From 2846af720af190cb75b45ebddbb0357dec1121ec Mon Sep 17 00:00:00 2001
From: Chafik Boukhalfa <chafik.boukhalfa@gmail.com>
Date: Tue, 13 Oct 2026 19:38:45 +0400
Subject: [PATCH] Research: cubic arbitrary-stride inversion without a length
 admission

---
 .../cyclic-stride-effective-depth.yml         |  41 ++
 .../LINUX314-MATH.json.gz                     | Bin 0 -> 220903 bytes
 .../LINUX314-PACKAGE.json                     |  66 +++
 .../LINUX314-PLATFORM.json                    |   9 +
 .../LINUX314-TERMINAL.json                    |   6 +
 .../LINUX314.log.txt                          |  24 ++
 .../MACOS311-MATH.json.gz                     | Bin 0 -> 220903 bytes
 .../MACOS311-PACKAGE.json                     |  66 +++
 .../MACOS311-PLATFORM.json                    |   9 +
 .../MACOS311-TERMINAL.json                    |   6 +
 .../MACOS311.log.txt                          |  24 ++
 .../README.md                                 |  46 ++
 .../SOURCE-GIT-CHECK.json                     | 101 +++++
 .../VALIDATION.json                           |  23 +
 .../EXPECTED.json.gz                          | Bin 0 -> 220903 bytes
 .../MANIFEST.json                             |  31 ++
 .../cyclic-stride-effective-depth/NOTICE.md   |  40 ++
 .../cyclic-stride-effective-depth/PROOF.md    | 238 +++++++++++
 .../cyclic-stride-effective-depth/README.md   |  47 ++
 .../SOURCE-PINS.json                          |  90 ++++
 .../cyclic-stride-effective-depth/cases.py    |  80 ++++
 .../cyclic-stride-effective-depth/common.py   |  29 ++
 .../cyclic-stride-effective-depth/controls.py | 102 +++++
 .../cyclic-stride-effective-depth/cubic.py    |  62 +++
 .../cyclic-stride-effective-depth/early.py    |  60 +++
 .../cyclic-stride-effective-depth/kernels.py  |  56 +++
 .../test_package.py                           | 127 ++++++
 .../cyclic-stride-effective-depth/uniform.py  |  29 ++
 .../uniform_kernels.py                        |  84 ++++
 .../vendor/PR862-NOTICE.md                    |  31 ++
 .../vendor/PR862-PROOF.md                     | 259 +++++++++++
 .../vendor/pr846/NOTICE.md                    |  22 +
 .../vendor/pr846/PROOF.md                     | 211 +++++++++
 .../vendor/pr846/SOURCE.json                  |  15 +
 .../vendor/pr846/stride.py                    | 168 ++++++++
 .../vendor/pr846/vendor/FACTORS.json          | 402 ++++++++++++++++++
 .../vendor/pr846/vendor/PR628-NOTICE.md       |  24 ++
 .../vendor/pr846/vendor/check.py              | 178 ++++++++
 .../vendor/pr846/vendor/interval_kernel.py    |  61 +++
 .../vendor/pr846/vendor/scan.py               | 123 ++++++
 .../vendor/pr846/verify.py                    | 234 ++++++++++
 .../cyclic-stride-effective-depth/verify.py   | 112 +++++
 42 files changed, 3336 insertions(+)
 create mode 100644 .github/workflows/cyclic-stride-effective-depth.yml
 create mode 100644 research/cyclic-stride-effective-depth-validation/LINUX314-MATH.json.gz
 create mode 100644 research/cyclic-stride-effective-depth-validation/LINUX314-PACKAGE.json
 create mode 100644 research/cyclic-stride-effective-depth-validation/LINUX314-PLATFORM.json
 create mode 100644 research/cyclic-stride-effective-depth-validation/LINUX314-TERMINAL.json
 create mode 100644 research/cyclic-stride-effective-depth-validation/LINUX314.log.txt
 create mode 100644 research/cyclic-stride-effective-depth-validation/MACOS311-MATH.json.gz
 create mode 100644 research/cyclic-stride-effective-depth-validation/MACOS311-PACKAGE.json
 create mode 100644 research/cyclic-stride-effective-depth-validation/MACOS311-PLATFORM.json
 create mode 100644 research/cyclic-stride-effective-depth-validation/MACOS311-TERMINAL.json
 create mode 100644 research/cyclic-stride-effective-depth-validation/MACOS311.log.txt
 create mode 100644 research/cyclic-stride-effective-depth-validation/README.md
 create mode 100644 research/cyclic-stride-effective-depth-validation/SOURCE-GIT-CHECK.json
 create mode 100644 research/cyclic-stride-effective-depth-validation/VALIDATION.json
 create mode 100644 research/cyclic-stride-effective-depth/EXPECTED.json.gz
 create mode 100644 research/cyclic-stride-effective-depth/MANIFEST.json
 create mode 100644 research/cyclic-stride-effective-depth/NOTICE.md
 create mode 100644 research/cyclic-stride-effective-depth/PROOF.md
 create mode 100644 research/cyclic-stride-effective-depth/README.md
 create mode 100644 research/cyclic-stride-effective-depth/SOURCE-PINS.json
 create mode 100644 research/cyclic-stride-effective-depth/cases.py
 create mode 100644 research/cyclic-stride-effective-depth/common.py
 create mode 100644 research/cyclic-stride-effective-depth/controls.py
 create mode 100644 research/cyclic-stride-effective-depth/cubic.py
 create mode 100644 research/cyclic-stride-effective-depth/early.py
 create mode 100644 research/cyclic-stride-effective-depth/kernels.py
 create mode 100644 research/cyclic-stride-effective-depth/test_package.py
 create mode 100644 research/cyclic-stride-effective-depth/uniform.py
 create mode 100644 research/cyclic-stride-effective-depth/uniform_kernels.py
 create mode 100644 research/cyclic-stride-effective-depth/vendor/PR862-NOTICE.md
 create mode 100644 research/cyclic-stride-effective-depth/vendor/PR862-PROOF.md
 create mode 100644 research/cyclic-stride-effective-depth/vendor/pr846/NOTICE.md
 create mode 100644 research/cyclic-stride-effective-depth/vendor/pr846/PROOF.md
 create mode 100644 research/cyclic-stride-effective-depth/vendor/pr846/SOURCE.json
 create mode 100644 research/cyclic-stride-effective-depth/vendor/pr846/stride.py
 create mode 100644 research/cyclic-stride-effective-depth/vendor/pr846/vendor/FACTORS.json
 create mode 100644 research/cyclic-stride-effective-depth/vendor/pr846/vendor/PR628-NOTICE.md
 create mode 100644 research/cyclic-stride-effective-depth/vendor/pr846/vendor/check.py
 create mode 100644 research/cyclic-stride-effective-depth/vendor/pr846/vendor/interval_kernel.py
 create mode 100644 research/cyclic-stride-effective-depth/vendor/pr846/vendor/scan.py
 create mode 100644 research/cyclic-stride-effective-depth/vendor/pr846/verify.py
 create mode 100644 research/cyclic-stride-effective-depth/verify.py

diff --git a/.github/workflows/cyclic-stride-effective-depth.yml b/.github/workflows/cyclic-stride-effective-depth.yml
new file mode 100644
index 0000000000..07bcfb457b
--- /dev/null
+++ b/.github/workflows/cyclic-stride-effective-depth.yml
@@ -0,0 +1,41 @@
+name: Cyclic stride effective depth
+
+on:
+  workflow_dispatch:
+  pull_request:
+    paths:
+      - 'research/cyclic-stride-effective-depth/**'
+      - '.github/workflows/cyclic-stride-effective-depth.yml'
+  push:
+    branches: ['codex/stride-effective-depth-20261013']
+    paths:
+      - 'research/cyclic-stride-effective-depth/**'
+      - '.github/workflows/cyclic-stride-effective-depth.yml'
+
+permissions:
+  contents: read
+
+jobs:
+  verify:
+    runs-on: ubuntu-latest
+    timeout-minutes: 30
+    strategy:
+      fail-fast: false
+      matrix:
+        python: ['3.11', '3.14']
+    steps:
+      - uses: actions/checkout@v4
+      - uses: actions/setup-python@v5
+        with:
+          python-version: ${{ matrix.python }}
+      - name: Effective depth, original and cubic kernels, and defect controls
+        run: python -I -O -B research/cyclic-stride-effective-depth/verify.py --out /tmp/stride-depth-math.json
+      - name: Package rejection controls and full read-only relocation
+        run: python -I -O -B research/cyclic-stride-effective-depth/test_package.py --out /tmp/stride-depth-package.json
+      - uses: actions/upload-artifact@v4
+        if: always()
+        with:
+          name: stride-depth-python-${{ matrix.python }}
+          path: |
+            /tmp/stride-depth-math.json
+            /tmp/stride-depth-package.json
diff --git a/research/cyclic-stride-effective-depth-validation/LINUX314-MATH.json.gz b/research/cyclic-stride-effective-depth-validation/LINUX314-MATH.json.gz
new file mode 100644
index 0000000000000000000000000000000000000000..8f7abb0657690786264ec60638a792766c5c8419
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diff --git a/research/cyclic-stride-effective-depth-validation/LINUX314-PACKAGE.json b/research/cyclic-stride-effective-depth-validation/LINUX314-PACKAGE.json
new file mode 100644
index 0000000000..a5a0fa2d79
--- /dev/null
+++ b/research/cyclic-stride-effective-depth-validation/LINUX314-PACKAGE.json
@@ -0,0 +1,66 @@
+{
+  "controls": [
+    {
+      "name": "missing manifest",
+      "rejected": true,
+      "terminal": "ValueError: missing package manifest"
+    },
+    {
+      "name": "altered implementation",
+      "rejected": true,
+      "terminal": "ValueError: package hash: early.py"
+    },
+    {
+      "name": "extra file",
+      "rejected": true,
+      "terminal": "ValueError: package file coverage"
+    },
+    {
+      "name": "missing source",
+      "rejected": true,
+      "terminal": "ValueError: package file coverage"
+    },
+    {
+      "name": "source hash independent of manifest",
+      "rejected": true,
+      "terminal": "ValueError: source SHA256: vendor/pr846/PROOF.md"
+    },
+    {
+      "name": "source Git object independent of manifest",
+      "rejected": true,
+      "terminal": "ValueError: source Git blob: vendor/pr846/stride.py"
+    },
+    {
+      "name": "duplicate manifest key",
+      "rejected": true,
+      "terminal": "ValueError: duplicate JSON key: files"
+    },
+    {
+      "name": "symlink injection",
+      "rejected": true,
+      "terminal": "ValueError: package symlink"
+    },
+    {
+      "name": "preloaded module substitution",
+      "rejected": true,
+      "terminal": "ValueError: local module binding: common"
+    },
+    {
+      "name": "false receipt boolean integer confusion",
+      "rejected": true,
+      "terminal": "ValueError: mathematical receipt mismatch"
+    },
+    {
+      "name": "existing output preserved",
+      "rejected": true
+    },
+    {
+      "name": "full read-only isolated optimized relocation",
+      "passed": true,
+      "receipt_sha256": "f30ad0e119c9928f30e2fc97e1911e053fe347bc4fe3f2b328b295141bda1552"
+    }
+  ],
+  "full_readonly_replay": true,
+  "rejected": 11,
+  "status": "PASS_STRIDE_DEPTH_PACKAGE"
+}
diff --git a/research/cyclic-stride-effective-depth-validation/LINUX314-PLATFORM.json b/research/cyclic-stride-effective-depth-validation/LINUX314-PLATFORM.json
new file mode 100644
index 0000000000..57a1c500f0
--- /dev/null
+++ b/research/cyclic-stride-effective-depth-validation/LINUX314-PLATFORM.json
@@ -0,0 +1,9 @@
+{
+  "bytecode_disabled": true,
+  "isolated": true,
+  "machine": "aarch64",
+  "manifest_sha256": "0b40eccb5d47d2ac152ed7fe47c259eb71bc17d3c818a71c63e2bc026e9377e3",
+  "optimized": true,
+  "platform": "Linux-6.17.0-1032-nvidia-aarch64-with-glibc2.39",
+  "python": "3.14.8"
+}
diff --git a/research/cyclic-stride-effective-depth-validation/LINUX314-TERMINAL.json b/research/cyclic-stride-effective-depth-validation/LINUX314-TERMINAL.json
new file mode 100644
index 0000000000..a481613cf4
--- /dev/null
+++ b/research/cyclic-stride-effective-depth-validation/LINUX314-TERMINAL.json
@@ -0,0 +1,6 @@
+{
+  "full_package_controls": true,
+  "manifest_sha256": "0b40eccb5d47d2ac152ed7fe47c259eb71bc17d3c818a71c63e2bc026e9377e3",
+  "math_sha256": "f30ad0e119c9928f30e2fc97e1911e053fe347bc4fe3f2b328b295141bda1552",
+  "status": "PASS_FROZEN_STRIDE_DEPTH_PACKAGE"
+}
diff --git a/research/cyclic-stride-effective-depth-validation/LINUX314.log.txt b/research/cyclic-stride-effective-depth-validation/LINUX314.log.txt
new file mode 100644
index 0000000000..5fc04dddac
--- /dev/null
+++ b/research/cyclic-stride-effective-depth-validation/LINUX314.log.txt
@@ -0,0 +1,24 @@
+PASS stride cases 423 scalar coefficient paths 90
+PASS depth controls 12 log boundaries 1545
+PASS cubic budget rows 216 identities 3
+PASS 300 original kernel cases with uniform depth admission
+PASS actual cubic stride radix 3 cumulative cases 16
+PASS actual cubic stride radix 4 cumulative cases 32
+PASS actual cubic stride radix 5 cumulative cases 48
+PASS actual cubic stride radix 7 cumulative cases 64
+PASS_CYCLIC_STRIDE_EFFECTIVE_DEPTH arrays=423 kernels=300 uniform=64 scalar_paths=90 budgets=216 kappa=unchanged
+PASS package control missing manifest
+PASS package control altered implementation
+PASS package control extra file
+PASS package control missing source
+PASS package control source hash independent of manifest
+PASS package control source Git object independent of manifest
+PASS package control duplicate manifest key
+PASS package control symlink injection
+PASS package control preloaded module substitution
+PASS package control false receipt boolean integer confusion
+PASS package control existing output preserved
+PASS full read-only isolated optimized relocation
+PASS_STRIDE_DEPTH_PACKAGE rejected=11 full_readonly_replay=True
+PASS_STRIDE_DEPTH_BINDINGS sources=12
+PASS_FROZEN_STRIDE_DEPTH_PACKAGE
diff --git a/research/cyclic-stride-effective-depth-validation/MACOS311-MATH.json.gz b/research/cyclic-stride-effective-depth-validation/MACOS311-MATH.json.gz
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diff --git a/research/cyclic-stride-effective-depth-validation/MACOS311-PACKAGE.json b/research/cyclic-stride-effective-depth-validation/MACOS311-PACKAGE.json
new file mode 100644
index 0000000000..a5a0fa2d79
--- /dev/null
+++ b/research/cyclic-stride-effective-depth-validation/MACOS311-PACKAGE.json
@@ -0,0 +1,66 @@
+{
+  "controls": [
+    {
+      "name": "missing manifest",
+      "rejected": true,
+      "terminal": "ValueError: missing package manifest"
+    },
+    {
+      "name": "altered implementation",
+      "rejected": true,
+      "terminal": "ValueError: package hash: early.py"
+    },
+    {
+      "name": "extra file",
+      "rejected": true,
+      "terminal": "ValueError: package file coverage"
+    },
+    {
+      "name": "missing source",
+      "rejected": true,
+      "terminal": "ValueError: package file coverage"
+    },
+    {
+      "name": "source hash independent of manifest",
+      "rejected": true,
+      "terminal": "ValueError: source SHA256: vendor/pr846/PROOF.md"
+    },
+    {
+      "name": "source Git object independent of manifest",
+      "rejected": true,
+      "terminal": "ValueError: source Git blob: vendor/pr846/stride.py"
+    },
+    {
+      "name": "duplicate manifest key",
+      "rejected": true,
+      "terminal": "ValueError: duplicate JSON key: files"
+    },
+    {
+      "name": "symlink injection",
+      "rejected": true,
+      "terminal": "ValueError: package symlink"
+    },
+    {
+      "name": "preloaded module substitution",
+      "rejected": true,
+      "terminal": "ValueError: local module binding: common"
+    },
+    {
+      "name": "false receipt boolean integer confusion",
+      "rejected": true,
+      "terminal": "ValueError: mathematical receipt mismatch"
+    },
+    {
+      "name": "existing output preserved",
+      "rejected": true
+    },
+    {
+      "name": "full read-only isolated optimized relocation",
+      "passed": true,
+      "receipt_sha256": "f30ad0e119c9928f30e2fc97e1911e053fe347bc4fe3f2b328b295141bda1552"
+    }
+  ],
+  "full_readonly_replay": true,
+  "rejected": 11,
+  "status": "PASS_STRIDE_DEPTH_PACKAGE"
+}
diff --git a/research/cyclic-stride-effective-depth-validation/MACOS311-PLATFORM.json b/research/cyclic-stride-effective-depth-validation/MACOS311-PLATFORM.json
new file mode 100644
index 0000000000..01a868eb85
--- /dev/null
+++ b/research/cyclic-stride-effective-depth-validation/MACOS311-PLATFORM.json
@@ -0,0 +1,9 @@
+{
+  "bytecode_disabled": true,
+  "isolated": true,
+  "machine": "arm64",
+  "manifest_sha256": "0b40eccb5d47d2ac152ed7fe47c259eb71bc17d3c818a71c63e2bc026e9377e3",
+  "optimized": true,
+  "platform": "macOS-27.0.1-arm64-arm-64bit",
+  "python": "3.11.15"
+}
diff --git a/research/cyclic-stride-effective-depth-validation/MACOS311-TERMINAL.json b/research/cyclic-stride-effective-depth-validation/MACOS311-TERMINAL.json
new file mode 100644
index 0000000000..a481613cf4
--- /dev/null
+++ b/research/cyclic-stride-effective-depth-validation/MACOS311-TERMINAL.json
@@ -0,0 +1,6 @@
+{
+  "full_package_controls": true,
+  "manifest_sha256": "0b40eccb5d47d2ac152ed7fe47c259eb71bc17d3c818a71c63e2bc026e9377e3",
+  "math_sha256": "f30ad0e119c9928f30e2fc97e1911e053fe347bc4fe3f2b328b295141bda1552",
+  "status": "PASS_FROZEN_STRIDE_DEPTH_PACKAGE"
+}
diff --git a/research/cyclic-stride-effective-depth-validation/MACOS311.log.txt b/research/cyclic-stride-effective-depth-validation/MACOS311.log.txt
new file mode 100644
index 0000000000..5fc04dddac
--- /dev/null
+++ b/research/cyclic-stride-effective-depth-validation/MACOS311.log.txt
@@ -0,0 +1,24 @@
+PASS stride cases 423 scalar coefficient paths 90
+PASS depth controls 12 log boundaries 1545
+PASS cubic budget rows 216 identities 3
+PASS 300 original kernel cases with uniform depth admission
+PASS actual cubic stride radix 3 cumulative cases 16
+PASS actual cubic stride radix 4 cumulative cases 32
+PASS actual cubic stride radix 5 cumulative cases 48
+PASS actual cubic stride radix 7 cumulative cases 64
+PASS_CYCLIC_STRIDE_EFFECTIVE_DEPTH arrays=423 kernels=300 uniform=64 scalar_paths=90 budgets=216 kappa=unchanged
+PASS package control missing manifest
+PASS package control altered implementation
+PASS package control extra file
+PASS package control missing source
+PASS package control source hash independent of manifest
+PASS package control source Git object independent of manifest
+PASS package control duplicate manifest key
+PASS package control symlink injection
+PASS package control preloaded module substitution
+PASS package control false receipt boolean integer confusion
+PASS package control existing output preserved
+PASS full read-only isolated optimized relocation
+PASS_STRIDE_DEPTH_PACKAGE rejected=11 full_readonly_replay=True
+PASS_STRIDE_DEPTH_BINDINGS sources=12
+PASS_FROZEN_STRIDE_DEPTH_PACKAGE
diff --git a/research/cyclic-stride-effective-depth-validation/README.md b/research/cyclic-stride-effective-depth-validation/README.md
new file mode 100644
index 0000000000..56324bc9e3
--- /dev/null
+++ b/research/cyclic-stride-effective-depth-validation/README.md
@@ -0,0 +1,46 @@
+# Frozen stride-depth validation
+
+Complete macOS arm64 / Python 3.11.15 and Linux aarch64 / Python 3.14.8
+replays passed with isolation, optimization and bytecode disabled. Their
+full mathematical receipts are byte-identical to each other and to the
+decompressed `../cyclic-stride-effective-depth/EXPECTED.json.gz`.
+
+Frozen package manifest SHA256:
+`0b40eccb5d47d2ac152ed7fe47c259eb71bc17d3c818a71c63e2bc026e9377e3`.
+Complete mathematical receipt SHA256:
+`f30ad0e119c9928f30e2fc97e1911e053fe347bc4fe3f2b328b295141bda1552`.
+
+Each platform recomputed the full mathematics three times: default replay,
+rejection of an altered expected receipt, and a read-only relocated replay
+in a path containing spaces. Eleven package rejection controls passed on
+each platform. Linux also began with the supplied package read-only, outside
+a Git checkout, under a durable supervisor that recorded exit status zero.
+
+The full `*-MATH.json.gz` receipts, package controls, platform information,
+terminal records and logs are retained here. Compare decompressed receipt
+bytes; container compression identity is not required. `VALIDATION.json`
+records their hashes. All 12 retained source files match their pinned Git
+objects and SHA256 hashes, as recorded in `SOURCE-GIT-CHECK.json`.
+
+Evidence includes 423 array cases, 90 scalar coefficient paths, all 300
+original PR846 kernel cases, 64 actual cubic-width kernel executions,
+216 integer budgets, three exact nonnegative polynomial identities,
+54 admission checks, 1,545 logarithm boundaries, 12 new mathematical
+controls and four inherited controls, 12 zero-base checks and 33 traffic
+cases through length 131,072. Scalar paths are not huge-array executions.
+Original and early-exit outputs agree exactly; the two traffic bills are
+40L and 44L respectively, with the additional head returns charged.
+
+The source record preserves preparation base
+`007664306fff84fbb1c9a1b4053b69f439764034`. Before publication the branch was
+fast-forwarded to `bbc600db2fc15007b7c914f4f2b2ed55fa909c87`, without changing
+any frozen package byte. Selected-record consistency passed there at
+`1661960547713947/1000000000000000000`; staged whitespace and byte checks
+also passed. Only two new research directories and their workflow are added.
+The full repository suite and hosted CI are separate from these local
+prepublication records. The scoped workflow repeats both Python versions.
+
+Finite exact checks support the written universal argument; they are not
+independent human review or proof-assistant verification. Clean scratch,
+phase access and remaining transform obligations remain explicit. No new
+kappa or bit-reversible/borrowed-array implementation is claimed.
diff --git a/research/cyclic-stride-effective-depth-validation/SOURCE-GIT-CHECK.json b/research/cyclic-stride-effective-depth-validation/SOURCE-GIT-CHECK.json
new file mode 100644
index 0000000000..9eb91072d8
--- /dev/null
+++ b/research/cyclic-stride-effective-depth-validation/SOURCE-GIT-CHECK.json
@@ -0,0 +1,101 @@
+{
+  "sources": [
+    {
+      "git_blob": "9229a4193b004d0dd939a502cfae75c27791dc76",
+      "git_bytes_equal": true,
+      "head": "4bdb3e755ea7118170a7a34764df955880dec282",
+      "path": "vendor/pr846/stride.py",
+      "sha256": "da56e45f414a633453e06e8d058bcec6c815c73950816fcf049d92c570f8928e",
+      "source_path": "research/cyclic-inverse-stride-bridge/stride.py"
+    },
+    {
+      "git_blob": "0cc3867b79082894a18732f7751bc30414da0822",
+      "git_bytes_equal": true,
+      "head": "4bdb3e755ea7118170a7a34764df955880dec282",
+      "path": "vendor/pr846/verify.py",
+      "sha256": "a56acd90c79597fa57602b053de09fa66930f262c39b849362b49b3cce707f2c",
+      "source_path": "research/cyclic-inverse-stride-bridge/verify.py"
+    },
+    {
+      "git_blob": "e1624e72ddffcd9fc2271691f79edcedeffc3c91",
+      "git_bytes_equal": true,
+      "head": "4bdb3e755ea7118170a7a34764df955880dec282",
+      "path": "vendor/pr846/SOURCE.json",
+      "sha256": "07bc9fe307f254b68810f664bcef0f30eabd130226d39804bb38517777e3a92b",
+      "source_path": "research/cyclic-inverse-stride-bridge/SOURCE.json"
+    },
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diff --git a/research/cyclic-stride-effective-depth/MANIFEST.json b/research/cyclic-stride-effective-depth/MANIFEST.json
new file mode 100644
index 0000000000..65e7eb0c80
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/MANIFEST.json
@@ -0,0 +1,31 @@
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+    "vendor/pr846/SOURCE.json": "07bc9fe307f254b68810f664bcef0f30eabd130226d39804bb38517777e3a92b",
+    "vendor/pr846/stride.py": "da56e45f414a633453e06e8d058bcec6c815c73950816fcf049d92c570f8928e",
+    "vendor/pr846/vendor/FACTORS.json": "643d60ac24d5e67f6c9305175bd1c99bf5e42edfef6d1995c051e0b98ba446a7",
+    "vendor/pr846/vendor/PR628-NOTICE.md": "abeacf1ad1491d5797cfcf1f3832799bbf3b24b99597229d7bc884059a51ec68",
+    "vendor/pr846/vendor/check.py": "a3e46b556894d71a1c83bbbac01c50c1a541285fc9422a3567d5eb1d1ee86e7d",
+    "vendor/pr846/vendor/interval_kernel.py": "9b7bb24640a9a7d3f83c7951dfef483ec933db4c50de6c0130b85f450eeddfbe",
+    "vendor/pr846/vendor/scan.py": "1a9c01ca1a29a9be1e756858161202621db29d5d332a4266b4017a9502a50d7e",
+    "vendor/pr846/verify.py": "a56acd90c79597fa57602b053de09fa66930f262c39b849362b49b3cce707f2c",
+    "verify.py": "bf72f42102a044d7db81d2e69a9e4b173113ff5474a1b628ecbaed242fe6ba96"
+  }
+}
diff --git a/research/cyclic-stride-effective-depth/NOTICE.md b/research/cyclic-stride-effective-depth/NOTICE.md
new file mode 100644
index 0000000000..57939c758d
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/NOTICE.md
@@ -0,0 +1,40 @@
+# Attribution and scope
+
+New effective-depth argument, cubic-stride integration and checks by Chafik
+Boukhalfa (@chafreaky), with substantial OpenAI Codex assistance, under the
+repository's Apache-2.0 license. No independent human/delegated-agent review
+or literature-priority claim is made.
+
+- Rohan Guliani (@Rohansguliani), PR846, head
+  `4bdb3e755ea7118170a7a34764df955880dec282`: arbitrary-stride Euclidean
+  chain/closure solver, fixed-tape FIFO accounting, uniform error argument
+  and kernel integration. Its solver, checker, proof, source record and
+  notice are retained unchanged. `early.py` is an explicitly attributed
+  adaptation of its `solve` function, adding a charged exact-zero base case.
+- Charles Bethin (@cbethin), PR764, head
+  `983d66f0f2e48ee500ec7eee2da9244c001c5064`: shared-suffix scan identity and
+  inverse-square error analysis, used through the retained PR846 proof.
+- Chafik Boukhalfa, PR628, head
+  `0768f97d06c623b8211e651543c66135796b62bf`: root-factor certificates,
+  preparation/phase budgets and exact arithmetic. The five files retained
+  by PR846 remain byte-identical and independently hash-bound here.
+- Chafik Boukhalfa, PR862, head
+  `57e0659150d323b784580110b36aa64a2dca2a33`: common-operator complement
+  bound and cubic preparation/execution precision. Its proof and notice
+  are retained unchanged.
+- Ali Romman (@aliromman), PR621, head
+  `520774462b19b29af41554b74def31f9d01e997e`: prime-power kernel and its
+  uniform inverse bound, retained through the cited preparation lineage.
+
+Original notices retain the OpenAI exact-DFT manuscript, classical
+Mahler/Wiener arguments, Pan factorization, fast elementary arithmetic,
+and the standard queue-from-two-stacks attribution. Their credits and
+licenses remain unchanged. Nearest-grid extinction and pointwise-zero
+substitution are elementary; no priority claim is made for those ideas.
+
+The new result removes the stride solver's length-dependent precision
+admission and combines its unchanged local error with PR862's cubic bound.
+It does not replace the inherited preparation theorem with the finite root
+fixtures or claim that the Python reference implements its asymptotic cost.
+Clean scratch, phase access and all remaining transform/physical obligations
+stay explicit. No new multiplication exponent is claimed.
diff --git a/research/cyclic-stride-effective-depth/PROOF.md b/research/cyclic-stride-effective-depth/PROOF.md
new file mode 100644
index 0000000000..6019f4d03f
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/PROOF.md
@@ -0,0 +1,238 @@
+# An effective depth bound for the rounded stride solver
+
+This research argument extends PR846 at head
+4bdb3e755ea7118170a7a34764df955880dec282. The original source is retained
+unchanged in vendor/pr846/stride.py. Its clean scratch and sequential-tape model
+remain unchanged. No borrowed-array, bit-reversible or kappa claim.
+
+Let beta=2^-g, g>=1, and let u=2^-W with W>=2g+8. Assume every input
+coefficient has modulus at most 1-beta. Add one exact zero test at each
+recursion level: when every coefficient is zero, return the data rounded
+once. This solves that level exactly up to one rounding. Count the test as
+one full coefficient scan and head return; traffic is at most 44L records instead of
+the original conservative 40L.
+
+For the early-terminated solver, the number K of non-base levels is bounded independently of L by
+
+    D(g,W)=2g+ceil(log2 W)+4.
+
+Consequently 4(K+1)u<=beta follows from W>=2g+8. This removes the original
+p>=log2(log2 L+2) admission condition, after imposing this sufficient local
+working-precision condition instead.
+
+## Decay of the rounded closure coefficients
+
+Write rho for the maximum coefficient modulus at a level. Every chain has
+at least two coefficients. Its product is initialized with the first exact
+coefficient and then rounded after each multiplication. For chain length
+ell>=2, its magnitude is bounded by
+
+    rho^ell+u*(1+rho+...+rho^(ell-2)) <= rho^2+u/(1-rho).
+
+Thus a scalar majorant for the next maximum is rho^2+u/(1-rho).
+
+First suppose beta<=m=1-rho<=1/2. Since u<=beta^2/16<=m^2/16,
+
+    1-rho_next >= m(2-m)-u/m >= (23/16)m > sqrt(2)m.
+
+Within at most 2g levels, the margin therefore reaches 1/2, unless recursion
+has already terminated. While rho<=1/2, the majorant is rho^2+2u.
+This stays below 1/2, and two further levels give rho<=1/4 (W>=10 is more
+than sufficient for these elementary inequalities).
+
+While rho>2sqrt(u), the same majorant satisfies rho_next<=3rho^2/2.
+For t=2rho, this implies t_next<=t^2, starting from t<=1/2. After at most
+ceil(log2 W) additional levels, rho<=2sqrt(u): otherwise repeated squaring
+would give rho<=2^(-2^j-1)<=u/2, a contradiction.
+
+One more level gives rho<=6u. At the following level the first multiplication
+in every chain has modulus at most 36u^2<u/2. Each component therefore rounds
+exactly to zero. Subsequent products in the chain remain exactly zero. All
+closure coefficients at that next level are zero, so the new base case
+terminates. Adding the four stage counts gives D(g,W).
+
+All inequalities concern complex modulus and use only the nearest-grid
+rounding error <=u. They do not assume real positive coefficients, a common
+phase, a common chain length or a dyadic initial coefficient. Exact zero in
+the final step uses the componentwise nearest rounding rule, not just a
+continuous norm upper bound. The strict threshold avoids tie ambiguity.
+
+## Exact equivalence with the unchanged solver
+
+At any level where all coefficients are exactly zero, the unchanged solver
+also returns the data rounded once, regardless of its remaining structural
+recursion. For stride zero this is its explicit division by one. Otherwise,
+pass A copies a chain-start datum or rounds a later datum; every chain
+product and every recursive coefficient remains zero. Its finite structural
+recursion is well-defined, and pass C returns rnd(x_i+0*seed)=rnd(x_i) at
+every coordinate. Reversal commutes with this pointwise rounding.
+
+Thus replacing the whole zero-coefficient subtree by the new base case
+preserves the exact rounded output. This is an algebraic identity for the
+instruction sequence, not an estimate and not an inference from finite tests.
+It applies separately at every zero subtree. Consequently the unchanged
+PR846 solver and the truncated solver have exactly identical outputs.
+
+The accuracy proof may analyze the truncated solver while executing the
+original unchanged solver. The latter retains its original 40L traffic
+bound: no zero scan is required at runtime. The optionally early-terminated
+version uses at most 44L records including the extra coefficient scans and
+head returns. Its effective nonzero depth is bounded by D; the unchanged
+implementation's structural depth may still depend on L.
+
+## Admission and inherited error proof
+
+For W>=2g+8, ceil(log2 W)<=W-1 and 2g<=W-8, hence
+
+    4(D+1)<=8W-16.
+
+Also W-g>=ceil(W/2)+4 and 2^ceil(W/2)>=W for W>=4, so
+
+    2^(W-g)>=16W>4(D+1).
+
+This proves 4(K+1)u<=beta, and u<=beta/3 is automatic. PR846's margin,
+weighted-data and shared-suffix error arguments now apply through at most
+D levels. The new terminal system has exact solution x; rounding it once
+has error at most u, matching the old terminal bound. Therefore the stated all-length error bound holds for both implementations,
+with K equal to the truncated depth, or conservatively min(D,1/beta) in its
+geometric error factor. The original structural depth is not used for admission.
+
+The algebraic source/input restoration properties are unchanged: this is
+still a clean-tape solver with the input unmodified. The optional early-exit check adds at most 2 sum L_k<=4L coefficient records
+to the original 40L traffic bound. Executing the unchanged implementation
+needs no extra scan and retains 40L.
+Any full-kernel composition must use a working precision satisfying both
+its existing error target and W>=2g+8. That added local condition is O(g)
+and preserves the previous polynomial-in-r precision order.
+
+## Cubic precision for the full kernel at arbitrary strides
+
+This section combines the local result above with PR862's retained
+common-operator argument. The target is the same prime-power kernel
+P(U), with U=tau Z_L^w and |tau|=1, as in PR846. All inverse factors use
+this one operator. Different noncommuting shifts are outside the argument.
+
+A stride solve on input norm at most X has error at most
+
+    (u/beta^2)(5+4X/beta)+u.
+
+After the outer scalar or inner unit rotation and its rounding, its error
+is at most 7(X+1)u/beta^3 for beta<=1/2. This is exactly the local bound
+required by PR862. Its proof only needs stable factors and that bound;
+it does not require the neighboring-stride implementation.
+
+Let n=r-1, let Q=(1/r) product(z-rho_j), and suppose
+
+    ||Q||_1<=M, ||Q(U)^-1||<=H,
+    B=max(1,2^n M H).
+
+The complementary-factor/Mahler argument in the retained PR862 proof
+bounds every inverse prefix and contiguous inverse subproduct by B.
+Starting with r x, ||x||<=D, the maximum prefix execution error E obeys
+
+    E <= t(r B D+E+1),  t=7nB beta^-3 u.
+
+For t<1 this yields E<=t(rBD+1)/(1-t). This does not repeatedly multiply
+the worst reciprocal root margin at every factor.
+
+For prime-power r>=3 and p,m>=0, retain the source preparation hypotheses
+and put
+
+    ell=ceil(log2 r), c=ceil(log2(r-1)), phi=EulerPhi(r),
+    d=1+2ell+r(r-1)phi, a0=2d+r+2,
+    g=d+ell+2r+2, beta=2^-g, h=a0+2r,
+    e=max(d+1,a0+1,p+2a0+m+3),
+    b=max((r-1)ell,e+r+3).
+
+PR628/862's charged approximate factorization and coefficient verification
+supply prepared roots with stable margin beta, ||Q||_1<=2^r,
+||Q(U)^-1||<=2^(a0+1), and source-scaled kernel perturbation at most
+2^(-p-2) for D<=2^m. Thus B<=2^h. These are certified premises, not
+conclusions drawn merely from a numerical root list.
+
+Choose
+
+    J=c+2h+3g+ell+m+5, W=p+J+2.
+
+This W exceeds 2g+8, so the stride admission holds for every L and every
+stride. Moreover n<=2^c and 7<=2^3 imply
+
+    t <= 2^(c+h+3+3g-W) <= 1/2,
+    rBD+1 <= 2^(ell+h+m+1),
+    E <= 2t(rBD+1) <= 2^(J-W)=2^(-p-2).
+
+Adding the independently charged kernel preparation perturbation gives
+error at most 2^(-p-1), within the target 2^-p. Approximate phase oracles
+still need their separately charged perturbation and access budget.
+
+A sufficient integer guard for data/state magnitudes is
+
+    G=m+ell+h+g+8.
+
+Indeed each factor input is bounded by rBD+E. PR846's weighted-data
+invariant bounds recursive data and closure values by O(rBD/beta);
+its corresponding recursive rounding error is at most one under the
+chosen W (h>=g). Chain values, products, scalar application and division
+fit this guard. Exact arithmetic accumulators and numerator/denominator
+products may use a constant-factor wider representation, as usual.
+
+Using phi<=r, ell<=r and c<=r gives the explicit cubic bounds
+
+    W <= p+m+9r^3, G <= m+6r^3, b <= p+m+6r^3.
+
+The exact nonnegative remainders in t=r-3 are
+
+| Bound | Remainder after removing p and m as appropriate |
+| --- | --- |
+| 9r^3 minus the working overhead | 65t+25t^2+2t^3 |
+| 6r^3 minus the integer guard overhead | 51+85t+30t^2+3t^3 |
+| 6r^3 minus the factor-precision overhead | 43+67t+22t^2+2t^3 |
+
+For the final row, the alternative (r-1)ell<=r^2 also fits the cap.
+`cubic.py` checks these coefficient identities and exact exponent
+inequalities; it does not infer an all-size bound from sampled radices.
+
+If the incoming data have fractional width w, define the word scale
+A=max(p,w)+2m+16r^3. With a supplied exact rational phase of representation
+size B_tau, the one-kernel bill is
+
+    soft-O(r^3(p+m+r^3))
+    + O(rL) M(A+B_tau)
+    + O(rL(A+B_tau)).
+
+The terms are factor preparation/coefficient verification, rounded
+execution, and sequential tape traffic. Here M is the cost of the charged
+integer arithmetic; the Python reference is not a measured implementation
+of asymptotically fast multiplication or factorization. Longer initial
+coefficient/phase descriptions retain their actual input-reading cost.
+Execution uses the inherited clean scratch tapes and O(L) data records;
+there is no borrowed-array or bit-reversible realization claim.
+
+The original solver retains its 40L traffic bound per stable factor.
+Exact inner rotations and scalar applications add their inherited O(L)
+work. The optional zero-truncated solver has at most 44L per factor.
+Neither bound has a sorting factor or a precision admission depending
+on L. Transform-level layout, phase collection, normalization, common
+labels and the multiplication reduction remain separate.
+
+## Evidence and attribution
+
+The checks compare 423 array cases exactly between the unchanged and
+truncated solver, with independent geometric oracles for the short cases.
+They include non-dyadic margins and four deeper arrays through L=5741.
+Ninety scalar coefficient paths test exact extinction for gaps through
+2^-256; they are not huge-array executions.
+
+All 300 original PR846 kernel cases pass using the uniform effective-depth
+admission, with exact output equivalence and their original residual/dense
+checks. An additional 64 executions at the actual cubic W cover radices
+3/4/5/7, every stride at lengths 1/2/5, two unit phases, and arbitrary odd
+lowest working bits. Roots are rounded to no more than the stated
+preparation budget. Every common-operator hypothesis and the independent
+kernel perturbation are checked before using the cubic budget.
+
+The source lineage is PR846 (Rohan Guliani), PR764 (Charles Bethin), PR628
+and PR862 (Chafik Boukhalfa), and PR621 (Ali Romman), together with their
+retained classical references. This argument and its exact integration
+checks do not claim independent human peer review or formal verification.
+No new kappa is claimed.
diff --git a/research/cyclic-stride-effective-depth/README.md b/research/cyclic-stride-effective-depth/README.md
new file mode 100644
index 0000000000..6f0079ca6f
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/README.md
@@ -0,0 +1,47 @@
+# Cubic precision at arbitrary strides, with no length admission
+
+PR846's clean-tape stride inverse had a precision admission depending on
+`log log L`. This package removes that condition. Rounded chain coefficients
+become exactly zero after at most `2g+ceil(log2 W)+4` effective levels when
+`beta=2^-g` and `W>=2g+8`. The unchanged solver's remaining zero-coefficient
+subtree is exactly pointwise rounding, so its original output and 40L
+traffic bound are preserved. An optional early-exit version is retained
+for direct comparison and charged at 44L, including head returns.
+
+Combining that result with PR862's existing common-operator inverse bound
+gives **working precision `W<=p+m+9r^3` for every cycle length and stride**,
+under the inherited certified prime-power preparation hypotheses. The
+input norm is at most `2^m`, and the target error is `2^-p`. Root preparation
+has an explicit cubic budget too. The complete proof and cost bill are in
+[PROOF.md](PROOF.md).
+
+This is the original clean-scratch tape model. It is separate from PR951's
+neighboring-cycle construction without a borrowed data array. It does not
+supply bit-reversible arithmetic, physical common labels, transform-level
+composition or a multiplication theorem. **No new kappa.**
+
+## Replay
+
+Python 3.11+, standard library only, offline:
+
+```sh
+python3 -I -O -B research/cyclic-stride-effective-depth/verify.py
+python3 -I -O -B research/cyclic-stride-effective-depth/test_package.py
+```
+
+The expected marker is `PASS_CYCLIC_STRIDE_EFFECTIVE_DEPTH`. The default
+replay checks the manifest, 12 retained source SHA256/Git objects, local
+module bindings and every field of the recomputed receipt. `--out` writes
+a fresh external copy after comparison. `--bindings-only` checks package
+bindings. `--write` is explicit authoring mode, not verification.
+
+Evidence includes 423 array comparisons, 90 scalar coefficient paths,
+300 original kernel cases, 64 actual cubic-budget kernel executions,
+216 exact integer budget rows, three nonnegative polynomial identities,
+54 admission checks, 1,545 logarithm boundaries, 12 new mathematical controls, four inherited controls,
+12 zero-base checks and 33 tape-traffic cases through length 131,072.
+Finite checks and the written all-size argument are distinct.
+
+The original PR846 solver and checker, its five PR628 inputs, and PR862's
+proof and notice are retained unchanged. [SOURCE-PINS.json](SOURCE-PINS.json)
+and [NOTICE.md](NOTICE.md) identify their exact versions and contributions.
diff --git a/research/cyclic-stride-effective-depth/SOURCE-PINS.json b/research/cyclic-stride-effective-depth/SOURCE-PINS.json
new file mode 100644
index 0000000000..8fb635094f
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/SOURCE-PINS.json
@@ -0,0 +1,90 @@
+{
+  "base_commit": "007664306fff84fbb1c9a1b4053b69f439764034",
+  "repository": "https://github.com/CrocSwap/integer-mult-bounds",
+  "sources": [
+    {
+      "git_blob": "9229a4193b004d0dd939a502cfae75c27791dc76",
+      "head": "4bdb3e755ea7118170a7a34764df955880dec282",
+      "path": "vendor/pr846/stride.py",
+      "sha256": "da56e45f414a633453e06e8d058bcec6c815c73950816fcf049d92c570f8928e",
+      "source_path": "research/cyclic-inverse-stride-bridge/stride.py"
+    },
+    {
+      "git_blob": "0cc3867b79082894a18732f7751bc30414da0822",
+      "head": "4bdb3e755ea7118170a7a34764df955880dec282",
+      "path": "vendor/pr846/verify.py",
+      "sha256": "a56acd90c79597fa57602b053de09fa66930f262c39b849362b49b3cce707f2c",
+      "source_path": "research/cyclic-inverse-stride-bridge/verify.py"
+    },
+    {
+      "git_blob": "e1624e72ddffcd9fc2271691f79edcedeffc3c91",
+      "head": "4bdb3e755ea7118170a7a34764df955880dec282",
+      "path": "vendor/pr846/SOURCE.json",
+      "sha256": "07bc9fe307f254b68810f664bcef0f30eabd130226d39804bb38517777e3a92b",
+      "source_path": "research/cyclic-inverse-stride-bridge/SOURCE.json"
+    },
+    {
+      "git_blob": "8987bfdfc4ba6635cc3e0e984ca24d91da429084",
+      "head": "4bdb3e755ea7118170a7a34764df955880dec282",
+      "path": "vendor/pr846/PROOF.md",
+      "sha256": "3937696d2156e000519fb185a8a55fa7fd55dc12266fe0fe806dba42b2acf4e1",
+      "source_path": "research/cyclic-inverse-stride-bridge/PROOF.md"
+    },
+    {
+      "git_blob": "4dcef892b50028a3c9bce2d84954bca2fce44879",
+      "head": "4bdb3e755ea7118170a7a34764df955880dec282",
+      "path": "vendor/pr846/NOTICE.md",
+      "sha256": "9403006a0d8193298ac9c5dd4036619190aea29bfe37b03d2d716d8be264fc75",
+      "source_path": "research/cyclic-inverse-stride-bridge/NOTICE.md"
+    },
+    {
+      "git_blob": "5ef6dd67b6c6cf949fedf16c0cd7b34ca4ccafc2",
+      "head": "4bdb3e755ea7118170a7a34764df955880dec282",
+      "path": "vendor/pr846/vendor/check.py",
+      "sha256": "a3e46b556894d71a1c83bbbac01c50c1a541285fc9422a3567d5eb1d1ee86e7d",
+      "source_path": "research/cyclic-inverse-stride-bridge/vendor/check.py"
+    },
+    {
+      "git_blob": "d3894bd2a1660229f1a7ed16c9d982689effc96f",
+      "head": "4bdb3e755ea7118170a7a34764df955880dec282",
+      "path": "vendor/pr846/vendor/FACTORS.json",
+      "sha256": "643d60ac24d5e67f6c9305175bd1c99bf5e42edfef6d1995c051e0b98ba446a7",
+      "source_path": "research/cyclic-inverse-stride-bridge/vendor/FACTORS.json"
+    },
+    {
+      "git_blob": "1182175a55cb871481de0f56ea9ae56e474705fe",
+      "head": "4bdb3e755ea7118170a7a34764df955880dec282",
+      "path": "vendor/pr846/vendor/interval_kernel.py",
+      "sha256": "9b7bb24640a9a7d3f83c7951dfef483ec933db4c50de6c0130b85f450eeddfbe",
+      "source_path": "research/cyclic-inverse-stride-bridge/vendor/interval_kernel.py"
+    },
+    {
+      "git_blob": "d620e493596f12578c1ce47f216c44c79b65de08",
+      "head": "4bdb3e755ea7118170a7a34764df955880dec282",
+      "path": "vendor/pr846/vendor/scan.py",
+      "sha256": "1a9c01ca1a29a9be1e756858161202621db29d5d332a4266b4017a9502a50d7e",
+      "source_path": "research/cyclic-inverse-stride-bridge/vendor/scan.py"
+    },
+    {
+      "git_blob": "22fbeb8ef0a28dd6b3feea718492091319ceb96e",
+      "head": "4bdb3e755ea7118170a7a34764df955880dec282",
+      "path": "vendor/pr846/vendor/PR628-NOTICE.md",
+      "sha256": "abeacf1ad1491d5797cfcf1f3832799bbf3b24b99597229d7bc884059a51ec68",
+      "source_path": "research/cyclic-inverse-stride-bridge/vendor/PR628-NOTICE.md"
+    },
+    {
+      "git_blob": "0196499f010857b58ea297611f44721a4b7a0ec6",
+      "head": "57e0659150d323b784580110b36aa64a2dca2a33",
+      "path": "vendor/PR862-PROOF.md",
+      "sha256": "bdb0518892c631f8b6ac149535ac5a92270def19fe7778ca737641cd56b7da0b",
+      "source_path": "research/cubic-cyclic-inverse/PROOF.md"
+    },
+    {
+      "git_blob": "737060bd4d9666a925dfd1e17fc6eaa53c289b8d",
+      "head": "57e0659150d323b784580110b36aa64a2dca2a33",
+      "path": "vendor/PR862-NOTICE.md",
+      "sha256": "08c25cd6248e8c4dc7008e14e9ea7690f9084e94f927bb602363773637a89921",
+      "source_path": "research/cubic-cyclic-inverse/NOTICE.md"
+    }
+  ]
+}
diff --git a/research/cyclic-stride-effective-depth/cases.py b/research/cyclic-stride-effective-depth/cases.py
new file mode 100644
index 0000000000..023e57a229
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/cases.py
@@ -0,0 +1,80 @@
+"""Exact small-cycle, long-chain and scalar-decay evidence."""
+from fractions import Fraction as F
+from common import need,original as st
+from early import solve as truncated
+from uniform import admission,error_bound
+
+
+def compare(length,stride,beta,bits,dense):
+    u=F(1,1<<bits)
+    phases=[st.ONE,(F(3,5),F(4,5)),(F(-5,13),F(12,13))]
+    b=[tuple((1-beta)*v for v in phases[i%3]) for i in range(length)]
+    x=[(F((5*i)%9-4,8)+u,F((7*i)%11-5,16)-u) for i in range(length)]
+    before=(tuple(x),tuple(b))
+    full_tape,short_tape=st.Tape(),st.Tape()
+    trace=[]
+    full=st.solve(x,b,stride,bits,full_tape)
+    short=truncated(x,b,stride,bits,short_tape,trace)
+    need(full==short,'exact output equivalence')
+    need(before==(tuple(x),tuple(b)),'input source and coefficients unchanged')
+    record=admission(beta,bits)
+    depth=max(row['level'] for row in trace)
+    need(depth<=record['effective_depth'],'effective depth')
+    need(full_tape.records<=40*length and short_tape.records<=44*length,'complete traffic bills')
+    for row in trace:
+        need(2*row['stride']<=row['length'],'chain length at least two')
+        need(all(st.norm2(v)<=(1-record['beta0'])**2 for v in row['b']),'certified margin throughout')
+    B=max(abs(v[0])+abs(v[1]) for v in x)
+    bound=error_bound(B,beta,bits)
+    actual=None
+    if dense:
+        exact=st.exact_cycle_solve(x,b,stride)
+        actual=max(st.norm2(st.sub(v,e)) for v,e in zip(full,exact))
+        need(actual<=bound*bound,'length-independent original-solver error bound')
+    return dict(length=length,stride=stride,beta=beta,**record,
+                structural_depth=st.levels(length,stride),effective_observed_depth=depth,
+                original_records=full_tape.records,truncated_records=short_tape.records,
+                exact_output_equal=True,inputs_unchanged=True,actual_error_squared=actual,
+                uniform_error_bound=bound,terminal_zero=all(v==st.ZERO for v in trace[-1]['b']))
+
+
+def scalar_decay():
+    rows=[]
+    for g in [1,4,16,64,256]:
+        beta=F(1,1<<g)
+        for extra in [0,64]:
+            bits=2*g+8+extra
+            record=admission(beta,bits)
+            for phase in [st.ONE,(F(3,5),F(4,5)),(F(-5,13),F(12,13))]:
+                for chain_length in [2,3,7]:
+                    value=tuple((1-beta)*v for v in phase)
+                    steps=0
+                    while value!=st.ZERO and steps<=record['effective_depth']:
+                        product=value
+                        for _ in range(chain_length-1):
+                            product=st.rounded(st.mul(value,product),bits)
+                        value=product
+                        steps+=1
+                        need(st.norm2(value)<=(1-beta)**2,'scalar path stable margin')
+                    need(value==st.ZERO and steps<=record['effective_depth'],'exact coefficient extinction')
+                    rows.append(dict(g=g,bits=bits,phase=phase,chain_length=chain_length,
+                                     steps=steps,depth_cap=record['effective_depth']))
+    return rows
+
+
+def main():
+    rows=[]
+    for g in [1,2,4]:
+        for length in range(1,15):
+            for stride in range(length):
+                rows.append(compare(length,stride,F(1,1<<g),2*g+8,True))
+    for length,stride in [(408,169),(985,408),(2378,985),(5741,2378)]:
+        rows.append(compare(length,stride,F(1,2),10,False))
+    for beta in [F(1,3),F(2,7),F(5,17),F(1,1000)]:
+        for length in [7,19]:
+            for stride in range(length):
+                rows.append(compare(length,stride,beta,40,True))
+    decay=scalar_decay()
+    print('PASS stride cases',len(rows),'scalar coefficient paths',len(decay),flush=True)
+    return dict(cases=rows,scalar_decay=decay,
+                scalar_scope='Finite scalar product paths, not executions of arrays with exponentially large length.')
diff --git a/research/cyclic-stride-effective-depth/common.py b/research/cyclic-stride-effective-depth/common.py
new file mode 100644
index 0000000000..7a1d32c4a8
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/common.py
@@ -0,0 +1,29 @@
+"""Load the immutable predecessor and expose exact arithmetic helpers."""
+from pathlib import Path
+import importlib.util
+import sys
+
+P=Path(__file__).resolve().parent
+
+
+def need(ok,message):
+    if not ok:
+        raise ValueError(message)
+
+
+def load(path,name):
+    path=path.resolve()
+    if name in sys.modules:
+        module=sys.modules[name]
+        need(Path(module.__file__).resolve()==path,'local module binding: '+name)
+        return module
+    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
+
+
+predecessor=load(P/'vendor/pr846/verify.py','effective_depth_predecessor')
+original=predecessor.st
+need(Path(original.__file__).resolve()==P/'vendor/pr846/stride.py','original solver binding')
diff --git a/research/cyclic-stride-effective-depth/controls.py b/research/cyclic-stride-effective-depth/controls.py
new file mode 100644
index 0000000000..9f34c9586f
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/controls.py
@@ -0,0 +1,102 @@
+"""Admission, rounding, mutation and traffic checks for effective depth."""
+from pathlib import Path
+from fractions import Fraction as F
+import importlib.util
+import tempfile
+from common import P,need,original as st
+from early import solve
+from uniform import admission,error_bound,ceil_log2
+
+
+def main():
+    controls=[]
+    calls=[('zero margin',lambda:admission(F(0),16)),
+           ('unit margin',lambda:admission(F(1),16)),
+           ('floating margin',lambda:admission(0.25,16)),
+           ('boolean margin',lambda:admission(True,16)),
+           ('width below uniform admission',lambda:admission(F(1,16),15)),
+           ('boolean width',lambda:admission(F(1,2),True)),
+           ('negative width',lambda:admission(F(1,2),-1)),
+           ('negative source bound',lambda:error_bound(F(-1),F(1,2),10)),
+           ('zero logarithm',lambda:ceil_log2(F(0)))]
+    for name,call in calls:
+        try:
+            call()
+        except ValueError as error:
+            controls.append(dict(name=name,rejected=str(error)))
+        else:
+            raise ValueError('admission accepted '+name)
+
+    source=(P/'early.py').read_text()
+    mutations={
+        'any zero used as all zero':('all(bi == st.ZERO for bi in b)','any(bi == st.ZERO for bi in b)'),
+        'coarse product destroys coupling':('p[i] = st.rounded(st.mul(b[i], prev[1]), bits)',
+                                           'p[i] = st.rounded(st.mul(b[i], prev[1]), 0)'),
+    }
+    length,stride,bits,beta=7,3,16,F(1,4)
+    x=[st.ONE]*length
+    b=[st.ZERO]+[(F(3,4),F(0))]*(length-1)
+    exact=st.exact_cycle_solve(x,b,stride)
+    bound=error_bound(F(1),beta,bits)
+    baseline=solve(x,b,stride,bits,st.Tape())
+    need(all(st.norm2(st.sub(a,e))<=bound*bound for a,e in zip(baseline,exact)),'mutation baseline')
+    with tempfile.TemporaryDirectory(prefix='stride math controls ') as directory:
+        for index,(name,(before,after)) in enumerate(mutations.items()):
+            need(source.count(before)==1,'unique mutation target')
+            path=Path(directory)/(str(index)+'.py')
+            path.write_text(source.replace(before,after))
+            spec=importlib.util.spec_from_file_location('stride_mutation_'+str(index),path)
+            mutant=importlib.util.module_from_spec(spec)
+            spec.loader.exec_module(mutant)
+            bad=mutant.solve(x,b,stride,bits,st.Tape())
+            need(any(st.norm2(st.sub(a,e))>bound*bound for a,e in zip(bad,exact)),
+                 'defect not detected: '+name)
+            controls.append(dict(name=name,detected=True))
+
+    u=F(1,1<<16)
+    need(st.rounded((u/2,F(0)),16)!=st.ZERO,'positive half-grid tie is not zero')
+    need(st.rounded((u/4,-u/4),16)==st.ZERO,'strict zero threshold')
+    controls.append(dict(name='strict threshold distinguishes half-grid tie',detected=True))
+    zeros=[]
+    for length in [1,2,7,19]:
+        for stride in [-1,0,length//2]:
+            x=[(F(1,3),F(-2,7))]*length
+            b=[st.ZERO]*length
+            trace=[]
+            tape=st.Tape()
+            short=solve(x,b,stride,10,tape,trace)
+            full=st.solve(x,b,stride,10,st.Tape())
+            need(short==full and len(trace)==1,'zero subtree rounds once')
+            zeros.append(dict(length=length,stride=stride,records=tape.records))
+
+    traffic=[]
+    for length in [1,2,7,64,1000,4096,65536,131072]:
+        for stride in sorted({0,1,length//2,length-1,97%length}):
+            full_tape,short_tape=st.Tape(),st.Tape()
+            st.solve([None]*length,[None]*length,stride,None,full_tape)
+            solve([None]*length,[None]*length,stride,None,short_tape)
+            need(full_tape.records<=40*length and short_tape.records<=44*length,'traffic with head returns')
+            traffic.append(dict(length=length,stride=stride,original_records=full_tape.records,
+                                truncated_records=short_tape.records))
+    logarithms=[]
+    for exponent in range(-257,258):
+        power=F(2)**exponent
+        for value,expected in [(power*(1-F(1,1<<20)),exponent),(power,exponent),
+                               (power*(1+F(1,1<<20)),exponent+1)]:
+            answer=ceil_log2(value)
+            need(answer==expected and F(2)**(answer-1)<value<=F(2)**answer,'exact logarithm boundary')
+            logarithms.append(dict(exponent=exponent,answer=answer))
+    admissions=[]
+    for g in [1,2,3,4,8,16,64,256,1024]:
+        for extra in [0,1,2,16,128,1024]:
+            admissions.append(admission(F(1,1<<g),2*g+8+extra))
+    u0=F(1,1024)
+    q1=F(1,4)+2*u0
+    q2=q1*q1+2*u0
+    need(F(23,16)**2>2 and q1<F(1,2) and q2<F(1,4) and 36*u0*u0<u0/2,
+         'rational decay constants')
+    identities=dict(growth_square_gap=F(23,16)**2-2,first_half_step=q1,second_half_step=q2,
+                    strict_zero_gap=u0/2-36*u0*u0)
+    print('PASS depth controls',len(controls),'log boundaries',len(logarithms),flush=True)
+    return dict(controls=controls,zero_bases=zeros,traffic=traffic,log_boundaries=logarithms,
+                admissions=admissions,decay_constants=identities)
diff --git a/research/cyclic-stride-effective-depth/cubic.py b/research/cyclic-stride-effective-depth/cubic.py
new file mode 100644
index 0000000000..61c213dd25
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/cubic.py
@@ -0,0 +1,62 @@
+"""Exact cubic budgets combining the retained partial-inverse and stride bounds."""
+from fractions import Fraction as F
+from math import gcd
+from common import need
+from uniform import ceil_log2
+
+
+def plan(r,phi,p,m):
+    need(all(type(v) is int for v in (r,phi,p,m)) and r>=3 and 1<=phi<=r and p>=0 and m>=0,
+         'integer cubic budget inputs')
+    ell,c=ceil_log2(r),ceil_log2(r-1)
+    d=1+2*ell+r*(r-1)*phi
+    a0=2*d+r+2
+    g=d+ell+2*r+2
+    h=a0+2*r
+    e=max(d+1,a0+1,p+2*a0+m+3)
+    factor_bits=max((r-1)*ell,e+r+3)
+    J=c+2*h+3*g+ell+m+5
+    W=p+J+2
+    guard=m+ell+h+g+8
+    depth=2*g+(W-1).bit_length()+4
+    need(W>=2*g+8 and ceil_log2(4*(depth+1))<=W-g,'uniform stride admission')
+    # n<=2^c, 7<=2^3 and r*2^(h+m)+1<=2^(ell+h+m+1).
+    # These exact exponent bounds avoid constructing enormous fractions.
+    t_exponent=c+h+3+3*g-W
+    need(r-1<=1<<c and r<=1<<ell and t_exponent<=-1,'common-error contraction')
+    error_exponent=1+t_exponent+ell+h+m+1
+    need(error_exponent==J-W==-p-2,'common-subproduct execution bound')
+    error=F(1,1<<(p+2))
+    need(W<=p+m+9*r**3 and guard<=m+6*r**3 and factor_bits<=p+m+6*r**3,
+         'cubic precision, preparation and guard')
+    return dict(r=r,phi=phi,p=p,m=m,ell=ell,c=c,d=d,a0=a0,g=g,h=h,J=J,W=W,
+                guard=guard,e=e,factor_bits=factor_bits,effective_depth=depth,t_exponent=t_exponent,execution_bound=error,
+                complete_word_bound=p+2*m+16*r**3)
+
+
+def main():
+    rows=[]
+    for r in [3,4,5,7,8,9,11,13,16,17,25,27,32,49,64,81,125,128]:
+        phi=sum(gcd(r,j)==1 for j in range(1,r))
+        for p in [0,1,32,128]:
+            for m in [0,1,16]:
+                record=plan(r,phi,p,m)
+                # Keep compact integer budgets; every admission and exponent inequality
+                # is checked exactly. The proof explains their universal
+                # implication for the rational perturbation bound.
+                rows.append({k:v for k,v in record.items() if k!='execution_bound'})
+    # Coefficients in ascending powers of t=r-3, obtained by exact shifts
+    # of the polynomial differences, rather than finite radix sampling.
+    def shifted(coefficients):
+        from math import comb
+        return [sum(coefficients[j]*comb(j,i)*3**(j-i) for j in range(i,len(coefficients)))
+                for i in range(len(coefficients))]
+    width=shifted([-24,-31,7,2])
+    guard=shifted([-15,-14,3,3])
+    factor=shifted([-14,-11,4,2])
+    need(width==[0,65,25,2] and guard==[51,85,30,3] and factor==[43,67,22,2],
+         'cubic polynomial identities')
+    need(all(v>=0 for v in width+guard+factor),'all-size nonnegative remainders')
+    print('PASS cubic budget rows',len(rows),'identities 3',flush=True)
+    return dict(rows=rows,working_width_remainder=width,integer_guard_remainder=guard,
+                factor_preparation_remainder=factor)
diff --git a/research/cyclic-stride-effective-depth/early.py b/research/cyclic-stride-effective-depth/early.py
new file mode 100644
index 0000000000..edb120c955
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/early.py
@@ -0,0 +1,60 @@
+"""Optional exact-zero truncation of PR846; extra scans and head returns charged."""
+from common import original as st
+
+
+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)
+    st.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
+    # Experimental coefficient scan and return to its initial head position.
+    tape.records += 2 * L
+    if not count_only and all(bi == st.ZERO for bi in b):
+        tape.records += 2 * L
+        return [st.rounded(xi, bits) for xi in x]
+    if w == 0:  # every point is its own cycle
+        tape.records += 2 * L
+        if count_only:
+            return [None] * L
+        return [st.rounded(st.div(xi, st.sub(st.ONE, bi)), bits) for xi, bi in zip(x, b)]
+    # Pass A: zero-seed chains along +w, with rounded chain products.
+    fifo = st.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] = st.rounded(st.add(x[i], st.mul(b[i], prev[0])), bits)
+            p[i] = st.rounded(st.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 = st.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] = st.rounded(st.add(x[i], st.mul(b[i], prev)), bits)
+        fifo.push(y[i])
+    return y
diff --git a/research/cyclic-stride-effective-depth/kernels.py b/research/cyclic-stride-effective-depth/kernels.py
new file mode 100644
index 0000000000..8e2f83b823
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/kernels.py
@@ -0,0 +1,56 @@
+"""All original PR846 kernels with uniform admission and exact output equivalence."""
+from fractions import Fraction as F
+from common import need,predecessor as source,original as st
+from early import solve as truncated
+from uniform import admission,error_bound
+
+
+def main():
+    predecessor=source.pinned_vendor()
+    rows=[]
+    original_bound,original_apply=source.cascade_bound,source.apply_kernel
+    comparisons=[]
+
+    def bound(state,length,stride):
+        beta,bits,r=state['beta'],state['bits'],state['radix']
+        record=admission(beta,bits)
+        beta0=record['beta0']
+        u=F(1,1<<bits)
+        error,norm=F(0),F(r)
+        for _ in state['factors']:
+            error=error/beta0+error_bound(norm+error,beta,bits)+u
+            norm/=beta0
+        rows.append(dict(radix=r,length=length,stride=stride,**record,
+                         uniform_execution_bound=error))
+        return error
+
+    def apply(*args,**kwargs):
+        result=original_apply(*args,**kwargs)
+        # Swap only this invocation's solve callback; always restore it.
+        saved=st.solve
+        try:
+            st.solve=truncated
+            other=list(args)
+            other[6]=st.Tape()
+            short=original_apply(*other,**kwargs)
+        finally:
+            st.solve=saved
+        need(result==short,'original kernel exact output equivalence')
+        comparisons.append(True)
+        return result
+
+    source.cascade_bound,source.apply_kernel=bound,apply
+    try:
+        result=source.kernels(predecessor)
+        need(len(result)==300 and len(comparisons)==300,'complete original kernel inventory')
+        kernel_rows=rows.copy()
+        inherited_controls=source.controls(predecessor)
+    finally:
+        source.cascade_bound,source.apply_kernel=original_bound,original_apply
+    # Float log summaries in the original verifier are presentation only.
+    # Preserve exact budgets instead for platform-independent receipts.
+    result=[{k:v for k,v in row.items() if k not in ('execution_bound_log2','total_bound_log2')}
+            for row in result]
+    print('PASS 300 original kernel cases with uniform depth admission',flush=True)
+    return dict(cases=result,uniform_budgets=kernel_rows,exact_output_equalities=300,
+                inherited_controls=inherited_controls,control_budgets=rows[len(kernel_rows):])
diff --git a/research/cyclic-stride-effective-depth/test_package.py b/research/cyclic-stride-effective-depth/test_package.py
new file mode 100644
index 0000000000..c7fc2ac21f
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/test_package.py
@@ -0,0 +1,127 @@
+"""Fail-closed package controls and a full read-only optimized relocation."""
+import argparse
+import gzip
+import hashlib
+import json
+from pathlib import Path
+import shutil
+import subprocess
+import sys
+import tempfile
+
+P = Path(__file__).resolve().parent
+sys.dont_write_bytecode = True
+sys.set_int_max_str_digits(100000)
+
+
+def require(value, message):
+    if not value:
+        raise ValueError(message)
+
+
+def run(root, *args, code=None):
+    command = [sys.executable, '-I', '-O', '-B']
+    command += ['-c', code] if code is not None else [str(root/'verify.py'), *map(str, args)]
+    return subprocess.run(command, cwd=root.parent, text=True, stdout=subprocess.PIPE,
+                          stderr=subprocess.STDOUT, timeout=2400)
+
+
+def rebind(root, name):
+    path = root/'MANIFEST.json'
+    record = json.loads(path.read_text())
+    record['files'][name] = hashlib.sha256((root/name).read_bytes()).hexdigest()
+    path.write_text(json.dumps(record, indent=2, sort_keys=True)+'\n')
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--out', type=Path)
+    args = parser.parse_args()
+    if args.out:
+        require(not args.out.exists() and not args.out.resolve().is_relative_to(P), 'fresh external report')
+    results = []
+    with tempfile.TemporaryDirectory(prefix='stride depth package controls ') as temporary:
+        base = Path(temporary)
+        def copy(name):
+            root = Path(shutil.copytree(P, base/name))
+            # The supplied package may itself be read-only. Only these
+            # disposable test fixtures must permit deliberate corruption.
+            root.chmod(0o755)
+            for path in root.rglob('*'):
+                path.chmod(0o755 if path.is_dir() else 0o644)
+            return root
+        def rejection(name, mutate, *, full=False, code=None):
+            root = copy(name)
+            mutate(root)
+            output = base/(name+' output.json')
+            result = run(root, '--out', output, code=code(root) if code else None) if full or code else run(root, '--bindings-only')
+            require(result.returncode != 0, 'mutation accepted: '+name)
+            require(not output.exists(), 'failed verification wrote a receipt: '+name)
+            results.append(dict(name=name, rejected=True, terminal=result.stdout.strip().splitlines()[-1]))
+            print('PASS package control', name, flush=True)
+
+        rejection('missing manifest', lambda root: (root/'MANIFEST.json').unlink())
+        rejection('altered implementation', lambda root: (root/'early.py').write_text((root/'early.py').read_text()+'\n# altered\n'))
+        rejection('extra file', lambda root: (root/'undeclared.txt').write_text('unbound\n'))
+        rejection('missing source', lambda root: (root/'vendor/pr846/PROOF.md').unlink())
+        def wrong_source(root):
+            file = root/'vendor/pr846/PROOF.md'
+            file.write_text(file.read_text()+'\nchanged\n')
+            rebind(root, 'vendor/pr846/PROOF.md')
+        rejection('source hash independent of manifest', wrong_source)
+        def wrong_blob(root):
+            file = root/'SOURCE-PINS.json'
+            value = json.loads(file.read_text())
+            value['sources'][0]['git_blob'] = '0'*40
+            file.write_text(json.dumps(value)+'\n')
+            rebind(root, 'SOURCE-PINS.json')
+        rejection('source Git object independent of manifest', wrong_blob)
+        rejection('duplicate manifest key', lambda root: (root/'MANIFEST.json').write_text('{"files":{},"files":{}}\n'))
+        rejection('symlink injection', lambda root: (root/'linked.txt').symlink_to(root/'README.md'))
+        rejection('preloaded module substitution', lambda root: None, code=lambda root:
+                  "import sys,types,runpy; m=types.ModuleType('common'); m.__file__='/not/the/package.py'; "
+                  "sys.modules['common']=m; sys.argv=["+repr(str(root/'verify.py'))+"] ; runpy.run_path(sys.argv[0],run_name='__main__')")
+        def wrong_receipt(root):
+            file = root/'EXPECTED.json.gz'
+            value = json.loads(gzip.decompress(file.read_bytes()))
+            require(value['cubic']['rows'][0]['p'] == 0, 'known zero budget field')
+            value['cubic']['rows'][0]['p'] = False
+            file.write_bytes(gzip.compress((json.dumps(value, indent=2, sort_keys=True)+'\n').encode(), mtime=0))
+            rebind(root, 'EXPECTED.json.gz')
+        rejection('false receipt boolean integer confusion', wrong_receipt, full=True)
+
+        root = copy('existing output')
+        existing = base/'existing.json'
+        existing.write_bytes(b'preserve this exact output\n')
+        result = run(root, '--out', existing)
+        require(result.returncode != 0 and existing.read_bytes() == b'preserve this exact output\n', 'existing output preservation')
+        results.append(dict(name='existing output preserved', rejected=True))
+        print('PASS package control existing output preserved', flush=True)
+
+        root = copy('read only isolated relocation')
+        for file in root.rglob('*'):
+            file.chmod(0o555 if file.is_dir() else 0o444)
+        root.chmod(0o555)
+        output = base/'relocated receipt.json'
+        try:
+            result = run(root, '--out', output)
+            require(result.returncode == 0, 'full relocated replay failed: '+result.stdout[-2000:])
+            require(output.read_bytes() == gzip.decompress((P/'EXPECTED.json.gz').read_bytes()), 'relocated receipt byte equality')
+            require(run(root, '--bindings-only').returncode == 0, 'read-only package unchanged after replay')
+        finally:
+            root.chmod(0o755)
+            for directory in root.rglob('*'):
+                if directory.is_dir():
+                    directory.chmod(0o755)
+        results.append(dict(name='full read-only isolated optimized relocation', passed=True,
+                            receipt_sha256=hashlib.sha256(output.read_bytes()).hexdigest()))
+        print('PASS full read-only isolated optimized relocation', flush=True)
+    report = dict(status='PASS_STRIDE_DEPTH_PACKAGE', controls=results, rejected=11, full_readonly_replay=True)
+    if args.out:
+        with args.out.open('x') as handle:
+            handle.write(json.dumps(report, indent=2, sort_keys=True)+'\n')
+    print('PASS_STRIDE_DEPTH_PACKAGE rejected=11 full_readonly_replay=True')
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/cyclic-stride-effective-depth/uniform.py b/research/cyclic-stride-effective-depth/uniform.py
new file mode 100644
index 0000000000..e7bd2cae98
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/uniform.py
@@ -0,0 +1,29 @@
+"""Sufficient precision admission independent of cyclic length."""
+from fractions import Fraction as F
+from common import need,original
+
+
+def ceil_log2(value):
+    need(type(value) in (F,int) and value>0,'positive exact logarithm')
+    value=F(value)
+    n,d=value.numerator,value.denominator
+    e=n.bit_length()-d.bit_length()
+    larger=n>(d<<e) if e>=0 else (n<<-e)>d
+    return e+int(larger)
+
+
+def admission(beta,bits):
+    need(type(beta) in (F,int) and 0<beta<=F(1,2),'certified stable margin')
+    need(type(bits) is int,'integer working precision')
+    g=ceil_log2(1/F(beta))
+    need(bits>=2*g+8,'uniform working precision W>=2g+8')
+    depth=2*g+(bits-1).bit_length()+4
+    beta0=F(1,1<<g)
+    need(4*(depth+1)<=(1<<(bits-g)),'uniform depth admission')
+    return dict(g=g,bits=bits,beta0=beta0,effective_depth=depth)
+
+
+def error_bound(B,beta,bits):
+    need(type(B) in (F,int) and B>=0,'nonnegative exact source bound')
+    record=admission(beta,bits)
+    return original.solve_error_bound(B,record['beta0'],bits,record['effective_depth'])
diff --git a/research/cyclic-stride-effective-depth/uniform_kernels.py b/research/cyclic-stride-effective-depth/uniform_kernels.py
new file mode 100644
index 0000000000..93188b9864
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/uniform_kernels.py
@@ -0,0 +1,84 @@
+"""Actual cubic-width executions with each common-operator premise checked."""
+from fractions import Fraction as F
+from math import gcd
+import json
+from common import P,need,predecessor as source,original as st
+from early import solve as truncated
+from cubic import plan
+
+
+def main():
+    checker=source.pinned_vendor()
+    fixtures=json.loads((P/'vendor/pr846/vendor/FACTORS.json').read_text())
+    rows=[]
+    for fixture in fixtures:
+        r=fixture['radix']
+        if r not in (3,4,5,7):
+            continue
+        phi=sum(gcd(r,j)==1 for j in range(1,r))
+        budget=plan(r,phi,128,4)
+        prepared=json.loads(json.dumps(fixture))
+        prepared['factor_bits']=min(fixture['factor_bits'],budget['factor_bits'])
+        prepared['working_bits']=budget['W']
+        prepared['roots']=[[str(v) for v in st.rounded(tuple(F(z) for z in pair),prepared['factor_bits'])]
+                           for pair in fixture['roots']]
+        state=checker.certify(prepared)
+        need(prepared['factor_bits']<=budget['factor_bits'],'bounded input root preparation')
+        beta=F(1,1<<budget['g'])
+        H=F(state['certificate']['inverse_norm_bound'])
+        mass=sum(abs(v[0])+abs(v[1]) for v in state['q'])
+        partial=max(F(1),(1<<(r-1))*mass*H)
+        need(state['beta']>=beta and mass<=1<<r and H<=1<<(budget['a0']+1),
+             'prepared polynomial and margin hypotheses')
+        need(partial<=1<<budget['h'],'certified complementary-factor norm')
+        bits=budget['W']
+        u=F(1,1<<bits)
+        bound=budget['execution_bound']
+        for tau_name,tau in [('1',st.ONE),('3/5+4i/5',(F(3,5),F(4,5)))]:
+            need(st.norm2(tau)==1,'unit phase')
+            for length in [1,2,5]:
+                for stride in range(length):
+                    x=[(a+u,b-u) for a,b in source.inputs(length,stride+length)]
+                    snapshot=tuple(x)
+                    need(all((v/u).denominator==1 and (v/u).numerator%2 for z in x for v in z),
+                         'arbitrary lowest working bits')
+                    original_tape=st.Tape()
+                    full=source.apply_kernel(checker,state,x,stride,tau,bits,original_tape)
+                    saved=st.solve
+                    try:
+                        st.solve=truncated
+                        early_tape=st.Tape()
+                        early=source.apply_kernel(checker,state,x,stride,tau,bits,early_tape)
+                    finally:
+                        st.solve=saved
+                    need(full==early and tuple(x)==snapshot,'exact equivalence and unchanged source')
+                    D=max(abs(z[0])+abs(z[1]) for z in x)
+                    need(D<=16,'declared source bound')
+                    perturbation=F(state['certificate']['factorization_inverse_error_bound'])*D
+                    need(perturbation<=F(1,1<<130) and bound+perturbation<=F(1,1<<128),
+                         'root preparation and cubic execution meet target')
+                    q=checker.phased_polynomial(state['q'],tau)
+                    residual=F(0)
+                    for i in range(length):
+                        value=st.ZERO
+                        for j,coefficient in enumerate(q):
+                            value=st.add(value,st.mul(coefficient,full[(i-stride*j)%length]))
+                        residual=max(residual,st.norm2(st.sub(value,x[i])))
+                    need(residual<=(mass*bound)**2,'expanded original polynomial residual')
+                    dense_error=None
+                    if length<=2 and r<=5:
+                        exact=checker.matrix_solve(x,q,stride)
+                        dense_error=max(st.norm2(st.sub(a,b)) for a,b in zip(full,exact))
+                        need(dense_error<=bound*bound,'independent dense cubic execution')
+                    rows.append(dict(radix=r,length=length,stride=stride,gauge=tau_name,
+                                     working_bits=bits,prepared_root_bits=prepared['factor_bits'],
+                                     root_preparation_cap=budget['factor_bits'],depth_cap=budget['effective_depth'],
+                                     common_partial_bound=partial,common_partial_cap=1<<budget['h'],
+                                     execution_bound=bound,factorization_bound=perturbation,
+                                     total_bound=bound+perturbation,residual_squared=residual,
+                                     dense_error_squared=dense_error,exact_output_equal=True,
+                                     source_unchanged=True,original_records=original_tape.records,
+                                     truncated_records=early_tape.records))
+        print('PASS actual cubic stride radix',r,'cumulative cases',len(rows),flush=True)
+    return dict(cases=rows,scope='Actual uniform-width execution at radices 3,4,5,7; '
+                'the all-size claim is the written common-operator argument.')
diff --git a/research/cyclic-stride-effective-depth/vendor/PR862-NOTICE.md b/research/cyclic-stride-effective-depth/vendor/PR862-NOTICE.md
new file mode 100644
index 0000000000..737060bd4d
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/vendor/PR862-NOTICE.md
@@ -0,0 +1,31 @@
+# Attribution and scope
+
+New contribution by Chafik Boukhalfa (@chafreaky), prepared with substantial
+OpenAI Codex assistance, under the repository's Apache-2.0 license.
+
+- Ali Romman (@aliromman), PR621 at
+  `520774462b19b29af41554b74def31f9d01e997e`: prime-power kernel, circle
+  lower bound and length-uniform inverse bound. Its notice is retained.
+- Chafik Boukhalfa (@chafreaky), PR628 at
+  `0768f97d06c623b8211e651543c66135796b62bf`: stable cyclic scan, rounded
+  execution, preparation and phase arguments, exact kernel fixtures and
+  earlier dirty-shear construction. Its notice is retained.
+- The classical Mahler-measure bound, periodic filtering and approximate
+  factorization methods are not claimed as new. The source-recovery and
+  complementary-factor accounting are the contribution of this package.
+- Pan's factorization bound is cited through Mehlhorn, Sagraloff and Wang,
+  *From Approximate Factorization to Root Isolation with Application to
+  Cylindrical Algebraic Decomposition*, Theorem 2 and Corollary 1,
+  https://arxiv.org/pdf/1301.4870, PDF page 8. The inherited fast elementary
+  function reference and original kernel attribution remain in PR628's note.
+
+PR846 by Rohan Guliani and PR764 by Charles Bethin are discussed only as
+related stride and rounding work. No source from either is consumed here;
+their arbitrary-stride or sharper local bounds are not prerequisites of
+this package's claims.
+
+The dense oracle is a separate implementation in the same package. No
+independent human or delegated-agent review, formal verification, practical
+speedup, literature priority, complete multiplication theorem or new kappa
+is claimed. The reserved low bits are part of the source contract; arbitrary
+dirty storage and common-label/Clifford realization are not supplied.
diff --git a/research/cyclic-stride-effective-depth/vendor/PR862-PROOF.md b/research/cyclic-stride-effective-depth/vendor/PR862-PROOF.md
new file mode 100644
index 0000000000..0196499f01
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/vendor/PR862-PROOF.md
@@ -0,0 +1,259 @@
+# A cubic precision guard for the prime-power cyclic inverse
+
+## Result and boundary
+
+For the PR621 prime-power kernel of order r>=3, the PR628 neighboring
+cyclic scan needs working precision
+
+    max(p,w) + O(r^3 + log(D+1)),
+
+including integer magnitude bits, to approximate the inverse to absolute
+error 2^-p on a source of norm <=D initially on the 2^-w grid. The same
+precision suffices for coarse-grid in-place source recovery. The previous
+uniform scan composition charged O(r^4) because it propagated every local
+error through a product of individual reciprocal margins. Here every
+partial inverse is bounded by the full inverse and complementary factors.
+
+Array execution still costs O(rL) rounded scalar operations. With the same
+charged approximate-factorization and coefficient-verification algorithms,
+the bit cost for a neighboring cyclic layout is
+
+    soft-O((Lr+r^3)(max(p,w)+r^3+log(D+1))).
+
+This is a standard-arithmetic, supplied-layout theorem. The worst-case
+factorization and fast arithmetic are inherited algorithms, not implemented
+by the Fraction reference. The restored-source version needs owned widened
+source words whose low working bits are initially zero; those bits are
+charged. No arbitrary dirty-bit, Clifford, common-label, complete-transform
+or multiplication-exponent claim follows.
+Phase-oracle costs and the cost of reading a longer phase description are
+additional to the displayed bit bound.
+
+## Full inverse controls every partial inverse
+
+Let U be an invertible isometry in the maximum norm: ||U^k||=1 for every
+integer k. A unit-weighted permutation, including a weighted cyclic shift,
+has this property. Let
+
+    f(z)=product_(j=1)^n (z-rho_j) = s Q(z),
+    M >= sum_j |Q_j|,       ||Q(U)^-1|| <= A.
+
+Assume Q(U) is invertible. All factors are polynomials in the **same U**.
+The assertion below does not apply to different noncommuting shifts.
+
+For a monic polynomial f define its Mahler measure
+
+    M(f)=product_j max(1,|rho_j|).
+
+For completeness, the elementary circle identity
+mean_t log|exp(it)-rho| = log max(1,|rho|) follows by integrating the
+power series of log(1-z) when |z|<1 and factoring out rho when |rho|>1.
+There are no unit-circle roots in this application. Summing the identities,
+then applying concavity of log and Parseval, gives
+
+    log M(f) = mean log|f(exp(it))|
+             <= (1/2) log(mean |f(exp(it))|^2)
+             = (1/2) log(sum_j |f_j|^2).
+
+Thus M(f)<=||f||_2<=||f||_1<=sM. For any subset J of the root occurrences
+(multiplicities included), the complementary factor has coefficient norm
+
+    ||product_(j notin J)(z-rho_j)||_1
+       <= product_(j notin J)(1+|rho_j|)
+       <= 2^n M(f) <= 2^n sM.
+
+As the factors commute,
+
+    product_(j in J)(U-rho_j I)^-1
+       = product_(j notin J)(U-rho_j I) Q(U)^-1 / s.
+
+Every such inverse, including every prefix and every contiguous suffix,
+therefore has norm <= B=max(1,2^n M A). Importantly, log B depends on the
+full inverse bound, not n times the worst individual reciprocal margin.
+
+For later source restoration, let K_j=1+|Re rho_j|+|Im rho_j|. Then
+K_j<=3 max(1,|rho_j|), so every forward subproduct has norm bounded by
+its product of K_j, which is <=F=3^n sM. This also bounds amplification of
+the rounding errors introduced during forward restoration.
+
+## Rounded execution without repeated reciprocal-margin multiplication
+
+Suppose every factor is implemented by PR628's stable scan with margin
+beta, and u=2^-W<=beta^2/2. Including the final outer scalar or inner unit
+shift, its local error for an input of norm <=X is at most
+
+    a (X+1)u,       a=7 beta^-3.
+
+Begin with s x and apply all n inverse factors. The exact prefix norm is
+<=sBD. Let E_j be the computed prefix error, E=max_(0<=j<=n) E_j, and
+delta_j the local error at stage j. Then
+
+    |delta_j| <= a(sBD+E+1)u.
+
+The error at any prefix is the sum of earlier delta_j propagated through
+contiguous inverse subproducts. Each propagation has norm <=B, hence
+
+    E <= nBa(sBD+E+1)u.
+
+For t=nBa u<1 this implies
+
+    E <= t(sBD+1)/(1-t).                         (1)
+
+This rearrangement is not an inductive assumption that the error is small:
+all finite computed prefixes exist by the stable closure condition, and
+their maximum obeys the displayed inequality. The strict contraction
+condition then bounds it. The verifier compares every prefix with an
+independent exact dense oracle in its small-cycle cases.
+
+## Recovering a coarse-grid source
+
+The cyclic inverse's first pass computes the boundary seed read-only. Its
+second pass reads each original source entry before overwriting that slot.
+An inner factor's unit shift is performed in place with one saved boundary
+entry; an outer factor needs a scalar pass. Thus no full-length intermediate
+array is required for evaluation on a neighboring cyclic layout.
+
+After adding the computed inverse to a distinct working-grid target,
+apply the forward factors in reverse order. Each forward pass saves the
+original cyclic predecessor before overwriting the current entry. It has
+error at most u per output entry. Divide by s and round at W bits. Relative
+to the original source, the resulting error is at most
+
+    R <= F E/s + nF u/s + u.                    (2)
+
+Alternatively, the finite checker uses the sharper exact recurrence
+R=E, then R=K_j R+u for j=n,...,1, then R=R/s+u. If R<2^(-w-1), rounding
+each component to the original grid recovers the source exactly. The target
+increment is independent of its initial value. The low W-w source bits
+are explicit reserved zeros; their capacity is 2L(W-w) bits. Integer guards,
+target widening and arithmetic workspace are also charged.
+
+For a fixed coarse-grid external shear, round the computed increment to
+the original grid before its exact addition to the target. The resulting
+function f(x) approximates Q(U)^-1 x, depends only on x, and gives the exact
+grid bijection (x,y)->(x,y+f(x)). Subtracting the identical f(x) is its inverse.
+All arrays then stay on the same coarse grid between calls. The separate
+composition proof gives a fixed (w,W) budget for an entire prescribed word;
+it does not require new low bits at every stage.
+
+## Explicit prime-power budgets
+
+Let r>=3 be a prime power, n=r-1, Dfield=phi(r), H=2^r-1 and
+ell=ceil(log2 r), c=ceil(log2 n). To avoid overloading D, write the source
+bound as D<=2^m with m>=0. PR621 supplies
+
+    delta = 1/(2rn H^(n Dfield-1)),
+    |P(z)|>=delta on |z|=1,
+    ||P(U)^-1|| <= delta^-1 + 2nH/(r delta^2).
+
+Its absolutely summable Laurent-series argument applies to the isometry U
+above, including varying unit phases. No phase table is constructed for
+this norm argument. Put
+
+    d=1+2ell+r n Dfield,           delta>=2^-d,
+    a0=2d+r+2,                    ||P(U)^-1||<=2^a0,
+    g=d+ell+2r+2,                 beta=2^-g,
+    h=a0+2r,
+    e=max(d+1,a0+1,p+2a0+m+3),    epsilon=2^-e,
+    b=max(n ell,e+r+3).
+
+As in PR628, approximate-factorization and exact coefficient verification
+produce dyadic roots for Q=(1/r)product(z-rho_j) with ||P-Q||_1<=epsilon.
+The first two e constraints imply |Q|>=delta/2 on the circle and
+||Q(U)^-1||<=2^(a0+1). The last one gives
+
+    D ||P(U)^-1-Q(U)^-1|| <= 2^(-p-2).
+
+Also ||Q||_1<=H/r+epsilon<=2^r. PR628's radial derivative argument applied
+to Q supplies the stated stable margin beta for its inner or reciprocal
+outer roots. Consequently the complementary-factor bound gives B<=2^h.
+The forward product bound gives F<=2^v for v=3r+ell.
+
+Define
+
+    J=c+2h+3g+ell+m+5,
+    W=J+max(p+2,w+v+3).                         (3)
+
+This W satisfies u<=beta^2/2 and t<=1/2. Since rBD+1<=2^(ell+h+m+1),
+equation (1) yields E<=2^(J-W)<=2^(-p-2). Combining with coefficient
+perturbation gives total inverse error <=2^(-p-1), leaving extra slack.
+Since 2^J>=n+1, equation (2) gives
+
+    R <= 2^(v+J+1-W) <= 2^(-w-2) < 2^(-w-1),
+
+which proves exact source recovery after coarse rounding.
+
+These are explicit integer exponent inequalities, not numerical fits.
+Using Dfield<=r, ell<=r, c<=r and r>=3 gives the convenient bounds
+
+    W <= max(p,w)+12r^3+m,
+    b <= p+6r^3+m.
+
+For example, J+v+3 <= 7r^3+35r+25+m <=12r^3+m. Likewise
+e+r+3<=p+4r^3+11r+14+m<=p+6r^3+m, and n ell<=r^2.
+Intermediate magnitude bits also have order O(r^3+m): exact prefixes are
+bounded by rBD, scan closure state adds at most O(g) magnitude bits, and
+forward subproducts add O(r) bits. A target additionally needs the bits
+for its own initial magnitude. Bounds (3) concern fractional precision;
+the complete word width includes these integer bits.
+
+## Preparation, phases, and scope
+
+The monic exact polynomial rP has coefficient norm <=H and logarithmic
+root bound O(r). Pan's approximate-factorization bound, as stated in
+Mehlhorn–Sagraloff–Wang, Theorem 2 and Corollary 1, applies at b>=n log n
+with cost soft-O(nb+n^2 Gamma) and O(b+n Gamma) coefficient precision.
+Here Gamma=O(r) and b=O(p+r^3+m). The same coefficient oracle and
+conservative exact product verification from PR628 cost
+soft-O(r^3(p+r^3+m)); the latter forms exact O(rb)-bit coefficients.
+Repeated roots need not be individually isolated. This inherits the cited
+algorithms; the finite fixture checker does not implement their worst-case
+runtime. [Approximate-factorization source](https://arxiv.org/pdf/1301.4870),
+PDF page 8.
+
+A supplied exact rational unit phase uses its actual numerator/denominator
+length in scalar costs. For an approximate phase oracle, bound the induced
+polynomial coefficient perturbation separately and allocate additional
+slack by replacing p with p+O(1). Its required precision is O(p+r^3+m)
+under the same bounded description assumptions as PR628. An arbitrarily
+long phase description remains an extra input cost.
+
+The proof applies to any implementation of each factor with the stated
+local error bound. The concrete in-place and restored-source schedule here
+covers neighboring cyclic layouts only. PR846's clean-tape arbitrary-stride
+solver has a compatible local bound and an additional depth admission
+condition; extending the present tape claim would require retaining that
+condition and its clean-scratch model. That solver is not part of this
+package or its certified source-restoration claim.
+
+The cubic improvement comes from bounding intermediate inverse products,
+not from improving PR621's circle lower bound. The full multiplication
+assembly, common-label and Clifford interfaces, arbitrary dirty storage,
+transform-level phase and normalization accounting remain separate.
+No new kappa, practical timing improvement, or literature-priority claim
+is made.
+
+## Executable evidence and attribution
+
+The exact verifier checks 5,220 integer budgets over all 116 prime-power
+radices from 3 through 512, five target precisions, three source precisions
+and three magnitude bounds. These support implementation of (3); the
+inequalities above are the all-size proof.
+
+There are 120 kernel executions on ten pinned PR628 fixtures, with both
+directions, varying exact rational unit phases and lengths 1,2,3,8,17,64.
+Sixty executions compare every prefix to independent dense solves. All
+120 check the expanded polynomial residual, pre-round recovery error,
+exact coarse-grid restoration and total true-kernel error <=2^-128.
+The finite tests use PR628's existing, conservative supplied inverse-norm
+certificates; they do not claim that those finite certificates are tighter
+than every previous finite execution budget. Six controls reject omitted
+complement growth, insufficient target/recovery/coefficient precision,
+nonshared shifts and lost prefix contraction.
+
+Credit Ali Romman for PR621, Chafik Boukhalfa for PR628, and the standard
+Mahler-measure, cyclic-filtering and approximate-factorization methods.
+This extension was prepared by Chafik Boukhalfa with substantial OpenAI
+Codex assistance. The independent dense checker is a separate implementation,
+not an independent human or delegated-agent review. Source pins are in
+`SOURCE.json`.
diff --git a/research/cyclic-stride-effective-depth/vendor/pr846/NOTICE.md b/research/cyclic-stride-effective-depth/vendor/pr846/NOTICE.md
new file mode 100644
index 0000000000..4dcef892b5
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/vendor/pr846/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-stride-effective-depth/vendor/pr846/PROOF.md b/research/cyclic-stride-effective-depth/vendor/pr846/PROOF.md
new file mode 100644
index 0000000000..8987bfdfc4
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/vendor/pr846/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-stride-effective-depth/vendor/pr846/SOURCE.json b/research/cyclic-stride-effective-depth/vendor/pr846/SOURCE.json
new file mode 100644
index 0000000000..e1624e72dd
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/vendor/pr846/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-stride-effective-depth/vendor/pr846/stride.py b/research/cyclic-stride-effective-depth/vendor/pr846/stride.py
new file mode 100644
index 0000000000..9229a4193b
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/vendor/pr846/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-stride-effective-depth/vendor/pr846/vendor/FACTORS.json b/research/cyclic-stride-effective-depth/vendor/pr846/vendor/FACTORS.json
new file mode 100644
index 0000000000..d3894bd2a1
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/vendor/pr846/vendor/FACTORS.json
@@ -0,0 +1,402 @@
+[
+  {
+    "radix": 3,
+    "embedding": 1,
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+        "-56200741848002868079111868432724886507129621548125121648642474266599250489673841691778046881437449293933434722648357280706913445930283045991236908716738562916199191451345481529063774448710946467551285759609323141571550416355308596877200595274065373565280076263027603986562578831336689401091825392816459762247/44942328371557897693232629769725618340449424473557664318357520289433168951375240783177119330601884005280028469967848339414697442203604155623211857659868531094441973356216371319075554900311523529863270738021251442209537670585615720368478277635206809290837627671146574559986811484619929076208839082406056034304"
+      ],
+      [
+        "15968822176892792285891476438139425317687583108455281192008770761103687215213981897594608046751009662616135817210933151016991807601573680585709456982599855455736137633949541869722566316681492518193528500018722529515726386747292338115274696063766393314538517893691137581991598848479112650484943606294759671/1404447761611184302913519680303925573139044514798677009948672509044786529730476274474284979081308875165000889686495260606709295068862629863225370551870891596701311667381761603721111090634735110308227210563164107569048052205800491261514946176100212790338675864723330454999587858894372783631526221325189251072",
+        "-266324790638046954468748086815326096854011998014910958879637384810491281671701735342579622902906841022217262017537242815542832447116697353634706329017518582629343845840130883918382746688279956080610789278337184153491906948023245930590272626869013417164765399270940382908317755754807962553226947387458433481969/179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216"
+      ],
+      [
+        "1060865642031292630770255648031651464303955219241748845301061404301449150107131123877033982501236564870546734157170738071205142825981730629211934466958373310795930234883644313113012070731987912222503770737372313560840648279391481374684780737322726034313967342539259478356716096981485614792922808816297076935/1404447761611184302913519680303925573139044514798677009948672509044786529730476274474284979081308875165000889686495260606709295068862629863225370551870891596701311667381761603721111090634735110308227210563164107569048052205800491261514946176100212790338675864723330454999587858894372783631526221325189251072",
+        "-246799233700152837800565619104070457222176842192478672102595825359550055541867252492449227477274629665694532828700249246838030620913792598556947021011742874894999694125075108903854687975235844168506798118891370394456073402356817011957930168970746212902758217822556747842079095849448037149783703853881509351465/179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216"
+      ],
+      [
+        "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-stride-effective-depth/vendor/pr846/vendor/PR628-NOTICE.md b/research/cyclic-stride-effective-depth/vendor/pr846/vendor/PR628-NOTICE.md
new file mode 100644
index 0000000000..22fbeb8ef0
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/vendor/pr846/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-stride-effective-depth/vendor/pr846/vendor/check.py b/research/cyclic-stride-effective-depth/vendor/pr846/vendor/check.py
new file mode 100644
index 0000000000..5ef6dd67b6
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/vendor/pr846/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-stride-effective-depth/vendor/pr846/vendor/interval_kernel.py b/research/cyclic-stride-effective-depth/vendor/pr846/vendor/interval_kernel.py
new file mode 100644
index 0000000000..1182175a55
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/vendor/pr846/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-stride-effective-depth/vendor/pr846/vendor/scan.py b/research/cyclic-stride-effective-depth/vendor/pr846/vendor/scan.py
new file mode 100644
index 0000000000..d620e49359
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/vendor/pr846/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-stride-effective-depth/vendor/pr846/verify.py b/research/cyclic-stride-effective-depth/vendor/pr846/verify.py
new file mode 100644
index 0000000000..0cc3867b79
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/vendor/pr846/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()
diff --git a/research/cyclic-stride-effective-depth/verify.py b/research/cyclic-stride-effective-depth/verify.py
new file mode 100644
index 0000000000..155506daeb
--- /dev/null
+++ b/research/cyclic-stride-effective-depth/verify.py
@@ -0,0 +1,112 @@
+"""Offline source-bound stride-depth replay, standard-library Python 3.11+."""
+import argparse
+import gzip
+import hashlib
+import importlib
+import json
+from pathlib import Path
+import re
+import sys
+
+P = Path(__file__).resolve().parent
+sys.dont_write_bytecode = True
+sys.set_int_max_str_digits(100000)
+
+
+def require(value, message):
+    if not value:
+        raise ValueError(message)
+
+
+def unique_object(pairs):
+    out = {}
+    for key, value in pairs:
+        require(key not in out, 'duplicate JSON key: '+key)
+        out[key] = value
+    return out
+
+
+def read_json(path):
+    data = gzip.decompress(path.read_bytes()) if path.suffix == '.gz' else path.read_bytes()
+    return json.loads(data, object_pairs_hook=unique_object)
+
+
+def canonical(value):
+    return json.dumps(value, sort_keys=True, separators=(',', ':'), default=str)
+
+
+def bindings(authoring=False):
+    manifest = P/'MANIFEST.json'
+    require(manifest.exists() or authoring, 'missing package manifest')
+    require(not any(path.is_symlink() for path in P.rglob('*')), 'package symlink')
+    if manifest.exists():
+        record = read_json(manifest)
+        require(set(record) == {'files'} and type(record['files']) is dict, 'manifest schema')
+        files = record['files']
+        actual = {str(path.relative_to(P)) for path in P.rglob('*') if path.is_file() and path != manifest}
+        require(set(files) == actual, 'package file coverage')
+        for name, sha in files.items():
+            require(not Path(name).is_absolute() and '..' not in Path(name).parts, 'relative package path')
+            require(type(sha) is str and re.fullmatch('[0-9a-f]{64}', sha), 'manifest hash type')
+            require(hashlib.sha256((P/name).read_bytes()).hexdigest() == sha, 'package hash: '+name)
+    pins = read_json(P/'SOURCE-PINS.json')
+    require(len(pins['sources']) == 12, 'source inventory')
+    require(len({row['path'] for row in pins['sources']}) == 12, 'unique source inventory')
+    for row in pins['sources']:
+        name = row['path']
+        require(not Path(name).is_absolute() and '..' not in Path(name).parts, 'relative source path')
+        raw = (P/name).read_bytes()
+        require(hashlib.sha256(raw).hexdigest() == row['sha256'], 'source SHA256: '+name)
+        require(hashlib.sha1(b'blob '+str(len(raw)).encode()+b'\0'+raw).hexdigest() == row['git_blob'],
+                'source Git blob: '+name)
+    return dict(files=12, source_pins_sha256=hashlib.sha256((P/'SOURCE-PINS.json').read_bytes()).hexdigest())
+
+
+def modules():
+    sys.path.insert(0, str(P))
+    names = ['common', 'uniform', 'early', 'cases', 'kernels', 'controls', 'cubic', 'uniform_kernels']
+    loaded = {}
+    for name in names:
+        loaded[name] = importlib.import_module(name)
+        require(Path(loaded[name].__file__).resolve() == P/(name+'.py'), 'local module binding: '+name)
+    return loaded
+
+
+def calculate():
+    loaded = modules()
+    result = dict(status='PASS_CYCLIC_STRIDE_EFFECTIVE_DEPTH', kappa='unchanged')
+    for name in ['cases', 'controls', 'cubic', 'kernels', 'uniform_kernels']:
+        result[name] = loaded[name].main()
+    result['scope'] = ('Effective nonzero depth independent of L, exact equivalence with the '
+                       'unchanged stride solver, and cubic prepared-kernel precision. '
+                       'Inherited clean scratch model; no dirty-array, bit-reversible or kappa claim.')
+    return result
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--write', type=Path, help='authoring: write a fresh external receipt')
+    parser.add_argument('--out', type=Path, help='verify expected mathematics and write a fresh external copy')
+    parser.add_argument('--bindings-only', action='store_true')
+    args = parser.parse_args()
+    require(sum(bool(v) for v in (args.write, args.out, args.bindings_only)) <= 1, 'choose one mode')
+    destination = args.write or args.out
+    if destination:
+        require(not destination.exists(), 'output already exists')
+        require(not destination.resolve().is_relative_to(P), 'output must be outside package')
+    pins = bindings(authoring=args.write is not None)
+    if args.bindings_only:
+        print('PASS_STRIDE_DEPTH_BINDINGS sources=12')
+        return
+    result = calculate()
+    result['sources'] = pins
+    if not args.write:
+        require(canonical(result) == canonical(read_json(P/'EXPECTED.json.gz')), 'mathematical receipt mismatch')
+    if destination:
+        with destination.open('x') as handle:
+            handle.write(json.dumps(result, indent=2, sort_keys=True, default=str)+'\n')
+    print('PASS_CYCLIC_STRIDE_EFFECTIVE_DEPTH arrays=423 kernels=300 uniform=64 scalar_paths=90 budgets=216 kappa=unchanged')
+
+
+if __name__ == '__main__':
+    main()