diff --git a/research/r437-copy-echo-budget/.gitattributes b/research/r437-copy-echo-budget/.gitattributes
new file mode 100644
index 0000000000..f0b04a4da0
--- /dev/null
+++ b/research/r437-copy-echo-budget/.gitattributes
@@ -0,0 +1,2 @@
+* -text
+inputs/* whitespace=cr-at-eol
diff --git a/research/r437-copy-echo-budget/PROOF.md b/research/r437-copy-echo-budget/PROOF.md
new file mode 100644
index 0000000000..e58dc87be2
--- /dev/null
+++ b/research/r437-copy-echo-budget/PROOF.md
@@ -0,0 +1,222 @@
+# One-child dirty echo boundary and the fixed two-child budget
+
+Chris R / Clear1nsight, with substantial OpenAI Codex assistance. Apache-2.0.
+This is a scoped obstruction and cost decision, not a new multiplication bound.
+
+## 1. Changed proposal and optimistic gate
+
+PR3982 supplies an all-class rank-seven COPY conjugator with 8036 ordered ADD
+slots per column, 196 ordinary interchanges and one reversed 7f child. Its
+ordinary tariff remains a positive exponent-gap term under the available B1/B2
+upper bounds. The new proposal tested here removes ordinary interchanges by
+using coupled dirty payload roles, coordinatewise reversible/Boolean echo,
+and the existing reversed child as the only backward address primitive.
+This is a changed consumer/operation contract, not a smaller permutation route.
+
+The completed old bill is
+
+ T_R(v,7f,w) + 196 b(v,f) + C_q v(8036f+196+3) + Gamma
+
+plus 63 actual dirty XOR readers and outer owner/producer/return costs when
+not already included. All of these are charges; none is a negative credit.
+For a candidate with child widths k_1 f,...,k_t f and no ordinary calls, use
+
+ sum_i T_R(v,k_i f,w) + C_scan(v; clone, coordinatewise gates, ordered ADDs,
+ readers, transfers, releases, disposal, head return) + Gamma_echo.       (1)
+
+Its reserve must sum the whole-run visited maxima on fixed reusable source,
+owned-output, helper, child, coefficient and parking inventories. Erasure does
+not refund previously visited cells, and disjoint occurrence inventories cannot
+be replaced by a maximum. These implementation-dependent scan and reserve terms
+are unknown here. Setting all of them to zero is the optimistic gate below,
+not a physical upper bound.
+
+The source profile has m=64, W=28416, paid rank mass1808352, hence deficit
+64W-1808352=10272. It has2592 COPY executions (nine captures in each of nine
+stages and32 replicas), already using one private transformed stream through
+63 readers. A fixed schedule replacing every one with two rank-seven children
+adds2592*7=18144 mass, giving1826496 and excess7872. Its optimistic moment is
+
+ M_two(c)= (1/W) sum_r n_r(r/64)^(1-c)
+                  + (2592/W)(7/64)^(1-c).
+
+For every c>=0, each term (r/64)^(1-c)>=r/64, so
+
+ M_two(c) >= M_two(0)=1826496/1818624=1+41/9472>1.       (2)
+
+Thus this particular fixed two-rank-seven replacement fails even before the
+remaining positive costs in(1). This does NOT follow for an adaptive schedule
+or for two children of different widths. If only a fraction theta pays one
+extra rank-seven child, the zero-exponent necessary condition is
+ theta<10272/(2592*7)=107/189. At positive c the necessary bound is
+ theta < (1-M_old(c))/((2592/W)(7/64)^(1-c)), when the numerator is positive.
+The obstruction below supplies a worst-case class, not its density. It does
+not license inserting theta=1 into an adaptive original good/fallback scheme.
+
+## 2. Actual address convention and the relaxed operation class
+
+Fix an odd prime q, w>=1, R=Z/q^w and A=q^w. Take the allowed subsequence
+n=64f with any f>=1 (delta=0); a uniform all-size claim must cover it. In one
+fixed physical prefix/fresh-row/spectator fiber, order the two banks as
+
+ head atoms 0,...,n-1; reflected tail atoms n,...,2n-1.
+
+Atom a=if+j in the head is paired with physical tail atom 2n-1-a. The standard
+reversed child W_l swaps a with2n-1-a for0<=a<l and fixes every other address
+coordinate. Its complete source-preserving payload operation is the pullback
+ F'(x)=F(W_l x), exposing one transformed output and no additional payload
+channels; borrowed sources and helpers are restored. A rank-k child in the incumbent notation has l=kf.
+
+An ordered ADD has its source physically before its target. Its address matrix
+is lower unitriangular: x_i <- x_i+a x_j for j<i, with a determined by fixed
+metadata. Products and inverses remain in the group U of lower unitriangular
+matrices over R. We generously allow ALL such matrices on the two banks,
+including cross-bank ordered additions, not just PR3982's same-bank alphabet.
+Optional invertible diagonal factors would not change the argument.
+
+The macro circuit is finite and its event schedule and address permutations
+are independent of payload values. It may have arbitrarily many payload roles,
+clones, dirty helpers and fresh constants. At any full physical address it may
+apply arbitrary Boolean functions of roles at that SAME address, including
+address-dependent masks, not merely XOR or linear mixing. It may apply any
+number of U pullbacks on a role, and at most one completed W_l pullback event,
+where l<n. Owned release/reset and required restoration are allowed. Every
+primitive preserves the prefix/fresh-row/spectator fiber.
+
+This is an address-dependency class, not a claim every Boolean gate has an
+admitted physical tariff. Allowing them for free only strengthens a negative
+result. A global payload-dependent branch that reads one address to control
+another is excluded: it adds cross-address dependency edges. A differently
+routed physical scan, table lookup, gather/scatter or arbitrary-address copy
+is also not silently counted as a coordinatewise operation. Such primitives
+change the hypothesis and must appear in(1). Fixed metadata may depend on q,w,
+header and column; it must not convey source payload information.
+
+Linearity of the body-address alphabet is essential. An affine translation
+z -> z+(y-x) reaches the separating y from x, so admitting arbitrary address
+translations destroys this particular orbit obstruction even with no child.
+An addition from a fixed nonzero header/address digit into a body digit may
+also act affinely on the fiber and is outside U here. No claim excludes every
+legal physical scan or proves that such an enlarged alphabet is expensive;
+a complete translated/affine mechanism would need its own full cost gate.
+
+## 3. The dependency-orbit theorem
+
+Define lead(z) as the first nonzero coordinate of z. If L is lower
+unitriangular, lead(Lz)=lead(z), with the leading value unchanged. This holds
+over R including nonunits and high carries: before the leading coordinate all
+summands are zero, and at it the diagonal entry is1.
+
+Let x=e_n, the first physical tail coordinate. For every V in U, Vx has zeros
+before n and value1 at n. If l<n, W_l fixes coordinate n. It moves any nonzero
+among the last l coordinates to the first l positions. Therefore
+
+ lead(W_l Vx) is either <l or n.
+
+Applying another U preserves that lead. Consequently the set
+
+ O_1(x) = Ux union {L W_l Vx : L,V in U}                  (3)
+
+contains no vector with lead n-1, because l<=n-1. This is an exact all-ring,
+all-f statement; it does not enumerate the huge address cube.
+
+Now unroll the macro circuit in time. A dependency path from a final output
+bit backwards to an initial payload bit stays at its address through a
+coordinatewise gate and changes address through the pullback's matrix at an
+address operation. A path can cross the ONE chronological child execution at
+most once, even if the role graph reuses roles or has apparent cycles. Collapse
+all its ordered-ADD matrices on either side to U. Every input address that can
+influence output at x is in(3). Fanout and Boolean cancellations cannot create
+an input dependency absent from this union. This proof covers arbitrary dirty
+helper contents: compare two runs with identical initial helpers and metadata,
+changing only one source bit outside(3). Their final output at x is identical.
+No restoration assumption is needed for the lower bound, so requiring all
+helpers and the borrowed source to be restored cannot invalidate it.
+
+## 4. A legal all-class rank-seven COPY outside the orbit
+
+Let Q=diag(0,...,0,1,...,1) on64 coordinates, with ones precisely at57,...,63.
+Let H=diag(h_i) be arbitrary ancestor units (possibly depending on column and
+fixed header). The required two-bank map in paired head/tail coordinates is
+
+ D_H(Q)=diag(H,I) [[I-Q,Q],[Q,I-Q]] diag(H^-1,I).
+
+Apply the same Q on each of f columns, and reflect the physical tail as above.
+At x=e_n only the tail atom paired to head n-1 is nonzero. Thus
+
+ D_H(Q)x = h_(63,f-1) e_(n-1) = y.
+
+The unit h makes lead(y)=n-1, so y is absent from(3). The two arbitrary Boolean
+sources zero and delta_y require different COPY-output bits at x. The circuit
+class in§2 produces the same bit, a contradiction. Thus at least two proper-
+width reversed-child events, or another address-mixing operation, are needed
+for a uniform all-class COPY in this class. A single wider child of width42f
+still fails; width n leaves this obstruction but is not a smaller recursive
+child at delta=0. No sufficiency claim is made for two events or for width n.
+
+This Q is NOT asserted to be one of the nine literal capture matrices P.
+Every literal rank-seven capture is a split projector over the admitted ring;
+a basis of im(P) followed by a basis of ker(P) gives P=T E T^-1. A permutation
+S sends the first seven basis axes to57,...,63. Then g=S T^-1 is invertible
+and g P g^-1=Q. The all-class contract ranges over precisely such GL64 low
+classes and lifts. The verifier reconstructs the actual capture frames from
+the pinned R437 word and builds these conjugacies in finite local-ring cases.
+The proof uses the quantified split-projector premise, not small-q admission.
+
+Choose any physical prefix, unused fresh rows and spectator address, hold it
+fixed, and put delta_y only there. The contradiction is a full-volume source
+counterexample with every other bit zero. Unequal physical prefixes are not
+merged. Adding untouched spectators, or the63 downstream dirty readers, does
+not create the missing source-to-private-output dependency.
+
+## 5. Why fanout and pooling alone do not pay the old term
+
+The inherited private result is already computed once and retained through all
+63 readers. Dividing its invoice by63 would only amortize a constant. If a
+new contract really supplies K identical requests from one unchanged source,
+source sharing can replace K evaluations by one, but it still costs
+ T_R+196 b+K X+Gamma+cleanup, with the shared result preserved until last use.
+For the available bound b(v,f)<=C v(fw)^p+E, write n~e^alpha,w~e^beta
+with f proportional to n. The ratio of its leading term to K v n scales as
+e^delta/K, delta=(alpha+beta)p-alpha. At a proposed R437 target selector
+alpha=1-c_R,beta=c_R and p=1-a_B, this is delta=c_R-a_B>0 for both
+pinned B1/B2 rates. This is a conditional target-scaling calculation; it does
+not establish a retuned R437 selector or its admission. The retained ordinary
+selector pattern is n=floor((e-k)/w), with k=o(e), as in PR3562 head
+abb93be8b9ceca07a2b17f3a97db7bb6b074271d, PROOF lines110–165 (there c=1/577). This explicitly distinguishes original ordinary width e
+from reduced width n. A fixed K
+(including63) cannot remove this upper-tariff gap. To infer absorption from
+that tariff would require growing K at least on the e^delta scale, while
+also paying readers, storage, source compatibility and preparation. This is
+not a lower bound on actual adapter runtime.
+
+For different independent source payloads, packing K fibers enlarges volume
+from v to Kv; the same volume-linear tariff gives196 C Kv(fw)^p, so dividing
+by K gives no reduction at all. Identical field labels are not identical
+physical sources. Different Q maps likewise need not yield the same private
+stream: delta sources separate distinct address permutations. No new equality
+of the actual R437 captures is asserted. PR3995's pooled owner IR still needs
+its replica-indexed physical contract; its finite profile does not establish
+source equality or free transport. PR3989 similarly demonstrates why a finite
+coupled identity's lifted transfer cannot be priced without its full consumer.
+
+## 6. Decision, evidence, and limits
+
+Stop this fixed one-child dirty-echo route: it fails the full-cube dependency
+contract even after relaxing pointwise operations. Stop the fixed two-rank-
+seven-per-COPY repair: its optimistic moment already exceeds1. These are two
+separate conclusions, not a universal lower bound of fourteen recursive ranks.
+A 7f+1f or other mixed-width construction, an adaptive schedule with an actual
+class-frequency proof, a legal backward address operation with a cheaper full
+tariff, or a changed producer/consumer can reopen the question. Each needs an
+exact complete bill and the appropriate whole-profile moment before expansion.
+
+Finite checks authenticate source bytes, all nine capture classes and high-lift
+conjugacies, reachable address orbits in small cubes, a separating arbitrary
+payload bit and arbitrary fixed dirty helpers, and exact profile/tariff
+arithmetic. Controls break the premise deliberately (a full-width child or
+an extra child permits the formerly absent dependency), detect wrong target
+orientation/weight, and reject invalid source/invoice inputs. The mathematical
+proof, finite checks and same-session AI cross-review are distinct evidence.
+No encoded supplier, formal-kernel proof, full native execution, selected cold
+replay, improved kappa or frontier change is claimed.
diff --git a/research/r437-copy-echo-budget/README.md b/research/r437-copy-echo-budget/README.md
new file mode 100644
index 0000000000..e28fd9202d
--- /dev/null
+++ b/research/r437-copy-echo-budget/README.md
@@ -0,0 +1,51 @@
+# R437 COPY: linear-address dirty echo boundary and fixed replacement budget
+
+This package resolves a changed COPY proposal following [PR3982](https://github.com/CrocSwap/integer-mult-bounds/pull/3982): remove its196 ordinary interchanges by coupling arbitrary dirty payload buffers, using linear ordered body-address ADDs and the existing reversed child. **A single proper-width reversed child cannot implement the all-class COPY contract in this linear-address, enlarged pointwise-gate model. A fixed two-rank-seven replacement also fails the exact paid moment before cleanup costs.** No new kappa is claimed.
+
+The proof combines local-ring projector conjugacy with time-unrolled circuit dependency and triangular-group address orbits. It permits arbitrary Boolean gates at the same physical address, not only linear XOR echoes. Any dependency path crosses the one chronological child event at most once; its address word lies in `U` or `U W_l U`. At output address `e_n`, those words cannot inspect a source bit with leading index `n-1` when `l<n`. The valid shifted rank-seven projector on indices57..63, with arbitrary diagonal ancestor units, requires exactly that bit. This holds over every admitted local ring and every positive tensor multiplicity; it is not a tiny-cube extrapolation.
+
+The source-bound profile has2592 COPY executions and deficit10272. Charging two rank-seven children on every COPY adds18144 rank mass. The optimistic moment is already **9513/9472 > 1 at exponent zero**, and increases for nonnegative saving exponent. All scans, cloning, readers, cleanup, preparation, stock and lifetime reserve costs are additional. This closes that fixed replacement proposal, not every possible way to escape the one-child obstruction.
+
+The distinction matters:
+
+- The dependency theorem requires a fixed payload-independent event/address schedule, coordinatewise payload gates, globally forward ADD pullbacks and at most one standard outer child. New address-dependent routing or global payload-controlled communication changes the alphabet. Affine address translations, including addition from a fixed nonzero header digit into a body coordinate, are also excluded: they destroy this orbit invariant and require a separate complete cost analysis.
+- The two-child cost result concerns **two rank-seven children per every COPY**. It is not a theorem that two arbitrary children have total rank at least14. Mixed widths, adaptive schedules and the original good/fallback construction remain outside that conclusion.
+- For an adaptive extra rank-seven child, the zero-exponent mass condition is `theta < 107/189`; the proof gives the stricter positive-exponent formula. The orbit witness does not establish the fraction of actual classes requiring extra work.
+- The existing63 readers already share one private residual. Further constant fanout changes constants; packing independent sources increases operand volume. Neither alone absorbs the positive upper-tariff exponent gap. An actually growing same-source cohort needs its own equality, consumer and lifetime proof.
+
+See [PROOF.md](PROOF.md) for the exact macro contract, full relative obligation ledger, all-f argument and optimistic recurrence. The one-event obstruction persists with arbitrary fixed dirty helpers even if restoration is waived, so mandatory restoration cannot evade it. Completed child semantics expose one transformed output and no extra payload channels. Prefixes, fresh rows and spectators remain separate physical fibers.
+
+## Evidence and reproduction
+
+The primary standard-library checker authenticates retained input bytes and reconstructs the nine actual capture frames. It explicitly builds36 GL64 conjugacies over Z/3, Z/9, Z/25 and Z/289 from those captures to the shifted projector. Small q is algebraic evidence only; it does not establish the large-q supplier admission. Four complete small-cube orbit closures test the obstruction, including nonunit high-lift coordinates; a second child and a full-width child provide premise-breaking controls. A complete81-address nonlinear dirty-helper example checks the output/source orientation and nontrivial ancestor unit. Exact rational arithmetic checks the paid profile, COPY/disposal census and B1/B2 tariff gaps.
+
+```sh
+python3 -I -B research/r437-copy-echo-budget/verify.py
+python3 -I -O -B research/r437-copy-echo-budget/verify.py
+python3 -I -B research/r437-copy-echo-budget/test_package.py
+python3 -B scripts/research_package.py --package research/r437-copy-echo-budget
+```
+
+The export test uses actual subprocesses on a Git-free read-only package and checks altered/missing/linked source and corrupted invoice failures. Runtime needs no network, Git, predecessor import, matrix library, GL enumeration, Lean or Docker. Finite checks, the written all-parameter proof and same-session AI cross-review are distinct evidence; no independent human or formal-kernel acceptance is asserted.
+
+## Provenance and research decision
+
+Canonical main20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision6. Explicit donor continuation refines `independent-moment`; the complete original card and changed premise are in [TASK.json](TASK.json). The [team claim](https://github.com/CrocSwap/integer-mult-bounds/pull/3967#issuecomment-6263066320) records the parallel round. The overlap helper had no usable remote PR/Discussion cache; live queries and the coordinating agent's discussion/PR review supply bounded coverage, not a novelty certificate.
+
+Exact snapshots are self-contained under `inputs/` and enumerated in [SOURCES.json](SOURCES.json): R437 word/license/notice via3950head36470149, current OQ4 EXPECTED at a24f9ff,3967 catalogue at32af0606, and3982 completed proof atf33a1708. The count2592 is read from the retained full catalogue receipt and agrees with9 captures ×9 stages ×32 replicas. This package does not rerun the7,814,016-event compiler. It independently checks the arithmetic and the new obstruction, preserving that census dependency explicitly.
+
+[PR3995](https://github.com/CrocSwap/integer-mult-bounds/pull/3995), headae29394a128ba2016919446a7377c7d75ff9c18e, supplies a replica-indexed pooled IR boundary, not shared source equality or a completed physical consumer. [PR3989](https://github.com/CrocSwap/integer-mult-bounds/pull/3989), head7b6dda7ab6ed10d608ecba357ac2f6924166d38b, gives a complementary transfer-lifting warning in the complex supplier. They are related-contract context, not executed dependencies here.
+
+Decision: stop the one-child fixed forward/pointwise echo and the fixed two-rank-seven repair. Do not expand either into an encoded implementation. A different operation alphabet or a genuinely cheaper mixed/adaptive complete profile can reopen the task.
+
+## Open next questions
+
+1. Can two proper-width children with different widths implement all-class COPY at a favorable full profile? Start with the complete two-bank map and one7f plus onef child, arbitrary dirty helpers and full restoration. No construction or fourteen-rank lower bound is supplied here.
+2. Can an adaptive child count be bounded over the actual GL64 classes and lifts? Derive the actual class frequency and complete producer cost, then test the positive-exponent moment. One bad class does not establish that density.
+3. Can a new legal backward address primitive or growing same-source cohort meet the missing tariff? Bind its actual full-volume source, coordinate mapping, consumer, restored helpers, head returns and visited backing. Constant fanout or a finite lifted identity alone is insufficient.
+
+## Credit and license
+
+Credit ai-nanachi's R437/#3647 work; Ali Romman/#1043 and Sagar Thakrar/#3700/OQ4; eumemic, DreamingOfClouds, Crow, DaysSky and chafreaky's retained local/hosting lineage. Credit zxv/#3690, David Arias, Jose Manuel/jmbarrancoml and the physical caller lineage; the donor's #3936/#3950/#3967/#3982 source-bound audits and construction. Rohan Arun/#3959 and its #3995 IR follow-up provide pooled-context work; Gonzalo Mier/#3975 and the credited #3989 line motivate the separate consumer-lifting caution. The full predecessor notice and license are retained byte-for-byte.
+
+New proof/checker by Chris R / Clear1nsight with substantial OpenAI Codex assistance. The coordinating agent supplied the exact profile/COPY-count pointers, and another agent reviewed the dependency theorem's orientation, all-f extension and scope. These are same-session AI checks. Apache-2.0. No merge, closure or selected-frontier change.
diff --git a/research/r437-copy-echo-budget/RESULT.json b/research/r437-copy-echo-budget/RESULT.json
new file mode 100644
index 0000000000..9ca694b1e7
--- /dev/null
+++ b/research/r437-copy-echo-budget/RESULT.json
@@ -0,0 +1,116 @@
+{
+  "budget": {
+    "W": 28416,
+    "adaptive_fraction_zero_necessary": "107/189",
+    "copies": 2592,
+    "deficit": 10272,
+    "excess": 7872,
+    "extra_mass": 18144,
+    "moment_zero": "9513/9472",
+    "new_mass": 1826496,
+    "old_mass": 1808352,
+    "profile": {
+      "1": 380160,
+      "2": 400864,
+      "26": 32,
+      "3": 22176,
+      "4": 26592,
+      "42": 384,
+      "5": 29376,
+      "6": 17376,
+      "7": 26496
+    },
+    "readers_each": 63,
+    "scope": "fixed two rank7 children per COPY only; not an adaptive or mixed-width impossibility",
+    "tariff_gaps": [
+      "39536042038446578551/1000000000000000000000000",
+      "36650955038446578551/1000000000000000000000000"
+    ]
+  },
+  "new_kappa": false,
+  "orbits": [
+    {
+      "k": 1,
+      "m": 2,
+      "missing": [
+        0,
+        1,
+        0,
+        0
+      ],
+      "modulus": 3,
+      "one": 57,
+      "pre": 3,
+      "two": 80
+    },
+    {
+      "k": 1,
+      "m": 2,
+      "missing": [
+        0,
+        1,
+        0,
+        0
+      ],
+      "modulus": 9,
+      "one": 4545,
+      "pre": 9,
+      "two": 6480
+    },
+    {
+      "k": 1,
+      "m": 3,
+      "missing": [
+        0,
+        0,
+        1,
+        0,
+        0,
+        0
+      ],
+      "modulus": 3,
+      "one": 495,
+      "pre": 9,
+      "two": 728
+    },
+    {
+      "k": 2,
+      "m": 3,
+      "missing": [
+        0,
+        0,
+        1,
+        0,
+        0,
+        0
+      ],
+      "modulus": 3,
+      "one": 657,
+      "pre": 9,
+      "two": 728
+    }
+  ],
+  "payload": {
+    "addresses": 81,
+    "dirty_helpers": 3,
+    "observed_difference": 0,
+    "output_address": [
+      0,
+      0,
+      1,
+      0
+    ],
+    "required_difference": 1,
+    "required_input": [
+      0,
+      2,
+      0,
+      0
+    ]
+  },
+  "source": {
+    "captures": 9,
+    "conjugacy_digest": "0a54fdd84a9b7f6b539c15bb42622931f84a3e9baa473ec73fb82bb13063e7dc",
+    "local_ring_cases": 36
+  }
+}
diff --git a/research/r437-copy-echo-budget/SOURCES.json b/research/r437-copy-echo-budget/SOURCES.json
new file mode 100644
index 0000000000..e23a813a2b
--- /dev/null
+++ b/research/r437-copy-echo-budget/SOURCES.json
@@ -0,0 +1,62 @@
+{
+  "schema_version": 1,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    }
+  ],
+  "retained": [
+    {
+      "path": "inputs/r437-word.json.gz",
+      "sha256": "e432182d5b62c8bbefeb871fce30317234d707e5e18d4295e65ac21f7fd8365e",
+      "git_blob": "c558a221dd44cbdaa732a401f9876de4c096d4a7",
+      "origin_commit": "364701494bc3d726806a15d4d1d4b547c1e0f9d5",
+      "origin_path": "research/r437-uniform-copy-reduction/inputs/r437-word.json.gz"
+    },
+    {
+      "path": "inputs/R437-LICENSE",
+      "sha256": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+      "git_blob": "261eeb9e9f8b2b4b0d119366dda99c6fd7d35c64",
+      "origin_commit": "364701494bc3d726806a15d4d1d4b547c1e0f9d5",
+      "origin_path": "research/r437-uniform-copy-reduction/R437-LICENSE"
+    },
+    {
+      "path": "inputs/R437-UPSTREAM-NOTICE.md",
+      "sha256": "8ed83f2f0bde43a41e44279f44d29efaeae3df44850b86f0e3578edf3ee104e4",
+      "git_blob": "9b519b904d456b0b5861f7469a66c365e77799a6",
+      "origin_commit": "364701494bc3d726806a15d4d1d4b547c1e0f9d5",
+      "origin_path": "research/r437-uniform-copy-reduction/R437-UPSTREAM-NOTICE.md"
+    },
+    {
+      "path": "inputs/r437-EXPECTED.json",
+      "sha256": "67d06bcb90d2307a11975d6b25107a695be477f168f2dc289b8484ba0b7e2de0",
+      "git_blob": "bc2585e1404402395a7c59ba6f6dde58b04e65a2",
+      "origin_commit": "a24f9ff95c12fd0b6792e3f710569237efd72db2",
+      "origin_path": "research/bit-p111-r437-gl464-470-20261018/EXPECTED.json"
+    },
+    {
+      "path": "inputs/pr3967-CATALOGUE.json",
+      "sha256": "fbd6b56ea4c488ea606668de84f45541ff941c44b5bfcd7314ba7b89cc845207",
+      "git_blob": "ec1daa219342c157deb61fdc6cf18e0faa1cc84a",
+      "origin_commit": "32af060614fefa9729c86370524c2e72265acd76",
+      "origin_path": "research/r437-oq4-caller-binding/CATALOGUE.json"
+    },
+    {
+      "path": "inputs/pr3982-PROOF.md",
+      "sha256": "27d54dcc80d2d9b6be05eab2202806c25833b680af89613fab94af911701c47a",
+      "git_blob": "da8c1c2bd289c590530f84e48b94752fbd64f8c5",
+      "origin_commit": "f33a1708675a675a5527a8b0dc80e27ec829f729",
+      "origin_path": "research/r437-parabolic-copy/PROOF.md"
+    }
+  ]
+}
diff --git a/research/r437-copy-echo-budget/TASK.json b/research/r437-copy-echo-budget/TASK.json
new file mode 100644
index 0000000000..3b1209f189
--- /dev/null
+++ b/research/r437-copy-echo-budget/TASK.json
@@ -0,0 +1,81 @@
+{
+  "schema_version": 1,
+  "task_id": "independent-moment",
+  "priority_revision": 6,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "original_card": {
+    "id": "independent-moment",
+    "question": "Does one changed supplier or assembly claim lack a source-bound independent check of a specific new implication?",
+    "starting_points": [
+      "certificates/selected-result.json",
+      "research/rank7-joint-endframes/README.md",
+      "docs/research/research-family-status.md"
+    ],
+    "methods": [
+      "Independently reconstruct a coordinated frame/circuit change and its single-change control, then derive the complete rank and physical ledger.",
+      "Implement certified moment inequalities through a different exact enclosure method."
+    ],
+    "acceptance_gate": "Demonstrate independence, source binding, positive cases and semantic failure controls. Name the existing checks and the missing coverage. More digits or another equivalent evaluation of an unchanged root is not by itself new verification coverage.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read #2979 joint rank-7 end-frame proposal and the revision-6 evidence status. Its claimed kappa is not selected. Compare coordinated changes with the losing single-change controls and locate the complete physical ledger; an aggregate moment alone does not validate the mechanism.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Prioritize a bounded independent audit of #2979: reconstruct one joint group and its single-lowering control, check the declared frames/chronology, then trace changed children, idle widths and cleanup buffers into the complete invoice and both supplier caps. Choose one still-missing implication after checking existing audits. Stop with exact unresolved dependencies if full admission exceeds the session; do not launch a new lowering census on an unvalidated premise or substitute extra root digits for mechanism checking.",
+    "bottleneck_link": "Independent pricing prevents unsupported frontier claims and can reveal genuine accounting slack.",
+    "allowed_variations": [
+      "Reconstruct either a source ledger or a numeric enclosure with a different method."
+    ],
+    "scope_change_triggers": [
+      "Importing the primary function and calling its output independent verification.",
+      "Promoting #2979 from merged source archive to validated supplier solely from author PASS receipts or aggregate pricing."
+    ],
+    "overlap_terms": [
+      "independent moment",
+      "rank histogram",
+      "exact enclosure",
+      "2979",
+      "rank7 joint endframes",
+      "paired extraction",
+      "joint lowering"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "All-size one-proper-child dependency obstruction for arbitrary dirty coordinatewise Boolean COPY echoes, plus exact fixed two-rank7 replacement budget.",
+  "scope_change_reason": "Explicit user-authorized next round changes the COPY consumer alphabet after3982: remove ordinary interchanges via coupled dirty echo rather than reduce their constant.",
+  "bottleneck_link": "Decide whether the196 ordinary upper-tariff term can disappear without repaying the saving through extra recursive children.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop fixed one-child pointwise/forward echo by orbit witness; stop fixed two-rank7-per-COPY repair by moment>1; do not infer mixed-width or adaptive impossibility.",
+  "outcome": "completed",
+  "evidence": "PROOF.md, RESULT.json, verify.py, test_package.py",
+  "author": "Chris R / Clear1nsight with OpenAI Codex assistance",
+  "ai_assistance": "OpenAI Codex research, proof and verifier authoring; separate same-session Codex dependency-orbit review.",
+  "open_questions": [
+    {
+      "question": "Can a mixed-width two-child COPY meet the full profile?",
+      "why_it_matters": "The one-event obstruction does not bound total recursive rank by14.",
+      "first_step": "Specify and exact-check one7f plus onef child with full dirty restoration, then compute complete moment.",
+      "blocked_by": "No mixed-width construction or sufficient tariff."
+    }
+  ],
+  "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/3967#issuecomment-6263066320",
+  "checkpoints": [
+    {
+      "utc": "2026-10-19T08:31:00Z",
+      "canonical_main": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+      "priority_revision": 6,
+      "status": "unchanged"
+    },
+    {
+      "utc": "2026-10-19T08:52:35.676676+00:00",
+      "canonical_main": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+      "priority_revision": 6,
+      "status": "unchanged; live COPY echo/commutator searches and local overlap checked; remote cache unavailable"
+    }
+  ]
+}
diff --git a/research/r437-copy-echo-budget/VALIDATION.json b/research/r437-copy-echo-budget/VALIDATION.json
new file mode 100644
index 0000000000..90c735db08
--- /dev/null
+++ b/research/r437-copy-echo-budget/VALIDATION.json
@@ -0,0 +1,54 @@
+{
+  "status": "PASS_LOCAL_REGISTERED_PACKAGE",
+  "checked_utc": "2026-10-19T08:52:35.676676+00:00",
+  "python": "3.12.3 (main, Aug 31 2026, 10:18:26) [GCC 13.3.0]",
+  "platform": "Linux-7.0.0-38-generic-x86_64-with-glibc2.39",
+  "registered_commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Exact source GL64 conjugacies, complete local-ring dependency orbits, dirty Boolean payload witness and full fixed replacement arithmetic."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "All checks remain active with assertions disabled."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/test_package.py"
+      ],
+      "timeout_seconds": 180,
+      "checks": "Actual Git-free read-only optimized export and four CLI source/invoice controls."
+    }
+  ],
+  "markers": [
+    "PASS_PACKAGE_METADATA_AND_PINS",
+    "PASS_SIX_RETAINED_EXACT_ORIGINAL_GIT_OBJECTS",
+    "PASS_COPY_ECHO_BUDGET normal and optimized",
+    "PASS_COPY_ECHO_EXPORT_AND_FOUR_REJECTIONS",
+    "PASS_REGISTERED_RESEARCH_PACKAGE outcome=completed"
+  ],
+  "scope": "All-parameter scoped proof with finite source/orbit controls and exact fixed-profile budget; no physical compiler or supplier admission.",
+  "scientific_hashes": {
+    "PROOF.md": "31b54cb4ea6b43670d97ab36e378c22a7a6e767692dceb92b4febdeda9929302",
+    "verify.py": "6010dd62d298c44599b1df8e2f42ead6a32ea52786807696fd3e667263e0deb6",
+    "RESULT.json": "df5477b70c08d87f16dce75cef53f1dd31e5e7cd728ac98936bcbc6abc0373c2",
+    "test_package.py": "359ad3dce3de72b8c961d57f199ad378c0d2347d784fbe38cd33809ecda1c111"
+  },
+  "documentation_scope_update": "Before publication, explicitly excluded affine translations and fixed-header-to-body additions; no verifier/result or claimed linear-U theorem changed."
+}
diff --git a/research/r437-copy-echo-budget/VERIFY.json b/research/r437-copy-echo-budget/VERIFY.json
new file mode 100644
index 0000000000..df5b99e8c7
--- /dev/null
+++ b/research/r437-copy-echo-budget/VERIFY.json
@@ -0,0 +1,40 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Exact source GL64 conjugacies, complete local-ring dependency orbits, dirty Boolean payload witness and full fixed replacement arithmetic."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "All checks remain active with assertions disabled."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/test_package.py"
+      ],
+      "timeout_seconds": 180,
+      "checks": "Actual Git-free read-only optimized export and four CLI source/invoice controls."
+    }
+  ]
+}
diff --git a/research/r437-copy-echo-budget/inputs/R437-LICENSE b/research/r437-copy-echo-budget/inputs/R437-LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/r437-copy-echo-budget/inputs/R437-LICENSE
@@ -0,0 +1,201 @@
+                                 Apache License
+                           Version 2.0, January 2004
+                        http://www.apache.org/licenses/
+
+   TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION
+
+   1. Definitions.
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+      "License" shall mean the terms and conditions for use, reproduction,
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+      form, that is based on (or derived from) the Work and for which the
+      editorial revisions, annotations, elaborations, or other modifications
+      represent, as a whole, an original work of authorship. For the purposes
+      of this License, Derivative Works shall not include works that remain
+      separable from, or merely link (or bind by name) to the interfaces of,
+      the Work and Derivative Works thereof.
+
+      "Contribution" shall mean any work of authorship, including
+      the original version of the Work and any modifications or additions
+      to that Work or Derivative Works thereof, that is intentionally
+      submitted to Licensor for inclusion in the Work by the copyright owner
+      or by an individual or Legal Entity authorized to submit on behalf of
+      the copyright owner. For the purposes of this definition, "submitted"
+      means any form of electronic, verbal, or written communication sent
+      to the Licensor or its representatives, including but not limited to
+      communication on electronic mailing lists, source code control systems,
+      and issue tracking systems that are managed by, or on behalf of, the
+      Licensor for the purpose of discussing and improving the Work, but
+      excluding communication that is conspicuously marked or otherwise
+      designated in writing by the copyright owner as "Not a Contribution."
+
+      "Contributor" shall mean Licensor and any individual or Legal Entity
+      on behalf of whom a Contribution has been received by Licensor and
+      subsequently incorporated within the Work.
+
+   2. Grant of Copyright License. Subject to the terms and conditions of
+      this License, each Contributor hereby grants to You a perpetual,
+      worldwide, non-exclusive, no-charge, royalty-free, irrevocable
+      copyright license to reproduce, prepare Derivative Works of,
+      publicly display, publicly perform, sublicense, and distribute the
+      Work and such Derivative Works in Source or Object form.
+
+   3. Grant of Patent License. Subject to the terms and conditions of
+      this License, each Contributor hereby grants to You a perpetual,
+      worldwide, non-exclusive, no-charge, royalty-free, irrevocable
+      (except as stated in this section) patent license to make, have made,
+      use, offer to sell, sell, import, and otherwise transfer the Work,
+      where such license applies only to those patent claims licensable
+      by such Contributor that are necessarily infringed by their
+      Contribution(s) alone or by combination of their Contribution(s)
+      with the Work to which such Contribution(s) was submitted. If You
+      institute patent litigation against any entity (including a
+      cross-claim or counterclaim in a lawsuit) alleging that the Work
+      or a Contribution incorporated within the Work constitutes direct
+      or contributory patent infringement, then any patent licenses
+      granted to You under this License for that Work shall terminate
+      as of the date such litigation is filed.
+
+   4. Redistribution. You may reproduce and distribute copies of the
+      Work or Derivative Works thereof in any medium, with or without
+      modifications, and in Source or Object form, provided that You
+      meet the following conditions:
+
+      (a) You must give any other recipients of the Work or
+          Derivative Works a copy of this License; and
+
+      (b) You must cause any modified files to carry prominent notices
+          stating that You changed the files; and
+
+      (c) You must retain, in the Source form of any Derivative Works
+          that You distribute, all copyright, patent, trademark, and
+          attribution notices from the Source form of the Work,
+          excluding those notices that do not pertain to any part of
+          the Derivative Works; and
+
+      (d) If the Work includes a "NOTICE" text file as part of its
+          distribution, then any Derivative Works that You distribute must
+          include a readable copy of the attribution notices contained
+          within such NOTICE file, excluding those notices that do not
+          pertain to any part of the Derivative Works, in at least one
+          of the following places: within a NOTICE text file distributed
+          as part of the Derivative Works; within the Source form or
+          documentation, if provided along with the Derivative Works; or,
+          within a display generated by the Derivative Works, if and
+          wherever such third-party notices normally appear. The contents
+          of the NOTICE file are for informational purposes only and
+          do not modify the License. You may add Your own attribution
+          notices within Derivative Works that You distribute, alongside
+          or as an addendum to the NOTICE text from the Work, provided
+          that such additional attribution notices cannot be construed
+          as modifying the License.
+
+      You may add Your own copyright statement to Your modifications and
+      may provide additional or different license terms and conditions
+      for use, reproduction, or distribution of Your modifications, or
+      for any such Derivative Works as a whole, provided Your use,
+      reproduction, and distribution of the Work otherwise complies with
+      the conditions stated in this License.
+
+   5. Submission of Contributions. Unless You explicitly state otherwise,
+      any Contribution intentionally submitted for inclusion in the Work
+      by You to the Licensor shall be under the terms and conditions of
+      this License, without any additional terms or conditions.
+      Notwithstanding the above, nothing herein shall supersede or modify
+      the terms of any separate license agreement you may have executed
+      with Licensor regarding such Contributions.
+
+   6. Trademarks. This License does not grant permission to use the trade
+      names, trademarks, service marks, or product names of the Licensor,
+      except as required for reasonable and customary use in describing the
+      origin of the Work and reproducing the content of the NOTICE file.
+
+   7. Disclaimer of Warranty. Unless required by applicable law or
+      agreed to in writing, Licensor provides the Work (and each
+      Contributor provides its Contributions) on an "AS IS" BASIS,
+      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or
+      implied, including, without limitation, any warranties or conditions
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+      PARTICULAR PURPOSE. You are solely responsible for determining the
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+      risks associated with Your exercise of permissions under this License.
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+      whether in tort (including negligence), contract, or otherwise,
+      unless required by applicable law (such as deliberate and grossly
+      negligent acts) or agreed to in writing, shall any Contributor be
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+   Licensed under the Apache License, Version 2.0 (the "License");
+   you may not use this file except in compliance with the License.
+   You may obtain a copy of the License at
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+       http://www.apache.org/licenses/LICENSE-2.0
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+   Unless required by applicable law or agreed to in writing, software
+   distributed under the License is distributed on an "AS IS" BASIS,
+   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+   See the License for the specific language governing permissions and
+   limitations under the License.
diff --git a/research/r437-copy-echo-budget/inputs/R437-UPSTREAM-NOTICE.md b/research/r437-copy-echo-budget/inputs/R437-UPSTREAM-NOTICE.md
new file mode 100644
index 0000000000..9b519b904d
--- /dev/null
+++ b/research/r437-copy-echo-budget/inputs/R437-UPSTREAM-NOTICE.md
@@ -0,0 +1,71 @@
+# Notice
+
+Apache-2.0 (`LICENSE`, copied from #493's package). Prepared by eumemic with substantial Anthropic Claude
+assistance.
+
+## New in this package
+
+- `code/gen_p3.py` (the generator of the punctured word: the L3 skeleton of #543 restricted to the live lines, the
+  pseudo-target totals, the phase-A mini-nodes and extra carriers), `code/pwordl.py` (word builder for a live subset
+  of the 120 lines; output in the bitE8/1 layout with the `ports_geom` and `punctured` fields introduced by #545),
+  `data/bitE8mut-p111.json.gz` (the word), `data/punct.json` (its frozen parameters), `discovery/choose.py` and
+  `discovery/search.py` (the nomination of P and of the parameters; not run by `verify.sh`).
+- `code/gen_p3.py` carries `derive`, `ex_part` and `plan_thirds` from #543's `code/gen_l3.py` (eumemic), and
+  `data/plan-l3.json` is #543's CP-SAT plan, unchanged. `code/nodeopt.py`, `code/planio.py`, `code/mutants.py`,
+  `code/mutrec_b.py` and `code/provenance.py` are #543's files, unchanged.
+- `code/xcheck_p.py`: #543's independent checker `code/xcheck.py` with nine listed edits for v <= 120 live labels
+  (root-line enumeration of the lattice, frame closure and target frames over the live labels) and a new part T (the
+  frame and COPY rank of every retained total at the scatter, and whether it is the hyperplane of a punctured line).
+- `code/mutants_p.py` (punctured-design controls), `code/adapt9p.py` (#510's nine-stage files parameterised for
+  v <= 120 and any copy ranks, the way #545's `adapt.py` parameterises #493's eleven-stage files), `code/cited.py`,
+  `code/receipt_kappa.py`, `code/summary.py`, `verify.sh`, `EXPECTED.json`, `DEPENDS.json`, `requirements.txt`,
+  `data/complex-b.json`, `data/selection9-p111.json` and `data/hosting11-p111.json` (frozen nominations of #442's
+  hosting programme, rediscovered by `verify.sh`), `.github/workflows/e8-bit-p111.yml`.
+
+## Vendored, byte-identical (git blob ids in `SOURCE-pr*.json`, checked by `code/provenance.py`)
+
+- **PR #510 (DreamingOfClouds, with Claude Opus 5.5 via Claude Code), commit 64edee1** (`vendor/pr510/`): the same 19
+  files as #543's package (#510's nine-stage chain with the files it vendors from #463, #443, #442 and #315, and #503's
+  R480 word). Their notices are listed in #543's NOTICE and in the files.
+- **PR #545 (DreamingOfClouds, with Claude Opus 5.5 via Claude Code; draft), commit 376426f** (`vendor/pr545/`):
+  `code/adapt.py` and `code/punct_cfg.py`, run unchanged on this package's word; `data/bitE8mut-v111-r489.json.gz`,
+  `data/hosting-v111-r489.json` and `EXPECTED.json` (a control).
+- **PR #548 (Dugongue), commit 66dbc2b** (`data/complex/pr548/`): `EXPECTED.json` and `evidence/EXACT-ASSEMBLY.json`.
+- **PR #544 (chafreaky, with OpenAI Codex assistance), commit 4858bb3** (`data/complex/pr544/`): `EXPECTED.json` and
+  the Linux exact-assembly receipt.
+- **PR #535 (icekylinx, with GPT-6 Astra assistance), commit 8b758a1** (`data/complex/pr535/`): `EXPECTED.json` and
+  `evidence/audit/EXACT-ASSEMBLY-PR493.json`.
+
+## Used from main, unchanged (pinned in `DEPENDS.json`)
+
+- `research/e8-bit-carrier-skeleton` (PR #493, eumemic): the skeleton modules and D4 partition the generator imports,
+  #370's checker (run here in #545's v <= 120 form), the official cohort checkers, `gram_frames.py`, the record
+  exporter, the geometries and the eleven-stage chain that #545's `adapt.py` parameterises, #442's hosting programme,
+  the R489 word, the control receipts and #315's engines.
+- `research/direct-delivery504-global/EXPECTED.json` (PR #531, icekylinx): its complex b and kappa.
+
+## Cited, not rebuilt
+
+Complex coarse savings in `data/complex-b.json`, each bound to its PR's receipt by `code/cited.py`: PR #548 (primary;
+open), PR #544 and PR #535 (open), PR #531 (merged), and PR #484, #471 and #458 (controls).
+
+## Ideas used
+
+- PR #545 (DreamingOfClouds): the punctured-star unit. Nine lines are neither sources nor targets; each retained total
+  is a pseudo-target that reads what the removed target would read, at frame rank 7, so the decoder constant is 63;
+  the rank-mass identity with v and the decoder constant as parameters; the parameterisation of the hub chain.
+- PR #543 (eumemic): the R420 L3 skeleton with keepers read by targets and rewritten in phase B.
+- PR #493 (eumemic), PR #503 and #510 (DreamingOfClouds), PR #425 (Zhihao Chen / jacklightChen), PR #370 (Dylan Crow),
+  PR #472, #445, #443 (chafreaky), PR #442 (DaysSky), PR #463 and #315 (DreamingOfClouds), PR #386 (Zhihao Chen): the
+  skeleton, the scalar-interface lemma, the format, and the pricing pipelines.
+- PR #523 (icekylinx): the puncture pattern on the complex side, cited by #545.
+- Jacob Sussman introduced the E8 unit.
+
+R437 composition: Ali Romman (@aliromman), with substantial OpenAI Codex assistance, authored the public1043 helper470 source-boundary deletion. Its exact canonical-main package and NOTICE are retained in public1043/. The new contribution here composes that credited deletion with the independently validated helper464 GL, then reconstructs actual owners, full invoice, independent checks and portable replay.
+
+Sagar Thakrar (thakrarsagar), PR3700 Open question4, proposed the exact
+window6 two-leftover coalescing used by this follow-up. No code from that
+PR is imported by the author pipeline; root independently replayed its
+hosting and price engines. Full owner/template/invoice regeneration and
+packaging used substantial OpenAI Codex assistance. All prior credits and
+Apache-2.0 notices remain in place.
diff --git a/research/r437-copy-echo-budget/inputs/pr3967-CATALOGUE.json b/research/r437-copy-echo-budget/inputs/pr3967-CATALOGUE.json
new file mode 100644
index 0000000000..ec1daa2193
--- /dev/null
+++ b/research/r437-copy-echo-budget/inputs/pr3967-CATALOGUE.json
@@ -0,0 +1,447 @@
+{
+  "changed_request_count": 7,
+  "changed_requests": [
+    {
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+    {
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+        [
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+        1,
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+      "identifier": 19854,
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+        210,
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+        1,
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+            29
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+        "owner",
+        null
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+    },
+    {
+      "current": [
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+        209,
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+        1,
+        "13c3d4f57cea761c1d7b88abb26da0ca4001059c0f047f09bf2c16e60ecef521",
+        1,
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+            36,
+            37
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+        null
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+      "old": [
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+        24888,
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+        210,
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+        1,
+        "13c3d4f57cea761c1d7b88abb26da0ca4001059c0f047f09bf2c16e60ecef521",
+        1,
+        [
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+          -1,
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+        "owner",
+        null
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+    },
+    {
+      "current": [
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+        6,
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+        2,
+        8,
+        "6c38408d019ad769697cc6a8a70093cc45251d8fc1244e8c303ef4e0becdbab2",
+        26,
+        [
+          "family",
+          98,
+          -1,
+          []
+        ],
+        "owner",
+        null
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+      "identifier": 27776,
+      "old": [
+        "IDLE",
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+        6,
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+        209,
+        2,
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+        "9710b96b4d5c382dbdb44bda4ccdb9e2f33217f4065ae9056958de95a9953db5",
+        34,
+        [
+          "family",
+          98,
+          -1,
+          []
+        ],
+        "owner",
+        null
+      ]
+    },
+    {
+      "current": [
+        "IDLE",
+        6,
+        6,
+        210,
+        210,
+        2,
+        8,
+        "3aeb89340b4aabdf7cdcbee3271e2bc7650188d764ee08ebbcef3be8bc3b6661",
+        42,
+        [
+          "family",
+          99,
+          -1,
+          []
+        ],
+        "owner",
+        null
+      ],
+      "identifier": 27777,
+      "old": [
+        "IDLE",
+        6,
+        6,
+        210,
+        210,
+        2,
+        8,
+        "9710b96b4d5c382dbdb44bda4ccdb9e2f33217f4065ae9056958de95a9953db5",
+        34,
+        [
+          "family",
+          99,
+          -1,
+          []
+        ],
+        "owner",
+        null
+      ]
+    }
+  ],
+  "current": {
+    "base_labels": 540,
+    "binding_sha256": "cbfea55f5cd0b74a2a0d4292ad0139d2b5bb73127951127698030bc90c73c532",
+    "canonical_ERASE_guards": 81,
+    "canonical_requests": 28233,
+    "chart_labels": 1235,
+    "full_paid_requests": 903456,
+    "rank_histogram": {
+      "1": 380160,
+      "2": 400864,
+      "26": 32,
+      "3": 22176,
+      "4": 26592,
+      "42": 384,
+      "5": 29376,
+      "6": 17376,
+      "7": 26496
+    },
+    "replicas": 32,
+    "request_kinds": {
+      "COPY": 81,
+      "IDLE": 1998,
+      "MOVE": 26154
+    }
+  },
+  "full_GL64_low_classes_executed": false,
+  "literal_opcode_counts": {
+    "0": 836928,
+    "1": 6971904,
+    "2": 2592,
+    "3": 2592
+  },
+  "old": {
+    "base_labels": 540,
+    "binding_sha256": "0fb70d6838583f2f439a8704eb5dba84ce9dbd231d91f140da13440b680d73aa",
+    "canonical_ERASE_guards": 81,
+    "canonical_requests": 28233,
+    "chart_labels": 1235,
+    "full_paid_requests": 903456,
+    "rank_histogram": {
+      "1": 380160,
+      "2": 400864,
+      "3": 22176,
+      "34": 64,
+      "4": 26592,
+      "42": 352,
+      "5": 29376,
+      "6": 17376,
+      "7": 26496
+    },
+    "replicas": 32,
+    "request_kinds": {
+      "COPY": 81,
+      "IDLE": 1998,
+      "MOVE": 26154
+    }
+  },
+  "rank_profile_delta": {
+    "26": 32,
+    "34": -64,
+    "42": 32
+  },
+  "scope": "Every literal source/controller site; all-low numerical emitter remains a quantified dependency",
+  "stale_digest_control": "REJECTED",
+  "status": "PASS_FULL_OQ4_SOURCE_TOPOLOGY_AND_CHRONOLOGY"
+}
diff --git a/research/r437-copy-echo-budget/inputs/pr3982-PROOF.md b/research/r437-copy-echo-budget/inputs/pr3982-PROOF.md
new file mode 100644
index 0000000000..da8c1c2bd2
--- /dev/null
+++ b/research/r437-copy-echo-budget/inputs/pr3982-PROOF.md
@@ -0,0 +1,275 @@
+# Parabolic COPY with batched pivot and graph routing
+
+Chris R / Clear1nsight, with substantial OpenAI Codex assistance. Apache-2.0.
+
+## Result and boundary
+
+For m=64, r=7 and every split rank-seven projector over Z/q^w, the construction
+below replaces PR3950's alternate generic SL64 word by **8,036 ordered scalar
+ADD slots per column and 196 ordinary width-f interchanges**, surrounding the
+same single reversed child of width 7f. PR3950's alternate has 41,832 of each.
+All slots, including zero coefficients, are charged. This is a uniform algebraic
+construction and a **conditional relative caller/cost theorem**; an encoded
+physical compiler has not been implemented. The original R437 good/fallback
+implementation is a different construction and its tariff is not altered.
+
+The change combines the parabolic stabilizer of a projector with small fixed
+permutation networks. It removes unnecessary internal image/kernel basis work
+and shares routing across commuting graph passes and mutually exclusive pivot
+stars. Generic conjugacy, local-ring split projectors and elementary matrices
+are credited predecessor mathematics, not new claims of this package.
+
+Even the reduced positive ordinary term is not absorbable from the pinned B1/B2
+upper tariffs at the R437 target scaling. The optimistic decision is therefore:
+retain this smaller constructive relative bill, then stop supplier expansion
+until a cheaper complete ordinary adapter or a different consumer is supplied.
+This does not lower-bound actual runtime or exclude the original algorithm.
+
+## 1. Domain and a principal chart that always exists
+
+Fix an odd prime q, w>=1, R=Z/q^w, A=q^w, m>=2 and 0<r<m. For the application,
+m=64 and r=7. Let P be a split rank-r projector and g invertible over R, and
+put Q=gPg^-1. Let H=diag(h_i) consist of arbitrary units; h_i may depend on
+physical header and column, but not on body addresses or fresh unused rows.
+The requested two-bank address map, with tail coordinates reflected, is
+
+    D_H(Q)=diag(H,I) [[I-Q,Q],[Q,I-Q]] diag(H^-1,I).
+
+Every Boolean source function is allowed, including arbitrary correlated dirty
+captures. Equal field-header values at distinct physical prefix ordinals remain
+distinct. There is no SOURCE projection or clean-data assumption.
+
+A principal r-by-r minor of Q is a unit. Indeed a split idempotent is similar
+to E=diag(I_r,0), so its characteristic polynomial is t^(m-r)(t-1)^r. Its
+coefficient of t^(m-r) is (-1)^r times both 1 and the sum of all principal
+r-minors. Reducing modulo q, their sum is 1. At least one is nonzero. This
+argument also works when q divides r; trace is not used as a rank certificate.
+
+Choose a sorted subset S=(s_0,...,s_(r-1)) with unit principal minor, so
+ i<=s_i<=m-r+i. One effective deterministic rule is the first unit subset in
+lexicographic order. This fixed-m search can be very expensive: at m=64,r=7
+there are binom(64,7) subsets. It is charged in gamma below, not called free
+or a practical large-instance implementation. No enumeration of GL64 is needed.
+The same S selected modulo q works at every high lift and precision.
+
+## 2. Fixed-support pivot stars
+
+For i=0,...,r-1 and every k in i+1,...,m-r+i, let e_ik=1 if s_i=k and 0
+otherwise. Use three stars, in chronological left-action order:
+
+    all E_(i,k)(e_ik), then all E_(k,i)(-e_ik),
+    then all E_(i,k)(e_ik).
+
+At most one flag is nonzero in this entire i-stage. Consequently this is either
+identity (s_i=i) or the signed swap (x_i,x_s)->(x_s,-x_i). Grouping the triples
+into stars is legal precisely because of that mutually exclusive flag property;
+it would be false with two active candidates. Earlier stages cannot move s_i,
+which is strictly larger than every earlier s_j. The product J therefore moves
+the original S to the first r coordinates and has determinant one.
+
+The schedule has 3r(m-r) scalar factors with support independent of S, q, w,
+header and column. No conditional physical permutation or low-class regrouping
+is assumed: only scalar coefficients carry the pivot flags. All zero factors
+remain emitted and paid. This addresses a source-dependent routing issue that
+merely choosing a matrix basis would leave unresolved.
+
+## 3. Quotient out the stabilizer
+
+Write Q'=JQJ^-1 in blocks [[A0,B],[C,D]], with A0 an invertible r-by-r matrix.
+Put X=C A0^-1 and Z=-B. Define
+
+    L_X=[[I,0],[X,I]], U_Z=[[I,Z],[0,I]], T0=L_X U_Z.
+
+Then Q'=T0 E T0^-1. Here is a proof over the local ring, without a probabilistic
+chart hypothesis. The first r columns of Q' are independent modulo q; since
+im(Q') is a free direct summand of rank r, they form an R-basis of that image.
+Thus D=C A0^-1 B=XB. Idempotence gives A0^2+BC=A0, hence A0+BX=I. Multiplying
+out T0 E T0^-1 gives blocks I-BX, B, X(I-BX), XB, exactly Q'. Both L_X and
+U_Z have determinant one; each is r(m-r) commuting elementary shears.
+Set T=J^-1 T0, so Q=TET^-1 and det(T)=1.
+
+This uses only the image/kernel splitting, rather than a full arbitrary GL64
+basis. The stabilizer degrees of freedom do not need to be implemented.
+
+## 4. Ancestor units and kernel-only normalization
+
+Let H'=J H J^-1=diag(d_i). This is diagonal because J is a signed
+permutation. Define the determinant-one diagonal
+
+    D*=diag(d_0,...,d_(r-1), (product_(i<r) d_i)^-1, 1,...,1),
+    Z_h=H'^-1 D*.
+
+Z_h is identity on every image coordinate; its arbitrary unit kernel entries
+are allowed. The desired head and tail conjugators are
+
+    U_t=T=J^-1 T0,
+    U_h=H T Z_h
+       =J^-1 (H' L_X H'^-1)(H' U_Z H'^-1) D*.
+
+Both have determinant one. The conjugated graph matrices have the same supports
+and their coefficients are multiplied by d_target/d_source. Kernel ancestor
+weights remain present in these coefficients; they have not been discarded.
+The right compensation Z_h fixes every image coordinate, so diag(Z_h,I)
+commutes with D(E). Therefore
+
+    K=diag(U_h,U_t),   K D(E) K^-1=D_H(Q).
+
+Normalizing an image column independently would generally change the weighted
+swap. Here only the kernel compensation is free. This image-only diagonal
+choice emerged in the coordinating agent's cross-review of the first all-axis
+normalization; it removes unnecessary kernel diagonal factors. The result and
+all checks remain in this same assisted research session.
+
+For any unit a, diag(a,a^-1) on coordinates (i,k) is four shears:
+
+    U(1) L(a-1) U(-a^-1) L(-a(a-1)).
+
+This displayed product uses ordinary matrix multiplication; chronological left
+actions run right to left. Its off-diagonal entries vanish and its diagonal is
+(a,a^-1). Pair each image coordinate i<r with the single kernel hub k=r.
+Their pair diagonals multiply to D*, giving 4r scalar factors, with two in each
+orientation. This identity is checked separately over all units of several
+local rings. Unlike implementing an entire arbitrary SL_m diagonal, no factors
+are emitted for the other kernel coordinates.
+
+## 5. Fixed permutation networks for ordered ADD
+
+An ADD is assumed available only when the source block precedes its target
+physically. Blocks contain f atoms; the coefficient for each column comes from
+unchanged saved metadata. Ordinary block interchange is a **completed**, same-q,
+source-preserving primitive that preserves the entire intervening spectator,
+returns its heads and restores owned work. Its full cost is b(v,f).
+
+A reverse individual shear can be implemented by swapping its two whole blocks,
+performing the ordered ADD on the relabelled coordinates, and swapping back.
+This is used for the diagonal factors and costs two ordinary calls per reverse
+factor. It is never identified with the central reversed child.
+
+A whole reverse pivot star has centre i and arms in i+1,...,m-r+i. Swap i with
+its largest arm, execute the entire relabelled star, and swap back. All arms
+then precede the centre (or the opposite relation in the reflected bank).
+There are exactly two ordinary calls per reverse star, regardless of the
+number of arms. The routing permutation is fixed, even if every coefficient
+is zero. The one-hot property is needed for the algebraic regrouping, not
+for the routing operation itself.
+
+For a reverse graph pass, put t=min(r,m-r). Swap i with m-t+i for i=0,...,t-1,
+execute its relabelled shears, then undo these t swaps. The images now lie
+entirely after the kernels (or entirely before them in reflected order), so
+every ADD is ordered. This costs 2t ordinary calls per reverse graph pass.
+It preserves the original layout at the pass boundary and acts pointwise on
+all external spectators, fresh rows and physical prefix ordinals.
+
+Within this permutation-only route-and-return model, t swaps per direction
+are optimal for reversing the complete image/kernel order: the smaller group
+must all leave its original side and a transposition moves at most one of its
+members to the other side. This narrow claim does not lower-bound all COPY
+implementations. Exhaustive permutations through dimension seven check it.
+
+Head/tail physical orders are opposite. Across both banks and both directions,
+there are 6r reverse pivot stars and four reverse graph passes. The diagonal
+appears only on the head, in both directions, and has 4r reverse scalar
+slots in total. Hence the exact fixed emitted counts are
+
+    scalar ADD factors = 4[3r(m-r)+2r(m-r)]+2[4r]
+                       =20r(m-r)+8r,
+    ordinary calls     =20r+8 min(r,m-r).
+
+At (m,r)=(64,7), these are **8,036** and **196**. Breakdown of ordinary calls:
+84 pivot, 56 graph, 56 diagonal. There is one reversed child, not one per
+prefix. At fixed header/column some coefficients can vanish; the invoice does
+not omit them. The interpreter checks every emitted ADD's physical direction
+and executes every ordinary swap, including routing back.
+
+## 6. Full caller, consumer and relative bill
+
+Use the complete packet shape and ownership contract of PR3950:
+
+    [Z physical prefixes][U unused fresh rows][B][A^n head][G][A^n tail][S],
+    n=64f+delta, 0<=delta<64, v=Z U B G S A^(2n).
+
+Clone the borrowed complete arbitrary source. Apply the complete address
+permutation using the two conjugation words and one reversed 7f child, return
+the owned result to the actual COPY private port, and preserve it through all
+63 literal XOR readers. Release every intermediate only after its last reader.
+At the equality guard, clear the private output and reset its port.
+
+Matrix words are chronological left actions on address vectors. Pullback
+implementation reverses the entire vector-event order. That reversal preserves
+the physical direction and coefficient of each emitted ADD; it is not the
+operation of negating a factor. Inverse conjugators separately reverse and
+negate their scalar words. The finite payload checker executes these primitive
+pullbacks and all 63 dirty consumers, rather than checking only matrix rank.
+
+The central child's spectators are B'=B, G'=G A^(114f+2delta), S'=S, with
+fresh rows U and all Z physical prefixes. Their volume equals v exactly and
+keeps the peeled atoms/external gap in their original coordinate order. The
+child is one complete call outside any prefix loop. For the inherited stock
+K0=269241600 |GL64(F_q)| q^(4096(w-1)) and
+D(n)=0 below64, otherwise1+D(42 floor(n/64)), the standard parent stock
+ t K0^D(n) leaves U=t*9475*K0^(D(n)-1). Since 7f<=42f<n, child divisibility
+follows. This is reuse of the stock implication; it does not prove the original
+producer's stock coefficient or authorize replacing its source program.
+
+Let t_R(v,7f,w) be the full completed reversed-child cost. Let gamma include
+principal-chart search, matrix/weight coefficient generation and recomputation,
+Counts, source/program authentication, numerical work erasure and metadata
+preparation not included in the completed primitives. Let C_scan cover actual
+linear copying, scalar ADD, transfers, rewinds, releases and disposal. Then
+
+    C_residual <= t_R(v,7f,w) + 196 b(v,f)
+                     + C_scan,q v(8036f+196+3) + gamma.          (1)
+
+There are at most 8036f+196+1 intermediate releases; this upper bound is paid in
+the scan term. Borrowed source is never cleared. For the literal COPY episode,
+append 63 X(v) for the completed XOR readers and the actual outer owner gather,
+route/scatter, directory producer and protected return charges if those have
+not already been charged by the caller. No reader recomputes the residual;
+no reader's arbitrary incoming target is assumed zero.
+
+Conditional on fixed reusable tape inventories, a conservative visited-cell
+reserve is
+
+    2v + S_R(v,7f,w) + S_B(v,f) + S_ADD(v) + M_coeff + F_q,
+
+where each backend reserve is a whole-run lifetime maximum on its fixed
+reusable inventory, including owned output, numerical work, parking and
+blank-but-visited cells. The two volume buffers must likewise use fixed reusable
+inventories; M_coeff and F_q include whole-run coefficient and finite-control
+backing. Per-call live occupancy does not establish these bounds. If a backend
+allocates disjoint tapes across occurrences, their lifetime backing must be
+summed instead. Sequential cleanup does not justify taking a maximum
+across disjoint tape inventories. There is no stored length-f array of matrices;
+coefficients can be recomputed from header and column, with recomputation time
+included in gamma. Successful cleanup is covered by the named contracts;
+interruption recovery is not proved.
+
+No hidden absorption is made in (1). Under the additionally admitted backend
+and root/history metadata premises one may substitute
+b(v,f)<=C_B v(fw)^p+E_B and absorb the justified scan/preparation terms. However
+the 196 C_B v(fw)^p contribution remains, as does 196 E_B until separately
+absorbed. This package does not prove those physical premises or implement a
+source-bound alternative SelectedBinding.
+
+## 7. Cost gate, evidence and remaining obligations
+
+For c_R=1736646042038446578551/10^24, n approximately e and w approximately
+e^c_R, the retained B1 and B2 rates give (1+c_R)(1-a_B)-1>0. Multiplying their
+upper bound by 196 instead of 41832 does not change its exponent. Thus this
+new constructive constant improvement still cannot infer an O_q(v(n+w^p))
+local allowance from those upper bounds. It is not a runtime lower bound.
+
+The source-bound finite evidence reconstructs all nine actual capture projectors
+from R437's root rows at a24f9ff, covering45 contexts at q=3,5,7,11 with high
+lifts and unit ancestors,192 GL2/weight contexts, an ambient64 nonzero graph
+with singular leading chart, complete small arbitrary payload/63 readers,
+and semantic failures. Separate principal-minor/chart and routing checks do not
+import the compiler. Tests at small q establish their stated algebraic cases;
+large-prime supplier cost theorems still require their admitted q domain.
+
+Still open: an encoded coefficient producer and physical primitive realization
+for this alternate schedule; a genuinely cheaper complete ordinary tariff or
+an actual jointly charged consumer; uniform source/return/refinement and visited
+reserve admission; full native/router/precision/recovery obligations. No selected
+exponent changes. The written proof has not received independent human or
+formal-kernel review; separate code is still from this assisted session.
diff --git a/research/r437-copy-echo-budget/inputs/r437-EXPECTED.json b/research/r437-copy-echo-budget/inputs/r437-EXPECTED.json
new file mode 100644
index 0000000000..bc2585e140
--- /dev/null
+++ b/research/r437-copy-echo-budget/inputs/r437-EXPECTED.json
@@ -0,0 +1,34 @@
+{
+  "word_gzip_sha256": "e432182d5b62c8bbefeb871fce30317234d707e5e18d4295e65ac21f7fd8365e",
+  "word_content_sha256": "da77deaf11368d016622348227657b98404b6afbc7e4f3d6fe9a246baae0dcce",
+  "records_sha256": "caeb2eb0e417992788c32c1f5812ada60b46d1dfdba9f4ac376299b2dede13cb",
+  "template_content_sha256": "c425a64739981b3ae3ed6b36d1a2f42c2d35e93f007649e89d4c677d8e158fe5",
+  "program_sha256": "a777373994bd84681d6c2be75251ba6a04a1521580910f23afe09bb50db632b2",
+  "profile": {
+    "1": 380160,
+    "2": 400864,
+    "3": 22176,
+    "4": 26592,
+    "5": 29376,
+    "6": 17376,
+    "7": 26496,
+    "26": 32,
+    "42": 384
+  },
+  "live": 28416,
+  "physical": 28425,
+  "rank_mass": 1808352,
+  "deficit": 10272,
+  "emitted_events": 7814016,
+  "hosted_residuals": 125856,
+  "BIT_c": "1736646042038446578551/1000000000000000000000000",
+  "BIT_c_decimal": "0.001736646042038446578551",
+  "next_grid": "217080755254805822319/125000000000000000000000",
+  "invoice": {
+    "prefix_bits": 177,
+    "charged_prefix": 192,
+    "two_level_bits": 393,
+    "charged_two_level": 512,
+    "ROW": 18355
+  }
+}
diff --git a/research/r437-copy-echo-budget/inputs/r437-word.json.gz b/research/r437-copy-echo-budget/inputs/r437-word.json.gz
new file mode 100644
index 0000000000..c558a221dd
Binary files /dev/null and b/research/r437-copy-echo-budget/inputs/r437-word.json.gz differ
diff --git a/research/r437-copy-echo-budget/test_package.py b/research/r437-copy-echo-budget/test_package.py
new file mode 100644
index 0000000000..36e263d878
--- /dev/null
+++ b/research/r437-copy-echo-budget/test_package.py
@@ -0,0 +1,29 @@
+#!/usr/bin/env python3
+# SPDX-License-Identifier: Apache-2.0
+"""Actual optimized export and source/invoice rejection subprocesses."""
+import json,os,shutil,subprocess,sys,tempfile
+from pathlib import Path
+HERE=Path(__file__).resolve().parent
+
+def need(ok,msg):
+    if not ok:raise ValueError(msg)
+def invoke(p,ok):
+    r=subprocess.run([sys.executable,'-I','-O','-B',str(p/'verify.py')],cwd=p.parent,
+                     stdout=subprocess.PIPE,stderr=subprocess.PIPE,text=True,timeout=60)
+    need((r.returncode==0)==ok,'export subprocess status '+r.stdout+r.stderr)
+    if ok:need('PASS_COPY_ECHO_BUDGET' in r.stdout,'positive scientific marker')
+with tempfile.TemporaryDirectory(prefix='copy-echo-export-') as t:
+    root=Path(t);p=root/'package';shutil.copytree(HERE,p,ignore=shutil.ignore_patterns('__pycache__'))
+    for f in p.rglob('*'):
+        if f.is_file():f.chmod(0o444)
+    invoke(p,True)
+    for f in p.rglob('*'):
+        if f.is_file():f.chmod(0o644)
+    source=p/'inputs/r437-word.json.gz';b=source.read_bytes()
+    source.write_bytes(b+b'changed');invoke(p,False);source.write_bytes(b)
+    source.unlink();invoke(p,False);source.write_bytes(b)
+    other=root/'source-copy.gz';other.write_bytes(b);source.unlink();source.symlink_to(other)
+    invoke(p,False);source.unlink();source.write_bytes(b)
+    result=p/'RESULT.json';saved=result.read_bytes();d=json.loads(saved);d['budget']['excess']=0
+    result.write_text(json.dumps(d));invoke(p,False);result.write_bytes(saved)
+print('PASS_COPY_ECHO_EXPORT_AND_FOUR_REJECTIONS')
diff --git a/research/r437-copy-echo-budget/verify.py b/research/r437-copy-echo-budget/verify.py
new file mode 100644
index 0000000000..2f952c8a40
--- /dev/null
+++ b/research/r437-copy-echo-budget/verify.py
@@ -0,0 +1,186 @@
+#!/usr/bin/env python3
+# SPDX-License-Identifier: Apache-2.0
+# Chris R / Clear1nsight, with substantial OpenAI Codex assistance.
+"""Exact local-ring source binding, orbit control and fixed-profile arithmetic."""
+from collections import deque
+from fractions import Fraction as F
+import gzip, hashlib, itertools, json
+from pathlib import Path
+HERE=Path(__file__).resolve().parent
+
+def need(ok,msg):
+    if not ok: raise ValueError(msg)
+
+def canonical(x): return json.dumps(x,sort_keys=True,separators=(',',':')).encode()
+def eye(n): return [[int(i==j) for j in range(n)] for i in range(n)]
+def mm(a,b,M):
+    out=[[0]*len(b[0]) for _ in a]
+    for i,row in enumerate(a):
+        for k,x in enumerate(row):
+            if x:
+                for j,y in enumerate(b[k]):
+                    if y:out[i][j]=(out[i][j]+x*y)%M
+    return out
+def inv(a,M):
+    from math import gcd
+    n=len(a);w=[row[:]+e for row,e in zip(a,eye(n))]
+    for j in range(n):
+        p=next((i for i in range(j,n) if gcd(w[i][j],M)==1),None)
+        need(p is not None,'unit basis pivot')
+        w[j],w[p]=w[p],w[j];u=pow(w[j][j],-1,M)
+        w[j]=[u*x%M for x in w[j]]
+        for i in range(n):
+            if i!=j:
+                u=w[i][j];w[i]=[(x-u*y)%M for x,y in zip(w[i],w[j])]
+    return [row[n:] for row in w]
+
+def projector(rows):
+    a=[[F(x) for x in row] for row in rows];n=len(a[0]);k=0;piv=[]
+    for j in range(n):
+        p=next((i for i in range(k,len(a)) if a[i][j]),None)
+        if p is None: continue
+        a[k],a[p]=a[p],a[k];u=a[k][j];a[k]=[x/u for x in a[k]]
+        for i in range(len(a)):
+            if i!=k:
+                u=a[i][j];a[i]=[x-u*y for x,y in zip(a[i],a[k])]
+        piv.append(j);k+=1
+    free=[j for j in range(n) if j not in piv]
+    need(k==7 and len(free)==1,'actual rank7 hyperplane')
+    z=[F(j==free[0]) for j in range(n)]
+    for i,j in enumerate(piv):z[j]=-a[i][free[0]]
+    norm=sum(x*x for x in z)
+    return [[F(i==j)-z[i]*z[j]/norm for j in range(n)] for i in range(n)]
+
+def authenticate():
+    s=json.loads((HERE/'SOURCES.json').read_text())
+    for r in s['retained']:
+        p=HERE/r['path'];need(p.is_file() and not p.is_symlink(),'missing/linked retained source')
+        b=p.read_bytes()
+        need(hashlib.sha256(b).hexdigest()==r['sha256'],'retained SHA256 '+r['path'])
+        need(hashlib.sha1(b'blob '+str(len(b)).encode()+b'\0'+b).hexdigest()==r['git_blob'],'retained Git blob')
+    return json.loads(gzip.decompress((HERE/'inputs/r437-word.json.gz').read_bytes()))
+
+def source_conjugacies(word):
+    cur=word['start'][:]
+    for sec in ['D','A']:
+        for frame,target,sources in word[sec]:
+            for role in [target]+sources:cur[role]=frame
+    need(word['ret']==list(range(470,479)),'actual nine captures')
+    need([cur[i] for i in word['ret']]==list(range(510,519)),'actual capture frames')
+    hashes=[]
+    for capture in word['ret']:
+        p=projector([word['lines'][i] for i in word['frames'][cur[capture]]])
+        for prime,w in [(3,1),(3,2),(5,2),(17,2)]:
+            M=prime**w;P=[[x.numerator*pow(x.denominator,-1,M)%M for x in row] for row in p]
+            need(mm(P,P,M)==P,'source idempotence over local ring')
+            # Pick seven image columns and one complementary kernel column.
+            T=None
+            for cols in itertools.combinations(range(8),7):
+                for k in range(8):
+                    cand=[[P[i][j] for j in cols]+[(int(i==k)-P[i][k])%M] for i in range(8)]
+                    try:Ti=inv(cand,M)
+                    except ValueError:continue
+                    T=cand;break
+                if T is not None:break
+            need(T is not None,'split basis found')
+            E=[[int(i==j and i<7) for j in range(8)] for i in range(8)]
+            need(mm(mm(T,E,M),Ti,M)==P,'actual capture split basis')
+            # g=S diag(T^-1,I56), conjugating the actual embedded capture to last7.
+            bigP=[[P[i][j] if i<8 and j<8 else 0 for j in range(64)] for i in range(64)]
+            G=eye(64)
+            for i in range(8):G[i][:8]=Ti[i]
+            for i in range(7):G[i],G[57+i]=G[57+i],G[i]
+            Q=mm(mm(G,bigP,M),inv(G,M),M)
+            want=[[int(i==j and i>=57) for j in range(64)] for i in range(64)]
+            need(Q==want,'actual GL64 class reaches shifted projector')
+            hashes.append(hashlib.sha256(canonical([capture,M,G])).hexdigest())
+    return dict(captures=9,local_ring_cases=len(hashes),conjugacy_digest=hashlib.sha256(canonical(hashes)).hexdigest())
+
+def lead(x):return next((i for i,a in enumerate(x) if a),len(x))
+def swap_outer(x,k):
+    y=list(x);N=len(x)
+    for i in range(k):y[i],y[N-1-i]=y[N-1-i],y[i]
+    return tuple(y)
+def closure(seeds,M):
+    seen=set(seeds);queue=deque(seeds)
+    while queue:
+        x=queue.popleft()
+        for i in range(len(x)):
+            for j in range(i):
+                if not x[j]:continue
+                y=list(x);y[i]=(y[i]+x[j])%M;y=tuple(y)
+                if y not in seen:seen.add(y);queue.append(y)
+    return seen
+
+def orbit_checks():
+    rows=[]
+    for m,M in [(2,3),(2,9),(3,3)]:
+        x=tuple(int(i==m) for i in range(2*m));y=tuple(int(i==m-1) for i in range(2*m))
+        pre=closure([x],M)
+        for k in range(1,m):
+            one=closure([swap_outer(z,k) for z in pre],M)|pre
+            need(y not in one,'one-child orbit misses required input')
+            need(all(lead(z)<k or lead(z)==m for z in one),'whole reachable support leading flag')
+            need(all(lead(tuple(a*t%M for a in y))==m-1 and tuple(a*t%M for a in y) not in one
+                     for t in range(1,M) if __import__('math').gcd(t,M)==1),'every unit ancestor weight')
+            two=closure([swap_outer(z,k) for z in one],M)|one
+            need(y in two,'extra-child premise control exposes missing dependency')
+            need(swap_outer(x,m)==y,'full-width premise control')
+            # High-lift nonunit off-leading values are included in complete closures.
+            rows.append(dict(m=m,modulus=M,k=k,pre=len(pre),one=len(one),two=len(two),missing=list(y)))
+    return rows
+
+def payload_control():
+    # A concrete nonlinear dirty echo is tested on the complete 3^4 address cube.
+    M=3;N=4;xs=list(itertools.product(range(M),repeat=N));idx={x:i for i,x in enumerate(xs)}
+    x=(0,0,1,0);y=(0,2,0,0);j=idx[x]
+    # Q=diag(0,1), H=diag(1,2), reflected physical tail.
+    target=lambda z:(z[0],2*z[2]%M,pow(2,-1,M)*z[1]%M,z[3])
+    need(target(x)==y and target(target(x))==x,'weighted target pullback orientation')
+    zero=[0]*len(xs);delta=[int(z==y) for z in xs]
+    helpers=[[((i*17+seed*13)%11)<5 for i in range(len(xs))] for seed in range(3)]
+    def run(source):
+        source=source[:];saved=source[:];h=[a[:] for a in helpers];private=source[:]
+        def pull(f):
+            nonlocal private
+            private=[private[idx[f(z)]] for z in xs]
+        # Pointwise nonlinear Boolean gates: helper bits are read-only and dirty.
+        private=[a ^ (b and c) for a,b,c in zip(private,h[0],h[1])]
+        pull(lambda z:(z[0],(z[1]+2*z[0])%M,z[2],(z[3]+z[2])%M))
+        pull(lambda z:swap_outer(z,1))
+        private=[a ^ b for a,b in zip(private,h[2])]
+        pull(lambda z:(z[0],z[1],(z[2]+z[0])%M,(z[3]+2*z[1])%M))
+        need(source==saved and h==helpers,'borrowed source and arbitrary helpers unchanged')
+        return private
+    a,b=run(zero),run(delta)
+    need(a[j]==b[j],'one-child dirty Boolean echo cannot distinguish witness source bit')
+    need(zero[idx[y]]!=delta[idx[y]],'required weighted COPY does distinguish it')
+    need(delta[idx[y]]!=delta[idx[(0,1,0,0)]],'dropped ancestor unit changes target bit')
+    return dict(addresses=len(xs),dirty_helpers=3,output_address=list(x),required_input=list(y),observed_difference=0,required_difference=1)
+
+def budget(word):
+    e=json.loads((HERE/'inputs/r437-EXPECTED.json').read_text())
+    cat=json.loads((HERE/'inputs/pr3967-CATALOGUE.json').read_text())
+    profile={int(k):v for k,v in e['profile'].items()};W=e['live'];m=64
+    copies=len(word['ret'])*9*32
+    need(copies==cat['literal_opcode_counts']['2']==cat['literal_opcode_counts']['3']==2592,'source-bound COPY and disposal census')
+    mass=sum(r*n for r,n in profile.items());need(mass==e['rank_mass']==1808352,'exact source rank mass')
+    need(W*m-mass==e['deficit']==10272,'exact source deficit')
+    newmass=mass+copies*7;need(newmass>m*W,'fixed two-rank7 optimistic gate fails')
+    c=F(e['BIT_c']);a=F(17,10000)
+    gaps=[c-a*(1-a),c-a*(1-a*a)]
+    need(all(g>0 for g in gaps),'both available upper-tariff gaps positive')
+    return dict(profile=e['profile'],W=W,copies=copies,readers_each=63,old_mass=mass,new_mass=newmass,
+        deficit=m*W-mass,extra_mass=copies*7,excess=newmass-m*W,moment_zero=str(F(newmass,m*W)),
+        adaptive_fraction_zero_necessary=str(F(m*W-mass,copies*7)),tariff_gaps=list(map(str,gaps)),
+        scope='fixed two rank7 children per COPY only; not an adaptive or mixed-width impossibility')
+
+def evaluate():
+    w=authenticate()
+    return dict(source=source_conjugacies(w),orbits=orbit_checks(),payload=payload_control(),budget=budget(w),new_kappa=False)
+
+def main():
+    got=evaluate();want=json.loads((HERE/'RESULT.json').read_text())
+    need(got==want,'committed exact result differs')
+    print('PASS_COPY_ECHO_BUDGET capture_conjugacies=36 one_child_orbits=4 dirty_boolean_payload=81 fixed_two_child_excess=7872 no_new_kappa')
+if __name__=='__main__':main()