diff --git a/research/bit-two-relation-any-registers/CERTIFICATE.json b/research/bit-two-relation-any-registers/CERTIFICATE.json
new file mode 100644
index 0000000000..50527cb795
--- /dev/null
+++ b/research/bit-two-relation-any-registers/CERTIFICATE.json
@@ -0,0 +1 @@
+{"schema_version":1,"encoding":"matrix bit 4*i+j is entry (i,j); subspace bit v marks binary vector 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72,16400,29440,9],[13072,16448,29520,9],[13072,16452,29524,9],[13072,16454,29526,9],[13072,16576,29648,9],[13072,16588,29660,9],[13072,17168,28672,9],[13072,17174,28678,9],[13072,17216,28752,9],[13072,17358,28894,9],[13072,18462,31502,9],[13072,18624,31696,9],[13072,20480,25360,129],[13072,20484,25364,129],[13072,20564,25412,129],[13072,20688,25536,129],[13072,20736,25104,129],[13072,20752,25088,129],[13072,20758,25094,129],[13072,20816,25152,129],[13072,21264,24576,129],[13072,21278,24590,129],[13072,21328,24640,129],[13072,21334,24646,129],[13072,22796,27164,129],[13072,22992,27328,129],[13072,23004,27340,129],[13072,23006,27342,129],[13074,16452,29526,9],[13074,23004,27342,129],[13078,16448,29526,9],[13084,23004,27328,129],[13120,16448,29440,9],[13124,16452,29440,9],[13136,20816,25088,129],[13138,16388,29526,9],[13142,16384,29526,9],[13142,20816,25094,129],[13146,16396,29526,9],[13206,16576,29526,9],[13210,16588,29526,9],[13248,16576,29440,9],[13250,22796,27342,129],[13260,16588,29440,9],[13260,22796,27328,129],[14594,22796,24590,129],[14604,22796,24576,129],[14666,22796,24646,129],[14668,22796,24640,129],[14746,23004,24646,129],[14748,23004,24640,129],[14802,23004,24590,129],[14812,23004,24576,129],[14864,20688,27328,129],[14878,20688,27342,129],[14996,20564,27328,129],[15002,20564,27342,129],[15040,20480,27328,129],[15044,20484,27328,129],[15050,20484,27342,129],[15054,20480,27342,129],[15104,16384,31488,9],[15104,16388,31492,9],[15104,16396,31500,9],[15104,16398,31502,9],[15104,16448,31552,9],[15104,16452,31556,9],[15104,16576,31680,9],[15104,16588,31692,9],[15104,16592,31696,9],[15104,18432,29440,9],[15104,18518,29526,9],[15104,19200,28672,9],[15104,19206,28678,9],[15104,19280,28752,9],[15104,19422,28894,9],[15104,20480,27392,129],[15104,20484,27396,129],[15104,20564,27476,129],[15104,20688,27600,129],[15104,20736,27136,129],[15104,20816,27216,129],[15104,20928,27328,129],[15104,20942,27342,129],[15104,22784,25088,129],[15104,22790,25094,129],[15104,22796,25100,129],[15104,23004,25308,129],[15104,23296,24576,129],[15104,23310,24590,129],[15104,23360,24640,129],[15104,23366,24646,129],[15106,16396,31502,9],[15114,16388,31502,9],[15114,22796,25094,129],[15116,22796,25088,129],[15118,16384,31502,9],[15120,16576,31696,9],[15132,16588,31696,9],[15178,16452,31502,9],[15182,16448,31502,9],[15248,16448,31696,9],[15248,20816,27328,129],[15252,16452,31696,9],[15258,16384,31642,9],[15258,16388,31646,9],[15258,16396,31638,9],[15258,16448,31706,9],[15258,16452,31710,9],[15258,16458,31696,9],[15258,16532,31502,9],[15258,16576,31578,9],[15258,16588,31574,9],[15258,18586,29440,9],[15258,18636,29526,9],[15258,19268,28894,9],[15258,19354,28672,9],[15258,19356,28678,9],[15258,19402,28752,9],[15258,20480,27546,129],[15258,20484,27550,129],[15258,20564,27598,129],[15258,20688,27466,129],[15258,20736,27290,129],[15258,20816,27338,129],[15258,20820,27342,129],[15258,20826,27328,129],[15258,22796,25238,129],[15258,22938,25088,129],[15258,22940,25094,129],[15258,23004,25158,129],[15258,23444,24590,129],[15258,23450,24576,129],[15258,23514,24640,129],[15258,23516,24646,129],[15262,20816,27342,129],[15296,20736,27328,129],[15298,16588,31502,9],[15310,16576,31502,9],[15310,20736,27342,129],[15312,16384,31696,9],[15316,16388,31696,9],[15322,23004,25094,129],[15324,16396,31696,9],[15324,23004,25088,129]]}
diff --git a/research/bit-two-relation-any-registers/PROOF.md b/research/bit-two-relation-any-registers/PROOF.md
new file mode 100644
index 0000000000..8dca10ed1d
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+++ b/research/bit-two-relation-any-registers/PROOF.md
@@ -0,0 +1,418 @@
+# Two exponent relations cannot amortize rank at any scalar width
+
+Prepared for **@CubeJerry** with substantial OpenAI Codex assistance.
+New work is Apache-2.0. This is a written reduction to a named inherited
+three-register theorem, not a proof-assistant formalization. The finite
+controls corroborate the new lemmas; they do not establish unbounded
+quantifiers by enumeration.
+
+## 1. Main theorem and exact inherited ingredients
+
+Let `H` be a torsion-free abelian group generated by finitely many
+exponents `A_1,...,A_k`. Let `L` be the kernel of `Z^k -> H` sending
+basis vector `j` to `A_j`. Write `[A][B]=[A+B]` in `R=F_2[H]`.
+A scalar call on any finite number `W` of registers is
+
+\[
+D_j=I+([A_j]-1)N_j,\qquad N_j=x_jy_j^T,\quad
+x_j,y_j\in\mathbb F_2^W,\quad y_j^Tx_j=1.
+\]
+
+**Theorem.** If `rank L <= 2` and the exact whole-matrix endpoint is
+
+\[
+\prod_{j=1}^kD_j=\operatorname{diag}([C_1],\ldots,[C_W]),          \tag{1}
+\]
+
+then there are pairwise disjoint subsets `S_1,...,S_W` of the call
+indices such that
+
+\[
+C_i=\sum_{j\in S_i}A_j.                                         \tag{2}
+\]
+
+The theorem allows arbitrary width and word length, repeated targets,
+identity targets, and arbitrary diagonal monomial helper outputs.
+Neither a frame-rank hypothesis, target invertibility, nor a fixed
+address dimension is needed. For rational matrix exponents it implies
+
+\[
+\sum_i\operatorname{rank}_{\mathbb Q}C_i
+\le\sum_j\operatorname{rank}_{\mathbb Q}A_j.                     \tag{3}
+\]
+
+The scalar field is binary; rational address entries are never reduced
+modulo two. Constant binary basis changes below are proof changes of
+coordinates, not interposed free address transformations or a physical
+implementation invoice.
+
+We use these precisely scoped inherited ingredients:
+
+* **Three-register disjoint subsets:** David Govea's CrocSwap/integer-mult-bounds#720, README
+  Sections 1--4, proves (2) at `W<=3`, for any torsion-free exponent
+  group and relation rank at most two. It permits identity and repeated
+  targets and arbitrary address ranks. Pad with identity coordinates
+  when `W<3`. We use this universal lemma, not CrocSwap/integer-mult-bounds#720's later finite
+  address-dimension-three word search.
+* **Off-diagonal fibers:** Nick Petalas's CrocSwap/integer-mult-bounds#5921 proves that a realized
+  nonzero off-diagonal pattern has exactly one binary rank-one
+  idempotent lift, except a single edge `E_ab`, whose two lifts are
+  `E_aa+E_ab` and `E_bb+E_ab`.
+* The cube-relation reduction below follows CrocSwap/integer-mult-bounds#720. The common-pattern
+  cyclic/acyclic organization follows woklerx's CrocSwap/integer-mult-bounds#1456. Their arguments
+  are recalled to make the new composition checkable.
+
+woklerx's CrocSwap/integer-mult-bounds#1846 already classifies all 4,964 four-register pattern
+planes as reducible. We independently replay that finite result, but
+the analytic scalar classification in Section 4 makes it unnecessary
+as a proof dependency of this theorem.
+
+## 2. Saturation, augmentation and the free-coordinate split
+
+The generated group `H` is finitely generated and torsion-free, hence
+free abelian. The sequence `0 -> L -> Z^k -> H -> 0` splits, so reduction
+modulo two identifies the kernel of `F_2^k -> H/2H` with `L/2L`.
+Expand the universal product `prod_j(I+(t_j-1)N_j)` at `t=1`.
+Its first-order off-diagonal coefficient vectors belong to `L/2L`
+by (1). Consequently the span of the occurring off-diagonal patterns
+has dimension at most two.
+
+Only subset sums of frames occur in the universal product. A difference
+of subset indicators is a *cube relation*, whose entries belong to
+`{-1,0,1}`. If these relations span at most one rational dimension, let
+`L_0` be their saturated span in `Z^k`. Then `L_0` is contained in `L`
+and has rank at most one. Two subset sums coincide modulo `L` exactly
+when they coincide modulo `L_0`, because their difference is itself a
+cube relation. Replacing `H` by the torsion-free lift `Z^k/L_0` preserves
+every coefficient fiber of the product and its diagonal monomial
+endpoint. It suffices to prove (2) in the lift and map back. Its
+first-order pattern span is at most one, handled in Section 3.
+
+Otherwise choose independent cube relations `r,s`. If their reductions
+modulo two are independent, an odd `2 x 2` minor is `+1` or `-1`: cube
+entries bound its absolute value by two. Thus they generate a saturated
+rank-two lattice and form a basis of `L`. If their reductions coincide,
+replace them by `(r+s)/2,(r-s)/2`. These are integral cube vectors
+with disjoint nonempty supports and a unit minor, again a basis of `L`.
+For this cube-valued basis there are binary off-diagonal matrices `B,C`
+with
+
+\[
+\operatorname{offdiag}(N_j)
+=(r_j\bmod2)B+(s_j\bmod2)C.                                    \tag{4}
+\]
+
+Pattern span at most one is covered by Section 3. If its dimension is
+two, `B,C` are independent. Let `S` index noncoordinate projectors and
+`T` coordinate projectors. For `j in T`, (4) forces `r_j=s_j=0`
+modulo two, hence integrally because they are cube-valued. Every
+relation is supported on `S`, and
+
+\[
+H\simeq (\mathbb Z^S/L)\oplus\mathbb Z^T=H_S\oplus\mathbb Z^T.   \tag{5}
+\]
+
+Thus the coordinate calls have independent free exponent coordinates.
+Projecting onto `H_S` removes all of them and preserves a diagonal
+endpoint. Section 5 will return them to their original target labels.
+This exact lattice split is essential; arbitrary coordinate gates
+cannot be added to a noncoordinate pool's invariant-subspace claim.
+
+## 3. A common pattern at arbitrary width and at most two relations
+
+If all patterns vanish, every projector is a coordinate projector and
+(2) is immediate. If the common nonzero pattern is acyclic, a common
+coordinate order makes every projector, including coordinate gates,
+triangular. A rank-one idempotent then has exactly one nonzero diagonal
+entry. Product diagonals give disjoint subsets.
+
+In the cyclic case the pattern uniquely determines a projector `N=xy^T`,
+as in CrocSwap/integer-mult-bounds#1456 or the fiber lemma. Let `I=supp(x) intersect supp(y)`.
+All directed cycles are in `I`, where the graph is complete off the
+diagonal. Its size `s` is odd by idempotence, and cyclicity forces
+`s>=3`. Order coordinates in `supp(x)\supp(y)`, then `I`, then
+`supp(y)\supp(x)`; all other coordinates are isolated. The common
+projector and every coordinate projector become block upper triangular.
+The only nonzero diagonal block of `N` is the all-ones idempotent `J_s`.
+Outside it, singleton targets are sums of their own coordinate-call
+frames. The core subword still has relation rank at most two: its
+relations embed into the original lattice by zero extension.
+
+It remains to generalize the common-J argument of CrocSwap/integer-mult-bounds#720 Section 3.
+
+**Common-J lemma.** For every odd `s>=3`, a word with projectors in
+`{J_s,E_11,...,E_ss}`, relation rank at most two and diagonal monomial
+endpoint has the same endpoint after deleting all J factors.
+
+Write augmentation variables `u_j=t_j-1`, and let `L_J` be their sum
+over J calls. If none occurs there is nothing to prove. Otherwise
+`L_J` is a nonzero universal linear form and first-order off-diagonal
+vanishing puts it in the at-most-two-dimensional linear relation space.
+For coordinate `i` set
+
+\[
+K_i=\operatorname{offdiag}(J_sE_{ii}+E_{ii}J_s),\qquad
+q_i=\sum_{j:N_j=E_{ii}}u_j\sum_{a<j:N_a=J_s}u_a.
+\]
+
+The stars `K_i` have the sole relation `sum_i K_i=0`, even modulo
+`span((J_s)_off)`. Indeed `sum_i a_i K_i=lambda (J_s)_off` says
+`a_i+a_j=lambda` for all distinct `i,j`. A triangle gives `lambda=0`,
+then all `a_i` are equal.
+
+In this matrix-space quotient the universal first-order term vanishes
+**identically before exponent substitution**. Therefore nonlinear
+terms of the Laurent substitution cannot repair the quadratic term.
+Precisely, the associated graded algebra at augmentation is
+`Sym(H/2H)`; the degree-two term of an expression with universally zero
+constant and linear terms is mapped by the linear substitution alone.
+Mixed J/coordinate pairs reduce, modulo `L_J`, to `sum_i K_i q_i`.
+J/J and coordinate/coordinate pairs vanish in the quotient. After
+eliminating `L_J`, the quadratic conditions are
+
+\[
+q_i+q_s=0\quad(1\le i<s)                                       \tag{6}
+\]
+
+modulo at most one further linear relation `M`.
+
+Eliminate the last of the `h` J variables by `L_J=0`. The `h-1`
+proper J prefix sums are linearly independent. Each nonzero quadratic
+in (6) is bilinear between J and coordinate variables and divisible
+by the same possible `M`. A nonzero bilinear quadratic with a linear
+divisor has coefficient-matrix rank one: in its factorization into
+linear forms, the bidegrees `(2,0)` and `(0,2)` force one factor to use
+only J variables and the other only coordinate variables. Its nonzero
+columns are proper J prefixes, so all must be the same prefix.
+Thus each nonzero quadratic puts its internal coordinate calls in
+one gap between J calls.
+
+All these gaps agree. If `q_s!=0`, its coordinate columns occur in
+every equation (6), forcing the same gap. If `q_s=0`, nonzero `q_i`
+have disjoint coordinate-variable supports. Two cannot share a nonzero
+linear divisor in those variables, so their common divisor is the same
+J prefix. With only one nonzero quadratic its own rank-one condition
+suffices. If all vanish identically after eliminating `L_J` and before
+imposing `M`, disjoint variable supports imply every `q_i=0`, so there
+are no internal coordinate calls. This includes `h=1`.
+
+Strip exterior invertible diagonal factors and combine consecutive J
+factors. The remaining word is
+
+\[
+U(b)D U(c),\quad U(z)=I+(z-1)J_s,\quad D=\operatorname{diag}(d_1,\ldots,d_s).
+\]
+
+Every letter is a group monomial. Its off-diagonal entry `i,j` is
+
+\[
+(b-1)d_j+(c-1)d_i+(b-1)(c-1)\sum_l d_l.                          \tag{7}
+\]
+
+If `b=1`, an off-diagonal zero forces `c=1`, and conversely. Otherwise
+compare columns `j,k` in a row distinct from both, possible since
+`s>=3`. The Laurent domain gives `d_j=d_k`, so `D=dI`. Oddness of `s`
+makes `J_s` idempotent, and the word is `d U(bc)`. Diagonality forces
+`bc=1`, leaving exactly `D`. Restoring exterior factors proves the
+common-J lemma. Together with the triangular tails this establishes
+(2) whenever the pattern span has dimension at most one, at any width.
+
+## 4. Joint-image classification for every two-pattern pool
+
+Let `P` be a two-dimensional space of off-diagonal binary matrices in
+`M_W(F_2)`. Its *noncoordinate pool* contains every rank-one idempotent
+whose nonzero pattern lies in `P`.
+
+**Joint-image lemma.** Either the pool's joint image has dimension at
+most **three**, or the pool consists of exactly the four lifts of two
+single edges on four distinct coordinates. That exceptional pool is
+coordinate triangular and has joint-image dimension four.
+
+We use the fiber lemma of CrocSwap/integer-mult-bounds#5921. A single edge `E_ab` has exactly the
+lifts
+
+\[
+E_{aa}+E_{ab}=e_a(e_a+e_b)^T,\qquad
+E_{bb}+E_{ab}=(e_a+e_b)e_b^T,                                  \tag{8}
+\]
+
+whose images span `e_a,e_b`. Every other realized nonzero pattern has
+one lift. For completeness, two distinct lifts differ by a nonzero
+diagonal matrix of rank at most two and trace zero, hence supported
+on exactly two diagonal positions. Both their image and covector
+spaces lie in those positions; the six binary `2 x 2` rank-one
+idempotents give exactly (8). This fiber theorem is inherited, not
+claimed as new.
+
+There are at most three realized nonzero patterns. If all three are
+realized and none is a single edge, only three projectors occur, so
+the image bound follows. Suppose exactly one is a single edge
+`p=E_ab`, with the other two patterns `q,r=q+p` represented uniquely
+by `M=xy^T`, `N=uv^T`. Then
+
+\[
+K=M+N=E_{ab}+D,\quad D\text{ diagonal},\quad
+\operatorname{rank}K\le2,\quad\operatorname{tr}D=0.              \tag{9}
+\]
+
+The diagonal support of `D` has size at most three since
+`rank D <= rank K+1`. It has even size by trace zero. Therefore
+`D=0` or `D=E_cc+E_dd` for distinct `c,d`.
+
+If `D=0`, either `x,u` agree or they are independent. In the latter
+case the rank-one sum `K` forces `y=v`; then `(x+u)y^T=E_ab` gives
+`x+u=e_a`. In either case adding the edge-lift image span
+`span(e_a,e_b)` to `im M+im N` gives dimension at most three.
+
+If `D=E_cc+E_dd`, the sets `{c,d}` and `{a,b}` must intersect.
+Otherwise columns `b,c,d` of `K` are the three independent vectors
+`e_a,e_c,e_d`. In every allowed case `rank K=2`: the principal minor
+on `c,d` is an identity matrix possibly with one off-diagonal entry,
+so has determinant one. Thus `im M+im N=im K`. Adding `e_a,e_b`
+is contained in the coordinate union `{a,b,c,d}`, now of size at
+most three.
+
+If all three patterns are realized and two are single edges, their
+sum has exactly two off-diagonal entries. A rank-one idempotent
+realizing that sum has support intersection of size one: a cyclic
+intersection has odd size at least three and at least six edges.
+For the two disjoint tails' sizes `a_0,b_0`, the edge count is
+`a_0+b_0+a_0 b_0=2`, forcing `(a_0,b_0)=(2,0)` or `(0,2)`.
+The two edges share a row or a column and use three coordinates.
+All five possible lifts are supported there, so again the joint
+image has dimension at most three. A two-dimensional space cannot
+contain three distinct matrix units, since they are linearly
+independent.
+
+If fewer than three patterns are realized, there are at most three
+projectors unless both realized patterns are single edges. Their four
+lifts have joint image equal to the coordinate span of their endpoints.
+This is at most three-dimensional unless all four endpoints differ.
+In that exceptional case the edges are disjoint, and a coordinate
+order making both edges upper triangular triangularizes the entire
+pool. This proves the lemma in every ambient dimension.
+
+The sharp example is `P=span(E_01,E_23)`. Its four lifts have independent
+joint image; the third pattern is unrealized. The joint-image maximum
+is therefore four, improving the cardinality-derived image bound five
+in CrocSwap/integer-mult-bounds#5921. That predecessor's sharp pool-cardinality maximum five is
+unchanged. The new statement identifies precisely every image-dimension-
+four case, rather than merely checking fixed small register counts.
+
+## 5. Restrict the image and reconstruct every original target
+
+In the remaining pattern-span-two case use the split (5), and project
+away all coordinate calls `T`. The remaining endpoint is diagonal,
+possibly with additional equal or zero target exponents. Apply the
+joint-image lemma to its complete pattern pool.
+
+For the exceptional disjoint-edge pool, coordinate triangularity
+assigns disjoint frame subsets directly. Otherwise let `V` be its
+joint image, of dimension at most three. Every projector preserves
+`V` and induces zero on the quotient. The endpoint also preserves
+`V tensor R`.
+
+The invariant subspace need not be spanned by original coordinate
+vectors. For each distinct projected target exponent `h`, let `E_h`
+be the span of original coordinate vectors with that value, and
+`pi_h` the coordinate projection. For `v in V`,
+
+\[
+Dv=\sum_h[h]\pi_hv\in V\otimes R.
+\]
+
+Mapping to `(F_2^W/V) tensor R` and using linear independence of
+monomials gives `pi_h v in V` for every `h`. Consequently
+
+\[
+V=\bigoplus_h(V\cap E_h).                                      \tag{10}
+\]
+
+Choose a binary basis within each `E_h` adapted to this intersection,
+and put the basis vectors of `V` first. The endpoint stays diagonal
+with its original monomial multiset, while all projectors are block
+upper triangular. Each restricted projector on `V` is rank one and
+idempotent: its nonzero image vector lies in `V` and is fixed by it.
+All quotient projectors vanish, so the quotient endpoint is identity.
+On each complementary equal-value block that endpoint is multiplication
+by `[h]`; hence every quotient label has `h=0`, since distinct group
+elements are distinct basis monomials.
+
+The relation lattice of the restricted word is unchanged; more generally,
+removing identity factors embeds its relation lattice by zero extension
+and cannot increase its rank above two. Apply the inherited
+three-register disjoint-subset lemma to `V`, padding with identity
+coordinates if needed. Match its target monomials, with multiplicity,
+to original projected target labels. The quotient labels all have
+zero exponent and receive empty subsets. Repeated target values pose
+no problem: choose any matching inside each equal-value class.
+
+Finally project (1) onto a free generator `t in T`. Its only surviving
+call is its original coordinate projector `E_ii`; thus its exponent
+belongs to exactly original target `i`. Append that index to target
+`i`'s assigned subset. The direct sum (5) makes the reconstructed
+exponent equal to the full original target. The assignments remain
+pairwise disjoint. This completes the theorem for every `W`.
+
+## 6. Rank consequence and historical boundary
+
+Apply rank subadditivity to (2) to obtain (3). With rational rank-one
+paid frames, any saving below total target rank therefore requires
+**at least three independent rational frame relations**, regardless
+of register count, address dimension or diagonal helper outputs.
+
+For eight calls and three invertible rational `3 x 3` data targets,
+frame span at least six gives relation rank at most two but total
+target rank at least nine. Such a word is impossible even with
+arbitrarily many dirty auxiliary inputs and arbitrary diagonal
+monomial auxiliary outputs.
+
+The four-register span-six case was already covered by CrocSwap/integer-mult-bounds#4119, CrocSwap/integer-mult-bounds#5075,
+CrocSwap/integer-mult-bounds#5231 and CrocSwap/integer-mult-bounds#5552. Spans at least seven were already excluded at arbitrary
+width by the one-relation work. The new span-six consequence extends
+the helper width, and follows from a theorem at every word length.
+JoyboyBrian's CrocSwap/integer-mult-bounds#5481 already excludes the five-call `2 x 2` block with
+arbitrary diagonal auxiliaries and closes eight-call frame spans at
+most four. Combining those earlier results with the present theorem
+leaves frame span five as the remaining eight-call case in this formal
+alphabet. This is not a claim that a span-five construction exists.
+
+The six-call span-three fixture in the verifier is a known sharpness
+and scope control, not a new construction or closure of CrocSwap/integer-mult-bounds#5481. This
+package supplies no new supplier, physical implementation, recursive
+invoice or multiplication exponent. **No new κ.**
+
+## 7. Verification and provenance
+
+The proof depends on CrocSwap/integer-mult-bounds#720's precise universal three-register lemma.
+That lemma is inherited, not newly proof-kernel verified here. The
+CrocSwap/integer-mult-bounds#5921 fiber lemma is credited and recalled; CrocSwap/integer-mult-bounds#1456 and CrocSwap/integer-mult-bounds#720 supply the
+common-pattern and augmentation framework. The final analytic proof
+does not depend on the finite 4,964-plane census, though our certificate
+independently replays CrocSwap/integer-mult-bounds#1846 and checks that the new structural statement
+agrees with every pool there.
+
+Further finite controls check all 108,165 generated pattern planes
+on five registers, 20,097 bounded common-J prefix placements, whole
+formal endpoints and meaningful rejected mutations. They corroborate
+specific lemmas; they do not prove arbitrary width, word length or
+rational-lattice statements by sampling. No unrelated full CrocSwap/integer-mult-bounds#468/#720
+address-dimension-three search is rerun.
+
+Base: `7c49fe32c55de02c2c873655c804b16e4c3fef0a`. Exact source pins are in
+SOURCES.json. The initially assigned five-call restored-helper question
+was found already solved by CrocSwap/integer-mult-bounds#5481; the lead explicitly reassigned the
+scope to the arbitrary-width two-relation question left in CrocSwap/integer-mult-bounds#1456,
+CrocSwap/integer-mult-bounds#1846 and CrocSwap/integer-mult-bounds#5921. The selected conditional bound and all five inherited
+assumptions remain unchanged.
+
+Credit eumemic's CrocSwap/integer-mult-bounds#288 formal model; Ali Romman's CrocSwap/integer-mult-bounds#468 relation and rank
+methods; David Govea's CrocSwap/integer-mult-bounds#720 theorem/common-J proof; woklerx's CrocSwap/integer-mult-bounds#1456
+one-relation result and CrocSwap/integer-mult-bounds#1846 classification; Nick Petalas's CrocSwap/integer-mult-bounds#5921 fiber
+and cardinality theorem; JoyboyBrian's CrocSwap/integer-mult-bounds#5481 five-call obstruction;
+Josie Elliston's CrocSwap/integer-mult-bounds#6055 independent short-word census; and the cited
+four-register span-six contributors. @CubeJerry's CrocSwap/integer-mult-bounds#6760 compression
+motivated the original assignment but is not a proof dependency of the
+final result. Substantial OpenAI Codex assistance includes separately
+assigned mathematical review. No human or proof-kernel acceptance is
+asserted.
diff --git a/research/bit-two-relation-any-registers/README.md b/research/bit-two-relation-any-registers/README.md
new file mode 100644
index 0000000000..f888fe1683
--- /dev/null
+++ b/research/bit-two-relation-any-registers/README.md
@@ -0,0 +1,196 @@
+# Two frame relations cannot share rank, at any register count
+
+**Scoped universal lower bound. No new κ.** In the formal binary
+scalar-idempotent model, a word with at most two independent integral
+relations among its address frames cannot save paid rank, regardless
+of its number of scalar registers or diagonal auxiliary outputs.
+
+More precisely, every target address in a diagonal monomial endpoint
+is a sum of a disjoint subset of the paid addresses. For rational
+matrix frames this gives
+
+\[
+\sum_i \operatorname{rank}_{\mathbb Q} C_i
+\le \sum_j \operatorname{rank}_{\mathbb Q} A_j.
+\]
+
+Thus rank-one calls need **at least three independent frame relations**
+to beat total target rank. In particular, eight calls for three
+invertible `3 x 3` targets cannot have frame span six, even with
+arbitrarily many dirty auxiliaries and arbitrary diagonal monomial
+auxiliary outputs. Combined with earlier results, frame span five is
+the remaining eight-call case in this formal alphabet.
+
+## What is new relative to the archive
+
+The result extends David Govea's three-register disjoint-subset theorem
+in CrocSwap/integer-mult-bounds#720 and woklerx's arbitrary-register
+one-relation bound in CrocSwap/integer-mult-bounds#1456. It resolves
+their missing width/two-relation combination by three compatible steps:
+
+1. A common cyclic pattern reduces to an odd all-ones core. The
+   quadratic prefix argument is extended to **every odd core size**,
+   proving that its noncoordinate calls cancel at a diagonal endpoint.
+2. A two-pattern noncoordinate pool has joint image dimension at most
+   **three**, except the four lifts of two disjoint single edges,
+   which are coordinate triangular. This is a written classification
+   for every width. Its sharp maximum image dimension four improves
+   the bound five obtained from the pool-cardinality theorem in
+   CrocSwap/integer-mult-bounds#5921; that predecessor's sharp
+   cardinality theorem remains unchanged.
+3. A saturated cube-valued relation basis separates intervening
+   coordinate calls as free exponent generators. Restrict the remaining
+   word to its small image, apply the inherited three-register theorem,
+   then reconstruct every original target, including repeated values.
+
+[PROOF.md](PROOF.md) supplies each step and its exact inherited hypotheses.
+The final analytic proof does not depend on a new finite scalar census.
+[REVIEW.md](REVIEW.md) records a separate written mathematical review and
+an alternate target-label argument. It checks the precise inherited
+CrocSwap/integer-mult-bounds#720 scope without claiming a new proof of that
+three-register theorem or a replay of its unrelated address search.
+We credit and independently replay woklerx's **existing** 4,964-plane
+classification in CrocSwap/integer-mult-bounds#1846 as corroboration.
+
+Several narrower results were already established and are not claimed
+as new here. JoyboyBrian's CrocSwap/integer-mult-bounds#5481 excluded
+five calls for three invertible `2 x 2` targets with arbitrary diagonal
+auxiliaries, closing the span-four handoff in
+CrocSwap/integer-mult-bounds#4118. Josie Elliston's
+CrocSwap/integer-mult-bounds#6055 independently enumerated the five-call
+case through five registers. The **four-register** span-six case was
+closed by CrocSwap/integer-mult-bounds#4119,
+CrocSwap/integer-mult-bounds#5075, CrocSwap/integer-mult-bounds#5231 and
+CrocSwap/integer-mult-bounds#5552. The new theorem is at arbitrary
+width, arbitrary word length, and arbitrary address dimension.
+
+## Exact evidence and verification boundary
+
+The universal proof is written. Its central inherited dependency is
+CrocSwap/integer-mult-bounds#720's three-register relation theorem, not
+that package's later address-dimension-three search. That theorem is
+**inherited, not newly formally verified here**. Nick Petalas's
+CrocSwap/integer-mult-bounds#5921 off-diagonal fiber lemma is credited
+and recalled in the proof. No unmerged source or network fetch is
+needed to run the standalone checker.
+
+The finite tests corroborate precise parts of the argument:
+
+| Check | Executed domain |
+| --- | --- |
+| Historical four-register alphabet | All 65,536 binary matrices; exactly 120 rank-one idempotents |
+| Historical pattern-plane replay | All 104 nonzero patterns and 4,964 generated planes |
+| Certificate coverage | Every plane once; every projector preserves its explicit proper-subspace witness |
+| Invariant block reduction | All 3,600 invariant projector/subspace pairs on four registers |
+| Sharp joint-image classification | Every generated pattern plane at widths 2, 3, 4, 5: 1, 129, 4,964, 108,165 planes |
+| Equality cases | Exactly 12 image-dimension-four pools at width four and 60 at width five; all are disjoint-edge pools |
+| Odd-core quadratic mechanism | 20,097 prefix-placement cases at core sizes 3, 5, 7 and J-call counts 2, 3, 4, one coordinate occurrence per role |
+| Whole formal endpoints | Odd-core cancellation and noncancellation words; a six-call rational span-three fixture |
+| Failure controls | Omitted plane, proper but noninvariant witness, deleted paid calls, missing helper cleanup, separated internal gaps |
+| Scope controls | Irreducible odd-width common-J alphabet, nontriangular three-coordinate subblock, intervening coordinate gates, and repeated-target eigenspaces |
+
+[CERTIFICATE.json](CERTIFICATE.json) records all 4,964 invariant-space
+witnesses. [verify.py](verify.py) regenerates the complete domain and
+checks each witness directly. Its structural census uses a separate
+outer-product enumeration at widths through five. The universal proof
+does not extrapolate those finite counts.
+
+The six-call fixture has integer frames `2 E11, 2 E22`, then
+`2 E11, 4 E22`, then `3 [[1,1],[1,1]], [[1,-1],[-1,1]]` on the three
+data coordinates. These span the symmetric `2 x 2` space and produce
+`2 I`, `diag(2,4)`, `[[4,2],[2,4]]`, while helpers remain identity.
+Their target relatives have commutator determinant one, so no common
+algebraic flag is silently assumed. This is a sharpness control,
+not a new construction. Even address entries remain nonzero; only
+scalar coefficients use characteristic two.
+
+## Reproduce
+
+From the repository root, Python 3.12 and its standard library suffice:
+
+```sh
+python3 -I -W error -B research/bit-two-relation-any-registers/verify.py
+```
+
+Expected first line:
+
+```
+PASS_TWO_RELATION_ANY_REGISTERS
+```
+
+The next JSON line gives every count above. The same complete check
+is registered normally and with Python assertions disabled:
+
+```sh
+python3 -I -O -W error -B research/bit-two-relation-any-registers/verify.py
+python3 -B scripts/research_package.py --package research/bit-two-relation-any-registers --check-only
+python3 -B scripts/research_package.py --package research/bit-two-relation-any-registers
+```
+
+Regenerate certificate bytes without changing the tracked file:
+
+```sh
+python3 -I -B research/bit-two-relation-any-registers/verify.py --emit-certificate /tmp/two-relation-certificate.json
+cmp research/bit-two-relation-any-registers/CERTIFICATE.json /tmp/two-relation-certificate.json
+```
+
+There is no random seed: domains are deterministic and fully exhausted
+at the stated widths and prefix bounds. This is not a word search over
+`120^5` cases. No native compiler, solver or third-party library is used.
+Metadata validation authenticates the canonical provenance in
+[SOURCES.json](SOURCES.json); the direct verifier is self-contained.
+[VERIFY.json](VERIFY.json) declares no external live scientific inputs.
+
+## Scope, assumptions and provenance
+
+Base: `7c49fe32c55de02c2c873655c804b16e4c3fef0a`; catalogue revision 8;
+seed 12, card `bit-new-register-model`. [TASK.json](TASK.json) retains
+the full original card, the explicit scope change and stopping rule.
+The original five-call question had already been solved; the contribution
+addresses the remaining arbitrary-width two-relation question identified
+across CrocSwap/integer-mult-bounds#1456,
+CrocSwap/integer-mult-bounds#1846 and CrocSwap/integer-mult-bounds#5921.
+
+The theorem requires a torsion-free exponent group, binary rank-one
+idempotent scalar calls, and exact whole-matrix diagonal monomial
+endpoints. Helpers may be arbitrary inputs and may return arbitrary
+diagonal monomials; exact identity restoration is a special case.
+Finite-address coincidences, approximate endpoints, discarded or
+zero-initialized-only inputs, nondiagonal outputs, nonabelian address
+operations and other scalar fields are outside the theorem.
+
+The selected conditional κ remains `0.003125804` at the pinned base.
+Native all-columns, owned erasure, compiler, precision/tape and analytic
+recovery assumptions are unchanged. No complete supplier, physical
+cost invoice or multiplication improvement follows from this obstruction.
+
+## Open next questions
+
+1. **Eight calls at frame span five.** This gives three independent
+   relations and is the remaining formal three-target case after the
+   older low-span results and this theorem. Begin with rank-three cube
+   relation lattices and the associated three-dimensional spaces of scalar
+   patterns. Unlike rank two, independent cube relations need not have a
+   unit minor; the free-coordinate split requires new saturation work.
+2. **Sharp joint-image dimension for three-dimensional pattern spaces.**
+   Three disjoint edges give image dimension six, while
+   CrocSwap/integer-mult-bounds#5921 gives the upper bound ten from
+   cardinality. Extend the diagonal-difference argument before launching
+   a large pool search. A better width reduction would help the span-five
+   question, but does not itself prove a three-relation word theorem.
+
+## Credit and license
+
+Prepared for **@CubeJerry** with substantial OpenAI Codex assistance
+for mathematical development, exact checking and separately assigned
+adversarial review. No independent human acceptance or proof-assistant
+certification is claimed. New files are **Apache-2.0**.
+
+Credit eumemic's CrocSwap/integer-mult-bounds#288 model; Ali Romman's
+CrocSwap/integer-mult-bounds#468 scalar/relation framework; David Govea's
+CrocSwap/integer-mult-bounds#720 theorem and common-J proof; woklerx's
+CrocSwap/integer-mult-bounds#1456 and CrocSwap/integer-mult-bounds#1846;
+Nick Petalas's CrocSwap/integer-mult-bounds#5921 fiber/cardinality theorem;
+and the predecessors identified above. @CubeJerry's
+CrocSwap/integer-mult-bounds#6760 compression motivated the original
+assignment but is not a dependency of the final theorem.
diff --git a/research/bit-two-relation-any-registers/REVIEW.md b/research/bit-two-relation-any-registers/REVIEW.md
new file mode 100644
index 0000000000..35fc0a940b
--- /dev/null
+++ b/research/bit-two-relation-any-registers/REVIEW.md
@@ -0,0 +1,232 @@
+# Independent review: two exponent relations at arbitrary scalar width
+
+Prepared for Jerry Chen / @CubeJerry with substantial OpenAI Codex assistance.
+Apache-2.0. This is an independent written mathematical review, not external
+human review or proof-kernel verification. No second copy of the author's
+finite census was run for this review.
+
+Reviewed objects:
+
+- `research/bit-two-relation-any-registers/PROOF.md`, Sections 1–7.
+- David Govea's CrocSwap/integer-mult-bounds#720, canonical base
+  `7c49fe32c55de02c2c873655c804b16e4c3fef0a`,
+  `research/three-register-no-target-flag/README.md`, especially Sections 1–4.
+
+The new theorem is precisely about binary rank-one idempotent scalar calls
+over the formal group algebra of a finitely generated torsion-free exponent
+group, with at most two independent integer relations and an exact diagonal
+monomial endpoint. It does not concern arbitrary scalar fields, torsion
+relations, address changes between calls, or finite-address coincidence.
+
+## Review conclusion
+
+I find the new arbitrary-width reduction valid. The common-J argument extends
+to every odd core size, the two-pattern image classification is exhaustive
+without an ambient-width bound, and original repeated or identity target
+labels can be recovered. The exact inherited #720 lemma has the needed
+torsion-free, at-most-two-relation scope; it has no frame-rank, target-rank,
+invertibility, flag or target-distinctness hypothesis.
+
+The conclusion remains a written theorem depending on that named inherited
+three-register result. This review did not rerun #720's finite classification
+or its unrelated address-dimension-three search. It also does not newly verify
+every older low-span result cited for the historical eight-call summary.
+
+## 1. Saturation and the cube lift
+
+Because the finitely generated exponent group H is torsion-free, the exact
+sequence from Z^k splits. Thus the mod-two linear relation space is exactly
+L/2L, rather than a larger kernel created by torsion. First-order off-diagonal
+coefficient vectors therefore span at most rank(L) directions.
+
+Every universal product entry is multilinear in the individual call variables:
+each call contributes either its constant matrix part or one occurrence of its
+own variable. Consequently a collision between two contributing exponent
+monomials has a relation vector in {-1,0,1}^k. Saturating the rational span of
+all such cube relations cannot introduce a new equality between subset sums,
+and removes none. If that span has rank at most one, the resulting torsion-free
+lift preserves all coefficient fibers. Each endpoint monomial has at least
+one subset representative, so the lifted diagonal endpoint is well defined
+as well; repeated or identity target values remain consistent. The returned
+disjoint subsets map back to the original exponents.
+
+If two independent cube relations exist, the claimed saturated basis argument
+is sound. Independent mod-two reductions give an odd 2-by-2 minor, necessarily
+plus or minus one because its absolute value is at most two. Coincident
+nonzero reductions give two integral half-sums with disjoint nonempty support
+and a unit minor. Their membership in L uses saturation, equivalently
+torsion-freeness of H. Since rank(L) is at most two, either basis spans all
+of L integrally.
+
+In the pattern-span-two case, a coordinate projector has zero pattern, hence
+both basis-relation coefficients there are zero modulo two. Cube-valued
+coefficients make them zero as integers. This proves the direct sum
+H=H_S plus Z^T and justifies removing coordinate calls before applying a
+noncoordinate invariant-image argument. Without this step, the claimed
+common image would not in general be invariant under arbitrary coordinate
+projectors.
+
+## 2. Arbitrary odd common-J cores
+
+A cyclic rank-one binary idempotent has an odd support intersection of size
+s at least three. Its off-diagonal graph gives upper-triangular tails around
+an all-ones J_s core. Coordinate projectors preserve that block order.
+The diagonal endpoint outside the core is already the disjoint product of
+the corresponding coordinate calls, and the core relation lattice embeds
+into the original lattice by zero extension.
+
+The following points make the degree-two argument valid rather than a
+first-order heuristic:
+
+1. The universal first-order off-diagonal expression is a multiple of the
+   constant matrix `(J_s)_off`. Quotienting that matrix direction kills the
+   entire first-order expression before any Laurent substitution. Therefore
+   quadratic terms introduced by nonlinear substitutions into a surviving
+   universal linear term cannot interfere: no such term survives.
+2. The associated graded augmentation ring of F2[H] is the polynomial ring
+   Sym(H/2H). After eliminating the nonzero J-sum relation L_J, at most one
+   further linear relation M remains. A nonzero homogeneous quadratic that
+   vanishes in the quotient is therefore divisible by M.
+3. For the star matrices K_i, an equation sum(a_i K_i)=lambda J_off means
+   a_i+a_j=lambda for every distinct i,j. A triangle forces lambda=0;
+   then all a_i are equal. Thus the only star relation, even after this
+   matrix quotient, is their total sum. This works for every s at least
+   three, not just s=3.
+
+Eliminating the last J variable leaves independent proper J-prefix forms
+P_1,...,P_(h-1). Every remaining quadratic q_i+q_s is bilinear between J
+variables X and coordinate-call variables Y. If a nonzero bilinear
+polynomial factors into two linear forms, its pure X and pure Y degree-two
+parts must vanish. Since a polynomial ring over F2 is a domain, one factor
+must be purely X and the other purely Y. Its coefficient matrix has rank one.
+All its nonzero columns are therefore the same prefix P_j; distinct proper
+prefixes are independent and cannot lie on one binary line.
+
+The common-divisor condition synchronizes the gap across all coordinates.
+If q_s is nonzero, its own columns occur in every q_i+q_s, so their rank-one
+prefix must agree. If q_s is zero, different nonzero q_i have disjoint
+Y-variable supports; they cannot share a nonzero Y divisor. Their common
+divisor is consequently the same X prefix. If just one quadratic is
+nonzero, its own rank-one condition is enough. If all these quadratics are
+identically zero before the final relation M, disjoint Y supports force all
+q_i to vanish identically and there are no internal coordinate calls.
+
+This reduces the word to U(b) D U(c) after removing exterior invertible
+diagonal factors. If either b or c equals one, an off-diagonal equation
+forces the other to equal one. Otherwise subtracting two columns in a third
+row gives `(b-1)(d_j-d_k)=0`. The Laurent domain and s at least three force
+D=dI. Since J_s is idempotent for odd s, the remaining word is d U(bc),
+whose off-diagonal equations force bc=1. Its endpoint is exactly the one
+obtained by deleting all J calls. No word-length or core-size enumeration
+is used in this conclusion.
+
+The phrase "all quadratics vanish" should consistently mean universal
+vanishing after L_J has been eliminated and before imposing M; vanishing
+modulo M alone would not imply the absence of internal coordinate calls.
+The current argument uses the former meaning.
+
+## 3. The joint-image classification is exhaustive
+
+I checked all structural cases independently of the finite 4- and 5-register
+data.
+
+Two distinct lifts of one nonzero off-diagonal pattern differ by a nonzero
+diagonal matrix of rank at most two and trace zero. It therefore has exactly
+two supported diagonal entries. Rank equality forces both images and both
+covectors into those two coordinates. The binary 2-by-2 rank-one idempotents
+give exactly the two stated lifts of a single edge; other realized nonzero
+patterns have one lift.
+
+A two-dimensional pattern plane has only three nonzero elements. With no
+edge ambiguity there are at most three projectors, giving image dimension
+at most three. With exactly one edge and all three patterns realized,
+write the other lifts as M,N and M+N=E_ab+D. Rank subadditivity bounds the
+diagonal support of D by three, and trace parity makes it zero or two.
+
+- If D=0 and the two image vectors are independent, rank(M+N)=1 forces
+  their covectors equal and their image sum to e_a. Adding e_a,e_b thus
+  still spans at most three dimensions. Dependent images are easier.
+- If D has support {c,d}, disjointness from {a,b} would produce three
+  independent columns, contrary to rank at most two. Its principal
+  {c,d} minor has determinant one, so rank(M+N)=2 and its image equals
+  the joint image of M,N. All image vectors then lie in the union of
+  at most three coordinate positions.
+
+With two edges and their sum realized, the latter has only two off-diagonal
+entries. Its support intersection cannot have cyclic size at least three,
+which would already create six edges. For intersection size one, the tail
+sizes satisfy a+b+ab=2, forcing (2,0) or (0,2). Thus the two edges share a
+row or a column and all lifts are supported on three coordinates.
+
+If only two edge patterns are realized, their four lifts span their
+endpoints. The only image-dimension-four case has four distinct endpoints.
+Those disjoint edges are simultaneously coordinate upper triangular, and
+their sum is unrealizable by the preceding two-edge count. This proves the
+claimed exception and rules out an unexamined high-width case. The sharp
+pool span(E_01,E_23) also prevents incorrectly strengthening the bound to
+three without the triangular exception.
+
+## 4. Independent target-label argument
+
+The author's invariant-eigenspace decomposition is valid: monomial
+independence applied to a constant vector in V separates each distinct
+target exponent's coordinate projection. Repeated target values require
+grouping equal monomials, which the proof does.
+
+There is a stronger direct argument in this setting. Because every N_j has
+image in V, a telescoping product identity gives
+
+    (D-I)(F2^W tensor R) contained in V tensor R.
+
+For original target label i with projected exponent h_i not zero,
+`(D-I)e_i=([h_i]-1)e_i`. Its image in the constant-vector quotient by V
+can vanish only if e_i lies in V: apply a binary linear functional nonzero
+on that quotient vector and use `[h_i]-1 != 0`. Consequently every original
+nonzero-target coordinate axis lies in V, and
+
+    V = span{e_i: h_i != 0} direct-sum (V intersect E_0).
+
+Choose those original nonzero target axes themselves in the restricted
+basis, followed by a basis of the zero-target intersection. This retains
+every nonzero target label directly, even when its exponent is repeated.
+All quotient target labels have zero exponent. Each restricted projector
+is still rank one and idempotent because its nonzero image vector lies in
+V and is fixed. Applying the inherited three-register lemma is therefore
+legitimate after padding if necessary. Any subsets assigned to artificial
+zero-target basis vectors can simply be discarded; their sums are zero.
+
+Finally, projecting onto a free coordinate-call generator leaves only its
+original coordinate projector. Formal monomial equality assigns that
+generator to exactly its original target label with coefficient one,
+and coefficient zero to all others. Reattaching these calls preserves
+disjointness and reconstructs the full exponent by the exact direct sum.
+No free scalar-basis transformation has been asserted as a physical gate.
+
+## 5. Quantifiers and consequences
+
+Every argument above is independent of W and k. The only finite width used
+as an inherited mathematical theorem is at most three, reached by an
+analytic image reduction. The disjoint-edge exception is solved directly
+by triangular diagonals rather than by the 4,964-plane census. Thus the
+finite 4- and 5-register checks corroborate the new structural lemma; they
+do not carry the arbitrary-width quantifier.
+
+For rational matrix exponents the group map is additive over the rationals,
+while scalar projectors remain binary. Rational address entries are not
+reduced modulo two. Disjoint subset assignment and ordinary matrix-rank
+subadditivity give total target rank no larger than total paid frame rank.
+In particular, three invertible 3-by-3 targets require at least nine total
+rank-one calls whenever the frame-relation rank is at most two, regardless
+of additional diagonal monomial helper outputs.
+
+This proves no kappa improvement, physical supplier admission, recursive
+cost model or unrestricted lower bound for three or more frame relations.
+The historical remaining span-five case uses the separately cited earlier
+low-span results as well as this theorem; it is not a new assertion that a
+span-five construction exists.
+
+## Author response and final proof pin
+
+The requested distinction between universal quadratic vanishing and vanishing
+modulo M is now explicit in PROOF.md. This clarifies the reviewed argument.
+Final proof SHA256: `ad05bccffaf0fe11120312d8514146e5433d7032bbc65bd9d57974e06a6c3f82`.
diff --git a/research/bit-two-relation-any-registers/SOURCES.json b/research/bit-two-relation-any-registers/SOURCES.json
new file mode 100644
index 0000000000..5b0cf1290d
--- /dev/null
+++ b/research/bit-two-relation-any-registers/SOURCES.json
@@ -0,0 +1,88 @@
+{
+  "schema_version": 1,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa",
+      "role": "Selected baseline at assignment"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81",
+      "role": "Original task and priority revision"
+    },
+    {
+      "path": "research/bit-fivecall-dirty-block/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "dc10506db48c67040c0743296cc0af13c85fc1c3",
+      "sha256": "9bae10083e68fb1bccf38827741100e394a5e66b19f7020d6c90fad8887dc50c",
+      "role": "Prior arbitrary-helper fivecall and span-four closures",
+      "originating_pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/5481"
+    },
+    {
+      "path": "research/bit-four-role-two-patterns/README.md",
+      "usage": "provenance",
+      "git_blob": "e79087d9cd8f4473e6e48fa4e521902b7809f3a4",
+      "sha256": "e4ee830e783b20bfb0886ae8c91133a6138157ed5dcaac90ce16ddc616c8d49f",
+      "role": "Existing finite4964 census, independently replayed but not a final proof dependency",
+      "originating_pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/1846"
+    },
+    {
+      "path": "research/bit-one-relation-any-registers/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "9b42c6cd2e93514bd8c5247e2ef2d516d6603ae6",
+      "sha256": "38805f881d0bd625418fd8bdc9410d06fe3873374db9678c722d13abd3bfe392",
+      "role": "Common-pattern framework and predecessor one-relation bound",
+      "originating_pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/1456"
+    },
+    {
+      "path": "research/bit-span4-five-call-two-relation/README.md",
+      "usage": "provenance",
+      "git_blob": "a8cf0a663e479b5570fc6e20a3eaff4db14e6ddf",
+      "sha256": "5af44cffa72cad900a0c367986cec6fd17448b7ceec2b8d145ee7554d62f465a",
+      "role": "Independent W<=5 fivecall corroboration, not a proof dependency",
+      "originating_pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/6055"
+    },
+    {
+      "path": "research/coordinate-kernel-span-six-closure/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "bf20f5d4f906dcb037503c027fe21b4e5e8deb11",
+      "sha256": "eb488595c5e931948ba737b9da2a7146ee7763af4ca163d826d7815610db5576",
+      "role": "Prior four-register span-six scope",
+      "originating_pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/5552"
+    },
+    {
+      "path": "research/pattern-pool-cardinality/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "4fce5d3b6a9c810cfd4d298b209dbd51cefa173c",
+      "sha256": "47ebe562180c7276bbf0636e9ba3aa1a84f8962ff8b60b3b2fb30cee4514e989",
+      "role": "Inherited all-width fiber lemma and previous image bound",
+      "originating_pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/5921"
+    },
+    {
+      "path": "research/three-register-no-target-flag/README.md",
+      "usage": "provenance",
+      "git_blob": "290bfb79db5f4177b8b03bc2027c3848671fac9e",
+      "sha256": "95d4483da9b19df051c7d13fe290eefb82da019ab4be8e66ad9216199ddda4eb",
+      "role": "Inherited universal three-register disjoint-subset lemma and common-J method",
+      "originating_pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/720"
+    }
+  ],
+  "cited_pending_sources": [
+    {
+      "pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/6760",
+      "head": "3b3d6670d509fd74c1780cfacdf550badf8871db",
+      "path": "research/restored-helper-compression/PROOF.md",
+      "git_blob": "2f2a51dbcac84a0fe9a133514a8620ca0aa542ec",
+      "sha256": "4d9edf19ed4eec2708cbfcd186e9d6085af43cc0213e14fb2bd54efdeda2a20b",
+      "role": "Historical assignment motivation only; not a dependency of the final theorem.",
+      "runtime_input": false
+    }
+  ],
+  "pending_source_contract": "This exact-head historical citation is not an input or source_pin. The final proof uses canonical sources listed above; no pending Git object is needed for replay or the mathematical proof."
+}
diff --git a/research/bit-two-relation-any-registers/TASK.json b/research/bit-two-relation-any-registers/TASK.json
new file mode 100644
index 0000000000..895671e2c0
--- /dev/null
+++ b/research/bit-two-relation-any-registers/TASK.json
@@ -0,0 +1,72 @@
+{
+  "schema_version": 1,
+  "task_id": "bit-new-register-model",
+  "priority_revision": 8,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "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_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Resolve the current selected review, separate complex/ordinary BIT/admitted BIT/outer budgets, and compare the card against live handoffs before starting. Historical ceilings require exact source/model equivalence.",
+      "scope": "Source-binding orientation only; not a scientific replay or all-size proof. Materialize selected inputs if this checkout is sparse."
+    },
+    "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": "Prove a disjoint-subset and rank-budget theorem for any finite number of binary scalar registers and any word length with at most two independent exponent relations and exact diagonal monomial endpoints.",
+  "scope_change_reason": "The assigned fivecall restored-helper question was found already solved by merged CrocSwap/integer-mult-bounds#5481, and the finite4964 classification already existed in CrocSwap/integer-mult-bounds#1846. The lead explicitly reassigned the task to the unresolved arbitrary-width two-relation question, using the inherited three-register theorem, a new analytic joint-image classification and an odd-core common-J generalization.",
+  "bottleneck_link": "A rank-saving binary formal primitive now requires at least three independent rational frame relations. Eight-call 3x3 constructions at span six are excluded with any number of arbitrary diagonal auxiliaries; older results leave only span five. No new kappa or physical supplier follows.",
+  "budget_minutes": 90,
+  "stop_rule": "Complete the universal written reduction and independent adversarial check, or preserve a precisely scoped gap. Bound controls to scalar widths2..5 and odd cores3,5,7; do not launch a raw ordered-word search or claim the rank-three relation case.",
+  "outcome": "completed",
+  "evidence": "PROOF.md establishes the arbitrary-width theorem via the inherited #720 lemma. The analytic pool lemma gives image<=3 except disjoint-edge pools, and the common-J proof covers every odd core. verify.py independently replays4964 historical pools, checks108165 five-register planes,20097 prefix cases, exact endpoints and rejection controls.",
+  "author": "@CubeJerry",
+  "ai_assistance": "Substantial OpenAI Codex assistance for mathematical development, exact Python verification, and separately assigned adversarial mathematical review. The reviewer supplied an alternate target-label argument. No human peer review or proof-kernel verification is claimed.",
+  "track_seed": 12,
+  "open_questions": [
+    {
+      "question": "Can eight calls realize three invertible3x3 targets at frame span five with arbitrary diagonal auxiliaries?",
+      "why_it_matters": "This is the remaining eight-call case after the prior span<=4 theorem and the new arbitrary-width two-relation bound.",
+      "first_step": "Analyze rank-three saturated cube-relation lattices at eight calls and their three-dimensional scalar pattern spaces, retaining chronological gates and exact endpoints.",
+      "blocked_by": "The rank-two cube-basis unit-minor argument need not extend to rank three; the free-coordinate split and three-relation word theorem need new proofs."
+    },
+    {
+      "question": "What is the sharp joint-image bound for a three-dimensional binary off-diagonal pattern space?",
+      "why_it_matters": "Three disjoint edges give dimension six, while the #5921 cardinality bound gives ten. Reducing this gap may bound the width in the remaining span-five case.",
+      "first_step": "Extend the diagonal-difference analysis to pattern sums and classify equality mechanisms before any large pool census.",
+      "blocked_by": "A scalar image bound alone does not establish endpoint compatibility or a three-relation disjoint-subset theorem."
+    }
+  ],
+  "coordination": "Lead rechecked archive provenance and formally expanded the original narrow assignment after identifying prior #5481/#1846 work. The lead reviewed the algebraic proof and a separate worker was assigned adversarial review; evidence and reviews are retained with the package."
+}
diff --git a/research/bit-two-relation-any-registers/VALIDATION.json b/research/bit-two-relation-any-registers/VALIDATION.json
new file mode 100644
index 0000000000..f5122a1251
--- /dev/null
+++ b/research/bit-two-relation-any-registers/VALIDATION.json
@@ -0,0 +1,211 @@
+{
+  "schema_version": 1,
+  "python": "3.12.14",
+  "platform": "Linux-6.18.57-x86_64-with-glibc2.39",
+  "status": "PASS_AUTHOR_LOCAL_VERIFICATION",
+  "normal_and_optimized_direct": "Both executed successfully after final verifier edit; identical full JSON counts.",
+  "registered_metadata": "PASS_PACKAGE_METADATA_AND_PINS",
+  "registered_replay": "PASS_REGISTERED_RESEARCH_PACKAGE research/bit-two-relation-any-registers outcome=completed",
+  "certificate_regeneration": "Exact byte match against committed-intended CERTIFICATE.json",
+  "relocated_optimized_replay": "PASS_TWO_RELATION_ANY_REGISTERS with only verify.py and CERTIFICATE.json copied; no source repository or pending PR objects used.",
+  "relocated_report": {
+    "census": {
+      "full_rank_pools": 12,
+      "idempotents": 120,
+      "image_covector_ranks": [
+        [
+          1,
+          2,
+          190
+        ],
+        [
+          1,
+          3,
+          4
+        ],
+        [
+          2,
+          1,
+          190
+        ],
+        [
+          2,
+          2,
+          3520
+        ],
+        [
+          2,
+          3,
+          108
+        ],
+        [
+          3,
+          1,
+          4
+        ],
+        [
+          3,
+          2,
+          108
+        ],
+        [
+          3,
+          3,
+          828
+        ],
+        [
+          4,
+          4,
+          12
+        ]
+      ],
+      "irreducible_pools": 0,
+      "patterns": 104,
+      "planes": 4964,
+      "pool_sizes": [
+        [
+          2,
+          3814
+        ],
+        [
+          3,
+          1012
+        ],
+        [
+          4,
+          114
+        ],
+        [
+          5,
+          24
+        ]
+      ]
+    },
+    "certificate_rejections": [
+      "omitted plane",
+      "proper but noninvariant witness"
+    ],
+    "common_j_controls": {
+      "odd_core_sizes": [
+        3,
+        5,
+        7
+      ],
+      "one_gap_positive_words": 3,
+      "quadratic_prefix_cases": 20097,
+      "separated_gap_negative_words": 3,
+      "width_five_irreducible_control": true
+    },
+    "exact_word_controls": {
+      "deleted_call_rejections": 6,
+      "helper_cleanup_rejection": 1,
+      "nonflag_commutator_determinant": 1,
+      "rational_frame_span": 3,
+      "sharp_six_call_words": 1
+    },
+    "joint_image_controls": [
+      {
+        "idempotents": 6,
+        "image_dimensions": [
+          [
+            2,
+            1
+          ]
+        ],
+        "patterns": 2,
+        "planes": 1,
+        "realized_two_edge_planes": 0,
+        "width": 2
+      },
+      {
+        "idempotents": 28,
+        "image_dimensions": [
+          [
+            1,
+            9
+          ],
+          [
+            2,
+            63
+          ],
+          [
+            3,
+            57
+          ]
+        ],
+        "patterns": 19,
+        "planes": 129,
+        "realized_two_edge_planes": 6,
+        "width": 3
+      },
+      {
+        "idempotents": 120,
+        "image_dimensions": [
+          [
+            1,
+            194
+          ],
+          [
+            2,
+            3818
+          ],
+          [
+            3,
+            940
+          ],
+          [
+            4,
+            12
+          ]
+        ],
+        "patterns": 104,
+        "planes": 4964,
+        "realized_two_edge_planes": 24,
+        "width": 4
+      },
+      {
+        "idempotents": 496,
+        "image_dimensions": [
+          [
+            1,
+            2515
+          ],
+          [
+            2,
+            97275
+          ],
+          [
+            3,
+            8315
+          ],
+          [
+            4,
+            60
+          ]
+        ],
+        "patterns": 471,
+        "planes": 108165,
+        "realized_two_edge_planes": 60,
+        "width": 5
+      }
+    ],
+    "structural_controls": {
+      "coordinate_split_control": true,
+      "invariant_block_cases": 3600,
+      "nontriangular_control": true,
+      "one_pattern_pools": 104,
+      "proper_subspaces": 65,
+      "repeated_target_control": true
+    }
+  },
+  "verification_boundary": "Finite controls corroborate the written universal proof; #720 lemma inherited, no human/proof-kernel verification claimed.",
+  "source_sha256": {
+    "verify.py": "83b590379b6d0532b76ddd0979811a39ea13156e73ff622e587fdea47614818b",
+    "CERTIFICATE.json": "78ec903f3952f3eb3343a9122a7d482a5ed9bc7d8c098a3c70218bb5267b652b",
+    "PROOF.md": "ad05bccffaf0fe11120312d8514146e5433d7032bbc65bd9d57974e06a6c3f82",
+    "README.md": "238ae1a6b4f11b73e3e51a632701541a2d824e58293d8e01b3371adb53309208",
+    "REVIEW.md": "b5fa222353c446a6124b3c30bcd50c5a0eacc6f2aa25b58eda499565af744e47"
+  },
+  "relocation_and_generation_seconds": 1.543514,
+  "review": "Separate worker reviewed the universal proof and exact #720 scope, supplied a distinct D-I target-label argument, found no defect, and requested one clarification now applied."
+}
diff --git a/research/bit-two-relation-any-registers/VERIFY.json b/research/bit-two-relation-any-registers/VERIFY.json
new file mode 100644
index 0000000000..8f9176c0aa
--- /dev/null
+++ b/research/bit-two-relation-any-registers/VERIFY.json
@@ -0,0 +1,34 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Complete historical4964-plane replay, new analytic joint-image classification at widths2..5 including108165 width-five planes,20097 odd-core prefix cases, exact formal endpoints, repeated-target and rejection controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Complete historical4964-plane replay, new analytic joint-image classification at widths2..5 including108165 width-five planes,20097 odd-core prefix cases, exact formal endpoints, repeated-target and rejection controls."
+    }
+  ]
+}
diff --git a/research/bit-two-relation-any-registers/verify.py b/research/bit-two-relation-any-registers/verify.py
new file mode 100644
index 0000000000..723bc5d6b3
--- /dev/null
+++ b/research/bit-two-relation-any-registers/verify.py
@@ -0,0 +1,466 @@
+#!/usr/bin/env python3
+"""Exact finite lemma and controls for the four-register two-relation proof.
+
+Prepared for @CubeJerry with substantial OpenAI Codex assistance. Apache-2.0.
+The universal group-lattice theorem is in PROOF.md; this is not formalization.
+Only Python's standard library is used. Checks remain active under -O.
+"""
+
+import argparse
+from collections import Counter
+from fractions import Fraction
+from itertools import combinations, permutations, product
+import json
+from pathlib import Path
+
+
+class VerificationError(Exception):
+    pass
+
+
+def need(condition, message):
+    if not condition:
+        raise VerificationError(message)
+
+
+def span(vectors):
+    result = {0}
+    for vector in vectors:
+        result |= {value ^ vector for value in tuple(result)}
+    return result
+
+
+def brank(vectors):
+    return len(span(vectors)).bit_length() - 1
+
+
+def subspaces(n):
+    masks = set()
+    for dimension in range(1, n):
+        for basis in combinations(range(1, 1 << n), dimension):
+            values = span(basis)
+            if len(values) == 1 << dimension:
+                masks.add(sum(1 << value for value in values))
+    return sorted(masks)
+
+
+def rows(matrix, n=4):
+    return tuple((matrix >> (n * i)) & ((1 << n) - 1) for i in range(n))
+
+
+def packed(matrix):
+    n = len(matrix)
+    return sum(row << (n * i) for i, row in enumerate(matrix))
+
+
+def apply(matrix, vector):
+    return sum(((row & vector).bit_count() & 1) << i
+               for i, row in enumerate(matrix))
+
+
+def multiply(left, right):
+    return tuple(xor_rows(row, right) for row in left)
+
+
+def xor_rows(mask, matrix):
+    result = 0
+    for i, row in enumerate(matrix):
+        if mask >> i & 1:
+            result ^= row
+    return result
+
+
+def invariant(matrix, space):
+    return all(space >> apply(matrix, v) & 1
+               for v in range(1 << len(matrix)) if space >> v & 1)
+
+
+def proper_space(space, n=4):
+    if type(space) is not int or not 0 < space < (1 << (1 << n)) - 1:
+        return False
+    values = [v for v in range(1 << n) if space >> v & 1]
+    return len(values) > 1 and 0 in values and all(
+        space >> (u ^ v) & 1 for u in values for v in values)
+
+
+def catalogue():
+    # Exhaust all 65,536 matrices, not merely a list of supplied outer products.
+    diagonal = sum(1 << (4 * i + i) for i in range(4))
+    found = {}
+    for matrix in range(1 << 16):
+        rr = rows(matrix)
+        nonzero = set(rr) - {0}
+        if len(nonzero) != 1:
+            continue  # A nonzero binary rank-one matrix has one nonzero row value.
+        if multiply(rr, rr) != rr:
+            continue
+        y = next(iter(nonzero))
+        x = sum((row != 0) << i for i, row in enumerate(rr))
+        need((x & y).bit_count() % 2 == 1, "idempotent self-pairing")
+        found[matrix] = (matrix & ~diagonal, x, y)
+    patterns = sorted({entry[0] for entry in found.values()} - {0})
+    planes = sorted({tuple(sorted((p, q, p ^ q)))
+                     for p, q in combinations(patterns, 2)})
+    need(len(found) == 120, "120 rank-one idempotents")
+    need(len(patterns) == 104, "104 nonzero patterns")
+    need(len(planes) == 4964, "4964 pattern planes")
+    return found, patterns, planes
+
+
+def pool(found, plane):
+    return sorted(m for m, (pattern, _, _) in found.items() if pattern in plane)
+
+
+def canonical_full_rank_pools():
+    result = set()
+    for p in permutations(range(4)):
+        result.add(tuple(sorted(
+            (1 << (4 * p[a] + p[a])) | (1 << (4 * p[b] + p[c]))
+            for a, b, c in ((0, 0, 1), (1, 0, 1), (2, 2, 3), (3, 2, 3)))))
+    need(len(result) == 12, "twelve disjoint-edge pools")
+    return result
+
+
+def make_certificate(found, planes, spaces):
+    records = []
+    for plane in planes:
+        matrices = [rows(m) for m in pool(found, plane)]
+        witness = next((s for s in spaces if all(invariant(m, s) for m in matrices)), None)
+        need(witness is not None, "an irreducible pool was found")
+        records.append([*plane, witness])
+    return {"schema_version": 1,
+            "encoding": "matrix bit 4*i+j is entry (i,j); subspace bit v marks binary vector v",
+            "records": records}
+
+
+def check_certificate(document, found, planes):
+    need(document.get("schema_version") == 1, "certificate schema")
+    records = document.get("records")
+    need(type(records) is list and len(records) == len(planes), "plane coverage count")
+    seen = set()
+    sizes, ranks = Counter(), Counter()
+    full_rank = set()
+    for record in records:
+        need(type(record) is list and len(record) == 4, "record shape")
+        p, q, r, space = record
+        plane = (p, q, r)
+        need(plane not in seen, "duplicate plane")
+        seen.add(plane)
+        need(proper_space(space), "witness is not a proper binary subspace")
+        matrices = pool(found, plane)
+        need(matrices, "empty pool")
+        for matrix in matrices:
+            need(invariant(rows(matrix), space), "noninvariant witness")
+        rx = brank(found[m][1] for m in matrices)
+        ry = brank(found[m][2] for m in matrices)
+        sizes[len(matrices)] += 1
+        ranks[rx, ry] += 1
+        if rx == ry == 4:
+            full_rank.add(tuple(matrices))
+    need(seen == set(planes), "complete plane domain differs")
+    need(dict(sizes) == {2: 3814, 3: 1012, 4: 114, 5: 24}, "pool-size census")
+    need(full_rank == canonical_full_rank_pools(), "full-rank pool characterization")
+    return {"idempotents": 120, "patterns": 104, "planes": len(seen),
+            "pool_sizes": sorted(sizes.items()),
+            "image_covector_ranks": [list(key) + [value] for key, value in sorted(ranks.items())],
+            "full_rank_pools": len(full_rank), "irreducible_pools": 0}
+
+
+def reject(function, description):
+    try:
+        function()
+    except VerificationError:
+        return description
+    raise VerificationError("negative control accepted: " + description)
+
+
+def inverse(matrix):
+    n = len(matrix)
+    work = [row | (1 << (n + i)) for i, row in enumerate(matrix)]
+    for j in range(n):
+        pivot = next((i for i in range(j, n) if work[i] >> j & 1), None)
+        need(pivot is not None, "basis is singular")
+        work[j], work[pivot] = work[pivot], work[j]
+        for i in range(n):
+            if i != j and work[i] >> j & 1:
+                work[i] ^= work[j]
+    return tuple(row >> n for row in work)
+
+
+def adapted_basis(space):
+    basis = []
+    for value in range(1, 16):
+        if space >> value & 1 and value not in span(basis):
+            basis.append(value)
+    dimension = len(basis)
+    for value in range(1, 16):
+        if value not in span(basis):
+            basis.append(value)
+    return tuple(sum(((basis[j] >> i) & 1) << j for j in range(4))
+                 for i in range(4)), dimension
+
+
+def structural_controls(found, patterns, spaces):
+    count = 0
+    for space in spaces:
+        basis, d = adapted_basis(space)
+        inv = inverse(basis)
+        for matrix in found:
+            rr = rows(matrix)
+            if not invariant(rr, space):
+                continue
+            block = multiply(multiply(inv, rr), basis)
+            need(all(not (block[i] & ((1 << d) - 1)) for i in range(d, 4)), "lower block")
+            left = tuple(block[i] & ((1 << d) - 1) for i in range(d))
+            right = tuple(block[i] >> d for i in range(d, 4))
+            need(bool(any(left)) != bool(any(right)), "exactly one active block")
+            active = left if any(left) else right
+            need(multiply(active, active) == active and brank(active) == 1, "active block projector")
+            count += 1
+    coordinates = [1 << (4 * i + i) for i in range(4)]
+    for pattern in patterns:
+        matrices = [rows(m) for m, entry in found.items() if entry[0] == pattern]
+        matrices += [rows(m) for m in coordinates]
+        need(any(all(invariant(m, s) for m in matrices) for s in spaces), "one-pattern reduction")
+
+    # P,Q,R from #720: irreducible on three active coordinates, reducible on four.
+    exceptional = ((3, 0, 0), (2, 2, 0), (0, 5, 5))
+    need(not any(all(invariant(m, s) for m in exceptional) for s in subspaces(3)),
+         "three-register exception must not be labelled triangular")
+    embedded = tuple(tuple(m) + (0,) for m in exceptional)
+    need(all(invariant(m, (1 << 0) | (1 << 8)) for m in embedded), "embedded invariant line")
+
+    overlapping = [(7, 7, 7, 0), (0, 14, 14, 14)]
+    with_coordinates = overlapping + [rows(m) for m in coordinates]
+    need(any(all(invariant(m, s) for m in overlapping) for s in spaces), "noncoordinate reduction")
+    need(not any(all(invariant(m, s) for m in with_coordinates) for s in spaces),
+         "coordinate calls cannot be retained in general pool lemma")
+
+    # Repeated target values can preserve a non-coordinate line; separating them cannot.
+    line = {0, 3}
+    def diagonal_preserves(groups):
+        return all((v & group) in line for v in line for group in groups)
+    need(diagonal_preserves((3, 4, 8)), "equal-target invariant decomposition")
+    need(not diagonal_preserves((1, 6, 8)), "separated-target rejection")
+    return {"proper_subspaces": len(spaces), "invariant_block_cases": count,
+            "one_pattern_pools": len(patterns), "nontriangular_control": True,
+            "coordinate_split_control": True, "repeated_target_control": True}
+
+
+def qrank(matrix):
+    matrix = [[Fraction(x) for x in row] for row in matrix]
+    if not matrix:
+        return 0
+    rank = 0
+    for j in range(len(matrix[0])):
+        pivot = next((i for i in range(rank, len(matrix)) if matrix[i][j]), None)
+        if pivot is None:
+            continue
+        matrix[rank], matrix[pivot] = matrix[pivot], matrix[rank]
+        scalar = matrix[rank][j]
+        matrix[rank] = [value / scalar for value in matrix[rank]]
+        for i in range(len(matrix)):
+            if i != rank:
+                scalar = matrix[i][j]
+                matrix[i] = [a - scalar * b for a, b in zip(matrix[i], matrix[rank])]
+        rank += 1
+        if rank == len(matrix):
+            break
+    return rank
+
+
+ZERO = (0, 0, 0, 0)
+
+
+def polynomial_product(left, right):
+    result = set()
+    for a in left:
+        for b in right:
+            exponent = tuple(x + y for x, y in zip(a, b))
+            result.symmetric_difference_update((exponent,))
+    return result
+
+
+def formal_word(calls, n):
+    result = [[{ZERO} if i == j else set() for j in range(n)] for i in range(n)]
+    for frame, x, y in calls:
+        need(len(frame) == 4 and frame != ZERO and (x & y).bit_count() % 2, "legal formal call")
+        factor = [[{ZERO} if i == j else set() for j in range(n)] for i in range(n)]
+        for i in range(n):
+            for j in range(n):
+                if x >> i & 1 and y >> j & 1:
+                    factor[i][j].symmetric_difference_update((ZERO, frame))
+        new = [[set() for _ in range(n)] for _ in range(n)]
+        for i in range(n):
+            for j in range(n):
+                for k in range(n):
+                    new[i][j].symmetric_difference_update(polynomial_product(result[i][k], factor[k][j]))
+        result = new
+    return result
+
+
+def endpoint(targets):
+    return [[{target} if i == j else set() for j in range(len(targets))]
+            for i, target in enumerate(targets)]
+
+
+def exact_word_controls():
+    # Six rank-one integer frames in Sym_2, span three; all address arithmetic stays over Z.
+    frames = [(2, 0, 0, 0), (0, 0, 0, 2), (2, 0, 0, 0),
+              (0, 0, 0, 4), (3, 3, 3, 3), (1, -1, -1, 1)]
+    calls = [(frame, 1 << (j // 2), 1 << (j // 2)) for j, frame in enumerate(frames)]
+    targets = [(2, 0, 0, 2), (2, 0, 0, 4), (4, 2, 2, 4), ZERO]
+    need(qrank(frames) == 3, "six-call frame span three")
+    need(all(qrank((f[:2], f[2:])) == 1 for f in frames), "rank-one rational frames")
+    need(all(qrank((c[:2], c[2:])) == 2 for c in targets[:3]), "invertible targets")
+    need(formal_word(calls, 4) == endpoint(targets), "six-call exact endpoint")
+    for j in range(6):
+        need(formal_word(calls[:j] + calls[j + 1:], 4) != endpoint(targets), "deleted-call rejection")
+    # C1^-1 C2=diag(1,2), C1^-1 C3=[[2,1],[1,2]]: commutator determinant one.
+    u = tuple(Fraction(v, 2) for v in targets[1])
+    v = tuple(Fraction(value, 2) for value in targets[2])
+    def product2(a, b):
+        return tuple(sum(a[2 * i + k] * b[2 * k + j] for k in range(2))
+                     for i in range(2) for j in range(2))
+    commutator = tuple(a - b for a, b in zip(product2(u, v), product2(v, u)))
+    need(commutator[0] * commutator[3] - commutator[1] * commutator[2] == 1,
+         "no-common-algebraic-flag witness")
+    helper = (1, 0, 0, 0)
+    restored = calls + [(helper, 8, 8), (tuple(-x for x in helper), 8, 8)]
+    need(formal_word(restored, 4) == endpoint(targets), "arbitrary helper restoration")
+    need(formal_word(restored[:-1], 4) != endpoint(targets), "missing helper cleanup")
+    return {"sharp_six_call_words": 1, "rational_frame_span": 3,
+            "deleted_call_rejections": 6, "helper_cleanup_rejection": 1,
+            "nonflag_commutator_determinant": 1}
+
+
+def joint_image_controls():
+    """Bounded corroboration of the written all-width lemma; not extrapolation."""
+    report = []
+    for n in range(2, 6):
+        diagonal = sum(1 << (n * i + i) for i in range(n))
+        by_pattern = {}
+        total = 0
+        for x in range(1, 1 << n):
+            for y in range(1, 1 << n):
+                if not (x & y).bit_count() % 2:
+                    continue
+                total += 1
+                matrix = sum((y if x >> i & 1 else 0) << (n * i) for i in range(n))
+                pattern = matrix & ~diagonal
+                if pattern:
+                    by_pattern.setdefault(pattern, []).append(x)
+        for pattern, images in by_pattern.items():
+            need(len(images) == (2 if pattern.bit_count() == 1 else 1), "inherited fiber control")
+        planes = {tuple(sorted((p, q, p ^ q))) for p, q in combinations(by_pattern, 2)}
+        histogram = Counter()
+        two_edges_realized = 0
+        for plane in planes:
+            images = [x for pattern in plane for x in by_pattern.get(pattern, [])]
+            dimension = brank(images)
+            need(dimension <= 4, "joint image exceeds four")
+            histogram[dimension] += 1
+            edges = [p for p in plane if p.bit_count() == 1]
+            if dimension == 4:
+                need(len(edges) == 2 and edges[0] ^ edges[1] not in by_pattern,
+                     "dimension four must have exactly two realized edge patterns")
+                endpoints = [coordinate for edge in edges
+                             for coordinate in divmod(edge.bit_length() - 1, n)]
+                need(len(set(endpoints)) == 4, "dimension-four edges must have disjoint endpoints")
+            if len(edges) == 2 and edges[0] ^ edges[1] in by_pattern:
+                a, b = divmod(edges[0].bit_length() - 1, n)
+                c, d = divmod(edges[1].bit_length() - 1, n)
+                need(a == c or b == d, "realized two-edge pattern must share row or column")
+                need(dimension <= 3, "two-edge realized pool dimension")
+                two_edges_realized += 1
+        if n >= 4:
+            need(max(histogram) == 4, "sharp image dimension four")
+        report.append({"width": n, "idempotents": total, "patterns": len(by_pattern),
+                       "planes": len(planes), "image_dimensions": sorted(histogram.items()),
+                       "realized_two_edge_planes": two_edges_realized})
+    return report
+
+
+def common_j_controls():
+    """Exact bounded checks for the new odd-core quadratic and endpoint steps."""
+    prefix_cases = 0
+    for s in (3, 5, 7):
+        # Stars have the one all-sum relation, independent of the J pattern.
+        stars = [tuple(((1 << s) - 1) ^ (1 << i) if row == i else (1 << i)
+                       for row in range(s)) for i in range(s)]
+        packed_stars = [packed(m) for m in stars]
+        j_off = packed(tuple(((1 << s) - 1) ^ (1 << i) for i in range(s)))
+        need(brank(packed_stars) == s - 1, "star relation rank")
+        need(j_off not in span(packed_stars), "J pattern independent of stars")
+        # One coordinate occurrence per role. Universal proof covers all multiplicities.
+        for h in (2, 3, 4):
+            prefixes = [0] + [(1 << gap) - 1 for gap in range(1, h)]
+            for placements in product(prefixes, repeat=s):
+                common = None
+                for i in range(s - 1):
+                    columns = [0] * s
+                    columns[i], columns[-1] = placements[i], placements[-1]
+                    nonzero = set(columns) - {0}
+                    if len(nonzero) > 1:
+                        common = set()
+                        break
+                    if nonzero:
+                        j_factor = next(iter(nonzero))
+                        c_factor = sum((bool(value)) << j for j, value in enumerate(columns))
+                        factors = {("J", j_factor), ("coordinate", c_factor)}
+                        common = factors if common is None else common & factors
+                divisible = common is None or bool(common)
+                single_gap = len(set(placements) - {0}) <= 1
+                need(divisible == single_gap, "shared linear divisor versus single gap")
+                prefix_cases += 1
+        j = (1 << s) - 1
+        b, a = (1, 0, 0, 0), (0, 0, 0, 1)
+        minus_b, minus_a = tuple(-v for v in b), tuple(-v for v in a)
+        positive = [(b, j, j), (a, 1, 1), (minus_a, 1, 1), (minus_b, j, j)]
+        negative = [(b, j, j), (a, 1, 1), (minus_b, j, j), (minus_a, 1, 1)]
+        need(formal_word(positive, s) == endpoint([ZERO] * s), "odd-core one-gap cancellation")
+        bad = formal_word(negative, s)
+        need(any(bad[i][k] for i in range(s) for k in range(s) if i != k),
+             "separated internal gaps rejected by exact endpoint")
+    # A five-coordinate common-J pool plus coordinates is irreducible. The
+    # arbitrary-width case needs the quadratic proof, not a four-dimensional image shortcut.
+    spaces5 = subspaces(5)
+    need(len(spaces5) == 372, "five-coordinate proper subspace count")
+    matrices = [(31,) * 5] + [tuple(1 << i if row == i else 0 for row in range(5)) for i in range(5)]
+    need(not any(all(invariant(m, space) for m in matrices) for space in spaces5),
+         "odd-width full common-pattern alphabet must remain irreducible")
+    return {"odd_core_sizes": [3, 5, 7], "quadratic_prefix_cases": prefix_cases,
+            "one_gap_positive_words": 3, "separated_gap_negative_words": 3,
+            "width_five_irreducible_control": True}
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--emit-certificate", type=Path)
+    args = parser.parse_args()
+    found, patterns, planes = catalogue()
+    spaces = subspaces(4)
+    need(len(spaces) == 65, "65 proper nonzero binary subspaces")
+    if args.emit_certificate:
+        document = make_certificate(found, planes, spaces)
+        args.emit_certificate.write_text(json.dumps(document, separators=(",", ":")) + "\n")
+    else:
+        document = json.loads(Path(__file__).with_name("CERTIFICATE.json").read_text())
+    census = check_certificate(document, found, planes)
+    controls = []
+    bad = dict(document, records=document["records"][:-1])
+    controls.append(reject(lambda: check_certificate(bad, found, planes), "omitted plane"))
+    bad_records = [list(row) for row in document["records"]]
+    first_pool = pool(found, tuple(bad_records[0][:3]))
+    bad_records[0][3] = next(s for s in spaces if not all(invariant(rows(m), s) for m in first_pool))
+    controls.append(reject(lambda: check_certificate(dict(document, records=bad_records), found, planes),
+                           "proper but noninvariant witness"))
+    report = {"census": census, "structural_controls": structural_controls(found, patterns, spaces),
+              "exact_word_controls": exact_word_controls(), "certificate_rejections": controls,
+              "joint_image_controls": joint_image_controls(), "common_j_controls": common_j_controls()}
+    print("PASS_TWO_RELATION_ANY_REGISTERS")
+    print(json.dumps(report, sort_keys=True))
+
+
+if __name__ == "__main__":
+    main()