diff --git a/research/right-vector-commutative-products/.gitattributes b/research/right-vector-commutative-products/.gitattributes
new file mode 100644
index 0000000000..84ccf846c9
--- /dev/null
+++ b/research/right-vector-commutative-products/.gitattributes
@@ -0,0 +1 @@
+source/* -text -eol
diff --git a/research/right-vector-commutative-products/CODE-REVIEW.md b/research/right-vector-commutative-products/CODE-REVIEW.md
new file mode 100644
index 0000000000..31a614f2bf
--- /dev/null
+++ b/research/right-vector-commutative-products/CODE-REVIEW.md
@@ -0,0 +1,26 @@
+# Separate static review and root integration check
+
+A separate OpenAI Codex reviewer inspected the retained source contract,
+PROOF.md, engine.py, oracle.py and both core payloads. No concrete correctness
+defect was found. It checked the common-row/column unit condition, componentwise
+unit choice in the three-factor word, all-finite-field quadratic-extension
+argument and the delta-two deduction. The BFS review covered the complete
+descriptor domain, sound singular-factor filtering, paid/free distance
+certificate, arbitrary-input witness interpreter and independent column ranks.
+The oracle's one-factor literal-image classification and projective/two-factor
+test were also checked.
+
+The reviewer flagged that BFS witnesses use execution order while oracle words
+use algebraic product order. README now explicitly distinguishes them. Neither
+algorithm or frozen core payload was changed for this documentation fix.
+It also required that prime-field software evidence remain separate from the
+quantified all-finite-field proof. Physical costs and broader algebra targets
+remain outside scope.
+
+The root independently read the proof and implementations, and compared every
+one of the 216 cost/delta records between the two actual core payloads. Full
+normal/optimized, public-byte, cold and hosted replay records are separate
+handoff evidence. Static reviewers did not rerun scientific suites. This is
+separate AI review, not independent human acceptance or formal verification.
+
+The integration review also identified that only the optimized command initially used -W error. The normal manifest command now explicitly uses that flag, and the root reran the normal warning-strict command before sealing. Earlier core/normal outputs remain preserved unchanged.
diff --git a/research/right-vector-commutative-products/EXPECTED-BFS.json b/research/right-vector-commutative-products/EXPECTED-BFS.json
new file mode 100644
index 0000000000..9664a969a7
--- /dev/null
+++ b/research/right-vector-commutative-products/EXPECTED-BFS.json
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{"components": [[1, 0, 1, 1], [1, 1, 1, 0], [1, 1, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 0, 1, 1], [1, 1, 1, 0], [1, 1, 1, 0]], "cost": 1, "delta": 1}, {"components": [[1, 1, 0, 1], [0, 1, 1, 0], [0, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [0, 1, 1, 0], [0, 1, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [0, 1, 1, 0], [1, 0, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [0, 1, 1, 0], [1, 0, 1, 1]], "cost": 3, "delta": 2}, {"components": [[1, 1, 0, 1], [0, 1, 1, 0], [1, 1, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [0, 1, 1, 0], [1, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [0, 1, 1, 1], [0, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [0, 1, 1, 1], [0, 1, 1, 1]], "cost": 1, "delta": 1}, {"components": [[1, 1, 0, 1], [0, 1, 1, 1], [1, 0, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [0, 1, 1, 1], [1, 0, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [0, 1, 1, 1], [1, 1, 0, 1]], "cost": 1, "delta": 1}, {"components": [[1, 1, 0, 1], [0, 1, 1, 1], [1, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 0, 0, 1], [0, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 0, 0, 1], [0, 1, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 0, 0, 1], [1, 0, 0, 1]], "cost": 1, "delta": 1}, {"components": [[1, 1, 0, 1], [1, 0, 0, 1], [1, 0, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 0, 0, 1], [1, 1, 0, 1]], "cost": 1, "delta": 1}, {"components": [[1, 1, 0, 1], [1, 0, 0, 1], [1, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 0, 1, 1], [0, 1, 1, 0]], "cost": 3, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 0, 1, 1], [0, 1, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 0, 1, 1], [1, 0, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 0, 1, 1], [1, 0, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 0, 1, 1], [1, 1, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 0, 1, 1], [1, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 1, 0, 1], [0, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 1, 0, 1], [0, 1, 1, 1]], "cost": 1, "delta": 1}, {"components": [[1, 1, 0, 1], [1, 1, 0, 1], [1, 0, 0, 1]], "cost": 1, "delta": 1}, {"components": [[1, 1, 0, 1], [1, 1, 0, 1], [1, 0, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 1, 0, 1], [1, 1, 0, 1]], "cost": 0, "delta": 0}, {"components": [[1, 1, 0, 1], [1, 1, 0, 1], [1, 1, 1, 0]], "cost": 1, "delta": 1}, {"components": [[1, 1, 0, 1], [1, 1, 1, 0], [0, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 1, 1, 0], [0, 1, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 1, 1, 0], [1, 0, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 1, 1, 0], [1, 0, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 0, 1], [1, 1, 1, 0], [1, 1, 0, 1]], "cost": 1, "delta": 1}, {"components": [[1, 1, 0, 1], [1, 1, 1, 0], [1, 1, 1, 0]], "cost": 1, "delta": 1}, {"components": [[1, 1, 1, 0], [0, 1, 1, 0], [0, 1, 1, 0]], "cost": 1, "delta": 1}, {"components": [[1, 1, 1, 0], [0, 1, 1, 0], [0, 1, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [0, 1, 1, 0], [1, 0, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [0, 1, 1, 0], [1, 0, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [0, 1, 1, 0], [1, 1, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [0, 1, 1, 0], [1, 1, 1, 0]], "cost": 1, "delta": 1}, {"components": [[1, 1, 1, 0], [0, 1, 1, 1], [0, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [0, 1, 1, 1], [0, 1, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [0, 1, 1, 1], [1, 0, 0, 1]], "cost": 3, "delta": 2}, {"components": [[1, 1, 1, 0], [0, 1, 1, 1], [1, 0, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [0, 1, 1, 1], [1, 1, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [0, 1, 1, 1], [1, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 0, 0, 1], [0, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 0, 0, 1], [0, 1, 1, 1]], "cost": 3, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 0, 0, 1], [1, 0, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 0, 0, 1], [1, 0, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 0, 0, 1], [1, 1, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 0, 0, 1], [1, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 0, 1, 1], [0, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 0, 1, 1], [0, 1, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 0, 1, 1], [1, 0, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 0, 1, 1], [1, 0, 1, 1]], "cost": 1, "delta": 1}, {"components": [[1, 1, 1, 0], [1, 0, 1, 1], [1, 1, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 0, 1, 1], [1, 1, 1, 0]], "cost": 1, "delta": 1}, {"components": [[1, 1, 1, 0], [1, 1, 0, 1], [0, 1, 1, 0]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 1, 0, 1], [0, 1, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 1, 0, 1], [1, 0, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 1, 0, 1], [1, 0, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 1, 0, 1], [1, 1, 0, 1]], "cost": 1, "delta": 1}, {"components": [[1, 1, 1, 0], [1, 1, 0, 1], [1, 1, 1, 0]], "cost": 1, "delta": 1}, {"components": [[1, 1, 1, 0], [1, 1, 1, 0], [0, 1, 1, 0]], "cost": 1, "delta": 1}, {"components": [[1, 1, 1, 0], [1, 1, 1, 0], [0, 1, 1, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 1, 1, 0], [1, 0, 0, 1]], "cost": 2, "delta": 2}, {"components": [[1, 1, 1, 0], [1, 1, 1, 0], [1, 0, 1, 1]], "cost": 1, "delta": 1}, {"components": [[1, 1, 1, 0], [1, 1, 1, 0], [1, 1, 0, 1]], "cost": 1, "delta": 1}, {"components": [[1, 1, 1, 0], [1, 1, 1, 0], [1, 1, 1, 0]], "cost": 0, "delta": 0}], "upper_word_matrices": 2550}
diff --git a/research/right-vector-commutative-products/PACKAGE-REVIEW.json b/research/right-vector-commutative-products/PACKAGE-REVIEW.json
new file mode 100644
index 0000000000..6c8accd610
--- /dev/null
+++ b/research/right-vector-commutative-products/PACKAGE-REVIEW.json
@@ -0,0 +1,22 @@
+{
+  "status": "PASS_SEPARATE_PROOF_CODE_INTEGRATION_REVIEW_AND_ROOT_REPLAY",
+  "reviewed_at": "2026-10-21T10:29:13.178798+00:00",
+  "file_sha256": {
+    ".gitattributes": "e33596b6cf4e4ef921155cb5808ca49593cb4cedc766277a833701dbe2e5dae1",
+    "CODE-REVIEW.md": "75deb8f45a64ba2550ec30a5570ac749b5629d00a0627753c3d668b80dfc95bd",
+    "EXPECTED-BFS.json": "313192ab25845980122d39e9d91c0901a08748722c18ab23691d64df5bc5e70c",
+    "EXPECTED-ORACLE.json": "b0eae660ae44d46d445de8b31f40827e03aa2b18eb2057f7e616644c28ea5f78",
+    "PROOF.md": "b401a15bc3a11e2040cdde9d99d4a5b8ec7b4f92643eac547f21b7c667fbf408",
+    "README.md": "6222431360182204f502334d1c707eefc39a174d06578ef75c52f37b5873055f",
+    "SOURCE-PINS.json": "b57337cf7d7d387d9cec638fddaefcedd06e6292a58ecf612cc90e794b08e063",
+    "SOURCES.json": "a2634e490a808b7d3120559803bd41dbc523f2cc8a734ba6a476e9624353037c",
+    "TASK.json": "d1bf25963bde13fa7287b495b69907cd2b3c5de1d134d3e584913acfe33db8c4",
+    "VERIFY.json": "8380496e1c8f003d24cca8e8cd1adfccbc165f2a6792f29e976e9eabad9604ea",
+    "engine.py": "c60ffb769596d8c93674556bd35087ae42d10e2076ab8a5d6e8d03a9e62a9791",
+    "oracle.py": "76a6ee7ca38ef938fa328f0f720006375e9d244ff33024aa292aeb567fcf9903",
+    "source/PROOF.md": "c1065c51c839d11b0d31bccb8d39ebc07a869eb0f29899038518e3090a58c895",
+    "source/README.md": "e00d4352a98b02c5046d8796ba4ef2afd3c6a78d66b4398d23794ebd8147580d",
+    "verify.py": "159302875cb4b599b0ac3f4a230a4e208addc235aa0ad220a80d6a39b506b215"
+  },
+  "review_scope": "Independent AI proof/BFS/oracle/static integration; full typed records match, normal and optimized warning-strict root replays; no human/kernel acceptance"
+}
diff --git a/research/right-vector-commutative-products/PROOF.md b/research/right-vector-commutative-products/PROOF.md
new file mode 100644
index 0000000000..f149515c16
--- /dev/null
+++ b/research/right-vector-commutative-products/PROOF.md
@@ -0,0 +1,306 @@
+# Invertible targets over products of fields in the right-vector model
+
+## Model, statements, and predecessor
+
+Let `K` be a field and let `R` be a commutative unital `K`-algebra. We use
+exactly the retained source's model at `n = 2`:
+
+```text
+P(u,y) = I + u y^T,       u in R^2, y in K^2,
+free factors: GL_2(K), usable anywhere in the word,
+cost(M): minimum number of paid P factors in a word equal to M,
+delta(M) = min_{T in GL_2(K)} dim_K ((M T - I) K^2).
+```
+
+The `K^2` in the definition of `delta` is the common two-dimensional input
+space, not an independently chosen input in each product component. The
+embedding of `K` into a product is diagonal. A free matrix and each right
+vector `y` must therefore be common to all components. In contrast, the two
+entries of `u` are arbitrary elements of `R`, so they may vary by component.
+
+The predecessor is Nick Petalas's PR #4276,
+`research/right-vector-two-factor-criterion`, at source head
+`59c7bac0122ef2b80793c30a3d52329db6bdf17d`. Its retained
+[PROOF.md](source/PROOF.md), Theorems A and B, supplies the one-factor and
+two-factor criteria. Its [README.md](source/README.md) asks whether `delta`
+is exact for invertible targets over commutative algebras. We rederive the
+needed criteria below to make this argument self-contained.
+
+For an invertible `M` over `R = K^k`, with `k >= 1` and components
+`M_i in GL_2(K)`, we prove:
+
+1. `cost(M) <= 2` if and only if there are nonzero
+   `phi in (K^2)^*` and `t in K^2` such that `phi M_i t != 0` for every `i`.
+2. Every invertible target over any finite product of fields that are
+   unital `K`-algebras has an explicit word with at most three paid factors,
+   using the fixed right-vector sequence `e1, e2, e1`.
+3. If `K` is finite of size `q`, every invertible target over `K^k` costs at
+   most two when `k <= q`. For every `k >= q+1` there is an invertible target
+   of cost exactly three and `delta = 2`.
+
+Thus commutativity alone does not make `cost = delta` on invertible targets
+in this model. The smallest field-size instance proved here is `F2^3`.
+No assertion is made that it is the smallest commutative algebra with this
+property among all algebras.
+
+## 1. Elementary one-factor and two-factor criteria
+
+Conjugation by a free matrix preserves the paid alphabet:
+
+```text
+S P(u,y) S^{-1} = P(Su, S^{-T}y).
+```
+
+Moving free factors to the right therefore rewrites a word as a product of
+the same number of paid factors followed by one `S in GL_2(K)`. Also, paid
+factors with proportional nonzero right vectors merge:
+
+```text
+P(u,y) P(v,c y) = P(u + c v + c u (y^T v), y),   c in K.
+```
+
+This follows by expansion. A zero right vector or zero left vector gives
+the identity.
+
+For later use, `cost(M) <= 1` holds exactly when `delta(M) <= 1`. A word
+`M = P(u,y) S` gives `(M S^{-1} - I) K^2` contained in `K u`.
+Conversely, if the image of `M T - I` has dimension one, choose a spanning
+vector `u` and write its two columns as `y_1 u, y_2 u`, with `y_j in K`.
+Then `M T = P(u,y)` and `M = P(u,y) T^{-1}`. If the image has dimension
+zero, `M = T^{-1}` is free. This also shows that `delta = 0` exactly when
+`cost = 0`, and hence that cost and delta agree whenever cost is at most one.
+
+We next prove the retained two-factor criterion:
+
+```text
+cost(M) <= 2
+iff some G,H in GL_2(K) give N = G M H with N12 in N11 R.
+```
+
+For sufficiency, write `N12 = A C`, where `A = N11`, and put
+`B = N21`, `D = N22 - B C`. Then
+
+```text
+N = [[A,0],[B,1]] [[1,C],[0,D]]
+  = P((A-1,B)^T,e1) P((C,D-1)^T,e2).
+```
+
+Consequently `M = G^{-1} N H^{-1}` has a word with two paid factors and
+free factors. This uses the explicit permission to place free maps anywhere.
+
+For necessity, normalize a word with at most two paid factors to
+`M = P(u1,y1) P(u2,y2) S`. If its right vectors are dependent, merge the
+factors, handle any identity factors, and pad with an identity paid factor
+whose right vector is independent of the remaining nonzero right vector.
+For a completely free word, pad with identities in any two independent
+directions. Thus we may arrange that `y1,y2` are a basis of `K^2`.
+Choose `G in GL_2(K)` with `G^{-T} y_j = e_j` and set
+`H = S^{-1} G^{-1}`. Then
+
+```text
+G M H = P(v1,e1) P(v2,e2)
+      = [[A,0],[B,1]] [[1,C],[0,D]]
+      = [[A,A C],[B,B C+D]]
+```
+
+for suitable entries in `R`. Its `(1,2)` entry belongs to `N11 R`, as
+required. This argument is the predecessor's criterion, not a new claim
+about the alphabet.
+
+## 2. The common row and column criterion
+
+First allow `R = F_1 x ... x F_k`, where each `F_i` is a field containing
+the given copy of `K`. An element `A = (A_i)` of `R` is a unit exactly when
+every `A_i` is nonzero. Also,
+
+```text
+Z in A R  iff  A_i = 0 implies Z_i = 0 for every i.
+```
+
+Let `M` be invertible. Its components `M_i` are invertible, and the same is
+true of every component of `N = G M H`, for `G,H in GL_2(K)`.
+If `N12 in N11 R` and a component `N11_i` were zero, then `N12_i` would
+also be zero. That component matrix would have a zero first row, which is
+impossible. Thus `N11` is a unit. Conversely, if `N11` is a unit, then
+`N12 in N11 R` automatically.
+
+Write `phi` for the first row of `G` and `t` for the first column of `H`.
+They are nonzero and
+
+```text
+N11_i = phi M_i t.
+```
+
+Every nonzero `K` row extends to a basis of the row space, and every nonzero
+`K` column extends to a basis of the column space. Thus every such pair
+`phi,t` occurs for some `G,H`. The preceding section now proves
+
+```text
+cost(M) <= 2  iff  there exist common nonzero K row phi and K column t
+                        with phi M_i t != 0 for every i.
+```
+
+In particular this holds for `R = K^k`. Invertibility matters: for a
+singular component, a zero first row need not give a contradiction, so the
+unit-entry reduction above is not asserted for arbitrary targets.
+
+## 3. A constructive three-factor word
+
+Let again `R = F_1 x ... x F_k`, and write an invertible target as
+
+```text
+M = [[a,b],[c,d]].
+```
+
+The second column `(b_i,d_i)^T` is nonzero in each field component. Define
+an allowed coefficient `r in R` componentwise by
+
+```text
+r_i = 0 if d_i != 0, and r_i = 1 if d_i = 0.
+D = d - r b.
+```
+
+If `d_i != 0`, then `D_i = d_i != 0`. If `d_i = 0`, then `b_i != 0` and
+`D_i = -b_i != 0`, including in characteristic two. Therefore `D` is a
+unit. Put
+
+```text
+f = D^{-1} (c - r a),
+e = a - b f.
+```
+
+Because `R` is commutative,
+
+```text
+e D = a(d-rb) - b(c-ra) = ad-bc = det(M).
+```
+
+The determinant of `M` and `D` are units, so `e` is a unit as well. Direct
+multiplication gives the promised word:
+
+```text
+M = [[1,0],[r,1]] [[1,b],[0,D]] [[e,0],[f,1]]
+  = P((0,r)^T,e1) P((b,D-1)^T,e2) P((e-1,f)^T,e1).
+```
+
+Indeed the right-hand product has entries `e+bf = a`, `b`,
+`r(e+bf)+Df = ra+(c-ra) = c`, and `rb+D = d`.
+All three factors are invertible, although the paid alphabet does not
+require this separately. An identity factor can be omitted, so the result
+is a bound of at most three paid factors. No free map is needed for this
+construction. The componentwise choice of `r` is placed in a left vector
+over `R`; it does not introduce componentwise free maps or right vectors.
+
+## 4. Finite-field threshold and sharp examples
+
+Suppose `K` has `q` elements and `R = K^k`. The projective line `P^1(K)`
+has `(q^2-1)/(q-1) = q+1` points: its points are the one-dimensional
+subspaces of `K^2`, each with `q-1` nonzero vectors. The dual row space has
+the same number of projective points.
+
+Fix any nonzero `t in K^2`. Each `M_i t` is nonzero. Exactly one projective
+row `phi` annihilates it, since its annihilator has dimension one. There
+are at most `k` forbidden projective rows. If `k <= q`, at least one of the
+`q+1` projective rows is not forbidden, and it gives `phi M_i t != 0` for
+every component. Section 2 proves `cost(M) <= 2`.
+
+This also gives `cost(M) = delta(M)` in this range. Section 1 decides the
+cases `delta <= 1`. If the cost is two, `delta` cannot be at most one, and
+the map defining delta has domain `K^2`, so delta is exactly two.
+
+To prove sharpness for every finite field, first construct a quadratic
+field extension. There are `q^2` monic quadratic polynomials over `K`.
+A reducible monic quadratic is determined uniquely by its unordered pair
+of roots, allowing a repeated root. There are
+
+```text
+q + q(q-1)/2 = q(q+1)/2
+```
+
+such pairs. Thus there are `q(q-1)/2 > 0` irreducible monic quadratics.
+Choose one, `h`, and form the field `L = K[X]/(h)`, which has dimension two
+over `K`. Fix a `K` basis of `L` and choose a nonzero representative
+`alpha_l` of each of its `q+1` one-dimensional `K` subspaces `l`.
+Let `A_l in GL_2(K)` be the matrix of multiplication by `alpha_l`.
+It is invertible because `alpha_l` is nonzero in a field.
+
+For a fixed nonzero `t in K^2`, let `beta` be the corresponding nonzero
+element of `L`. The vectors `A_l t` correspond to `alpha_l beta`.
+Multiplication by `beta` is a `K`-linear bijection, so it permutes the
+one-dimensional `K` subspaces of `L`. Hence these `q+1` outputs meet every
+projective line exactly once. For every nonzero row `phi`, one of the
+outputs therefore lies in the line `ker(phi)`.
+
+The target with components `(A_l)_l` has no common pair `phi,t` satisfying
+Section 2. It costs more than two. Section 3 supplies a word with at most
+three factors, so its cost is exactly three. For any `k > q+1`, append
+identity components. The original `q+1` components still prevent a common
+pair, and the three-factor construction still applies.
+
+Finally, these examples have `delta = 2`. Delta is at most two because its
+defining map has a two-dimensional domain. If delta were at most one,
+Section 1 would give a word with at most one paid factor, contradicting
+the proved cost of three. This proves the stated threshold with no search
+or asymptotic assumption.
+
+## 5. The explicit F2^3 target
+
+Take `h(X) = X^2+X+1` over `F2`, with basis `(1,theta)` of its quotient
+field, and use the line representatives `1,theta,1+theta`. The target has
+components
+
+```text
+M1 = [[1,0],[0,1]],
+M2 = [[0,1],[1,1]],
+M3 = [[1,1],[1,0]].
+```
+
+For each of the three possible nonzero input columns, the outputs are:
+
+| `t` | `M1 t` | `M2 t` | `M3 t` |
+|---|---|---|---|
+| `(1,0)^T` | `(1,0)^T` | `(0,1)^T` | `(1,1)^T` |
+| `(0,1)^T` | `(0,1)^T` | `(1,1)^T` | `(1,0)^T` |
+| `(1,1)^T` | `(1,1)^T` | `(1,0)^T` | `(0,1)^T` |
+
+Every row of this table contains every nonzero vector, so each nonzero
+row `phi` annihilates at least one output. The target therefore costs
+more than two. Here is a three-factor word, written entirely in `R=F2^3`:
+
+```text
+zero = (0,0,0),  one = (1,1,1),
+r = (0,0,1),    b = (0,1,1),    f = (0,1,0),
+M = P((zero,r)^T,e1) P((b,zero)^T,e2) P((zero,f)^T,e1).
+```
+
+Its first component is the identity. Its second component is
+`[[1,1],[0,1]] [[1,0],[1,1]] = M2`, and its third component is
+`[[1,0],[1,1]] [[1,1],[0,1]] = M3`. Thus this target has cost exactly three
+and delta exactly two. These identities and the three-line table are
+mathematical checks, not claims that a computational replay was run here.
+
+## Scope, credit, and remaining trust
+
+The model and column bound originate in Wentao Li's #4099, building on
+#2513; #4243 supplies the diagonal/free-basis development. The general-target
+question and earlier noncommutative failures are from #4265. Nick Petalas's
+#4276 supplies the retained criteria and the explicit commutative-product
+question. Those contributions retain their credit. The scoped deductions
+here are the product-field reduction, constructive three-factor bound, and
+sharp finite-field component threshold. Projective-line counting, quadratic
+extensions, determinant algebra, and rank-one factor manipulations are
+classical facts and are not claimed as new general mathematics.
+
+This is a proof in the stated formal alphabet. It assumes that arbitrary
+left vectors over the whole product algebra are permitted at one paid
+factor and that `GL_2(K)` maps are free anywhere. No physical implementation,
+component-selection cost, routing, storage, precision, restoration, native
+supplier invoice, or recursive multiplication cost has been supplied;
+those costs remain UNKNOWN. The argument does not classify singular targets
+or general commutative algebras, and does not change selected kappa.
+
+The proof was prepared with substantial OpenAI Codex assistance, including
+separate AI mathematical assessment. It is not independent human acceptance
+or formal proof-kernel verification. Finite enumerations, if supplied
+elsewhere in this package, remain separate evidence and do not substitute
+for the quantified argument above. No source files were changed for it.
diff --git a/research/right-vector-commutative-products/README.md b/research/right-vector-commutative-products/README.md
new file mode 100644
index 0000000000..fc9889d416
--- /dev/null
+++ b/research/right-vector-commutative-products/README.md
@@ -0,0 +1,126 @@
+# Commutative products can require a third right-vector factor
+
+**Scoped proof with a complete small-model certificate. No new kappa.**
+This answers Open question 3 of Nick Petalas's
+[PR4276](https://github.com/CrocSwap/integer-mult-bounds/pull/4276):
+commutativity does not make `cost(M)=delta(M)` for every invertible target.
+
+The exact source model has two roles, paid factors `I+u y^T` with `u in R^2`
+and `y in K^2`, and arbitrary `GL2(K)` maps free anywhere. `K` embeds
+diagonally into a product algebra. The free maps and right vectors must be
+common to all components; the `R` left coefficients may vary by component.
+No scratch or borrowed helper is used.
+
+## Result
+
+For an invertible target over `R=K^k`, two paid factors suffice exactly when
+some common nonzero base-field row `phi` and column `t` make every
+`phi M_i t` nonzero. Every invertible target over a finite product of field
+`K`-algebras has an explicit three-factor word, with right vectors
+`e1,e2,e1`. If `K` has `q` elements, the component threshold is sharp:
+
+- `k<=q`: every invertible target has `cost=delta<=2`.
+- `k>=q+1`: an invertible target can have `cost=3` and `delta=2`.
+
+[PROOF.md](PROOF.md) gives the complete quantified argument, including the
+quadratic-extension construction for **every finite field**, not just prime
+fields. It rederives the predecessor criteria, which retain their credit.
+Projective counting, field extensions and determinant algebra are classical;
+this is their scoped application to the retained research question.
+
+The small counterexample is `R=F2^3`, with component matrices
+
+```text
+I = [[1,0],[0,1]],  J = [[0,1],[1,1]],  J^2 = [[1,1],[1,0]].
+```
+
+For each nonzero `t`, these three matrices send `t` to all three nonzero
+binary columns. Every nonzero row kills one of them, so all nine common
+pivots fail. An explicit attaining word is in the proof. In particular the
+predecessor's suggested invertible `F3^3` search cannot refute the claim:
+three components are still below its four-component obstruction threshold.
+This does not classify singular targets or every commutative algebra.
+
+## Exact finite evidence
+
+`engine.py` independently enumerates the literal paid alphabet and computes
+the complete invertible `F2^3` metric by BFS. It imports no predecessor code
+and uses no projective criterion. All 216 targets are reached using 22 distinct
+invertible paid factors. Filtering singular factors is sound for an
+invertible endpoint over a product of fields.
+
+| Paid cost | delta | Targets |
+|---:|---:|---:|
+| 0 | 0 | 6 |
+| 1 | 1 | 54 |
+| 2 | 2 | 144 |
+| 3 | 2 | 12 |
+
+The distance certificate checks all 4,752 paid edges and all 1,296 free edges,
+132 conjugations, all 216 attaining matrix words and 13,824 arbitrary-input
+evaluations. Delta is computed from the literal six-by-two binary column
+matrix at all six free endpoints. `EXPECTED-BFS.json` retains every target,
+distance, delta endpoint and attaining word.
+
+The separate `oracle.py` computes the complete same 216 cost/delta records
+from projective pivots and common literal-vector images, with no BFS import.
+All records agree. It checks sharp witnesses over prime fields
+`2,3,5,7,11,13,17,19`, all 512 possible nonzero first-column triples for
+`F3^3`, and 2,550 literal three-factor identities over `F2,F3,F5,F7`.
+`EXPECTED-ORACLE.json` retains the complete regenerated data. Prime-field
+replay corroborates the written all-finite-field theorem; it does not replace
+that proof or claim an extension-field software census.
+
+**Word conventions:** BFS records use execution order: apply the recorded
+free map first, then left-apply each paid factor, giving `P_r...P_1 S`.
+Oracle `word` arrays use algebraic product order `F1 F2 F3`; a column-vector
+execution applies `F3` first. These are separately checked witnesses and must
+not be interchanged. The independent review identified this presentation
+distinction; it is explicit here without changing either computation.
+
+Controls detect omitted factors, wrong component and free chart, reversed
+products, a dropped obstruction component, treating componentwise free maps
+as the common free model, composite moduli incorrectly labelled fields,
+singular targets outside the upper theorem, and typed/coverage evidence
+corruptions. Explicit checks remain active under optimized Python.
+
+## Reproduce
+
+Python 3.12 standard library only:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/right-vector-commutative-products
+```
+
+Both registered normal and optimized warning-strict commands freshly execute
+both implementations, compare **all typed regenerated data** with the
+retained artifacts, and compare all 216 cross-engine cost/delta records.
+The command prints a compact receipt; the full data remains in the two
+EXPECTED files. Source contracts are exact unchanged exports of PR4276 head
+`59c7bac0122ef2b80793c30a3d52329db6bdf17d`, retained through input ancestor
+`326b25e5da8330cc3a44f84e8dcde0e4ff50c5e0` with Git/SHA256 pins.
+
+## Open next questions
+
+1. Specify a product of commutative local algebras and prove the unit/right-ideal
+   reduction before searching. This could separate shared residue-field charts
+   from nilpotent effects; the present theorem only classifies field products.
+2. Bind one literal native producer/consumer to the stipulated `R`-left-vector
+   and common-free-map interface, with its complete invoice before construction.
+   Preparation, routing, control, precision, restoration and stock costs are
+   **UNKNOWN**. This formal three-factor word supplies none of them.
+
+`TASK.json` records the changed original card, bounded decision and structured
+follow-ups. No native caller, supplier, outer multiplication theorem,
+independent human acceptance or formal proof-kernel review is supplied. The
+selected frontier is unchanged; maintainer edits are disabled.
+
+## Credit
+
+Nick Petalas's PR4276 supplies the exact criteria and open question, following
+PR4265 and PR4243. Wentao Li's PR4099 supplies the right-vector alphabet and
+column bound, building on PR2513. Their source documents and attribution are
+retained unchanged. New work by ToikY with substantial OpenAI Codex assistance:
+separate proof author, BFS implementation, projective oracle and static review
+contexts. These are AI-assisted checks, not independent human review. New
+material Apache-2.0; no merge or selected-frontier change.
diff --git a/research/right-vector-commutative-products/SOURCE-PINS.json b/research/right-vector-commutative-products/SOURCE-PINS.json
new file mode 100644
index 0000000000..5b753efb0b
--- /dev/null
+++ b/research/right-vector-commutative-products/SOURCE-PINS.json
@@ -0,0 +1,18 @@
+{
+  "original_head": "59c7bac0122ef2b80793c30a3d52329db6bdf17d",
+  "retained_ancestor": "326b25e5da8330cc3a44f84e8dcde0e4ff50c5e0",
+  "files": [
+    {
+      "path": "source/PROOF.md",
+      "bytes": 3969,
+      "sha256": "c1065c51c839d11b0d31bccb8d39ebc07a869eb0f29899038518e3090a58c895",
+      "git_blob": "71888afa0db63d6d332cbbdc408abf1aa879681c"
+    },
+    {
+      "path": "source/README.md",
+      "bytes": 6479,
+      "sha256": "e00d4352a98b02c5046d8796ba4ef2afd3c6a78d66b4398d23794ebd8147580d",
+      "git_blob": "8952fc9ae8a6e1144e5a0008141121aee66aec69"
+    }
+  ]
+}
diff --git a/research/right-vector-commutative-products/SOURCES.json b/research/right-vector-commutative-products/SOURCES.json
new file mode 100644
index 0000000000..37e265a842
--- /dev/null
+++ b/research/right-vector-commutative-products/SOURCES.json
@@ -0,0 +1,38 @@
+{
+  "schema_version": 1,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "64bfc5af6709ed9811171e0a2108f05e1219cb65",
+      "sha256": "352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96"
+    },
+    {
+      "path": "docs/research/research-family-status.md",
+      "usage": "provenance",
+      "git_blob": "abdd2792e6a5596b3fa6be3dc9db882bd8e3ef89",
+      "sha256": "0e61c47dfbc6f16e5f64594fcc7d907d4187568e38dddf10d473f0992419c44f"
+    },
+    {
+      "path": "research/right-vector-commutative-products/source/PROOF.md",
+      "usage": "input",
+      "git_blob": "71888afa0db63d6d332cbbdc408abf1aa879681c",
+      "sha256": "c1065c51c839d11b0d31bccb8d39ebc07a869eb0f29899038518e3090a58c895",
+      "pin": "head:326b25e5da8330cc3a44f84e8dcde0e4ff50c5e0:research/right-vector-commutative-products/source/PROOF.md"
+    },
+    {
+      "path": "research/right-vector-commutative-products/source/README.md",
+      "usage": "input",
+      "git_blob": "8952fc9ae8a6e1144e5a0008141121aee66aec69",
+      "sha256": "e00d4352a98b02c5046d8796ba4ef2afd3c6a78d66b4398d23794ebd8147580d",
+      "pin": "head:326b25e5da8330cc3a44f84e8dcde0e4ff50c5e0:research/right-vector-commutative-products/source/README.md"
+    }
+  ]
+}
diff --git a/research/right-vector-commutative-products/TASK.json b/research/right-vector-commutative-products/TASK.json
new file mode 100644
index 0000000000..1088fe8aa2
--- /dev/null
+++ b/research/right-vector-commutative-products/TASK.json
@@ -0,0 +1,97 @@
+{
+  "schema_version": 1,
+  "task_id": "bit-new-register-model",
+  "priority_revision": 7,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "original_card": {
+    "id": "bit-new-register-model",
+    "question": "Can an explicitly larger or nonabelian register model escape the scoped three-register obstruction?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/bit-three-register-lower-bound/README.md",
+      "certificates/selected-result.json",
+      "research/bit-gl3-mixed-targets-length10/README.md",
+      "docs/research/community-efficient1000-review.md"
+    ],
+    "methods": [
+      "Construct the smallest exact identity in a newly specified legal operation alphabet.",
+      "Derive a target-specific relaxation bound for that same explicit alphabet."
+    ],
+    "acceptance_gate": "Name the changed hypothesis, all counted roles and a uniform identity; finite column tests alone do not certify it. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read Model section 1 of the three-register bound and identify the exact excluded free operations.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "First compare the alphabet, role count and length with the fully replayed #1974 census and #2027 mixed-root tariff. Identify a changed premise and a legal native implementation; then price one smallest completed instance before searching larger words.",
+    "bottleneck_link": "A small-width completed saving changes the bit primitive; a finite-address identity alone is insufficient.",
+    "allowed_variations": [
+      "Choose a fixed larger register count or a specified nonabelian operation family."
+    ],
+    "scope_change_triggers": [
+      "Remaining inside the already-excluded three-register abelian model.",
+      "Repeating the two-register GL3 census or a covered three-register length without changing a hypothesis; a formal group-algebra word without native conversion is a partial result."
+    ],
+    "overlap_terms": [
+      "three-register",
+      "nonabelian",
+      "group algebra"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Answer PR4276 OpenQ3 in the formal right-vector model: invertible commutative field products can have cost3 withdelta2; sharp q+1 component threshold, constructive3 upperbound, complete216-target F2^3 metric.",
+  "scope_change_reason": "Take the named live commutative-algebra handoff after the earlier noncommutative failures, changing the algebra hypothesis rather than another seed or larger same-algebra census.",
+  "bottleneck_link": "A common free chart across components is an actual constraint even over a commutative algebra; removing noncommutativity does not restore the general cost=delta claim.",
+  "budget_minutes": 60,
+  "stop_rule": "One scoped counterexample/threshold proof and bounded exact checks; science stops2026-10-21T10:52:48UTC, final15minutes verification/handoff. No native or large algebra campaign.",
+  "outcome": "completed",
+  "evidence": "Self-contained quantified proof; full216-target independent BFS and projective cost/delta cross-check; all4752paid/1296free edges,132conjugations,13824arbitraryinputs;8prime-field sharp witnesses and2550literal upper-word matrices; exactsource pins.",
+  "author": "ToikY with substantial OpenAI Codex assistance",
+  "ai_assistance": "Separate proof author, BFS implementation, projective oracle and independent static review contexts. Not independent human or formal-kernel acceptance.",
+  "open_questions": [
+    {
+      "question": "Does an analogous common-unit-pivot criterion classify invertible targets over products of commutative local algebras?",
+      "why_it_matters": "This separates shared base-field charts from nilpotent or residue-field effects; general commutativity is already insufficient.",
+      "first_step": "Specify one product of local algebras and its residue fields, refresh overlap and prove how units/right ideals reduce before any census.",
+      "blocked_by": "This package proves the sharp equal-field product threshold, not all commutative algebras."
+    },
+    {
+      "question": "Can a literal native caller realize the allowed R-left coefficients and common free GL2(K) maps at a complete competitive bill?",
+      "why_it_matters": "The formal query count gives no physical supplier admission.",
+      "first_step": "Name one producer/consumer and full preparation, routing, control, precision, cleanup and stock ledger before implementation.",
+      "blocked_by": "All physical costs and supplier/outer obligations remain UNKNOWN; no candidate kappa is derived."
+    }
+  ],
+  "canonical_claim": "https://github.com/CrocSwap/integer-mult-bounds/discussions/4333",
+  "predecessor_head": "59c7bac0122ef2b80793c30a3d52329db6bdf17d",
+  "scope_key": "f09e0798647413fe5aac",
+  "freshness": {
+    "open_draft_prs": 442,
+    "live_threads": [
+      1166,
+      4333
+    ],
+    "active_discussions_list": true,
+    "source_handoff": "1166#19085611",
+    "broader_coverage": "partial; no novelty inferred",
+    "submission_upstream_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+    "submission_utc": "2026-10-21T10:29:13.178798+00:00",
+    "submission_update_ordered_delta": 100,
+    "source_head_unchanged": true,
+    "predecessor_claim_reply": "1166#19094593"
+  },
+  "family_decision": "NARROW to one algebraic acceptance-boundary result; no physical construction campaign",
+  "obligation_ledger": {
+    "formal_model": "source stipulates arbitrary R-left coefficients and common GL2(K) free maps",
+    "formal_cost": "proved exact3 on sharp examples; proved<=2 for k<=q; constructive<=3 for fieldproducts",
+    "finite_metric": "all216invertible F2^3 targets checked independently",
+    "physical_preparation_routing_control_precision_restoration_stock": "UNKNOWN",
+    "native_supplier_outer": "not admitted",
+    "human_kernel": "not claimed"
+  }
+}
diff --git a/research/right-vector-commutative-products/VERIFY.json b/research/right-vector-commutative-products/VERIFY.json
new file mode 100644
index 0000000000..a5b5b526d9
--- /dev/null
+++ b/research/right-vector-commutative-products/VERIFY.json
@@ -0,0 +1,37 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/right-vector-commutative-products/source/PROOF.md",
+    "research/right-vector-commutative-products/source/README.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Fresh complete216-target BFS with all paid/free edges and arbitrary-input words; independent projective/explicit-word oracle and full typed payload comparisons."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Same fresh complete metric/cross-engine/corruption controls under optimized warning-strict Python; finite evidence is separate from the quantified written theorem."
+    }
+  ]
+}
diff --git a/research/right-vector-commutative-products/engine.py b/research/right-vector-commutative-products/engine.py
new file mode 100644
index 0000000000..5e21c2d145
--- /dev/null
+++ b/research/right-vector-commutative-products/engine.py
@@ -0,0 +1,306 @@
+#!/usr/bin/env python3
+"""Exact independent paid-factor BFS on GL_2(F_2)^3.
+
+Credit: Nick Petalas, PRs #4276/#4265/#4243; Wentao Li, PR #4099's
+right-vector alphabet/column bound; the retained PR #2513 lineage.  This
+implementation was prepared by ToikY with substantial OpenAI Codex assistance.
+It imports no predecessor verifier and uses no projective pivot criterion.
+
+Only the invertible target domain is searched.  Each component of a word is
+a product of square matrices over F_2.  Invertibility of the final component
+forces every component factor to be invertible, so filtering singular paid
+factors cannot remove a word attaining a target in this domain.  Free matrices
+are constant triples from GL_2(F_2).  BFS starts at all six free endpoints;
+the certificate checks every paid edge and every zero-cost free edge as well
+as every attaining word.  Thus the distance certificate also directly covers
+free maps anywhere in a word.
+
+The two original target roles are arbitrary inputs, not borrowed helpers.
+No scratch/helper is allocated.  All six output bits are checked against the
+specified target map; no role is silently treated as disposable or restored.
+"""
+
+from collections import Counter, deque
+import hashlib
+import json
+import sys
+
+
+IDENTITY = 9
+CANONICAL = (9, 14, 7)  # I, [[0,1],[1,1]], [[1,1],[1,0]].
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def entries(code):
+    """Row-major entries, with entry i represented by bit i."""
+    return tuple((code >> index) & 1 for index in range(4))
+
+
+def pack(values):
+    return sum((value & 1) << index for index, value in enumerate(values))
+
+
+def multiply_component(left, right):
+    a = entries(left)
+    b = entries(right)
+    return pack(
+        sum(a[2 * row + mid] * b[2 * mid + col] for mid in range(2)) % 2
+        for row in range(2)
+        for col in range(2)
+    )
+
+
+def invertible(code):
+    a, b, c, d = entries(code)
+    return (a * d) ^ (b * c) == 1
+
+
+def literal_factor(descriptor):
+    """Descriptor is [u0,u1,y0,y1], u entries are three-bit ring values."""
+    u0, u1, y0, y1 = descriptor
+    require(0 <= u0 < 8 and 0 <= u1 < 8, "ring coefficient outside F2^3")
+    require(y0 in (0, 1) and y1 in (0, 1), "right vector outside F2^2")
+    return tuple(
+        pack((1 ^ (((u0 >> j) & 1) * y0),
+              ((u0 >> j) & 1) * y1,
+              ((u1 >> j) & 1) * y0,
+              1 ^ (((u1 >> j) & 1) * y1)))
+        for j in range(3)
+    )
+
+
+def factor_catalogue():
+    representatives = {}
+    admitted_descriptors = 0
+    for u0 in range(8):
+        for u1 in range(8):
+            for y0 in range(2):
+                for y1 in range(2):
+                    descriptor = (u0, u1, y0, y1)
+                    state = literal_factor(descriptor)
+                    if all(invertible(component) for component in state):
+                        admitted_descriptors += 1
+                        representatives.setdefault(state, descriptor)
+    return representatives, admitted_descriptors
+
+
+def apply_free(free_code, roles):
+    """Direct input interpreter, independent of the matrix product table."""
+    a, b, c, d = entries(free_code)
+    first, second = roles
+    return ((first if a else 0) ^ (second if b else 0),
+            (first if c else 0) ^ (second if d else 0))
+
+
+def apply_paid(descriptor, roles):
+    u0, u1, y0, y1 = descriptor
+    first, second = roles
+    readout = (first if y0 else 0) ^ (second if y1 else 0)
+    return first ^ (u0 & readout), second ^ (u1 & readout)
+
+
+def apply_target(state, roles):
+    first, second = roles
+    out_first = out_second = 0
+    for j, code in enumerate(state):
+        a, b, c, d = entries(code)
+        x, y = (first >> j) & 1, (second >> j) & 1
+        out_first |= ((a * x) ^ (b * y)) << j
+        out_second |= ((c * x) ^ (d * y)) << j
+    return out_first, out_second
+
+
+def evaluate_word(free_code, descriptors, roles):
+    result = apply_free(free_code, roles)
+    for descriptor in descriptors:
+        result = apply_paid(descriptor, result)
+    return result
+
+
+def word_agrees(state, free_code, descriptors):
+    return all(
+        evaluate_word(free_code, descriptors, (first, second))
+        == apply_target(state, (first, second))
+        for first in range(8)
+        for second in range(8)
+    )
+
+
+def delta_by_columns(state, free_codes):
+    """Rank of the literal six-by-two binary matrix (M T-I)|K^2."""
+    ranks = []
+    for free in free_codes:
+        t00, t01, t10, t11 = entries(free)
+        columns = [0, 0]
+        for component, code in enumerate(state):
+            a, b, c, d = entries(code)
+            # Deliberately do not call multiply_component or use its table.
+            m00 = (a * t00) ^ (b * t10) ^ 1
+            m01 = (a * t01) ^ (b * t11)
+            m10 = (c * t00) ^ (d * t10)
+            m11 = (c * t01) ^ (d * t11) ^ 1
+            columns[0] |= (m00 << (2 * component)) | (m10 << (2 * component + 1))
+            columns[1] |= (m01 << (2 * component)) | (m11 << (2 * component + 1))
+        if columns == [0, 0]:
+            rank = 0
+        elif 0 in columns or columns[0] == columns[1]:
+            rank = 1
+        else:
+            rank = 2
+        ranks.append(rank)
+    return min(ranks), ranks
+
+
+def recover_word(state, predecessors, catalogue):
+    backwards = []
+    while predecessors[state] is not None:
+        previous, factor = predecessors[state]
+        backwards.append(catalogue[factor])
+        state = previous
+    return state[0], tuple(reversed(backwards))
+
+
+def negative_controls(state, free_code, word, free_codes):
+    require(word_agrees(state, free_code, word), "positive word rejected")
+    dropped = []
+    for omitted in range(len(word)):
+        damaged = word[:omitted] + word[omitted + 1:]
+        require(not word_agrees(state, free_code, damaged), "dropped factor undetected")
+        dropped.append({"omitted_step": omitted, "rejected": True})
+
+    wrong_component = None
+    for step, descriptor in enumerate(word):
+        for first, second in ((0, 1), (0, 2), (1, 2)):
+            changed = list(descriptor)
+            for coordinate in (0, 1):
+                value = changed[coordinate]
+                if ((value >> first) ^ (value >> second)) & 1:
+                    changed[coordinate] ^= (1 << first) | (1 << second)
+            if tuple(changed) == descriptor:
+                continue
+            candidate = word[:step] + (tuple(changed),) + word[step + 1:]
+            require(all(invertible(c) for c in literal_factor(changed)),
+                    "component permutation damaged factor admissibility")
+            if not word_agrees(state, free_code, candidate):
+                wrong_component = {"step": step, "swapped_components": [first, second],
+                                   "replacement_descriptor": changed, "rejected": True}
+                break
+        if wrong_component is not None:
+            break
+    require(wrong_component is not None, "no discriminating component mutant")
+
+    altered_free = next(code for code in free_codes if code != free_code)
+    require(not word_agrees(state, altered_free, word), "wrong free chart undetected")
+    return {"positive_word_accepted": True, "dropped_factors": dropped,
+            "wrong_component": wrong_component,
+            "wrong_free_chart": {"replacement_free_code": altered_free, "rejected": True}}
+
+
+def run():
+    free_codes = tuple(code for code in range(16) if invertible(code))
+    require(len(free_codes) == 6, "GL2(F2) cardinality")
+    component_table = tuple(tuple(multiply_component(a, b) for b in range(16))
+                            for a in range(16))
+
+    def multiply(left, right):
+        return tuple(component_table[a][b] for a, b in zip(left, right))
+
+    catalogue, admitted_descriptors = factor_catalogue()
+    factors = tuple(sorted(catalogue))
+    free_states = tuple((code,) * 3 for code in free_codes)
+    states = tuple((a, b, c) for a in free_codes for b in free_codes for c in free_codes)
+    state_set = set(states)
+    require(len(states) == 216, "invertible target domain cardinality")
+
+    distances = {state: 0 for state in free_states}
+    predecessors = {state: None for state in free_states}
+    pending = deque(free_states)
+    while pending:
+        previous = pending.popleft()
+        for factor in factors:
+            state = multiply(factor, previous)
+            require(state in state_set, "invertible paid transition left target domain")
+            if state not in distances:
+                distances[state] = distances[previous] + 1
+                predecessors[state] = (previous, factor)
+                pending.append(state)
+    require(set(distances) == state_set, "BFS did not cover the entire invertible domain")
+
+    paid_edges = free_edges = conjugations = 0
+    for state in states:
+        require((distances[state] == 0) == (state in free_states), "wrong zero-cost set")
+        for factor in factors:
+            require(distances[multiply(factor, state)] <= distances[state] + 1,
+                    "paid-edge distance certificate failed")
+            paid_edges += 1
+        for free in free_states:
+            require(distances[multiply(free, state)] == distances[state],
+                    "zero-cost free-edge distance certificate failed")
+            free_edges += 1
+    for free in free_states:
+        inverse = next(other for other in free_states
+                       if multiply(free, other) == (IDENTITY,) * 3)
+        for factor in factors:
+            require(multiply(multiply(free, factor), inverse) in catalogue,
+                    "paid alphabet not closed under free conjugation")
+            conjugations += 1
+
+    records = []
+    for state in states:
+        free_code, word = recover_word(state, predecessors, catalogue)
+        require(len(word) == distances[state], "predecessor word length mismatch")
+        matrix_replay = (free_code,) * 3
+        for descriptor in word:
+            matrix_replay = multiply(literal_factor(descriptor), matrix_replay)
+        require(matrix_replay == state, "predecessor matrix replay failed")
+        require(word_agrees(state, free_code, word), "arbitrary-input word replay failed")
+        delta, ranks = delta_by_columns(state, free_codes)
+        require(delta <= distances[state], "delta lower bound violated")
+        records.append({"state": list(state), "components": [list(entries(code)) for code in state],
+                        "cost": distances[state], "delta": delta,
+                        "delta_ranks_in_free_order": ranks, "free_code": free_code,
+                        "word_in_execution_order": [list(d) for d in word]})
+
+    canonical_record = next(record for record in records if tuple(record["state"]) == CANONICAL)
+    require(canonical_record["cost"] == 3 and canonical_record["delta"] == 2,
+            "canonical target is not cost3/delta2")
+    canonical_word = tuple(tuple(d) for d in canonical_record["word_in_execution_order"])
+    controls = negative_controls(CANONICAL, canonical_record["free_code"],
+                                 canonical_word, free_codes)
+    canonical_bytes = json.dumps(records, sort_keys=True, separators=(",", ":")).encode("ascii")
+    joint_histogram = Counter((row["cost"], row["delta"]) for row in records)
+    return {
+        "schema_version": 1,
+        "claim_type": "complete_finite_invertible_target_metric",
+        "field": "F2", "ring": "F2^3", "roles": 2,
+        "scratch_roles": 0, "borrowed_helper_roles": 0,
+        "component_encoding": "row-major bit i is matrix entry i; triple order is ring component order",
+        "factor_encoding": "[u0,u1,y0,y1]; u0/u1 are three-bit ring values; y0/y1 are F2",
+        "word_convention": "apply free_code first, then left-apply listed factors in execution order",
+        "free_codes": list(free_codes),
+        "paid_descriptor_domain": 256,
+        "invertible_paid_descriptors": admitted_descriptors,
+        "distinct_invertible_paid_factors": len(factors),
+        "paid_factors": [{"state": list(f), "descriptor": list(catalogue[f])} for f in factors],
+        "target_count": len(states),
+        "cost_delta_histogram": [{"cost": cost, "delta": delta, "count": count}
+                                 for (cost, delta), count in sorted(joint_histogram.items())],
+        "records": records,
+        "records_sha256": hashlib.sha256(canonical_bytes).hexdigest(),
+        "canonical_target": canonical_record,
+        "checks": {"all_paid_edges": paid_edges, "all_free_edges": free_edges,
+                   "free_conjugations": conjugations, "attaining_matrix_words": len(records),
+                   "attaining_word_arbitrary_inputs": 64 * len(records),
+                   "canonical_arbitrary_inputs": 64, "delta_endpoints_per_target": len(free_codes)},
+        "controls": controls,
+        "scope": "Formal paid-factor alphabet only; no physical caller, supplier, precision, stock, or kappa claim.",
+    }
+
+
+if __name__ == "__main__":
+    require(len(sys.argv) == 1, "engine CLI takes no arguments")
+    print(json.dumps(run(), sort_keys=True, separators=(",", ":")))
diff --git a/research/right-vector-commutative-products/oracle.py b/research/right-vector-commutative-products/oracle.py
new file mode 100644
index 0000000000..bc8f946513
--- /dev/null
+++ b/research/right-vector-commutative-products/oracle.py
@@ -0,0 +1,153 @@
+#!/usr/bin/env python3
+"""Independent projective/word oracle; no BFS or predecessor code imports.
+
+ToikY with substantial OpenAI Codex assistance, Apache-2.0. The right-vector
+alphabet and predecessor question retain Nick Petalas/Wentao Li's credit.
+Finite checks corroborate PROOF.md; they do not prove its quantified claims.
+"""
+import itertools
+import json
+
+
+def require(value, message):
+    if not value:
+        raise ValueError(message)
+
+
+def prime(p):
+    return type(p) is int and p >= 2 and all(p % d for d in range(2, int(p**0.5) + 1))
+
+
+def multiply(a, b, p):
+    return ((a[0]*b[0]+a[1]*b[2]) % p, (a[0]*b[1]+a[1]*b[3]) % p,
+            (a[2]*b[0]+a[3]*b[2]) % p, (a[2]*b[1]+a[3]*b[3]) % p)
+
+
+def determinant(a, p):
+    return (a[0]*a[3]-a[1]*a[2]) % p
+
+
+def apply(a, t, p):
+    return ((a[0]*t[0]+a[1]*t[1]) % p, (a[2]*t[0]+a[3]*t[1]) % p)
+
+
+def lines(p):
+    require(prime(p), "the finite oracle implements prime fields only")
+    return [(1, x) for x in range(p)] + [(0, 1)]
+
+
+def zero_census(components, p):
+    return [sum((phi[0]*apply(m, t, p)[0]+phi[1]*apply(m, t, p)[1]) % p == 0
+                for m in components)
+            for phi in lines(p) for t in lines(p)]
+
+
+def triple_word(components, p):
+    """Common paid rows e1/e2/e1; compare literal products, not receipts."""
+    require(prime(p), "prime-field implementation boundary")
+    word = [[], [], []]
+    for m in components:
+        a, b, c, d = m
+        require(determinant(m, p) != 0, "invertible-target premise")
+        r = 0 if d else 1
+        pivot = (d-r*b) % p
+        require(pivot != 0, "nonunit third-factor pivot")
+        f = (c-r*a)*pow(pivot, -1, p) % p
+        e = determinant(m, p)*pow(pivot, -1, p) % p
+        factors = [(1, 0, r, 1), (1, b, 0, pivot), (e, 0, f, 1)]
+        require(multiply(multiply(factors[0], factors[1], p), factors[2], p) == tuple(m), "complete three-factor product")
+        for stage, factor in zip(word, factors):
+            stage.append(factor)
+    for stage, right in zip(word, [(1, 0), (0, 1), (1, 0)]):
+        for m in stage:
+            if right == (1, 0):
+                require(m[1] == 0 and m[3] == 1, "e1 factor shape")
+            else:
+                require(m[0] == 1 and m[2] == 0, "e2 factor shape")
+    return word
+
+
+def product(word, p):
+    result = [(1, 0, 0, 1)] * len(word[0])
+    for stage in word:
+        result = [multiply(a, b, p) for a, b in zip(result, stage)]
+    return result
+
+
+def classify(components, p):
+    if all(m == components[0] for m in components):
+        return 0, 0
+    # A single paid factor leaves a common K input line unchanged after
+    # the free endpoint. Equality here is literal vector equality.
+    one = any(len({apply(m, t, p) for m in components}) == 1 for t in lines(p))
+    if one:
+        return 1, 1
+    two = 0 in zero_census(components, p)
+    return (2 if two else 3), 2
+
+
+def run():
+    binary = [m for m in itertools.product(range(2), repeat=4) if determinant(m, 2)]
+    records = []
+    for components in itertools.product(binary, repeat=3):
+        cost, delta = classify(components, 2)
+        word = triple_word(components, 2)
+        require(product(word, 2) == list(components), "full product-field endpoint")
+        records.append(dict(components=[list(m) for m in components], cost=cost, delta=delta))
+
+    prime_rows = []
+    for p in [2, 3, 5, 7, 11, 13, 17, 19]:
+        irreducible = [(a, b) for a in range(p) for b in range(p)
+                       if all((x*x+a*x+b) % p for x in range(p))]
+        require(len(irreducible) == p*(p-1)//2, "quadratic counting argument")
+        a, b = irreducible[0]
+        alpha = (0, -b % p, 1, -a % p)
+        components = [(1, (-t*b) % p, t, (1-t*a) % p) for t in range(p)] + [alpha]
+        require(all(determinant(m, p) for m in components), "extension representative invertibility")
+        zeros = zero_census(components, p)
+        require(set(zeros) == {1}, "one forbidden component at every projective pair")
+        word = triple_word(components, p)
+        require(product(word, p) == components, "attaining three-factor witness")
+        prime_rows.append(dict(p=p,components=p+1,polynomial=[b,a,1],projective_pairs=(p+1)**2,zero_components_per_pair=1,word=[[[*m] for m in stage] for stage in word]))
+
+    # The predecessor's proposed F3^3 test cannot refute the invertible claim:
+    # check all possible nonzero first-column triples, not sampled matrices.
+    vectors = [v for v in itertools.product(range(3), repeat=2) if any(v)]
+    column_triples = 0
+    for images in itertools.product(vectors, repeat=3):
+        require(any(all((phi[0]*v[0]+phi[1]*v[1]) % 3 for v in images) for phi in lines(3)), "F3^3 common-row guarantee")
+        column_triples += 1
+
+    upper_cases = 0
+    for p in [2, 3, 5, 7]:
+        for matrix in itertools.product(range(p), repeat=4):
+            if determinant(matrix, p):
+                require(product(triple_word([matrix], p), p) == [matrix], "literal upper word")
+                upper_cases += 1
+
+    canonical = [(1,0,0,1),(0,1,1,1),(1,1,1,0)]
+    require(classify(canonical, 2) == (3,2), "canonical obstruction")
+    require(0 not in zero_census(canonical, 2), "all nine pivots obstructed")
+    require(classify(canonical[:2], 2)[0] <= 2, "removing a component removes this floor")
+    word = triple_word(canonical, 2)
+    require(product(word[:2], 2) != canonical, "dropped paid factor must fail")
+    require(product(list(reversed(word)), 2) != canonical, "reversed paid word must fail")
+    require(len(set(canonical)) > 1, "componentwise-free relaxation is outside the common-free model")
+    for bad in [4,6,9]:
+        try:
+            lines(bad)
+        except ValueError:
+            pass
+        else:
+            raise ValueError("composite-modulus arithmetic was mislabelled a field")
+    try:
+        triple_word([(1,0,0,0)],2)
+    except ValueError:
+        pass
+    else:
+        raise ValueError("singular target admitted outside the stated upper theorem")
+    return dict(records=records,prime_witnesses=prime_rows,first_column_triples_F3=column_triples,upper_word_matrices=upper_cases,canonical=dict(components=canonical,cost=3,delta=2,zero_census=zero_census(canonical,2),word=word),controls=dict(drop_component=True,drop_factor=True,reverse_factors=True,componentwise_free_is_changed=True,composite_moduli_rejected=[4,6,9],singular_outside_premise=True))
+
+
+if __name__ == "__main__":
+    print(json.dumps(run(),sort_keys=True))
diff --git a/research/right-vector-commutative-products/source/PROOF.md b/research/right-vector-commutative-products/source/PROOF.md
new file mode 100644
index 0000000000..71888afa0d
--- /dev/null
+++ b/research/right-vector-commutative-products/source/PROOF.md
@@ -0,0 +1,93 @@
+# Proofs
+
+`K` is a field and `R` an associative unital `K`-algebra with `K` central. A
+paid factor is `P(u, y) = I + u y^T` with `u` in `R^n` and `y` in `K^n`; a free
+map is any `S` in `GL_n(K)`, used anywhere in a word. `cost(M)` is the least
+number of paid factors in a word equal to `M` (infinite if none exists), and
+
+```text
+delta(M) = min over T in GL_n(K) of dim_K (M T - I) K^n.
+```
+
+Facts used from #4243 and #4099: `cost(M) >= delta(M)`, and every word can be
+rewritten as `Q S` with `Q` a product of the same number of paid factors,
+because `S P(u, y) S^{-1} = P(S u, S^{-T} y)`.
+
+## Lemma 1 (merging)
+
+`P(u, y) P(v, c y) = P(u + c v + c u (y^T v), y)` for `c` in `K`. Expand:
+`(I + u y^T)(I + c v y^T) = I + u y^T + c v y^T + c u (y^T v) y^T`, and
+`y^T v` is an element of `R`. Also `P(u, 0) = P(0, y) = I`.
+
+## Theorem A (one factor, every n, every R)
+
+`cost(M) <= 1` if and only if `delta(M) <= 1`, and then `cost(M) = delta(M)`.
+
+*Proof.* `cost >= delta` gives one direction. If `delta(M) = 0`, then
+`M T = I` and `M = T^{-1}` costs 0. If `delta(M) = 1`, then
+`(M T - I) K^n = K w` for some `w` in `R^n`, so `(M T - I) e_j = c_j w` with
+`c` in `K^n`, that is `M T = I + w c^T = P(w, c)`, and `M = P(w, c) T^{-1}`.
+
+## Theorem B (two factors, n = 2, every R)
+
+`cost(M) <= 2` if and only if some `G, H` in `GL_2(K)` give `N = G M H` with
+`N12` in the right ideal `N11 R`.
+
+*Proof.* (If.) Write `N12 = N11 C` and put `A = N11`, `B = N21`,
+`D = N22 - N21 C`. Then
+
+```text
+F1 = P((A - 1, B), e1) = [[A, 0], [B, 1]],
+F2 = P((C, D - 1), e2) = [[1, C], [0, D]],
+F1 F2 = [[A, A C], [B, B C + D]] = N,
+```
+
+so `M = (G^{-1} F1 G)(G^{-1} F2 G) G^{-1} H^{-1}`, and each conjugate is a
+paid factor.
+
+(Only if.) Write `M = P(u1, y1) P(u2, y2) S`, padding with `P(0, y) = I`. If
+`y1, y2` are dependent, Lemma 1 merges the two factors into one with a nonzero
+`y` (or both are `I`); replace the second by `P(0, y2')` with `y2'`
+independent. So `y1, y2` is a basis. Choose `G` with `G^{-T} y_i = e_i`. Then
+`G M S^{-1} G^{-1} = P(u1', e1) P(u2', e2)`, which is
+`[[A, 0], [B, 1]] [[1, C], [0, D]] = [[A, A C], [B, B C + D]]` for some
+`A, B, C, D` in `R`. Its `(1, 2)` entry is `A C`, in `A R`. Take
+`H = S^{-1} G^{-1}`.
+
+## Corollary C (matrix algebras)
+
+Let `R = M_m(K)` and view `M` as a `2m x 2m` matrix. Then `cost(M) <= 2` if
+and only if there are nonzero `phi` in `(K^2)^*` and `t` in `K^2` with
+
+```text
+rank (phi ⊗ I_m) M = rank (phi ⊗ I_m) M (t ⊗ I_m).
+```
+
+*Proof.* In `M_m(K)`, `Z` lies in `A M_m(K)` exactly when the column space of
+`Z` lies in that of `A`. Let `phi` be the first row of `G` and `t, s` the
+columns of `H`. Then `N11 = (phi ⊗ I) M (t ⊗ I)`,
+`N12 = (phi ⊗ I) M (s ⊗ I)`, and since `H` is invertible the column space of
+`[N11 N12]` is that of `(phi ⊗ I) M`. So `col N12 ⊆ col N11` is the stated
+rank equality. Every nonzero `phi` is a first row and every nonzero `t` a first
+column.
+
+## Theorem D (local algebras, n = 2)
+
+Let the non-units of `R` form an ideal `J`. Every target `M` with an entry
+outside `J` (in particular every invertible `M`) has `cost(M) = delta(M)`.
+
+*Proof.* Permutation matrices `G, H` move a unit to position `(1, 1)`. Then
+`N11 R = R`, so Theorem B gives `cost(M) <= 2`. If `delta(M) <= 1`, Theorem A
+gives equality. Otherwise `2 = delta(M) <= cost(M) <= 2`.
+
+## Corollary E (quotients and F2[S3])
+
+If `rho: R -> R'` is a unital `K`-algebra homomorphism, then
+`cost_R(M) >= cost_R'(rho(M))`, because `rho` maps paid factors to paid factors
+and free maps to free maps. Since `delta(M) <= n` always, every target over
+`F2[S3]` whose image under the surjection `rho: F2[S3] -> M_2(F_2)` is one of
+the 72 cost-3 targets of #4265 costs at least 3 while `delta <= 2`. `rho`
+extends the isomorphism `S3 = GL_2(F_2)` given by the action on the three
+nonzero vectors of `F_2^2`; `verify.py` checks that it is a surjective unital
+algebra map and lifts a three-factor word to an explicit target of cost
+exactly 3.
diff --git a/research/right-vector-commutative-products/source/README.md b/research/right-vector-commutative-products/source/README.md
new file mode 100644
index 0000000000..8952fc9ae8
--- /dev/null
+++ b/research/right-vector-commutative-products/source/README.md
@@ -0,0 +1,124 @@
+# Exact two-factor criterion for right-vector targets
+
+**Scoped proofs with finite certificates. No new κ.** Selected κ stays
+`0.001672660197076918`.
+
+Paid factors are `I + u y^T` with `u` in `R^n` and `y` in `K^n`; free maps are
+`GL_n(K)` changes of basis anywhere in a word (#4099, #4243). #4243 proved
+`cost(M) >= delta(M) = min_T dim_K (M T - I) K^n` and equality on diagonal
+targets. #4265 showed equality fails over `M2(F2)` at `n = 2` (72 targets cost
+3 against `delta = 2`) and asked what the exact cost is over noncommutative
+algebras, whether the failure lifts to `F2[S3]`, and whether `delta` is exact
+over commutative or local algebras.
+
+## Result
+
+Proofs are in [PROOF.md](PROOF.md).
+
+1. **One factor, every `n` and every algebra:** `cost(M) <= 1` exactly when
+   `delta(M) <= 1`, and then the two are equal.
+2. **Two factors at `n = 2`, every algebra:** `cost(M) <= 2` exactly when some
+   `G, H` in `GL_2(K)` give `N = G M H` with `N12` in the right ideal `N11 R`.
+   With item 1, this decides `cost(M)` in {0, 1, 2, ≥3} for every target.
+3. **Matrix algebras.** Over `R = M_m(K)` the condition reads: some nonzero
+   `phi`, `t` in `K^2` give
+   `rank (phi ⊗ I_m) M = rank (phi ⊗ I_m) M (t ⊗ I_m)`. This answers #4265's
+   question 3 at `n = 2`.
+4. **Local algebras.** If the non-units of `R` form an ideal `J`, every target
+   with an entry outside `J`, in particular every invertible target, costs
+   exactly `delta(M)`. This answers #4265's question 1 at `n = 2` for local
+   algebras and invertible targets. It cannot be extended to every target:
+   over the commutative local algebra `F2[x,y]/(x,y)^2`, 12 targets in
+   `M_2(J)` are unreachable while `delta <= 2`, for example
+   `[[x + y, y], [y, x]]`.
+5. **`F2[S3]`.** The failure lifts. A unital surjection
+   `F2[S3] -> M_2(F_2)` maps words to words, so every lift of a #4265
+   failure costs at least 3 while `delta <= 2`. `verify.py` builds one target
+   of cost exactly 3. Over all `2^24` targets at `n = 2`, exactly **120,060**
+   fail the criterion, so they cost at least 3 against `delta <= 2`. Of these,
+   18,432 = 72 · 4^4 are lifts of `M2(F2)` failures. The rest fail because one
+   pair `(G, H)` must work in both blocks.
+6. **A smaller noncommutative failure.** Over the 3-dimensional algebra
+   `UT2(F2)` of upper triangular matrices, 12 of 4,096 targets cost 3 with
+   `delta = 2`.
+
+| Algebra `R` | Targets (n = 2) | Cost histogram 0/1/2/3 | Unreachable | Criterion failures |
+|---|---:|---|---:|---:|
+| `F2[x,y]/(x,y)^2` (local) | 4,096 | 6 / 549 / 3,529 / 0 | 12 | 12 |
+| `F2 x F2` | 256 | 6 / 117 / 133 / 0 | 0 | 0 |
+| `F2[C3]` | 4,096 | 6 / 549 / 3,541 / 0 | 0 | 0 |
+| `UT2(F2)` | 4,096 | 6 / 549 / 3,529 / 12 | 0 | 12 |
+| `M2(F2)` | 65,536 | 6 / 2,277 / 63,181 / 72 | 0 | 72 |
+| `F2[S3]` (criterion only) | 16,777,216 | — | — | 120,060 |
+
+In every exhaustively searched algebra, the criterion holds exactly on the
+targets of cost at most 2, and every criterion failure costs 3 or is
+unreachable. The 72 `M2(F2)` failures agree with #4265.
+
+Claim types: items 1–4 and the lift in item 5 are proofs for every algebra in
+scope. The counts in the table and items 5–6 are finite certificates. Cost 3 as
+an upper bound is only checked by search on the listed algebras; no general
+three-factor theorem is claimed.
+
+## Method
+
+`verify.py` builds each algebra from a structure table and checks that it is
+associative with `K` central. It precomputes right-ideal membership `z ∈ a R`
+by `F_p`-rank. Stage `main` computes the exact minimum on every target by
+breadth-first search over words `Q S`, computes `delta` over every `T`, and
+asserts: the criterion equals `cost <= 2`, Theorem A, the local statement on
+`F2[x,y]/(x,y)^2` (non-units checked to form an ideal), the rank form over
+`M2(F2)`, and the expected counts. It also checks that `rho` is a surjective
+unital algebra map, lifts a three-factor `M2(F2)` word to `F2[S3]`, and
+asserts that the lifted target fails the criterion. Stage `s3` evaluates the
+criterion on all `2^24` targets over `F2[S3]`. The script writes nothing.
+
+## Reproduce
+
+From the repository root:
+
+```sh
+~/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/right-vector-two-factor-criterion/verify.py --stage main
+~/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/right-vector-two-factor-criterion/verify.py --stage s3
+~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/right-vector-two-factor-criterion
+```
+
+Expected final lines:
+
+```text
+PASS_RIGHT_VECTOR_TWO_FACTOR_CRITERION F2[x,y]/(x,y)^2:4096:12 F2xF2:256:0 F2[C3]:4096:0 UT2(F2):4096:12 M2(F2):65536:72 m2_rank_form=ok f2s3_lift=cost3 no_new_kappa
+PASS_RIGHT_VECTOR_TWO_FACTOR_S3 targets=16777216 three_or_more=120060 no_new_kappa
+```
+
+Each stage takes under 40 seconds on one core (CPython 3.12, NumPy 2.5.4,
+macOS arm64). The asserts are the checks, so do not run with `-O`.
+
+## Limits
+
+This is a formal alphabet. Native frames, hosting, routing, precision and
+recursive cost are not priced, and no supplier changes. Theorems B–D are for
+`n = 2` only. Whether every reachable target at `n = 2` costs at most 3 is
+not proved. The `F2[S3]` count gives cost at least 3; exact costs there are not
+searched. Independent human and formal proof-kernel review are unclaimed.
+
+## Open next questions
+
+1. **Three factors.** Is every reachable target at `n = 2` reachable with
+   three factors, over every algebra? Characterise the three-factor set as
+   Theorem B does for two; with it, the cost at `n = 2` would be exact.
+2. **Larger `n`.** With `y_1, ..., y_k` a basis, `k` factors are block column
+   operations. Write the `n = 3`, `k = 2` condition as a flag of right
+   ideals and test it by search over `F2[C2]` or `UT2(F2)`.
+3. **Commutative non-local algebras.** Is `delta` exact for invertible
+   targets over every commutative algebra at `n = 2`? A product of local
+   algebras needs one `(G, H)` that works in every factor. Count criterion
+   failures over `F3 x F3 x F3` first.
+
+## Credit
+
+Produced by Nick Petalas. The alphabet, normal form and lower bound come from
+#4243 (`research/right-vector-diagonal-free-basis`); the open questions and the
+`M2(F2)` counts come from #4265 (`research/right-vector-general-target`). The
+column bound and right-vector alphabet are Wentao Li's
+`research/f2-right-vector-diagonal-minimum` (#4099), building on
+`research/bit-cyclic-root-barrier` (#2513). Apache-2.0.
diff --git a/research/right-vector-commutative-products/verify.py b/research/right-vector-commutative-products/verify.py
new file mode 100644
index 0000000000..b575ed4247
--- /dev/null
+++ b/research/right-vector-commutative-products/verify.py
@@ -0,0 +1,74 @@
+#!/usr/bin/env python3
+"""Fresh full BFS and separate projective/word replay, with exact typed data."""
+import hashlib
+import importlib.util
+import json
+from pathlib import Path
+import sys
+
+HERE = Path(__file__).resolve().parent
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def canonical(value):
+    return json.dumps(value,sort_keys=True,separators=(",",":"),allow_nan=False)
+
+
+def load(name, file):
+    spec = importlib.util.spec_from_file_location(name,HERE/file)
+    module = importlib.util.module_from_spec(spec)
+    sys.modules[name] = module
+    spec.loader.exec_module(module)
+    return module
+
+
+def compare(actual, expected):
+    require(canonical(actual) == canonical(expected), "full typed regenerated payload differs")
+
+
+def main():
+    pins = json.loads((HERE/"SOURCE-PINS.json").read_text())
+    for row in pins["files"]:
+        raw = (HERE/row["path"]).read_bytes()
+        require(len(raw)==row["bytes"],"source size")
+        require(hashlib.sha256(raw).hexdigest()==row["sha256"],"source SHA256")
+        require(hashlib.sha1(b"blob "+str(len(raw)).encode()+b"\0"+raw).hexdigest()==row["git_blob"],"source Git blob")
+    bfs = load("paid_bfs","engine.py").run()
+    oracle = load("projective_oracle","oracle.py").run()
+    compare(bfs,json.loads((HERE/"EXPECTED-BFS.json").read_text()))
+    compare(oracle,json.loads((HERE/"EXPECTED-ORACLE.json").read_text()))
+    key = lambda row: canonical(row["components"])
+    bm = {key(row):(row["cost"],row["delta"]) for row in bfs["records"]}
+    om = {key(row):(row["cost"],row["delta"]) for row in oracle["records"]}
+    require(len(bm)==len(om)==216 and bm==om,"complete cost/delta cross-engine agreement")
+    # These mutations exercise evidence scope/type boundaries rather than
+    # asserting that a checksum alone is a mathematical certificate.
+    for changed in [dict(bfs,target_count=215),dict(bfs,scratch_roles=False),dict(bfs,records=bfs["records"][:-1])]:
+        try:
+            compare(changed,bfs)
+        except ValueError:
+            pass
+        else:
+            raise ValueError("typed evidence corruption accepted")
+    payload = dict(status="PASS_RIGHT_VECTOR_COMMUTATIVE_PRODUCTS",targets=216,
+                   cost_delta_histogram=bfs["cost_delta_histogram"],
+                   source_inputs=len(pins["files"]),bfs_checks=bfs["checks"],
+                   bfs_controls=bfs["controls"],oracle_controls=oracle["controls"],
+                   bfs_full_payload_sha256=hashlib.sha256(canonical(bfs).encode()).hexdigest(),
+                   oracle_full_payload_sha256=hashlib.sha256(canonical(oracle).encode()).hexdigest(),
+                   prime_fields=[r["p"] for r in oracle["prime_witnesses"]],
+                   projective_pairs=sum(r["projective_pairs"] for r in oracle["prime_witnesses"]),
+                   first_column_triples_F3=oracle["first_column_triples_F3"],
+                   literal_upper_word_matrices=oracle["upper_word_matrices"],
+                   canonical_cost=3,canonical_delta=2,typed_evidence_controls=3,
+                   finite_scope="complete invertible F2^3 metric plus listed prime-field witnesses; quantified proof is separate",
+                   physical_costs="unknown",new_kappa=None)
+    print(canonical(payload))
+
+
+if __name__ == "__main__":
+    main()