diff --git a/research/clean-checkpoint/CLAIM.txt b/research/clean-checkpoint/CLAIM.txt
new file mode 100644
index 0000000000..231890181c
--- /dev/null
+++ b/research/clean-checkpoint/CLAIM.txt
@@ -0,0 +1,16 @@
+Track: recursive interfaces; OS-random assignment seed 54.
+Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60.
+Scope: one explicitly clean length-L checkpoint at the root of PR628's dirty product-shear recursion, removing the final right-subtree cancellation and pricing the extra clean storage and both restoration passes. Investigate the optimal root split with unchanged balanced dirty subtrees, n=2..256; seed-selected demonstration n=54.
+Method: exact integer/noncommutative execution, complete leaf/resource recurrences, wrong-dirt and omitted-restoration controls. No kappa or rounded-accuracy improvement claim.
+Deliverable: standalone research note, executable checker and deterministic certificate.
+Candidates: (1) this clean checkpoint, (2) weighted dirty split tree, (3) stride-boundary fusion. Pick (1) because #663 audits the all-dirty dispatcher, #665 concerns spectral hosting, and neither supplies this changed-storage interface.
+Coordination: the open-issue query found only unpinned #650 (Discussions request), with no comments; no pinned sprint coordination issue was available. Claim recorded locally. Reviewed recent open PR metadata through #668 plus targeted clean/dirty, checkpoint/commutator and restoration/recursive searches.
+Expected finish: 2026-10-13 12:07 Asia/Shanghai (04:07 UTC). Last 10 minutes reserved for verification/packaging.
+
+RESULT — completed 2026-10-13, final verification/packaging phase.
+One clean-root adapter, exact optimal-split proof, full work/storage recurrence, 32,640 split checks, 498 noncommutative cases and 216 rounded dense comparisons. Generic n54: 3,136 -> 2,112 leaves; one clean plus five dirty arrays replaces six dirty arrays. No new kappa. Result: README.md; full proof: PROOF.md; certificate: EXPECTED.json.
+Fresh overlap review through #675 found concurrent #673 (head 0e35f56cab3cae1eb0a99447135569d687c5695a), whose distinct multiway echo is credited. Its phase-alias finding was incorporated as a root guard with five alias controls and a removed-guard rejection test; its original patch is pinned. This result retains balanced dirty subtrees and the same total scratch-array count, a different storage tradeoff.
+Verification: both local Python versions, isolated optimized execution, standalone relocation, four package corruption controls and selected-record integrity pass. Supported-Python root make verify passed seven historical groups then stopped at gen4 on missing SymPy; no full-suite pass claimed. Exact details: VALIDATION.md.
+Initial local handoff: local branch research/seed54-clean-checkpoint, clean patch plus standalone package; GitHub submission unavailable because the connected account has no push permission or existing accessible fork and the initially checked in-app browser was signed out. No coordination post or PR was created; no existing PR was changed. Selected frontier and upstream/ remain unchanged.
+
+Publication preparation update — 2026-10-13. A user-authorized continuation found the Chrome session signed in as FanPUYen. Refreshed recent PR metadata through #709 and the targeted clean-checkpoint search (no matches). PR689 by Dillon Amburgey refines balanced ternary placement in PR673; this result instead changes the root storage contract while retaining balanced binary dirty children. No PR689 code is consumed. Publication status is tracked in the external handoff record.
diff --git a/research/clean-checkpoint/EXPECTED.json b/research/clean-checkpoint/EXPECTED.json
new file mode 100644
index 0000000000..517dc8170c
--- /dev/null
+++ b/research/clean-checkpoint/EXPECTED.json
@@ -0,0 +1,9259 @@
+{
+  "assignment_seed": 54,
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "controls": [
+    "dirty",
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+    "target_alias",
+    "shortage",
+    "phase_alias_0",
+    "phase_alias_1",
+    "phase_alias_2",
+    "phase_alias_3",
+    "phase_alias_4",
+    "omitted_checkpoint_restoration",
+    "wrong_product_order",
+    "wrong_inverse_chronology",
+    "dirty_checkpoint_leaks_to_target",
+    "unpaid_inverse_count"
+  ],
+  "noncommutative_cases": [
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+      "commit": "0768f97d06c623b8211e651543c66135796b62bf",
+      "git_blob": "22fbeb8ef0a28dd6b3feea718492091319ceb96e",
+      "path": "vendor/pr628-NOTICE.md",
+      "sha256": "abeacf1ad1491d5797cfcf1f3832799bbf3b24b99597229d7bc884059a51ec68",
+      "source_path": "research/prime-power-cyclic-scan/NOTICE.md"
+    },
+    {
+      "commit": "e0e546fd01b396478c68b6b6c5a16d87515983ba",
+      "git_blob": "aeed8aeb2667d684786b72617df359f3e7941e96",
+      "path": "vendor/pr663-README.md",
+      "sha256": "0fed6e3a82c7c158614c696504d25b5693a0e8b6fe24a2140ced4e8cae2f7582",
+      "source_path": "research/dirty-commutator-audit/README.md"
+    },
+    {
+      "commit": "e0e546fd01b396478c68b6b6c5a16d87515983ba",
+      "git_blob": "f70d7122899b217709597a655443b05ef3c5a338",
+      "path": "vendor/pr663-NOTICE.md",
+      "sha256": "39c6f175693297f1e0b24264747aec8dbee09833cb52c585142a7920d80e9d52",
+      "source_path": "research/dirty-commutator-audit/NOTICE.md"
+    },
+    {
+      "commit": "0768f97d06c623b8211e651543c66135796b62bf",
+      "git_blob": "d620e493596f12578c1ce47f216c44c79b65de08",
+      "path": "vendor/pr628-scan.py",
+      "sha256": "1a9c01ca1a29a9be1e756858161202621db29d5d332a4266b4017a9502a50d7e",
+      "source_path": "research/prime-power-cyclic-scan/scan.py"
+    },
+    {
+      "commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+      "git_blob": "261eeb9e9f8b2b4b0d119366dda99c6fd7d35c64",
+      "path": "LICENSE",
+      "sha256": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+      "source_path": "LICENSE"
+    },
+    {
+      "commit": "0e35f56cab3cae1eb0a99447135569d687c5695a",
+      "git_blob": "6323498dfb2dfbb7c2252f4fa6fa6a505493651b",
+      "path": "vendor/pr673-phase-alias-guard.patch",
+      "sha256": "b752822b2c032eaa4158766922c572c51c67f955e899570df6798c6906f27d9e",
+      "source_path": "research/cyclic-dirty-commutator-audit/phase-alias-guard.patch"
+    }
+  ],
+  "status": "PASS_CLEAN_CHECKPOINT"
+}
diff --git a/research/clean-checkpoint/LICENSE b/research/clean-checkpoint/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/clean-checkpoint/LICENSE
@@ -0,0 +1,201 @@
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diff --git a/research/clean-checkpoint/NOTICE.md b/research/clean-checkpoint/NOTICE.md
new file mode 100644
index 0000000000..8c41f77fac
--- /dev/null
+++ b/research/clean-checkpoint/NOTICE.md
@@ -0,0 +1,34 @@
+# Attribution and disclosure
+
+Prepared for GitHub contributor **FanPUYen**, with substantial **OpenAI Codex**
+assistance in mathematical derivation, implementation, verification and writing.
+Separate Codex agents reviewed the mathematics and checker and built the rounded
+evidence. These are automated reviews, not independent human peer review.
+
+Chafik Boukhalfa / **chafreaky** contributed PR628's cyclic rounded leaves,
+dirty product-shear dispatcher, restoration proof and precision/cost estimates.
+Its original notice is retained verbatim in `vendor/pr628-NOTICE.md`.
+**@0xkilgore** and OpenAI Codex contributed PR663's exact balanced count and
+independent nonlinear-restoration audit; its notice is retained verbatim in
+`vendor/pr663-NOTICE.md`. That concurrent result is a credited dependency,
+not displaced by the clean-storage tradeoff here.
+
+**@androolloyd** and OpenAI Codex contributed concurrent PR673's phase-array
+alias admission finding and repair. The adapter adopts the corresponding
+identity exclusion; its original patch is retained and pinned. PR673's distinct
+multiway echo schedule is credited as concurrent work, not incorporated or
+independently re-verified here.
+
+PR628 credits **Ali Romman** for PR621's prime-power circle bound, OpenAI's
+exact-DFT manuscript, classical q-Pochhammer/Newton identities and the cited
+factorization, elementary-function and recursive-filtering literature.
+Those credits are preserved. This contribution claims no literature priority
+for compute/use/uncompute or reversible checkpointing.
+
+The selected #626/#543 construction and its Jeremy Kothe, Chafik Boukhalfa,
+eumemic, DreamingOfClouds, Rohan Arun and earlier contributor lineage provide
+the repository context. They are not claimed as implementations of this new
+interface, and their conditional frontier is not modified.
+
+New code and writing use the repository's **Apache-2.0** license. All vendored
+files are unchanged, with their source-specific notices and exact pins retained.
diff --git a/research/clean-checkpoint/PROOF.md b/research/clean-checkpoint/PROOF.md
new file mode 100644
index 0000000000..363edd94ac
--- /dev/null
+++ b/research/clean-checkpoint/PROOF.md
@@ -0,0 +1,318 @@
+# One paid clean checkpoint at the root
+
+This note changes only the root interface of the balanced dirty product-shear
+recursion in [PR628's pinned proof](vendor/pr628-DIRTY-PROOF.md) and
+[dispatcher](vendor/pr628-dirty.py). It uses the exact dirty recurrence and
+restoration argument recorded independently in [PR663](vendor/pr663-README.md).
+The new resource is one explicitly supplied zero-filled length-`L` array.
+All subtrees retain the balanced dirty recursion. Optimality below concerns
+the root split in this family, with unit leaf-call prices.
+
+## Interface and identity
+
+Arrays belong to an abelian group, such as complex dyadic arrays under exact
+addition. A leaf adds a deterministic function of its source to its target,
+restores its source, and has a target-independent increment. It is also
+zero-preserving: its increment at source zero is exactly zero. Its negative
+call implements its exact inverse. Coefficients, phases, rounding rules and
+other read-only parameters remain fixed between paired calls. All source,
+target, checkpoint and borrowed arrays are distinct; no ancestor source or
+target may serve as deeper scratch.
+The phase array is also distinct from every mutable array. This explicit root
+guard follows @androolloyd's concurrent PR673 admission finding: initial unit
+modulus alone does not imply read-only phase access when Python objects alias.
+The pinned original repair is retained in `vendor/pr673-phase-alias-guard.patch`.
+
+For a dirty child, write `B(x;t)` for its increment, where `t` contains the
+initial deeper scratch. The usual parent with checkpoint dirt `z0` executes
+
+```text
+z += B(x;t)
+y += A(z;t)
+z -= B(x;t)
+y -= A(z;t)
+```
+
+Inductively each child restores its source and deeper scratch. The two `B`
+calls therefore see identical inputs, restoring `z` exactly. The increment is
+
+\[
+ A(z_0+B(x;t);t)-A(z_0;t).
+\]
+
+This proves restoration and target independence without assuming linearity.
+It also proves zero preservation inductively: `B(0;t)=0` makes the displayed
+difference zero for every `z0,t`. Thus arbitrary deeper dirt is allowed even
+though the increment can depend on that dirt.
+
+Now supply `z0=0` and omit the last call. The remaining word is
+
+```text
+U:     z += B(x;t)
+V:     y += A(z;t)
+U^-1:  z -= B(x;t)
+```
+
+It restores `x,z,t` and adds `A(B(x;t);t)` to `y`. The omitted call would have
+source zero; since `A(0;t)=0`, it leaves every data array unchanged. Consequently
+the three-call and four-call words **at the same root split** have identical
+final data arrays, including bitwise equality on a fixed dyadic grid. The
+negative word is the reverse sequence of inverse calls, chronologically
+`U, V^-1, U^-1`; each child's inverse must itself reverse its instruction word.
+This yields exact whole-word inversion on the allowed zero-checkpoint states.
+
+PR628's rounded leaf has zero increment at source zero: every causal state,
+closure numerator, closure seed and final multiplied value is then zero.
+Its stable denominator remains governed by the inherited leaf assumptions.
+This observation does not independently re-prove its nonzero-source accuracy.
+
+For linear ideal leaves the increment is the ordered product of all factors,
+with the first factor in the list applied first. No commutation is needed.
+Changing the root split can change the nonlinear rounded increment; it preserves
+the ideal ordered product, not necessarily the rounded output of the original
+balanced-root program.
+
+Cleanliness is essential. For identity children, `x=0,z0=1`, the three-call
+word increments `y` by one while restoring `z0=1`; the intended product is zero.
+Supplying a zero array is a stronger storage assumption than borrowing arbitrary
+dirt. Neither the identity nor this implementation erases borrowed information.
+
+## Exact counts and the optimal root split
+
+Count every positive and inverse leaf invocation, including invocations whose
+net effect happens to be zero. The balanced dirty count is
+
+\[
+ D(1)=1,\qquad D(m)=2D(\lfloor m/2\rfloor)+2D(\lceil m/2\rceil).
+\]
+
+For `p=2^floor(log2(m))`, PR663's formula is
+
+\[
+ D(m)=3pm-2p^2.
+\]
+
+Indeed the ordinary balanced factor tree has `2p-m` leaves at depth `log2(p)`
+and `2m-2p` at the next depth; each descent doubles execution multiplicity.
+For a root split `k+(n-k)`, with `1<=k<n`, the clean count is therefore
+
+\[
+ F_n(k)=2D(k)+D(n-k).
+\]
+
+**Theorem.** For `n>=3`, let `q` be the unique power of two satisfying
+`3q<=n<6q`. All minimizing splits are the integers in
+
+\[
+ \boxed{\max(q,n-4q)\le k\le\min(2q,n-2q)},
+\]
+
+and their common count is
+
+\[
+ \boxed{N(n)=6q(n-2q)}.
+\]
+
+For `n=2`, the sole split is `k=1` and `N(2)=3`.
+
+To prove the theorem, the count formula gives
+
+\[
+ D(m+1)-D(m)=3\,2^{\lfloor\log_2m\rfloor}.
+\]
+
+Hence `F_n(k+1)-F_n(k)` is nondecreasing in `k`: it is twice this slope at
+`k`, minus the slope at `n-k-1`. The objective is discretely convex.
+On the stated interval, `q<=k<=2q` and `2q<=n-k<=4q`, so
+
+\[
+ 2D(k)+D(n-k)
+ =2(3qk-2q^2)+(6q(n-k)-8q^2)=6q(n-2q).
+\]
+
+The formula for `D` is valid at these closed dyadic endpoints. If the interval
+has a left neighbor, its slope into the interval is strictly negative: either
+`k` has just reached `q`, or `n-k` has just dropped to `4q`. If it has a right
+neighbor, the outgoing slope is strictly positive: either `k` has reached
+`2q`, or `n-k` has dropped to `2q`. Convexity excludes every other minimizer.
+
+Let `k*=min(2q,n-2q)`, the largest minimizing split. All the minimizing splits
+have `k<=n/2`, so this choice minimizes the largest child size. With
+`h(m)=ceil(log2(m))` and `h(1)=0`, scratch consists of
+
+\[
+ 1\text{ clean array}+h(\max(k,n-k))\text{ dirty arrays}.
+\]
+
+The children run sequentially and restore the same dirty pool before reuse.
+For `3q<=n<=4q`, `n-k*=2q`, so the total is `1+log2(2q)=h(n)`.
+For `4q<n<6q`, `2q<n-k*=n-2q<4q`, so the total is again `h(n)`.
+The `n=2` case has one clean array and no dirty arrays. Thus the chosen optimum
+replaces one dirty array with a clean array without increasing the total array
+count. Some other cost-optimal splits require one more total array; the tie
+break matters.
+
+The relative count has the uniform bounds
+
+\[
+ \frac35\le\frac{N(n)}{D(n)}\le\frac34\qquad(n\ge2).
+\]
+
+For `3q<=n<=4q`, compare `N=6qn-12q^2` with `D=6qn-8q^2`; their ratio
+increases from `3/5` to `3/4`. For `4q<=n<6q`, use `D=12qn-32q^2`;
+the ratio decreases from `3/4` toward `3/5`. Equality at the lower bound
+occurs at `n=3*2^j`, and at the upper bound at powers of two, including `n=2`.
+These are leaf-call savings, not automatically wall-clock or precision savings
+by the same percentage.
+
+At the seed-selected generic case `n=54`, `q=16` and all optimal splits have
+`16<=k<=22`. Choosing `22+32` gives `D(22)=544`, `D(32)=1024`, and
+
+| Root and subtree policy | Leaf calls | Clean arrays | Dirty arrays |
+| --- | ---: | ---: | ---: |
+| Balanced dirty, `27+27` | 3136 | 0 | 6 |
+| Clean root, balanced `27+27` | 2352 | 1 | 5 |
+| Clean root, optimal `22+32` | 2112 | 1 | 5 |
+
+The final row saves 1024 calls against the dirty baseline, or `16/49`
+(approximately 32.653%), and another 240 against the balanced clean root.
+This is a generic factor-count example, not a claimed prime-power kernel at
+radix 55.
+
+## Charged work and storage
+
+The unit recurrence alone is not a complete machine cost. For each factor `j`,
+let `t_j` bound the cost of **either signed leaf** at the selected precision
+and trajectory magnitude. It must include the causal, output and restoration
+passes; source/target grid scans; coefficient stability validation; scalar
+arithmetic; and every phase/coefficient access performed inside that leaf.
+The inherited prototype's three arithmetic passes give `3L` scalar steps per
+leaf, but its additional validation scans must also be charged.
+
+For a contiguous factor list `I`, define the weighted leaf charge
+
+\[
+ T_D([j])=t_j,\qquad
+ T_D(I)=2T_D(I_{\rm left})+2T_D(I_{\rm right}),\qquad
+ T_C(I,k)=2T_D(I[:k])+T_D(I[k:]).
+\]
+
+This counts both executions of the left subtree, including uncomputation.
+With common leaf price `t`, it becomes `T_C=N t`. For unequal `t_j`, the
+displayed recurrence still charges the work, but the unit-count split theorem
+does not assert a weighted-time optimum. If signed prices differ, one may use
+their maximum for `t_j`, or sum the actual signed invocation prices instead.
+
+Dispatch costs are separate. Let `g(I,h)` charge an invocation's array-alias
+and length checks, its own full unit-phase scan, factor slicing, descriptors,
+counters and recursive control; let `g_C` charge the clean adapter's analogous
+work apart from the separately listed zero check. The complete recursive
+upper charge can be written without suppressing these costs:
+
+\[
+\begin{aligned}
+ R_D([j],h)&=t_j+g([j],h),\\
+ R_D(I,h)&=2R_D(I_{\rm left},h-1)
+          +2R_D(I_{\rm right},h-1)+g(I,h),\\
+ R_C(I,k,h)&=2R_D(I[:k],h)+R_D(I[k:],h)+g_C.
+\end{aligned}
+\]
+
+Here `h` is the supplied dirty-pool depth; leaves may receive unused remaining
+arrays. With `N` leaves there are exactly `4N/3` recursive invocations including
+the clean root: that root has three children and every other internal node has
+four. In particular, repeated dispatch phase scans cost `O(NL)` phase-validation
+operations rather than a single uncharged scan. A conservative bound for the
+prototype's descriptor, list and controller operations is `O(N log n)` word
+operations; this is additional to arithmetic, phase access and data movement.
+
+The full charge is
+
+\[
+ P_{\rm factors}+P_{\rm phases}+Z_{\rm provision}+Z_{\rm verify}
+ +R_C+R_{\rm routing},
+\]
+
+where preparation, enclosure certification and phase generation/storage retain
+their supplier costs. `Z_provision` pays for obtaining a genuinely clean length-`L`
+array; if ordinary allocation and explicit initialization are available, this
+includes `L` word writes. The adapter's zero verification is another length-`L`
+scan. Such writes do not establish reversible erasure of arbitrary borrowed
+data, and a model lacking clean-array provisioning does not meet this contract.
+Routing includes actual phase/source/target access and any charged stride or
+layout movement not already counted inside the leaf or preparation price; count
+each cost once, and assume no free global phase table or free routing.
+
+Besides the input and target, mutable data-array storage is `L` clean words
+plus `L*h(max(k,n-k))` dirty words. At the selected optimum this has the same
+total array count as the dirty baseline, but a stronger cleanliness requirement.
+One also pays the inherited scalar arithmetic workspace (including the chosen
+multiplication routine's workspace), indexing and recursive controller storage,
+and read-only factor/phase descriptions. A materialized length-`L` phase table,
+as accepted by the prototype, is additional read-only storage. Its allocation
+and preparation do not disappear from the cost by calling it a parameter.
+Factor lists can be described by intervals; the Python prototype instead copies
+lists and retains ordinary recursion frames, whose storage and work remain paid.
+An interval dispatcher uses `O(log n)` frames with `O(log n+log L)` indexing
+bits per frame. With the minimal supplied dirty pool, the prototype additionally
+retains `O(n+log^2 n)` list references along the active stack. Supplying extra
+unused dirty arrays increases both their allocated storage and the array-check
+costs; the stated array counts use the minimal pool. Precision-`w` scalar words
+must also include the integer magnitude bits allowed by the inherited trajectory
+bound, not just `w` fractional bits.
+
+## Conditional pathwise execution-error bound
+
+Retain the inherited estimates for both signs of every leaf: the ideal linear
+map has norm at most `K=r/beta`, and the local rounding error is at most
+`a*u*(B+1)` when the source norm is at most `B`, where
+
+\[
+ a=13(r+1)/\beta^3,\quad u=2^{-w},\quad G=2+K.
+\]
+
+Let `D0` bound all initial array norms, including the zero checkpoint, and let
+`M_j` be the largest norm after `j` completed leaves. Assuming `a*u<=1`,
+
+\[
+ M_{j+1}\le(1+K)M_j+au(M_j+1),\qquad
+ M_j+1\le G^j(D_0+1).
+\]
+
+In the second inequality `M_j+1` means one plus `M_j`. Compare the complete
+computed instruction word with the corresponding ideal word from the same
+initial arrays. If `e_j` is their completed-leaf state error, then
+
+\[
+ e_{j+1}\le(1+K)e_j+auG^j(D_0+1),\qquad
+ e_N\le Na(D_0+1)uG^{N-1}.
+\]
+
+This is a pathwise estimate, so it applies to the changed root word with its
+**actual** `N=2D(k)+D(n-k)`. The ideal clean word computes the ordered product
+and cancels all borrowed initial dirt. A sufficient execution precision is
+
+\[
+ w\ge p+1+\left\lceil\log_2(Na(D_0+1))\right\rceil
+       +(N-1)\left\lceil\log_2G\right\rceil,
+\]
+
+together with `a*u<=1` and `u<=beta^2/2`. It yields execution error at most
+`2^(-p-1)`. Coefficient/phase preparation must independently supply the other
+half of the error budget, scaled to the original source norm as in PR628.
+No independence of rounded output from deeper dirt is asserted. Reduced leaf
+count permits substitution of a smaller `N` in this inherited sufficient bound;
+it does not change its asymptotic precision order or independently establish
+the supplier's leaf-accuracy assumptions or fast-arithmetic implementation.
+
+## Scope of the result
+
+This is an algebraic root-interface result with explicit storage and cost
+obligations, supported by a written all-`n` argument and finite executable
+checks. The requested `n=2..256` enumeration is bounded verification of the
+split theorem, not an unrestricted search. Mathematical review here was by
+another Codex agent; no human review or formal proof-assistant certification
+is claimed. There is no new kappa, common-label Clifford realization,
+bit-reversible arithmetic compiler, complete multiplication reduction, or
+all-size fixed-tape theorem. The clean-array supply, physical routing and
+inherited analytic interfaces remain explicit obligations.
diff --git a/research/clean-checkpoint/README.md b/research/clean-checkpoint/README.md
new file mode 100644
index 0000000000..5ea5a92dd8
--- /dev/null
+++ b/research/clean-checkpoint/README.md
@@ -0,0 +1,114 @@
+# Research note: a paid clean root checkpoint
+
+One genuinely clean array removes the last right-subtree cancellation from
+PR628's dirty product-shear recursion. An exact root-split formula then uses
+**25–40% fewer primitive leaf calls**, while retaining the same total number
+of scratch arrays: one clean array replaces one arbitrary dirty array.
+Both child subtrees keep the existing balanced dirty implementation.
+
+For the seed-selected **54-factor** example, the split is **22 + 32**:
+**2,112 calls instead of 3,136**, with one clean and five dirty arrays instead
+of six dirty arrays. This generic example is not a prime-power radix-55 kernel.
+The neighboring 63-factor case is also checked algebraically, without claiming
+that its kernel preparation or full transform has been implemented here.
+
+This is a **research note and scoped interface construction; no new κ**.
+Clean storage must be provided and paid for. The adapter cannot erase borrowed
+dirt; a counterexample demonstrates why. Physical allocation, phase preparation,
+source access, routing, repeated validation scans and precision remain charged
+in [PROOF.md](PROOF.md). The selected construction is unchanged.
+
+## Reproduce
+
+Python 3.9+; standard library only; no downloads or optimizer needed:
+
+```sh
+python3 -B research/clean-checkpoint/verify.py
+```
+
+Expected output:
+
+```text
+PASS_CLEAN_CHECKPOINT: 32640 splits; 498 noncommutative cases; n=54: 3136 -> 2112 leaves, 1 clean + 5 dirty arrays
+```
+
+The command checks immutable predecessor bytes, recomputes the entire receipt
+and compares it with [EXPECTED.json](EXPECTED.json). `--write /path/result.json`
+instead writes a fresh receipt. Checks stay active under `python3 -I -O -B`.
+The package can be copied elsewhere and run without the repository or Git.
+
+## What is proved and what is checked
+
+For the inherited balanced dirty count `D`, a clean root at split `k` costs
+`2 D(k) + D(n-k)`. The first child is computed and uncomputed; the second child
+is used once. If `n >= 3` and `q` is the power of two with `3q <= n < 6q`,
+every cost-optimal split lies in
+
+```text
+max(q, n-4q) <= k <= min(2q, n-2q),
+N(n) = 6q(n-2q).
+```
+
+Choose the largest such `k` to minimize the largest child and avoid an extra
+scratch array. The proof covers every factor count; the finite checks corroborate
+it. The optimum is only over **one clean root and balanced all-dirty subtrees**,
+with unit leaf prices. Weighted leaf prices have their own complete recurrence;
+neither a globally optimal reversible schedule nor a runtime speedup is claimed.
+
+The bounded evidence consists of:
+
+- All 32,640 root splits for `n=2..256`, checking the optimal interval, exact
+  counts, space equality and relative count bounds.
+- 498 exact free noncommutative linear-operator cases: every split for `n=2..32`,
+  plus the selected splits at 54 and 63. All initial source, target and dirty
+  columns are tracked. Pinned dispatcher traces match an independent compiler;
+  restoration, factor order, inversion and weighted leaf bills are checked.
+- 216 executions using PR628's actual rounded leaf definitions, with 2–8
+  factors, lengths 1/2/5, both directions and two dirty initializations. Independent
+  dense rational solves check the target against the inherited actual-call-count
+  error bound. All 216 have nonzero rounding error. Exact restoration, standalone
+  negative execution, target independence, same-split four-call equality and
+  `3L` arithmetic-pass steps per leaf are checked.
+- Fourteen algebraic/interface controls plus two rounded controls: dirty checkpoint,
+  source/target aliasing, insufficient scratch, omitted restoration, wrong factor
+  order, incorrect inverse chronology, dirt leakage, unpriced uncomputation and
+  five phase-array aliases (source, target, clean checkpoint, each dirty level).
+  Package/source corruption controls are recorded in [VALIDATION.md](VALIDATION.md).
+
+The shorter word and the four-call word have identical rounded output **at the
+same root split**. Changing the split need not preserve rounded bits; the ideal
+ordered product and the conditional accuracy bound are the common comparison.
+Finite replay is not formal verification of the full multiplication theorem.
+
+## Provenance and coordination
+
+Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+The selected record at that base is #626/#543 with
+`κ = 51110230636731/31250000000000000` (0.001635527380375392).
+Its record, actual assembly, source manifest, construction and maintainer review
+were followed before choosing this distinct interface task.
+
+Direct dependencies:
+
+- Chafik Boukhalfa / chafreaky's [PR628](https://github.com/CrocSwap/integer-mult-bounds/pull/628)
+  at `0768f97d06c623b8211e651543c66135796b62bf`: dirty rounded leaves,
+  balanced dispatcher and conditional error/cost interface.
+- @0xkilgore's concurrent [PR663](https://github.com/CrocSwap/integer-mult-bounds/pull/663)
+  at `e0e546fd01b396478c68b6b6c5a16d87515983ba`: exact all-dirty call count
+  and restoration audit. This contribution changes the storage interface and
+  instruction word; it does not repeat that unchanged-dispatcher audit.
+- @androolloyd's concurrent [PR673](https://github.com/CrocSwap/integer-mult-bounds/pull/673)
+  at `0e35f56cab3cae1eb0a99447135569d687c5695a` was discovered during the final
+  overlap refresh. Its multiway dirty echo changes a different schedule and can
+  use more dirty arrays; no superiority over that interface is claimed here.
+  Its phase-alias finding motivated an explicit root admission guard, credited
+  and tested here; the original repair patch is pinned in `vendor/` as a reference.
+  Its broader echo and arithmetic evidence were not re-audited in this package.
+
+[SOURCE.json](SOURCE.json) records Git commits, paths, blob IDs and SHA-256
+for every vendored predecessor file. [NOTICE.md](NOTICE.md) retains attribution
+and substantial AI-assistance disclosure. No upstream or selected-result file
+is edited. [CLAIM.txt](CLAIM.txt) records track, OS-random seed 54, three candidate
+tasks, selection, bounded coverage and the local coordination claim. No pinned
+sprint issue was available. [VALIDATION.md](VALIDATION.md) separates completed
+checks from unverified repository-wide and platform obligations.
diff --git a/research/clean-checkpoint/SOURCE.json b/research/clean-checkpoint/SOURCE.json
new file mode 100644
index 0000000000..9e58fc48b5
--- /dev/null
+++ b/research/clean-checkpoint/SOURCE.json
@@ -0,0 +1,61 @@
+{
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "sources": [
+    {
+      "path": "vendor/pr628-dirty.py",
+      "source_path": "research/prime-power-cyclic-scan/dirty.py",
+      "commit": "0768f97d06c623b8211e651543c66135796b62bf",
+      "sha256": "09e7f41141594e6540828330d27be27c083ee6569660ae42c0cd519b09fc8d21",
+      "git_blob": "2b9471083706a37519a1af7598e012f5fecf2636"
+    },
+    {
+      "path": "vendor/pr628-DIRTY-PROOF.md",
+      "source_path": "research/prime-power-cyclic-scan/DIRTY-PROOF.md",
+      "commit": "0768f97d06c623b8211e651543c66135796b62bf",
+      "sha256": "bb7c2fdc1c4673d56531fd56fb8790c81a16813ac152c10f1afabc9efdd70d61",
+      "git_blob": "caf4aafc784a630d61ca7874a0bd19cbcb34e585"
+    },
+    {
+      "path": "vendor/pr628-NOTICE.md",
+      "source_path": "research/prime-power-cyclic-scan/NOTICE.md",
+      "commit": "0768f97d06c623b8211e651543c66135796b62bf",
+      "sha256": "abeacf1ad1491d5797cfcf1f3832799bbf3b24b99597229d7bc884059a51ec68",
+      "git_blob": "22fbeb8ef0a28dd6b3feea718492091319ceb96e"
+    },
+    {
+      "path": "vendor/pr663-README.md",
+      "source_path": "research/dirty-commutator-audit/README.md",
+      "commit": "e0e546fd01b396478c68b6b6c5a16d87515983ba",
+      "sha256": "0fed6e3a82c7c158614c696504d25b5693a0e8b6fe24a2140ced4e8cae2f7582",
+      "git_blob": "aeed8aeb2667d684786b72617df359f3e7941e96"
+    },
+    {
+      "path": "vendor/pr663-NOTICE.md",
+      "source_path": "research/dirty-commutator-audit/NOTICE.md",
+      "commit": "e0e546fd01b396478c68b6b6c5a16d87515983ba",
+      "sha256": "39c6f175693297f1e0b24264747aec8dbee09833cb52c585142a7920d80e9d52",
+      "git_blob": "f70d7122899b217709597a655443b05ef3c5a338"
+    },
+    {
+      "path": "vendor/pr628-scan.py",
+      "source_path": "research/prime-power-cyclic-scan/scan.py",
+      "commit": "0768f97d06c623b8211e651543c66135796b62bf",
+      "sha256": "1a9c01ca1a29a9be1e756858161202621db29d5d332a4266b4017a9502a50d7e",
+      "git_blob": "d620e493596f12578c1ce47f216c44c79b65de08"
+    },
+    {
+      "path": "LICENSE",
+      "source_path": "LICENSE",
+      "commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+      "sha256": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+      "git_blob": "261eeb9e9f8b2b4b0d119366dda99c6fd7d35c64"
+    },
+    {
+      "path": "vendor/pr673-phase-alias-guard.patch",
+      "source_path": "research/cyclic-dirty-commutator-audit/phase-alias-guard.patch",
+      "commit": "0e35f56cab3cae1eb0a99447135569d687c5695a",
+      "sha256": "b752822b2c032eaa4158766922c572c51c67f955e899570df6798c6906f27d9e",
+      "git_blob": "6323498dfb2dfbb7c2252f4fa6fa6a505493651b"
+    }
+  ]
+}
diff --git a/research/clean-checkpoint/VALIDATION.md b/research/clean-checkpoint/VALIDATION.md
new file mode 100644
index 0000000000..ab641ce253
--- /dev/null
+++ b/research/clean-checkpoint/VALIDATION.md
@@ -0,0 +1,88 @@
+# Validation record
+
+Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+Platform: macOS arm64. Focused package requires Python 3.9+ and only the
+standard library. Local interpreters are CPython 3.9.6 and 3.12.14.
+
+## Contribution checks
+
+The following commands pass and compare the complete deterministic receipt:
+
+```sh
+python3 -B research/clean-checkpoint/verify.py
+python3 -I -O -B research/clean-checkpoint/verify.py
+python3 -I -O -B research/clean-checkpoint/test_package.py
+```
+
+The exact package replay passes on both listed interpreters. Coverage:
+32,640 root splits; 498 noncommutative symbolic cases; 216 rounded executions
+with independent dense rational comparisons and nonzero rounding error in every
+case; 14 algebraic/interface rejection controls and two rounded controls.
+
+`test_package.py` copies the package into a path with spaces, without repository
+or Git dependencies, and verifies it under isolated optimized Python. It then
+requires failures for four corruptions: a changed pinned source, a changed
+expected call count, removal of the clean-checkpoint guard, and removal of the
+phase-array alias guard. These check actual subprocess exit status and diagnostic
+markers. The temporary copies are restored or deleted; source files stay intact.
+
+The first implementation review caught and fixed two checker defects before
+packaging: Python assertions in the rounded harness would disappear under `-O`,
+and a same-order sign-negation control at seven factors happened to invert the
+ideal odd-degree product. Explicit checks now remain active under optimization;
+the inverse-order control uses two factors where the mutation is nontrivial.
+These were checker-development findings, not counterexamples to the clean-root
+identity. A separate Codex reviewer subsequently passed the optimized package.
+
+Concurrent PR673 arrived during the final overlap check. Its phase-alias defect
+motivated a guard before any mutable operation. Five phase aliases are rejected,
+and removing the guard makes the package control fail. Its original patch and
+head are pinned; no broader PR673 replay or review is claimed.
+
+## Repository checks and limits
+
+`make selected-record-check` passes at the base, checking the selected κ,
+original source inventory and proof supplements. Only the new research package
+is changed; selected sources and `upstream/` are preserved. Staged whitespace,
+vendored Git-blob/SHA-256 bindings and clean-patch applicability are checked in
+the final handoff.
+
+The repository-wide `make verify` was launched in a separate detached checkout
+at the exact base, to avoid regenerating unrelated artifacts in the contribution.
+The initial system-Python 3.9 attempt passed the two opening arithmetic audits
+but failed in the historical copied-centers producer at `int.bit_count`; the root
+suite requires Python 3.11+. It was retried with the available Python 3.12.14.
+The supported-interpreter run passed `verify-community`, `verify-producers`,
+`verify-partial-gauge`, `verify-three-stage-cover`, `verify-paired-cube`,
+`verify-recycled-bit` and `verify-entrance-banks`, including their exact words,
+assembly checks and controls. It then stopped in `gen4-bank-verify` at
+`research/five-stage-gen4-banks/verify.py:52` with
+`ModuleNotFoundError: No module named 'sympy'` (make exit 2). No root-wide pass
+is claimed. Later root targets, including `frontier626-verify`, were not reached.
+No third-party dependency was installed merely to broaden this additive
+standard-library package's scope.
+
+The complete expected receipt has SHA-256
+`9c7b5aa11a2592547f2432f0d2a6c434b8ab160126c71c07c23a586f54d8bdbd`.
+All eight vendored/source-license files were independently compared to the
+exact Git objects in SOURCE.json, in addition to the offline checker bindings.
+
+The initial overlap review covered recent open PRs through #675. The
+user-authorized publication continuation refreshed metadata through #709 and
+repeated the targeted `clean checkpoint` search (no matches). PR689 by Dillon
+Amburgey optimizes balanced ternary placement in PR673's all-dirty echo; our
+result instead changes the root storage contract and retains balanced binary
+dirty children. No PR689 code is consumed. PR673 remains explicitly credited
+and distinguished in README.md. No pinned sprint coordination issue was
+available at the initial claim. Main still resolves to the exact base.
+
+The initial deliverable was a local patch and standalone package because the
+connected account lacked upstream push permission and the checked in-app
+browser was signed out. The user then authorized publication using Chrome,
+where the FanPUYen session is signed in. Final publication status belongs to
+the external handoff record, separate from the local verification evidence.
+
+No selected-supplier cold replay, remote CI, Linux replay, Lean proof or complete
+multiplication theorem is claimed for this additive interface package. The
+uniform rounded-leaf estimates and preparation/tape/hosting obligations retain
+the explicit limits stated in PROOF.md.
diff --git a/research/clean-checkpoint/checkpoint.py b/research/clean-checkpoint/checkpoint.py
new file mode 100644
index 0000000000..f188e573cf
--- /dev/null
+++ b/research/clean-checkpoint/checkpoint.py
@@ -0,0 +1,68 @@
+"""A paid clean root checkpoint over unchanged balanced dirty subtrees.
+
+Research adapter, not a supplier or tape compiler. Apache-2.0.
+Prepared for FanPUYen with substantial OpenAI Codex assistance.
+"""
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def dirty_count(n):
+    need(type(n) is int and n >= 1, 'positive integer factor count')
+    q = 1 << (n.bit_length() - 1)
+    return 3 * q * n - 2 * q * q
+
+
+def optimal_interval(n):
+    """All optimal root splits, with balanced dirty subtrees on each side."""
+    need(type(n) is int and n >= 2, 'at least two factors')
+    if n == 2:
+        return 1, 1
+    q = 1 << ((n // 3).bit_length() - 1)
+    return max(q, n - 4 * q), min(2 * q, n - 2 * q)
+
+
+def plan(n, split=None):
+    lo, hi = optimal_interval(n)
+    k = hi if split is None else split
+    need(type(k) is int and 0 < k < n, 'interior split')
+    return dict(factors=n, split=k, optimal_splits=[lo, hi],
+                leaf_calls=2 * dirty_count(k) + dirty_count(n - k),
+                balanced_dirty_leaf_calls=dirty_count(n),
+                clean_arrays=1,
+                dirty_arrays=(max(k, n - k) - 1).bit_length(),
+                balanced_dirty_arrays=(n - 1).bit_length())
+
+
+def compose_clean(compose, c, x, y, factors, clean, scratch, phases,
+                  beta, bits, d, split=None, sign=1, stats=None):
+    """Apply a three-call root with the supplied target-additive dispatcher.
+
+    Preconditions inherited from that dispatcher include immutable parameters,
+    exact inverse signs, source/scratch restoration and zero-preserving leaves.
+    Initial clean storage is an additional caller obligation, never an erasure
+    of borrowed data. The adapter checks all root array exclusions before work.
+    """
+    p = plan(len(factors), split)
+    need(sign in (-1, 1) and d in (-1, 1), 'orientation and sign')
+    arrays = [x, y, clean] + list(scratch)
+    need(len({id(a) for a in arrays}) == len(arrays), 'distinct root arrays')
+    # Credit PR673's concurrent admission finding: phases cannot alias an
+    # array mutated by the leaf, even if its initial entries are unit phases.
+    need(all(phases is not a for a in arrays), 'phase array must be distinct')
+    need(len(x) > 0 and all(len(a) == len(x) for a in arrays), 'array lengths')
+    need(len(phases) == len(x) and all(c.norm2(t) == 1 for t in phases),
+         'unit phase array')
+    need(len(scratch) >= p['dirty_arrays'], 'dirty depth')
+    need(all(v == c.ZERO for v in clean), 'checkpoint must initially be zero')
+    k = p['split']
+    operations = [(x, clean, factors[:k], 1),
+                  (clean, y, factors[k:], 1),
+                  (x, clean, factors[:k], -1)]
+    if sign == -1:
+        operations = [(a, b, fs, -s) for a, b, fs, s in reversed(operations)]
+    for a, b, fs, s in operations:
+        compose(a, b, fs, scratch, phases, beta, bits, d, s, stats)
diff --git a/research/clean-checkpoint/rounded_check.py b/research/clean-checkpoint/rounded_check.py
new file mode 100644
index 0000000000..13eb9d2bd2
--- /dev/null
+++ b/research/clean-checkpoint/rounded_check.py
@@ -0,0 +1,224 @@
+"""Finite rounded-leaf evidence for a clean root; no general accuracy theorem.
+
+Uses only selected definitions of the hash-pinned PR628 dirty prototype.
+The error envelope is inherited conditionally from its leaf estimates.
+Prepared for FanPUYen with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+import ast
+from fractions import Fraction as F
+import importlib.util
+import itertools
+import json
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+
+
+def need(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def load_file(name, path):
+    spec = importlib.util.spec_from_file_location(name, path)
+    module = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(module)
+    return module
+
+
+def implementation():
+    c = load_file('pinned_scan', HERE / 'vendor/pr628-scan.py')
+    path = HERE / 'vendor/pr628-dirty.py'
+    tree = ast.parse(path.read_text(), filename=str(path))
+    names = {'scaled', 'conj', 'ordered', 'lift', 'leaf', 'compose'}
+    definitions = [node for node in tree.body
+                   if isinstance(node, ast.FunctionDef) and node.name in names]
+    c.need({node.name for node in definitions} == names, 'vendored definitions')
+    namespace = dict(F=F, c=c, itertools=itertools)
+    exec(compile(ast.Module(body=definitions, type_ignores=[]), str(path), 'exec'),
+         namespace)
+    return c, namespace['compose']
+
+
+def dense_operator(c, x, factors, phases, direction):
+    """Gaussian elimination on each cyclic factor, independent of the lifts."""
+    out = x[:]
+    length = len(x)
+    for inside, b, scale in factors:
+        root = b if inside else c.div(c.ONE, b)
+        matrix = [[c.ZERO for _ in range(length)] + [out[i]]
+                  for i in range(length)]
+        for i in range(length):
+            matrix[i][i] = c.neg(root)
+            j = (i - direction) % length
+            matrix[i][j] = c.add(matrix[i][j], phases[i])
+        for col in range(length):
+            pivot = next(row for row in range(col, length)
+                         if matrix[row][col] != c.ZERO)
+            matrix[col], matrix[pivot] = matrix[pivot], matrix[col]
+            divisor = matrix[col][col]
+            matrix[col] = [c.div(v, divisor) for v in matrix[col]]
+            for row in range(length):
+                if row == col:
+                    continue
+                coefficient = matrix[row][col]
+                matrix[row] = [c.sub(v, c.mul(coefficient, w))
+                               for v, w in zip(matrix[row], matrix[col])]
+        out = [(row[-1][0] * scale, row[-1][1] * scale) for row in matrix]
+    return out
+
+
+def leaves(n):
+    return 1 if n == 1 else 2 * (leaves(n // 2) + leaves(n - n // 2))
+
+
+def ceil_log2(value):
+    """Nonnegative ceiling, computed with integers only."""
+    bits = max(0, value.numerator.bit_length() - value.denominator.bit_length())
+    return bits + ((value.denominator << bits) < value.numerator)
+
+
+def precision(calls, magnitude):
+    beta, r = F(1, 2), 3
+    a, growth = 13 * (r + 1) / beta ** 3, 2 + F(r) / beta
+    numerator = calls * a * (magnitude + 1) * growth ** (calls - 1)
+    bits = ceil_log2(max(a, 2 / beta ** 2, numerator * (1 << 32)))
+    bound = numerator / (1 << bits)
+    need(a / (1 << bits) <= 1 and F(1, 1 << bits) <= beta ** 2 / 2,
+         'chosen precision satisfies inherited leaf and closure hypotheses')
+    need(bound <= F(1, 1 << 32), 'chosen precision meets the error target')
+    return bits, bound
+
+
+def four_call(compose, x, y, factors, clean, scratch, phases, bits, d, k):
+    for source, target, fs, sign in (
+        (x, clean, factors[:k], 1), (clean, y, factors[k:], 1),
+        (x, clean, factors[:k], -1), (clean, y, factors[k:], -1),
+    ):
+        compose(source, target, fs, scratch, phases, F(1, 2), bits, d, sign)
+
+
+def controls(c, compose, adapter):
+    factors = [(True, (F(1, 4), F(0)), F(1))] * 2
+    x, y, dirty_checkpoint = [c.ONE], [c.ZERO], [c.ONE]
+    before = [v[:] for v in (x, y, dirty_checkpoint)]
+    try:
+        adapter(compose, c, x, y, factors, dirty_checkpoint, [], [c.ONE],
+                F(1, 2), 128, 1, split=1)
+    except ValueError as error:
+        rejected = str(error)
+    else:
+        raise AssertionError('nonzero checkpoint accepted')
+    need([x, y, dirty_checkpoint] == before,
+         'nonzero checkpoint rejection must precede mutation')
+    # Deliberately bypass the clean precondition: A(z + Bx) retains A(z).
+    compose(x, dirty_checkpoint, factors[:1], [], [c.ONE], F(1, 2), 128, 1)
+    compose(dirty_checkpoint, y, factors[1:], [], [c.ONE], F(1, 2), 128, 1)
+    compose(x, dirty_checkpoint, factors[:1], [], [c.ONE], F(1, 2), 128, 1, -1)
+    ideal = dense_operator(c, x, factors, [c.ONE], 1)
+    need(c.norm2(c.sub(y[0], ideal[0])) > 1 and dirty_checkpoint == [c.ONE],
+         'bypassing the clean precondition must exhibit contamination')
+    contamination = c.sub(y[0], ideal[0])
+    # Deliberately omit the third call: the checkpoint must fail restoration.
+    clean, target = [c.ZERO], [c.ZERO]
+    compose(x, clean, factors[:1], [], [c.ONE], F(1, 2), 128, 1)
+    compose(clean, target, factors[1:], [], [c.ONE], F(1, 2), 128, 1)
+    need(clean != [c.ZERO] and x == [c.ONE],
+         'omitting the restore call must leave a nonzero checkpoint')
+    return [dict(name='nonzero_checkpoint', rejected=rejected,
+                 unchanged_on_rejection=True,
+                 bypass_error=[str(v) for v in contamination],
+                 bypass_squared_error_greater_than_one=True),
+            dict(name='omitted_restore', detected_nonzero_checkpoint=True,
+                 checkpoint=[str(v) for v in clean[0]])]
+
+
+def run():
+    c, compose = implementation()
+    checkpoint = load_file('clean_checkpoint', HERE / 'checkpoint.py')
+    adapter = checkpoint.compose_clean
+    units = [c.ONE, (F(0), F(1)), (F(3, 5), F(4, 5)),
+             (F(-5, 13), F(12, 13))]
+    cases, nonzero_errors = [], 0
+    for n in range(2, 9):
+        factors = [(i % 2 == 0, (F(1, 4), F((-1) ** i, 8)),
+                    F(3) if i == 0 else F(1)) for i in range(n)]
+        for k in sorted({1, n // 2, n - 1}):
+            calls = 2 * leaves(k) + leaves(n - k)
+            depth = (max(k, n - k) - 1).bit_length()
+            for length, direction, dirt in itertools.product((1, 2, 5), (-1, 1), (0, 7)):
+                x0 = [(F(i % 5 - 2, 4), F(i % 3 - 1, 8)) for i in range(length)]
+                y0 = [(F(dirt + i, 4), F(dirt - i, 8)) for i in range(length)]
+                s0 = [[(F(dirt + 2 * i + j, 4), F(dirt - 3 * i - j, 8))
+                       for i in range(length)] for j in range(depth)]
+                phases = [units[(i * i + 3 * i + 1) % 4] for i in range(length)]
+                magnitude = max(abs(a) + abs(b) for v in [x0, y0] + s0 for a, b in v)
+                bits, bound = precision(calls, magnitude)
+                x, y, clean = x0[:], y0[:], [c.ZERO] * length
+                scratch = [v[:] for v in s0]
+                stats = dict(leaves=0, scalar_steps=0)
+                adapter(compose, c, x, y, factors, clean, scratch, phases,
+                        F(1, 2), bits, direction, split=k, stats=stats)
+                need(x == x0 and scratch == s0 and clean == [c.ZERO] * length,
+                     'positive word restores source, scratch and checkpoint')
+                need(stats == dict(leaves=calls, scalar_steps=3 * length * calls),
+                     'observed leaf and scalar-step counts match recurrence')
+                need(checkpoint.plan(n, k)['leaf_calls'] == calls,
+                     'planner leaf count matches independent recurrence')
+                net = [c.sub(v, w) for v, w in zip(y, y0)]
+                exact = dense_operator(c, x0, factors, phases, direction)
+                error2 = max(c.norm2(c.sub(v, w)) for v, w in zip(net, exact))
+                need(error2 <= bound ** 2, 'dense oracle error within inherited bound')
+                nonzero_errors += error2 > 0
+                output = y[:]
+                adapter(compose, c, x, y, factors, clean, scratch, phases,
+                        F(1, 2), bits, direction, split=k, sign=-1)
+                need(x == x0 and y == y0 and scratch == s0 and clean == [c.ZERO] * length,
+                     'negative word exactly inverts the positive word')
+                # From an arbitrary original target, the inverse has net -net.
+                adapter(compose, c, x, y, factors, clean, scratch, phases,
+                        F(1, 2), bits, direction, split=k, sign=-1)
+                need([c.sub(v, w) for v, w in zip(y0, y)] == net,
+                     'standalone negative word has the opposite target increment')
+                need(x == x0 and scratch == s0 and clean == [c.ZERO] * length,
+                     'standalone negative word restores all auxiliary arrays')
+                shifted = [c.add(v, (F(11), F(-3))) for v in y0]
+                shifted0 = shifted[:]
+                adapter(compose, c, x, shifted, factors, clean, scratch, phases,
+                        F(1, 2), bits, direction, split=k)
+                need([c.sub(v, w) for v, w in zip(shifted, shifted0)] == net,
+                     'target increment is independent of the initial target')
+                need(x == x0 and scratch == s0 and clean == [c.ZERO] * length,
+                     'shifted-target word restores all auxiliary arrays')
+                four = y0[:]
+                four_call(compose, x, four, factors, clean, scratch, phases,
+                          bits, direction, k)
+                need(four == output and x == x0 and scratch == s0,
+                     'same-split four-call word has identical result and restoration')
+                need(clean == [c.ZERO] * length, 'four-call word restores checkpoint')
+                cases.append(dict(factors=n, split=k, length=length,
+                                  direction=direction, dirt=dirt, leaf_calls=calls,
+                                  dirty_arrays=depth, clean_arrays=1, bits=bits,
+                                  initial_magnitude=str(magnitude),
+                                  inherited_error_bound=str(bound)))
+    need(nonzero_errors > 0, 'finite cases must exercise nonzero rounding error')
+    return dict(status='passed', cases=cases, case_count=len(cases),
+                nonzero_rounding_error_cases=nonzero_errors,
+                checks=['dense rational oracle within inherited bound',
+                        'exact source, dirty scratch and clean restoration',
+                        'exact inverse and standalone negative word',
+                        'target independence', 'same-split four-call identity',
+                        'leaf and scalar-step counts'],
+                controls=controls(c, compose, adapter),
+                error_envelope=dict(N='actual three-call leaf count',
+                                    a='13*(r+1)/beta^3', G='2+r/beta',
+                                    beta='1/2', r=3,
+                                    formula='N*a*(D+1)*2^(-bits)*G^(N-1)',
+                                    precision_target='2^-32'),
+                scope='Finite evidence with pinned PR628 rounded leaves; inherited '
+                      'leaf error estimates are assumed, not proved here. No supplier, '
+                      'bit-reversible tape compiler, or multiplication theorem.')
+
+
+if __name__ == '__main__':
+    print(json.dumps(run(), sort_keys=True, indent=2))
diff --git a/research/clean-checkpoint/test_package.py b/research/clean-checkpoint/test_package.py
new file mode 100644
index 0000000000..ded2492142
--- /dev/null
+++ b/research/clean-checkpoint/test_package.py
@@ -0,0 +1,60 @@
+#!/usr/bin/env python3
+"""Portable-package and optimized-mode corruption controls (standard library)."""
+import argparse
+import json
+from pathlib import Path
+import shutil
+import subprocess
+import sys
+import tempfile
+
+HERE = Path(__file__).resolve().parent
+
+
+def need(ok, why):
+    if not ok:
+        raise ValueError(why)
+
+
+def main():
+    parser = argparse.ArgumentParser()
+    parser.add_argument('--work', type=Path, help='existing parent for temporary copies')
+    args = parser.parse_args()
+    with tempfile.TemporaryDirectory(prefix='checkpoint-controls-', dir=args.work) as temp:
+        copied = Path(temp) / 'relocated package with spaces'
+        shutil.copytree(HERE, copied, ignore=shutil.ignore_patterns('__pycache__'))
+        def run(marker, success):
+            result = subprocess.run([sys.executable, '-I', '-O', '-B',
+                                     str(copied/'verify.py')],
+                                    cwd=temp, capture_output=True, text=True)
+            output = result.stdout+result.stderr
+            need((result.returncode == 0) == success and marker in output,
+                 'unexpected subprocess result: '+output)
+        run('PASS_CLEAN_CHECKPOINT', True)
+        source = copied/'vendor/pr628-dirty.py'
+        original = source.read_bytes()
+        source.write_bytes(original+b'\n# corrupted source\n')
+        run('source SHA256', False)
+        source.write_bytes(original)
+        expected = copied/'EXPECTED.json'
+        original_receipt = expected.read_bytes()
+        data = json.loads(original_receipt)
+        data['selected']['leaf_calls'] -= 1
+        expected.write_text(json.dumps(data))
+        run('complete certificate mismatch', False)
+        expected.write_bytes(original_receipt)
+        adapter = copied/'checkpoint.py'
+        text = adapter.read_text()
+        guard = "need(all(v == c.ZERO for v in clean), 'checkpoint must initially be zero')"
+        need(text.count(guard) == 1, 'unique clean guard mutation site')
+        adapter.write_text(text.replace(guard, 'pass  # deliberately removed clean guard'))
+        run('accepted interface control: dirty', False)
+        guard = "need(all(phases is not a for a in arrays), 'phase array must be distinct')"
+        need(text.count(guard) == 1, 'unique phase guard mutation site')
+        adapter.write_text(text.replace(guard, 'pass  # deliberately removed phase guard'))
+        run('accepted interface control: phase_alias_0', False)
+    print('PASS_PACKAGE_CONTROLS: relocation; source, receipt, clean-guard and phase-guard corruptions rejected under -I -O')
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/clean-checkpoint/vendor/pr628-DIRTY-PROOF.md b/research/clean-checkpoint/vendor/pr628-DIRTY-PROOF.md
new file mode 100644
index 0000000000..caf4aafc78
--- /dev/null
+++ b/research/clean-checkpoint/vendor/pr628-DIRTY-PROOF.md
@@ -0,0 +1,128 @@
+# Cyclic inverse shears with dirty array restoration
+
+This extends the classical rounded cyclic-inverse primitive of PR628. It removes
+its need for a clean length-L intermediate array. It does not supply a common-label
+Clifford realization, reversible bit arithmetic, transform layout, or the complete
+integer-multiplication reduction. No new kappa follows from this lemma.
+
+## Contract
+
+Input x, target y and h borrowed arrays are complex dyadic grids of spacing u=2^-w.
+Every array has length L>=1 and may initially contain arbitrary data. The arrays
+are distinct. Their initial infinity norms are at most D. Read-only phases tau_i
+have modulus one. Phase generation/storage and coefficient preparation are paid;
+there is no free phase table. The primitive adds an approximation to Q(W)^-1 x
+to y, restores x and every borrowed array exactly, and needs only a constant number
+of clean scalar arithmetic registers besides loop counters and factor descriptors.
+Here (Wx)_i=tau_i*x_(i-d), initially for d=+/-1 in a cyclic layout, and
+Q(z)=r^-1 product_(j=1)^n(z-rho_j), n=r-1, with stable reciprocal/inner factors
+of norm at most 1-beta. The final increment can depend on the initial dirt within
+the stated approximation error. Exact independence of the increment from dirt is
+not claimed for rounded arithmetic.
+
+## One factor
+
+First consider T x = y defined by y_i=x_i+b_i*y_(i-d), |b_i|<=1-beta.
+Choose the cyclic order in direction d. Let z be the zero-boundary causal solution
+and p_i the product of b from the first position through i. Let c be the complete
+product. Then y_i=z_i+p_i*s, s=z_last/(1-c). The stable cyclic equation itself
+implies ||y||<=B/beta when ||x||<=B, so |s|<=B/beta.
+
+On the dyadic grid, perform the causal pass directly in x, skipping the first
+position: x_i += round(b_i*x_previous). These are integer lifts. Reversing their
+order and subtracting the same rounded source products restores x exactly,
+including arbitrary nonzero initial values. During the first pass also accumulate
+the rounded scalar product c_hat. Compute s_hat=round(z_hat_last/(1-c_hat)).
+In a second pass accumulate rounded prefix products p_hat_i and add
+round(m_i*(z_hat_i+round(p_hat_i*s_hat))) to the appropriate target position.
+Finally undo the causal lifts. Each target position is written exactly once;
+its existing value never enters the computed increment. Scalar arithmetic and
+coefficient scratch can be cleared classically; no reversible arithmetic gate
+claim is made. Three passes suffice, excluding validation scans and paid
+coefficient access. O(1) clean scalar words and O(log L) indexing bits suffice.
+
+Nearest complex rounding has norm error at most u. Assuming u<=beta^2/2,
+|z_hat-z|<=u/beta, |p_hat-p|<=u/beta, |c_hat-c|<=u/beta, and
+|1-c_hat|>=beta/2. Using s=z_last/(1-c), rather than separately bounding the
+numerator and denominator, gives
+
+    |s_hat-s| <= u + (2/beta)*(u/beta + (B/beta)*(u/beta))
+              <= 5*(B+1)*u/beta^3.
+
+Also |p_hat|<=1+u/beta<=3/2. Hence the corrected output before m multiplication
+has error at most 12*(B+1)*u/beta^3. If |m_i|<=r, the final rounded multiplication
+has error at most a*(B+1)*u, a=13*(r+1)/beta^3. None of these constants depends
+on L.
+
+For an outer root, b_i=rho^-1*tau_i, m_i=-rho^-1. For an inner root, scan in
+direction -d with b_i=rho*conjugate(tau_(i+d)); the value at position i is
+multiplied by conjugate(tau_i) and added at i-d. This is
+(W-rho I)^-1=(I-rho W^-1)^-1 W^-1. The scalar r is included in one leaf's
+multiplier. Each exact factor has operator norm at most K=r/beta. No permutation
+of the mutable source array is required for the inner factor.
+
+## Product of factors
+
+For exact linear maps A,B and a dirty scratch array z, the sequence
+
+    z += B*x; y += A*z; z -= B*x; y -= A*z
+
+adds AB*x to y and restores z. Split the factor list in halves recursively;
+use one distinct borrowed array per recursion depth. A negative call executes
+the exact inverse instruction word (reverse operation order and invert signs).
+It is not obtained by negating signs in the original order.
+
+The rounded construction still restores every borrowed array and x exactly.
+Inductively each call preserves its input and all deeper scratch, and its increment
+is independent of its target. The paired calls computing B therefore see the same
+input and deeper initial dirt. Never use an ancestor input or target as deeper
+scratch: the ancestor target may have changed between the paired calls. This is
+an explicit alias exclusion in the executable contract.
+
+There are h=ceil(log2 n) borrowed arrays and at most N=4*n^2 leaf calls. Pad to the
+next power of two for this count if desired (identity leaves require no rounded
+inverse). The implementation splits without padding and checks the actual count.
+
+For the ideal elementary shears, the joint max norm grows by at most C=1+K per
+leaf. With a*u<=1 the computed trajectory satisfies
+
+    M_j+1 <= G^j*(D+1), G=C+1.
+
+Comparing the complete rounded word with the ideal word on the same initial
+arrays, and summing the propagated leaf errors, gives
+
+    E_N <= N*a*(D+1)*u*G^(N-1).
+
+The ideal word cancels all borrowed initial data exactly. This proves the target
+error bound for every length and every initial dirt bounded by D, while actual
+restoration is exact without an error allowance. Choosing
+
+    w >= p+1+ceil(log2(N*a*(D+1)))+(N-1)*ceil(log2 G)
+
+and enforcing a*u<=1 and u<=beta^2/2 gives execution error <=2^(-p-1).
+PR628's coefficient/phase perturbation bound, budgeted to <=2^(-p-1) for the
+original x, then gives total <=2^-p. That perturbation bound scales with ||x||;
+for general input bound X use p+ceil(log2(max(1,X))) in its preparation budget.
+The initial borrowed dirt does not enter the ideal operator perturbation bound.
+
+PR628 has log2(beta^-1)=O(r^3). Therefore these deliberately loose budgets need
+w=O(p+log(D+1)+r^5) bits, and O(r^2) passes. Inside a leaf, the causal state and closure scalars add at most another
+O(log(beta^-1)+log r) magnitude bits. Maximum transient magnitudes thus add
+O(log(D+1)+r^5) integer bits, already covered by this word-size bound. Standard
+fast fixed-precision arithmetic yields soft-O((L*r^2+r^3)*(p+log(D+1)+r^5))
+bit work plus charged phase-oracle and routing costs. Factor preparation and
+coefficient enclosures are inherited from PR628. Borrowed storage is
+O(L log r) words; clean arithmetic storage is O(w+log L) bits up to the chosen
+multiplication implementation's workspace, plus O(r) read-only factor descriptors.
+A machine with only a fixed number of dirty data arrays is not covered by this
+contract. Arbitrary stride still incurs PR628's explicitly charged routing.
+
+## Executable evidence
+
+`dirty.py` uses exact rational arithmetic and explicit nearest dyadic rounding.
+It checks generic variable-coefficient leaves, dense independently assembled
+factor inverses, both orientations, nonzero targets, several initial dirty arrays,
+exact source/scratch restoration, exact whole-word inversion, and the leaf count.
+Finite checks support the implementation; the inequalities above are the all-L
+argument. The exact-rational prototype does not itself implement the claimed
+fast fixed-precision arithmetic cost.
diff --git a/research/clean-checkpoint/vendor/pr628-NOTICE.md b/research/clean-checkpoint/vendor/pr628-NOTICE.md
new file mode 100644
index 0000000000..22fbeb8ef0
--- /dev/null
+++ b/research/clean-checkpoint/vendor/pr628-NOTICE.md
@@ -0,0 +1,24 @@
+# Attribution and scope
+
+Ali Romman's PR621, exact head 520774462b19b29af41554b74def31f9d01e997e,
+supplies the prime-power kernel circle bound. Its kernel attribution is
+OpenAI's exact-DFT preprint at adc7f1241b42e322a6451854ab7e4b4c146bf78a
+(`loc:newton`, `loc:fourier-factorization`) and classical q-Pochhammer/Newton
+factorization, including Alexey Kuznetsov's cited q-calculus work. This
+package does not claim to originate those constructions.
+
+Pan's approximate-factorization theorem is used in the form stated by
+Mehlhorn, Sagraloff and Wang; Brent's work supplies fast elementary-function
+evaluation. Causal/anticausal recursive filtering is standard, as illustrated
+by the Briand-Davy source listed in SOURCE.json. Sources are cited for these
+specific tools, not as proofs of this package's complete claim.
+
+The scan rounding analysis, explicit budget composition, interval and
+factor certificates, routed reference execution, and packaging were
+prepared with substantial OpenAI Codex assistance for chafreaky. The dense
+matrix solver and algebraic coefficient crosschecks provide independent
+computational paths, not human peer review or formal Lean verification.
+
+No new multiplication exponent, literature priority, maintainer acceptance,
+physical tape execution, or production performance result is claimed.
+The code and documents use the repository's Apache-2.0 license.
diff --git a/research/clean-checkpoint/vendor/pr628-dirty.py b/research/clean-checkpoint/vendor/pr628-dirty.py
new file mode 100644
index 0000000000..2b94710837
--- /dev/null
+++ b/research/clean-checkpoint/vendor/pr628-dirty.py
@@ -0,0 +1,133 @@
+"""Integer lifting for cyclic inverse shears using borrowed dirty arrays.
+Exact rational prototype. Arithmetic workspace and phase access are charged in the proof;
+this is not a common-label Clifford word or a bit-reversible arithmetic compiler.
+"""
+from pathlib import Path
+from fractions import Fraction as F
+import importlib.util,json,argparse,itertools
+P=Path(__file__).resolve().parent
+s=importlib.util.spec_from_file_location('weighted',P/'weighted.py');w=importlib.util.module_from_spec(s);s.loader.exec_module(w);c=w.c
+
+def scaled(z,q):return(z[0]*q,z[1]*q)
+def conj(z):return(z[0],-z[1])
+def ordered(n,d):return range(n)if d==1 else range(n-1,-1,-1)
+
+def lift(x,y,coefficient,beta,bits,d,sign=1,destination=None,multiplier=None,stats=None):
+    """y += sign*F(x), restoring x exactly; no length-dependent clean data array."""
+    n=len(x);c.need(x is not y and n==len(y)>0,'distinct equally sized arrays')
+    c.need(d in(-1,1)and sign in(-1,1),'orientation and sign')
+    c.need(0<beta<=1 and F(1,1<<bits)<=beta*beta/2,'closure precision')
+    c.need(all(c.norm2(coefficient(i))<=(1-beta)**2 for i in range(n)),'stable coefficients')
+    c.need(all(c.rounded(z,bits)==z for z in itertools.chain(x,y)),'fixed grid data')
+    destination=destination or (lambda i:i);multiplier=multiplier or (lambda i:c.ONE)
+    power=c.ONE;previous=None
+    for i in ordered(n,d):
+        if previous is not None:x[i]=c.add(x[i],c.rounded(c.mul(coefficient(i),x[previous]),bits))
+        power=c.rounded(c.mul(coefficient(i),power),bits);previous=i
+    denominator=c.sub(c.ONE,power);c.need(c.norm2(denominator)>=(beta/2)**2,'closure denominator')
+    seed=c.rounded(c.div(x[previous],denominator),bits)
+    power=c.ONE
+    for i in ordered(n,d):
+        power=c.rounded(c.mul(coefficient(i),power),bits)
+        value=c.add(x[i],c.rounded(c.mul(power,seed),bits))
+        value=c.rounded(c.mul(multiplier(i),value),bits)
+        j=destination(i);y[j]=c.add(y[j],scaled(value,sign))
+    # The scalar seed and coefficient scratch may be cleared by ordinary classical
+    # arithmetic. The dirty data itself is restored by exact inverse integer lifts.
+    first=0 if d==1 else n-1
+    for i in ordered(n,-d):
+        if i!=first:x[i]=c.sub(x[i],c.rounded(c.mul(coefficient(i),x[i-d]),bits))
+    if stats is not None:stats['leaves']+=1;stats['scalar_steps']+=3*n
+
+def leaf(x,y,factor,phases,beta,bits,d,sign,stats):
+    inside,b,scale=factor;n=len(x)
+    if inside:
+        lift(x,y,lambda i:c.mul(b,conj(phases[(i+d)%n])),beta,bits,-d,sign,
+             lambda i:(i-d)%n,lambda i:scaled(conj(phases[i]),scale),stats)
+    else:
+        lift(x,y,lambda i:c.mul(b,phases[i]),beta,bits,d,sign,
+             multiplier=lambda i:scaled(c.neg(b),scale),stats=stats)
+
+def compose(x,y,factors,scratch,phases,beta,bits,d,sign=1,stats=None):
+    """Balanced commutator; negative call is the exact inverse instruction word."""
+    c.need(sign in(-1,1)and d in(-1,1),'orientation and sign')
+    c.need(factors and len(scratch)>=(len(factors)-1).bit_length(),'dirty depth')
+    arrays=[x,y]+scratch;c.need(len({id(v)for v in arrays})==len(arrays),'all ancestor arrays distinct')
+    c.need(len(x)>0 and all(len(v)==len(x)for v in arrays)and len(phases)==len(x),'array lengths')
+    c.need(all(c.norm2(t)==1 for t in phases),'unit phases')
+    if len(factors)==1:return leaf(x,y,factors[0],phases,beta,bits,d,sign,stats)
+    mid=len(factors)//2;B,A=factors[:mid],factors[mid:];z=scratch[0];rest=scratch[1:]
+    operations=[(x,z,B,1),(z,y,A,1),(x,z,B,-1),(z,y,A,-1)]
+    if sign==-1:operations=[(src,dst,fs,-sgn)for src,dst,fs,sgn in reversed(operations)]
+    for src,dst,fs,sgn in operations:compose(src,dst,fs,rest,phases,beta,bits,d,sgn,stats)
+
+def exact_operator(x,factors,phases,d):
+    # Dense rational oracle constructed independently from the lifting recurrence.
+    n=len(x);out=x[:]
+    for inside,b,scale in factors:
+        a=[[c.ZERO for _ in range(n)]for _ in range(n)]
+        if inside:
+            for i in range(n):a[i][i]=c.neg(b);a[i][(i-d)%n]=c.add(a[i][(i-d)%n],phases[i])
+        else:
+            rho=c.div(c.ONE,b)
+            for i in range(n):a[i][i]=c.neg(rho);a[i][(i-d)%n]=c.add(a[i][(i-d)%n],phases[i])
+        out=[scaled(z,scale)for z in w.solve(a,out)]
+    return out
+
+def certificate(n,beta,r,D,bits):
+    N=4*n*n;a=13*(r+1)/beta**3;G=2+F(r)/beta;u=F(1,1<<bits)
+    c.need(a*u<=1 and u<=beta**2/2,'uniform precision')
+    return N*a*(D+1)*u*G**(N-1)
+
+def main():
+    ap=argparse.ArgumentParser();ap.add_argument('--out',type=Path,required=True);args=ap.parse_args()
+    units=[c.ONE,(F(0),F(1)),(F(3,5),F(4,5)),(F(-5,13),F(12,13))]
+    cases=[];controls=[];generic=[]
+    for beta in(F(1,2),F(1,16),F(1,4096)):
+        for n in(1,2,3,8,17,64):
+            bs=[scaled(units[(i*i+i+1)%4],(1-beta)/(1+i%2))for i in range(n)]
+            original=[(F(i%5-2,4),F(i%7-3,8))for i in range(n)]
+            for d in(-1,1):
+                x=original[:];y=[(F(19+i,4),F(3-i,8))for i in range(n)];y0=y[:];bits=128
+                lift(x,y,lambda i:bs[i],beta,bits,d);c.need(x==original,'exact source restoration')
+                net=[c.sub(z,t)for z,t in zip(y,y0)];B=max(abs(a)+abs(b)for a,b in original);bound=13*(B+1)*F(1,1<<bits)/beta**3
+                matrix=[[c.ZERO for _ in range(n)]for _ in range(n)]
+                for i in range(n):matrix[i][i]=c.ONE;matrix[i][(i-d)%n]=c.sub(matrix[i][(i-d)%n],bs[i])
+                w.residual(original,net,matrix,(2-beta)*bound)
+                if n<=8:c.need(all(c.norm2(c.sub(z,t))<=bound**2 for z,t in zip(net,w.solve(matrix,original))),'leaf dense')
+                lift(x,y,lambda i:bs[i],beta,bits,d,-1);c.need(y==y0 and x==original,'exact leaf cancellation')
+                generic.append(dict(beta=str(beta),length=n,direction=d,dense=n<=8))
+    for nf in(1,2,3,5,8):
+        factors=[(i%2==0,(F(1,4),F((-1)**i,8)),F(3)if i==0 else F(1))for i in range(nf)]
+        for n in(1,2,3,8):
+            for d in(-1,1):
+                for dirt in(0,7,-13):
+                    x=[(F(i%5-2,4),F(i%3-1,8))for i in range(n)];x0=x[:]
+                    y=[(F(dirt+i,4),F(dirt-i,8))for i in range(n)];y0=y[:]
+                    scratch=[[(F(dirt+2*i+j,4),F(dirt-3*i-j,8))for i in range(n)]for j in range((nf-1).bit_length())];s0=[z[:]for z in scratch]
+                    phases=[units[(i*i+3*i+1)%4]for i in range(n)];bits=4096;stats=dict(leaves=0,scalar_steps=0)
+                    D=max(abs(a)+abs(b)for v in[x,y]+scratch for a,b in v);bound=certificate(nf,F(1,2),3,D,bits)
+                    compose(x,y,factors,scratch,phases,F(1,2),bits,d,stats=stats)
+                    c.need(x==x0 and scratch==s0,'exact dirty restoration')
+                    expected=exact_operator(x0,factors,phases,d);net=[c.sub(z,t)for z,t in zip(y,y0)]
+                    c.need(all(c.norm2(c.sub(z,t))<=bound**2 for z,t in zip(net,expected)),'commutator dense')
+                    calls=stats['leaves'];c.need(calls<=4*nf*nf,'quadratic leaf count')
+                    compose(x,y,factors,scratch,phases,F(1,2),bits,d,-1)
+                    c.need(x==x0 and scratch==s0 and y==y0,'exact whole word inverse')
+                    cases.append(dict(factors=nf,length=n,direction=d,dirt=dirt,leaf_calls=calls,dirty_arrays=len(scratch),bound=str(bound)))
+        print('PASS factors',nf,flush=True)
+    def reject(name,fn):
+        try:fn()
+        except ValueError as e:controls.append(dict(name=name,rejected=str(e)))
+        else:raise ValueError('control accepted '+name)
+    x=[c.ONE,c.ZERO];y=[c.ZERO,c.ONE];fs=[(True,(F(1,4),F(0)),F(1))]*2
+    reject('borrowed_outer_target',lambda:compose(x,y,fs,[y],[c.ONE]*2,F(1,2),128,1))
+    reject('borrowed_source',lambda:compose(x,y,fs,[x],[c.ONE]*2,F(1,2),128,1))
+    reject('missing_dirty_level',lambda:compose(x,y,fs,[],[c.ONE]*2,F(1,2),128,1))
+    reject('nonunit_phase',lambda:compose(x,y,fs,[[c.ZERO]*2],[c.ONE,(F(2),F(0))],F(1,2),128,1))
+    reject('underprecision',lambda:lift(x,y,lambda i:c.ZERO,F(1,16),4,1))
+    reject('unstable_site',lambda:lift(x,y,lambda i:c.ONE,F(1,2),128,1))
+    reject('off_grid_source',lambda:lift([(F(1,3),F(0))]*2,y,lambda i:c.ZERO,F(1,2),128,1))
+    args.out.write_text(json.dumps(dict(status='passed',generic=generic,cases=cases,controls=controls,scope='Classical fixed-grid dirty-array shear; excludes bit-reversible arithmetic and full multiplication reduction'),indent=2)+'\n')
+    print('PASS',len(generic),'leaf cases',len(cases),'compositions',len(controls),'controls')
+if __name__=='__main__':main()
diff --git a/research/clean-checkpoint/vendor/pr628-scan.py b/research/clean-checkpoint/vendor/pr628-scan.py
new file mode 100644
index 0000000000..d620e49359
--- /dev/null
+++ b/research/clean-checkpoint/vendor/pr628-scan.py
@@ -0,0 +1,123 @@
+"""Exact tests of a rounded two-pass periodic first-order filter.
+
+No floating arithmetic, numerical root finder, or kernel builder is used.
+This certifies the listed finite cases; PROOF.md gives the general bound.
+"""
+from fractions import Fraction as F
+from pathlib import Path
+import argparse
+import json
+
+ZERO = (F(0), F(0))
+ONE = (F(1), F(0))
+
+def add(a, b): return (a[0]+b[0], a[1]+b[1])
+def neg(a): return (-a[0], -a[1])
+def sub(a, b): return add(a, neg(b))
+def mul(a, b): return (a[0]*b[0]-a[1]*b[1], a[0]*b[1]+a[1]*b[0])
+def norm2(a): return a[0]*a[0]+a[1]*a[1]
+def div(a, b):
+    d = norm2(b)
+    if not d: raise ValueError('zero divisor')
+    c = mul(a, (b[0], -b[1]))
+    return (c[0]/d, c[1]/d)
+
+def need(ok, message):
+    if not ok: raise ValueError(message)
+
+def rounded(a, bits):
+    scale = 1 << bits
+    def component(v):
+        w = v * scale + F(1, 2)
+        return F(w.numerator // w.denominator, scale)
+    return (component(a[0]), component(a[1]))
+
+def scan(x, b, beta, bits, direction):
+    need(len(x) > 0, 'empty cycle')
+    need(direction in (-1, 1), 'invalid direction')
+    need(0 < beta <= 1, 'invalid margin')
+    need(norm2(b) <= (1-beta)**2, 'uncertified stable factor')
+    u = F(1, 1 << bits)
+    need(u <= beta*beta/2, 'precision too low for closure')
+    order = list(range(len(x))) if direction == 1 else list(reversed(range(len(x))))
+    z, power = ZERO, ONE
+    for i in order:
+        z = rounded(add(x[i], mul(b, z)), bits)
+        power = rounded(mul(b, power), bits)
+    den = sub(ONE, power)
+    need(norm2(den) >= (beta/2)**2, 'closure lower bound failed')
+    seed = rounded(div(z, den), bits)
+    y, previous = [ZERO]*len(x), seed
+    for i in order:
+        previous = rounded(add(x[i], mul(b, previous)), bits)
+        y[i] = previous
+    return y
+
+def dense_solve(x, b, direction):
+    """Independent exact Gaussian elimination on the cyclic matrix."""
+    n = len(x)
+    a = [[ZERO for _ in range(n)] + [x[i]] for i in range(n)]
+    for i in range(n):
+        a[i][i] = add(a[i][i], ONE)
+        j = (i-direction) % n
+        a[i][j] = sub(a[i][j], b)
+    for k in range(n):
+        pivot = next(i for i in range(k, n) if a[i][k] != ZERO)
+        a[k], a[pivot] = a[pivot], a[k]
+        p = a[k][k]
+        a[k] = [div(v, p) for v in a[k]]
+        for i in range(n):
+            if i == k: continue
+            p = a[i][k]
+            a[i] = [sub(v, mul(p, w)) for v, w in zip(a[i], a[k])]
+    return [row[-1] for row in a]
+
+def residual_bound(x, y, b, beta, direction, bound):
+    # Neumann inverse norm <= 1/beta converts residual to forward error.
+    for i in range(len(x)):
+        residual = sub(sub(y[i], mul(b, y[(i-direction) % len(x)])), x[i])
+        need(norm2(residual) <= (beta*bound)**2, 'residual error bound failed')
+
+def main():
+    parser = argparse.ArgumentParser()
+    parser.add_argument('--out', type=Path, required=True)
+    args = parser.parse_args()
+    cases = []
+    for beta in (F(1, 2), F(1, 16), F(1, 4096)):
+        for unit in (ONE, (F(-1), F(0)), (F(3, 5), F(4, 5))):
+            b = mul((1-beta, F(0)), unit)
+            for n in (1, 2, 3, 8, 17, 64):
+                x = [(F((i*7)%13-6, 8), F((i*11)%17-8, 16)) for i in range(n)]
+                B = max(abs(v[0])+abs(v[1]) for v in x)
+                for direction in (-1, 1):
+                    bits = 64
+                    bound = 6*(B+1)*F(1, 1 << bits)/beta**3
+                    y = scan(x, b, beta, bits, direction)
+                    residual_bound(x, y, b, beta, direction, bound)
+                    if n <= 8:
+                        exact = dense_solve(x, b, direction)
+                        need(all(norm2(sub(v,w)) <= bound**2 for v,w in zip(y,exact)), 'dense comparison failed')
+                    cases.append(dict(beta=str(beta), b=[str(v) for v in b], length=n,
+                                      direction=direction, bits=bits, forward_error_bound=str(bound),
+                                      dense_crosscheck=n<=8, residual_check=True))
+    controls = []
+    for name, x, b, beta, bits, direction in (
+        ('unstable', [ONE], ONE, F(1,2), 64, 1),
+        ('underprecision', [ONE], (F(15,16),F(0)), F(1,16), 3, 1),
+        ('empty', [], ZERO, F(1,2), 64, 1),
+        ('bad_direction', [ONE], ZERO, F(1,2), 64, 0),
+    ):
+        try: scan(x,b,beta,bits,direction)
+        except ValueError as error: controls.append(dict(name=name,rejected=str(error)))
+        else: raise ValueError('negative control accepted: '+name)
+    x = [ONE, ZERO, ONE]
+    b, beta = (F(1,2),F(0)), F(1,2)
+    y = scan(x,b,beta,64,1)
+    y[0] = add(y[0],ONE)
+    try: residual_bound(x,y,b,beta,1,F(1,1<<40))
+    except ValueError as error: controls.append(dict(name='corrupt_output',rejected=str(error)))
+    else: raise ValueError('corrupt output accepted')
+    args.out.write_text(json.dumps(dict(status='passed',cases=cases,controls=controls),indent=2)+'\n')
+    print(json.dumps(dict(status='passed',cases=len(cases),dense_cases=sum(c['dense_crosscheck'] for c in cases),controls=len(controls))))
+
+if __name__ == '__main__': main()
diff --git a/research/clean-checkpoint/vendor/pr663-NOTICE.md b/research/clean-checkpoint/vendor/pr663-NOTICE.md
new file mode 100644
index 0000000000..f70d712289
--- /dev/null
+++ b/research/clean-checkpoint/vendor/pr663-NOTICE.md
@@ -0,0 +1,26 @@
+# Attribution, license and assistance
+
+Prepared for GitHub contributor **@0xkilgore**, with substantial **OpenAI Codex**
+assistance. Codex drafted the independent verifier, proof, evidence and packaging;
+a separate Codex agent independently derived the restoration and call-count
+arguments and checked the error-budget substitution. This is AI-assisted review,
+not independent human peer review.
+
+The rounded dirty cyclic inverse, balanced commutator, its restoration contract,
+and the original conservative error budget are from **Chafik Boukhalfa
+(@chafreaky), PR628**, at `0768f97d06c623b8211e651543c66135796b62bf`.
+`vendor/pr628_dirty.py` preserves the submitted `dirty.py` bytes. Its author's
+own substantial OpenAI Codex assistance disclosure is retained verbatim in
+`vendor/PR628-NOTICE.md`. The audit does not claim to originate that construction.
+
+The analytic context retains **Ali Romman's PR621** at
+`520774462b19b29af41554b74def31f9d01e997e`, OpenAI's exact-DFT preprint at
+`adc7f1241b42e322a6451854ab7e4b4c146bf78a`, and the classical
+Newton/q-Pochhammer factorization cited there (including Alexey Kuznetsov).
+These analytic results are not premises of the abstract restoration identity.
+The current accepted construction's cumulative community attribution and
+selected certificate are unchanged.
+
+All new code and text use the repository's **Apache-2.0** license. Vendored
+PR628 code retains its Apache-2.0 provenance and notice. No priority claim,
+new multiplication saving, or completed physical implementation is asserted.
diff --git a/research/clean-checkpoint/vendor/pr663-README.md b/research/clean-checkpoint/vendor/pr663-README.md
new file mode 100644
index 0000000000..aeed8aeb26
--- /dev/null
+++ b/research/clean-checkpoint/vendor/pr663-README.md
@@ -0,0 +1,214 @@
+# Independent audit: PR628's balanced dirty commutator
+
+**Result:** the pinned commutator restores its input and borrowed arrays under
+its stated leaf contract, including nonlinear rounded leaves. No defect was
+found in this scoped obligation. Its exact leaf count is
+
+\[
+ C(n)=3qn-2q^2<\frac98n^2,\qquad q=2^{\lfloor\log_2n\rfloor}.
+\]
+
+This tightens the submitted allowance `4*n*n` without changing the instruction
+word, and can replace that allowance in the execution-error budget. It does
+not change the asymptotic precision order or establish a new multiplication
+exponent. The constant `9/8` is the sharp supremum for this splitting rule.
+
+The contribution adds an independent free-term audit of the **actual pinned
+recursive dispatcher**, a written all-`n` argument, and exact accounting.
+PR628 already supplies dense rational accuracy comparisons and instruction
+mutations. This audit's different evidence is that leaf functions are left
+uninterpreted: they need not be linear, continuous, zero-preserving, or close
+to a linear map for the restoration check. A second engine checks the intended
+product in a free noncommutative algebra, without assuming that factors commute.
+No literature-priority claim is made for the commutator or recurrence.
+
+Interpreting input generators as initial arrays and function atoms as deterministic
+leaf outputs preserves every symbolic cancellation. Thus each checked symbolic
+identity holds for every such leaf instantiation at that tested factor count.
+
+## Reproduce
+
+From the repository root, with Python 3.9+ and no third-party packages:
+
+```sh
+python3 -B research/dirty-commutator-audit/verify.py --check
+```
+
+Expected output:
+
+```text
+PASS: 64 symbolic words; 10000 recurrence cases; 7 rejected controls; nonlinear dirt counterexample. n=62: 3904 leaves, 6 scratch arrays.
+```
+
+The command authenticates the vendored source and compares its complete result
+with [EXPECTED.json](EXPECTED.json). `--write /tmp/dirty-audit.json` writes a
+new certificate. The checker uses explicit exceptions; checks remain enabled
+under optimized Python. It does not import or execute the producer's arithmetic,
+root preparation, matrix solver, or top-level module initialization. It extracts
+only `compose` from the pinned Python AST and intercepts leaf operations.
+
+## Mathematical obligation and proof
+
+Let a primitive on distinct arrays `x,y` add a deterministic function `F(x)` to
+`y`, restore `x` exactly, and have a target-independent increment. Its negative
+call must implement the exact inverse. Coefficients, phases, and any other
+read-only parameters must remain fixed between paired calls. Arrays below are
+elements of an abelian group; rounded dyadic arrays satisfy this requirement.
+
+For a recursive child let `F(x;t)` denote its increment, where `t` contains
+the deeper scratch arrays' **initial** contents. Inductively assume it restores
+`x,t` and does not depend on its initial target. The parent performs
+
+```text
+z += B(x;t)
+y += A(z;t)
+z -= B(x;t)
+y -= A(z;t)
+```
+
+Every child restores its deeper scratch. The paired `B` calls see identical
+`x,t`, so `z` returns to its initial value `z0`. The net increment is
+
+\[
+ F_n(x;z_0,t)=A(z_0+B(x;t);t)-A(z_0;t).
+\]
+
+It is independent of the initial `y`; all other arrays are restored exactly.
+This proves the recursive contract by induction, **without linearity**. With
+linear ideal factors, the same induction gives exactly `A B x`, with no
+dependence on the initial scratch. The factor list executes in list order,
+so the resulting operator is `F_(n-1) ... F_1 F_0`.
+
+The inverse word reverses the whole instruction sequence and inverts every
+sign. The pinned `compose` does this at each recursive call. Just negating signs
+in their original order is invalid: for `B(x)=x`, `A(z)=z^2`, and `x=z0=1`, the
+positive increment is `3`, the true inverse increment is `-3`, and same-order
+sign negation gives `+1`.
+
+All source, target and scratch arrays must be pairwise distinct. In particular,
+an ancestor target cannot be borrowed as deeper scratch: it can change between
+the paired calls. The pinned implementation explicitly enforces this exclusion.
+
+Exact restoration does **not** imply independence from initial scratch for
+nonlinear leaves. The same two-factor example with `x=1` gives increments `1`
+and `3` at `z0=0` and `z0=1`, respectively, while restoring the source and scratch
+in both executions. This concrete example corroborates PR628's stated caveat;
+it is not a counterexample to that PR's claim.
+
+## Complete call recurrence
+
+Count all forward and inverse primitive leaf calls, without cancellation or
+special treatment of identity factors. The exact recurrence is
+
+\[
+ C(1)=1,\qquad C(n)=2C(\lfloor n/2\rfloor)+2C(\lceil n/2\rceil).
+\]
+
+Write `q=2^k <= n < 2q`. Repeated balanced splitting gives `2q-n` ordinary
+factor-tree leaves at depth `k` and `2n-2q` at depth `k+1`. To see that these are
+the only depths, after `k` splits the group sizes are one or two; each group
+of size two splits once more. Each descent in the commutator doubles a factor
+leaf's execution multiplicity. Therefore
+
+\[
+ C(n)=(2q-n)q+(2n-2q)2q=3qn-2q^2.
+\]
+
+The exact identity
+
+\[
+ \frac98n^2-C(n)=\frac{(3n-4q)^2}{8}
+\]
+
+proves the strict upper bound: `3n=4q` is impossible for integer `n` when `q`
+is a power of two. Along the integers nearest `4q/3`, the square is `1`, so
+`C(n)/n^2` tends to `9/8`. The recursion uses exactly
+`ceil(log2(n))` distinct scratch arrays, with zero when `n=1`.
+
+| Factor count | Exact leaf calls | Submitted allowance `4n²` | Scratch arrays |
+| ---: | ---: | ---: | ---: |
+| 1 | 1 | 4 | 0 |
+| 3 | 10 | 36 | 2 |
+| 7 | 52 | 196 | 3 |
+| 15 | 232 | 900 | 4 |
+| 31 | 976 | 3844 | 5 |
+| 62 | 3904 | 15376 | 6 |
+| 63 | 4000 | 15876 | 6 |
+| 64 | 4096 | 16384 | 6 |
+
+The seed-selected `n=62` case tests the generic dispatcher. It does not assert
+a prime-power kernel at radix 63. The neighboring `n=63` case is the factor
+count for a radix-64 kernel.
+
+## Consequence for the inherited execution-error budget
+
+Retain PR628's assumptions on **both signs** of each leaf: ideal operator norm
+at most `K=r/beta`, and local error at most `a*u*(B+1)`, where
+`a=13*(r+1)/beta^3`, `u=2^-w`, and `B` bounds the source norm. Let `M_j` be
+the maximum array norm after `j` completed leaves and `D` its initial bound.
+Under `a*u <= 1`, with `G=2+K`,
+
+\[
+ M_{j+1}\le(1+K)M_j+au(M_j+1),\qquad
+ M_j+1\le G^j(D+1).
+\]
+
+The second expression means **one plus `M_j`**, not the next indexed norm.
+If `e_j` compares computed and ideal completed-leaf states, then
+
+\[
+ e_{j+1}\le(1+K)e_j+auG^j(D+1),\qquad
+ e_N\le Na(D+1)uG^{N-1}.
+\]
+
+Thus the submitted sufficient execution precision remains valid with the
+**exact** `N=C(n)`:
+
+\[
+ w\ge p+1+\lceil\log_2(Na(D+1))\rceil
+       +(N-1)\lceil\log_2G\rceil,
+\]
+
+together with `a*u<=1` and `u<=beta^2/2`. This is an implication conditional
+on the inherited leaf estimates, not a new audit of those estimates. It gives
+execution error at most `2^(-p-1)`; coefficient/phase preparation must still
+supply the other half of the total error budget, scaled to the original input.
+The existing `O(p+log(D+1)+r^5)` precision and `O(r^2)` pass orders remain.
+This package does not modify PR628's generator or its certificates.
+
+## Evidence, coverage and limits
+
+- **Exact symbolic checks, `1 <= n <= 64`:** source/scratch restoration,
+  target independence, whole-word inversion, equality of pinned and independently
+  compiled instruction traces, and the noncommuting ideal linear product.
+- **Exact integer checks, `1 <= n <= 10000`:** recursive and closed counts agree,
+  and the sharp-envelope square identity holds.
+- **Seven negative controls:** omitted final subtraction, omitted scratch
+  restoration, missing inverse reversal, borrowing an ancestor source or target,
+  reusing a scratch depth, and insufficient scratch depth.
+- **Nonlinear example:** exact dirty restoration with different target increments
+  for two initial scratch values.
+
+The finite symbolic replay checks these concrete instruction words in an exact
+term model; it is not a formal proof of the all-`n` theorem. The induction above
+is the written all-`n` argument, independently reviewed by another Codex agent.
+No human peer review, Lean certification, or full multiplication theorem follows.
+The actual rounded leaf's accuracy and restoration, preparation complexity,
+phase access, bit arithmetic, routing, and full-transform composition retain
+their separate obligations. Selected supplier assumptions remain unchanged.
+
+This is a bounded audit, not a parameter search. There is no random search:
+seed 62 selected the object and uneven-split case; the enumerations above were
+completed in full with no adaptive early stop.
+
+## Pins and attribution
+
+Repository base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. Subject:
+[Chafik Boukhalfa's PR628](https://github.com/CrocSwap/integer-mult-bounds/pull/628),
+head `0768f97d06c623b8211e651543c66135796b62bf`, specifically
+[`DIRTY-PROOF.md`](https://github.com/CrocSwap/integer-mult-bounds/blob/0768f97d06c623b8211e651543c66135796b62bf/research/prime-power-cyclic-scan/DIRTY-PROOF.md)
+and [`dirty.py`](https://github.com/CrocSwap/integer-mult-bounds/blob/0768f97d06c623b8211e651543c66135796b62bf/research/prime-power-cyclic-scan/dirty.py).
+The latter is retained byte-for-byte for offline replay. SHA-256 pins,
+the accepted-frontier snapshot, and analytic predecessor information are in
+[SOURCE.json](SOURCE.json). [CLAIM.md](CLAIM.md) records track selection and overlap
+review. See [NOTICE.md](NOTICE.md) for source credits and assistance disclosure.
diff --git a/research/clean-checkpoint/vendor/pr673-phase-alias-guard.patch b/research/clean-checkpoint/vendor/pr673-phase-alias-guard.patch
new file mode 100644
index 0000000000..6323498dfb
--- /dev/null
+++ b/research/clean-checkpoint/vendor/pr673-phase-alias-guard.patch
@@ -0,0 +1,10 @@
+--- a/research/prime-power-cyclic-scan/dirty.py
++++ b/research/prime-power-cyclic-scan/dirty.py
+@@ -53,6 +53,7 @@
+     c.need(sign in(-1,1)and d in(-1,1),'orientation and sign')
+     c.need(factors and len(scratch)>=(len(factors)-1).bit_length(),'dirty depth')
+     arrays=[x,y]+scratch;c.need(len({id(v)for v in arrays})==len(arrays),'all ancestor arrays distinct')
++    c.need(all(phases is not v for v in arrays),'phase array aliases mutable data')
+     c.need(len(x)>0 and all(len(v)==len(x)for v in arrays)and len(phases)==len(x),'array lengths')
+     c.need(all(c.norm2(t)==1 for t in phases),'unit phases')
+     if len(factors)==1:return leaf(x,y,factors[0],phases,beta,bits,d,sign,stats)
diff --git a/research/clean-checkpoint/verify.py b/research/clean-checkpoint/verify.py
new file mode 100644
index 0000000000..c29cc149f4
--- /dev/null
+++ b/research/clean-checkpoint/verify.py
@@ -0,0 +1,230 @@
+#!/usr/bin/env python3
+"""Exact bounded replay of a clean root checkpoint; see README and PROOF.
+
+No third-party dependencies. Checks remain active under python -O.
+"""
+import argparse
+import ast
+from collections import Counter
+from functools import lru_cache
+import hashlib
+import importlib.util
+import json
+from pathlib import Path
+from types import SimpleNamespace
+
+HERE = Path(__file__).resolve().parent
+
+
+def module(name):
+    spec = importlib.util.spec_from_file_location(name, HERE / (name + '.py'))
+    out = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(out)
+    return out
+
+
+cp = module('checkpoint')
+need = cp.need
+
+
+def digest(data):
+    return hashlib.sha256(data).hexdigest()
+
+
+def sources():
+    record = json.loads((HERE / 'SOURCE.json').read_text())
+    for entry in record['sources']:
+        data = (HERE / entry['path']).read_bytes()
+        need(digest(data) == entry['sha256'], 'source SHA256: ' + entry['path'])
+        blob = b'blob ' + str(len(data)).encode() + b'\0' + data
+        need(hashlib.sha1(blob).hexdigest() == entry['git_blob'],
+             'source Git blob: ' + entry['path'])
+    return record
+
+
+def inverse(word):
+    return [(f, x, y, -s) for f, x, y, s in reversed(word)]
+
+
+def reference(fs, x, y, scratch):
+    if len(fs) == 1:
+        return [(fs[0], x, y, 1)]
+    k = len(fs) // 2
+    a = reference(fs[:k], x, scratch[0], scratch[1:])
+    b = reference(fs[k:], scratch[0], y, scratch[1:])
+    return a + b + inverse(a) + inverse(b)
+
+
+def pinned_namespace():
+    tree = ast.parse((HERE / 'vendor/pr628-dirty.py').read_text())
+    defs = [n for n in tree.body if isinstance(n, ast.FunctionDef)
+            and n.name == 'compose']
+    need(len(defs) == 1, 'unique pinned dispatcher')
+    ns = {'c': SimpleNamespace(need=need, norm2=lambda t: t*t, ZERO=0)}
+    exec(compile(ast.Module(body=defs, type_ignores=[]), 'pinned-compose', 'exec'), ns)
+    return ns
+
+
+def trace(ns, n, k, sign=1, mutation=None):
+    p = cp.plan(n, k)
+    arrays = [[0] for _ in range(3 + p['dirty_arrays'])]
+    if mutation == 'dirty':
+        arrays[2][0] = 1
+    if mutation == 'source_alias':
+        arrays[2] = arrays[0]
+    if mutation == 'target_alias':
+        arrays[-1] = arrays[1]
+    if mutation == 'shortage':
+        arrays.pop()
+    phases = [1]
+    if mutation and mutation.startswith('phase_alias_'):
+        role = int(mutation.rsplit('_', 1)[1])
+        phases = arrays[role]
+        phases[:] = [1]
+    labels = {id(a): i for i, a in enumerate(arrays)}
+    word = []
+    def leaf(x, y, f, phases, beta, bits, d, s, stats):
+        word.append((f, labels[id(x)], labels[id(y)], s))
+    ns['leaf'] = leaf
+    cp.compose_clean(ns['compose'], ns['c'], arrays[0], arrays[1],
+                     list(range(n)), arrays[2], arrays[3:], phases, 1, 1, 1,
+                     split=k, sign=sign)
+    return word
+
+
+def initial(size, clean=True):
+    return [{} if clean and i == 2 else {(i, ()): 1} for i in range(size)]
+
+
+def replay(word, state):
+    """Free noncommutative linear operators on independent initial columns."""
+    result = [dict(v) for v in state]
+    for f, x, y, s in word:
+        for (variable, monomial), a in list(result[x].items()):
+            key = variable, (f,) + monomial
+            value = result[y].get(key, 0) + s * a
+            if value:
+                result[y][key] = value
+            else:
+                result[y].pop(key, None)
+    return result
+
+
+@lru_cache(None)
+def recursive_count(n):
+    return 1 if n == 1 else 2 * (recursive_count(n//2) + recursive_count(n-n//2))
+
+
+def run():
+    src = sources()
+    ns = pinned_namespace()
+    plans, candidates = [], 0
+    for n in range(2, 257):
+        need(recursive_count(n) == cp.dirty_count(n), 'inherited count')
+        costs = [2*recursive_count(k)+recursive_count(n-k) for k in range(1, n)]
+        candidates += len(costs)
+        optimum = min(costs)
+        winners = [k for k in range(1, n) if costs[k-1] == optimum]
+        p = cp.plan(n)
+        need(p['optimal_splits'] == [winners[0], winners[-1]], 'optimal interval')
+        need(winners == list(range(winners[0], winners[-1]+1)), 'no hidden gaps')
+        need(p['leaf_calls'] == optimum, 'minimum cost')
+        need(1+p['dirty_arrays'] == p['balanced_dirty_arrays'], 'total array count')
+        need(5*optimum >= 3*recursive_count(n), 'three-fifths bound')
+        need(4*optimum <= 3*recursive_count(n), 'three-quarters bound')
+        plans.append(p)
+    algebra = []
+    cases = [(n, k) for n in range(2, 33) for k in range(1, n)]
+    cases += [(54, cp.plan(54)['split']), (63, cp.plan(63)['split'])]
+    for n, k in cases:
+        p = cp.plan(n, k)
+        size = 3 + p['dirty_arrays']
+        word = trace(ns, n, k)
+        a = reference(list(range(k)), 0, 2, list(range(3, size)))
+        b = reference(list(range(k, n)), 2, 1, list(range(3, size)))
+        need(word == a+b+inverse(a), 'pinned vs independent compiler')
+        need(len(word) == p['leaf_calls'], 'all forward and reverse calls priced')
+        state = initial(size)
+        expected = initial(size)
+        expected[1][(0, tuple(reversed(range(n))))] = 1
+        actual = replay(word, state)
+        need(actual == expected, 'noncommuting product / all dirty columns restored')
+        neg = trace(ns, n, k, -1)
+        need(neg == inverse(word), 'inverse chronology')
+        need(replay(neg, actual) == state, 'exact inverse')
+        need(replay(a+b+inverse(a)+inverse(b), state) == actual,
+             'same-split omitted zero-source cancellation')
+        # Nonuniform leaf weights are recounted from the actual emitted word;
+        # the unit-weight optimum is not asserted for this weighted bill.
+        counts = Counter(f for f, _, _, _ in word)
+        wa = sum(1+f%7 for f, _, _, _ in a)
+        wb = sum(1+f%7 for f, _, _, _ in b)
+        need(sum(counts[f]*(1+f%7) for f in counts) == 2*wa+wb, 'weighted bill')
+        algebra.append(dict(n=n, split=k, leaves=len(word),
+                            trace_sha256=digest(json.dumps(word).encode())))
+    controls = []
+    for name in ('dirty', 'source_alias', 'target_alias', 'shortage',
+                 'phase_alias_0', 'phase_alias_1', 'phase_alias_2',
+                 'phase_alias_3', 'phase_alias_4'):
+        try:
+            trace(ns, 7, 3, mutation=name)
+        except ValueError as error:
+            if name.startswith('phase_alias_'):
+                need(str(error) == 'phase array must be distinct', 'wrong phase rejection')
+            controls.append(name)
+        else:
+            raise ValueError('accepted interface control: ' + name)
+    n, k = 7, 3
+    p = cp.plan(n, k)
+    size = 3+p['dirty_arrays']
+    a = reference(list(range(k)), 0, 2, list(range(3, size)))
+    b = reference(list(range(k, n)), 2, 1, list(range(3, size)))
+    word = a+b+inverse(a)
+    state = initial(size)
+    good = replay(word, state)
+    mutants = {
+        'omitted_checkpoint_restoration': a+b,
+        'wrong_product_order': [(n-1-f,x,y,s) for f,x,y,s in word],
+    }
+    for name, mutant in mutants.items():
+        actual = replay(mutant, state)
+        need(actual != good, 'vacuous mutation: ' + name)
+        controls.append(name)
+    # An even two-factor product detects same-order sign inversion. For an
+    # odd-length ideal product that mutation can accidentally give the inverse.
+    short_word = trace(ns, 2, 1)
+    short_state = initial(3)
+    short_good = replay(short_word, short_state)
+    wrong_inverse = [(f,x,y,-s) for f,x,y,s in short_word]
+    need(replay(wrong_inverse, short_good) != short_state, 'wrong inverse control')
+    controls.append('wrong_inverse_chronology')
+    need(replay(word, initial(size, clean=False))[1] != good[1], 'dirty counterexample')
+    controls.append('dirty_checkpoint_leaks_to_target')
+    need(len(a)+len(b) != len(word), 'unpaid inverse control')
+    controls.append('unpaid_inverse_count')
+    rounded = module('rounded_check').run()
+    return dict(status='PASS_CLEAN_CHECKPOINT', assignment_seed=54,
+                base_commit=src['base_commit'], source_pins=src['sources'],
+                root_search=dict(min_n=2, max_n=256, candidates=candidates,
+                                 stopping_rule='all interior root splits; balanced subtrees'),
+                plans=plans, noncommutative_cases=algebra,
+                controls=controls, rounded=rounded,
+                selected=cp.plan(54),
+                scope='Paid clean root only; no new kappa, global scheduler optimum, or full theorem')
+
+
+def main():
+    parser = argparse.ArgumentParser()
+    parser.add_argument('--write', type=Path, help='write a fresh receipt instead of checking EXPECTED')
+    args = parser.parse_args()
+    result = run()
+    if args.write:
+        args.write.write_text(json.dumps(result, indent=2, sort_keys=True)+'\n')
+    else:
+        need(result == json.loads((HERE/'EXPECTED.json').read_text()), 'complete certificate mismatch')
+    print('PASS_CLEAN_CHECKPOINT: 32640 splits; 498 noncommutative cases; '
+          'n=54: 3136 -> 2112 leaves, 1 clean + 5 dirty arrays')
+
+
+if __name__ == '__main__':
+    main()