, b, *_ = e; q = Q(a, b); rows[t] = {j: q * x for j, x in rows[t].items()}
+    else:
+        _, t, s, a, b, *_ = e; q = Q(a, b)
+        for j, x in list(rows[s].items()):
+            z = rows[t].get(j, Q(0)) + q * x
+            if z: rows[t][j] = z
+            else: rows[t].pop(j, None)
+
+
+def inverse(e, F):
+    e = copy.deepcopy(e)
+    if e[0] == 'add': e[3] *= -1
+    else: e[2], e[3] = e[3], e[2]
+    e[-1] = list(F); return e
+
+
+def responses(word):
+    """dirty-response columns col[h] = {target: coefficient} of the unmodified word (after the frame-zero preconditioner)"""
+    R = word['report']['R']; n = 2 * V + R
+    rows = [{r: Q(1)} for r in range(n)]
+    for e in word['A'] + word['cut'] + word['B']: apply_op(e, rows)
+    dirty = [{} for _ in range(n)]
+    for r in range(V, 2 * V): dirty[r] = {j: x for j, x in rows[r].items() if j >= 2 * V}
+    for e in word['precondition']: apply_op(e, dirty)
+    col = [{} for _ in range(R)]
+    for r in range(V, 2 * V):
+        for j, x in dirty[r].items(): col[j - 2 * V][r - V] = x
+    return col
+
+
+def first_frames(word):
+    first = {}
+    for seq in (word['A'], word['B']):
+        for e in seq:
+            F = basis(e[-1])
+            for r in [e[1]] + ([e[2]] if e[0] == 'add' else []):
+                if r >= 2 * V: first.setdefault(r - 2 * V, F)
+    return first
+
+
+def solve_relation(col, donors, p):
+    """exact rational c with col[p] = sum_d c_d col[d]; None if p is not in the span"""
+    ts = sorted({t for d in donors for t in col[d]} | set(col[p]))
+    idx = {t: i for i, t in enumerate(ts)}
+    Mx = [[Q(0)] * (len(donors) + 1) for _ in ts]
+    for ci, d in enumerate(donors):
+        for t, x in col[d].items(): Mx[idx[t]][ci] = x
+    for t, x in col[p].items(): Mx[idx[t]][-1] = x
+    piv = []; r = 0
+    for c in range(len(donors)):
+        pr = next((i for i in range(r, len(Mx)) if Mx[i][c]), None)
+        if pr is None: continue
+        Mx[r], Mx[pr] = Mx[pr], Mx[r]; q = Mx[r][c]; Mx[r] = [x / q for x in Mx[r]]
+        for i in range(len(Mx)):
+            if i != r and Mx[i][c]:
+                f = Mx[i][c]; Mx[i] = [a - f * b for a, b in zip(Mx[i], Mx[r])]
+        piv.append(c); r += 1
+    if any(Mx[i][-1] for i in range(r, len(Mx))): return None
+    assert r == len(donors), 'donors must have independent response columns'
+    return {donors[c]: Mx[i][-1] for i, c in enumerate(piv) if Mx[i][-1]}
+
+
+def span(U):
+    s = {0}
+    for b in U: s |= {x ^ b for x in s}
+    return s
+
+
+def perp(x): return basis([z for z in range(1, 512) if not (x & z).bit_count() % 2])
+
+
+def profile(word):
+    """local paid profile of a gk word by direct simulation of the frame chains (second implementation: audit_gk.py)"""
+    R = word['report']['R']; n = 2 * V + R; P = word['ports']
+    cur = {}
+    for r in range(n):
+        cur[r] = (P[r],) if r < V else ((basis(word['start_extra'][str(r)]) if str(r) in word.get('start_extra', {}) else ()) if r >= 2 * V else ())
+    paid = Counter()
+    def climb(r, F):
+        F = basis(F); o = cur[r]
+        assert basis(list(o) + list(F)) == F, ('frame violation', r, o, F)
+        if o != F: paid[len(F) - len(o)] += 1; cur[r] = F
+    lev = {int(t): basis(E) for t, E in word.get('target_levels', {}).items()}
+    for seq, part in ((word['A'], 'A'), (word['B'], 'B')):
+        if part == 'B':
+            for r, f in zip(word['centers'], word['center_frames']):
+                assert cur[r] == basis(f), 'centre frame'
+                served = sorted({lev.get(e[1] - V, ()) for e in word['cut'] if e[2] == r}, key=lambda E: -len(E))
+                o = basis(f)
+                for Lv in served:
+                    assert span(Lv) <= span(o), 'served levels must descend inside the centre frame'
+                    if Lv != o: paid[len(o) - len(Lv)] += 1; o = Lv
+            for t in range(V): assert cur[V + t] == lev.get(t, ()), 'target level at the centre scatter'
+        for e in seq:
+            for r in [e[1]] + ([e[2]] if e[0] == 'add' else []): climb(r, e[-1])
+    for r in range(n): climb(r, FULL if (r < V or r >= 2 * V) else perp(P[r - V]))
+    return paid
+
+
+def global_profile(local, widths):
+    """unhosted nine-stage profile: local x (stages*replicas) + the 240*8 idle children per rank + bank pads"""
+    live, pads = unhosted_stock(widths)
+    g = Counter({r: z * L * T for r, z in local.items()})
+    for r in (16, 8, 32, 8, 48, 8, 48, 32, 16): g[r] += 240 * T
+    g.update(pads)
+    return live, g
+
+
+def build(word, sel, check=True):    # check=False only for the mutation controls that must be caught by the certifying scripts
+    if not sel: return word
+    w = copy.deepcopy(word); R = w['report']['R']
+    col = responses(word); first = first_frames(word)
+    gauge = {int(h): basis(E) for h, E in sel.get('gauge', {}).items()}
+    classes = sel.get('kernel', [])
+    pivots = {p for c in classes for p in c['pivots']}; donors = {d for c in classes for d in c['donors']}
+    assert not (pivots & donors) and not (pivots | donors) & set(gauge) and len(pivots) == sum(len(c['pivots']) for c in classes)
+    # --- structural conditions -------------------------------------------------------------------------------------
+    for h, E in gauge.items():
+        assert not check or span(E) <= span(first[h]), ('gauge entrance outside the first frame', h)
+    levels = defaultdict(list)
+    for h, E in gauge.items():
+        for t in col[h]: levels[t].append(E)
+    for t, Es in levels.items():                                   # pairwise nested comps per target
+        for a in Es:
+            for b in Es: assert not check or span(a) <= span(b) or span(b) <= span(a), ('non-nested gauges at target', t)
+    lev = {t: max(Es, key=len) for t, Es in levels.items()}
+    for k, r in enumerate(w['centers']):                              # chain per centre, inside the centre frame
+        served = {lev.get(e[1] - V, ()) for e in w['cut'] if e[2] == r}
+        for a in served:
+            for b in served: assert not check or span(a) <= span(b) or span(b) <= span(a), ('centre chain', k)
+        top = max(served, key=len)
+        assert not check or span(top) <= span(basis(w['center_frames'][k])), ('level outside centre frame', k)
+    # --- gates ----------------------------------------------------------------------------------------------------------
+    gates = []
+    for h, E in sorted(gauge.items(), key=lambda kv: (len(kv[1]), kv[0])):
+        for t, x in sorted(col[h].items()):
+            c = -x; gates.append(['add', V + t, 2 * V + h, c.numerator, c.denominator, list(E)])
+    pre = []
+    start = {2 * V + h: list(E) for h, E in gauge.items()}
+    for cl in sorted(classes, key=lambda c: (len(c['E']), c['E'])):
+        E = basis(cl['E'])
+        for x in [*cl['donors'], *cl['pivots']]: assert span(E) <= span(first[x]), ('kernel entrance outside the first frame', x)
+        for p in cl['pivots']:
+            rel = solve_relation(col, cl['donors'], p)
+            assert rel is not None, ('pivot response not in the donor span', p)
+            start[2 * V + p] = list(E)
+            for d, c in rel.items():
+                q = -c; pre.append(['add', 2 * V + d, 2 * V + p, q.numerator, q.denominator, list(E)])
+    w['A'] = gates + pre + w['A']
+    w['cleanup'] = [inverse(e, FULL) for e in reversed(w['A'] + w['B']) if not V <= e[1] < 2 * V]
+    w['start_extra'] = {str(r): list(E) for r, E in sorted(start.items())}
+    w['target_levels'] = {str(t): list(E) for t, E in sorted(lev.items()) if E}
+    if not check: return w
+    prof = profile(w); widths = widths_of(w)
+    live, g = global_profile(prof, widths)
+    w['report']['local_profile'] = {str(k): z for k, z in sorted(prof.items())}
+    w['report']['global_profile'] = {str(k): z for k, z in sorted(g.items())}
+    w['report']['live'] = live
+    w['report']['entrance_rank'] = sum(L - x for x in widths); w['report']['entering_helpers'] = sum(1 for x in widths if x < L)
+    w['report']['selection_sha256'] = hashlib.sha256(json.dumps(sel, sort_keys=True).encode()).hexdigest()
+    return w
+
+
+def dump(w): return gzip.compress(json.dumps(w, sort_keys=True, separators=(',', ':')).encode(), mtime=0)
+
+
+if __name__ == '__main__':
+    src, selp, out = map(Path, sys.argv[1:4])
+    sel = json.loads(selp.read_text())
+    if not sel: out.write_bytes(src.read_bytes()); print('empty selection: copied byte for byte'); sys.exit(0)
+    word = json.load(gzip.open(src)); w = build(word, sel); out.write_bytes(dump(w))
+    print('gk word: entering helpers', w['report']['entering_helpers'], 'entrance rank', w['report']['entrance_rank'],
+          'local profile', w['report']['local_profile'], 'unhosted live', w['report']['live'])
diff --git a/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/integrate_gk.py b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/integrate_gk.py
new file mode 100644
index 0000000000..df261063e1
--- /dev/null
+++ b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/integrate_gk.py
@@ -0,0 +1,123 @@
+# GK: modified copy of PR #433's complex_integrate.py (icekylinx with Codex assistance; PR #425) by DreamingOfClouds with Claude Opus 5.5
+# (Anthropic), via Claude Code.  Every change is marked "GK:".  On a word without start_extra / target_levels the behaviour
+# and every written file equal PR #433's (regression anchor in verify.py).
+import sys as _sys, os as _os; _sys.path.insert(0, _os.path.dirname(_os.path.abspath(__file__)))
+# Adapted from PR425 for heterogeneous centres by icekylinx, with Codex assistance.
+import copy,gzip,hashlib,importlib.util,json,math
+from collections import Counter
+from fractions import Fraction as Q
+from pathlib import Path
+OUT=Path('split-centres');cp=OUT/'CANONICAL-WORD.json.gz'
+c=json.load(gzip.open(cp));audit=json.loads((OUT/'SCALAR-FRAME-AUDIT.json').read_text())
+phase=json.loads((OUT/'CLIFFORD-REVIEW.json').read_text());sha=lambda p:hashlib.sha256(Path(p).read_bytes()).hexdigest()
+assert phase['canonical_word_sha256']==audit['canonical_word_sha256']==sha(cp)
+assert phase['all_child_equalities_include_global_phase']
+programs=json.load(gzip.open(OUT/'LOCAL-PROGRAMS.json.gz'));checks=json.loads((OUT/'LOCAL-CHECKS.json').read_text())
+frames=c['frames'];v,R,h=c['v'],c['R'],c['h'];n=2*v+R;private_centres=len(c['centers'])
+for z in checks.values():z['paid_profile']={int(r):n for r,n in z['paid_profile'].items()}
+old=Path('work/complex-nine-native');template=json.load(gzip.open(old/'GLOBAL-TEMPLATE.json.gz'))
+data_phase=json.loads((old/'DATA-PHASES.json').read_text())
+assert template['exact_data_phase_recipe_sha256']==sha(old/'DATA-PHASES.json')
+assert data_phase['status']=='PASS_EXACT_Z4_DATA_ROUTE_IDLE_AND_TERMINAL_PHASES'
+m,T,L,k=73,8,9,4
+# GK: residual widths 9 - dim(start frame); banks tile per width class (all 9: PR #433's eight residuals + one pad)
+from gk_common import tile,route
+widths=[h-len(frames[c['start'][2*v+i]]) for i in range(R)]
+assign,nb,pads=tile([(rep,i,widths[i]) for rep in range(T) for i in range(R)]);entering0=any(w_<h for w_ in widths)
+W=2*v*k*T+L*nb;physical=W+R+private_centres+2
+template['version']='hamming-mutable-basis-nine-v1';template['live']=W;template['physical']=physical
+template['local_programs']=programs;paid=Counter();assignments=set();event_count=0;scalar_primitives=0
+for call in template['calls']:
+    stage,rep,pair=call['stage'],call['replica'],call['pair'];ori=call['orientation']
+    owners=[[rep*2*k*v+(2*pair+r//v)*v+r%v,[1,stage,r%v]] for r in range(2*v)]
+    for r in range(R):
+        bank,off_,w_,pure_,sp_=assign[rep,r];owners.append([2*v*k*T+stage*nb+bank,route(off_,w_,pure_,sp_)])   # GK: [2,slot] in pure width-9 banks, [2,offset,width,spectators] for entering helpers
+        assignments.add((stage,rep,r,bank,off_))
+    assert len(owners)==n and len({(f,json.dumps(route)) for f,route in owners})==n
+    assert all(f<W for f,route in owners)   # GK: every owner lies in the live stock (none in the private namespace)
+    fam=Counter(f for f,route in owners)   # GK: an entering helper is the only owner of its bank family in the invocation (PR #458's rule, banks included); the data families are the 240 distinct active ones
+    assert all(fam[f]==1 for f,route in owners[:2*v]) and all(fam[owners[2*v+r][0]]==1 for r in range(R) if widths[r]<h),('entering helper shares its bank family',stage,rep)
+    call['owners']=owners;call['local_program_sha256']=checks[ori]['sha256']
+    for r,z in checks[ori]['paid_profile'].items():paid[r]+=z
+    event_count+=checks[ori]['events'];scalar_primitives+=checks[ori]['real_component_primitive_bound']
+assert len(assignments)==L*T*R and len(template['calls'])==72
+for item in template['idle_children']:paid[item['rank']]+=item['replicas']
+for r_,z_ in pads.items():paid[r_]+=z_*L   # GK: bank pads (rank 1 each at width 9 and 8)
+assert dict(paid)=={int(r):z for r,z in json.load(gzip.open(OUT/'FULL-WORD.json.gz'))['report']['global_profile'].items()}
+mass=sum(r*z for r,z in paid.items());assert m*W-mass==2496
+if entering0:template['route_semantics']['helper']+='; entering helpers: paid wrappers R_E^-1 before and L_E^-1 after the invocation; an entering helper is the only owner of its bank family in the invocation, route [2, offset, width, spectator cells] with the spectators the first cells of the bank beyond every residual cell'   # GK
+template['bank_padding']['families']=list(range(7680,W))
+template['cover_final_correction']['families']=[0,W]
+template['cover_final_correction']['charged_calls_per_cover']=W
+for e in template['execution_schedule']:
+    if e['kind']=='bank_padding_cover_sweep':e['families']=[7680,W]
+    elif e['kind']=='translate_every_live_family':e.update(template['cover_final_correction'])
+template['private_namespace']={'centers':[W,W+private_centres],'reserved_tableau':[W+private_centres,W+private_centres+R],
+                               'coefficient':[physical-2,physical]}
+template['exact_local_clifford_recipe_sha256']=sha(OUT/'CLIFFORD-RECIPES.json')
+if (OUT/'RESIDUAL-ADAPTERS.json').exists():template['residual_adapter_sha256']=sha(OUT/'RESIDUAL-ADAPTERS.json')   # GK: paid wrappers of entering helpers (Lemma A)
+template['expanded_scalar_interface']='See PROOF.txt: mutable source basis, frame-zero output preconditioner, explicit dirty compensator, and full-frame inverse source/helper cleanup.'
+blob=json.dumps(template,sort_keys=True,separators=(',',':')).encode()
+(OUT/'GLOBAL-TEMPLATE.json.gz').write_bytes(gzip.compress(blob,mtime=0))
+
+# Fresh finite overcharge, with the changed stock, scalar counts, coefficient
+# lowering, and private work reserve. No unchanged local invoice is assumed.
+compiler=64*(m+1)**3;macros=max(z['scalar_macros'] for z in checks.values());children=sum(checks['forward']['paid_profile'].values())
+adapters=16*(macros+children+n+private_centres)+64*m
+local_bound=max(z['real_component_primitive_bound'] for z in checks.values())+adapters*compiler
+bridges=[e for e in template['recipe'] if e['kind']=='bridge'];bridge_adds=2*v*T*len(bridges)
+terminal_shifts=k*v*T;terminal_moves=2*k*v*T
+entering=sum(1 for w_ in widths if w_<h)   # GK: entrance adapters (Lemma A): a monomial on 9 bits is at most 81+45+9=135 elementary factors (matrix, quadratic form, translation); charge 144 at entry and at exit of every entering helper in every invocation
+nonlocal_items=4*v*T*L+2*v*T*L+2*L*T*R+sum(e['replicas'] for e in template['idle_children'])+L*nb+bridge_adds+terminal_shifts+terminal_moves+W+physical+2*L*T*entering*144
+outside=8*nonlocal_items+64*m
+pervertex=scalar_primitives+bridge_adds+terminal_shifts+terminal_moves+W+(72*adapters+outside)*compiler+64*physical
+r=m//2;group=2**(r*r)*math.prod(2**(2*i)-1 for i in range(1,r+1));stock=group*physical
+smass=group*mass;logical=group*pervertex+64;router=compiler*(logical+1)*(stock+1)**2
+GG,EE,C0=1<<30000,1<<100000,1<<210000;BB=smass+EE
+literal=2*GG*stock**2+8*smass+4*stock+4+32*m
+maxchild=max(paid);depth=2;assert m**depth>2*maxchild**depth
+assert local_bound<1<<48 and logical<1<<6050 and router<GG and literal<EE and BB<1<<100001
+assert 32*m*BB*BB<C0 and 2*BB+18<C0
+row=depth*stock.bit_length()+9909+252;assert row<20161
+
+def load(name):
+    sp=importlib.util.spec_from_file_location('hamming_integrate_'+name,Path('work/pr361/code/engines')/(name+'.py'))
+    mod=importlib.util.module_from_spec(sp);sp.loader.exec_module(mod);return mod
+Outer=load('outer');b=Q(audit['prospective_complex_b']);bit=Q(json.loads(Path('BIT-PRICE.json').read_text())['lower'])
+eta=Q(1,10**12);beta=Q(1,10**9);grid=Q(1,10**18)
+s=Q(int(((1-beta)*min(b,bit)-grid)*10**18),10**18);assert s<bit
+chain=[Q(384599,10**10)];cutoffs=[]
+for _ in range(3):
+    a=chain[-1];z=(1-s)*s+s*a;gap=min(s-z,1-z-s,1-z-s*(1-a),1-s)
+    assert a<z<s<1-z and gap>0
+    bound=max(math.ceil(36/gap**2),math.ceil(2*(4+512)/gap))
+    assert bound*gap**2>=36 and bound*gap>=2*(4+512)
+    cutoffs.append(str(bound));chain.append(z)
+bridge={'semantic':{'C0':{'base':2,'exponent':210000},'strict_literal_gap':1},
+        'rows':{'degree_gap':Q(10**6)-Q(51*20161,25)}}
+q=chain[-1]*(1-2*eta);ticks=(1-eta)*q/(1+q)/grid;kap=Q((ticks.numerator-1)//ticks.denominator,10**18)
+assembly=Outer.assembly(chain[-1],b,bridge,kap,eta=eta,beta=beta)
+try:Outer.assembly(chain[-1],b,bridge,kap+grid,eta=eta,beta=beta)
+except AssertionError:pass
+else:raise AssertionError('next kappa grid accepted')
+result={'status':'PASS_NEW_STRUCTURAL_PROGRAM_AND_FINITE_ASSEMBLY_PENDING_REPLAY',
+        'canonical_word_sha256':sha(cp),'scalar_audit_sha256':sha(OUT/'SCALAR-FRAME-AUDIT.json'),
+        'exact_phase_receipt_sha256':sha(OUT/'CLIFFORD-REVIEW.json'),'global_template_sha256':hashlib.sha256(blob).hexdigest(),
+        'local_program_checks':checks,'R':R,'m':m,'live':W,'physical':physical,
+        'bank_assignments':len(assignments),'rank_mass':mass,'deficit':2496,
+        'global_profile':dict(sorted(paid.items())),'emitted_local_events_all_calls':event_count,
+        'finite_invoice':{'local_primitive_bound':local_bound,'local_adapters':adapters,
+                          'pervertex':pervertex,'physical_stock_bits':stock.bit_length(),
+                          'router_bits':router.bit_length(),'literal_bits':literal.bit_length(),
+                          'physical_row_coefficient':row,'retained_row_coefficient':20161,
+                          'compiler':compiler,'G_exponent':30000,'E_exponent':100000,'C0_exponent':210000,
+                          'required_C0_bits':(32*m*BB*BB).bit_length(),'halving_degree':depth,
+                          'halving_integer_inequality':[m**depth,2*maxchild**depth]},
+        'complex_b':str(b),'bit_c':str(bit),'prospective_kappa':str(kap),'kappa_float':float(kap),
+        'log2_kappa':math.log2(float(kap)),
+        'kappa_gain_percent':float(100*(kap/Q(998390813251625,10**18)-1)),
+        'strict_outer_constraints':{name:str(z) for name,z in assembly['strict_constraints'].items()},
+        'next_kappa_grid_rejected':True,'bootstrap_cutoffs_log2':cutoffs,
+        'scope':'New finite scalar interface with a written conjugation proof, conditional on inherited all-size compiler, precision/tape and analytic transfer. Independent emitted-program replay and fresh package replay still required before final promotion.'}
+(OUT/'INTEGRATION.json').write_text(json.dumps(result,indent=2)+'\n')
+print('PASS prospective kappa',float(kap),'gain',result['kappa_gain_percent'],'percent',flush=True)
diff --git a/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/onehost_gk.py b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/onehost_gk.py
new file mode 100644
index 0000000000..220cc7a2f5
--- /dev/null
+++ b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/onehost_gk.py
@@ -0,0 +1,111 @@
+"""Whole-residual idle-axis hosting with at most ONE helper per physical family per invocation (the rule of PR #458), plus the exact
+host chart with explicit spectator coordinates.  Apache-2.0.  Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.
+A copy (port) of an idle child hosts a PATTERN: residuals of pairwise distinct stages inside the child's window, offsets 0, w1, w1+w2, ...
+(a residual is whole: one slot in one family).  The children of one (pair, side) have pairwise disjoint windows (checked), so the
+families they share never host two helpers of one invocation, and the integer programme needs only the distinct-stage condition.
+Chart: for the host's port q the parent's idle basis is B_q = U_q (+) I (PR #433 geometry; the construction is that of PR #458).  A hosted
+residual of width w occupies the pending vectors b_off..b_(off+w-1) of its child; the other e = 9 - w coordinates of the local nine are
+SPECTATORS: the first e basis indices of B_q not occupied by the residual, in increasing order.  The paid wrappers (adapters_gk.py) act on
+these nine coordinates and restore the spectators identically."""
+import math
+from collections import Counter
+
+H, M = 9, 73
+
+
+def cost(r, m): return r * math.log(m / r) if r else 0.0
+
+
+def patterns(rank, stages, widths):
+    out = [()]
+    ws = sorted(widths, reverse=True)
+    def rec(i, cur, tot):
+        for j in range(i, len(stages)):
+            for w in ws:
+                if tot + w <= rank:
+                    c = cur + [(stages[j], w)]; out.append(tuple(c)); rec(j + 1, c, tot + w)
+    rec(0, [], 0)
+    return out
+
+
+def solve(win, copies, avail, m, pad_value, time_limit=600):
+    """win {idx: (pair, rank, stages)}, avail {(stage, width): helpers per replica}.  Maximise saved idle cost (+ pad_value per hosted residual)."""
+    import numpy as np
+    from scipy.optimize import milp, LinearConstraint, Bounds
+    widths = sorted({w for (_, w) in avail}); var = []
+    for idx, (p, r, st) in sorted(win.items()):
+        for pat in patterns(r, sorted(st), widths): var.append((idx, pat))
+    n = len(var); c = np.zeros(n)
+    for j, (idx, pat) in enumerate(var):
+        r = win[idx][1]; c[j] = -(cost(r, m) - cost(r - sum(w for _, w in pat), m) + pad_value * len(pat))
+    rows, lo, hi = [], [], []
+    by = {}
+    for j, (idx, pat) in enumerate(var): by.setdefault(idx, []).append(j)
+    for idx, js in by.items():
+        row = np.zeros(n); row[js] = 1; rows.append(row); lo.append(copies); hi.append(copies)
+    for (s, w), a in sorted(avail.items()):
+        row = np.zeros(n)
+        for j, (idx, pat) in enumerate(var): row[j] = sum(1 for sw in pat if sw == (s, w))
+        rows.append(row); lo.append(0); hi.append(a)
+    res = milp(c, constraints=LinearConstraint(np.array(rows), lo, hi), integrality=np.ones(n), bounds=Bounds(np.zeros(n), np.full(n, np.inf)),
+               options=dict(time_limit=time_limit, disp=False))
+    assert res.x is not None, res.message
+    x = [int(round(t)) for t in res.x]
+    return [[idx, [list(sw) for sw in pat], x[j]] for j, (idx, pat) in enumerate(var) if x[j]], -res.fun
+
+
+def save(path, key, sel, val):
+    import json
+    from pathlib import Path
+    p = Path(path); d = json.loads(p.read_text()) if p.exists() else {}
+    d[key] = dict(rule='one-helper-per-family', patterns=sorted(sel), proxy_saving=val)
+    p.write_text(json.dumps(d, indent=1, sort_keys=True) + '\n')
+
+
+def load(path, key):
+    import json
+    from pathlib import Path
+    d = json.loads(Path(path).read_text())[key]; assert d['rule'] == 'one-helper-per-family'
+    return d['patterns'], d['proxy_saving']
+
+
+def check_disjoint_groups(win, pair_of, side_of):
+    """children of one (pair, side) share their physical families; their windows must be disjoint"""
+    seen = {}
+    for idx, (p, r, st) in win.items():
+        for s in st:
+            k = (pair_of[idx], side_of[idx], s); assert k not in seen, ('windows of two children of one family group overlap', idx, seen[k], k); seen[k] = idx
+
+
+def expand(win, sel, widths, V):
+    """-> slots [idx, port, off, w, s, h] (one replica), rem {(idx, port): remaining pending rank}"""
+    R = len(widths); pool = {}
+    for h in sorted(range(R)): pool.setdefault(widths[h], []).append(h)
+    pool = {(s, w): list(hs) for s in range(9) for w, hs in pool.items()}
+    slots = []; rem = {}; used = Counter()
+    for idx, pat, copies in sorted(sel, key=lambda e: (e[0], e[1])):
+        r = win[idx][1]
+        for _ in range(copies):
+            cp = used[idx]; used[idx] += 1; assert cp < V; off = 0
+            assert len({s for s, w in pat}) == len(pat)
+            for s, w in pat:
+                assert s in win[idx][2] and off + w <= r
+                slots.append([idx, cp, off, w, s, pool[s, w].pop(0)]); off += w
+            rem[idx, cp] = r - off
+    assert all(n == V for n in used.values()) and len(used) == len(win)
+    return slots, rem
+
+
+def idle_basis(q):
+    """PR #433 / #458: the parent's orthonormal idle basis U_q (+) I for port q (H = 9 local coordinates, M = 73)"""
+    z = next(i for i in range(H) if not (q >> i & 1)); a = q ^ (1 << z); order = list(range(H)); order[0], order[z] = order[z], order[0]
+    return [(1 << i) ^ (a if a >> i & 1 else 0) for i in order] + [1 << i for i in range(H, M)]
+
+
+def spectators(q, pending, off, w):
+    """indices (into B_q) of the e = 9 - w spectator coordinates of a residual at pending[off:off+w]"""
+    B = idle_basis(q); occ = [B.index(x) for x in pending[off:off + w]]
+    assert len(set(occ)) == w
+    sp = [j for j in range(M) if j not in occ][:H - w]
+    assert len(sp) == H - w
+    return sp
diff --git a/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/replay_gk.py b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/replay_gk.py
new file mode 100644
index 0000000000..b2e08a30c9
--- /dev/null
+++ b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/code/replay_gk.py
@@ -0,0 +1,271 @@
+# GK: modified copy of PR #433's complex_replay.py (icekylinx with Codex assistance; PR #425) by DreamingOfClouds with Claude Opus 5.5
+# (Anthropic), via Claude Code.  Every change is marked "GK:".  On a word without start_extra / target_levels the behaviour
+# and every written file equal PR #433's (regression anchor in verify.py).
+import sys as _sys, os as _os; _sys.path.insert(0, _os.path.dirname(_os.path.abspath(__file__)))
+# Adapted for heterogeneous centres by icekylinx with Codex assistance.
+"""Independent read-back of the serialized local/global instruction stream."""
+import copy,gzip,hashlib,importlib.util,json,math
+from collections import Counter
+from fractions import Fraction as Q
+from pathlib import Path
+
+OUT=Path('split-centres');P=OUT/'GLOBAL-TEMPLATE.json.gz'
+ir=json.load(gzip.open(P));c=json.load(gzip.open(OUT/'CANONICAL-WORD.json.gz'))
+integration=json.loads((OUT/'INTEGRATION.json').read_text())
+assert hashlib.sha256(gzip.decompress(P.read_bytes())).hexdigest()==integration['global_template_sha256']
+assert c['ports']==json.load(gzip.open('work/higher-stage-e8/verified/gcert1-e8-retimed.json.gz'))['ports']
+v,R,h=c['v'],c['R'],c['h'];n=2*v+R;frames=c['frames']
+from gk_common import tile   # GK
+widths=[h-len(frames[c['start'][2*v+i]]) for i in range(R)]
+assign,nb,pads=tile([(rep,i,widths[i]) for rep in range(8) for i in range(R)]);W=7680+9*nb
+levels={int(t):f for t,f in c.get('target_levels',{}).items()}
+entering=[r for r,f in enumerate(c['start']) if r>=2*v and f!=0];plain={r for r in range(n) if r not in set(entering)}   # GK
+def contains(a,b):
+    for x in a:
+        for y in b:
+            if x>>(y.bit_length()-1)&1:x^=y
+        if x:return False
+    return True
+def local(events,back):
+    rows={('L',r):{r:Q(1)} for r in range(n)}
+    norm={('L',r):Q(1) for r in range(n)};prefix=Q(1);cur=c['start'].copy();private={}
+    hist=Counter();phases=[];entry=[];exit=[];centers=[];matrix_hash=hashlib.sha256();chain={};wrappers=[]
+    allowed=['zero_frame_precondition','entry_adapters','A','centers','B','final','exit_adapters','full_frame_cleanup','done']
+    for event in events:
+        tag=event[0]
+        if tag=='phase':
+            phases.append(event[1]);assert phases==allowed[:len(phases)]
+            if event[1]=='A':assert entry==[(r,f) for r,f in enumerate(c['start']) if r in plain] and sorted(r for k_,r,f_ in wrappers if k_=='pre')==entering   # GK
+            if event[1]=='centers':assert all(cur[r]==levels.get(r-v,0) for r in range(v,2*v))   # GK
+            if event[1]=='full_frame_cleanup':assert cur==c['final'] and exit==list(enumerate(c['final']))
+            assert not private
+        elif tag=='residual_adapter':   # GK: paid wrappers of entering helpers: R_E^-1 before the invocation, L_E^-1 after the full-frame cleanup
+            _,kind,r,f=event;assert r in entering and f==c['start'][r] and kind in ('pre','post')
+            assert (kind=='pre' and phases[-1]=='entry_adapters') or (kind=='post' and phases[-1]=='full_frame_cleanup')
+            assert (kind,r) not in {(k,x) for k,x,_ in wrappers};wrappers.append((kind,r,f))
+        elif tag=='adapter':
+            assert event[3]==(c['start'] if event[1]=='old_to_new' else c['final'])[event[2]]
+            (entry if event[1]=='old_to_new' else exit).append((event[2],event[3]))
+        elif tag=='climb':
+            _,part,r,a,b,rank=event
+            assert not private and phases[-1]==part and cur[r]==a
+            assert contains(frames[a],frames[b]) and rank==len(frames[b])-len(frames[a])>0
+            hist[rank]+=1;cur[r]=b
+        elif tag=='center':
+            _,k,a,b,rank=event
+            assert phases[-1]=='centers' and private=={('C',k):a}
+            cf=c['center_frames'][k];cr=len(frames[cf])
+            # GK: stepwise descent cf -> served levels -> lowest (forward), reversed ascent (backward); each step is a nested child
+            assert contains(frames[a],frames[b]) if back else contains(frames[b],frames[a])
+            assert rank==abs(len(frames[a])-len(frames[b]))>0
+            chain.setdefault(k,[]).append((a,b))
+            private['C',k]=b;hist[rank]+=1;centers.append(k)
+        elif tag=='scalar':
+            _,e,f=event;op=e[0];target=tuple(e[1]);source=tuple(e[2]) if op in ('copy','add') else None
+            if target[0]=='C' and op=='clear':
+                assert private.get(target,f)==f
+                private.pop(target,None)
+            else:
+                for r in (target,source):
+                    if r is None:continue
+                    if r[0]=='L':assert cur[r[1]]==f,('common frame',r,cur[r[1]],f)
+                    else:
+                        if r==target and r not in private:private[r]=f
+                        assert private[r]==f
+            if op=='clear':rows[target]={};norm[target]=Q(0)
+            elif op=='copy':rows[target]=dict(rows[source]);norm[target]=norm[source]
+            elif op=='scale':
+                q=Q(e[2],e[3]);a,b=abs(q.numerator),q.denominator
+                assert a&(a-1)==0 and b&(b-1)==0
+                rows[target]={j:q*x for j,x in rows[target].items()};norm[target]*=abs(q)
+            else:
+                q=Q(e[3],e[4]);a,b=abs(q.numerator),q.denominator
+                assert source!=target and b&(b-1)==0
+                # Independent symbolic replay of the cleared coefficient work.
+                w=0
+                for bit in bin(a)[2:]:w=2*w+int(bit)
+                assert Q(w,b)*(1 if q>0 else -1)==q
+                prefix=max(prefix,Q(a)*norm[source])
+                norm[target]=norm.get(target,Q(0))+abs(q)*norm[source]
+                row=rows.setdefault(target,{})
+                for j,x in list(rows[source].items()):
+                    z=row.get(j,Q(0))+q*x
+                    if z:row[j]=z
+                    else:row.pop(j,None)
+            prefix=max(prefix,norm[target])
+        else:raise AssertionError(tag)
+    assert phases==allowed and set(centers)==set(range(len(c['centers']))) and not private
+    assert sorted(r for k,r,_ in wrappers if k=='pre')==entering and sorted(r for k,r,_ in wrappers if k=='post')==entering   # GK
+    for k,st in chain.items():   # GK: the steps of each centre chain link up and end at the centre frame
+        cf=c['center_frames'][k]
+        if back:assert st[-1][1]==cf and all(st[i][1]==st[i+1][0] for i in range(len(st)-1))
+        else:assert st[0][0]==cf and all(st[i][1]==st[i+1][0] for i in range(len(st)-1))
+    assert all(not row for reg,row in rows.items() if reg[0]=='C')
+    for r in range(n):
+        expected={r:Q(1)}
+        if back and r<v:expected[v+r]=Q(-1)
+        elif not back and v<=r<2*v:expected[r-v]=Q(1)
+        assert rows['L',r]==expected
+    # A 192-bit local payload allowance vastly dominates even coefficient work.
+    assert prefix<2**192
+    return hist,{'all_exact_columns':n,'prefix_l1_upper':str(prefix),
+                 'prefix_bits_upper':prefix.numerator.bit_length()-prefix.denominator.bit_length()+1,
+                 'charged_payload_bits':192,'private_work_erased':True}
+
+profiles={};checks={}
+for name in ('forward','backward'):
+    profiles[name],checks[name]=local(ir['local_programs'][name],name=='backward')
+    assert dict(profiles[name])=={int(r):z for r,z in c['local_profile'].items()}
+    print('independent emitted replay PASS',name,flush=True)
+controls=[]
+for name,mut in [
+    ('wrong_scale',lambda es:next(e for e in es if e[0]=='scalar' and e[1][0]=='scale')[1].__setitem__(2,3)),
+    ('missing_center',lambda es:es.pop(next(i for i,e in enumerate(es) if e[0]=='center'))),
+    ('wrong_common_frame',lambda es:next(e for e in es if e[0]=='scalar' and e[2]!=0).__setitem__(2,0)),
+    ('wrong_child_rank',lambda es:next(e for e in es if e[0]=='climb').__setitem__(5,99)),
+]+([
+    ('missing_pre_wrapper',lambda es:es.pop(next(i for i,e in enumerate(es) if e[0]=='residual_adapter' and e[1]=='pre'))),
+    ('missing_post_wrapper',lambda es:es.pop(next(i for i,e in enumerate(es) if e[0]=='residual_adapter' and e[1]=='post'))),
+    ('wrapper_with_wrong_frame',lambda es:next(e for e in es if e[0]=='residual_adapter').__setitem__(3,0)),
+    ('wrapper_in_wrong_phase',lambda es:es.insert(es.index(['phase','A']),es.pop(next(i for i,e in enumerate(es) if e[0]=='residual_adapter' and e[1]=='post')))),
+    ('wrapper_on_a_data_register',lambda es:next(e for e in es if e[0]=='residual_adapter').__setitem__(2,3)),
+    ('wrapper_on_a_target_register',lambda es:next(e for e in es if e[0]=='residual_adapter').__setitem__(2,v+3)),
+    ('wrapper_on_a_centre_register',lambda es:next(e for e in es if e[0]=='residual_adapter').__setitem__(2,['C',0])),
+] if entering else [])+([
+    ('centre_step_not_nested',lambda es:next(e for e in es if e[0]=='center').__setitem__(3,next(f for f in range(len(frames)) if not contains(frames[f],frames[next(e for e in es if e[0]=='center')[2]]) and not contains(frames[next(e for e in es if e[0]=='center')[2]],frames[f])))),
+    ('centre_descent_step_skipped',lambda es:es.pop(next(i for i,e in enumerate(es) if e[0]=='center' and e[3]!=0))),
+] if levels else []):
+    es=copy.deepcopy(ir['local_programs']['forward']);mut(es)
+    try:local(es,False)
+    except (AssertionError,KeyError):controls.append(name)
+    else:raise AssertionError('mutation accepted: '+name)
+
+def residual_cells(route):
+    """cells of a bank family occupied by a helper residual (independent of tile): a width-9 slot or [offset, offset + width)"""
+    return set(range(route[1]*9,route[1]*9+9)) if len(route)==2 else set(range(route[1],route[1]+route[2]))
+def bank_rule(owners,occ,sole=True):
+    """GK: the one-helper rule of PR #458 on EVERY register class of one invocation (owners: 240 active data owners, then the 560 helpers).
+    data class (family < 7680): only the active wires, each the sole owner of its family; bank class (7680 <= family < W): helpers only; an ENTERING
+    helper (width < 9) is the sole owner of its bank family, its route is [2, offset, width, spectators] with the spectators the first
+    cells of the bank beyond every residual cell of it (occ[family] = those cells, over all invocations); width-9 helpers share a pure bank in
+    disjoint 9-cell slots (no wrapper acts on them); nobody owns a family of the private namespace (family >= W).  sole=False skips only the
+    sole-owner assertion (used by a control to show that the footprints themselves are fine)."""
+    fam=Counter(f for f,_ in owners);foot={}
+    for i,(f,route) in enumerate(owners):
+        assert f<W,('owner in the private namespace',f)
+        if f<7680:
+            assert i<240 and route[0]==1 and fam[f]==1,('data class: only the active wires, each alone on its family',i,f)
+        else:
+            assert i>=240 and route[0]==2,('bank class: helpers only',i,f);w=widths[i-240]
+            if w==9: assert len(route)==2,('width-9 helper in a pure bank',i)
+            else:
+                if sole: assert fam[f]==1,('entering helper shares its bank family in one invocation',i,f)
+                assert len(route)==4 and route[2]==w and len(set(route[3]))==9-w==len(route[3]),('entering helper without a spectator record',i)
+                assert route[3]==list(range(len(occ[f]),len(occ[f])+9-w)) and len(occ[f])+9-w<=73,('spectators are not the first cells beyond the residuals',i,f)
+            cells=residual_cells(route)|(set(route[3]) if len(route)==4 else set())
+            assert len(cells)==9 and not cells&foot.setdefault(f,set()),('footprints overlap on a bank family',i,f);foot[f]|=cells
+def global_check(t):
+    assert t['live']==W and t['physical']==W+R+len(c['centers'])+2   # GK
+    occ={}
+    for call in t['calls']:
+        for fam,route in call['owners'][240:]:occ.setdefault(fam,set()).update(residual_cells(route))
+    assert all(cs==set(range(len(cs))) for cs in occ.values()),'bank residuals are packed from cell 0'
+    # Eight data-wire scalar columns, independently recomposed.
+    matrix=[[int(i==j) for j in range(8)] for i in range(8)]
+    for e in t['recipe']:
+        if e['kind']=='local':
+            a=2*e['pair'];b=a+1;dst,src,q=(a,b,-1) if e['reverse'] else (b,a,1)
+            matrix[dst]=[x+q*y for x,y in zip(matrix[dst],matrix[src])]
+        elif e['kind']=='bridge':
+            i,j,q=e['dst'],e['src'],e['coefficient']
+            matrix[2*i]=[x+q*y for x,y in zip(matrix[2*i],matrix[2*j])]
+            matrix[2*j+1]=[x-q*y for x,y in zip(matrix[2*j+1],matrix[2*i+1])]
+        else:
+            assert e['kind']=='terminal';old=copy.deepcopy(matrix)
+            for i in range(4):matrix[2*i]=old[2*i+1];matrix[2*i+1]=[-x for x in old[2*i]]
+    assert matrix==[[int(i==j) for j in range(8)] for i in range(8)]
+    seen=set();hist=Counter();pairs={e['stage']:e for e in t['recipe'] if e['kind']=='local'}
+    for call in t['calls']:
+        stage,rep,pair=call['stage'],call['replica'],call['pair']
+        assert (stage,rep) not in seen;seen.add((stage,rep))
+        assert pair==pairs[stage]['pair']
+        assert call['orientation']==('backward' if pairs[stage]['reverse'] else 'forward')
+        assert len(call['owners'])==800
+        for r,(fam,route) in enumerate(call['owners']):
+            if r<240:assert fam==rep*960+(2*pair+r//120)*120+r%120 and route==[1,stage,r%120]
+            else:
+                bk,off_,w_,pure_,sp_=assign[rep,r-240];assert fam==7680+stage*nb+bk and route==([2,off_//9] if (w_==9 and pure_) else [2,off_,w_,list(sp_)])   # GK
+        assert len(set((f,json.dumps(r)) for f,r in call['owners']))==800
+        bank_rule(call['owners'],occ)   # GK: the rule on every register class, stated independently of tile()
+        for r,z in profiles[call['orientation']].items():hist[r]+=z
+    assert seen=={(s,r) for s in range(9) for r in range(8)}
+    for child in t['idle_children']:hist[child['rank']]+=child['replicas']
+    assert t['bank_padding']=={'families':list(range(7680,W)),'axis':72,'child_rank':1}
+    for r_,z_ in pads.items():hist[r_]+=z_*9   # GK
+    assert t['cover_final_correction']['families']==[0,W]
+    assert t['cover_final_correction']['charged_calls_per_cover']==W
+    assert t['execution_schedule'][-1]=={'kind':'translate_every_live_family',**{k:v for k,v in t['cover_final_correction'].items() if k!='kind'}}
+    assert 73*W-sum(r*z for r,z in hist.items())==2496
+    return hist
+H=global_check(ir)
+for name,mut in [('missing_padding',lambda t:t['bank_padding']['families'].pop()),
+                 ('aliased_owner',lambda t:t['calls'][0]['owners'][241].__setitem__(1,[2,0])),
+                 ('missing_cover_phase',lambda t:t['cover_final_correction'].__setitem__('charged_calls_per_cover',0))]:
+    t=copy.deepcopy(ir);mut(t)
+    try:global_check(t)
+    except (AssertionError,KeyError):controls.append(name)
+    else:raise AssertionError('global mutation accepted: '+name)
+# GK: controls of the rule itself, one per register class, run on bank_rule (not on the equality with tile()); only for words with entering
+# helpers (the anchor's INDEPENDENT-REPLAY.json must stay byte-identical to PR #433's)
+occ_all={}
+for call in ir['calls']:
+    for fam_,route_ in call['owners'][240:]:occ_all.setdefault(fam_,set()).update(residual_cells(route_))
+def rule_control(name,mut,must):
+    o=copy.deepcopy(ir['calls'][0]['owners']);mut(o)
+    try:bank_rule(o,occ_all)
+    except AssertionError as e:
+        assert must in str(e),(name,str(e));controls.append(name)
+    else:raise AssertionError('rule mutation accepted: '+name)
+bank_rule(ir['calls'][0]['owners'],occ_all)
+ent=[r for r in range(240,800) if widths[r-240]<9]
+if entering:
+    a_,b_=ent[0],ent[1]
+    def to_one_bank(o):   # bank class: b joins a's bank family at the next free cells, with a spectator of its own
+        fa=o[a_][0];free=len(occ_all[fa]);o[b_]=[fa,[2,free,widths[b_-240],[free+widths[b_-240]]]]
+    rule_control('two_entering_helpers_on_one_bank_family',to_one_bank,'shares its bank family')
+    rule_control('bank_spectator_is_a_residual_cell',lambda o:o[a_][1].__setitem__(3,[o[a_][1][1]]+o[a_][1][3][1:]),'first cells beyond')
+    rule_control('bank_spectator_shifted',lambda o:o[a_][1].__setitem__(3,[x+1 for x in o[a_][1][3]]),'first cells beyond')
+    rule_control('bank_spectator_record_dropped',lambda o:o[a_].__setitem__(1,o[a_][1][:3]),'without a spectator record')
+    rule_control('bank_spectator_count_wrong',lambda o:o[a_][1].__setitem__(3,o[a_][1][3]*2),'without a spectator record')
+    rule_control('second_owner_on_an_active_data_family',lambda o:o.__setitem__(241,[o[0][0],[1,0,0]]),'data class')
+    rule_control('helper_owner_on_an_active_data_family',lambda o:o.__setitem__(250,[o[3][0],[2,0]]),'data class')
+    rule_control('active_wire_on_a_bank_family',lambda o:o.__setitem__(5,[7680,[1,0,5]]),'bank class')
+    C_=len(c['centers'])
+    rule_control('owner_in_the_private_namespace_centres',lambda o:o.__setitem__(300,[W+3,[2,0]]),'private namespace')
+    rule_control('owner_in_the_private_namespace_reserved_tableau',lambda o:o.__setitem__(300,[W+C_+5,[2,0]]),'private namespace')
+    rule_control('owner_in_the_private_namespace_coefficient',lambda o:o.__setitem__(300,[W+C_+R+1,[2,0]]),'private namespace')
+    t=copy.deepcopy(ir);t['calls'][0]['owners'][0][0]+=960   # copies: an active owner of replica 0's invocation on a family of replica 1
+    try:global_check(t)
+    except (AssertionError,KeyError):controls.append('active_owner_on_the_family_of_another_replica')
+    else:raise AssertionError('global mutation accepted: active_owner_on_the_family_of_another_replica')
+assert dict(H)=={int(r):z for r,z in integration['global_profile'].items()}
+
+def load(name):
+    sp=importlib.util.spec_from_file_location('replay_hamming_'+name,Path('work/pr361/code/engines')/(name+'.py'))
+    m=importlib.util.module_from_spec(sp);sp.loader.exec_module(m);return m
+M,B=load('moment'),load('base_two_moment');b=Q(integration['complex_b']);grid=Q(1,10**18)
+assert M.moment(H,73,W,b,True)[1]<1<M.moment(H,73,W,b+grid,True)[0]
+def other(a):
+    lo,hi=B.moment(73,W,list(H.items()),a)
+    l2,h2=B.moment(73,W,[(1,32*73**2*sum(H.values()))],a)
+    return lo+l2/10**16,hi+h2/10**16
+assert other(b)[1]<1<other(b+grid)[0]
+result={'status':'PASS_INDEPENDENT_NEW_STRUCTURAL_IR_REPLAY',
+        'template_sha256':integration['global_template_sha256'],'local':checks,
+        'mutation_controls_rejected':controls,'all_eight_global_scalar_columns':True,
+        'bank_owners_and_padding_exact':True,'paid_profile':dict(sorted(H.items())),
+        'two_rational_moment_engines':True,'complex_b':str(b),
+        'next_b_grid_rejected':True,
+        'scope':'Independent finite replay; retains the written generic scalar-interface lemma and inherited all-size/analytic assumptions.'}
+(OUT/'INDEPENDENT-REPLAY.json').write_text(json.dumps(result,indent=2)+'\n')
+print('PASS independent local/global replay and two exact moments',flush=True)
diff --git a/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/data/selections/gauge80-kernel42.json b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/data/selections/gauge80-kernel42.json
new file mode 100644
index 0000000000..cf624433f0
--- /dev/null
+++ b/research/coordinated-gauge-async/vendor/complex/e8-complex-gauge-kernel/data/selections/gauge80-kernel42.json
@@ -0,0 +1,470 @@
+{
+ "gauge": {
+  "134": [
+   510
+  ],
+  "135": [
+   510
+  ],
+  "136": [
+   510
+  ],
+  "137": [
+   510
+  ],
+  "138": [
+   510
+  ],
+  "139": [
+   510
+  ],
+  "164": [
+   510
+  ],
+  "165": [
+   510
+  ],
+  "166": [
+   510
+  ],
+  "167": [
+   510
+  ],
+  "190": [
+   408,
+   102
+  ],
+  "191": [
+   408,
+   102
+  ],
+  "192": [
+   510
+  ],
+  "193": [
+   510
+  ],
+  "194": [
+   510
+  ],
+  "195": [
+   510
+  ],
+  "220": [
+   510
+  ],
+  "221": [
+   510
+  ],
+  "222": [
+   510
+  ],
+  "223": [
+   510
+  ],
+  "246": [
+   408,
+   102
+  ],
+  "247": [
+   408,
+   102
+  ],
+  "248": [
+   510
+  ],
+  "249": [
+   510
+  ],
+  "250": [
+   510
+  ],
+  "251": [
+   510
+  ],
+  "274": [
+   408,
+   102
+  ],
+  "275": [
+   408,
+   102
+  ],
+  "276": [
+   510
+  ],
+  "277": [
+   510
+  ],
+  "278": [
+   510
+  ],
+  "279": [
+   510
+  ],
+  "304": [
+   510
+  ],
+  "305": [
+   510
+  ],
+  "306": [
+   510
+  ],
+  "307": [
+   510
+  ],
+  "326": [
+   408
+  ],
+  "327": [
+   408
+  ],
+  "330": [
+   408
+  ],
+  "331": [
+   408
+  ],
+  "408": [
+   408
+  ],
+  "409": [
+   408
+  ],
+  "414": [
+   408
+  ],
+  "415": [
+   408
+  ],
+  "464": [
+   408
+  ],
+  "465": [
+   408
+  ],
+  "470": [
+   408
+  ],
+  "471": [
+   408
+  ],
+  "492": [
+   408
+  ],
+  "493": [
+   408
+  ],
+  "494": [
+   408
+  ],
+  "495": [
+   408
+  ],
+  "532": [
+   510
+  ],
+  "533": [
+   510
+  ],
+  "534": [
+   510
+  ],
+  "535": [
+   510
+  ],
+  "536": [
+   408,
+   102
+  ],
+  "537": [
+   408,
+   102
+  ],
+  "538": [
+   408,
+   102
+  ],
+  "539": [
+   408,
+   102
+  ],
+  "540": [
+   291,
+   163,
+   102,
+   24
+  ],
+  "541": [
+   291,
+   163,
+   102,
+   24
+  ],
+  "542": [
+   291,
+   163,
+   102,
+   24
+  ],
+  "543": [
+   291,
+   163,
+   102,
+   24
+  ],
+  "544": [
+   510
+  ],
+  "545": [
+   510
+  ],
+  "546": [
+   510
+  ],
+  "547": [
+   510
+  ],
+  "548": [
+   408,
+   102
+  ],
+  "549": [
+   408,
+   102
+  ],
+  "550": [
+   408,
+   102
+  ],
+  "551": [
+   408,
+   102
+  ],
+  "552": [
+   510
+  ],
+  "553": [
+   510
+  ],
+  "554": [
+   510
+  ],
+  "555": [
+   510
+  ],
+  "556": [
+   510
+  ],
+  "557": [
+   510
+  ],
+  "558": [
+   510
+  ],
+  "559": [
+   510
+  ]
+ },
+ "kernel": [
+  {
+   "E": [
+    504
+   ],
+   "donors": [
+    334,
+    335,
+    362,
+    363
+   ],
+   "pivots": [
+    390,
+    391,
+    418,
+    419
+   ]
+  },
+  {
+   "E": [
+    390
+   ],
+   "donors": [
+    324,
+    325,
+    328,
+    329,
+    380,
+    384,
+    440,
+    444
+   ],
+   "pivots": [
+    381,
+    385,
+    441,
+    445,
+    498,
+    499,
+    500,
+    501
+   ]
+  },
+  {
+   "E": [
+    6
+   ],
+   "donors": [
+    128,
+    129
+   ],
+   "pivots": [
+    156,
+    157
+   ]
+  },
+  {
+   "E": [
+    414
+   ],
+   "donors": [
+    218,
+    219,
+    302,
+    303
+   ],
+   "pivots": [
+    358,
+    359,
+    386,
+    387
+   ]
+  },
+  {
+   "E": [
+    384
+   ],
+   "donors": [
+    162,
+    163,
+    240,
+    241,
+    300,
+    352,
+    353,
+    356,
+    360,
+    361,
+    412,
+    416,
+    417,
+    438,
+    442
+   ],
+   "pivots": [
+    301,
+    357,
+    413,
+    436,
+    437,
+    439,
+    443,
+    496,
+    497,
+    502,
+    503
+   ]
+  },
+  {
+   "E": [
+    126
+   ],
+   "donors": [
+    160,
+    161
+   ],
+   "pivots": [
+    242,
+    243
+   ]
+  },
+  {
+   "E": [
+    102
+   ],
+   "donors": [
+    188,
+    189
+   ],
+   "pivots": [
+    244,
+    245
+   ]
+  },
+  {
+   "E": [
+    30
+   ],
+   "donors": [
+    132,
+    133
+   ],
+   "pivots": [
+    268,
+    269
+   ]
+  },
+  {
+   "E": [
+    96
+   ],
+   "donors": [
+    186,
+    187
+   ],
+   "pivots": [
+    216,
+    217
+   ]
+  },
+  {
+   "E": [
+    120
+   ],
+   "donors": [
+    158,
+    159
+   ],
+   "pivots": [
+    214,
+    215
+   ]
+  },
+  {
+   "E": [
+    480
+   ],
+   "donors": [
+    272,
+    273,
+    298
+   ],
+   "pivots": [
+    299
+   ]
+  },
+  {
+   "E": [
+    24
+   ],
+   "donors": [
+    130,
+    131
+   ],
+   "pivots": [
+    212,
+    213
+   ]
+  }
+ ]
+}
\ No newline at end of file
diff --git a/research/coordinated-gauge-async/vendor/complex/split-centres-global/code/complex_phases_full.py b/research/coordinated-gauge-async/vendor/complex/split-centres-global/code/complex_phases_full.py
new file mode 100644
index 0000000000..b5a5bd64b4
--- /dev/null
+++ b/research/coordinated-gauge-async/vendor/complex/split-centres-global/code/complex_phases_full.py
@@ -0,0 +1,39 @@
+# Adapted from PR425 for heterogeneous centres by icekylinx with Codex assistance.
+"""Exact phase-aware lowering of every used edge of the new structural word."""
+import gzip,hashlib,json,sys,time
+from collections import Counter
+from pathlib import Path
+sys.path.insert(0,str(Path('work/odd-stages').resolve()))
+from clifford_exact import *
+
+OUT=Path('split-centres');P=OUT/'CANONICAL-WORD.json.gz'
+c=json.load(gzip.open(P));n=c['h'];frames=c['frames'];start=time.monotonic();ops=[]
+for U in frames:
+    op=canonical_frame(n,U)
+    L=[u|(u<<n) for u in U]+[v<<n for v in nullspace(U,n)]
+    pre=[x|(z<<n) for x,z,p in inverse(op.tab)[n:]]
+    assert rank(L)==n and rank(L+pre)==n
+    assert rank([x for x,z,p in op.tab[n:]])==len(U)
+    ops.append(op)
+edges=set(map(tuple,c['edges']))|{(a,b) for f in c['center_frames'] for a,b in ((f,0),(0,f))};recipes=[];hist=Counter()
+for i,(a,b) in enumerate(sorted(edges)):
+    op=ops[a].adjoint().then(ops[b]);rec,compiled=lower_child(op)
+    assert rec['rank']==abs(len(frames[a])-len(frames[b]))
+    hist[rec['rank']]+=1;recipes.append({'old':a,'new':b,**rec})
+    if i%200==0:print('exact phase edges',i,'/',len(edges),'seconds',time.monotonic()-start,flush=True)
+adapters=[];F=fourier(n)
+for i,q in enumerate(c['ports']):
+    for role,old,f in [('source',cgate(n,q),c['start'][i]),
+                       ('target',cgate(n,q).adjoint().then(F),c['final'][c['v']+i])]:
+        rec,compiled=lower_child(old.adjoint().then(ops[f]));assert rec['rank']==0
+        adapters.append({'port':i,'role':role,'frame':f,**rec})
+raw=json.dumps({'children':recipes,'endpoint_adapters':adapters},sort_keys=True,separators=(',',':')).encode()
+(OUT/'CLIFFORD-RECIPES.json').write_bytes(raw)
+result={'status':'PASS_NEW_WORD_EXACT_LOCAL_PHASE_RECIPES',
+        'canonical_word_sha256':hashlib.sha256(P.read_bytes()).hexdigest(),
+        'frames':len(frames),'directed_edges':len(edges),'ranks':dict(hist),
+        'endpoint_adapters':len(adapters),'all_child_equalities_include_global_phase':True,
+        'recipe_sha256':hashlib.sha256(raw).hexdigest(),'seconds':time.monotonic()-start,
+        'scope':'Every local frame and center transition. Expanded scalar-interface integration and global emitted program are separate obligations.'}
+(OUT/'CLIFFORD-REVIEW.json').write_text(json.dumps(result,indent=2)+'\n')
+print('PASS exact local phases',result['seconds'],'seconds',flush=True)
diff --git a/research/coordinated-gauge-async/vendor/complex/split-centres-global/vendor/clifford_exact.py b/research/coordinated-gauge-async/vendor/complex/split-centres-global/vendor/clifford_exact.py
new file mode 100644
index 0000000000..166841a037
--- /dev/null
+++ b/research/coordinated-gauge-async/vendor/complex/split-centres-global/vendor/clifford_exact.py
@@ -0,0 +1,149 @@
+"""Small exact Gaussian-dyadic Clifford compiler, independent of gx internals.
+Paulis use i^p X^x Z^z. State vectors use integer Gaussian numerators / 2^d.
+The tableau is supplemented by a literal circuit, so global phases are kept.
+"""
+from dataclasses import dataclass
+
+def parity(x):return x.bit_count()&1
+def rank(rows):
+ p={}
+ for x in rows:
+  while x:
+   b=x.bit_length()-1
+   if b not in p:p[b]=x;break
+   x^=p[b]
+ return len(p)
+def transpose(rows,n):return tuple(sum(((rows[j]>>i)&1)<<j for j in range(n)) for i in range(n))
+def mv(rows,x):return sum(parity(r&x)<<i for i,r in enumerate(rows))
+def inv(rows,n):
+ a=[r|(1<<(n+i)) for i,r in enumerate(rows)]
+ for j in range(n):
+  k=next(i for i in range(j,n) if a[i]>>j&1);a[j],a[k]=a[k],a[j]
+  for i in range(n):
+   if i!=j and a[i]>>j&1:a[i]^=a[j]
+ assert [r&((1<<n)-1) for r in a]==[1<<i for i in range(n)]
+ return tuple(r>>n for r in a)
+def basis_extend(cols,n):
+ out=[]
+ for x in list(cols)+[1<<i for i in range(n)]:
+  if rank(out+[x])>len(out):out.append(x)
+ assert len(out)==n
+ return tuple(out)
+def nullspace(rows,n):
+ a=list(rows);p=[];k=0
+ for j in range(n):
+  t=next((i for i in range(k,len(a)) if a[i]>>j&1),None)
+  if t is None:continue
+  a[t],a[k]=a[k],a[t]
+  for i in range(len(a)):
+   if i!=k and a[i]>>j&1:a[i]^=a[k]
+  p.append(j);k+=1
+ return tuple((1<<j)|sum(((a[i]>>j)&1)<<v for i,v in enumerate(p)) for j in range(n) if j not in p)
+def pmul(a,b):
+ x,z,p=a;y,w,q=b
+ return x^y,z^w,(p+q+2*parity(z&y))%4
+def identity(n):return tuple([(1<<i,0,0) for i in range(n)]+[(0,1<<i,0) for i in range(n)])
+def apply(T,p):
+ n=len(T)//2;x,z,ph=p;out=(0,0,ph)
+ for i in range(n):
+  if x>>i&1:out=pmul(out,T[i])
+ for i in range(n):
+  if z>>i&1:out=pmul(out,T[n+i])
+ return out
+def compose(A,B):return tuple(apply(A,p) for p in B)
+def inverse(T):
+ n=len(T)//2;cols=[x|(z<<n) for x,z,p in T];rows=transpose(cols,2*n);cols_inv=transpose(inv(rows,2*n),2*n);out=[]
+ for j,w in enumerate(cols_inv):
+  x=w&((1<<n)-1);z=w>>n;u,v,p=apply(T,(x,z,0));assert (u,v)==identity(n)[j][:2];out.append((x,z,(-p)%4))
+ return tuple(out)
+def qvalue(Q,alpha,x,n):
+ return (sum(alpha[i] for i in range(n) if x>>i&1)+2*sum(parity((Q[i]>>(i+1))&(x>>(i+1))) for i in range(n) if x>>i&1))%4
+def mul_i(a,p):
+ x,y=a;p%=4
+ return [(x,y),(-y,x),(-x,-y),(y,-x)][p]
+@dataclass
+class Op:
+ n:int
+ tab:tuple
+ circuit:list
+ def then(self,b):
+  assert self.n==b.n
+  return Op(self.n,compose(b.tab,self.tab),self.circuit+b.circuit)
+ def adjoint(self):
+  circ=[]
+  for e in reversed(self.circuit):
+   if e[0]=='P':circ.append(('P',inv(e[1],self.n)))
+   elif e[0]=='D':circ.append(('D',e[1],tuple((-x)%4 for x in e[2])))
+   elif e[0]=='C':circ.append(('C',e[1],-e[2]))
+   elif e[0]=='phase':circ.append(('phase',(-e[1])%4))
+   else:circ.append(e)
+  return Op(self.n,inverse(self.tab),circ)
+ def zero_state(self,basis=0):
+  N=1<<self.n;vec=[(0,0)]*N;vec[basis]=(1,0);den=0
+  for e in self.circuit:
+   if e[0]=='P':
+    v=[(0,0)]*N
+    for x,a in enumerate(vec):v[mv(e[1],x)]=a
+    vec=v
+   elif e[0]=='X':vec=[vec[x^e[1]] for x in range(N)]
+   elif e[0]=='D':vec=[mul_i(a,qvalue(e[1],e[2],x,self.n)) for x,a in enumerate(vec)]
+   elif e[0]=='phase':vec=[mul_i(a,e[1]) for a in vec]
+   elif e[0]=='C':
+    q,sgn=e[1:];v=[]
+    for x,(a,b) in enumerate(vec):
+     c,d=vec[x^q]
+     # C=(1+i)I/2+(1-i)X/2; inverse conjugates both coefficients.
+     v.append((a-sgn*b+c+sgn*d,b+sgn*a+d-sgn*c))
+    vec=v;den+=1
+   else:raise AssertionError(e)
+   while den and all((a&1)==(b&1)==0 for a,b in vec):vec=[(a//2,b//2) for a,b in vec];den-=1
+  return vec,den
+def ident(n):return Op(n,identity(n),[])
+def perm(n,G):
+ cols=transpose(G,n);gi=inv(G,n)
+ return Op(n,tuple([(a,0,0) for a in cols]+[(0,a,0) for a in gi]),[('P',tuple(G))])
+def diag(n,Q,alpha=None):
+ Q=tuple(Q);assert transpose(Q,n)==Q
+ alpha=tuple(alpha) if alpha is not None else tuple((Q[i]>>i)&1 for i in range(n))
+ assert all(alpha[i]%2==(Q[i]>>i)&1 for i in range(n))
+ return Op(n,tuple([(1<<i,Q[i],alpha[i]) for i in range(n)]+[(0,1<<i,0) for i in range(n)]),[('D',Q,alpha)])
+def cgate(n,q,sgn=1):
+ return Op(n,tuple([(1<<i,0,0) for i in range(n)]+[(q if q>>i&1 else 0,1<<i,(3*sgn)%4 if q>>i&1 else 0) for i in range(n)]),[('C',q,sgn)])
+def xgate(n,b):return Op(n,tuple([(1<<i,0,0) for i in range(n)]+[(0,1<<i,2*((b>>i)&1)) for i in range(n)]),[('X',b)])
+def fourier(n,axes=None):
+ out=ident(n)
+ for i in range(n) if axes is None else axes:out=out.then(cgate(n,1<<i))
+ return out
+def hadamard(n,axes):
+ axes=list(axes);Q=tuple(1<<i if i in axes else 0 for i in range(n));S=diag(n,Q)
+ return S.then(fourier(n,axes)).then(S)
+def canonical_frame(n,U):
+ r=len(U)
+ if r==0:return ident(n)
+ if r==n:return fourier(n)
+ G=transpose(basis_extend(U,n),n);gi=inv(G,n);Q=tuple((1<<i)^sum(parity(gi_i&gi_j)<<j for j,gi_j in enumerate(transpose(gi,n))) for i,gi_i in enumerate(transpose(gi,n)))
+ K=perm(n,G).then(diag(n,Q))
+ return K.adjoint().then(fourier(n,range(r))).then(K)
+def monomial(op):
+ n=op.n;T=op.tab;assert all(x==0 and p%2==0 for x,z,p in T[n:]);G=transpose([x for x,z,p in T[:n]],n);b=mv(G,sum((p//2)<<i for i,(x,z,p) in enumerate(T[n:])))
+ gt=transpose(G,n);cols=[mv(gt,z) for x,z,p in T[:n]];Q=transpose(cols,n);alpha=[(p-2*parity(b&z))%4 for x,z,p in T[:n]]
+ out=diag(n,Q,alpha).then(perm(n,G)).then(xgate(n,b));assert out.tab==T
+ vec,den=op.zero_state();assert den==0 and all(a==(0,0) for i,a in enumerate(vec) if i!=b);phase=[(1,0),(0,1),(-1,0),(0,-1)].index(vec[b]);out=out.then(Op(n,identity(n),[('phase',phase)]));assert out.zero_state()==op.zero_state()
+ return out,dict(matrix_rows=G,offset=b,quadratic_rows=Q,linear_Z4=alpha,phase=phase)
+def lower_child(op):
+ n=op.n;Bcols=[x for x,z,p in op.tab[n:]];r=rank(Bcols)
+ if not r:
+  mon,rec=monomial(op);return dict(rank=0,left=rec,right=None),mon
+ G=transpose(basis_extend(Bcols,n),n);A=inv(G,n);PA=perm(n,A);part=op.then(PA);Bcols=[x for x,z,p in part.tab[n:]];Dcols=[z for x,z,p in part.tab[n:]]
+ picks=[];selected=[]
+ for j,x in enumerate(Bcols):
+  if rank(selected+[x])>len(selected):picks.append(j);selected.append(x)
+ assert len(picks)==r and all(x>>r==0 for x in Bcols)
+ Br=transpose(selected,r);Bi=inv(Br,r);Dr=transpose([Dcols[j]&((1<<r)-1) for j in picks],r)
+ Qsmall=tuple(sum(parity(row&col)<<j for j,col in enumerate(transpose(Bi,r))) for row in Dr);Q=Qsmall+(0,)*(n-r);DQ=diag(n,Q)
+ V=part.then(DQ).then(hadamard(n,range(r)));assert all(x==0 for x,z,p in V.tab[n:]);Vmon,Vrec=monomial(V)
+ S=diag(n,tuple(1<<i if i<r else 0 for i in range(n)))
+ left=S.adjoint().then(DQ.adjoint()).then(PA.adjoint());right=Vmon.then(S.adjoint()).then(xgate(n,(1<<r)-1))
+ lmon,lrec=monomial(left);rmon,rrec=monomial(right);compiled=rmon.then(fourier(n,range(r))).then(lmon)
+ assert compiled.tab==op.tab and compiled.zero_state()==op.zero_state()
+ return dict(rank=r,left=lrec,right=rrec),compiled
diff --git a/research/coordinated-gauge-async/verify.py b/research/coordinated-gauge-async/verify.py
new file mode 100644
index 0000000000..76d1284267
--- /dev/null
+++ b/research/coordinated-gauge-async/verify.py
@@ -0,0 +1,52 @@
+"""Offline, stdlib-only reconstruction of the asynchronous complex derivative.
+The unchanged PR493 bit operator is a cited dependency, not freshly verified here.
+"""
+from pathlib import Path
+import argparse,hashlib,json,os,shutil,subprocess,sys,time,gzip
+if not __debug__:raise RuntimeError('Assertions must remain enabled; do not use -O or PYTHONOPTIMIZE.')
+sys.dont_write_bytecode=True
+if hasattr(sys,'set_int_max_str_digits'):sys.set_int_max_str_digits(0)
+P=Path(__file__).resolve().parent
+a=argparse.ArgumentParser();a.add_argument('--work',type=Path,required=True);args=a.parse_args()
+W=args.work.resolve();assert P not in W.parents and W!=P,'Output must be outside this package.';assert not W.exists(),'Choose a new output directory; existing outputs are preserved.'
+mf=P/'MANIFEST.json'
+if mf.exists():
+ for n,h in json.loads(mf.read_text()).items():assert hashlib.sha256((P/n).read_bytes()).hexdigest()==h,n
+W.mkdir(parents=True);V=W/'base';V.mkdir();shutil.copytree(P/'inputs/work',V/'work')
+shutil.copy2(P/'inputs/BASE-WORD.json.gz',V/'BASE-WORD.json.gz')
+bit=json.loads((P/'inputs/PR493-EXPECTED.json').read_text())['eleven']
+(V/'BIT-PRICE.json').write_text(json.dumps({'lower':bit['bit_lower'],'scope':'PR493 cited published supplier'}))
+C=P/'vendor/complex';G=C/'e8-complex-gauge-kernel';A=W/'audit';A.mkdir();H=W/'hosted'
+env=dict(os.environ,PYTHONUTF8='1',PYTHONDONTWRITEBYTECODE='1',PYTHONINTMAXSTRDIGITS='0',TEMP=str(W),TMP=str(W))
+env.pop('PYTHONOPTIMIZE',None)
+steps=[]
+def run(name,script,*argv):
+ start=time.monotonic();print('RUN',name,flush=True)
+ with (W/(name+'.log')).open('w',encoding='utf-8') as log:
+  result=subprocess.run([sys.executable,'-B',str(P/'code/run_portable.py'),str(script),*map(str,argv)],cwd=V,env=env,stdout=log,stderr=subprocess.STDOUT)
+ steps.append({'stage':name,'exit':result.returncode,'seconds':time.monotonic()-start})
+ (W/'STEPS.json').write_text(json.dumps(steps,indent=2)+'\n')
+ if result.returncode:
+  print((W/(name+'.log')).read_text()[-5000:]);raise SystemExit(result.returncode)
+ print('PASS',name,flush=True)
+run('retiming',P/'code/build_retimed.py',G,V/'BASE-WORD.json.gz',P/'inputs/PR495-FULL-WORD.json.gz',V/'retimed')
+run('construct',P/'code/build_rank2_collapse.py',G/'code',V/'retimed/RETIMED-BASE.json.gz',G/'data/selections/gauge80-kernel42.json',V/'split-centres')
+run('scalar',G/'code/audit_gk.py')
+run('emit-local',G/'code/emit_gk.py')
+run('phases',C/'split-centres-global/code/complex_phases_full.py')
+run('adapters',G/'code/adapters_gk.py',V/'split-centres/CANONICAL-WORD.json.gz',V/'split-centres/RESIDUAL-ADAPTERS.json',V/'LEMMA-A.json')
+run('parent-integration',G/'code/integrate_gk.py')
+run('parent-replay',G/'code/replay_gk.py')
+run('emit-async',P/'code/emit_async.py',G,V,P/'inputs/ASYNC-OWNERS.json.gz',P/'inputs/ASYNC-6.json',H)
+run('independent-async',P/'code/audit_async.py',V,H,A/'ASYNC-AUDIT.json')
+run('invoice',P/'code/audit_async_invoice.py',A,V,H)
+run('two-moments-assembly',P/'code/assemble.py',A,V,H,P/'inputs/PR493-EXPECTED.json')
+x=json.loads((A/'EXACT-ASSEMBLY.json').read_text());audit=json.loads((A/'ASYNC-AUDIT.json').read_text())
+assert all(z['exit']==0 for z in steps)
+result={'status':'PASS_CONDITIONAL_ASYNCHRONOUS_COMPLEX_DERIVATIVE','kappa':x['kappa'],'kappa_decimal':x['kappa_decimal'],'complex_lower':x['complex_lower'],'bit_lower':x['bit_lower'],'template_sha256':x['complex_template_sha256'],'strict_constraints':len(x['strict_outer_constraints']),'scope':'New complex program independently replayed. The unchanged PR493 bit operator and inherited all-size compiler, common-label, Lemma A realization, restored-row, precision/tape and analytic-transfer contracts remain dependencies. No unconditional multiplication bound or fresh Lean certification is claimed.','stages':steps,'package_manifest_sha256':hashlib.sha256(mf.read_bytes()).hexdigest() if mf.exists() else None}
+expected=P/'EXPECTED.json'
+if expected.exists():
+ e=json.loads(expected.read_text())
+ for key in ['kappa','complex_lower','bit_lower','template_sha256']:assert result[key]==e[key],key
+(W/'RESULT.json').write_text(json.dumps(result,indent=2)+'\n')
+print(result['status'],'kappa',result['kappa_decimal'],flush=True)
diff --git a/research/coordinated-gauge-async/verify.sh b/research/coordinated-gauge-async/verify.sh
new file mode 100644
index 0000000000..275570d71e
--- /dev/null
+++ b/research/coordinated-gauge-async/verify.sh
@@ -0,0 +1,10 @@
+#!/usr/bin/env bash
+set -euo pipefail
+here="$(cd -- "$(dirname -- "$0")" && pwd)"
+if [ "$#" -eq 0 ]; then
+  out="$(mktemp -d)/run"
+  echo "Output: $out"
+  exec python3 -B "$here/verify.py" --work "$out"
+else
+  exec python3 -B "$here/verify.py" "$@"
+fi