diff --git a/.github/workflows/cyclic-inverse-stride-bridge.yml b/.github/workflows/cyclic-inverse-stride-bridge.yml
new file mode 100644
index 0000000000..d30c9e27c6
--- /dev/null
+++ b/.github/workflows/cyclic-inverse-stride-bridge.yml
@@ -0,0 +1,27 @@
+name: Cyclic inverse stride bridge
+on:
+  pull_request:
+    paths:
+      - 'research/cyclic-inverse-stride-bridge/**'
+      - '.github/workflows/cyclic-inverse-stride-bridge.yml'
+  push:
+    paths:
+      - 'research/cyclic-inverse-stride-bridge/**'
+      - '.github/workflows/cyclic-inverse-stride-bridge.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  verify:
+    runs-on: ubuntu-latest
+    strategy:
+      matrix:
+        python: ['3.11', '3.14']
+    steps:
+      - uses: actions/checkout@v4
+      - uses: actions/setup-python@v5
+        with:
+          python-version: ${{ matrix.python }}
+      - run: python -B -I -O research/cyclic-inverse-stride-bridge/verify.py --check
diff --git a/research/cyclic-inverse-stride-bridge/EXPECTED.json b/research/cyclic-inverse-stride-bridge/EXPECTED.json
new file mode 100644
index 0000000000..5ec1bed50c
--- /dev/null
+++ b/research/cyclic-inverse-stride-bridge/EXPECTED.json
@@ -0,0 +1,3936 @@
+{
+ "controls": [
+  "rotated_closure_reseed_rejected",
+  "low_precision_refused",
+  "unit_coefficient_rejected",
+  "wrong_inner_shift_rejected"
+ ],
+ "generic": {
+  "cases": 2052,
+  "max_error_to_bound_squared": 0.00032978173727428434
+ },
+ "kernels": [
+  {
+   "dense": true,
+   "embedding": 1,
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+   "gauge": "1",
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+   "gauge": "1",
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+    "records": 1703948,
+    "records_per_entry": 13.000091552734375,
+    "stride": 2
+   },
+   {
+    "length": 131072,
+    "levels": 2,
+    "pr628_sort_route_records": 8912896,
+    "records": 1703997,
+    "records_per_entry": 13.000465393066406,
+    "stride": 3
+   },
+   {
+    "length": 131072,
+    "levels": 4,
+    "pr628_sort_route_records": 8912896,
+    "records": 1706775,
+    "records_per_entry": 13.021659851074219,
+    "stride": 97
+   },
+   {
+    "length": 131072,
+    "levels": 2,
+    "pr628_sort_route_records": 8912896,
+    "records": 2796190,
+    "records_per_entry": 21.333236694335938,
+    "stride": 43690
+   },
+   {
+    "length": 131072,
+    "levels": 1,
+    "pr628_sort_route_records": 8912896,
+    "records": 2097152,
+    "records_per_entry": 16.0,
+    "stride": 65536
+   },
+   {
+    "length": 131072,
+    "levels": 3,
+    "pr628_sort_route_records": 8912896,
+    "records": 3866635,
+    "records_per_entry": 29.500083923339844,
+    "stride": 65537
+   },
+   {
+    "length": 131072,
+    "levels": 9,
+    "pr628_sort_route_records": 8912896,
+    "records": 3733510,
+    "records_per_entry": 28.484420776367188,
+    "stride": 81006
+   },
+   {
+    "length": 131072,
+    "levels": 1,
+    "pr628_sort_route_records": 8912896,
+    "records": 2228230,
+    "records_per_entry": 17.000045776367188,
+    "stride": 131071
+   }
+  ]
+ }
+}
diff --git a/research/cyclic-inverse-stride-bridge/NOTICE.md b/research/cyclic-inverse-stride-bridge/NOTICE.md
new file mode 100644
index 0000000000..4dcef892b5
--- /dev/null
+++ b/research/cyclic-inverse-stride-bridge/NOTICE.md
@@ -0,0 +1,22 @@
+# Notice and attribution
+
+New material in this directory is by Rohan Guliani (@Rohansguliani) and is
+released under Apache-2.0. Substantial AI assistance was used in the
+derivation, implementation and writing. No human peer review or
+proof-assistant verification is claimed.
+
+Predecessor work:
+
+* **#621, Ali Romman (aliromman).** Kernel and the `L`-uniform circle and
+  inverse bounds (`research/prime-power-cyclic-inverse`).
+* **#628, Chafik Boukhalfa (chafreaky).** Certified approximate
+  factorization, preparation and phase budgets, the two-scan primitive, the
+  dirty-shear composition and the sort-route layout charge. The files in
+  `vendor/` are byte-identical to #628 head `0768f97d`. Its notice is
+  retained in `vendor/PR628-NOTICE.md`.
+* **#764, Charles Bethin (cbethin).** The shared-suffix identity
+  `1-Q >= beta G` and the inverse-square scan bound, used in Lemma 1. No
+  #764 code is consumed.
+
+The two-stack queue and Euclidean stride reduction are standard techniques.
+No literature-priority claim is made.
diff --git a/research/cyclic-inverse-stride-bridge/PROOF.md b/research/cyclic-inverse-stride-bridge/PROOF.md
new file mode 100644
index 0000000000..8987bfdfc4
--- /dev/null
+++ b/research/cyclic-inverse-stride-bridge/PROOF.md
@@ -0,0 +1,211 @@
+# Linear-traffic rounded inversion for every cyclic stride
+
+## Result and scope
+
+PR628 inverts PR621's kernel `P(tau Z_L^w)` for neighbouring strides with
+precision `p+O(r^4)` that does not depend on `L`. For other strides it routes
+the cycles into contiguous order with two merge sorts, at a cost of
+`O(L(B+log L)log L)`. That sort cost brings back a `log L` traffic factor.
+This note removes the sort. The solver below handles every stride `w` with
+`O(L)` sequential record moves and `O(L)` rounded operations per factor. Its
+error bound does not depend on `L`.
+
+**Theorem 1 (one stable factor, any stride).** Let `L>=1`, let `w` be any
+integer, and let `0<beta<=1/2`. Assume `|b_i|<=1-beta` and `|x_i|<=B`. The
+algorithm `solve` of `stride.py` returns `yhat`. Here `y` is the unique
+solution of `y_i = x_i + b_i y_((i-w) mod L)`. Let `K` be the number of
+non-base recursion levels, so `K <= floor(log2 L)+1`. Every complex
+operation is rounded once to a grid of spacing `u`. Assume the admission
+conditions `u<=beta/3` and `4(K+1)u<=beta`. Then
+
+    max_i |yhat_i - y_i| <= kappa (u/beta)(5 + 4B/beta) + u,
+    kappa = min(max(K,1), 1/beta).
+
+Traffic is at most `40L` record moves: a level of length `L_k`, with
+stride `w_k <= L_k/2`, costs at most `13L_k+5w_k` plus `4L_k` for an
+optional reversal, and `sum_k L_k <= 2L`. The solver performs `O(L)`
+rounded multiply-adds. All of this runs on a fixed number of sequential
+tapes.
+
+The bound stays finite as `L` grows because `kappa <= 1/beta`. The only
+dependence on `L` is in the admission condition `4(K+1)u<=beta`. That
+condition is `w_bits >= g+2+log2(K+1)`, where `beta=2^-g`. Any execution
+budget for target precision `p >= log2(log2 L + 2)` already meets it.
+
+## Algorithm
+
+Normalize `w` modulo `L`. If `2w>L`, reverse the arrays. Reversal turns the
+stride into `L-w`, which is at most `L/2`. A tape reads one array backwards
+while it writes another forwards, which costs `2L` record moves each way.
+If `w=0`, return `x_i/(1-b_i)` rounded.
+
+Otherwise the residue classes modulo `w` form chains. Chain `c` contains
+`c, c+w, c+2w, ...`. Each chain has at least two members because `w<=L/2`.
+
+1. **Pass A.** Scan `i=0..L-1`. Read the lag-`w` predecessor from a FIFO
+   that starts with `w` start markers. Compute the zero-seed value
+   `zhat_i = rnd(x_i + b_i zhat_(i-w))` and the chain product
+   `pihat_i = rnd(b_i pihat_(i-w))`. At a chain start, `zhat=x` and
+   `pihat=b` exactly.
+2. **Closure.** The last `w` positions `e=L-w+k` are the chain ends. The
+   start of chain `c` has predecessor `c-w+L`, which is end `k=c`. Hence the
+   end values satisfy
+
+       Y_k = zhat_(L-w+k) + pihat_(L-w+k) Y_((k+L) mod w).
+
+   This is the same problem on `w` entries with stride `(-L) mod w`.
+   Solve it recursively.
+3. **Pass C.** Scan `i=0..L-1` again. This time the FIFO starts with
+   `Y_0..Y_(w-1)`. Compute `yhat_i = rnd(x_i + b_i * pop())` and push it.
+
+Each level at least halves the length, so `sum_k L_k <= 2L` and
+`K <= floor(log2 L)+1`. A FIFO built from two stacks moves each record at
+most four times: a push, a transfer pop, a transfer push and a pop. A stack
+on a tape keeps its head at the top. Pass A, the closure handoff, pass C and
+the two reversals therefore cost at most a fixed multiple of `L_k` record
+moves at level `k`. In the finite replay through `L=131072` the maximum is
+29.68 per entry. For comparison, PR628's two sorts make
+`4L ceil(log2 L)` record moves. A record holds the `O(B)` bits of one or
+two numbers. Moving it costs `O(B)` tape cells. Loop counters need only
+amortized `O(1)` bit changes per step, plus `O(log L)` per level for the
+values `L_k` and `w_k`. Stride arithmetic over all levels costs
+`O(log^2 L)` bit operations.
+
+## Proof of Theorem 1
+
+**Notation.** `Shat_k` is the computed system at level `k`, with data
+`xhat^k` and coefficients `bhat^k`. `Ytilde^k` is its exact solution, and
+`m_k = min_i (1-|bhat^k_i|)` is its margin. `S*` is the exact Schur
+complement of `Shat_k` on the chain ends. The solution of `S*` is the
+restriction of `Ytilde^k` to those ends.
+
+**Lemma 1 (PR764's shared-suffix identity).** Let `q_1..q_l` lie in
+`[0,1-m]`. Put `P_j = q_(j+1)...q_l` and `G = sum_j P_j`. Then
+
+    sum_j (1-q_j) P_j = 1 - q_1...q_l,   hence   G <= (1-|pi|)/m.
+
+The quantity `pi` is the chain product. A rounded recursion multiplies its
+previous error by `b` and adds at most `u`, so after a chain its error is at
+most `uG`. Therefore
+
+    |zhat_e - z_e| <= u(1-|pi_e|)/m_k,   |pihat_e - pi_e| <= u(1-|pi_e|)/m_k.
+
+**Lemma 2 (margins never shrink below beta).** From Lemma 1,
+`1-|pihat| >= (1-|pi|)(1-u/m_k)`. A chain has at least two members, so
+`1-|pi| >= 1-(1-m_k)^2 = m_k(2-m_k)`. Hence `m_(k+1) >= f(m_k)` with
+`f(m)=(2-m)(m-u)`. The function `f` is concave on `[beta,1]`. At the
+endpoints,
+
+    f(beta) = (2-beta)(beta-u) >= (3/2)(2beta/3) = beta   when beta<=1/2 and u<=beta/3,
+    f(1) = 1-u >= beta.
+
+Therefore `m_k >= beta` at every level. The bound does not depend on depth.
+The replay checks `|bhat|<=1-beta` exactly at every level of all 2052
+generic cases.
+
+**Lemma 3 (weighted data).** Suppose `|xhat^k_i| <= A_k(1-|bhat^k_i|)`.
+Then the telescoping identity gives `|z_e| <= A_k(1-|pi_e|)`. Lemma 1 adds
+at most `u(1-|pi_e|)/m_k`, and Lemma 2 compares `1-|pi|` with `1-|pihat|`.
+Together they give the same invariant at level `k+1` with
+
+    A_(k+1) = (A_k + u/beta)/(1-u/beta),   A_0 = B/beta.
+
+The admission condition `4(K+1)u <= beta` gives `A_k <= 2A_0+1` at every
+level.
+
+**Lemma 4 (weighted inverse).** Suppose `|v_i| <= C(1-|b_i|)`. Then the
+solution of `Y = v + b W Y` satisfies `|Y_i| <= C`. The Neumann series has
+weights `P_j(1-q_j)`, and these telescope to at most one. In particular,
+`|Ytilde^k| <= A_k`.
+
+**Error recursion.** `Shat_(k+1)` differs from `S*` only by the pass-A
+rounding. The two solutions satisfy
+
+    (I - Bhat W)(Ytilde^(k+1) - Y*) = Delta x + Delta B * W Y*.
+
+By Lemma 1, each entry of the right side is at most
+`(u/beta)(1+A_k)(1-|pi_e|)`, and `1-|pi_e| <= (1-|pihat_e|)/(1-u/beta)`.
+Lemma 4 then bounds the solution difference by
+`(u/beta)(1+A_k)/(1-u/beta) <= 2(u/beta)(1+A_k)`. Pass C seeds every chain
+with the closure value. The seed error is multiplied by the chain prefix
+product. That product contains at least one coefficient, so its modulus is
+at most `1-beta`. Pass C's own rounding adds at most `u/beta`. Let
+`e_k = |Yhat^k - Ytilde^k|`. Then
+
+    e_k <= (1-beta)(e_(k+1) + 2(u/beta)(1+A_k)) + u/beta,
+    e_K <= u   (base case: one exactly rounded division, |1-b|>=beta).
+
+Sum this recursion. At level 0 the data are exact, so `Ytilde^0 = y`. With
+`A_k <= 2B/beta+1` and `sum_(k<K)(1-beta)^k <= min(K,1/beta)`, the theorem
+follows.
+
+## Theorem 2: the full kernel at any stride
+
+This theorem keeps PR628's certified factorization `Q`, its bound
+`||P(U)^-1-Q(U)^-1|| <= 2^(-p-1)`, its preparation charges and its phase
+charges. Every factor of `Q(U)^-1` with `U = tau Z_L^w` uses one call to
+Theorem 1:
+
+* **Outer root.** `(U-rho)^-1 = -rho^-1 (I - (tau/rho) Z^w)^-1`. Apply one
+  stride-`w` solve, then one rounded multiplication by `-1/rho`.
+* **Inner root.** `(U-rho)^-1 = conj(tau)(I - rho conj(tau) Z^-w)^-1 Z^-w`.
+  Apply an exact rotation by `w`, one stride-`(-w)` solve and one rounded
+  multiplication by `conj(tau)`.
+
+A tape performs the rotation by copying the tail and then the head, which
+costs `O(L)` record moves. Let `E_j` be the execution error after `j`
+factors, and let `N_j = r beta^-j` bound the exact prefix norm. Then
+
+    E_j <= E_(j-1)/beta + bound_1(N_(j-1)+E_(j-1)) + u,
+
+where `bound_1` is the bound of Theorem 1. The replay evaluates this
+recurrence exactly for each certified kernel. It requires
+`E_n + perturbation <= 2^-target`. In closed form, `kappa <= 1/beta` turns
+the per-factor constant into at most `C=(5+4B)/beta^3+2u`. PR628's `T_j`
+argument then gives
+
+    w_bits = p + (n+2) g + O(log r),   g = log2(1/beta) = O(r^3).
+
+This has the same order as PR628's neighbouring budget and contains no `L`
+term. For strides with small `K`, `kappa=K` applies instead, and the budget
+improves to `p+(n+1)g+O(log(rK))`. That form follows PR764's `beta^-2`
+shape. The replay uses the exact `min`.
+
+## What is paid, and the verdict on PR621's gain
+
+PR621's gain is an inverse whose cost per entry is independent of the
+cyclic length: `L`-uniform precision and work linear in `L`. The table
+charges each cost that the K492 review lists. All four apply to one kernel
+inverse at an arbitrary stride on fixed sequential tapes.
+
+| Cost | Charge | Source |
+|---|---|---|
+| Preparation | `soft-O(r^3(p+r^4))` bit operations, independent of `L`, with Pan's factorization and fast elementary functions; phases are paid per oracle call | PR628 (cited theorems, not implemented) |
+| Rounded execution | precision `p+O(r^4)`, uniform in `L`; per factor `O(L)` rounded operations | Theorems 1–2 here; PR764's identity |
+| Restoration | the input tape is never written; all scratch tapes are erased at `O(LB)` | this note, sequential-tape model |
+| Tape | at most `40L` record moves per factor at any stride, versus PR628's `O(L(B+log L)log L)` sorts | Theorem 1 here |
+
+The total for one kernel inverse is
+
+    soft-O(r^3(p+r^4)) + O(r L) M(p+r^4) + O(r L (p+r^4)),
+
+where `M` is the cost of integer multiplication. The per-entry cost does not
+depend on `L`, so **the gain survives payment at the level of one kernel
+inverse.** The only `L`-dependent term is the admission requirement
+`p >= log2(log2 L+2)`.
+
+## Not closed here
+
+* PR628's dirty-array contract allows `ceil(log2(r-1))` borrowed arrays and
+  only `O(1)` clean scalars. That contract arises in the exact-width word
+  model. The stride solver here uses blank scratch tape, which is free in
+  the sequential-tape model and is erased at linear cost. It has not been
+  recast for borrowed arrays. That case still uses PR628's
+  `O(p+log(D+1)+r^5)` precision and neighbouring-stride leaves.
+* The guard `O(r^4)` is a proof bound. Numerical probes for `r<=32` put
+  `min_|z|=1 |P(z)|` near `2^-10`, while PR621's bound gives up to
+  `2^-27880`. A sharper lower bound would shrink every budget above.
+  This package does not certify those probe values.
+* Composition across the complete transform, the transform's phase
+  collection, normalization and the integer-multiplication reduction remain
+  open. No integer kappa, complex b, or BIT c changes.
diff --git a/research/cyclic-inverse-stride-bridge/README.md b/research/cyclic-inverse-stride-bridge/README.md
new file mode 100644
index 0000000000..f33410800c
--- /dev/null
+++ b/research/cyclic-inverse-stride-bridge/README.md
@@ -0,0 +1,81 @@
+# Cyclic inverse stride bridge
+
+This package continues the open obligation recorded for #621 in
+[the K492 review](../../docs/research/community-k492-review.md). That
+obligation is to pay for efficient preparation, rounded execution,
+restoration and tape before using #621's length-independent inverse bound.
+
+#628 pays for preparation and neighbouring-stride execution. #764 sharpens
+the scan rounding. Both PRs are still open and are pinned in
+[SOURCE.json](SOURCE.json). For any stride other than ±1, #628's route
+depends on two merge sorts costing `O(L(B+log L)log L)`. Those sorts bring
+back a `log L` factor that the inverse bound was supposed to remove.
+
+[PROOF.md](PROOF.md) replaces the sorts with a Euclidean chain/closure
+recursion. A FIFO built from two stacks gives every pass a lag-`w`
+predecessor in natural order. Theorem 1 gives one stable factor at any
+stride with:
+
+* at most `40L` sequential record moves;
+* `O(L)` rounded operations;
+* an error bound `min(K,1/beta)(u/beta)(5+4B/beta)+u` that holds for every
+  `L`.
+
+The proof reuses #764's shared-suffix identity. It also shows that margins
+do not shrink across recursion levels. Theorem 2 composes the solver with
+#628's certified factorization. The kernel precision is
+`p+(n+2)g+O(log r)=p+O(r^4)`, with no `L` term. One admission condition,
+`p >= log2(log2 L+2)`, depends on `L`. The table in PROOF.md charges every
+listed cost.
+
+**Verdict:** consider one inverse of `P(tau Z_L^w)` on fixed sequential
+tapes, with all four listed costs paid. Its cost per entry still does not
+depend on `L`, so #621's gain survives. Three things are not claimed:
+
+* a borrowed-array (dirty) form of the stride solver;
+* composition across the complete transform;
+* any change to integer kappa, complex b, or BIT c.
+
+## Replay
+
+Python 3.11 or later with the standard library only. The replay takes
+about 100 seconds:
+
+```sh
+python3 -B -I -O research/cyclic-inverse-stride-bridge/verify.py --check
+```
+
+Expected: `PASS_CYCLIC_INVERSE_STRIDE_BRIDGE generic=2052 kernel_cases=300
+controls=4 max_records_per_entry=29.671875`.
+
+The replay covers the following checks:
+
+* **Generic cases (2052).** `L=1..18`, every stride in `[-L,L)`,
+  `beta in {1/4,1/8,1/32}` and two grids. Each case is compared exactly
+  with an independent per-cycle geometric-closure oracle against Theorem 1.
+  The margin `|bhat|<=1-beta` is checked at every recursion level.
+* **Traffic.** Count-only runs reach `L=131072`. Strides include `0`, `±1`,
+  `L/2`, `L/2+1`, golden-ratio strides and the prime `L=99991`. Every run is
+  checked against `40L` record moves. The maximum observed is
+  29.68 moves per entry, compared with #628's
+  sort route of `4L ceil(log2 L)`.
+* **Kernel cases (300).** These use all ten #628 certified kernels, radix 3
+  through 16. Gauges are `1` and `3/5+4i/5`. Lengths are 1, 2, 3, 5 and 8,
+  with all strides for `L<=5`. Each case evaluates Theorem 2's exact budget
+  against the target together with #628's perturbation bound. Each also
+  checks the expanded polynomial residual. 81 cases are compared with an
+  independent dense Gauss–Jordan solve.
+* **Controls (4).**
+  * a rotated closure reseed;
+  * a precision below the admission condition;
+  * a unit coefficient;
+  * a wrong inner-shift direction.
+
+  The replay rejects all four.
+
+The vendored #628 files must match the hashes in SOURCE.json byte for byte.
+The verifier refuses drift. `--write` regenerates `EXPECTED.json`.
+
+Finite cases support the reference implementation. The written proof
+carries the all-`L` claims. The Python lists model sequential tapes and
+count record moves; they are not a literal Turing machine.
diff --git a/research/cyclic-inverse-stride-bridge/SOURCE.json b/research/cyclic-inverse-stride-bridge/SOURCE.json
new file mode 100644
index 0000000000..e1624e72dd
--- /dev/null
+++ b/research/cyclic-inverse-stride-bridge/SOURCE.json
@@ -0,0 +1,15 @@
+{
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "pr621": {"url": "https://github.com/CrocSwap/integer-mult-bounds/pull/621", "head": "520774462b19b29af41554b74def31f9d01e997e", "author": "Ali Romman (aliromman)", "use": "circle lower bound and L-uniform inverse bound; on main at research/prime-power-cyclic-inverse"},
+  "pr628": {"url": "https://github.com/CrocSwap/integer-mult-bounds/pull/628", "head": "0768f97d06c623b8211e651543c66135796b62bf", "author": "Chafik Boukhalfa (chafreaky)", "use": "certified factors, preparation and perturbation budget, dirty-shear composition, sort-route layout charge; vendored files below are byte-identical to that head"},
+  "pr764": {"url": "https://github.com/CrocSwap/integer-mult-bounds/pull/764", "head": "983d66f0f2e48ee500ec7eee2da9244c001c5064", "author": "Charles Bethin (cbethin)", "use": "shared-suffix identity 1-Q >= beta*G, reused in Lemma 1; no code consumed"},
+  "vendored_sha256": {
+    "check.py": "a3e46b556894d71a1c83bbbac01c50c1a541285fc9422a3567d5eb1d1ee86e7d",
+    "FACTORS.json": "643d60ac24d5e67f6c9305175bd1c99bf5e42edfef6d1995c051e0b98ba446a7",
+    "interval_kernel.py": "9b7bb24640a9a7d3f83c7951dfef483ec933db4c50de6c0130b85f450eeddfbe",
+    "scan.py": "1a9c01ca1a29a9be1e756858161202621db29d5d332a4266b4017a9502a50d7e",
+    "PR628-NOTICE.md": "abeacf1ad1491d5797cfcf1f3832799bbf3b24b99597229d7bc884059a51ec68"
+  },
+  "two_stack_queue": "standard amortized queue from two stacks (folklore; e.g. Okasaki, Purely Functional Data Structures, 1998)",
+  "unimplemented": "Pan worst-case factorizer, fast elementary functions, a literal Turing-machine tape, borrowed-array (dirty) form of the stride solver, transform-level composition and the multiplication reduction."
+}
diff --git a/research/cyclic-inverse-stride-bridge/stride.py b/research/cyclic-inverse-stride-bridge/stride.py
new file mode 100644
index 0000000000..9229a4193b
--- /dev/null
+++ b/research/cyclic-inverse-stride-bridge/stride.py
@@ -0,0 +1,168 @@
+"""Rounded stride-w cyclic first-order solver with linear sequential traffic.
+
+Solves y_i = x_i + b_i * y_((i-w) mod L) for |b_i| <= 1-beta by a
+Euclidean chain/closure recursion. Every array pass is in natural order
+(or its reverse) and reads the lag-w predecessor from a FIFO built from two
+stacks, which a fixed number of sequential tapes implement with amortized
+O(1) record moves. See PROOF.md. Exact rational arithmetic is used; the only
+inexactness is the explicit nearest-dyadic rounding on a grid of spacing u.
+"""
+from fractions import Fraction as F
+
+ZERO = (F(0), F(0))
+ONE = (F(1), F(0))
+
+
+def add(a, b): return (a[0] + b[0], a[1] + b[1])
+def sub(a, b): return (a[0] - b[0], a[1] - b[1])
+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):
+    """Nearest point of the grid 2^-bits in each component, ties upward."""
+    if bits is None:
+        return a
+    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]))
+
+
+class Tape:
+    """Counts sequential record traffic. One unit is one B-bit record moved
+    past a tape head (read or write). Stacks are tapes whose head sits at
+    the top; a FIFO is two stacks; reversal copies one record per unit."""
+
+    def __init__(self):
+        self.records = 0
+        self.levels = []
+
+
+class Fifo:
+    def __init__(self, tape, initial):
+        self.tape = tape
+        self.inbox = []
+        self.outbox = []
+        for v in initial:
+            self.push(v)
+
+    def push(self, v):
+        self.tape.records += 1
+        self.inbox.append(v)
+
+    def pop(self):
+        if not self.outbox:
+            while self.inbox:  # one transfer per record, amortized O(1)
+                self.tape.records += 2
+                self.outbox.append(self.inbox.pop())
+        self.tape.records += 1
+        return self.outbox.pop()
+
+
+def solve(x, b, w, bits, tape, trace=None, level=0):
+    """Return the rounded solution. trace collects per-level (L, w, data,
+    coefficients) for invariant checks. Values may be None in count-only
+    mode (structure and traffic only)."""
+    L = len(x)
+    need(L >= 1 and len(b) == L, 'shape')
+    w %= L
+    if 2 * w > L:  # reverse orientation: stride L-w <= L/2
+        tape.records += 2 * L  # read backwards, write forwards
+        y = solve(x[::-1], b[::-1], L - w, bits, tape, trace, level)
+        tape.records += 2 * L
+        return y[::-1]
+    tape.levels.append((L, w))
+    if trace is not None:
+        trace.append(dict(level=level, length=L, stride=w, x=x, b=b))
+    count_only = x[0] is None
+    if w == 0:  # every point is its own cycle
+        tape.records += 2 * L
+        if count_only:
+            return [None] * L
+        return [rounded(div(xi, sub(ONE, bi)), bits) for xi, bi in zip(x, b)]
+    # Pass A: zero-seed chains along +w, with rounded chain products.
+    fifo = Fifo(tape, [None] * w)
+    z = [None] * L
+    p = [None] * L
+    for i in range(L):
+        tape.records += 2  # read x_i, b_i
+        prev = fifo.pop()
+        if count_only:
+            fifo.push(None)
+            continue
+        if prev is None:
+            z[i], p[i] = x[i], b[i]
+        else:
+            z[i] = rounded(add(x[i], mul(b[i], prev[0])), bits)
+            p[i] = rounded(mul(b[i], prev[1]), bits)
+        fifo.push((z[i], p[i]))
+    # Closure system on the w chain ends: Y_k = c_k + a_k Y_((k+L) mod w).
+    c, a = z[L - w:], p[L - w:]
+    tape.records += 2 * w  # the ends are the tail of the pass-A output
+    Y = solve(c, a, (-L) % w, bits, tape, trace, level + 1)
+    # Pass C: reseed chain starts from the closure values (Y_k feeds start k).
+    fifo = Fifo(tape, Y)
+    y = [None] * L
+    for i in range(L):
+        tape.records += 3  # read x_i, b_i; write y_i
+        prev = fifo.pop()
+        if not count_only:
+            y[i] = rounded(add(x[i], mul(b[i], prev)), bits)
+        fifo.push(y[i])
+    return y
+
+
+def levels(L, w):
+    """Number of non-base recursion levels K (stride halves every level)."""
+    K = 0
+    while True:
+        w %= L
+        if 2 * w > L:
+            w = L - w
+        if w == 0:
+            return K
+        L, w, K = w, (-L) % w, K + 1
+
+
+def exact_cycle_solve(x, b, w):
+    """Independent oracle: per-cycle finite geometric closure."""
+    L = len(x)
+    y = [None] * L
+    for i in range(L):
+        total, weight, j = ZERO, ONE, i
+        while True:
+            total = add(total, mul(weight, x[j]))
+            weight = mul(weight, b[j])
+            j = (j - w) % L
+            if j == i:
+                break
+        y[i] = div(total, sub(ONE, weight))
+    return y
+
+
+def solve_error_bound(B, beta, bits, K):
+    """PROOF.md, Theorem 1: kappa*(u/beta)*(5+4B/beta)+u, kappa=min(K,1/beta)."""
+    u = F(1, 1 << bits)
+    kappa = min(F(max(K, 1)), 1 / beta)
+    return kappa * (u / beta) * (5 + 4 * B / beta) + u
+
+
+def admitted(beta, bits, K):
+    u = F(1, 1 << bits)
+    return beta <= F(1, 2) and u <= beta / 3 and u * 4 * (K + 1) <= beta
diff --git a/research/cyclic-inverse-stride-bridge/vendor/FACTORS.json b/research/cyclic-inverse-stride-bridge/vendor/FACTORS.json
new file mode 100644
index 0000000000..d3894bd2a1
--- /dev/null
+++ b/research/cyclic-inverse-stride-bridge/vendor/FACTORS.json
@@ -0,0 +1,402 @@
+[
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+        "28326730094049526360308020645864856724598891128838513609367588904514833835204347361667185978800994436970191543152173345365891935121884286237303162650500563575271973969509154285033455092089962932786873166080400502570268738193931444116625301385853996639117280974700258567439169279262226550039720244993440086577/179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216"
+      ],
+      [
+        "-212793201622109962448330100692808593439645159644577548679667883054110536914612983822159045347472254754395342771708966598767283083523909873556284381276124835470115664134649247694090104295004718385271153849431718533152316853933967919972046963990390480364949318130626453951329110678301062337081449200475601735875/179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216",
+        "-57964249005932469131896837719919287256295100537523457777437606447238450243954529094264672294496860934864424519386491456408615251057823684209861120741434670159618673936343232127524840096572869938406481577342691505214852241781904532730223280532007263456890157126325151281429988053858853115810915974956182248013/179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216"
+      ],
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+      ],
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+        "-56200741848002868079111868432724886507129621548125121648642474266599250489673841691778046881437449293933434722648357280706913445930283045991236908716738562916199191451345481529063774448710946467551285759609323141571550416355308596877200595274065373565280076263027603986562578831336689401091825392816459762247/44942328371557897693232629769725618340449424473557664318357520289433168951375240783177119330601884005280028469967848339414697442203604155623211857659868531094441973356216371319075554900311523529863270738021251442209537670585615720368478277635206809290837627671146574559986811484619929076208839082406056034304"
+      ],
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+        "15968822176892792285891476438139425317687583108455281192008770761103687215213981897594608046751009662616135817210933151016991807601573680585709456982599855455736137633949541869722566316681492518193528500018722529515726386747292338115274696063766393314538517893691137581991598848479112650484943606294759671/1404447761611184302913519680303925573139044514798677009948672509044786529730476274474284979081308875165000889686495260606709295068862629863225370551870891596701311667381761603721111090634735110308227210563164107569048052205800491261514946176100212790338675864723330454999587858894372783631526221325189251072",
+        "-266324790638046954468748086815326096854011998014910958879637384810491281671701735342579622902906841022217262017537242815542832447116697353634706329017518582629343845840130883918382746688279956080610789278337184153491906948023245930590272626869013417164765399270940382908317755754807962553226947387458433481969/179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216"
+      ],
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+        "1060865642031292630770255648031651464303955219241748845301061404301449150107131123877033982501236564870546734157170738071205142825981730629211934466958373310795930234883644313113012070731987912222503770737372313560840648279391481374684780737322726034313967342539259478356716096981485614792922808816297076935/1404447761611184302913519680303925573139044514798677009948672509044786529730476274474284979081308875165000889686495260606709295068862629863225370551870891596701311667381761603721111090634735110308227210563164107569048052205800491261514946176100212790338675864723330454999587858894372783631526221325189251072",
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+      ],
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+        "74864797941206513281047175374406371423373655545874336711328481273747977974984455616940936075339467433634151221026156653070150698647294564580345247390350645817572015177013890581338145235391049860293804890641833349803840457797326162410138640935908664989180260619163393823243511563841284652103776498758852275761/179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216",
+        "158507016237416030279508951176317259509379692049347194102510424353036380843727700204504205980624447511489013333078714007658126019484831026660454241743943509290714726174111157436372813963025263036212831092360360315950519574485871493725846468575071813444421951074798721371147434849601216160281721594412331444679/179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216"
+      ],
+      [
+        "126287595802823388378145790573347116516250877023657551588569070335106577368621778092749228800089231992059072999369812292461321688715048725080489109408634374024142218044000898630952519239661477382516567361587110636505950019813691734756152201601098996057298499115007197346303456331071237494888612444968034206643/89884656743115795386465259539451236680898848947115328636715040578866337902750481566354238661203768010560056939935696678829394884407208311246423715319737062188883946712432742638151109800623047059726541476042502884419075341171231440736956555270413618581675255342293149119973622969239858152417678164812112068608",
+        "-143526397691224128056251209454968080411106577138808675995376721394908697117978740021220231248704896684803444156609685788793425078524968774144075750535259582404256698559171991950020896681595928832179228331976729295065837037698098662420392076528307364731612360503267270299597486337931330781272500922478650324859/179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216"
+      ]
+    ],
+    "beta": "1/512",
+    "factor_bits": 1024,
+    "working_bits": 768,
+    "target_bits": 128
+  }
+]
diff --git a/research/cyclic-inverse-stride-bridge/vendor/PR628-NOTICE.md b/research/cyclic-inverse-stride-bridge/vendor/PR628-NOTICE.md
new file mode 100644
index 0000000000..22fbeb8ef0
--- /dev/null
+++ b/research/cyclic-inverse-stride-bridge/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/cyclic-inverse-stride-bridge/vendor/check.py b/research/cyclic-inverse-stride-bridge/vendor/check.py
new file mode 100644
index 0000000000..5ef6dd67b6
--- /dev/null
+++ b/research/cyclic-inverse-stride-bridge/vendor/check.py
@@ -0,0 +1,178 @@
+"""Rational proof of mixed inside/outside cyclic factor inversion."""
+from pathlib import Path
+from fractions import Fraction as F
+import argparse,importlib.util,json,math,sys
+sys.set_int_max_str_digits(100000)
+P=Path(__file__).resolve().parent
+def module(name):
+    s=importlib.util.spec_from_file_location(name,P/(name+'.py'));m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m
+c=module('scan');ik=module('interval_kernel')
+
+def prime_power(r):
+    if type(r)!=int or r<3:return False
+    p=next((d for d in range(2,math.isqrt(r)+1) if r%d==0),r)
+    while r%p==0:r//=p
+    return r==1
+
+def certify(data):
+    r,k=data['radix'],data['embedding']
+    c.need(prime_power(r) and type(k)==int and 0<k<r and math.gcd(r,k)==1,'radix/embedding')
+    ci=ik.kernel(r,k)
+    beta=F(data['beta']);bits=data['working_bits'];target=data['target_bits'];fb=data['factor_bits']
+    c.need(0<beta<=1 and type(bits)==int and bits>0 and type(target)==int and target>0,'precision budget')
+    c.need(type(fb)==int and 0<fb<=1024,'factor bit declaration')
+    roots=[tuple(F(v) for v in z) for z in data['roots']]
+    c.need(len(roots)==r-1 and all(len(z)==2 for z in roots),'factor count')
+    q=[c.ONE];factors=[]
+    for z in roots:
+        c.need(all(v.denominator&(v.denominator-1)==0 and v.denominator<=1<<fb for v in z),'dyadic root')
+        inside=c.norm2(z)<1
+        b=z if inside else c.div(c.ONE,z)
+        c.need(c.norm2(b)<=(1-beta)**2,'stable margin')
+        factors.append((inside,b))
+        new=[c.ZERO]*(len(q)+1)
+        for j,v in enumerate(q):
+            new[j]=c.sub(new[j],c.mul(z,v));new[j+1]=c.add(new[j+1],v)
+        q=new
+    q=[(v[0]/r,v[1]/r) for v in q]
+    epsilon=F(0)
+    for box,z in zip(ci,q):
+        epsilon+=sum(max(abs(t-v) for t in iv) for iv,v in zip(box,z))
+    A=F(r)*beta**-(r-1);t=A*epsilon
+    c.need(t<1,'coefficient residual')
+    perturbation=A*t/(1-t)
+    u=F(1,1<<bits);c.need(u<=beta**2/2,'closure precision')
+    error=F(0);amplification=F(r);budgets=[]
+    for inside,b in factors:
+        # Includes final rounded multiplication by -b for outer roots.
+        error=error/beta+7*(amplification+error+1)*u/beta**3
+        amplification/=beta;budgets.append(error)
+    c.need(error+perturbation<=F(1,1<<target),'target error')
+    return dict(radix=r,embedding=k,factors=factors,beta=beta,bits=bits,q=q,budgets=budgets,
+                certificate=dict(radix=r,embedding=k,inner_factors=sum(t[0] for t in factors),outer_factors=sum(not t[0] for t in factors),
+                coefficient_error_bound=str(epsilon),inverse_norm_bound=str(A),factorization_inverse_error_bound=str(perturbation),
+                rounded_execution_error_bound=str(error),total_error_bound=str(error+perturbation),target_bits=target,
+                working_bits=bits,factor_bits=fb,beta=str(beta),all_cycle_lengths=True))
+
+def apply(x,state,direction,tau=c.ONE,omit_inner_shift=False):
+    c.need(c.norm2(tau)==1,'unit gauge')
+    r=state['radix'];y=[(v[0]*r,v[1]*r) for v in x]
+    for inside,b in state['factors']:
+        conjugate=(tau[0],-tau[1])
+        coefficient=c.mul(b,conjugate if inside else tau)
+        y=c.scan(y,coefficient,state['beta'],state['bits'],-direction if inside else direction)
+        if inside:
+            if not omit_inner_shift:y=[y[(i+direction)%len(y)] for i in range(len(y))]
+            y=[c.rounded(c.mul(conjugate,v),state['bits']) for v in y]
+        else:y=[c.rounded(c.neg(c.mul(b,v)),state['bits']) for v in y]
+    return y
+
+def phased_polynomial(q,tau):
+    power=c.ONE;out=[]
+    for v in q:out.append(c.mul(v,power));power=c.mul(power,tau)
+    return out
+
+def matrix_solve(x,q,direction):
+    n=len(x);a=[[c.ZERO for _ in range(n)]+[x[i]] for i in range(n)]
+    for i in range(n):
+        for j,v in enumerate(q):
+            column=(i-direction*j)%n;a[i][column]=c.add(a[i][column],v)
+    for k in range(n):
+        pivot=next(i for i in range(k,n) if a[i][k]!=c.ZERO);a[k],a[pivot]=a[pivot],a[k]
+        p=a[k][k];a[k]=[c.div(v,p) for v in a[k]]
+        for i in range(n):
+            if i==k:continue
+            p=a[i][k];a[i]=[c.sub(v,c.mul(p,w)) for v,w in zip(a[i],a[k])]
+    return [row[-1] for row in a]
+
+def residual_check(x,y,state,direction,tau=c.ONE):
+    q=phased_polynomial(state['q'],tau);bound=sum(abs(v[0])+abs(v[1]) for v in q)*state['budgets'][-1]
+    for i in range(len(x)):
+        total=c.ZERO
+        for j,v in enumerate(q):total=c.add(total,c.mul(v,y[(i-direction*j)%len(x)]))
+        c.need(c.norm2(c.sub(total,x[i]))<=bound**2,'expanded residual')
+
+def kernel_crosschecks():
+    # Independent closed coefficients in Q(i) and Q(sqrt(2),i).
+    exact4=[(F(1),F(0)),(F(1,2),F(1,2)),(F(1,4),F(1,4)),(F(1,4),F(0))]
+    for box,z in zip(ik.kernel(4,1),exact4):
+        c.need(all(iv[0]<=v<=iv[1] for iv,v in zip(box,z)),'Gaussian kernel crosscheck')
+    raw=[[1,0,0,0],[F(1,2)]*4,[0,0,F(1,2),F(1,2)],[0,0,0,F(1,2)],
+         [0,0,0,F(1,4)],[F(1,8),F(-1,8),F(1,8),F(1,8)],[F(1,8),0,0,F(1,8)],[F(1,8),0,0,0]]
+    scale=1<<4096;v=math.isqrt((1<<8192)//2);s=(F(v,scale),F(v+1,scale))
+    c.need(2*s[0]**2<1<2*s[1]**2,'independent sqrt enclosure')
+    for box,a in zip(ik.kernel(8,1),raw):
+        for iv,constant,multiplier in ((box[0],a[0],a[1]-a[3]),(box[1],a[2],a[1]+a[3])):
+            values=[constant+multiplier*t for t in s]
+            c.need(iv[0]<=min(values)<=max(values)<=iv[1],'radix8 independent coefficient enclosure')
+    return ['radix4_exact_Gaussian_coefficients','radix8_independent_algebraic_coefficients']
+
+def irrational_gauge():
+    # Enclose tau=exp(2*pi*i/7), then project a dyadic stereographic
+    # parameter to an exactly rational point on the unit circle.
+    box=ik.root_interval(7,1)
+    choices=[]
+    for center in (c.ONE,(F(-1),F(0)),(F(0),F(1)),(F(0),F(-1))):
+        conjugate=(center[0],-center[1])
+        rotated=ik.cmul(box,(ik.point(conjugate[0]),ik.point(conjugate[1])))
+        choices.append((rotated[0][0],center,rotated))
+    _,center,v=max(choices,key=lambda x:x[0])
+    c.need(v[0][0]>=F(1,2),'gauge chart')
+    parameters=[im/(1+re) for im in v[1] for re in v[0]]
+    mid=(min(parameters)+max(parameters))/2
+    t=c.rounded((mid,F(0)),1024)[0]
+    sigma=c.mul(center,((1-t*t)/(1+t*t),2*t/(1+t*t)))
+    c.need(c.norm2(sigma)==1,'exact rational unit phase')
+    error=sum(max(abs(endpoint-coordinate) for endpoint in iv) for iv,coordinate in zip(box,sigma))
+    c.need(error<F(1,1<<1020),'phase approximation error')
+    return sigma,error,dict(target='exp(2*pi*i/7)',rational_unit=[str(v) for v in sigma],error_bound=str(error),parameter_bits=1024)
+
+def main():
+    parser=argparse.ArgumentParser();parser.add_argument('--out',required=True,type=Path);args=parser.parse_args()
+    crosschecks=kernel_crosschecks();sigma,gauge_error,gauge_record=irrational_gauge()
+    data=json.loads((P/'FACTORS.json').read_text());certificates=[];cases=[]
+    for row in data:
+        state=certify(row)
+        # The approximation of the target phase is charged as an operator
+        # coefficient perturbation, using both phases' unit modulus.
+        derivative=sum(j*(abs(v[0])+abs(v[1])) for j,v in enumerate(state['q']))
+        combined=F(state['certificate']['coefficient_error_bound'])+derivative*gauge_error
+        A=F(state['certificate']['inverse_norm_bound']);c.need(A*combined<1,'gauged residual contraction')
+        gauged_error=A*A*combined/(1-A*combined)+state['budgets'][-1]
+        c.need(gauged_error<=F(1,1<<row['target_bits']),'irrational gauge target')
+        state['certificate']['irrational_gauge_total_error_bound']=str(gauged_error)
+        certificates.append(state['certificate'])
+        for n in (1,2,3,8,17,64):
+            for direction in (-1,1):
+                for gauge_name,tau in (('1',c.ONE),('i',(F(0),F(1))),('3/5+4i/5',(F(3,5),F(4,5))),('certified_exp(2pi_i/7)',sigma)):
+                    x=[(F((5*i)%7-3,8),F((7*i)%11-5,16)) for i in range(n)]
+                    y=apply(x,state,direction,tau);residual_check(x,y,state,direction,tau)
+                    if n<=3:
+                        exact=matrix_solve(x,phased_polynomial(state['q'],tau),direction)
+                        c.need(all(c.norm2(c.sub(v,w))<=state['budgets'][-1]**2 for v,w in zip(y,exact)),'independent dense comparison')
+                    cases.append(dict(radix=row['radix'],embedding=row['embedding'],length=n,direction=direction,gauge=gauge_name,dense_comparison=n<=3,residual=True))
+        print(json.dumps(dict(radix=row['radix'],embedding=row['embedding'],inner=state['certificate']['inner_factors'],status='passed')),flush=True)
+    controls=[]
+    mixed=next(row for row,cert in zip(data,certificates) if cert['inner_factors']>0)
+    for name in ('unit_root','missing_root','wrong_embedding','low_precision','bad_margin'):
+        row=json.loads(json.dumps(mixed))
+        if name=='unit_root':row['roots'][0]=['1','0']
+        if name=='missing_root':row['roots'].pop()
+        if name=='wrong_embedding':row['embedding']=row['radix']
+        if name=='low_precision':row['working_bits']=8
+        if name=='bad_margin':row['beta']='1'
+        try:certify(row)
+        except ValueError as error:controls.append(dict(name=name,rejected=str(error)))
+        else:raise ValueError('bad control accepted '+name)
+    state=certify(mixed);x=[c.ONE,c.ZERO,(F(1,2),F(1,4))]
+    bad=apply(x,state,1,omit_inner_shift=True)
+    try:residual_check(x,bad,state,1)
+    except ValueError as error:controls.append(dict(name='omitted_inner_shift',rejected=str(error)))
+    else:raise ValueError('inner-shift mutation accepted')
+    try:apply(x,state,1,(F(1),F(1)))
+    except ValueError as error:controls.append(dict(name='nonunit_gauge',rejected=str(error)))
+    else:raise ValueError('nonunit gauge accepted')
+    args.out.write_text(json.dumps(dict(status='passed',kernel_crosschecks=crosschecks,irrational_gauge=gauge_record,certificates=certificates,cases=cases,controls=controls),indent=2)+'\n')
+    print(json.dumps(dict(status='passed',certificates=len(certificates),cases=len(cases),dense_cases=sum(row['dense_comparison'] for row in cases),controls=len(controls))))
+
+if __name__=='__main__':main()
diff --git a/research/cyclic-inverse-stride-bridge/vendor/interval_kernel.py b/research/cyclic-inverse-stride-bridge/vendor/interval_kernel.py
new file mode 100644
index 0000000000..1182175a55
--- /dev/null
+++ b/research/cyclic-inverse-stride-bridge/vendor/interval_kernel.py
@@ -0,0 +1,61 @@
+"""Outward-rounded rational enclosures for prime-power cyclic kernels."""
+from fractions import Fraction as F
+import math
+from functools import lru_cache
+
+BITS=1536
+SCALE=1<<BITS
+def hull(lo,hi):
+    a=lo*SCALE;b=hi*SCALE
+    return (F(a.numerator//a.denominator,SCALE),F(-((-b.numerator)//b.denominator),SCALE))
+def point(x):return (F(x),F(x))
+def add(a,b):return hull(a[0]+b[0],a[1]+b[1])
+def neg(a):return (-a[1],-a[0])
+def sub(a,b):return add(a,neg(b))
+def mul(a,b):
+    v=[x*y for x in a for y in b];return hull(min(v),max(v))
+def scale(a,s):return mul(a,point(s))
+def cadd(a,b):return (add(a[0],b[0]),add(a[1],b[1]))
+def cneg(a):return (neg(a[0]),neg(a[1]))
+def cmul(a,b):return (sub(mul(a[0],b[0]),mul(a[1],b[1])),add(mul(a[0],b[1]),mul(a[1],b[0])))
+ZERO=(point(0),point(0));ONE=(point(1),point(0))
+
+def atan_inverse(q):
+    # Alternating series, with an exact bound by the next omitted term.
+    total=F(0);power=F(1,q);j=0
+    while True:
+        total+=(-1 if j%2 else 1)*power/(2*j+1)
+        power/=q*q;j+=1
+        remainder=power/(2*j+1)
+        if remainder<F(1,1<<(BITS+16)):break
+    other=total+(-1 if j%2 else 1)*remainder
+    return hull(min(total,other),max(total,other))
+
+@lru_cache(maxsize=None)
+def pi_interval():return sub(scale(atan_inverse(5),16),scale(atan_inverse(239),4))
+
+def root_interval(r,k):
+    if r<3 or math.gcd(r,k)!=1 or not 0<k<r:raise ValueError('invalid primitive embedding')
+    signed=k if 2*k<=r else k-r
+    theta=scale(pi_interval(),F(2*signed,r))
+    if max(abs(theta[0]),abs(theta[1]))>4:raise ValueError('angle bound')
+    itheta=(point(0),theta)
+    total=term=ONE
+    # Taylor remainder for complex exp: <=exp(4)*4^(N+1)/(N+1)! <128*...
+    n=0
+    while True:
+        n+=1;term=cmul(term,itheta);term=(scale(term[0],F(1,n)),scale(term[1],F(1,n)));total=cadd(total,term)
+        remainder=F(128*4**(n+1),math.factorial(n+1))
+        if remainder<F(1,1<<(BITS+16)):break
+    widen=(-remainder,remainder)
+    return (add(total[0],widen),add(total[1],widen))
+
+@lru_cache(maxsize=None)
+def kernel(r,k):
+    z=root_interval(r,k)
+    powers=[ONE]
+    for _ in range(r-1):powers.append(cmul(powers[-1],z))
+    # f_j=product_(s=j+1)^(r-1)(1-zeta^s), P=f/r.
+    a=[ZERO]*r;a[-1]=ONE
+    for j in range(r-2,-1,-1):a[j]=cmul(a[j+1],cadd(ONE,cneg(powers[j+1])))
+    return [(scale(x,F(1,r)),scale(y,F(1,r))) for x,y in a]
diff --git a/research/cyclic-inverse-stride-bridge/vendor/scan.py b/research/cyclic-inverse-stride-bridge/vendor/scan.py
new file mode 100644
index 0000000000..d620e49359
--- /dev/null
+++ b/research/cyclic-inverse-stride-bridge/vendor/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/cyclic-inverse-stride-bridge/verify.py b/research/cyclic-inverse-stride-bridge/verify.py
new file mode 100644
index 0000000000..0cc3867b79
--- /dev/null
+++ b/research/cyclic-inverse-stride-bridge/verify.py
@@ -0,0 +1,234 @@
+"""Replay for the stride bridge. Standard library only.
+
+    python3 -B research/cyclic-inverse-stride-bridge/verify.py --check
+    python3 -B research/cyclic-inverse-stride-bridge/verify.py --write   # regenerate EXPECTED.json
+"""
+from fractions import Fraction as F
+from pathlib import Path
+import argparse
+import hashlib
+import importlib.util
+import json
+import math
+import sys
+
+sys.set_int_max_str_digits(100000)
+HERE = Path(__file__).resolve().parent
+
+
+def load(path, name):
+    spec = importlib.util.spec_from_file_location(name, path)
+    module = importlib.util.module_from_spec(spec)
+    sys.modules[name] = module
+    spec.loader.exec_module(module)
+    return module
+
+
+st = load(HERE / 'stride.py', 'bridge_stride')
+need = st.need
+
+
+def pinned_vendor():
+    source = json.loads((HERE / 'SOURCE.json').read_text())
+    for name, digest in source['vendored_sha256'].items():
+        need(hashlib.sha256((HERE / 'vendor' / name).read_bytes()).hexdigest() == digest,
+             'vendored source drift: ' + name)
+    # check.py loads scan.py and interval_kernel.py from its own directory.
+    return load(HERE / 'vendor' / 'check.py', 'pr628_check')
+
+
+def unit(t):
+    t = F(t)
+    return ((1 - t * t) / (1 + t * t), 2 * t / (1 + t * t))
+
+
+def coefficients(L, beta, seed):
+    out = []
+    for i in range(L):
+        scale = (F(1), F(1, 2), F(0), F(1))[(i * 7 + seed) % 4]
+        out.append(tuple((1 - beta) * scale * v for v in unit(F((i * 5 + seed) % 13 - 6, 5))))
+    return out
+
+
+def inputs(L, seed):
+    return [(F((5 * i + seed) % 9 - 4, 8), F((7 * i + 3 * seed) % 11 - 5, 16)) for i in range(L)]
+
+
+def generic():
+    cases = 0
+    worst = F(0)
+    for g in (2, 3, 5):
+        beta = F(1, 1 << g)
+        for bits in (g + 6, 2 * g + 9):
+            for L in range(1, 19):
+                for w in range(-L, L):
+                    K = st.levels(L, w)
+                    need(st.admitted(beta, bits, K), 'admission')
+                    b, x = coefficients(L, beta, L + w), inputs(L, L * w)
+                    trace = []
+                    y = st.solve(x, b, w, bits, st.Tape(), trace)
+                    for row in trace:  # margin is preserved at every level
+                        need(all(st.norm2(c) <= (1 - beta) ** 2 for c in row['b']), 'level margin')
+                        need(2 * row['stride'] <= row['length'], 'stride halving')
+                    exact = st.exact_cycle_solve(x, b, w)
+                    bound = st.solve_error_bound(F(1), beta, bits, K)
+                    for v, e in zip(y, exact):
+                        err2 = st.norm2(st.sub(v, e))
+                        need(err2 <= bound ** 2, 'generic error bound L=%d w=%d' % (L, w))
+                        worst = max(worst, err2 / bound ** 2)
+                    cases += 1
+    return dict(cases=cases, max_error_to_bound_squared=float(worst))
+
+
+def traffic():
+    rows = []
+    worst = 0
+    for L in (1, 2, 3, 7, 64, 1000, 4096, 65536, 99991, 131072):
+        phi = int(L * 0.6180339887) or 1
+        for w in sorted({0, 1, 2, 3, L // 2, L // 2 + 1, L - 1, phi, (L // 3) or 1, 97 % L}):
+            tape = st.Tape()
+            st.solve([None] * L, [None] * L, w, None, tape)
+            K = st.levels(L, w)
+            need(K <= max(1, L).bit_length(), 'depth bound')
+            need(sum(n for n, _ in tape.levels) <= 2 * L, 'geometric level sizes')
+            ratio = F(tape.records, L)
+            need(ratio <= 40, 'linear traffic L=%d w=%d' % (L, w))
+            worst = max(worst, ratio)
+            sort_route = 4 * L * max(1, math.ceil(math.log2(L))) if L > 1 else 0
+            rows.append(dict(length=L, stride=w, levels=K, records=tape.records,
+                             records_per_entry=float(ratio), pr628_sort_route_records=sort_route))
+    return dict(max_records_per_entry=float(worst), rows=rows)
+
+
+def apply_kernel(c, state, x, w, tau, bits, tape, mutate=None):
+    r = state['radix']
+    L = len(x)
+    y = [(v[0] * r, v[1] * r) for v in x]
+    taubar = (tau[0], -tau[1])
+    for inside, b in state['factors']:
+        if inside:  # (U-rho)^-1 = (I - rho U^-1)^-1 U^-1, U = tau Z^w
+            shift = w if mutate != 'wrong_inner_shift' else -w
+            v = [y[(i + shift) % L] for i in range(L)]
+            y = st.solve(v, [st.mul(b, taubar)] * L, -w, bits, tape)
+            y = [st.rounded(st.mul(taubar, t), bits) for t in y]
+        else:  # (U-rho)^-1 = -rho^-1 (I - rho^-1 U)^-1, here b = 1/rho
+            y = st.solve(y, [st.mul(b, tau)] * L, w, bits, tape)
+            y = [st.rounded(st.mul((-b[0], -b[1]), t), bits) for t in y]
+    return y
+
+
+def cascade_bound(state, L, w):
+    """Execution budget for the factor cascade, Theorem 2 of PROOF.md."""
+    beta, bits, r = state['beta'], state['bits'], state['radix']
+    K = max(st.levels(L, w), st.levels(L, -w))
+    need(st.admitted(beta, bits, K), 'cascade admission')
+    u = F(1, 1 << bits)
+    error, norm = F(0), F(r)
+    for _ in state['factors']:
+        # one solve plus one rounded scalar multiplication of modulus <= 1
+        error = error / beta + st.solve_error_bound(norm + error, beta, bits, K) + u
+        norm /= beta
+    return error
+
+
+def kernels(c):
+    data = json.loads((HERE / 'vendor' / 'FACTORS.json').read_text())
+    rows = []
+    for row in data:
+        state = c.certify(row)
+        perturbation = F(state['certificate']['factorization_inverse_error_bound'])
+        for tau_name, tau in (('1', st.ONE), ('3/5+4i/5', (F(3, 5), F(4, 5)))):
+            for L in (1, 2, 3, 5, 8):
+                strides = range(L) if L <= 5 else (1, 3, 5, 7)
+                for w in strides:
+                    x = inputs(L, w + L)
+                    tape = st.Tape()
+                    y = apply_kernel(c, state, x, w, tau, state['bits'], tape)
+                    bound = cascade_bound(state, L, w)
+                    need(bound + perturbation <= F(1, 1 << row['target_bits']), 'target budget')
+                    q = c.phased_polynomial(state['q'], tau)
+                    dense = L <= 2 or (L == 3 and row['radix'] <= 9 and tau_name == '1')
+                    if dense:  # independent exact Gauss-Jordan solve of Q(tau Z^w)
+                        exact = c.matrix_solve(x, q, w)
+                        need(all(st.norm2(st.sub(v, e)) <= bound ** 2 for v, e in zip(y, exact)),
+                             'dense kernel comparison r=%d L=%d w=%d' % (row['radix'], L, w))
+                    # expanded residual: ||Q(U)y - x|| <= ||Q||_1 * execution bound
+                    scale = sum(abs(v[0]) + abs(v[1]) for v in q) * bound
+                    for i in range(L):
+                        total = st.ZERO
+                        for j, v in enumerate(q):
+                            total = st.add(total, st.mul(v, y[(i - w * j) % L]))
+                        need(st.norm2(st.sub(total, x[i])) <= scale ** 2, 'kernel residual')
+                    rows.append(dict(radix=row['radix'], embedding=row['embedding'], gauge=tau_name,
+                                     length=L, stride=w, dense=dense,
+                                     execution_bound_log2=round(math.log2(bound), 3),
+                                     total_bound_log2=round(math.log2(bound + perturbation), 3),
+                                     target_bits=row['target_bits']))
+    return rows
+
+
+def controls(c):
+    out = []
+    # 1. Reseeding with the wrong closure rotation must break the solution.
+    L, w, beta, bits = 7, 3, F(1, 4), 16
+    b, x = coefficients(L, beta, 1), inputs(L, 2)
+    good = st.solve(x, b, w, bits, st.Tape())
+    original = st.Fifo.__init__
+
+    def rotated(self, tape, initial):
+        initial = list(initial)
+        original(self, tape, initial[1:] + initial[:1])
+    st.Fifo.__init__ = rotated
+    try:
+        bad = st.solve(x, b, w, bits, st.Tape())
+    finally:
+        st.Fifo.__init__ = original
+    exact = st.exact_cycle_solve(x, b, w)
+    bound = st.solve_error_bound(F(1), beta, bits, st.levels(L, w))
+    need(all(st.norm2(st.sub(v, e)) <= bound ** 2 for v, e in zip(good, exact)), 'control baseline')
+    need(any(st.norm2(st.sub(v, e)) > bound ** 2 for v, e in zip(bad, exact)), 'rotated reseed accepted')
+    out.append('rotated_closure_reseed_rejected')
+    # 2. Precision below the admission is refused.
+    need(not st.admitted(F(1, 8), 4, 3), 'low precision admitted')
+    out.append('low_precision_refused')
+    # 3. Margin-free coefficient (|b|=1) is not a contraction: the oracle divides by zero.
+    try:
+        st.exact_cycle_solve([st.ONE], [st.ONE], 0)
+    except ValueError:
+        out.append('unit_coefficient_rejected')
+    else:
+        raise ValueError('unit coefficient accepted')
+    # 4. Wrong inner shift direction in the kernel cascade is caught by the dense solve.
+    data = json.loads((HERE / 'vendor' / 'FACTORS.json').read_text())
+    state = next(s for s in map(c.certify, data) if any(inside for inside, _ in s['factors']))
+    x = inputs(3, 3)
+    y = apply_kernel(c, state, x, 1, st.ONE, state['bits'], st.Tape(), mutate='wrong_inner_shift')
+    exact = c.matrix_solve(x, c.phased_polynomial(state['q'], st.ONE), 1)
+    bound = cascade_bound(state, 3, 1)
+    need(any(st.norm2(st.sub(v, e)) > bound ** 2 for v, e in zip(y, exact)), 'wrong inner shift accepted')
+    out.append('wrong_inner_shift_rejected')
+    return out
+
+
+def main():
+    parser = argparse.ArgumentParser()
+    mode = parser.add_mutually_exclusive_group(required=True)
+    mode.add_argument('--check', action='store_true')
+    mode.add_argument('--write', action='store_true')
+    args = parser.parse_args()
+    c = pinned_vendor()
+    result = dict(status='passed', generic=generic(), traffic=traffic(),
+                  kernels=kernels(c), controls=controls(c))
+    text = json.dumps(result, indent=1, sort_keys=True) + '\n'
+    target = HERE / 'EXPECTED.json'
+    if args.write:
+        target.write_text(text)
+    else:
+        need(target.read_text() == text, 'replay differs from EXPECTED.json')
+    print('PASS_CYCLIC_INVERSE_STRIDE_BRIDGE generic=%d kernel_cases=%d controls=%d max_records_per_entry=%s'
+          % (result['generic']['cases'], len(result['kernels']), len(result['controls']),
+             result['traffic']['max_records_per_entry']))
+
+
+if __name__ == '__main__':
+    main()