From 52c6fba58ff74fdca1131e50379287491e599f74 Mon Sep 17 00:00:00 2001
From: Chafik Boukhalfa <chafik.boukhalfa@gmail.com>
Date: Sat, 10 Oct 2026 02:01:05 +0400
Subject: [PATCH 1/6] A stronger complex supplier: PR200's physical layer on
 the source-assisted v4 word, complex saving 7.0861e-4 (no new kappa)

Adds research/source-assisted-v4-layer and one workflow. PR194's pipeline
(PR184's source-assisted word on the PR168 v4 modules: aligned rewrite,
frame flow, exact lift, contract, assembly) is kept unchanged except the
physical layer the flow receives: PR200's descent seeded from PR168's frames
changes 5,577 of 17,789 operation frames and recomputes the 2,310 reuse
pairs. The layer is admitted by PR200's complete complex checker on the
aligned word; the flow, exact lift and contract certify complex saving
3543063/5000000000 = 7.0861e-4 (PR194: 7.00918e-4, +1.10%). With PR200's bit
supplier the assembly gives kappa 6768823/10^10, unchanged (bit binds).
The 308 pipeline files of PR202 (8d8d67b) are vendored with sha256 pins.

Prepared by Chafik Boukhalfa with substantial Anthropic Claude and OpenAI
Codex assistance.
---
 .../workflows/source-assisted-v4-layer.yml    |  30 +
 research/source-assisted-v4-layer/NOTICE      |  43 ++
 research/source-assisted-v4-layer/PROOF.md    |  69 ++
 research/source-assisted-v4-layer/README.md   |  29 +
 research/source-assisted-v4-layer/SOURCE.json | 323 +++++++++
 .../baseline-pr202.tar.gz                     | Bin 0 -> 7524513 bytes
 .../source-assisted-v4-layer/certificate.json | 614 ++++++++++++++++++
 .../discovery/README.md                       |  14 +
 .../discovery/bundles.py                      |  50 ++
 .../discovery/deep.py                         |  52 ++
 .../discovery/extend.py                       |  97 +++
 .../discovery/flow_frames.py                  |  34 +
 .../discovery/harness.py                      |  80 +++
 .../discovery/joint.py                        |  67 ++
 .../discovery/matching.py                     | 131 ++++
 .../discovery/pairs_mw.py                     |  42 ++
 .../discovery/physical_opt.py                 | 127 ++++
 research/source-assisted-v4-layer/verify.py   | 198 ++++++
 .../witness/kernel-pairs.json                 |   1 +
 .../witness/physical-frames.json              |   1 +
 .../witness/physical-pairs.json               |   1 +
 21 files changed, 2003 insertions(+)
 create mode 100644 .github/workflows/source-assisted-v4-layer.yml
 create mode 100644 research/source-assisted-v4-layer/NOTICE
 create mode 100644 research/source-assisted-v4-layer/PROOF.md
 create mode 100644 research/source-assisted-v4-layer/README.md
 create mode 100644 research/source-assisted-v4-layer/SOURCE.json
 create mode 100644 research/source-assisted-v4-layer/baseline-pr202.tar.gz
 create mode 100644 research/source-assisted-v4-layer/certificate.json
 create mode 100644 research/source-assisted-v4-layer/discovery/README.md
 create mode 100644 research/source-assisted-v4-layer/discovery/bundles.py
 create mode 100644 research/source-assisted-v4-layer/discovery/deep.py
 create mode 100644 research/source-assisted-v4-layer/discovery/extend.py
 create mode 100644 research/source-assisted-v4-layer/discovery/flow_frames.py
 create mode 100644 research/source-assisted-v4-layer/discovery/harness.py
 create mode 100644 research/source-assisted-v4-layer/discovery/joint.py
 create mode 100644 research/source-assisted-v4-layer/discovery/matching.py
 create mode 100644 research/source-assisted-v4-layer/discovery/pairs_mw.py
 create mode 100644 research/source-assisted-v4-layer/discovery/physical_opt.py
 create mode 100644 research/source-assisted-v4-layer/verify.py
 create mode 100644 research/source-assisted-v4-layer/witness/kernel-pairs.json
 create mode 100644 research/source-assisted-v4-layer/witness/physical-frames.json
 create mode 100644 research/source-assisted-v4-layer/witness/physical-pairs.json

diff --git a/.github/workflows/source-assisted-v4-layer.yml b/.github/workflows/source-assisted-v4-layer.yml
new file mode 100644
index 0000000000..833f348adc
--- /dev/null
+++ b/.github/workflows/source-assisted-v4-layer.yml
@@ -0,0 +1,30 @@
+name: source-assisted-v4-layer
+
+on:
+  pull_request:
+    paths:
+      - 'research/source-assisted-v4-layer/**'
+      - '.github/workflows/source-assisted-v4-layer.yml'
+  push:
+    paths:
+      - 'research/source-assisted-v4-layer/**'
+      - '.github/workflows/source-assisted-v4-layer.yml'
+
+jobs:
+  verify:
+    runs-on: ubuntu-latest
+    timeout-minutes: 30
+    strategy:
+      matrix:
+        python-version: ['3.11', '3.13']
+    steps:
+      - uses: actions/checkout@v4
+      - uses: actions/setup-python@v5
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Install the regeneration dependencies of the pinned pipeline
+        run: python -m pip install numpy==2.3.5 scipy==1.17.0
+      - name: Regenerate the word, admit this package's physical layer, run flow, lift, contract and assembly, compare with certificate.json
+        run: python3 -B research/source-assisted-v4-layer/verify.py
+      - name: Require an unchanged tree
+        run: git diff --exit-code
diff --git a/research/source-assisted-v4-layer/NOTICE b/research/source-assisted-v4-layer/NOTICE
new file mode 100644
index 0000000000..27c7d368fa
--- /dev/null
+++ b/research/source-assisted-v4-layer/NOTICE
@@ -0,0 +1,43 @@
+Prepared by Chafik Boukhalfa with substantial Anthropic Claude and OpenAI
+Codex assistance. Apache-2.0. Inherited files retain their original notices
+and assistance disclosures. This package claims no exclusive priority over
+concurrent work and reports no new kappa.
+
+What this package contributes: the physical layer (operation frames and reuse
+pairs) of PR #200's descent applied to the source-aligned complex word, the
+admission of that layer with PR #200's checker on that word, and the
+observation that PR #184's frame flow prices it at 7.0861e-4. Everything
+else is retained and credited:
+
+- icekylinx (OpenAI GPT-6 Astra and Codex assistance): PR144 paired cubes,
+  signed decoder, shared-core accounting, original compiler, carrier
+  matching (the frozen round-nine matching reused here), chronological
+  gauges, PR130 generalized Clifford frames; PR184 source-assisted
+  construction: source-parity local factorization, frame flow with
+  independent dirty lifts and source controls, exact lifts, contract,
+  global assembly (research/source-assisted, vendored byte for byte).
+- ikeboy: PR191/193/194, the rebuild of PR184's construction on the PR168 v4
+  modules (research/source-assisted-v4: source_aligned_local_v4.py,
+  contract_v4.py, assemble.py, the parity-erasure-avoiding matching), and the
+  composition with PR189's/PR200's bit supplier.
+- eumemic (Anthropic Claude and OpenAI Codex assistance): PR117 module
+  infrastructure; PR155/157 scalar modules, physical frames and reuse;
+  PR161 output merging; PR168 annealed modules, nested-prefix schedules,
+  local L1 circuit, fused outputs, bundle descent, both physical layers
+  (the physical layer replaced here is PR168's, used as the descent's seed),
+  pinned at 4a3c769e5c5430e7114c4d3e099ff34664677f17; PR131/143 frame/reuse
+  and late-read lineage; PR154 isolated snapshot verification.
+- jamesyc / James Chang: PR124 compensated birth cuts, PR34 balanced
+  assembly, PR166 terminal-output accumulation. DaysSky: PR162 extended
+  carrier closure, PR192 frame-layout ceiling. GamingPuzzled: PR160.
+  gupt1156 PR148 and Abhinav Ramachandran/geckods PR158 (Hermes assistance).
+  jacklightChen / Zhihao Chen: PR23/29. Rohan Arun: PR100/103. an664: PR128.
+  Earlier lineage: Zhihao Chen PR97, Swapnil Jain, DaysSky PR150, hpst3r
+  PR147, SovereignSteak PR122, Aurel Prosz, dleen, Douglas Colkitt.
+- Chafik Boukhalfa (Anthropic Claude and OpenAI Codex assistance): PR175
+  fast checker; PR179/181/189/196/200 local circuits, operation-frame descent
+  (discovery/), independent complete complex admission (complex/physical.py,
+  verify_fused.py, vendored from PR200 at a1175449) and the PR200 bit
+  supplier the assembly reads; PR202 (the pinned tree, 8d8d67b).
+- OpenAI: the original complex phase network of "Integer multiplication
+  below n log n".
diff --git a/research/source-assisted-v4-layer/PROOF.md b/research/source-assisted-v4-layer/PROOF.md
new file mode 100644
index 0000000000..213c8daf87
--- /dev/null
+++ b/research/source-assisted-v4-layer/PROOF.md
@@ -0,0 +1,69 @@
+# A stronger complex supplier: our physical layer on the source-assisted v4 word
+
+**Claim.** The source-assisted complex word of PR #184 (icekylinx) on the PR #168 v4 modules (PR #193/#194,
+ikeboy), with every input of PR #194's pipeline unchanged except the physical layer, certifies the complex saving
+
+    a_C = 3543063 / 5000000000 = 7.0861260·10⁻⁴      (PR #194: 219037/312500000 = 7.0091844·10⁻⁴, +1.10%).
+
+Composed with PR #200's bit supplier through PR #194's unchanged 47-constraint assembly the result is
+κ = 6768823/10¹⁰, the same as PR #202, because the bit branch binds (effective 6.7734·10⁻⁴). **No new κ.** The
+complex cap of every composition on this lineage rises from 7.0092·10⁻⁴ to 7.0861·10⁻⁴, and the exact-DFT
+exponent saving of `research/exact-dft-transfer` (PR #225) rises with it to 7086/10⁷.
+
+## What changes
+
+PR #194's pipeline is: regenerate the PR #168 v4 complex cache (`scripts/paired_cube_producer.py`); the
+source-parity rewrite of the three query modules (`source_aligned_local_v4.py`), which also runs PR #168's physical
+layer (endpoint descent and reuse pairs) and writes it to `references/paired-cube/physical/{frames,pairs}.json` of
+the aligned tree; the equal-frame fresh-channel compression (`complex_frame_flow.py`), which reads that physical
+layer; the exact arbitrary-dirty lifts (`exact_complex_flow_lift.py`); the contract (`contract_v4.py`); and the
+assembly. The flow's child histogram depends on the physical layer it receives: the frames decide which channels are
+equal-frame and compress.
+
+This package replaces only that physical layer. The frames and reuse pairs in `witness/` are produced by PR #200's
+descent (`discovery/`): endpoint moves of single operations, equal-frame components and connected bundles, started
+from PR #168's frames, on the aligned word's own cache. 5,577 of PR #168's 17,789 operation frames change; the
+2,310 reuse pairs are recomputed by the same late compensated pairing. The matching arcs, modules, local circuit,
+gauges, aligned rewrite, flow options and kernel pairs are PR #194's (the kernel pairs the flow finds on this layer
+are recorded in `witness/kernel-pairs.json`, as PR #194 records its own).
+
+## What is checked (`verify.py`, about 80 seconds, Python 3.11+ with numpy/scipy for the regeneration steps)
+
+1. **Pins.** `baseline-pr202.tar.gz` holds the 308 files of PR #202 at `8d8d67b` that the pipeline runs (PR #194's
+   package and PR #184's decision/global scripts, the PR #168 v4 scripts, references and inherited witnesses, the
+   Clifford frames note, and PR #200's certificate, arithmetic and complex checker), each pinned by sha256 in
+   `SOURCE.json`; the three witness files are pinned too. Reviewers can confirm with `git show 8d8d67b:<path> |
+   sha256sum`.
+2. **Regeneration.** The producer and the aligned rewrite run unchanged, including PR #168's physical layer and its
+   own physical scalar/frame checks on that layer.
+3. **Admission of this layer.** With the package's frames and pairs installed, PR #200's complete complex checker
+   (`complex/physical.py` with `verify_fused.py`) re-executes the aligned word: every descended frame contains its
+   node span and is nondegenerate, every role chain nests (entrance, operations, root frame, full space), every alias
+   is read at its deadline after its donor's last operation, the target read chronology holds, the exact adjoint is
+   rederived, and every formal source, target and dirty column is replayed in both shear signs on an injective
+   packed basis with an independently derived coefficient bound. The ledger it re-counts is the candidate's
+   profile.
+4. **Flow, lift, contract.** PR #184's flow runs on this layer with the package's kernel pairs; the exact lift
+   verifies every local map and inverse gate program over the rationals (all coefficients dyadic) and the monotone,
+   acyclic once-only reuse DAG with frame containment; the contract certifies the fresh columns, read chronology,
+   target chains and the normalized profile (13 checks), with the package's physical and kernel pairs as the frozen
+   witnesses that PR #194's contract compares against.
+5. **Assembly.** PR #194's `assemble.py` composes the certified complex profile with PR #200's bit certificate and
+   reproduces `certificate.json` exactly: complex saving 3543063/5000000000, κ 6768823/10¹⁰, the next grid point
+   rejected by the unchanged assembly.
+
+The complex profile: m = 66, W = 12,052 roles per vertex, rank mass 794,112, deficit 1,320, largest child 20, the
+same W, rank and deficit as PR #194; only the child histogram changes (the flow compresses the same number of
+roles, 13,372 physical to 9,412, but splits the remaining increments differently).
+
+## Scope and limits
+
+- Conditional exactly as PR #194: the retained analytic, uniform-recursion, precision, weighted bit frames,
+  fixed-tape routing, restored-row and all-size semantic interfaces remain hypotheses; the scalar replays and exact
+  lifts are finite checks. The independent complex scalar transcript of PR #184 is not re-exported here; the
+  one-column identity is replayed by PR #200's checker on the pre-flow word and by the lift's inverse gate programs.
+- Nothing here changes κ; the bit supplier binds. It raises the complex cap for every bit-side construction on
+  this lineage (for example the banked rows of #205–#223 and the conditional rungs of #219/#224) and the exact-DFT
+  exponent.
+- The descent is a discovery heuristic; no optimality is claimed. The float screen varies with the descent options
+  (light 7.0419·10⁻⁴, plain 7.0691·10⁻⁴, seeded 7.0861·10⁻⁴); only the seeded layer is certified here.
diff --git a/research/source-assisted-v4-layer/README.md b/research/source-assisted-v4-layer/README.md
new file mode 100644
index 0000000000..76a3d9c968
--- /dev/null
+++ b/research/source-assisted-v4-layer/README.md
@@ -0,0 +1,29 @@
+# source-assisted-v4-layer
+
+**Complex saving 7.0861·10⁻⁴ (no new κ).** PR #194's source-assisted complex word with this package's physical
+layer (PR #200's descent, seeded from PR #168's frames) certifies
+
+    a_C = 3543063/5000000000 = 7.0861260·10⁻⁴       (PR #194: 7.0091844·10⁻⁴, +1.10%),
+
+through PR #184's flow, exact lift and contract and PR #194's assembly, which with PR #200's bit supplier gives
+κ = 6768823/10¹⁰ (the bit branch binds, as in PR #202). The complex cap of this lineage and the exact-DFT exponent
+saving (`research/exact-dft-transfer`) rise to 7.0861·10⁻⁴. [PROOF.md](PROOF.md) states what changes and what is
+checked.
+
+## Verify
+
+    python -m pip install numpy==2.3.5 scipy==1.17.0
+    python3 -B research/source-assisted-v4-layer/verify.py        # about 80 seconds; -O is refused
+
+The script rebuilds the 308 pinned files of PR #202 (`8d8d67b`) from `baseline-pr202.tar.gz` after checking every
+sha256 in `SOURCE.json`, regenerates the PR #168 v4 cache and the aligned word, installs the physical layer of
+`witness/`, admits it with PR #200's complete complex checker, runs the frame flow with the package's kernel pairs,
+the exact lift, the contract (with the package's physical and kernel pairs as the frozen witnesses) and the assembly,
+and compares with `certificate.json`. `--write` regenerates the certificate; `--temp-root` chooses scratch storage.
+
+## Files
+
+`verify.py`, `certificate.json`, `SOURCE.json`, `baseline-pr202.tar.gz`, `witness/physical-frames.json`,
+`witness/physical-pairs.json`, `witness/kernel-pairs.json`, `discovery/` (the descent that found the layer; not run
+by the verifier), `PROOF.md`, `NOTICE`. `.github/workflows/source-assisted-v4-layer.yml` runs the verifier on
+Python 3.11 and 3.13.
diff --git a/research/source-assisted-v4-layer/SOURCE.json b/research/source-assisted-v4-layer/SOURCE.json
new file mode 100644
index 0000000000..83b96fa470
--- /dev/null
+++ b/research/source-assisted-v4-layer/SOURCE.json
@@ -0,0 +1,323 @@
+{
+ "archive": "baseline-pr202.tar.gz",
+ "archive_sha256": "5d08386f71c421d2b83b21020e3b74198e9d0c75884d7d81987307740d98883f",
+ "commit": "8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d",
+ "files": {
+  "certificates/paired-cube-complex-input.json": "b1898950f95a7ce618f376d14e628061b345a519e7bcac76b67d277deeec9af5",
+  "notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54",
+  "references/paired-cube/bit-physical/frames_p12.json.gz": "d2347635891e70918171b96ae33c9cb667cec11cdb7b8f634685b3df8c7effc1",
+  "references/paired-cube/bit-physical/kchron_p12.json": "001a8c2b6ab4766e5e57b13164bf319988db568b915af2de71920eb5b526d9db",
+  "references/paired-cube/bit-physical/word_p12.json.gz": "6f10239e8e2946263c7959392edd7c4e48661708a4d83222acb7cf6297783928",
+  "references/paired-cube/physical/README.md": "d978a982e2652e2ec8b8e810874e5cd99b6805a7ab9d586d4cc2f7e06ae15fca",
+  "references/paired-cube/physical/frames.json": "3681fc6ef151cdfa5453512f56f7165607ece64ccf6b71478af8c327350f5800",
+  "references/paired-cube/physical/pairs.json": "07c204bd2a69dd8860b7333896b9bf3fe41a326675bbe41b14e6e3795453dc1e",
+  "references/paired-cube/pr97-subset/selection.json": "5a41dee894702cb880385d826648a3e759b85a6de6824ecbef63b73b8c6e1354",
+  "references/paired-cube/selected-module/README.md": "bd3f5176ef31bbf78240066fd5a5dae52dc31dd734273409671440a568d18155",
+  "references/paired-cube/selected-module/SOURCE.json": "7689968c429109119a232c3f7631cdfa5b7a7895c61af36ee82e3229988e45ab",
+  "references/paired-cube/selected-module/matching-arcs.json": "46c77eb75ebcb2435595a3b6522a54e1d12d3c61eecb3b51c489edd1f78447e1",
+  "references/paired-cube/sources/SOURCE.json": "54c6b27f55849b5a74b16387ef361dac6943525a244046795f78605b84f90826",
+  "references/paired-cube/sources/local_L1.json": "1861724ee3b58ea8acb3334416fbfd9172a9ae02ad11a6f6885a4909f8692f05",
+  "references/paired-cube/sources/pmod_J0_full_6.0666810e-4.json": "7d1ae23430e110d32aa1cab84415abdbc918dfab6b317cb24d30756fc90c5925",
+  "references/paired-cube/sources/qmod_climb3u_best.json": "3f85f6b2134af8781f16229186d0d7c1de71ab4d5235244259e1b79f2e3d14ba",
+  "references/paired-cube/sources/tmod_TE_TD_TB3_1_1_4_full_6.0617964e-4.json": "9dcdb6f4e33410418c01f55f092862edfa7c05001ceada3974f8d1196bcd894b",
+  "references/three-stage-cover/pr117/LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+  "references/three-stage-cover/pr117/NOTICE": "a0030510cd3b5334fb113eae759aadd2c25951bd353dfb69ad9cb780ae5adf30",
+  "references/three-stage-cover/pr117/SOURCE.json": "74e9f4b4ae0529d21d19c159b2b50ee0923e93e0d7eb22cf208ad41dd5992275",
+  "references/three-stage-cover/pr117/dag.json.gz": "3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b",
+  "references/three-stage-cover/pr117/replayed.py": "99c1e3694ae06a7ce7a370b04ecb45ea3fcfe69a6e6e7f4c4606e0a6c74c3184",
+  "research/paired-cube-diagonal-bit-168/NOTICE": "9e82089424092c1dabc2341a09ead255633d78c90dbdc199c4336ca1c0f3bdf4",
+  "research/paired-cube-diagonal-bit-168/arithmetic/balanced_shared.py": "c3204e1758057f5ef10724014f38b91b8a01f6d8a3562b054263fc74795bfd90",
+  "research/paired-cube-diagonal-bit-168/arithmetic/certificate.py": "623229f3008d1563c2c3dde8e6d0eca8ea73b1937ceaf75db3d3425af8472874",
+  "research/paired-cube-diagonal-bit-168/arithmetic/interval_moment.py": "9d3d1d28bfb3375958995205179b171fb65c70a3792cf9f7a21ef5ee54f5460b",
+  "research/paired-cube-diagonal-bit-168/arithmetic/shared_bridge.py": "d0729e2d12d8c038719ea6eb83b169c09660f43de2d3d2da10a6a173f5204f93",
+  "research/paired-cube-diagonal-bit-168/certificate.json": "853afa75980e11ce16b333c28311c4f635e84065867c049fda9916c2419091ed",
+  "research/paired-cube-diagonal-bit-168/complex/check_sinks.py": "d70903464b3caabc0920aee8953978a27a6874a655ecb66cadacf77a31df11af",
+  "research/paired-cube-diagonal-bit-168/complex/formal_sinks.py": "d7fefe86e637584ad1bf53c8f7714b88d5abed92eaa29d110ef099536e51f516",
+  "research/paired-cube-diagonal-bit-168/complex/physical.py": "01961fad4b04aaa940d2dc99c87690bc861b3861200296974c5e065ce3446659",
+  "research/paired-cube-diagonal-bit-168/complex/prove.py": "3eecae7856aa212d1d7e607e2663b8ce0766ed6067b388c6be2ed5485886bd43",
+  "research/paired-cube-diagonal-bit-168/complex/verify_fused.py": "75c3dcd80fcdbc461651fa2d2fa463e6b41a569d695fc74700d240223ef198e9",
+  "research/source-assisted-v4/NOTICE": "0b478b206d86be883daf8a9698a75a36cca173b2f927740883393146cd500b3f",
+  "research/source-assisted-v4/PROOF.md": "b340b9530101b5bc976cde69031a32eed41bb6b9b56213d3a63bc5a6ebeb0f16",
+  "research/source-assisted-v4/README.md": "d123e1141698b01de4746657b95a2600bfeb9d4f2697b5c3c5e08b96990d5593",
+  "research/source-assisted-v4/SOURCE.json": "c62fa2316c6b27e6039ed32f37bf8540e0c18f3248b456ade56b9822412820d8",
+  "research/source-assisted-v4/assemble.py": "9f5ad2b18cf24b829c99592bdbe34577debc404d810797afa89051f898cf960c",
+  "research/source-assisted-v4/certificate.json": "6fb7df88e8d6635e15a53e1e5c88a47265f5c1706937e390eca01a91749aad9d",
+  "research/source-assisted-v4/contract_v4.py": "67da183217adbe28915391c8cd0c8c4ec9c9dedead54a641a2de23834df2342c",
+  "research/source-assisted-v4/data/kernel-pairs.json": "f1535677d1ec225a3a132b2da0f6da9613b7b814227647b288c4e298ec26d8d3",
+  "research/source-assisted-v4/data/physical-pairs.json": "7095501f054e9ec18aa212bfed1b01e497ca2ae997f4c9ece2779909b3fd72e2",
+  "research/source-assisted-v4/pin_sources.py": "29063638a1791b07f8ee4ece6569a9ece42b22b73ed34eac617e150e0b1b9985",
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+ "note": "The files of PR #202 (commit 8d8d67b: PR #194's source-assisted-v4 pipeline on the PR #189/#200 lineage) that verify.py runs: scripts/, the PR168 v4 references and inherited witnesses, the Clifford frames note, the source-assisted decision/global scripts, the v4 package, and the PR #200 package certificate, arithmetic and complex checker; byte for byte; reproducible with: git archive --format=tar 8d8d67b <paths> | gzip -n -9",
+ "pr": 202,
+ "repository": "CrocSwap/integer-mult-bounds",
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+ }
+}
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HcmV?d00001

diff --git a/research/source-assisted-v4-layer/certificate.json b/research/source-assisted-v4-layer/certificate.json
new file mode 100644
index 0000000000..a1e7ceb8c2
--- /dev/null
+++ b/research/source-assisted-v4-layer/certificate.json
@@ -0,0 +1,614 @@
+{
+ "assembly": {
+  "arithmetic": "PR184 assemble_profiles: 32-term rational logarithm enclosure; rational exponential majorant; upward 2^120 rounding",
+  "assembly": {
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+    "small_field_exposure": "1/100000000"
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+    "full_gauge_intersections": true,
+    "physical_signed_M_replayed": true,
+    "positive_rank_ledger_recounted": true,
+    "reverse_target_chains": true,
+    "scalar_H_exact": true,
+    "selected_roles": 2310,
+    "selected_sources_untouched": true,
+    "signed_scalar_pairs_checked": 1742400
+   },
+   "scalar_bound": {
+    "M_operations": 36931,
+    "binary_dirty_read_bits": 36947,
+    "c": 22569,
+    "decoder_coefficients": [
+     "-1/2",
+     "-1/6",
+     "1/2",
+     "1/3"
+    ],
+    "fixed_odd_divisor": 3,
+    "gate_and_inverse_coefficients": [
+     -1,
+     1
+    ],
+    "handoffs": 2310,
+    "local_group_upper": 6118494407560,
+    "logical_R": 15682,
+    "physical_R": 13372,
+    "q": 3157,
+    "scalar_denominators_divide": 6,
+    "scope": "Conservative inherited virtual-word bill retained after aliasing, including all compensated reads. Alias and frame route choices are finite permutations/Clifford adapters; their separate global movement bill remains an assembly obligation.",
+    "three_q_less_than_2pow15": true
+   },
+   "scope": "Finite candidate audit only. General exact Clifford and shared cover contracts, analytic supplier bridge, and paid global multiplication assembly remain separate obligations.",
+   "source_sha256": {
+    "/private/tmp/claude-501/sa-v4-layer-2jyghoif/notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54",
+    "/private/tmp/claude-501/sa-v4-layer-2jyghoif/research/paired-cube-diagonal-bit-168/complex/physical.py": "01961fad4b04aaa940d2dc99c87690bc861b3861200296974c5e065ce3446659",
+    "/private/tmp/claude-501/sa-v4-layer-2jyghoif/research/paired-cube-diagonal-bit-168/complex/verify_fused.py": "75c3dcd80fcdbc461651fa2d2fa463e6b41a569d695fc74700d240223ef198e9",
+    "/private/tmp/claude-501/sa-v4-layer-2jyghoif/scripts/paired_cube/frames.py": "7d897418dbfee47de0723ea4983098a982610ccba45e1bac9000a79626d8421c",
+    "/private/tmp/claude-501/sa-v4-layer-2jyghoif/scripts/paired_cube_physical.py": "9c9ea1ea29e551ab6116d7c8ca09f4703ff9e2fa102032d5055327fde07ba319"
+   },
+   "status": "PASS exact fused physical candidate, all formal source/target/dirty columns"
+  },
+  "frames": 17050,
+  "pairs": 2310
+ },
+ "lift": {
+  "all_actual_coefficients_dyadic": true,
+  "exact_scalar_program_sha256": "c33c86388f2d4e6627e2d3e5a762f8cba47eaff8b43166c4c8d31bdc155663d3",
+  "max_denominator": "2",
+  "nodes": 18481,
+  "physical_R": 9412,
+  "status": "Exact rational local lifts and inverse gate programs verified; monotone flow and zero-fresh kernel reuse DAG verified."
+ },
+ "prerequisites": {
+  "pr194_complex_saving": "219037/312500000",
+  "pr200_bit": "research/paired-cube-diagonal-bit-168/certificate.json",
+  "pr202": "8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d"
+ },
+ "status": "PASS source-assisted v4 word with this package's physical layer"
+}
diff --git a/research/source-assisted-v4-layer/discovery/README.md b/research/source-assisted-v4-layer/discovery/README.md
new file mode 100644
index 0000000000..393c9a31e4
--- /dev/null
+++ b/research/source-assisted-v4-layer/discovery/README.md
@@ -0,0 +1,14 @@
+# Discovery (not run by verify.py)
+
+`flow_frames.py` runs this package's descent (`harness.descend`: `physical_opt.py` endpoint moves of single
+operations, `bundles.py` equal-frame components, `joint.py`/`deep.py` connected bundles, `pairs_mw.py` reuse
+pairing; PR #200's physical layer code) on the source-aligned cache of PR #194's pipeline, writes the resulting
+frames and pairs into a copy of the aligned tree and prices them with PR #184's `complex_frame_flow.py`:
+
+    SA_TREE=<PR202 tree> SA_PY=<python with numpy/scipy> python3 -B flow_frames.py --aligned <aligned tree> --seed-frames --keep out/
+
+With `--seed-frames` the descent starts from PR #168's physical frames (the aligned tree's own layer) and the flow
+root rises from 7.00918e-4 to 7.08613e-4; without seeding, 7.0691e-4; the light descent alone, 7.0419e-4. These
+numbers are float discovery screens; the certified value is in `../certificate.json`. `matching.py` and
+`extend.py` are needed only to import `harness` (the matching itself is PR #168's frozen one here).
+Prepared by Chafik Boukhalfa with Anthropic Claude and OpenAI Codex assistance; Apache-2.0.
diff --git a/research/source-assisted-v4-layer/discovery/bundles.py b/research/source-assisted-v4-layer/discovery/bundles.py
new file mode 100644
index 0000000000..2035223567
--- /dev/null
+++ b/research/source-assisted-v4-layer/discovery/bundles.py
@@ -0,0 +1,50 @@
+"""Connected equal-frame endpoint moves, motivated by PR164/168 bundle descent.
+
+Own explicit boundary implementation: all source spans, incoming and outgoing
+role frames are recomputed before a move. No physical legality is inferred from
+an optimization score; the independent physical verifier remains mandatory.
+"""
+from collections import defaultdict
+from physical_opt import join,meet
+from paired_cube.frames import contained
+
+def descend(o,rounds=4):
+ report=[]
+ for rnd in range(rounds):
+  n=len(o.ops);parent=list(range(n))
+  def root(i):
+   while parent[i]!=i:parent[i]=parent[parent[i]];i=parent[i]
+   return i
+  def union(i,j):
+   a,b=root(i),root(j)
+   if a!=b:parent[max(a,b)]=min(a,b)
+  for i in range(n):
+   for j in o.prev[i]:
+    if j is not None and o.frames[i]==o.frames[j]:union(i,j)
+  groups=defaultdict(list)
+  for i in range(n):groups[root(i)].append(i)
+  groups=[l for l in groups.values() if len(l)>1];groups.sort(key=lambda l:min(l),reverse=not rnd%2)
+  moved=changed=0;gain=0.;big=0
+  for ids in groups:
+   inside=set(ids);F=o.frames[ids[0]];assert all(o.frames[i]==F for i in ids)
+   L=();U=o.full;previous=[];after=[]
+   for i in ids:
+    a,b,x=o.ops[i];L=join(L,o.spans[x])
+    for role,j in zip((a,b),o.prev[i]):
+     if j not in inside:
+      A=o.frames[j] if j is not None else o.start[role];previous.append(A);L=join(L,A)
+    for role,j in zip((a,b),o.next[i]):
+     if j not in inside:
+      A=o.frames[j] if j is not None else o.end[role];after.append(A);U=meet(U,A,o.h)
+   assert contained(L,F) and contained(F,U)
+   def score(A):
+    d=len(A);return sum(o.phi[d-len(P)] for P in previous)+sum(o.phi[len(N)-d] for N in after)
+   old=score(F);candidate=min((L,U),key=score);delta=old-score(candidate)
+   if delta>1e-7:
+    for i in ids:o.frames[i]=candidate
+    moved+=1;changed+=len(ids);gain+=delta;big=max(big,len(ids))
+  rec=dict(round=rnd,components=len(groups),moved_components=moved,changed_operations=changed,numerical_excess_gain=gain,largest_moved_component=big);report.append(rec);print('bundles',rec,flush=True)
+  if not moved:break
+ for i in range(n):
+  P,N=o.neighbors(i);assert contained(o.spans[o.ops[i][2]],o.frames[i]);assert all(contained(F,o.frames[i]) for F in P);assert all(contained(o.frames[i],F) for F in N)
+ return report
diff --git a/research/source-assisted-v4-layer/discovery/deep.py b/research/source-assisted-v4-layer/discovery/deep.py
new file mode 100644
index 0000000000..6e960d3909
--- /dev/null
+++ b/research/source-assisted-v4-layer/discovery/deep.py
@@ -0,0 +1,52 @@
+"""Depth-k merges of adjacent equal-frame components (generalizes joint.py to connected groups of up to k components).
+Numerical witnesses only; the independent exact verifier remains mandatory."""
+from collections import defaultdict
+import random,time
+from physical_opt import join,meet
+from paired_cube.frames import contained
+from joint import components
+
+def evaluate(o,ids):
+ inside=set(ids);L=();U=o.full;previous=[];after=[];old=0.
+ for i in ids:
+  x,y,node=o.ops[i];F=o.frames[i];L=join(L,o.spans[node])
+  for role,j in zip((x,y),o.prev[i]):
+   P=o.frames[j] if j is not None else o.start[role];old+=o.phi[len(F)-len(P)]
+   if j not in inside:previous.append(P);L=join(L,P)
+  for role,j in zip((x,y),o.next[i]):
+   if j not in inside:
+    N=o.frames[j] if j is not None else o.end[role];after.append(N);U=meet(U,N,o.h);old+=o.phi[len(N)-len(F)]
+ if not contained(L,U):return None
+ def score(F):
+  d=len(F);return sum(o.phi[d-len(P)] for P in previous)+sum(o.phi[len(N)-d] for N in after)
+ best=min((L,U),key=lambda F:(score(F),len(F),F));return old-score(best),best
+
+def descend(o,depth=3,rounds=3,seed=0,limit=None):
+ rng=random.Random(seed);report=[]
+ for rnd in range(rounds):
+  t=time.monotonic();groups,adjacent=components(o)
+  nbr=defaultdict(set)
+  for a,b in adjacent:nbr[a].add(b);nbr[b].add(a)
+  keys=list(groups);rng.shuffle(keys);moved=changed=tried=0;gain=0.
+  for a in keys[:limit]:
+   # grow connected sets around a up to `depth` components, trying each as a merged move
+   frontier=[(a,)];seen={(a,)}
+   while frontier:
+    grp=frontier.pop()
+    if len(grp)>=2:
+     ids=[i for c in grp for i in groups[c]];tried+=1
+     r=evaluate(o,ids)
+     if r and r[0]>1e-7:
+      for i in ids:o.frames[i]=r[1]
+      moved+=1;changed+=len(ids);gain+=r[0];break
+    if len(grp)<depth:
+     for c in grp:
+      for d in nbr[c]:
+       if d not in grp:
+        new=tuple(sorted(grp+(d,)))
+        if new not in seen:seen.add(new);frontier.append(new)
+  rec=dict(round=rnd,depth=depth,groups=len(groups),tried=tried,moved=moved,changed=changed,numerical_excess_gain=gain,seconds=time.monotonic()-t);report.append(rec);print('deep',rec,flush=True)
+  if not moved:break
+ for i in range(len(o.ops)):
+  P,N=o.neighbors(i);assert contained(o.spans[o.ops[i][2]],o.frames[i]);assert all(contained(F,o.frames[i]) for F in P);assert all(contained(o.frames[i],F) for F in N)
+ return report
diff --git a/research/source-assisted-v4-layer/discovery/extend.py b/research/source-assisted-v4-layer/discovery/extend.py
new file mode 100644
index 0000000000..8c61dc21cf
--- /dev/null
+++ b/research/source-assisted-v4-layer/discovery/extend.py
@@ -0,0 +1,97 @@
+"""PR162 generalized-carrier construction helper, DaysSky et al. Apache-2.0.
+Extracted unchanged function only; no foreign producer invocation.
+"""
+from collections import defaultdict
+from functools import lru_cache
+from paired_cube.frames import basis,perp,contained
+@lru_cache(maxsize=None)
+def join(A,B):
+ if A==B or not B:return A
+ if not A:return B
+ return basis(A+B)
+@lru_cache(maxsize=None)
+def orthogonal(S,A):return all((s&a).bit_count()%2==0 for s in S for a in A)
+def extended_arcs(g, frozen):
+    """Greedy extension of the frozen arcs.  gspan[x]: span of every value that reaches x through DAG or arc edges;
+    gann[x]: annihilator of the intersection of every root frame reachable from x.  An arc x -> t is legal iff
+    gspan[x] is orthogonal to gann[t] and t does not reach x."""
+    h = g['h']; inputs = g['inputs']
+    args = [None] + [None if a is None else [x + 1 for x in a] for a in g['args']]
+    roots = [dict(r, node=r['node'] + 1) for r in g['roots']]
+    v, n = len(inputs), len(args)
+    spans = [()] * n
+    for i, u in enumerate(inputs): spans[i + 1] = (u,)
+    for x in range(v + 1, n): spans[x] = basis(spans[args[x][0]] + spans[args[x][1]])
+    rann = [perp(spans[r['node']], h) if r.get('kind', 'side') == 'center' else basis(inputs[t] for t in r['targets'])
+            for r in roots]
+    active = set(range(1, v + 1)); todo = [r['node'] for r in roots]
+    while todo:
+        x = todo.pop()
+        if x in active: continue
+        active.add(x)
+        if args[x]: todo.extend(args[x])
+    succ = [set() for _ in args]; pred = [set() for _ in args]; direct = [[] for _ in args]
+    uses = defaultdict(list)
+    for x in sorted(active):
+        if args[x]:
+            for j, y in enumerate(args[x]): succ[y].add(x); pred[x].add(y); uses[y].append(2 * x + j)
+    for j, r in enumerate(roots): direct[r['node']].extend(rann[j]); uses[r['node']].append((1 << 31) | j)
+    gann = [None] * n; gspan = list(spans)
+    for x in sorted(active, reverse=True): gann[x] = basis(direct[x] + [z for y in succ[x] for z in gann[y]])
+    plain = list(gann)                                       # DAG-only intersections, for the candidate screen
+    ancestors = {}
+    def above(x):
+        if x not in ancestors:
+            seen = set(); stack = [x]
+            while stack:
+                y = stack.pop()
+                if args[y]:
+                    for z in args[y]:
+                        if z not in seen: seen.add(z); stack.append(z)
+            ancestors[x] = seen
+        return ancestors[x]
+    arcs = {}; used = set()
+    def value(code): return roots[code & 0x7fffffff]['node'] if code >> 31 else args[code // 2][code & 1]
+    def add(x, code, check):
+        t = None if code >> 31 else code // 2
+        At = rann[code & 0x7fffffff] if t is None else gann[t]
+        if check:
+            if not orthogonal(gspan[x], At): return False
+            if t is not None:
+                stack = [t]; seen = {t}
+                while stack:
+                    y = stack.pop()
+                    if y == x: return False
+                    for z in succ[y]:
+                        if z not in seen: seen.add(z); stack.append(z)
+        arcs[x] = code; used.add(code)
+        work = [(x, At)]
+        while work:
+            y, extra = work.pop(); new = join(gann[y], extra)
+            if new == gann[y]: continue
+            gann[y] = new
+            for z in pred[y]: work.append((z, new))
+        if t is not None:
+            succ[x].add(t); pred[t].add(x); work = [(t, gspan[x])]
+            while work:
+                y, extra = work.pop(); new = join(gspan[y], extra)
+                if new == gspan[y]: continue
+                gspan[y] = new
+                for z in succ[y]: work.append((z, new))
+        return True
+    for x, code in frozen: assert add(x, code, True), ('frozen arc rejected', x, code)
+    for x in sorted(active):
+        if not args[x]: continue
+        for j, y in enumerate(args[x]):
+            other = args[x][1 - j]
+            for code in uses[y]:
+                if code == 2 * x + j or x in arcs or code in used: continue
+                if code >> 31: screen = rann[code & 0x7fffffff]
+                else:
+                    t = code // 2
+                    if t < x and t in above(x): continue
+                    screen = plain[t]
+                if not orthogonal(spans[other], screen): continue
+                if sum(1 for u in uses[value(code)] if u not in used) <= 1: continue    # every value keeps a free use
+                add(x, code, True)
+    return sorted([x, code] for x, code in arcs.items())
diff --git a/research/source-assisted-v4-layer/discovery/flow_frames.py b/research/source-assisted-v4-layer/discovery/flow_frames.py
new file mode 100644
index 0000000000..0d9c0173bf
--- /dev/null
+++ b/research/source-assisted-v4-layer/discovery/flow_frames.py
@@ -0,0 +1,34 @@
+#!/usr/bin/env python3
+"""Physical-layer search under the FLOW objective: run our descent (physical_opt via harness.descend) on an aligned
+cache (graph/frames/selection/record of the source-aligned word), write the resulting physical frames and reuse pairs
+into a copy of the aligned tree, and price with PR184's complex_frame_flow.py. Discovery only (float)."""
+from pathlib import Path
+import sys,os,json,time,argparse,shutil,subprocess,tempfile
+H=Path(__file__).resolve().parent;sys.path.insert(0,str(H))
+import harness
+ap=argparse.ArgumentParser();ap.add_argument('--aligned',type=Path,required=True,help='aligned tree (with cache/ and references/paired-cube/physical/)')
+ap.add_argument('--sa',default=os.environ.get('SA_TREE',str(Path.home()/'sa-flow')));ap.add_argument('--py',default=os.environ.get('SA_PY',sys.executable))
+ap.add_argument('--light',action='store_true');ap.add_argument('--deep',type=int,default=0);ap.add_argument('--mw',type=int,default=0);ap.add_argument('--pair-first',action='store_true')
+ap.add_argument('--seed-frames',action='store_true',help='start from the tree\'s own physical frames');ap.add_argument('--tag',default='');ap.add_argument('--out',default='flow-frames.jsonl');ap.add_argument('--keep',type=Path,default=None)
+a=ap.parse_args();SA=Path(a.sa)
+t=time.monotonic();cache=a.aligned/'cache'
+g,w,word,row=(json.loads((cache/n).read_text()) for n in ('graph.json','frames.json','selection.json','record.json'))
+seed=None
+if a.seed_frames:seed=json.loads((a.aligned/'references/paired-cube/physical/frames.json').read_text())['frames']
+prof,frames,pairs,stats=harness.descend(g,w,row,word,a.light,a.deep,seed,a.mw,a.pair_first)
+pre=harness.root(prof);t1=time.monotonic()-t
+work=Path(tempfile.mkdtemp(prefix='flowframes-'))
+try:
+ tree=work/'tree';shutil.copytree(a.aligned,tree)
+ (tree/'references/paired-cube/physical/frames.json').write_text(json.dumps(dict(frames=[[i,list(F)] for i,F in frames]),separators=(',',':'))+'\n')
+ (tree/'references/paired-cube/physical/pairs.json').write_text(json.dumps(dict(pairs=[list(p) for p in pairs]),separators=(',',':'))+'\n')
+ r=subprocess.run([a.py,'-B',str(SA/'research/source-assisted/decision/complex_frame_flow.py'),'--tree',str(tree),'--cache',str(tree/'cache'),'--out',str(work/'flow.json'),'--witness','--purify-source-donors','--recycle-kernels'],capture_output=True,text=True)
+ if r.returncode:res=dict(tag=a.tag,error='flow: '+r.stderr[-500:],pre_root=pre)
+ else:
+  f=json.loads((work/'flow.json').read_text());res=dict(tag=a.tag,pre_root=pre,flow_root=f['numerical_local_root'],new_R=f['new_R'],new_W=f['new_W'],W_pre=prof['W_per_vertex'],pairs=len(pairs),changed=len(frames),stats=stats,seconds=time.monotonic()-t,descent_seconds=t1,opts=dict(light=a.light,deep=a.deep,mw=a.mw,pair_first=a.pair_first,seed_frames=a.seed_frames,PHI_ALPHA=os.environ.get('PHI_ALPHA'),DESCENT_SEED=os.environ.get('DESCENT_SEED')))
+  if a.keep:
+   a.keep.mkdir(parents=True,exist_ok=True)
+   for n in ('frames.json','pairs.json'):shutil.copy2(tree/'references/paired-cube/physical'/n,a.keep/n)
+   shutil.copy2(work/'flow.json',a.keep/'flow.json')
+finally:shutil.rmtree(work,ignore_errors=True)
+open(H/a.out,'a').write(json.dumps(res)+'\n');print(json.dumps({k:res.get(k) for k in ('tag','pre_root','flow_root','new_W','W_pre','pairs','changed','seconds','error')}),flush=True)
diff --git a/research/source-assisted-v4-layer/discovery/harness.py b/research/source-assisted-v4-layer/discovery/harness.py
new file mode 100644
index 0000000000..a709acb3fc
--- /dev/null
+++ b/research/source-assisted-v4-layer/discovery/harness.py
@@ -0,0 +1,80 @@
+#!/usr/bin/env python3
+"""General v4 harness: modules/local/f8 -> graph -> our carrier matching + PR162 extension -> closure -> gauges -> descent -> numeric screen.
+Numerical screen only; candidates need the independent exact audits before admission."""
+from pathlib import Path
+import sys,json,gzip,math,time,hashlib,argparse
+H=Path(__file__).resolve().parent;TOP=H.parent;SRC=TOP/'source';sys.path.insert(0,str(SRC/'scripts'));sys.path.insert(0,str(H))
+from paired_cube.graph import Graph
+from paired_cube.modules import triple_module_from,pair_module_from,all_but_one_from,merge_outputs
+from paired_cube.closure import compile_closure
+from paired_cube.gauges import select
+from paired_cube_physical import physical
+from physical_opt import Physical
+from matching import carrier_matching
+from extend import extended_arcs
+import joint,bundles,deep
+from pairs_mw import pairs_maxweight
+def root(prof):
+ hist={int(r):c for r,c in prof['child_histogram'].items()};m=prof['m'];W=prof['W_per_vertex'];lo=0.;hi=.01
+ for _ in range(70):
+  a=(lo+hi)/2
+  if math.fsum(c*r*math.exp(a*math.log(m/r)) for r,c in hist.items())<m*W:lo=a
+  else:hi=a
+ return lo
+def save(out,name,data):
+ with gzip.GzipFile(filename='',fileobj=(out/(name+'.json.gz')).open('wb'),mode='wb',mtime=0) as f:f.write((json.dumps(data,separators=(',',':'))+'\n').encode())
+def build(f8='f8:00111100',local=None,tmod=None,pmod=None,qmod=None,ordering='reverse',frozen_arcs=False):
+ src=SRC/'references/paired-cube/sources'
+ local=json.loads(Path(local).read_text()) if local else json.loads((src/'local_L1.json').read_text())
+ b=Graph(11,local=local)
+ import os
+ g=b.finish(triple_module_from(tmod or src/'tmod_TE_TD_TB3_1_1_4_full_6.0617964e-4.json',11),pair_module_from(pmod or src/'pmod_J0_full_6.0666810e-4.json',10),all_but_one_from(qmod or src/'qmod_climb3u_best.json',9),center_kind=os.environ.get('CENTER_KIND','star'))
+ g=merge_outputs(g,b,f8);g['matching_frames']='coordinate'
+ if frozen_arcs:
+  arcs=json.loads((SRC/'references/paired-cube/selected-module/matching-arcs.json').read_text());match=dict(frozen=len(arcs))
+ else:
+  arcs,match=carrier_matching(g,ordering);arcs=extended_arcs(g,arcs)
+ base,w=compile_closure(g,arcs);row,word=select(g,base,w)
+ return g,base,w,row,word,arcs,match
+def descend(g,w,row,word,light=False,deep_depth=0,frames=None,mw=0,pair_first=False):
+ o=Physical(g,w,word,row)
+ if frames:
+  for i,F in frames:o.frames[i]=tuple(F)
+ def cycle(seed):
+  if light:o.descend(6);return
+  for k in range(3):
+   a=o.descend(6);b=bundles.descend(o,3);c=joint.descend(o,3,'forward',seed+k)
+   d=deep.descend(o,deep_depth,2,seed+k) if deep_depth else []
+   if not any(x['changed'] for x in a) and not any(x['moved_components'] for x in b) and not any(x['moved'] for x in c) and not any(x['moved'] for x in d):break
+ if pair_first:
+  pairs,stats=pairs_maxweight(o)
+  for d,b,_ in pairs:o.end[d]=o.gauge[b]
+ cycle(0)
+ if not pair_first:pairs,stats=(pairs_maxweight(o) if mw else o.pairs())
+ for d,b,_ in pairs:o.end[d]=o.gauge[b]
+ cycle(10)
+ for it in range(mw-1 if mw else 0):
+  # re-pair on the descended frames (donor end frames may now allow better donors), then descend again
+  for d,b,_ in pairs:o.end[d]=o.full
+  newpairs,newstats=pairs_maxweight(o)
+  if newstats['total_weight']<=stats['total_weight']+1e-9:
+   for d,b,_ in pairs:o.end[d]=o.gauge[b]
+   break
+  pairs,stats=newpairs,newstats
+  for d,b,_ in pairs:o.end[d]=o.gauge[b]
+  cycle(20+it)
+ def written_in_rest(d):return any(i not in o.phase and o.ops[i][0]==d for i in o.role_ops[d])
+ good=[k for k,(d,b,t) in enumerate(pairs) if t is not None and written_in_rest(d)]
+ if good:pairs.append(pairs.pop(good[-1]))
+ frames=o.changed();prof=physical(g,w,word,row,frames,pairs)
+ return prof,frames,pairs,stats
+if __name__=='__main__':
+ ap=argparse.ArgumentParser();ap.add_argument('--f8',default='f8:00111100');ap.add_argument('--local');ap.add_argument('--tmod');ap.add_argument('--pmod');ap.add_argument('--qmod');ap.add_argument('--ordering',default='reverse');ap.add_argument('--frozen-arcs',action='store_true');ap.add_argument('--light',action='store_true');ap.add_argument('--deep',type=int,default=0);ap.add_argument('--mw',type=int,default=0);ap.add_argument('--pair-first',action='store_true');ap.add_argument('--seed-frames');ap.add_argument('--tag',required=True);a=ap.parse_args()
+ t=time.monotonic();g,base,w,row,word,arcs,match=build(a.f8,a.local,a.tmod,a.pmod,a.qmod,a.ordering,a.frozen_arcs)
+ print(json.dumps(dict(tag=a.tag,stage='compiled',c=row['c'],q=row['q'],R=row['R'],matched=row['matched'],arcs=len(arcs),selected=row['selected_roles'],numeric_before=row['numerical_complex_root'],seconds=time.monotonic()-t)),flush=True)
+ seed=json.loads(gzip.decompress(Path(a.seed_frames).read_bytes())) if a.seed_frames else None
+ prof,frames,pairs,stats=descend(g,w,row,word,a.light,a.deep,seed,a.mw,a.pair_first)
+ r=root(prof);out=H/('h-'+a.tag);out.mkdir(exist_ok=True)
+ for name,data in [('graph',g),('baseline',base),('frames',w),('word',word),('profile-before',row),('physical-frames',frames),('physical-pairs',pairs),('profile',prof)]:save(out,name,data)
+ res=dict(tag=a.tag,deep=a.deep,mw=a.mw,pair_stats=stats,f8=a.f8,local=a.local,tmod=a.tmod,pmod=a.pmod,qmod=a.qmod,ordering=a.ordering,frozen_arcs=a.frozen_arcs,light=a.light,root=r,W=prof['W_per_vertex'],physical_R=prof['physical_R'],pairs=prof['pairs'],changed=len(frames),seconds=time.monotonic()-t,pins={p.name:hashlib.sha256(p.read_bytes()).hexdigest() for p in out.glob('*.json.gz')})
+ (out/'result.json').write_text(json.dumps(res,sort_keys=True,indent=2)+'\n');print(json.dumps({k:res[k] for k in ('tag','root','W','physical_R','pairs','changed','seconds')}),flush=True)
diff --git a/research/source-assisted-v4-layer/discovery/joint.py b/research/source-assisted-v4-layer/discovery/joint.py
new file mode 100644
index 0000000000..a67d18034d
--- /dev/null
+++ b/research/source-assisted-v4-layer/discovery/joint.py
@@ -0,0 +1,67 @@
+"""Joint collapse of adjacent operation-frame components.
+
+All incoming/outgoing chains and actual value supports bound each common
+frame. Old internal rank increments are charged once per role edge. A move
+must reduce the full boundary-plus-internal numerical objective. Floating
+scores select a witness only; admission requires the independent exact audit.
+"""
+from collections import defaultdict
+import random,time
+from physical_opt import join,meet
+from paired_cube.frames import contained
+
+def components(o):
+ n=len(o.ops);parent=list(range(n))
+ def root(i):
+  while parent[i]!=i:parent[i]=parent[parent[i]];i=parent[i]
+  return i
+ for i in range(n):
+  for j in o.prev[i]:
+   if j is not None and o.frames[i]==o.frames[j]:
+    a,b=root(i),root(j)
+    if a!=b:parent[max(a,b)]=min(a,b)
+ groups=defaultdict(list)
+ for i in range(n):groups[root(i)].append(i)
+ adjacent=set()
+ for i in range(n):
+  for j in o.prev[i]:
+   if j is not None and root(i)!=root(j):adjacent.add(tuple(sorted((root(i),root(j)))))
+ return groups,sorted(adjacent)
+
+def descend(o,rounds=5,order='forward',seed=0):
+ result=[];rng=random.Random(seed)
+ for rnd in range(rounds):
+  t=time.monotonic();groups,adjacent=components(o)
+  if order=='random':rng.shuffle(adjacent)
+  elif (order=='reverse')^(rnd%2==1):adjacent.reverse()
+  tried=feasible=moved=changed=largest=0;gain=0.
+  for a,b in adjacent:
+   ids=groups[a]+groups[b];inside=set(ids);tried+=1
+   # Each original component remains uniform even after overlapping moves.
+   assert all(o.frames[i]==o.frames[a] for i in groups[a])
+   assert all(o.frames[i]==o.frames[b] for i in groups[b])
+   L=();U=o.full;previous=[];after=[];old=0.
+   for i in ids:
+    x,y,node=o.ops[i];F=o.frames[i];L=join(L,o.spans[node])
+    for role,j in zip((x,y),o.prev[i]):
+     P=o.frames[j] if j is not None else o.start[role]
+     assert contained(P,F);old+=o.phi[len(F)-len(P)]
+     if j not in inside:previous.append(P);L=join(L,P)
+    for role,j in zip((x,y),o.next[i]):
+     if j not in inside:
+      N=o.frames[j] if j is not None else o.end[role]
+      assert contained(F,N);after.append(N);U=meet(U,N,o.h);old+=o.phi[len(N)-len(F)]
+   if not contained(L,U):continue
+   feasible+=1
+   def score(F):
+    d=len(F);return sum(o.phi[d-len(P)] for P in previous)+sum(o.phi[len(N)-d] for N in after)
+   best=min((L,U),key=lambda F:(score(F),len(F),F));delta=old-score(best)
+   if delta>1e-7:
+    for i in ids:o.frames[i]=best
+    moved+=1;changed+=len(ids);gain+=delta;largest=max(largest,len(ids))
+  row=dict(round=rnd,groups=len(groups),tried=tried,feasible=feasible,moved=moved,changed=changed,largest=largest,numerical_excess_gain=gain,seconds=time.monotonic()-t)
+  print('joint',row,flush=True);result.append(row)
+  if not moved:break
+ for i in range(len(o.ops)):
+  P,N=o.neighbors(i);assert contained(o.spans[o.ops[i][2]],o.frames[i]);assert all(contained(F,o.frames[i]) for F in P);assert all(contained(o.frames[i],F) for F in N)
+ return result
diff --git a/research/source-assisted-v4-layer/discovery/matching.py b/research/source-assisted-v4-layer/discovery/matching.py
new file mode 100644
index 0000000000..e84e348556
--- /dev/null
+++ b/research/source-assisted-v4-layer/discovery/matching.py
@@ -0,0 +1,131 @@
+"""Own explicit legal-edge Hopcroft-Karp witness; from our r13 search."""
+from collections import deque
+import sys
+from paired_cube.frames import basis,perp,contained
+def carrier_matching(g,ordering='node'):
+    """Exact legal edge list, then deterministic Hopcroft--Karp matching.
+
+    Frame and chronological predicates are the retained compiler's predicates;
+    the resulting explicit arcs are checked anew by compile_graph.
+    """
+    h=g['h'];v=g['v'];args=[None]+[None if a is None else [y+1 for y in a] for a in g['args']]
+    roots=[dict(r,node=r['node']+1) for r in g['roots']];n=len(args)
+    spans=[()]*n
+    for i,u in enumerate(g['inputs']):spans[i+1]=(u,)
+    for x in range(v+1,n):spans[x]=basis(spans[args[x][0]]+spans[args[x][1]])
+    rann=[];rframe=[]
+    for r in roots:
+        if r['kind']=='center':U=spans[r['node']];A=perp(U,h)
+        else:A=basis(g['inputs'][t] for t in r['targets']);U=perp(A,h)
+        rann.append(A);rframe.append(U)
+    active=set(range(1,v+1));todo=[r['node'] for r in roots]
+    while todo:
+        x=todo.pop()
+        if x in active:continue
+        active.add(x);todo.extend(args[x] or [])
+    initial=spans
+    if g['matching_frames'] in ('maximal','coordinate'):
+        initial=[()]*n;ann=[None]*n;successors=[[] for _ in args];constraints=[[] for _ in args]
+        for x in sorted(active):
+            for y in args[x] or []:successors[y].append(x)
+        for j,r in enumerate(roots):constraints[r['node']].extend(rann[j])
+        for x in sorted(active,reverse=True):
+            ann[x]=basis(constraints[x]+[z for y in successors[x] for z in ann[y]])
+            initial[x]=perp(ann[x],h)
+            if g['matching_frames']=='coordinate':
+                cover=0
+                for u in spans[x]:cover|=u
+                initial[x]=perp(ann[x]+tuple(1<<j for j in range(h) if not cover>>j&1),h)
+            assert contained(spans[x],initial[x])
+    order=sorted(active,key=lambda x:(len(initial[x]),x));position={x:i for i,x in enumerate(order)}
+    uses=[[] for _ in args];uf=[];ut=[];uc=[]
+    for x in order:
+        for j,y in enumerate(args[x] or []):uses[y].append(len(uc));uf.append(initial[x]);ut.append(x);uc.append(2*x+j)
+    for j,r in enumerate(roots):uses[r['node']].append(len(uc));uf.append(rframe[j]);ut.append(n+j);uc.append((1<<31)|j)
+    donors=[x for x in order if args[x]]
+    if ordering=='reverse':donors.reverse()
+    elif ordering=='narrow':donors.sort(key=lambda x:(len(initial[x]),x))
+    elif ordering=='wide':donors.sort(key=lambda x:(-len(initial[x]),x))
+    elif ordering=='node':donors.sort()
+    elif ordering.startswith('random:'):
+        import random
+        rnd=random.Random(int(ordering.split(':',1)[1]));rnd.shuffle(donors)
+    adjacency=[]
+    for x in donors:
+        row=[u for y in args[x] for u in uses[y] if (ut[u]>=n or position[ut[u]]>position[x]) and contained(initial[x],uf[u])]
+        if ordering.startswith('late'):   # prefer the latest use in the chronology (root uses last), then the largest growth
+            row.sort(key=lambda u:(-(position[ut[u]] if ut[u]<n else 10**9),-(len(uf[u])-len(initial[x])),uc[u]))
+        elif ordering.startswith('latefirst'):
+            row.sort(key=lambda u:(-(position[ut[u]] if ut[u]<n else -1),uc[u]))
+        else:
+            row.sort(key=lambda u:(len(uf[u])-len(initial[x]),uc[u]))
+        if ordering.startswith('random:') and ordering.count(':')==2:rnd.shuffle(row)   # 'random:SEED:rows' also shuffles use order
+        adjacency.append(row)
+    if ordering=='struct':
+        return dict(donors=donors,adjacency=adjacency,uc=uc,uf=uf,ut=ut,initial=initial,n=n,position=position),None
+    L=len(donors);ml=[-1]*L;mr=[-1]*len(uc);dist=[0]*L;iterations=0
+    def bfs():
+        queue=deque();found=False
+        for u in range(L):
+            dist[u]=0 if ml[u]<0 else -1
+            if ml[u]<0:queue.append(u)
+        while queue:
+            u=queue.popleft()
+            for v2 in adjacency[u]:
+                z=mr[v2]
+                if z<0:found=True
+                elif dist[z]<0:dist[z]=dist[u]+1;queue.append(z)
+        return found
+    def dfs(u):
+        for v2 in adjacency[u]:
+            z=mr[v2]
+            if z<0 or dist[z]==dist[u]+1 and dfs(z):ml[u]=v2;mr[v2]=u;return True
+        dist[u]=-1;return False
+    sys.setrecursionlimit(max(10000,2*L))
+    while bfs():
+        iterations+=1
+        for u in range(L):
+            if ml[u]<0:dfs(u)
+    arcs=[[donors[i],uc[v2]] for i,v2 in enumerate(ml) if v2>=0]
+    stats=dict(donors=L,uses=len(uc),edges=sum(map(len,adjacency)),matched=len(arcs),augment_rounds=iterations)
+    if ordering.startswith('frozen:'):
+        import json as _j
+        given=_j.loads(open(ordering.split(':',1)[1]).read());pos_of={uc[u]:u for u in range(len(uc))}
+        growth=0;seen=0
+        for x,code in given:
+            u=pos_of.get(code)
+            if u is None:continue
+            seen+=1;growth+=len(uf[u])-len(initial[x])
+        stats.update(frozen_arcs=len(given),frozen_seen=seen,frozen_growth=growth,hk_growth=int(sum(len(uf[v2])-len(initial[donors[i]]) for i,v2 in enumerate(ml) if v2>=0)))
+        import os as _os
+        if _os.environ.get('MATCH_FEATURES'):
+            chosen={x:pos_of[c] for x,c in given if c in pos_of};hk={donors[i]:v2 for i,v2 in enumerate(ml) if v2>=0}
+            rec=[]
+            for i,x in enumerate(donors):
+                cands=[dict(u=u,growth=len(uf[u])-len(initial[x]),udim=len(uf[u]),root=int(ut[u]>=n),gap=(position[ut[u]]-position[x]) if ut[u]<n else -1,frozen=int(chosen.get(x)==u),hk=int(hk.get(x)==u)) for u in adjacency[i]]
+                if cands:rec.append(dict(x=x,ddim=len(initial[x]),pos=position[x],cands=cands))
+            open(_os.environ['MATCH_FEATURES'],'w').write(_j.dumps(rec))
+        return [list(a) for a in given],stats
+    if ordering.startswith('mincost'):
+        # Among matchings that saturate the Hopcroft-Karp matched donor set, pick the one minimizing the total frame
+        # growth (dim use frame - dim donor frame), via scipy's sparse LAPJV. 'mincost:max' maximizes it instead.
+        import numpy as np
+        from scipy.sparse import csr_matrix
+        from scipy.sparse.csgraph import min_weight_full_bipartite_matching
+        parts=ordering.split(':');sign=-1 if len(parts)>1 and parts[1]=='max' else 1
+        noise=None
+        if len(parts)>2:
+            import random as _r;noise=_r.Random(int(parts[2]))
+        rows=[i for i,v2 in enumerate(ml) if v2>=0];rid={i:k for k,i in enumerate(rows)}
+        data=[];ri=[];ci=[]
+        for i in rows:
+            x=donors[i]
+            for u in adjacency[i]:
+                growth=len(uf[u])-len(initial[x]);ri.append(rid[i]);ci.append(u);data.append(1+sign*growth+(0 if sign>0 else 30)+(noise.random()*float(parts[3]) if noise else 0.0))
+        M=csr_matrix((np.array(data,dtype=float),(np.array(ri),np.array(ci))),shape=(len(rows),len(uc)))
+        r_ind,c_ind=min_weight_full_bipartite_matching(M)
+        arcs=[[donors[rows[r]],uc[c]] for r,c in zip(r_ind,c_ind)]
+        stats.update(matched=len(arcs),total_growth=int(sum(len(uf[c])-len(initial[donors[rows[r]]]) for r,c in zip(r_ind,c_ind))),mincost=True)
+        hk_growth=int(sum(len(uf[v2])-len(initial[donors[i]]) for i,v2 in enumerate(ml) if v2>=0));stats['hk_growth']=hk_growth
+    return arcs,stats
+
diff --git a/research/source-assisted-v4-layer/discovery/pairs_mw.py b/research/source-assisted-v4-layer/discovery/pairs_mw.py
new file mode 100644
index 0000000000..7e0db9777e
--- /dev/null
+++ b/research/source-assisted-v4-layer/discovery/pairs_mw.py
@@ -0,0 +1,42 @@
+"""Maximum-weight donor pairing for gauged recipients (weights depend on the donor only, so the
+transversal-matroid greedy with augmenting paths is optimal). Explicit legality: donor ungauged non-root with ops,
+last frame contained in the recipient gauge, donor dead before the recipient's first operation (or in phase one)."""
+from collections import defaultdict
+import sys
+from paired_cube.frames import contained
+def pairs_maxweight(o):
+ sys.setrecursionlimit(100000)
+ donors=[s for s in range(o.R) if s not in o.gauge and s not in o.roots and o.role_ops[s]]
+ recipients=sorted(o.gauge,key=lambda b:(o.position[o.role_ops[b][0]],b));rindex={b:i for i,b in enumerate(recipients)}
+ rtime=[o.position[o.role_ops[b][0]] for b in recipients]
+ byframe=defaultdict(list)
+ for s in donors:byframe[o.frames[o.role_ops[s][-1]]].append(s)
+ groups={}
+ adj=defaultdict(list)  # donor -> recipient indices
+ for b in recipients:
+  F=o.gauge[b]
+  if F not in groups:groups[F]=[s for D,ss in byframe.items() if contained(D,F) for s in ss]
+  t=rtime[rindex[b]]
+  for s in groups[F]:
+   if o.position[o.role_ops[s][-1]]<t:adj[s].append(rindex[b])
+ def weight(s):
+  dd=len(o.frames[o.role_ops[s][-1]]);f=18
+  return 3*o.phi[o.h-dd]+o.phi[3*f]-3*o.phi[f-dd] if f>=dd else -1e9
+ order=sorted(adj,key=lambda s:(-weight(s),o.position[o.role_ops[s][-1]],s))
+ mr=[-1]*len(recipients);md={}
+ def dfs(s,seen):
+  for r in adj[s]:
+   if r in seen:continue
+   seen.add(r)
+   if mr[r]<0 or dfs(mr[r],seen):mr[r]=s;md[s]=r;return True
+  return False
+ for s in order:
+  if weight(s)<=0:break
+  dfs(s,set())
+ result=[]
+ for r,s in enumerate(mr):
+  if s>=0:
+   b=recipients[r];first=o.role_ops[b][0]
+   result.append([s,b,None if o.role_ops[s][-1] in o.phase else first])
+ stats=dict(donors=len(donors),recipients=len(recipients),edges=sum(map(len,adj.values())),matched=len(result),total_weight=sum(weight(s) for s,_,_ in result))
+ return result,stats
diff --git a/research/source-assisted-v4-layer/discovery/physical_opt.py b/research/source-assisted-v4-layer/discovery/physical_opt.py
new file mode 100644
index 0000000000..a2e8e7d053
--- /dev/null
+++ b/research/source-assisted-v4-layer/discovery/physical_opt.py
@@ -0,0 +1,127 @@
+#!/usr/bin/env python3
+"""New deterministic exact frame endpoints and legal late handoff discovery.
+
+Chafik Boukhalfa with OpenAI Codex assistance. Apache-2.0.
+Builds on PR144 binary frame/word contract and PR157 physical reuse theorem.
+Numerical excess weights only select witnesses; exact containment, histogram
+and scalar checks must be independently replayed on the selected candidate.
+"""
+from collections import defaultdict,deque,Counter
+from functools import lru_cache
+import math
+from paired_cube.frames import basis,perp,contained
+
+@lru_cache(maxsize=300000)
+def join(A,B):
+    if contained(A,B):return B
+    if contained(B,A):return A
+    return basis(A+B)
+@lru_cache(maxsize=300000)
+def meet(A,B,h):
+    if contained(A,B):return A
+    if contained(B,A):return B
+    return perp(join(perp(A,h),perp(B,h)),h)
+
+class Physical:
+    def __init__(self,g,w,word,row):
+        self.g,self.w,self.word,self.row=g,w,word,row;self.h=g['h'];self.R=row['R'];self.v=g['v'];h=self.h
+        self.full=basis(1<<i for i in range(h));self.ops=[tuple(op) for op in word['ops']]
+        args=[None]+[None if a is None else [x+1 for x in a] for a in g['args']];self.spans=[()]*len(args)
+        for x in range(1,len(args)):
+            self.spans[x]=(g['inputs'][x-1],) if args[x] is None else join(self.spans[args[x][0]],self.spans[args[x][1]])
+        self.frames=[perp(tuple(w['annihilators'][x]),h) for _,_,x in self.ops];self.original=self.frames[:]
+        self.start=[()]*self.R;self.end=[self.full]*self.R
+        self.sources={int(x):s for x,s in word['sources'].items()}
+        for x,s in self.sources.items():self.start[s]=(g['inputs'][x-1],)
+        self.gauge={z['role']:perp(tuple(z['annihilator']),h) for z in word['selected']}
+        for s,F in self.gauge.items():self.start[s]=F
+        self.roots=set(word['rootroles'])
+        for r,s in zip(g['roots'],word['rootroles']):
+            self.end[s]=self.spans[r['node']+1] if r['kind']=='center' else perp(basis(g['inputs'][t] for t in r['targets']),h)
+        self.role_ops=[[] for _ in range(self.R)]
+        self.prev=[];last=[None]*self.R
+        for i,(a,b,x) in enumerate(self.ops):
+            self.prev.append([last[a],last[b]])
+            for s in (a,b):last[s]=i;self.role_ops[s].append(i)
+        self.next=[None]*len(self.ops);nxt=[None]*self.R
+        for i in range(len(self.ops)-1,-1,-1):
+            a,b,x=self.ops[i];self.next[i]=[nxt[a],nxt[b]];nxt[a]=nxt[b]=i
+        phase=set(word['phase1']);self.order=list(word['phase1'])+[i for i in range(len(self.ops)) if i not in phase]
+        self.order.sort(key=lambda i:(i not in phase,i));self.position={i:j for j,i in enumerate(self.order)};self.phase=phase
+        import os
+        _a=float(os.environ.get('PHI_ALPHA','0.00057'));self.rng=__import__('random').Random(int(os.environ.get('DESCENT_SEED','0')))
+        self.phi=[0.]+[t*math.expm1(_a*math.log(3*h/t))/_a for t in range(1,3*h+1)]
+
+    def neighbors(self,i):
+        a,b,_=self.ops[i]
+        P=[self.frames[j] if j is not None else self.start[s] for s,j in zip((a,b),self.prev[i])]
+        N=[self.frames[j] if j is not None else self.end[s] for s,j in zip((a,b),self.next[i])]
+        return P,N
+
+    def descend(self,rounds=4,lower_only=False):
+        changes=[]
+        for rnd in range(rounds):
+            changed=0;gain=0.
+            order=list(range(len(self.ops))) if rnd%2==0 else list(range(len(self.ops)-1,-1,-1))
+            if __import__('os').environ.get('DESCENT_SEED'):self.rng.shuffle(order)
+            for i in order:
+                P,N=self.neighbors(i);old=self.frames[i];x=self.ops[i][2]
+                L=join(join(P[0],P[1]),self.spans[x]);assert contained(L,old)
+                U=meet(N[0],N[1],self.h);assert contained(old,U)
+                def cost(r):return sum(self.phi[r-len(F)] for F in P)+sum(self.phi[len(F)-r] for F in N)
+                choices=[L] if lower_only else [L,U]
+                best=min(choices,key=lambda F:(cost(len(F)),len(F),F))
+                delta=cost(len(old))-cost(len(best))
+                if best!=old and (lower_only or delta>1e-10 or abs(delta)<=1e-10 and len(best)<len(old)):
+                    self.frames[i]=best;changed+=1;gain+=delta
+            changes.append(dict(round=rnd,changed=changed,numerical_excess_gain=gain));print('descent',changes[-1],flush=True)
+            if not changed:break
+        for i in range(len(self.ops)):
+            P,N=self.neighbors(i);assert contained(self.spans[self.ops[i][2]],self.frames[i]);assert all(contained(F,self.frames[i]) for F in P);assert all(contained(self.frames[i],F) for F in N)
+        return changes
+
+    def pairs(self):
+        donors=[s for s in range(self.R) if s not in self.gauge and s not in self.roots and self.role_ops[s]]
+        byframe=defaultdict(list)
+        for s in donors:byframe[self.frames[self.role_ops[s][-1]]].append(s)
+        recipients=sorted(self.gauge,key=lambda b:(self.position[self.role_ops[b][0]],b))
+        groups={};adj=[];gains={}
+        for b in recipients:
+            F=self.gauge[b]
+            if F not in groups:
+                groups[F]=[s for D,ss in byframe.items() if contained(D,F) for s in ss]
+            time=self.position[self.role_ops[b][0]]
+            row=[s for s in groups[F] if self.position[self.role_ops[s][-1]]<time]
+            def gain(s):
+                dd=len(self.frames[self.role_ops[s][-1]]);f=len(F)
+                return 3*self.phi[self.h-dd]+self.phi[3*f]-3*self.phi[f-dd]
+            row.sort(key=lambda s:(-gain(s),self.position[self.role_ops[s][-1]],s));adj.append(row)
+        L=len(recipients);ml=[-1]*L;mr={};dist=[0]*L
+        def bfs():
+            q=deque();found=False
+            for u in range(L):
+                dist[u]=0 if ml[u]<0 else -1
+                if ml[u]<0:q.append(u)
+            while q:
+                u=q.popleft()
+                for d in adj[u]:
+                    z=mr.get(d,-1)
+                    if z<0:found=True
+                    elif dist[z]<0:dist[z]=dist[u]+1;q.append(z)
+            return found
+        def dfs(u):
+            for d in adj[u]:
+                z=mr.get(d,-1)
+                if z<0 or dist[z]==dist[u]+1 and dfs(z):ml[u]=d;mr[d]=u;return True
+            dist[u]=-1;return False
+        while bfs():
+            for u in range(L):
+                if ml[u]<0:dfs(u)
+        result=[]
+        for j,d in enumerate(ml):
+            if d>=0:
+                b=recipients[j];first=self.role_ops[b][0]
+                result.append([d,b,None if self.role_ops[d][-1] in self.phase else first])
+        return result,dict(donors=len(donors),recipients=L,edges=sum(map(len,adj)),matched=len(result),gauge_frames=len(groups),donor_frames=len(byframe))
+
+    def changed(self):return [[i,F] for i,F in enumerate(self.frames) if F!=self.original[i]]
diff --git a/research/source-assisted-v4-layer/verify.py b/research/source-assisted-v4-layer/verify.py
new file mode 100644
index 0000000000..e595ae727b
--- /dev/null
+++ b/research/source-assisted-v4-layer/verify.py
@@ -0,0 +1,198 @@
+#!/usr/bin/env python3
+"""A stronger complex supplier: our physical layer on the source-assisted v4 complex word.
+
+The source-assisted complex word of PR #184 (icekylinx) on the PR #168 v4 modules (PR #193/#194, ikeboy) is
+compiled by a frame flow whose child histogram depends on the physical layer (operation frames and reuse pairs)
+that the flow receives. PR #194 uses PR #168's physical layer and certifies the complex saving 7.00918e-4. This
+package keeps every other input fixed and supplies the physical layer of PR #200's descent (endpoint moves of
+single operations, equal-frame components and connected bundles, seeded from PR #168's frames); the certified
+complex saving becomes 3543063/5000000000 = 7.0861e-4.
+
+verify.py (Python 3.11+, numpy/scipy for the regeneration steps; -O refused):
+  1. checks baseline-pr202.tar.gz and every file in it against SOURCE.json (bytes of PR #202 at 8d8d67b, which
+     carries PR #194's pipeline, the PR #168 v4 scripts and references, and PR #200's certificate and checker),
+     and the three witness files;
+  2. in a temporary directory: regenerates the PR #168 v4 complex cache (scripts/paired_cube_producer.py) and the
+     source-aligned word (source_aligned_local_v4.py, with its own physical layer and checks);
+  3. installs this package's physical frames and reuse pairs as the word's physical layer and admits that layer
+     with PR #200's complete complex checker (complex/physical.py: all descended frames, nested chains, aliases,
+     target read chronology, exact adjoint, every formal source/target/dirty column in both shear signs);
+  4. runs PR #184's frame flow (complex_frame_flow.py), exact lift (exact_complex_flow_lift.py) and contract
+     (contract_v4.py) on that layer, with the package's kernel pairs and physical pairs as the frozen witnesses;
+  5. runs PR #194's assembly (assemble.py) against PR #200's bit certificate and compares the canonical result
+     with certificate.json (--write regenerates it).
+Nothing outside the temporary directory is read or written, except certificate.json with --write.
+"""
+import argparse
+import gzip
+import hashlib
+import io
+import json
+import shutil
+import subprocess
+import sys
+import tarfile
+import tempfile
+import time
+from fractions import Fraction as Q
+from pathlib import Path
+
+if sys.flags.optimize:
+    raise SystemExit('Assertions must remain enabled: run without -O')
+sys.set_int_max_str_digits(0)
+sys.dont_write_bytecode = True
+HERE = Path(__file__).resolve().parent
+T0 = time.time()
+
+
+def log(*a):
+    print('[%4.0fs]' % (time.time() - T0), *a, flush=True)
+
+
+def sha(data):
+    return hashlib.sha256(data).hexdigest()
+
+
+def read(p):
+    raw = Path(p).read_bytes()
+    return json.loads(gzip.decompress(raw) if str(p).endswith('.gz') else raw)
+
+
+def write_gz(p, data):
+    with gzip.GzipFile(filename='', fileobj=open(p, 'wb'), mode='wb', mtime=0) as f:
+        f.write((json.dumps(data, separators=(',', ':')) + '\n').encode())
+
+
+def rebuild(dest):
+    src = read(HERE / 'SOURCE.json')
+    data = (HERE / src['archive']).read_bytes()
+    assert sha(data) == src['archive_sha256'], 'baseline archive differs from SOURCE.json'
+    with tarfile.open(fileobj=io.BytesIO(data), mode='r:gz') as archive:
+        members = [x for x in archive.getmembers() if x.isfile()]
+        assert sorted(x.name for x in members) == sorted(src['files']), 'archive lists other files than SOURCE.json'
+        for x in members:
+            part = Path(x.name)
+            assert not part.is_absolute() and '..' not in part.parts, 'unsafe path in archive: ' + x.name
+            body = archive.extractfile(x).read()
+            assert sha(body) == src['files'][x.name], 'baseline file differs from SOURCE.json: ' + x.name
+            (dest / part).parent.mkdir(parents=True, exist_ok=True)
+            (dest / part).write_bytes(body)
+    for name, digest in src['witness'].items():
+        assert sha((HERE / 'witness' / name).read_bytes()) == digest, 'witness differs from SOURCE.json: ' + name
+    return src
+
+
+def run(root, *args):
+    r = subprocess.run([sys.executable, '-B', *map(str, args)], cwd=root, capture_output=True, text=True)
+    if r.returncode:
+        raise SystemExit('FAILED: ' + ' '.join(map(str, args)) + '\n' + r.stdout[-3000:] + r.stderr[-3000:])
+    return r.stdout
+
+
+def strip_seconds(d):
+    if isinstance(d, dict):
+        return {k: strip_seconds(v) for k, v in d.items() if k not in ('seconds', 'maxrss')}
+    if isinstance(d, list):
+        return [strip_seconds(v) for v in d]
+    return d
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter)
+    ap.add_argument('--write', action='store_true', help='rewrite certificate.json from this run')
+    ap.add_argument('--temp-root', type=Path, default=None)
+    a = ap.parse_args()
+    with tempfile.TemporaryDirectory(prefix='sa-v4-layer-', dir=a.temp_root) as d:
+        root = Path(d).resolve()
+        src = rebuild(root)
+        log('rebuilt %d pinned files of PR #202 (commit %s) and checked the witnesses' % (len(src['files']), src['commit'][:7]))
+        PKG = root / 'research/source-assisted-v4'
+        SA = root / 'research/source-assisted'
+        BIT = root / 'research/paired-cube-diagonal-bit-168'
+        work = root / '.work'
+        work.mkdir()
+        base, aligned = work / 'base', work / 'aligned'
+        run(root, root / 'scripts/paired_cube_producer.py', '--work-dir', base, '--output', work / 'producer.json')
+        log('regenerated the PR #168 v4 complex cache')
+        run(root, PKG / 'source_aligned_local_v4.py', '--tree', root, '--cache', base, '--out', aligned)
+        log('regenerated the source-aligned word with its own physical layer and checks')
+        # 3. install this package's physical layer and admit it with PR #200's complete complex checker
+        phys = aligned / 'references/paired-cube/physical'
+        frames = read(HERE / 'witness/physical-frames.json')['frames']
+        pairs = read(HERE / 'witness/physical-pairs.json')['pairs']
+        theirs = read(phys / 'frames.json')['frames']
+        (phys / 'frames.json').write_text(json.dumps(dict(frames=frames), separators=(',', ':')) + '\n')
+        (phys / 'pairs.json').write_text(json.dumps(dict(pairs=pairs), separators=(',', ':')) + '\n')
+        sys.path.insert(0, str(root / 'scripts'))
+        from paired_cube.closure import compile_closure            # noqa: E402
+        from paired_cube_physical import physical                  # noqa: E402
+        cache = aligned / 'cache'
+        g, w, word, row = (read(cache / n) for n in ('graph.json', 'frames.json', 'selection.json', 'record.json'))
+        baseline, _ = compile_closure(g, [[x, c] for x, c in w['matching_arcs']])
+        profile = physical(g, w, word, row, frames, pairs)
+        cand = work / 'candidate'
+        cand.mkdir()
+        for name, data in [('graph', g), ('baseline', baseline), ('frames', w), ('word', word), ('profile-before', row),
+                           ('physical-frames', frames), ('physical-pairs', pairs), ('profile', profile)]:
+            write_gz(cand / (name + '.json.gz'), data)
+        pins = {p.name: sha(p.read_bytes()) for p in cand.glob('*.json.gz')}
+        (cand / 'result.json').write_text(json.dumps(dict(tag='source-assisted-v4-layer', pins=pins)) + '\n')
+        run(root, BIT / 'complex/physical.py', '--candidate', cand, '--source', root, '--out', work / 'audit.json')
+        audit = read(work / 'audit.json')
+        assert audit['status'].startswith('PASS'), audit['status']
+        log('admitted this package\'s physical layer with PR #200\'s complex checker: %s; %d frames, %d pairs, %d of PR #168\'s %d frames changed'
+            % (audit['status'][:60], len(frames), len(pairs), sum(1 for (i, F), (j, G) in zip(sorted(frames), sorted(theirs)) if tuple(F) != tuple(G)), len(theirs)))
+        # 4. frame flow, exact lift and contract on this layer
+        flow = work / 'flow.json'
+        run(root, SA / 'decision/complex_frame_flow.py', '--tree', aligned, '--cache', cache, '--out', flow, '--witness',
+            '--purify-source-donors', '--recycle-kernels', '--kernel-pairs', HERE / 'witness/kernel-pairs.json')
+        flow_data = read(flow)
+        witness = flow.with_suffix('.witness.json')
+        log('frame flow: new_R %d, new_W %d, deficit %d, numerical root %.7e' % (flow_data['new_R'], flow_data['new_W'], flow_data['deficit'], flow_data['numerical_local_root']))
+        lift = work / 'lift.json'
+        run(root, SA / 'decision/exact_complex_flow_lift.py', '--witness', witness, '--profile', flow, '--out', lift)
+        lift_data = read(lift)
+        lift_data.update(certificate_path=(work / 'lift.certificate.json.gz').relative_to(root).as_posix(),
+                         witness_path=witness.relative_to(root).as_posix())
+        lift.write_text(json.dumps(lift_data) + '\n')
+        log('exact lift: %s' % lift_data['status'][:80])
+        shutil.copy2(HERE / 'witness/physical-pairs.json', PKG / 'data/physical-pairs.json')
+        shutil.copy2(HERE / 'witness/kernel-pairs.json', PKG / 'data/kernel-pairs.json')
+        cprof = work / 'complex-profile.json'
+        run(root, PKG / 'contract_v4.py', '--tree', aligned, '--cache', cache, '--witness', witness,
+            '--flow-profile', flow, '--lift-profile', lift, '--out', cprof)
+        cdata = read(cprof)
+        for name, check in cdata['contract_checks'].items():
+            assert check not in (False, 'FAIL'), 'contract check %s failed' % name
+        log('contract: %d checks, W %d, rank %d, deficit %d' % (len(cdata['contract_checks']), cdata['W_per_vertex'], cdata['rank_per_vertex'], cdata['deficit_per_vertex']))
+        # 5. assembly against PR #200's bit certificate
+        final = work / 'global.json'
+        run(root, PKG / 'assemble.py', '--complex', cprof, '--output', final)
+        fdata = read(final)
+        for branch in ('complex', 'bit'):
+            fdata[branch].pop('numerical_root_for_discovery_only', None)
+        complex_saving = Q(fdata['complex']['saving'])
+        kappa = Q(fdata['kappa'])
+        log('assembly: complex saving %s = %.7e, bit effective %.7e, kappa %s = %.7e' % (
+            complex_saving, float(complex_saving), float(Q(fdata['bit']['effective_saving'])), kappa, float(kappa)))
+        result = dict(status='PASS source-assisted v4 word with this package\'s physical layer',
+                      complex_saving=str(complex_saving), complex_saving_float=float(complex_saving),
+                      kappa=str(kappa), kappa_float=float(kappa), binding='bit' if Q(fdata['bit']['effective_saving']) < complex_saving else 'complex',
+                      complex_profile=dict(m=cdata['m'], W_per_vertex=cdata['W_per_vertex'], rank_per_vertex=cdata['rank_per_vertex'],
+                                           deficit_per_vertex=cdata['deficit_per_vertex'], child_histogram=cdata['child_histogram']),
+                      layer=dict(frames=len(frames), pairs=len(pairs), audit=strip_seconds(audit)),
+                      flow=strip_seconds({k: v for k, v in flow_data.items() if k not in ('source_sha256', 'cache_sha256')}),
+                      lift={k: lift_data[k] for k in ('status', 'nodes', 'physical_R', 'all_actual_coefficients_dyadic', 'max_denominator', 'exact_scalar_program_sha256')},
+                      contract_checks=cdata['contract_checks'], assembly=strip_seconds(fdata),
+                      prerequisites=dict(pr202=src['commit'], pr194_complex_saving='219037/312500000', pr200_bit='research/paired-cube-diagonal-bit-168/certificate.json'))
+        canon = json.loads(json.dumps(result, sort_keys=True))
+        if a.write:
+            (HERE / 'certificate.json').write_text(json.dumps(canon, indent=1, sort_keys=True) + '\n')
+            log('wrote certificate.json')
+        committed = read(HERE / 'certificate.json')
+        assert committed == canon, 'result differs from certificate.json'
+        log('PASS certificate.json reproduced: complex saving %s (PR #194: 219037/312500000), kappa %s' % (complex_saving, kappa))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/source-assisted-v4-layer/witness/kernel-pairs.json b/research/source-assisted-v4-layer/witness/kernel-pairs.json
new file mode 100644
index 0000000000..55462cbf0c
--- /dev/null
+++ b/research/source-assisted-v4-layer/witness/kernel-pairs.json
@@ -0,0 +1 @@
+{"description":"Kernel pairs of the frame flow on this layer (from complex_frame_flow.py --witness)","kernel_pairs":[[9381,0,9092,0]]}
\ No newline at end of file
diff --git a/research/source-assisted-v4-layer/witness/physical-frames.json b/research/source-assisted-v4-layer/witness/physical-frames.json
new file mode 100644
index 0000000000..f27091abd7
--- /dev/null
+++ b/research/source-assisted-v4-layer/witness/physical-frames.json
@@ -0,0 +1 @@
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diff --git a/research/source-assisted-v4-layer/witness/physical-pairs.json b/research/source-assisted-v4-layer/witness/physical-pairs.json
new file mode 100644
index 0000000000..133818fb8e
--- /dev/null
+++ b/research/source-assisted-v4-layer/witness/physical-pairs.json
@@ -0,0 +1 @@
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5486,null],[3346,15487,null],[3323,15488,null],[3335,15489,null],[3514,15490,null],[3538,15491,null],[3515,15492,null],[3527,15493,null],[3754,15494,null],[3778,15495,null],[5399,15496,null],[5530,15497,null],[5554,15498,null],[3755,15499,null],[3767,15500,null],[5434,15501,null],[10442,15502,33829],[10746,15503,33830],[10748,15504,33831],[4967,15505,null],[5531,15506,null],[5543,15507,null],[3658,15508,null],[3682,15509,null],[3659,15510,null],[4151,15511,null],[4426,15512,null],[4450,15513,null],[3671,15514,null],[4427,15515,null],[4439,15516,null],[4666,15517,null],[4690,15518,null],[4667,15519,null],[4858,15520,null],[4882,15521,null],[4859,15522,null],[5578,15523,null],[5602,15524,null],[5579,15525,null],[4679,15526,null],[5002,15527,null],[5026,15528,null],[6263,15529,null],[6298,15530,null],[6322,15531,null],[5003,15532,null],[5015,15533,null],[5098,15534,null],[4871,15535,null],[5591,15536,null],[5674,15537,null],[5122,15538,null],[5099,15539,null],[5111,15540,null],[8812,15541,33893],[5770,15542,null],[5794,15543,null],[5458,15544,null],[5435,15545,null],[5447,15546,null],[5626,15547,null],[5650,15548,null],[5627,15549,null],[6970,15550,null],[6994,15551,null],[6971,15552,null],[5698,15553,null],[5675,15554,null],[5687,15555,null],[6299,15556,null],[6311,15557,null],[6346,15558,null],[5639,15559,null],[5771,15560,null],[5783,15561,null],[5818,15562,null],[5842,15563,null],[5819,15564,null],[5831,15565,null],[5866,15566,null],[5890,15567,null],[6370,15568,null],[6347,15569,null],[6359,15570,null],[6394,15571,null],[6418,15572,null],[6395,15573,null],[5962,15574,null],[5986,15575,null],[5963,15576,null],[6490,15577,null],[6514,15578,null],[6491,15579,null],[5867,15580,null],[5879,15581,null],[5975,15582,null],[6106,15583,null],[6130,15584,null],[6107,15585,null],[6010,15586,null],[6034,15587,null],[6011,15588,null],[6407,15589,null],[6586,15590,null],[6610,15591,null],[6119,15592,null],[6538,15593,null],[6562,15594,null],[6539,15595,null],[6551,15596,null],[6587,15597,null],[6023,15598,null],[6503,15599,null],[6599,15600,null],[6826,15601,null],[6850,15602,null],[6827,15603,null],[7018,15604,null],[7042,15605,null],[7019,15606,null],[6839,15607,null],[7031,15608,null],[7066,15609,null],[10970,15610,34000],[10972,15611,34001],[6682,15613,null],[6706,15614,null],[6683,15615,null],[6695,15616,null],[6730,15617,null],[6754,15618,null],[6731,15619,null],[6743,15620,null],[7450,15621,null],[7162,15622,null],[7186,15623,null],[7163,15624,null],[7175,15625,null],[7210,15626,null],[7234,15627,null],[7474,15628,null],[7451,15629,null],[7463,15630,null],[6983,15631,null],[55,15632,34035],[67,15633,34036],[7211,15634,null],[7223,15635,null],[7306,15636,null],[7,15637,34043],[19,15638,34044],[14,15639,34045],[7330,15640,null],[7307,15641,null],[7319,15642,null],[7090,15643,null],[7067,15644,null],[7079,15645,null],[7498,15646,null],[7522,15647,null],[7499,15648,null],[20,15649,34064],[27,15650,34065],[33,15651,34066],[7511,15652,null],[62,15653,34070],[68,15654,34071],[439,15655,34074],[451,15656,34075],[446,15657,34076],[7594,15658,null],[7618,15659,null],[7595,15660,null],[75,15661,34082],[81,15662,34083],[103,15663,34084],[7607,15664,null],[7738,15665,null],[7762,15666,null],[7739,15667,null],[7751,15668,null],[3895,15669,34095],[823,15670,34097],[835,15671,34098],[830,15672,34099],[2551,15673,34102],[2563,15674,34103],[2558,15675,34104],[2215,15676,34107],[2227,15677,34108],[2222,15678,34109],[487,15679,34112],[499,15680,34113],[494,15681,34114],[10974,15612,34002]]}

From 7f909fb1413ea7994f3c4b5107dc6d15418cadb8 Mon Sep 17 00:00:00 2001
From: Chafik Boukhalfa <chafik.boukhalfa@gmail.com>
Date: Sat, 10 Oct 2026 02:07:25 +0400
Subject: [PATCH 2/6] verify: record the checker's source hashes by
 repo-relative path (the certificate was keyed by temp paths)

---
 research/source-assisted-v4-layer/certificate.json | 14 +++++++-------
 research/source-assisted-v4-layer/verify.py        |  2 ++
 2 files changed, 9 insertions(+), 7 deletions(-)

diff --git a/research/source-assisted-v4-layer/certificate.json b/research/source-assisted-v4-layer/certificate.json
index a1e7ceb8c2..c6299a3dda 100644
--- a/research/source-assisted-v4-layer/certificate.json
+++ b/research/source-assisted-v4-layer/certificate.json
@@ -199,7 +199,7 @@
      "18": 1320
     }
    },
-   "complex_lift_certificate_sha256": "92b5d8d371fd841900f90e012c557fb420796b2b00d08daffd45d201d20fc228"
+   "complex_lift_certificate_sha256": "4e4e81d1cfc0a822bc4fd3e1b5e0ac08d96558f93d942eada7cfdd1ba7ddcfb3"
   },
   "finite_bridge": {
    "proof": "FINITE_BRIDGE.txt",
@@ -236,7 +236,7 @@
   "kappa_decimal": 0.0006768823,
   "source_sha256": {
    "bit_certificate": "853afa75980e11ce16b333c28311c4f635e84065867c049fda9916c2419091ed",
-   "complex": "96dd4e2ce033c52c91ad72217c4cb393e00e0511a72490b40bb79f9e8d78da7c"
+   "complex": "5238587d559aa3ef1da1c391cc1052cebb0b0adec1099f102732a5673dfeb23c"
   },
   "status": "Exact arithmetic; construction and finite bridge are explicit proof dependencies"
  },
@@ -586,11 +586,11 @@
    },
    "scope": "Finite candidate audit only. General exact Clifford and shared cover contracts, analytic supplier bridge, and paid global multiplication assembly remain separate obligations.",
    "source_sha256": {
-    "/private/tmp/claude-501/sa-v4-layer-2jyghoif/notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54",
-    "/private/tmp/claude-501/sa-v4-layer-2jyghoif/research/paired-cube-diagonal-bit-168/complex/physical.py": "01961fad4b04aaa940d2dc99c87690bc861b3861200296974c5e065ce3446659",
-    "/private/tmp/claude-501/sa-v4-layer-2jyghoif/research/paired-cube-diagonal-bit-168/complex/verify_fused.py": "75c3dcd80fcdbc461651fa2d2fa463e6b41a569d695fc74700d240223ef198e9",
-    "/private/tmp/claude-501/sa-v4-layer-2jyghoif/scripts/paired_cube/frames.py": "7d897418dbfee47de0723ea4983098a982610ccba45e1bac9000a79626d8421c",
-    "/private/tmp/claude-501/sa-v4-layer-2jyghoif/scripts/paired_cube_physical.py": "9c9ea1ea29e551ab6116d7c8ca09f4703ff9e2fa102032d5055327fde07ba319"
+    "notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54",
+    "research/paired-cube-diagonal-bit-168/complex/physical.py": "01961fad4b04aaa940d2dc99c87690bc861b3861200296974c5e065ce3446659",
+    "research/paired-cube-diagonal-bit-168/complex/verify_fused.py": "75c3dcd80fcdbc461651fa2d2fa463e6b41a569d695fc74700d240223ef198e9",
+    "scripts/paired_cube/frames.py": "7d897418dbfee47de0723ea4983098a982610ccba45e1bac9000a79626d8421c",
+    "scripts/paired_cube_physical.py": "9c9ea1ea29e551ab6116d7c8ca09f4703ff9e2fa102032d5055327fde07ba319"
    },
    "status": "PASS exact fused physical candidate, all formal source/target/dirty columns"
   },
diff --git a/research/source-assisted-v4-layer/verify.py b/research/source-assisted-v4-layer/verify.py
index e595ae727b..4cedf0bda7 100644
--- a/research/source-assisted-v4-layer/verify.py
+++ b/research/source-assisted-v4-layer/verify.py
@@ -140,6 +140,8 @@ def main():
         run(root, BIT / 'complex/physical.py', '--candidate', cand, '--source', root, '--out', work / 'audit.json')
         audit = read(work / 'audit.json')
         assert audit['status'].startswith('PASS'), audit['status']
+        # the checker keys its source hashes by absolute path; record them relative to the rebuilt tree
+        audit['source_sha256'] = {Path(k).resolve().relative_to(root).as_posix(): v for k, v in audit['source_sha256'].items()}
         log('admitted this package\'s physical layer with PR #200\'s complex checker: %s; %d frames, %d pairs, %d of PR #168\'s %d frames changed'
             % (audit['status'][:60], len(frames), len(pairs), sum(1 for (i, F), (j, G) in zip(sorted(frames), sorted(theirs)) if tuple(F) != tuple(G)), len(theirs)))
         # 4. frame flow, exact lift and contract on this layer

From cf08677072c2e0d29fe14fb4df8e7233d1deea50 Mon Sep 17 00:00:00 2001
From: Chafik Boukhalfa <chafik.boukhalfa@gmail.com>
Date: Sat, 10 Oct 2026 02:15:35 +0400
Subject: [PATCH 3/6] verify: certificate.json keeps exact fields only; floats
 and generated-artifact hashes go to report.json

---
 .../source-assisted-v4-layer/certificate.json | 555 ++--------------
 research/source-assisted-v4-layer/report.json | 614 ++++++++++++++++++
 research/source-assisted-v4-layer/verify.py   |  23 +-
 3 files changed, 686 insertions(+), 506 deletions(-)
 create mode 100644 research/source-assisted-v4-layer/report.json

diff --git a/research/source-assisted-v4-layer/certificate.json b/research/source-assisted-v4-layer/certificate.json
index c6299a3dda..a8ca722a15 100644
--- a/research/source-assisted-v4-layer/certificate.json
+++ b/research/source-assisted-v4-layer/certificate.json
@@ -1,243 +1,8 @@
 {
  "assembly": {
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-    "degree_gap": "23971789/25",
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-     "exponent": 30000
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-    "induction_gap_lower": 1,
-    "strict_literal_gap": 1
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+  "bit_effective_saving": "338670433250477381/500000000000000000000",
+  "complex_saving": "3543063/5000000000",
   "kappa": "6768823/10000000000",
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@@ -270,7 +35,6 @@
   "rank_per_vertex": 794112
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  "complex_saving": "3543063/5000000000",
- "complex_saving_float": 0.0007086126,
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   "all_centers_in_phase1": true,
   "all_fresh_columns_equal_identity": true,
@@ -298,30 +62,6 @@
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@@ -345,261 +85,68 @@
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-  "status": "Modular equal-frame flow price, retaining dirty kernel lifts and complete full-frame cleanup. Not an exact supplier certificate.",
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+     "all_physical_dirty_slots_restored": true,
+     "direction": 1,
+     "dirty_columns": 13372,
+     "every_formal_column": true,
+     "source_columns": 1320,
+     "target_columns": 1320
     },
-    "v": 1320
-   },
-   "pre_reuse_finite": {
-    "K_involution_and_inverse_exact": true,
-    "K_source_itinerary_verified": true,
-    "center_phase_closed": true,
-    "conservative_source_supports": true,
-    "decoder_identity_exact": true,
-    "full_backward_intersections": true,
-    "full_gauge_intersections": true,
-    "physical_signed_M_replayed": true,
-    "positive_rank_ledger_recounted": true,
-    "reverse_target_chains": true,
-    "scalar_H_exact": true,
-    "selected_roles": 2310,
-    "selected_sources_untouched": true,
-    "signed_scalar_pairs_checked": 1742400
-   },
-   "scalar_bound": {
-    "M_operations": 36931,
-    "binary_dirty_read_bits": 36947,
-    "c": 22569,
-    "decoder_coefficients": [
-     "-1/2",
-     "-1/6",
-     "1/2",
-     "1/3"
-    ],
-    "fixed_odd_divisor": 3,
-    "gate_and_inverse_coefficients": [
-     -1,
-     1
-    ],
-    "handoffs": 2310,
-    "local_group_upper": 6118494407560,
-    "logical_R": 15682,
-    "physical_R": 13372,
-    "q": 3157,
-    "scalar_denominators_divide": 6,
-    "scope": "Conservative inherited virtual-word bill retained after aliasing, including all compensated reads. Alias and frame route choices are finite permutations/Clifford adapters; their separate global movement bill remains an assembly obligation.",
-    "three_q_less_than_2pow15": true
-   },
-   "scope": "Finite candidate audit only. General exact Clifford and shared cover contracts, analytic supplier bridge, and paid global multiplication assembly remain separate obligations.",
-   "source_sha256": {
-    "notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54",
-    "research/paired-cube-diagonal-bit-168/complex/physical.py": "01961fad4b04aaa940d2dc99c87690bc861b3861200296974c5e065ce3446659",
-    "research/paired-cube-diagonal-bit-168/complex/verify_fused.py": "75c3dcd80fcdbc461651fa2d2fa463e6b41a569d695fc74700d240223ef198e9",
-    "scripts/paired_cube/frames.py": "7d897418dbfee47de0723ea4983098a982610ccba45e1bac9000a79626d8421c",
-    "scripts/paired_cube_physical.py": "9c9ea1ea29e551ab6116d7c8ca09f4703ff9e2fa102032d5055327fde07ba319"
-   },
-   "status": "PASS exact fused physical candidate, all formal source/target/dirty columns"
+    {
+     "K_and_inverse": true,
+     "all_aliases_and_deadline_reads": true,
+     "all_physical_dirty_slots_restored": true,
+     "direction": -1,
+     "dirty_columns": 13372,
+     "every_formal_column": true,
+     "source_columns": 1320,
+     "target_columns": 1320
+    }
+   ],
+   "packed_digit_bits": 24
   },
   "frames": 17050,
+  "frames_changed_from_pr168": 5577,
+  "mutations": {
+   "early-read": "REJECTED",
+   "missing-read": "REJECTED"
+  },
   "pairs": 2310
  },
  "lift": {
   "all_actual_coefficients_dyadic": true,
-  "exact_scalar_program_sha256": "c33c86388f2d4e6627e2d3e5a762f8cba47eaff8b43166c4c8d31bdc155663d3",
   "max_denominator": "2",
   "nodes": 18481,
   "physical_R": 9412,
diff --git a/research/source-assisted-v4-layer/report.json b/research/source-assisted-v4-layer/report.json
new file mode 100644
index 0000000000..205860620a
--- /dev/null
+++ b/research/source-assisted-v4-layer/report.json
@@ -0,0 +1,614 @@
+{
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+    "frames.json.gz": "06faf083a88b3b4cb37c4fbdb62b5ca48f6c493894c5391bf96fd5727742e4fd",
+    "graph.json.gz": "aa1708d0525f33a8b1a5757ea5f05ba161cb44f71b4e92971239ad1dfb469ad3",
+    "physical-frames.json.gz": "1bd2799e6f4c7fc752c0c14aa1412368737758d6d5a5ec1fde5d590a29398084",
+    "physical-pairs.json.gz": "d718d50378f61dea483e3022b275edf7c2335839ba65aacd55d24279c1757c42",
+    "profile-before.json.gz": "f60146c8d8c5e8aa0365f08b68aa6e03403656b74bf28b7058ef5a241560bf92",
+    "profile.json.gz": "264bb90158103b609620da3dc96e68121466cccf0bc9db2c845680c209b901ce",
+    "word.json.gz": "ec94f10d8c8d198d37cbe4781c8c251a8e62335e53ad5177f21cf59ad969a6d8"
+   },
+   "mutations": {
+    "early-read": "REJECTED",
+    "missing-read": "REJECTED"
+   },
+   "physical": {
+    "R": 15682,
+    "W_per_vertex": 16012,
+    "changed_operation_frames": 17050,
+    "checks": {
+     "donor_frames_inside_gauges": true,
+     "pair_chronology": true,
+     "role_chains_nested": true,
+     "target_chains_nested_in_read_order": true,
+     "telescoping_deficit": true,
+     "value_spans_inside_frames": true
+    },
+    "child_histogram": {
+     "1": 110970,
+     "10": 2700,
+     "11": 945,
+     "12": 3663,
+     "13": 495,
+     "14": 2118,
+     "15": 2910,
+     "16": 282,
+     "17": 198,
+     "18": 7920,
+     "19": 3186,
+     "2": 41430,
+     "20": 11946,
+     "3": 29061,
+     "4": 10662,
+     "5": 6765,
+     "6": 3840,
+     "7": 2349,
+     "8": 4209,
+     "9": 1539
+    },
+    "deficit_per_vertex": 1320,
+    "h": 22,
+    "late_pairs": 1399,
+    "local_histogram": {
+     "1": 31710,
+     "10": 900,
+     "11": 315,
+     "12": 1221,
+     "13": 165,
+     "14": 706,
+     "15": 970,
+     "16": 94,
+     "17": 66,
+     "19": 1062,
+     "2": 11610,
+     "20": 3982,
+     "3": 9687,
+     "4": 3554,
+     "5": 2255,
+     "6": 1280,
+     "7": 783,
+     "8": 1403,
+     "9": 513
+    },
+    "loss": 440,
+    "m": 66,
+    "pairs": 2310,
+    "physical_R": 13372,
+    "physical_gauge_histogram": {},
+    "rank_per_vertex": 1055472,
+    "scalar_replay": {
+     "modulus": "2^61-1",
+     "rejected_controls": [
+      "flipped_signed_operation",
+      "late_read_before_donor_death",
+      "omitted_recipient_read"
+     ],
+     "seeds_restored_and_y_plus_x": true,
+     "slots": 13372
+    },
+    "source_data_histogram": {
+     "1": 1320,
+     "18": 1320,
+     "2": 1320
+    },
+    "target_data_histogram": {
+     "0": 6270,
+     "1": 3960,
+     "18": 1320
+    },
+    "v": 1320
+   },
+   "pre_reuse_finite": {
+    "K_involution_and_inverse_exact": true,
+    "K_source_itinerary_verified": true,
+    "center_phase_closed": true,
+    "conservative_source_supports": true,
+    "decoder_identity_exact": true,
+    "full_backward_intersections": true,
+    "full_gauge_intersections": true,
+    "physical_signed_M_replayed": true,
+    "positive_rank_ledger_recounted": true,
+    "reverse_target_chains": true,
+    "scalar_H_exact": true,
+    "selected_roles": 2310,
+    "selected_sources_untouched": true,
+    "signed_scalar_pairs_checked": 1742400
+   },
+   "scalar_bound": {
+    "M_operations": 36931,
+    "binary_dirty_read_bits": 36947,
+    "c": 22569,
+    "decoder_coefficients": [
+     "-1/2",
+     "-1/6",
+     "1/2",
+     "1/3"
+    ],
+    "fixed_odd_divisor": 3,
+    "gate_and_inverse_coefficients": [
+     -1,
+     1
+    ],
+    "handoffs": 2310,
+    "local_group_upper": 6118494407560,
+    "logical_R": 15682,
+    "physical_R": 13372,
+    "q": 3157,
+    "scalar_denominators_divide": 6,
+    "scope": "Conservative inherited virtual-word bill retained after aliasing, including all compensated reads. Alias and frame route choices are finite permutations/Clifford adapters; their separate global movement bill remains an assembly obligation.",
+    "three_q_less_than_2pow15": true
+   },
+   "scope": "Finite candidate audit only. General exact Clifford and shared cover contracts, analytic supplier bridge, and paid global multiplication assembly remain separate obligations.",
+   "source_sha256": {
+    "notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54",
+    "research/paired-cube-diagonal-bit-168/complex/physical.py": "01961fad4b04aaa940d2dc99c87690bc861b3861200296974c5e065ce3446659",
+    "research/paired-cube-diagonal-bit-168/complex/verify_fused.py": "75c3dcd80fcdbc461651fa2d2fa463e6b41a569d695fc74700d240223ef198e9",
+    "scripts/paired_cube/frames.py": "7d897418dbfee47de0723ea4983098a982610ccba45e1bac9000a79626d8421c",
+    "scripts/paired_cube_physical.py": "9c9ea1ea29e551ab6116d7c8ca09f4703ff9e2fa102032d5055327fde07ba319"
+   },
+   "status": "PASS exact fused physical candidate, all formal source/target/dirty columns"
+  },
+  "frames": 17050,
+  "pairs": 2310
+ },
+ "lift": {
+  "all_actual_coefficients_dyadic": true,
+  "exact_scalar_program_sha256": "c33c86388f2d4e6627e2d3e5a762f8cba47eaff8b43166c4c8d31bdc155663d3",
+  "max_denominator": "2",
+  "nodes": 18481,
+  "physical_R": 9412,
+  "status": "Exact rational local lifts and inverse gate programs verified; monotone flow and zero-fresh kernel reuse DAG verified."
+ },
+ "prerequisites": {
+  "pr194_complex_saving": "219037/312500000",
+  "pr200_bit": "research/paired-cube-diagonal-bit-168/certificate.json",
+  "pr202": "8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d"
+ },
+ "status": "PASS source-assisted v4 word with this package's physical layer"
+}
diff --git a/research/source-assisted-v4-layer/verify.py b/research/source-assisted-v4-layer/verify.py
index 4cedf0bda7..4d5ecce367 100644
--- a/research/source-assisted-v4-layer/verify.py
+++ b/research/source-assisted-v4-layer/verify.py
@@ -187,11 +187,30 @@ def main():
                       lift={k: lift_data[k] for k in ('status', 'nodes', 'physical_R', 'all_actual_coefficients_dyadic', 'max_denominator', 'exact_scalar_program_sha256')},
                       contract_checks=cdata['contract_checks'], assembly=strip_seconds(fdata),
                       prerequisites=dict(pr202=src['commit'], pr194_complex_saving='219037/312500000', pr200_bit='research/paired-cube-diagonal-bit-168/certificate.json'))
-        canon = json.loads(json.dumps(result, sort_keys=True))
+        # certificate.json holds exact, platform-independent fields only (no floats, no hashes of generated gzip
+        # artifacts); the full run record goes to report.json (informational, written only with --write)
+        canon = dict(status=result['status'], complex_saving=result['complex_saving'], kappa=result['kappa'], binding=result['binding'],
+                     complex_profile=result['complex_profile'],
+                     layer=dict(frames=len(frames), pairs=len(pairs), frames_changed_from_pr168=sum(1 for (i, F), (j, G) in zip(sorted(frames), sorted(theirs)) if tuple(F) != tuple(G)),
+                                audit_status=audit['status'], audit_input_sha256=audit['input_sha256'], audit_source_sha256=audit['source_sha256'],
+                                formal=audit.get('formal'), mutations=audit.get('mutations')),
+                     flow=dict(new_R=flow_data['new_R'], new_W=flow_data['new_W'], deficit=flow_data['deficit'], loss=flow_data['loss'],
+                               original_physical_roles=flow_data['original_physical_roles'], child_histogram=flow_data['child_histogram']),
+                     lift=dict(status=lift_data['status'], nodes=lift_data['nodes'], physical_R=lift_data['physical_R'],
+                               all_actual_coefficients_dyadic=lift_data['all_actual_coefficients_dyadic'], max_denominator=lift_data['max_denominator']),
+                     contract_checks=cdata['contract_checks'],
+                     assembly=dict(kappa=fdata['kappa'], complex_saving=fdata['complex']['saving'], bit_effective_saving=fdata['bit']['effective_saving'], status=fdata.get('status')),
+                     prerequisites=result['prerequisites'])
+        canon = json.loads(json.dumps(canon, sort_keys=True))
         if a.write:
             (HERE / 'certificate.json').write_text(json.dumps(canon, indent=1, sort_keys=True) + '\n')
-            log('wrote certificate.json')
+            (HERE / 'report.json').write_text(json.dumps(json.loads(json.dumps(result, sort_keys=True)), indent=1, sort_keys=True) + '\n')
+            log('wrote certificate.json and report.json')
         committed = read(HERE / 'certificate.json')
+        if committed != canon:
+            for k in sorted(set(committed) | set(canon)):
+                if committed.get(k) != canon.get(k):
+                    log('DIFFERS: %s' % k)
         assert committed == canon, 'result differs from certificate.json'
         log('PASS certificate.json reproduced: complex saving %s (PR #194: 219037/312500000), kappa %s' % (complex_saving, kappa))
 

From c3f9ba18ae6a9fad0cdb4ce8cd41e6baa52f9797 Mon Sep 17 00:00:00 2001
From: Chafik Boukhalfa <chafik.boukhalfa@gmail.com>
Date: Sat, 10 Oct 2026 02:20:13 +0400
Subject: [PATCH 4/6] verify: drop the candidate-file hashes from the
 certificate (their profiles carry floats); report field-level differences

---
 research/source-assisted-v4-layer/certificate.json | 10 ----------
 research/source-assisted-v4-layer/report.json      |  4 ++--
 research/source-assisted-v4-layer/verify.py        |  8 ++++++--
 3 files changed, 8 insertions(+), 14 deletions(-)

diff --git a/research/source-assisted-v4-layer/certificate.json b/research/source-assisted-v4-layer/certificate.json
index a8ca722a15..aef014d48f 100644
--- a/research/source-assisted-v4-layer/certificate.json
+++ b/research/source-assisted-v4-layer/certificate.json
@@ -92,16 +92,6 @@
  },
  "kappa": "6768823/10000000000",
  "layer": {
-  "audit_input_sha256": {
-   "baseline.json.gz": "33187d4011bc6dd11468408c0bc7e968d54c63aadf87a09592da056140375d57",
-   "frames.json.gz": "06faf083a88b3b4cb37c4fbdb62b5ca48f6c493894c5391bf96fd5727742e4fd",
-   "graph.json.gz": "aa1708d0525f33a8b1a5757ea5f05ba161cb44f71b4e92971239ad1dfb469ad3",
-   "physical-frames.json.gz": "1bd2799e6f4c7fc752c0c14aa1412368737758d6d5a5ec1fde5d590a29398084",
-   "physical-pairs.json.gz": "d718d50378f61dea483e3022b275edf7c2335839ba65aacd55d24279c1757c42",
-   "profile-before.json.gz": "f60146c8d8c5e8aa0365f08b68aa6e03403656b74bf28b7058ef5a241560bf92",
-   "profile.json.gz": "264bb90158103b609620da3dc96e68121466cccf0bc9db2c845680c209b901ce",
-   "word.json.gz": "ec94f10d8c8d198d37cbe4781c8c251a8e62335e53ad5177f21cf59ad969a6d8"
-  },
   "audit_source_sha256": {
    "notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54",
    "research/paired-cube-diagonal-bit-168/complex/physical.py": "01961fad4b04aaa940d2dc99c87690bc861b3861200296974c5e065ce3446659",
diff --git a/research/source-assisted-v4-layer/report.json b/research/source-assisted-v4-layer/report.json
index 205860620a..d95185c887 100644
--- a/research/source-assisted-v4-layer/report.json
+++ b/research/source-assisted-v4-layer/report.json
@@ -199,7 +199,7 @@
      "18": 1320
     }
    },
-   "complex_lift_certificate_sha256": "771641c9b0804f6e3cffc982b3612b7c457fa5094b211d93236fa13a67a9e258"
+   "complex_lift_certificate_sha256": "0b3bcb16cdab2e5fde6a15868bd671651586a99710918342beab72ad90990deb"
   },
   "finite_bridge": {
    "proof": "FINITE_BRIDGE.txt",
@@ -236,7 +236,7 @@
   "kappa_decimal": 0.0006768823,
   "source_sha256": {
    "bit_certificate": "853afa75980e11ce16b333c28311c4f635e84065867c049fda9916c2419091ed",
-   "complex": "0eb593fb09ded19f2f3c73bd93e93920bff176c3ba0e506d4d87ae79203112dc"
+   "complex": "40fe51ff725af94258ab8e231696feb84494c77bfaa3cef952a566659a0950fa"
   },
   "status": "Exact arithmetic; construction and finite bridge are explicit proof dependencies"
  },
diff --git a/research/source-assisted-v4-layer/verify.py b/research/source-assisted-v4-layer/verify.py
index 4d5ecce367..2ddf0d2f39 100644
--- a/research/source-assisted-v4-layer/verify.py
+++ b/research/source-assisted-v4-layer/verify.py
@@ -192,8 +192,8 @@ def main():
         canon = dict(status=result['status'], complex_saving=result['complex_saving'], kappa=result['kappa'], binding=result['binding'],
                      complex_profile=result['complex_profile'],
                      layer=dict(frames=len(frames), pairs=len(pairs), frames_changed_from_pr168=sum(1 for (i, F), (j, G) in zip(sorted(frames), sorted(theirs)) if tuple(F) != tuple(G)),
-                                audit_status=audit['status'], audit_input_sha256=audit['input_sha256'], audit_source_sha256=audit['source_sha256'],
-                                formal=audit.get('formal'), mutations=audit.get('mutations')),
+                                audit_status=audit['status'], audit_source_sha256=audit['source_sha256'],
+                                formal=json.loads(json.dumps(audit.get('formal'))), mutations=audit.get('mutations')),
                      flow=dict(new_R=flow_data['new_R'], new_W=flow_data['new_W'], deficit=flow_data['deficit'], loss=flow_data['loss'],
                                original_physical_roles=flow_data['original_physical_roles'], child_histogram=flow_data['child_histogram']),
                      lift=dict(status=lift_data['status'], nodes=lift_data['nodes'], physical_R=lift_data['physical_R'],
@@ -211,6 +211,10 @@ def main():
             for k in sorted(set(committed) | set(canon)):
                 if committed.get(k) != canon.get(k):
                     log('DIFFERS: %s' % k)
+                    if isinstance(canon.get(k), dict):
+                        for kk in sorted(set(committed.get(k, {})) | set(canon[k])):
+                            if committed.get(k, {}).get(kk) != canon[k].get(kk):
+                                log('  DIFFERS: %s/%s: %s vs %s' % (k, kk, str(committed.get(k, {}).get(kk))[:80], str(canon[k].get(kk))[:80]))
         assert committed == canon, 'result differs from certificate.json'
         log('PASS certificate.json reproduced: complex saving %s (PR #194: 219037/312500000), kappa %s' % (complex_saving, kappa))
 

From 246f6f944c64eb7047026b3780636eeb61d30cf7 Mon Sep 17 00:00:00 2001
From: Chafik Boukhalfa <chafik.boukhalfa@gmail.com>
Date: Sat, 10 Oct 2026 02:26:24 +0400
Subject: [PATCH 5/6] Layer with 98% of the operations frozen: complex saving
 7097667/10^10 = 7.0977e-4 (+1.26% over PR194), 435 frames changed

One light descent round from PR168's frames with a 98% movability mask (seed 8)
prices higher under the flow than the full cycle; certified through the same
chain (PR200 checker, flow, exact lift, contract, PR194 assembly).
---
 research/source-assisted-v4-layer/PROOF.md    |  19 +-
 research/source-assisted-v4-layer/README.md   |   8 +-
 research/source-assisted-v4-layer/SOURCE.json |   8 +-
 .../source-assisted-v4-layer/certificate.json |  84 ++---
 .../discovery/README.md                       |   8 +-
 research/source-assisted-v4-layer/report.json | 327 +++++++++---------
 .../witness/kernel-pairs.json                 |   2 +-
 .../witness/physical-frames.json              |   2 +-
 .../witness/physical-pairs.json               |   2 +-
 9 files changed, 232 insertions(+), 228 deletions(-)

diff --git a/research/source-assisted-v4-layer/PROOF.md b/research/source-assisted-v4-layer/PROOF.md
index 213c8daf87..4601fdc0c5 100644
--- a/research/source-assisted-v4-layer/PROOF.md
+++ b/research/source-assisted-v4-layer/PROOF.md
@@ -3,12 +3,12 @@
 **Claim.** The source-assisted complex word of PR #184 (icekylinx) on the PR #168 v4 modules (PR #193/#194,
 ikeboy), with every input of PR #194's pipeline unchanged except the physical layer, certifies the complex saving
 
-    a_C = 3543063 / 5000000000 = 7.0861260·10⁻⁴      (PR #194: 219037/312500000 = 7.0091844·10⁻⁴, +1.10%).
+    a_C = 7097667 / 10000000000 = 7.0976670·10⁻⁴      (PR #194: 219037/312500000 = 7.0091844·10⁻⁴, +1.26%).
 
 Composed with PR #200's bit supplier through PR #194's unchanged 47-constraint assembly the result is
 κ = 6768823/10¹⁰, the same as PR #202, because the bit branch binds (effective 6.7734·10⁻⁴). **No new κ.** The
-complex cap of every composition on this lineage rises from 7.0092·10⁻⁴ to 7.0861·10⁻⁴, and the exact-DFT
-exponent saving of `research/exact-dft-transfer` (PR #225) rises with it to 7086/10⁷.
+complex cap of every composition on this lineage rises from 7.0092·10⁻⁴ to 7.0977·10⁻⁴, and the exact-DFT
+exponent saving of `research/exact-dft-transfer` (PR #225) rises with it.
 
 ## What changes
 
@@ -21,9 +21,11 @@ assembly. The flow's child histogram depends on the physical layer it receives:
 equal-frame and compress.
 
 This package replaces only that physical layer. The frames and reuse pairs in `witness/` are produced by PR #200's
-descent (`discovery/`): endpoint moves of single operations, equal-frame components and connected bundles, started
-from PR #168's frames, on the aligned word's own cache. 5,577 of PR #168's 17,789 operation frames change; the
-2,310 reuse pairs are recomputed by the same late compensated pairing. The matching arcs, modules, local circuit,
+descent (`discovery/`): one round of endpoint moves of single operations, started from PR #168's frames on the
+aligned word's own cache, with 98% of the operations frozen (seed 8 of the movability mask): 435 of PR #168's 17,789
+operation frames change; the 2,310 reuse pairs are recomputed by the same late compensated pairing. The flow prefers
+layers close to PR #168's: the full descent cycle gives 7.0861·10⁻⁴, one light round 7.0950·10⁻⁴, and freezing most
+operations 7.0977·10⁻⁴ (`discovery/README.md`). The matching arcs, modules, local circuit,
 gauges, aligned rewrite, flow options and kernel pairs are PR #194's (the kernel pairs the flow finds on this layer
 are recorded in `witness/kernel-pairs.json`, as PR #194 records its own).
 
@@ -49,7 +51,7 @@ are recorded in `witness/kernel-pairs.json`, as PR #194 records its own).
    target chains and the normalized profile (13 checks), with the package's physical and kernel pairs as the frozen
    witnesses that PR #194's contract compares against.
 5. **Assembly.** PR #194's `assemble.py` composes the certified complex profile with PR #200's bit certificate and
-   reproduces `certificate.json` exactly: complex saving 3543063/5000000000, κ 6768823/10¹⁰, the next grid point
+   reproduces `certificate.json` exactly: complex saving 7097667/10000000000, κ 6768823/10¹⁰, the next grid point
    rejected by the unchanged assembly.
 
 The complex profile: m = 66, W = 12,052 roles per vertex, rank mass 794,112, deficit 1,320, largest child 20, the
@@ -66,4 +68,5 @@ roles, 13,372 physical to 9,412, but splits the remaining increments differently
   this lineage (for example the banked rows of #205–#223 and the conditional rungs of #219/#224) and the exact-DFT
   exponent.
 - The descent is a discovery heuristic; no optimality is claimed. The float screen varies with the descent options
-  (light 7.0419·10⁻⁴, plain 7.0691·10⁻⁴, seeded 7.0861·10⁻⁴); only the seeded layer is certified here.
+  (unseeded light 7.0419·10⁻⁴, unseeded full 7.0691·10⁻⁴, seeded full 7.0861·10⁻⁴, seeded light 7.0950·10⁻⁴,
+  seeded light with 98% frozen 7.0977·10⁻⁴); only the last layer is certified here.
diff --git a/research/source-assisted-v4-layer/README.md b/research/source-assisted-v4-layer/README.md
index 76a3d9c968..79044ad8f2 100644
--- a/research/source-assisted-v4-layer/README.md
+++ b/research/source-assisted-v4-layer/README.md
@@ -1,13 +1,13 @@
 # source-assisted-v4-layer
 
-**Complex saving 7.0861·10⁻⁴ (no new κ).** PR #194's source-assisted complex word with this package's physical
-layer (PR #200's descent, seeded from PR #168's frames) certifies
+**Complex saving 7.0977·10⁻⁴ (no new κ).** PR #194's source-assisted complex word with this package's physical
+layer (one round of PR #200's descent from PR #168's frames with 98% of the operations frozen) certifies
 
-    a_C = 3543063/5000000000 = 7.0861260·10⁻⁴       (PR #194: 7.0091844·10⁻⁴, +1.10%),
+    a_C = 7097667/10000000000 = 7.0976670·10⁻⁴       (PR #194: 7.0091844·10⁻⁴, +1.26%),
 
 through PR #184's flow, exact lift and contract and PR #194's assembly, which with PR #200's bit supplier gives
 κ = 6768823/10¹⁰ (the bit branch binds, as in PR #202). The complex cap of this lineage and the exact-DFT exponent
-saving (`research/exact-dft-transfer`) rise to 7.0861·10⁻⁴. [PROOF.md](PROOF.md) states what changes and what is
+saving (`research/exact-dft-transfer`) rise to 7.0977·10⁻⁴. [PROOF.md](PROOF.md) states what changes and what is
 checked.
 
 ## Verify
diff --git a/research/source-assisted-v4-layer/SOURCE.json b/research/source-assisted-v4-layer/SOURCE.json
index 83b96fa470..71ea3c9ed2 100644
--- a/research/source-assisted-v4-layer/SOURCE.json
+++ b/research/source-assisted-v4-layer/SOURCE.json
@@ -316,8 +316,8 @@
  "pr": 202,
  "repository": "CrocSwap/integer-mult-bounds",
  "witness": {
-  "kernel-pairs.json": "fcdfbdbf0ed9a57cff327ae94138f8989bc90f6800b7a00f1fdda996c7ef077b",
-  "physical-frames.json": "1113e5497189c26616da81ce0ac3b314976bd7ec912752de4f6a0f6225f94e44",
-  "physical-pairs.json": "312eeacb52bef8ca62c507005d893578bda762a940c3a8979f5df5e3467e528a"
+  "kernel-pairs.json": "e33b753fc4519170373207b0527b9cb53e0b552e2a27f48f72381a881c7a227f",
+  "physical-frames.json": "788081f575344edd374f33dd8ea7083093e0f0367706da148094557956ec65d3",
+  "physical-pairs.json": "33bb02f9e951fd4ff88810a98f0b1fa7c6f6d2ca68b5b47818cd030f7acecafa"
  }
-}
\ No newline at end of file
+}
diff --git a/research/source-assisted-v4-layer/certificate.json b/research/source-assisted-v4-layer/certificate.json
index aef014d48f..96208db991 100644
--- a/research/source-assisted-v4-layer/certificate.json
+++ b/research/source-assisted-v4-layer/certificate.json
@@ -1,7 +1,7 @@
 {
  "assembly": {
   "bit_effective_saving": "338670433250477381/500000000000000000000",
-  "complex_saving": "3543063/5000000000",
+  "complex_saving": "7097667/10000000000",
   "kappa": "6768823/10000000000",
   "status": "Exact arithmetic; construction and finite bridge are explicit proof dependencies"
  },
@@ -9,32 +9,32 @@
  "complex_profile": {
   "W_per_vertex": 12052,
   "child_histogram": {
-   "1": 87210,
-   "10": 2700,
-   "11": 945,
-   "12": 3663,
-   "13": 495,
-   "14": 2118,
-   "15": 2910,
-   "16": 282,
-   "17": 198,
+   "1": 87528,
+   "10": 2514,
+   "11": 1059,
+   "12": 3423,
+   "13": 504,
+   "14": 2175,
+   "15": 1923,
+   "16": 243,
+   "17": 123,
    "18": 7920,
-   "19": 3186,
-   "2": 41430,
+   "19": 4209,
+   "2": 40644,
    "20": 66,
-   "3": 29061,
-   "4": 10662,
-   "5": 6765,
-   "6": 3840,
-   "7": 2349,
-   "8": 4209,
-   "9": 1539
+   "3": 29964,
+   "4": 11502,
+   "5": 5937,
+   "6": 4380,
+   "7": 1929,
+   "8": 4428,
+   "9": 1218
   },
   "deficit_per_vertex": 1320,
   "m": 66,
   "rank_per_vertex": 794112
  },
- "complex_saving": "3543063/5000000000",
+ "complex_saving": "7097667/10000000000",
  "contract_checks": {
   "all_centers_in_phase1": true,
   "all_fresh_columns_equal_identity": true,
@@ -63,26 +63,26 @@
  },
  "flow": {
   "child_histogram": {
-   "1": 87210,
-   "10": 2700,
-   "11": 945,
-   "12": 3663,
-   "13": 495,
-   "14": 2118,
-   "15": 2910,
-   "16": 282,
-   "17": 198,
+   "1": 87528,
+   "10": 2514,
+   "11": 1059,
+   "12": 3423,
+   "13": 504,
+   "14": 2175,
+   "15": 1923,
+   "16": 243,
+   "17": 123,
    "18": 7920,
-   "19": 3186,
-   "2": 41430,
+   "19": 4209,
+   "2": 40644,
    "20": 66,
-   "3": 29061,
-   "4": 10662,
-   "5": 6765,
-   "6": 3840,
-   "7": 2349,
-   "8": 4209,
-   "9": 1539
+   "3": 29964,
+   "4": 11502,
+   "5": 5937,
+   "6": 4380,
+   "7": 1929,
+   "8": 4428,
+   "9": 1218
   },
   "deficit": 1320,
   "loss": 440,
@@ -101,7 +101,7 @@
   },
   "audit_status": "PASS exact fused physical candidate, all formal source/target/dirty columns",
   "formal": {
-   "absolute_coefficient_bound": 462930,
+   "absolute_coefficient_bound": 408294,
    "base_exceeds_twice_bound": true,
    "columns": [
     {
@@ -127,8 +127,8 @@
    ],
    "packed_digit_bits": 24
   },
-  "frames": 17050,
-  "frames_changed_from_pr168": 5577,
+  "frames": 17787,
+  "frames_changed_from_pr168": 435,
   "mutations": {
    "early-read": "REJECTED",
    "missing-read": "REJECTED"
@@ -138,7 +138,7 @@
  "lift": {
   "all_actual_coefficients_dyadic": true,
   "max_denominator": "2",
-  "nodes": 18481,
+  "nodes": 18532,
   "physical_R": 9412,
   "status": "Exact rational local lifts and inverse gate programs verified; monotone flow and zero-fresh kernel reuse DAG verified."
  },
diff --git a/research/source-assisted-v4-layer/discovery/README.md b/research/source-assisted-v4-layer/discovery/README.md
index 393c9a31e4..d41b01d298 100644
--- a/research/source-assisted-v4-layer/discovery/README.md
+++ b/research/source-assisted-v4-layer/discovery/README.md
@@ -7,8 +7,10 @@ frames and pairs into a copy of the aligned tree and prices them with PR #184's
 
     SA_TREE=<PR202 tree> SA_PY=<python with numpy/scipy> python3 -B flow_frames.py --aligned <aligned tree> --seed-frames --keep out/
 
-With `--seed-frames` the descent starts from PR #168's physical frames (the aligned tree's own layer) and the flow
-root rises from 7.00918e-4 to 7.08613e-4; without seeding, 7.0691e-4; the light descent alone, 7.0419e-4. These
-numbers are float discovery screens; the certified value is in `../certificate.json`. `matching.py` and
+With `--seed-frames` the descent starts from PR #168's physical frames (the aligned tree's own layer). Float screens:
+unseeded light 7.0419e-4, unseeded full 7.0691e-4, seeded full 7.08613e-4, seeded light (`--light`) 7.0950e-4,
+seeded light with a movability mask (`DESCENT_LIGHT_ROUNDS=1 DESCENT_FREEZE_FRAC=0.98 DESCENT_SEED=8`, i.e. 98% of
+the operations frozen; `physical_opt.py`) 7.09767e-4, the layer certified in `../certificate.json`. The flow prefers
+layers close to PR #168's; the sweep over freeze fractions 0 to 0.98 is monotone. `matching.py` and
 `extend.py` are needed only to import `harness` (the matching itself is PR #168's frozen one here).
 Prepared by Chafik Boukhalfa with Anthropic Claude and OpenAI Codex assistance; Apache-2.0.
diff --git a/research/source-assisted-v4-layer/report.json b/research/source-assisted-v4-layer/report.json
index d95185c887..c4fa671314 100644
--- a/research/source-assisted-v4-layer/report.json
+++ b/research/source-assisted-v4-layer/report.json
@@ -20,7 +20,7 @@
     },
     "C1": 1,
     "a_bit": "338670433250477381/500000000000000000000",
-    "a_complex": "3543063/5000000000",
+    "a_complex": "7097667/10000000000",
     "alpha_squared_power": "1128909749279055278683696000159127/1667795568088256895719968174600000000",
     "beta": "1/1000000000",
     "c": "16933583823853435799522619/25000000000000000000000000000",
@@ -31,12 +31,12 @@
     "lambda_": "49966132957013622695150477381/50000000000000000000000000000",
     "lambda_prime": "24983066478676146564200477381/25000000000000000000000000000",
     "q": "16933521323853435799522619/25000000000000000000000000000",
-    "sigma": "4996456937/5000000000",
+    "sigma": "9992902333/10000000000",
     "tau": "499661329566749522619/500000000000000000000"
    },
    "recurrence": {
     "internal": "499661329566749522619/500000000000000000000",
-    "leaf": "4996456937003543063/5000000000000000000",
+    "leaf": "9992902333007097667/10000000000000000000",
     "reservations": "24983066416176146564200477381/25000000000000000000000000000"
    },
    "strict_constraints": {
@@ -55,11 +55,11 @@
     "c_below_one": "24983066416176146564200477381/25000000000000000000000000000",
     "c_positive": "16933583823853435799522619/25000000000000000000000000000",
     "cell_above_band": "1666666641661022159558715521400159127/3335591136176513791439936349200000000",
-    "compact_leaf": "781793658431249200477381/25000000000000000000000000000",
+    "compact_leaf": "810646158402396700477381/25000000000000000000000000000",
     "compact_phase_layer_above_kappa": "5512191445341671259318410221/83389778404412844785998408730000000000",
     "compact_reservations": "1/400000000",
-    "complex_above_bit": "15635866749522619/500000000000000000000",
-    "complex_below_one_over32": "152706937/5000000000",
+    "complex_above_bit": "16212916749522619/500000000000000000000",
+    "complex_below_one_over32": "305402333/10000000000",
     "coordinate_movement_above_kappa": "229283250477381/500000000000000000000",
     "delta_below_one_eighth": "99999999/800000000",
     "delta_positive": "1/800000000",
@@ -69,17 +69,17 @@
     "gamma_sublinear": "1/200000000",
     "guard_width": "5644590441284478599840873/8338977840441284478599840873",
     "lambda_above_internal": "338670433250477381/50000000000000000000000000000",
-    "lambda_above_sigma": "1563587013622695150477381/50000000000000000000000000000",
+    "lambda_above_sigma": "1621292013622695150477381/50000000000000000000000000000",
     "lambda_above_tau": "338670433250477381/50000000000000000000000000000",
     "lambda_prime_above_lambda": "338670433250477381/50000000000000000000000000000",
     "lambda_prime_below_one": "16933521323853435799522619/25000000000000000000000000000",
-    "leaf_saving_above_bit": "15635866395216319/500000000000000000000",
+    "leaf_saving_above_bit": "16212916394639269/500000000000000000000",
     "literal_scalar_guard": "1",
     "phase_boundary": "903126131627676134690061080159127/1334236454470605516575974539680000000",
     "phase_local": "4515664014049742438588219800159127/6671182272353027582879872698400000000",
     "prime_interval_packing": "5644590441284478599840873/8338977840441284478599840873",
     "q_below_internal": "338670433250477381/25000000000000000000000000000",
-    "q_below_leaf": "781793658431249200477381/25000000000000000000000000000",
+    "q_below_leaf": "810646158402396700477381/25000000000000000000000000000",
     "q_below_reservations": "1/400000000",
     "q_positive": "16933521323853435799522619/25000000000000000000000000000",
     "record_suffix": "5644590441284478599840873/8338977840441284478599840873",
@@ -140,37 +140,37 @@
    "counts": {
     "W": "12052",
     "delta": "1320",
-    "edge_count": "211548",
+    "edge_count": "211689",
     "histogram": {
-     "1": "87210",
-     "10": "2700",
-     "11": "945",
-     "12": "3663",
-     "13": "495",
-     "14": "2118",
-     "15": "2910",
-     "16": "282",
-     "17": "198",
+     "1": "87528",
+     "10": "2514",
+     "11": "1059",
+     "12": "3423",
+     "13": "504",
+     "14": "2175",
+     "15": "1923",
+     "16": "243",
+     "17": "123",
      "18": "7920",
-     "19": "3186",
-     "2": "41430",
+     "19": "4209",
+     "2": "40644",
      "20": "66",
-     "3": "29061",
-     "4": "10662",
-     "5": "6765",
-     "6": "3840",
-     "7": "2349",
-     "8": "4209",
-     "9": "1539"
+     "3": "29964",
+     "4": "11502",
+     "5": "5937",
+     "6": "4380",
+     "7": "1929",
+     "8": "4428",
+     "9": "1218"
     },
     "m": 66,
     "mass": "794112",
     "maxchild": 20
    },
-   "ideal_moment_upper": "176218413830235110364374820995641064426111/176218413857197867102603509595485841129472",
-   "saving": "3543063/5000000000",
-   "strict_gap_lower": "26962756738228688599844776703361/176218413857197867102603509595485841129472",
-   "strict_gap_lower_decimal": 1.5300760089736423e-10
+   "ideal_moment_upper": "352436827651667193920281873504433433238375/352436827714395734205207019190971682258944",
+   "saving": "7097667/10000000000",
+   "strict_gap_lower": "62728540284925145686538249020569/352436827714395734205207019190971682258944",
+   "strict_gap_lower_decimal": 1.779852028850926e-10
   },
   "construction_receipts": {
    "complex_endpoint_checks": {
@@ -199,7 +199,7 @@
      "18": 1320
     }
    },
-   "complex_lift_certificate_sha256": "0b3bcb16cdab2e5fde6a15868bd671651586a99710918342beab72ad90990deb"
+   "complex_lift_certificate_sha256": "a2a9ffa85df587ab61a2effe12b54e6321f9b9103617bfc494827a7d0e9c5027"
   },
   "finite_bridge": {
    "proof": "FINITE_BRIDGE.txt",
@@ -236,7 +236,7 @@
   "kappa_decimal": 0.0006768823,
   "source_sha256": {
    "bit_certificate": "853afa75980e11ce16b333c28311c4f635e84065867c049fda9916c2419091ed",
-   "complex": "40fe51ff725af94258ab8e231696feb84494c77bfaa3cef952a566659a0950fa"
+   "complex": "65c3f8a88e44545d8984b4e0af7e86a096547e876c5797089fef7bdead8c1853"
   },
   "status": "Exact arithmetic; construction and finite bridge are explicit proof dependencies"
  },
@@ -244,33 +244,33 @@
  "complex_profile": {
   "W_per_vertex": 12052,
   "child_histogram": {
-   "1": 87210,
-   "10": 2700,
-   "11": 945,
-   "12": 3663,
-   "13": 495,
-   "14": 2118,
-   "15": 2910,
-   "16": 282,
-   "17": 198,
+   "1": 87528,
+   "10": 2514,
+   "11": 1059,
+   "12": 3423,
+   "13": 504,
+   "14": 2175,
+   "15": 1923,
+   "16": 243,
+   "17": 123,
    "18": 7920,
-   "19": 3186,
-   "2": 41430,
+   "19": 4209,
+   "2": 40644,
    "20": 66,
-   "3": 29061,
-   "4": 10662,
-   "5": 6765,
-   "6": 3840,
-   "7": 2349,
-   "8": 4209,
-   "9": 1539
+   "3": 29964,
+   "4": 11502,
+   "5": 5937,
+   "6": 4380,
+   "7": 1929,
+   "8": 4428,
+   "9": 1218
   },
   "deficit_per_vertex": 1320,
   "m": 66,
   "rank_per_vertex": 794112
  },
- "complex_saving": "3543063/5000000000",
- "complex_saving_float": 0.0007086126,
+ "complex_saving": "7097667/10000000000",
+ "complex_saving_float": 0.0007097667,
  "contract_checks": {
   "all_centers_in_phase1": true,
   "all_fresh_columns_equal_identity": true,
@@ -305,106 +305,105 @@
    "1:3": 2016,
    "1:5": 154,
    "1:6": 88,
-   "2:10": 95,
+   "2:10": 74,
    "2:11": 5,
-   "2:12": 54,
+   "2:12": 57,
    "2:13": 11,
-   "2:14": 73,
-   "2:15": 30,
-   "2:16": 50,
-   "2:17": 66,
-   "2:19": 1,
+   "2:14": 92,
+   "2:15": 45,
+   "2:16": 37,
+   "2:17": 41,
    "2:3": 1548,
    "2:4": 503,
    "2:5": 396,
    "2:6": 94,
    "2:7": 23,
-   "2:8": 219,
+   "2:8": 241,
    "2:9": 5
   },
   "child_histogram": {
-   "1": 87210,
-   "10": 2700,
-   "11": 945,
-   "12": 3663,
-   "13": 495,
-   "14": 2118,
-   "15": 2910,
-   "16": 282,
-   "17": 198,
+   "1": 87528,
+   "10": 2514,
+   "11": 1059,
+   "12": 3423,
+   "13": 504,
+   "14": 2175,
+   "15": 1923,
+   "16": 243,
+   "17": 123,
    "18": 7920,
-   "19": 3186,
-   "2": 41430,
+   "19": 4209,
+   "2": 40644,
    "20": 66,
-   "3": 29061,
-   "4": 10662,
-   "5": 6765,
-   "6": 3840,
-   "7": 2349,
-   "8": 4209,
-   "9": 1539
+   "3": 29964,
+   "4": 11502,
+   "5": 5937,
+   "6": 4380,
+   "7": 1929,
+   "8": 4428,
+   "9": 1218
   },
   "deficit": 1320,
-  "edge_basis_values": 42250,
-  "edge_distinct_values": 42250,
+  "edge_basis_values": 42301,
+  "edge_distinct_values": 42301,
   "edge_nullity_histogram": {
-   "0": 41199
+   "0": 41194
   },
   "edge_removed_values": 0,
-  "entropy": 1860968.6507875004,
-  "entropy_ratio": 0.8204358133457282,
+  "entropy": 1857937.9138443007,
+  "entropy_ratio": 0.8190996677164191,
   "field_prime": 1000003,
-  "frames": 18481,
+  "frames": 18532,
   "h": 22,
-  "incoming_dependencies": 2,
-  "kernel_reuses": 1,
+  "incoming_dependencies": 1,
+  "kernel_reuses": 0,
   "local_histogram": {
-   "1": 23790,
-   "10": 900,
-   "11": 315,
-   "12": 1221,
-   "13": 165,
-   "14": 706,
-   "15": 970,
-   "16": 94,
-   "17": 66,
-   "19": 1062,
-   "2": 11610,
+   "1": 23896,
+   "10": 838,
+   "11": 353,
+   "12": 1141,
+   "13": 168,
+   "14": 725,
+   "15": 641,
+   "16": 81,
+   "17": 41,
+   "19": 1403,
+   "2": 11348,
    "20": 22,
-   "3": 9687,
-   "4": 3554,
-   "5": 2255,
-   "6": 1280,
-   "7": 783,
-   "8": 1403,
-   "9": 513
+   "3": 9988,
+   "4": 3834,
+   "5": 1979,
+   "6": 1460,
+   "7": 643,
+   "8": 1476,
+   "9": 406
   },
   "loss": 440,
   "new_R": 9412,
   "new_W": 12052,
-  "numerical_local_root": 0.0007086126653756381,
+  "numerical_local_root": 0.0007097667761553668,
   "original_physical_roles": 13372,
-  "original_role_crossings": 46211,
-  "outgoing_dependencies": 8094,
+  "original_role_crossings": 46261,
+  "outgoing_dependencies": 8093,
   "retirement_histogram": {
-   "1:13": 66,
+   "1:12": 19,
    "1:20": 198,
-   "1:3": 101,
-   "2:13": 22,
-   "2:14": 23,
-   "2:15": 66,
-   "2:16": 22,
-   "2:17": 418,
-   "2:18": 166,
+   "1:3": 413,
+   "2:12": 28,
+   "2:14": 1,
+   "2:15": 22,
+   "2:16": 27,
+   "2:17": 179,
+   "2:18": 165,
    "2:19": 3410,
    "2:20": 1320,
    "2:21": 2640,
-   "2:3": 961
+   "2:3": 990
   },
   "source_control_erasures": 0,
   "status": "Modular equal-frame flow price, retaining dirty kernel lifts and complete full-frame cleanup. Not an exact supplier certificate.",
   "v": 1320,
-  "vectors": 26199
+  "vectors": 26200
  },
  "kappa": "6768823/10000000000",
  "kappa_float": 0.0006768823,
@@ -418,7 +417,7 @@
    },
    "foreign_producer_replays": 0,
    "formal": {
-    "absolute_coefficient_bound": 462930,
+    "absolute_coefficient_bound": 408294,
     "base_exceeds_twice_bound": true,
     "columns": [
      {
@@ -448,10 +447,10 @@
     "baseline.json.gz": "33187d4011bc6dd11468408c0bc7e968d54c63aadf87a09592da056140375d57",
     "frames.json.gz": "06faf083a88b3b4cb37c4fbdb62b5ca48f6c493894c5391bf96fd5727742e4fd",
     "graph.json.gz": "aa1708d0525f33a8b1a5757ea5f05ba161cb44f71b4e92971239ad1dfb469ad3",
-    "physical-frames.json.gz": "1bd2799e6f4c7fc752c0c14aa1412368737758d6d5a5ec1fde5d590a29398084",
-    "physical-pairs.json.gz": "d718d50378f61dea483e3022b275edf7c2335839ba65aacd55d24279c1757c42",
+    "physical-frames.json.gz": "9b5c2b2b2a3dc9be05a334bf2e9edd71bb4a93127296995ca25d8b61915f43f0",
+    "physical-pairs.json.gz": "ff154310a86fce1f675dcbd4ed5289269b5439b087635e589a5123fd2896615c",
     "profile-before.json.gz": "f60146c8d8c5e8aa0365f08b68aa6e03403656b74bf28b7058ef5a241560bf92",
-    "profile.json.gz": "264bb90158103b609620da3dc96e68121466cccf0bc9db2c845680c209b901ce",
+    "profile.json.gz": "dcdb30264c13ec9dfd5e190f5514686f106268306468e8e603381a84b8d3fcd8",
     "word.json.gz": "ec94f10d8c8d198d37cbe4781c8c251a8e62335e53ad5177f21cf59ad969a6d8"
    },
    "mutations": {
@@ -461,7 +460,7 @@
    "physical": {
     "R": 15682,
     "W_per_vertex": 16012,
-    "changed_operation_frames": 17050,
+    "changed_operation_frames": 17787,
     "checks": {
      "donor_frames_inside_gauges": true,
      "pair_chronology": true,
@@ -471,50 +470,50 @@
      "value_spans_inside_frames": true
     },
     "child_histogram": {
-     "1": 110970,
-     "10": 2700,
-     "11": 945,
-     "12": 3663,
-     "13": 495,
-     "14": 2118,
-     "15": 2910,
-     "16": 282,
-     "17": 198,
+     "1": 111288,
+     "10": 2514,
+     "11": 1059,
+     "12": 3423,
+     "13": 504,
+     "14": 2175,
+     "15": 1923,
+     "16": 243,
+     "17": 123,
      "18": 7920,
-     "19": 3186,
-     "2": 41430,
+     "19": 4209,
+     "2": 40644,
      "20": 11946,
-     "3": 29061,
-     "4": 10662,
-     "5": 6765,
-     "6": 3840,
-     "7": 2349,
-     "8": 4209,
-     "9": 1539
+     "3": 29964,
+     "4": 11502,
+     "5": 5937,
+     "6": 4380,
+     "7": 1929,
+     "8": 4428,
+     "9": 1218
     },
     "deficit_per_vertex": 1320,
     "h": 22,
-    "late_pairs": 1399,
+    "late_pairs": 1664,
     "local_histogram": {
-     "1": 31710,
-     "10": 900,
-     "11": 315,
-     "12": 1221,
-     "13": 165,
-     "14": 706,
-     "15": 970,
-     "16": 94,
-     "17": 66,
-     "19": 1062,
-     "2": 11610,
+     "1": 31816,
+     "10": 838,
+     "11": 353,
+     "12": 1141,
+     "13": 168,
+     "14": 725,
+     "15": 641,
+     "16": 81,
+     "17": 41,
+     "19": 1403,
+     "2": 11348,
      "20": 3982,
-     "3": 9687,
-     "4": 3554,
-     "5": 2255,
-     "6": 1280,
-     "7": 783,
-     "8": 1403,
-     "9": 513
+     "3": 9988,
+     "4": 3834,
+     "5": 1979,
+     "6": 1460,
+     "7": 643,
+     "8": 1476,
+     "9": 406
     },
     "loss": 440,
     "m": 66,
@@ -594,14 +593,14 @@
    },
    "status": "PASS exact fused physical candidate, all formal source/target/dirty columns"
   },
-  "frames": 17050,
+  "frames": 17787,
   "pairs": 2310
  },
  "lift": {
   "all_actual_coefficients_dyadic": true,
-  "exact_scalar_program_sha256": "c33c86388f2d4e6627e2d3e5a762f8cba47eaff8b43166c4c8d31bdc155663d3",
+  "exact_scalar_program_sha256": "2524002e0f4407f1843a70a24d0459b02844c17fd809aa2b9b3784f3f6222379",
   "max_denominator": "2",
-  "nodes": 18481,
+  "nodes": 18532,
   "physical_R": 9412,
   "status": "Exact rational local lifts and inverse gate programs verified; monotone flow and zero-fresh kernel reuse DAG verified."
  },
diff --git a/research/source-assisted-v4-layer/witness/kernel-pairs.json b/research/source-assisted-v4-layer/witness/kernel-pairs.json
index 55462cbf0c..e889b7b95c 100644
--- a/research/source-assisted-v4-layer/witness/kernel-pairs.json
+++ b/research/source-assisted-v4-layer/witness/kernel-pairs.json
@@ -1 +1 @@
-{"description":"Kernel pairs of the frame flow on this layer (from complex_frame_flow.py --witness)","kernel_pairs":[[9381,0,9092,0]]}
\ No newline at end of file
+{"description":"Kernel pairs of the frame flow on this layer (from complex_frame_flow.py --witness)","kernel_pairs":[]}
\ No newline at end of file
diff --git a/research/source-assisted-v4-layer/witness/physical-frames.json b/research/source-assisted-v4-layer/witness/physical-frames.json
index f27091abd7..eb7d5841d2 100644
--- a/research/source-assisted-v4-layer/witness/physical-frames.json
+++ b/research/source-assisted-v4-layer/witness/physical-frames.json
@@ -1 +1 @@
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diff --git a/research/source-assisted-v4-layer/witness/physical-pairs.json b/research/source-assisted-v4-layer/witness/physical-pairs.json
index 133818fb8e..f25f6d01d4 100644
--- a/research/source-assisted-v4-layer/witness/physical-pairs.json
+++ b/research/source-assisted-v4-layer/witness/physical-pairs.json
@@ -1 +1 @@
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From 109a857a329d18ed5552d5573f17ddfa886ae57f Mon Sep 17 00:00:00 2001
From: Chafik Boukhalfa <chafik.boukhalfa@gmail.com>
Date: Sat, 10 Oct 2026 02:40:40 +0400
Subject: [PATCH 6/6] Pairs only: PR168's frames unchanged, PR200's
 maximum-weight reuse pairing; complex saving 3549537/5000000000 = 7.0991e-4
 (+1.28% over PR194)

The gain is the reuse pairing, not the frames: with PR168's operation frames
byte for byte and pairs_mw.py's maximum-weight late compensated pairing the
flow root is 7.09907e-4; single frame moves never help (flow_moves.py screen).
Same verification chain; docs rewritten for the pairs-only claim.
---
 research/source-assisted-v4-layer/PROOF.md    |  75 +++---
 research/source-assisted-v4-layer/README.md   |  23 +-
 research/source-assisted-v4-layer/SOURCE.json |   4 +-
 .../source-assisted-v4-layer/certificate.json |  58 ++---
 .../discovery/README.md                       |  24 +-
 research/source-assisted-v4-layer/report.json | 219 +++++++++---------
 .../witness/physical-frames.json              |   2 +-
 .../witness/physical-pairs.json               |   2 +-
 8 files changed, 205 insertions(+), 202 deletions(-)

diff --git a/research/source-assisted-v4-layer/PROOF.md b/research/source-assisted-v4-layer/PROOF.md
index 4601fdc0c5..ebc8e84f6a 100644
--- a/research/source-assisted-v4-layer/PROOF.md
+++ b/research/source-assisted-v4-layer/PROOF.md
@@ -1,33 +1,38 @@
-# A stronger complex supplier: our physical layer on the source-assisted v4 word
+# A stronger complex supplier: PR200's reuse pairing on the source-assisted v4 word
 
 **Claim.** The source-assisted complex word of PR #184 (icekylinx) on the PR #168 v4 modules (PR #193/#194,
-ikeboy), with every input of PR #194's pipeline unchanged except the physical layer, certifies the complex saving
+ikeboy), with every input of PR #194's pipeline unchanged **except the 2,310 reuse pairs** of its physical layer
+(PR #168's operation frames are kept byte for byte), certifies the complex saving
 
-    a_C = 7097667 / 10000000000 = 7.0976670·10⁻⁴      (PR #194: 219037/312500000 = 7.0091844·10⁻⁴, +1.26%).
+    a_C = 3549537/5000000000 = 7.0990740·10⁻⁴      (PR #194: 219037/312500000 = 7.0091844·10⁻⁴, +1.28%).
 
 Composed with PR #200's bit supplier through PR #194's unchanged 47-constraint assembly the result is
 κ = 6768823/10¹⁰, the same as PR #202, because the bit branch binds (effective 6.7734·10⁻⁴). **No new κ.** The
-complex cap of every composition on this lineage rises from 7.0092·10⁻⁴ to 7.0977·10⁻⁴, and the exact-DFT
+complex cap of every composition on this lineage rises from 7.0092·10⁻⁴ to 7.0990740·10⁻⁴, and the exact-DFT
 exponent saving of `research/exact-dft-transfer` (PR #225) rises with it.
 
 ## What changes
 
 PR #194's pipeline is: regenerate the PR #168 v4 complex cache (`scripts/paired_cube_producer.py`); the
 source-parity rewrite of the three query modules (`source_aligned_local_v4.py`), which also runs PR #168's physical
-layer (endpoint descent and reuse pairs) and writes it to `references/paired-cube/physical/{frames,pairs}.json` of
-the aligned tree; the equal-frame fresh-channel compression (`complex_frame_flow.py`), which reads that physical
-layer; the exact arbitrary-dirty lifts (`exact_complex_flow_lift.py`); the contract (`contract_v4.py`); and the
-assembly. The flow's child histogram depends on the physical layer it receives: the frames decide which channels are
-equal-frame and compress.
+layer (endpoint descent and late compensated reuse pairs) and writes it to
+`references/paired-cube/physical/{frames,pairs}.json` of the aligned tree; the equal-frame fresh-channel
+compression (`complex_frame_flow.py`), which reads that physical layer; the exact arbitrary-dirty lifts
+(`exact_complex_flow_lift.py`); the contract (`contract_v4.py`); and the assembly.
 
-This package replaces only that physical layer. The frames and reuse pairs in `witness/` are produced by PR #200's
-descent (`discovery/`): one round of endpoint moves of single operations, started from PR #168's frames on the
-aligned word's own cache, with 98% of the operations frozen (seed 8 of the movability mask): 435 of PR #168's 17,789
-operation frames change; the 2,310 reuse pairs are recomputed by the same late compensated pairing. The flow prefers
-layers close to PR #168's: the full descent cycle gives 7.0861·10⁻⁴, one light round 7.0950·10⁻⁴, and freezing most
-operations 7.0977·10⁻⁴ (`discovery/README.md`). The matching arcs, modules, local circuit,
-gauges, aligned rewrite, flow options and kernel pairs are PR #194's (the kernel pairs the flow finds on this layer
-are recorded in `witness/kernel-pairs.json`, as PR #194 records its own).
+The flow's child histogram depends on the reuse pairs it receives: a pair identifies a dead ungauged donor's
+register with a gauged recipient, and the flow's equal-frame compression follows those identifications. This
+package replaces only the pairing. PR #168's pairing is its maximum matching of donors to recipients in a fixed
+order; the pairing in `witness/physical-pairs.json` is PR #200's late compensated pairing (`discovery/`,
+`physical_opt.py:pairs`) with the maximum-weight pairing of `pairs_mw.py` (transversal-matroid greedy with augmenting paths, donor weight the gain heuristic of PR #200): the same legality (donor ungauged, non-root, dead before the recipient's
+first operation, donor end frame inside the recipient gauge), a maximum matching, but a different selection among
+the maximum matchings. The operation frames are PR #168's own (0 of 17,789 changed); the matching arcs, modules,
+local circuit, gauges, aligned rewrite and flow options are PR #194's.
+
+Discovery record (`discovery/README.md`, float flow roots): PR #168's pairing 7.00918·10⁻⁴; PR #200's default
+pairing 7.09765·10⁻⁴; the maximum-weight pairing of `pairs_mw.py` (transversal-matroid greedy with augmenting paths, donor weight the gain heuristic of PR #200) 7.0990740·10⁻⁴. Random pairings fall to 6.6·10⁻⁴, and single operation-frame moves on
+top of PR #168's frames never raise the flow root (a screen over every endpoint move), so the frames are left
+untouched.
 
 ## What is checked (`verify.py`, about 80 seconds, Python 3.11+ with numpy/scipy for the regeneration steps)
 
@@ -37,36 +42,32 @@ are recorded in `witness/kernel-pairs.json`, as PR #194 records its own).
    `SOURCE.json`; the three witness files are pinned too. Reviewers can confirm with `git show 8d8d67b:<path> |
    sha256sum`.
 2. **Regeneration.** The producer and the aligned rewrite run unchanged, including PR #168's physical layer and its
-   own physical scalar/frame checks on that layer.
-3. **Admission of this layer.** With the package's frames and pairs installed, PR #200's complete complex checker
-   (`complex/physical.py` with `verify_fused.py`) re-executes the aligned word: every descended frame contains its
-   node span and is nondegenerate, every role chain nests (entrance, operations, root frame, full space), every alias
-   is read at its deadline after its donor's last operation, the target read chronology holds, the exact adjoint is
-   rederived, and every formal source, target and dirty column is replayed in both shear signs on an injective
-   packed basis with an independently derived coefficient bound. The ledger it re-counts is the candidate's
-   profile.
+   own physical scalar/frame checks on that layer; the frames this package installs are byte for byte that layer's.
+3. **Admission of this pairing.** With the package's pairs installed, PR #200's complete complex checker
+   (`complex/physical.py` with `verify_fused.py`) re-executes the aligned word: every frame contains its node span
+   and is nondegenerate, every role chain nests (entrance, operations, root frame, full space), every alias is read
+   at its deadline after its donor's last operation with the donor's end frame inside the recipient gauge, the
+   target read chronology holds, the exact adjoint is rederived, and every formal source, target and dirty column
+   is replayed in both shear signs on an injective packed basis with an independently derived coefficient bound.
 4. **Flow, lift, contract.** PR #184's flow runs on this layer with the package's kernel pairs; the exact lift
    verifies every local map and inverse gate program over the rationals (all coefficients dyadic) and the monotone,
    acyclic once-only reuse DAG with frame containment; the contract certifies the fresh columns, read chronology,
    target chains and the normalized profile (13 checks), with the package's physical and kernel pairs as the frozen
    witnesses that PR #194's contract compares against.
 5. **Assembly.** PR #194's `assemble.py` composes the certified complex profile with PR #200's bit certificate and
-   reproduces `certificate.json` exactly: complex saving 7097667/10000000000, κ 6768823/10¹⁰, the next grid point
-   rejected by the unchanged assembly.
+   reproduces `certificate.json` exactly (exact fields only; the full run record is `report.json`): complex saving
+   3549537/5000000000, κ 6768823/10¹⁰, the next grid point rejected by the unchanged assembly.
 
-The complex profile: m = 66, W = 12,052 roles per vertex, rank mass 794,112, deficit 1,320, largest child 20, the
-same W, rank and deficit as PR #194; only the child histogram changes (the flow compresses the same number of
-roles, 13,372 physical to 9,412, but splits the remaining increments differently).
+The complex profile keeps PR #194's m = 66, W = 12,052 roles per vertex, rank mass 794,112 and deficit 1,320; only
+the child histogram changes (the flow compresses the same 13,372 physical roles to 9,412 but splits the remaining
+increments differently).
 
 ## Scope and limits
 
 - Conditional exactly as PR #194: the retained analytic, uniform-recursion, precision, weighted bit frames,
   fixed-tape routing, restored-row and all-size semantic interfaces remain hypotheses; the scalar replays and exact
-  lifts are finite checks. The independent complex scalar transcript of PR #184 is not re-exported here; the
-  one-column identity is replayed by PR #200's checker on the pre-flow word and by the lift's inverse gate programs.
+  lifts are finite checks. PR #184's independent complex scalar transcript is not re-exported here; the one-column
+  identity is replayed by PR #200's checker on the pre-flow word and by the lift's inverse gate programs.
 - Nothing here changes κ; the bit supplier binds. It raises the complex cap for every bit-side construction on
-  this lineage (for example the banked rows of #205–#223 and the conditional rungs of #219/#224) and the exact-DFT
-  exponent.
-- The descent is a discovery heuristic; no optimality is claimed. The float screen varies with the descent options
-  (unseeded light 7.0419·10⁻⁴, unseeded full 7.0691·10⁻⁴, seeded full 7.0861·10⁻⁴, seeded light 7.0950·10⁻⁴,
-  seeded light with 98% frozen 7.0977·10⁻⁴); only the last layer is certified here.
+  this lineage and the exact-DFT exponent.
+- The pairing is a discovery heuristic; no optimality among maximum matchings is claimed.
diff --git a/research/source-assisted-v4-layer/README.md b/research/source-assisted-v4-layer/README.md
index 79044ad8f2..4ca3773c66 100644
--- a/research/source-assisted-v4-layer/README.md
+++ b/research/source-assisted-v4-layer/README.md
@@ -1,13 +1,13 @@
 # source-assisted-v4-layer
 
-**Complex saving 7.0977·10⁻⁴ (no new κ).** PR #194's source-assisted complex word with this package's physical
-layer (one round of PR #200's descent from PR #168's frames with 98% of the operations frozen) certifies
+**Complex saving 7.0990740·10⁻⁴ (no new κ).** PR #194's source-assisted complex word with PR #200's reuse pairing in
+place of PR #168's (operation frames unchanged) certifies
 
-    a_C = 7097667/10000000000 = 7.0976670·10⁻⁴       (PR #194: 7.0091844·10⁻⁴, +1.26%),
+    a_C = 3549537/5000000000 = 7.0990740·10⁻⁴       (PR #194: 7.0091844·10⁻⁴, +1.28%),
 
 through PR #184's flow, exact lift and contract and PR #194's assembly, which with PR #200's bit supplier gives
 κ = 6768823/10¹⁰ (the bit branch binds, as in PR #202). The complex cap of this lineage and the exact-DFT exponent
-saving (`research/exact-dft-transfer`) rise to 7.0977·10⁻⁴. [PROOF.md](PROOF.md) states what changes and what is
+saving (`research/exact-dft-transfer`) rise to 7.0990740·10⁻⁴. [PROOF.md](PROOF.md) states what changes and what is
 checked.
 
 ## Verify
@@ -17,13 +17,14 @@ checked.
 
 The script rebuilds the 308 pinned files of PR #202 (`8d8d67b`) from `baseline-pr202.tar.gz` after checking every
 sha256 in `SOURCE.json`, regenerates the PR #168 v4 cache and the aligned word, installs the physical layer of
-`witness/`, admits it with PR #200's complete complex checker, runs the frame flow with the package's kernel pairs,
-the exact lift, the contract (with the package's physical and kernel pairs as the frozen witnesses) and the assembly,
-and compares with `certificate.json`. `--write` regenerates the certificate; `--temp-root` chooses scratch storage.
+`witness/` (PR #168's frames, this package's pairs), admits it with PR #200's complete complex checker, runs the
+frame flow with the package's kernel pairs, the exact lift, the contract (with the package's physical and kernel
+pairs as the frozen witnesses) and the assembly, and compares with `certificate.json`. `--write` regenerates the
+certificate; `--temp-root` chooses scratch storage.
 
 ## Files
 
-`verify.py`, `certificate.json`, `SOURCE.json`, `baseline-pr202.tar.gz`, `witness/physical-frames.json`,
-`witness/physical-pairs.json`, `witness/kernel-pairs.json`, `discovery/` (the descent that found the layer; not run
-by the verifier), `PROOF.md`, `NOTICE`. `.github/workflows/source-assisted-v4-layer.yml` runs the verifier on
-Python 3.11 and 3.13.
+`verify.py`, `certificate.json` (exact fields), `report.json` (full run record), `SOURCE.json`,
+`baseline-pr202.tar.gz`, `witness/physical-frames.json`, `witness/physical-pairs.json`, `witness/kernel-pairs.json`,
+`discovery/` (the pairing and the sweep that found it; not run by the verifier), `PROOF.md`, `NOTICE`.
+`.github/workflows/source-assisted-v4-layer.yml` runs the verifier on Python 3.11 and 3.13.
diff --git a/research/source-assisted-v4-layer/SOURCE.json b/research/source-assisted-v4-layer/SOURCE.json
index 71ea3c9ed2..dca4742101 100644
--- a/research/source-assisted-v4-layer/SOURCE.json
+++ b/research/source-assisted-v4-layer/SOURCE.json
@@ -317,7 +317,7 @@
  "repository": "CrocSwap/integer-mult-bounds",
  "witness": {
   "kernel-pairs.json": "e33b753fc4519170373207b0527b9cb53e0b552e2a27f48f72381a881c7a227f",
-  "physical-frames.json": "788081f575344edd374f33dd8ea7083093e0f0367706da148094557956ec65d3",
-  "physical-pairs.json": "33bb02f9e951fd4ff88810a98f0b1fa7c6f6d2ca68b5b47818cd030f7acecafa"
+  "physical-frames.json": "1a6c215cdc58216f7ec96fb898a471e25d094fd86de085f6c06e86fd8997bfad",
+  "physical-pairs.json": "e6409fffbf5f3b1ac6434acf2e426503de3a8a18537b8fee761fc4fc931fb896"
  }
 }
diff --git a/research/source-assisted-v4-layer/certificate.json b/research/source-assisted-v4-layer/certificate.json
index 96208db991..796efbb0d4 100644
--- a/research/source-assisted-v4-layer/certificate.json
+++ b/research/source-assisted-v4-layer/certificate.json
@@ -1,7 +1,7 @@
 {
  "assembly": {
   "bit_effective_saving": "338670433250477381/500000000000000000000",
-  "complex_saving": "7097667/10000000000",
+  "complex_saving": "3549537/5000000000",
   "kappa": "6768823/10000000000",
   "status": "Exact arithmetic; construction and finite bridge are explicit proof dependencies"
  },
@@ -9,32 +9,32 @@
  "complex_profile": {
   "W_per_vertex": 12052,
   "child_histogram": {
-   "1": 87528,
-   "10": 2514,
+   "1": 87534,
+   "10": 2415,
    "11": 1059,
    "12": 3423,
    "13": 504,
    "14": 2175,
-   "15": 1923,
+   "15": 1815,
    "16": 243,
-   "17": 123,
+   "17": 117,
    "18": 7920,
-   "19": 4209,
-   "2": 40644,
+   "19": 4323,
+   "2": 40641,
    "20": 66,
    "3": 29964,
-   "4": 11502,
-   "5": 5937,
-   "6": 4380,
-   "7": 1929,
-   "8": 4428,
-   "9": 1218
+   "4": 11517,
+   "5": 5928,
+   "6": 4473,
+   "7": 1926,
+   "8": 4434,
+   "9": 1212
   },
   "deficit_per_vertex": 1320,
   "m": 66,
   "rank_per_vertex": 794112
  },
- "complex_saving": "7097667/10000000000",
+ "complex_saving": "3549537/5000000000",
  "contract_checks": {
   "all_centers_in_phase1": true,
   "all_fresh_columns_equal_identity": true,
@@ -63,26 +63,26 @@
  },
  "flow": {
   "child_histogram": {
-   "1": 87528,
-   "10": 2514,
+   "1": 87534,
+   "10": 2415,
    "11": 1059,
    "12": 3423,
    "13": 504,
    "14": 2175,
-   "15": 1923,
+   "15": 1815,
    "16": 243,
-   "17": 123,
+   "17": 117,
    "18": 7920,
-   "19": 4209,
-   "2": 40644,
+   "19": 4323,
+   "2": 40641,
    "20": 66,
    "3": 29964,
-   "4": 11502,
-   "5": 5937,
-   "6": 4380,
-   "7": 1929,
-   "8": 4428,
-   "9": 1218
+   "4": 11517,
+   "5": 5928,
+   "6": 4473,
+   "7": 1926,
+   "8": 4434,
+   "9": 1212
   },
   "deficit": 1320,
   "loss": 440,
@@ -101,7 +101,7 @@
   },
   "audit_status": "PASS exact fused physical candidate, all formal source/target/dirty columns",
   "formal": {
-   "absolute_coefficient_bound": 408294,
+   "absolute_coefficient_bound": 603018,
    "base_exceeds_twice_bound": true,
    "columns": [
     {
@@ -127,8 +127,8 @@
    ],
    "packed_digit_bits": 24
   },
-  "frames": 17787,
-  "frames_changed_from_pr168": 435,
+  "frames": 17789,
+  "frames_changed_from_pr168": 0,
   "mutations": {
    "early-read": "REJECTED",
    "missing-read": "REJECTED"
diff --git a/research/source-assisted-v4-layer/discovery/README.md b/research/source-assisted-v4-layer/discovery/README.md
index d41b01d298..339adb1c76 100644
--- a/research/source-assisted-v4-layer/discovery/README.md
+++ b/research/source-assisted-v4-layer/discovery/README.md
@@ -1,16 +1,18 @@
 # Discovery (not run by verify.py)
 
-`flow_frames.py` runs this package's descent (`harness.descend`: `physical_opt.py` endpoint moves of single
-operations, `bundles.py` equal-frame components, `joint.py`/`deep.py` connected bundles, `pairs_mw.py` reuse
-pairing; PR #200's physical layer code) on the source-aligned cache of PR #194's pipeline, writes the resulting
-frames and pairs into a copy of the aligned tree and prices them with PR #184's `complex_frame_flow.py`:
+The physical layer of the certified package is PR #168's operation frames with PR #200's late compensated reuse
+pairing (`physical_opt.py:pairs`, the maximum-weight pairing of `pairs_mw.py` (transversal-matroid greedy with augmenting paths, donor weight the gain heuristic of PR #200)). `flow_frames.py` runs `harness.descend` on the source-aligned cache of
+PR #194's pipeline with all frame moves frozen (`DESCENT_LIGHT_ROUNDS=1 DESCENT_FREEZE_FRAC=1.0 --light
+--seed-frames`), recomputes the pairs and prices the layer with PR #184's `complex_frame_flow.py`:
 
-    SA_TREE=<PR202 tree> SA_PY=<python with numpy/scipy> python3 -B flow_frames.py --aligned <aligned tree> --seed-frames --keep out/
+    SA_TREE=<PR202 tree> SA_PY=<python with numpy/scipy> DESCENT_LIGHT_ROUNDS=1 DESCENT_FREEZE_FRAC=1.0 \
+      python3 -B flow_frames.py --aligned <aligned tree> --seed-frames --light --mw 1 --keep out/
 
-With `--seed-frames` the descent starts from PR #168's physical frames (the aligned tree's own layer). Float screens:
-unseeded light 7.0419e-4, unseeded full 7.0691e-4, seeded full 7.08613e-4, seeded light (`--light`) 7.0950e-4,
-seeded light with a movability mask (`DESCENT_LIGHT_ROUNDS=1 DESCENT_FREEZE_FRAC=0.98 DESCENT_SEED=8`, i.e. 98% of
-the operations frozen; `physical_opt.py`) 7.09767e-4, the layer certified in `../certificate.json`. The flow prefers
-layers close to PR #168's; the sweep over freeze fractions 0 to 0.98 is monotone. `matching.py` and
-`extend.py` are needed only to import `harness` (the matching itself is PR #168's frozen one here).
+Float flow roots: PR #168's pairing 7.00918e-4; PR #200's default pairing (donors ordered by the gain heuristic)
+7.09765e-4; donors ordered by widest end frame with recipients in reverse chronology (`PAIR_ORDER=wide
+PAIR_RECIPIENTS=reverse`) 7.09882e-4; the maximum-weight pairing (`--mw 1`, `pairs_mw.py`, every donor-weight variant
+tried) 7.09907e-4, the certified layer; random donor orders 6.4e-4 to 6.6e-4. With frames also descended: full cycle
+7.08613e-4, one light round 7.0950e-4, 98% of operations frozen 7.09767e-4; a screen over every single endpoint move
+on top of PR #168's frames (`flow_moves.py`) finds no improving move, so the frames are left untouched. The matching
+itself is PR #168's frozen carrier matching (`matching.py`, `extend.py` are needed only to import `harness`).
 Prepared by Chafik Boukhalfa with Anthropic Claude and OpenAI Codex assistance; Apache-2.0.
diff --git a/research/source-assisted-v4-layer/report.json b/research/source-assisted-v4-layer/report.json
index c4fa671314..c779422124 100644
--- a/research/source-assisted-v4-layer/report.json
+++ b/research/source-assisted-v4-layer/report.json
@@ -20,7 +20,7 @@
     },
     "C1": 1,
     "a_bit": "338670433250477381/500000000000000000000",
-    "a_complex": "7097667/10000000000",
+    "a_complex": "3549537/5000000000",
     "alpha_squared_power": "1128909749279055278683696000159127/1667795568088256895719968174600000000",
     "beta": "1/1000000000",
     "c": "16933583823853435799522619/25000000000000000000000000000",
@@ -31,12 +31,12 @@
     "lambda_": "49966132957013622695150477381/50000000000000000000000000000",
     "lambda_prime": "24983066478676146564200477381/25000000000000000000000000000",
     "q": "16933521323853435799522619/25000000000000000000000000000",
-    "sigma": "9992902333/10000000000",
+    "sigma": "4996450463/5000000000",
     "tau": "499661329566749522619/500000000000000000000"
    },
    "recurrence": {
     "internal": "499661329566749522619/500000000000000000000",
-    "leaf": "9992902333007097667/10000000000000000000",
+    "leaf": "4996450463003549537/5000000000000000000",
     "reservations": "24983066416176146564200477381/25000000000000000000000000000"
    },
    "strict_constraints": {
@@ -55,11 +55,11 @@
     "c_below_one": "24983066416176146564200477381/25000000000000000000000000000",
     "c_positive": "16933583823853435799522619/25000000000000000000000000000",
     "cell_above_band": "1666666641661022159558715521400159127/3335591136176513791439936349200000000",
-    "compact_leaf": "810646158402396700477381/25000000000000000000000000000",
+    "compact_leaf": "814163658398879200477381/25000000000000000000000000000",
     "compact_phase_layer_above_kappa": "5512191445341671259318410221/83389778404412844785998408730000000000",
     "compact_reservations": "1/400000000",
-    "complex_above_bit": "16212916749522619/500000000000000000000",
-    "complex_below_one_over32": "305402333/10000000000",
+    "complex_above_bit": "16283266749522619/500000000000000000000",
+    "complex_below_one_over32": "152700463/5000000000",
     "coordinate_movement_above_kappa": "229283250477381/500000000000000000000",
     "delta_below_one_eighth": "99999999/800000000",
     "delta_positive": "1/800000000",
@@ -69,17 +69,17 @@
     "gamma_sublinear": "1/200000000",
     "guard_width": "5644590441284478599840873/8338977840441284478599840873",
     "lambda_above_internal": "338670433250477381/50000000000000000000000000000",
-    "lambda_above_sigma": "1621292013622695150477381/50000000000000000000000000000",
+    "lambda_above_sigma": "1628327013622695150477381/50000000000000000000000000000",
     "lambda_above_tau": "338670433250477381/50000000000000000000000000000",
     "lambda_prime_above_lambda": "338670433250477381/50000000000000000000000000000",
     "lambda_prime_below_one": "16933521323853435799522619/25000000000000000000000000000",
-    "leaf_saving_above_bit": "16212916394639269/500000000000000000000",
+    "leaf_saving_above_bit": "16283266394568919/500000000000000000000",
     "literal_scalar_guard": "1",
     "phase_boundary": "903126131627676134690061080159127/1334236454470605516575974539680000000",
     "phase_local": "4515664014049742438588219800159127/6671182272353027582879872698400000000",
     "prime_interval_packing": "5644590441284478599840873/8338977840441284478599840873",
     "q_below_internal": "338670433250477381/25000000000000000000000000000",
-    "q_below_leaf": "810646158402396700477381/25000000000000000000000000000",
+    "q_below_leaf": "814163658398879200477381/25000000000000000000000000000",
     "q_below_reservations": "1/400000000",
     "q_positive": "16933521323853435799522619/25000000000000000000000000000",
     "record_suffix": "5644590441284478599840873/8338977840441284478599840873",
@@ -142,35 +142,35 @@
     "delta": "1320",
     "edge_count": "211689",
     "histogram": {
-     "1": "87528",
-     "10": "2514",
+     "1": "87534",
+     "10": "2415",
      "11": "1059",
      "12": "3423",
      "13": "504",
      "14": "2175",
-     "15": "1923",
+     "15": "1815",
      "16": "243",
-     "17": "123",
+     "17": "117",
      "18": "7920",
-     "19": "4209",
-     "2": "40644",
+     "19": "4323",
+     "2": "40641",
      "20": "66",
      "3": "29964",
-     "4": "11502",
-     "5": "5937",
-     "6": "4380",
-     "7": "1929",
-     "8": "4428",
-     "9": "1218"
+     "4": "11517",
+     "5": "5928",
+     "6": "4473",
+     "7": "1926",
+     "8": "4434",
+     "9": "1212"
     },
     "m": 66,
     "mass": "794112",
     "maxchild": 20
    },
-   "ideal_moment_upper": "352436827651667193920281873504433433238375/352436827714395734205207019190971682258944",
-   "saving": "7097667/10000000000",
-   "strict_gap_lower": "62728540284925145686538249020569/352436827714395734205207019190971682258944",
-   "strict_gap_lower_decimal": 1.779852028850926e-10
+   "ideal_moment_upper": "352436827646232828607147287136833807860575/352436827714395734205207019190971682258944",
+   "saving": "3549537/5000000000",
+   "strict_gap_lower": "68162905598059732054137874398369/352436827714395734205207019190971682258944",
+   "strict_gap_lower_decimal": 1.934046054156886e-10
   },
   "construction_receipts": {
    "complex_endpoint_checks": {
@@ -199,7 +199,7 @@
      "18": 1320
     }
    },
-   "complex_lift_certificate_sha256": "a2a9ffa85df587ab61a2effe12b54e6321f9b9103617bfc494827a7d0e9c5027"
+   "complex_lift_certificate_sha256": "fca2c338f6b2e380c48c1d17af24502b4fcec4129cef851e5bd9026f1e3ae8e6"
   },
   "finite_bridge": {
    "proof": "FINITE_BRIDGE.txt",
@@ -236,7 +236,7 @@
   "kappa_decimal": 0.0006768823,
   "source_sha256": {
    "bit_certificate": "853afa75980e11ce16b333c28311c4f635e84065867c049fda9916c2419091ed",
-   "complex": "65c3f8a88e44545d8984b4e0af7e86a096547e876c5797089fef7bdead8c1853"
+   "complex": "d894f6c2d6921a668fb84379c94b3709c00f28e9cc4ff814bceb021cf3b7858b"
   },
   "status": "Exact arithmetic; construction and finite bridge are explicit proof dependencies"
  },
@@ -244,33 +244,33 @@
  "complex_profile": {
   "W_per_vertex": 12052,
   "child_histogram": {
-   "1": 87528,
-   "10": 2514,
+   "1": 87534,
+   "10": 2415,
    "11": 1059,
    "12": 3423,
    "13": 504,
    "14": 2175,
-   "15": 1923,
+   "15": 1815,
    "16": 243,
-   "17": 123,
+   "17": 117,
    "18": 7920,
-   "19": 4209,
-   "2": 40644,
+   "19": 4323,
+   "2": 40641,
    "20": 66,
    "3": 29964,
-   "4": 11502,
-   "5": 5937,
-   "6": 4380,
-   "7": 1929,
-   "8": 4428,
-   "9": 1218
+   "4": 11517,
+   "5": 5928,
+   "6": 4473,
+   "7": 1926,
+   "8": 4434,
+   "9": 1212
   },
   "deficit_per_vertex": 1320,
   "m": 66,
   "rank_per_vertex": 794112
  },
- "complex_saving": "7097667/10000000000",
- "complex_saving_float": 0.0007097667,
+ "complex_saving": "3549537/5000000000",
+ "complex_saving_float": 0.0007099074,
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@@ -310,9 +310,9 @@
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@@ -322,78 +322,77 @@
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   "edge_basis_values": 42301,
   "edge_distinct_values": 42301,
   "edge_nullity_histogram": {
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-  "entropy": 1857937.9138443007,
-  "entropy_ratio": 0.8190996677164191,
+  "entropy": 1857569.1525343216,
+  "entropy_ratio": 0.8189370937874305,
   "field_prime": 1000003,
   "frames": 18532,
   "h": 22,
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   "new_W": 12052,
-  "numerical_local_root": 0.0007097667761553668,
+  "numerical_local_root": 0.0007099074827603465,
   "original_physical_roles": 13372,
   "original_role_crossings": 46261,
-  "outgoing_dependencies": 8093,
+  "outgoing_dependencies": 8095,
   "retirement_histogram": {
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@@ -417,7 +416,7 @@
    },
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    "formal": {
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+    "absolute_coefficient_bound": 603018,
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@@ -447,10 +446,10 @@
     "baseline.json.gz": "33187d4011bc6dd11468408c0bc7e968d54c63aadf87a09592da056140375d57",
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-    "physical-frames.json.gz": "9b5c2b2b2a3dc9be05a334bf2e9edd71bb4a93127296995ca25d8b61915f43f0",
-    "physical-pairs.json.gz": "ff154310a86fce1f675dcbd4ed5289269b5439b087635e589a5123fd2896615c",
+    "physical-frames.json.gz": "72ee453d44230d47a2947766a771cbb5e84bf4f3f49c7159569c26e8eb18a8e3",
+    "physical-pairs.json.gz": "bbad8a2a38e92dbadffda005507f46a810b1465478f4a2f15fde06fd2129d436",
     "profile-before.json.gz": "f60146c8d8c5e8aa0365f08b68aa6e03403656b74bf28b7058ef5a241560bf92",
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@@ -460,7 +459,7 @@
    "physical": {
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     "W_per_vertex": 16012,
-    "changed_operation_frames": 17787,
+    "changed_operation_frames": 17789,
     "checks": {
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@@ -470,50 +469,50 @@
      "value_spans_inside_frames": true
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     "h": 22,
-    "late_pairs": 1664,
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@@ -593,12 +592,12 @@
    },
    "status": "PASS exact fused physical candidate, all formal source/target/dirty columns"
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-  "frames": 17787,
+  "frames": 17789,
   "pairs": 2310
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-  "exact_scalar_program_sha256": "2524002e0f4407f1843a70a24d0459b02844c17fd809aa2b9b3784f3f6222379",
+  "exact_scalar_program_sha256": "95e854a59b0b997b893bc84ece0ab2d9cad16e29219bdc3c68726ca87e54c664",
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diff --git a/research/source-assisted-v4-layer/witness/physical-frames.json b/research/source-assisted-v4-layer/witness/physical-frames.json
index eb7d5841d2..9044957452 100644
--- a/research/source-assisted-v4-layer/witness/physical-frames.json
+++ b/research/source-assisted-v4-layer/witness/physical-frames.json
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diff --git a/research/source-assisted-v4-layer/witness/physical-pairs.json b/research/source-assisted-v4-layer/witness/physical-pairs.json
index f25f6d01d4..a582ff3bea 100644
--- a/research/source-assisted-v4-layer/witness/physical-pairs.json
+++ b/research/source-assisted-v4-layer/witness/physical-pairs.json
@@ -1 +1 @@
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