Show the no-edge irreducible bound fails in pattern dimension 4.

research/pattern-dim4-irreducible-image/PROOF.md

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+# The no-edge irreducible bound does not stay 3 in pattern dimension 4
+
+Prepared by unpaidmoon with substantial Cursor Grok assistance. New work is
+Apache-2.0. This is a written proof for one explicit pool, checked again by
+finite enumeration. **No new κ.** The selected frontier stays
+`781451/250000000`.
+
+## 1. Statement
+
+Work over F2. For a matrix N, write off(N) for the matrix obtained by
+deleting its diagonal. Let P be a linear space of zero-diagonal W-by-W
+matrices, and define the complete noncoordinate pool
+
+    C(P) = {N = x y^T : y^T x = 1, off(N) in P, off(N) != 0}.
+
+The condition y^T x = 1 makes N a rank-one idempotent. Put
+
+    V(P) = sum_{N in C(P)} im(N),
+
+and let e(P) be the number of matrix units E_ab, a != b, contained in P.
+A subspace W ⊆ V(P) is invariant when N(W) ⊆ W for every N in C(P). The
+action is irreducible when no subspace 0 < W < V(P) is invariant.
+
+**Theorem.** For every ambient width W ≥ 4 there is a pattern space P of
+dimension 4 with e(P) = 0 whose complete pool acts irreducibly on a joint
+image of dimension 4. Explicitly, on coordinates Z/4Z,
+
+    N_i = e_i (e_i + e_{i+1} + e_{i+2})^T,    i in Z/4Z,
+
+and P is the span of their four off-diagonal patterns, padded by zero rows
+and columns when W > 4. Thus C(P) = {N_0, N_1, N_2, N_3}, dim V(P) = 4, and
+the action is irreducible.
+
+Under the same hypotheses used for pattern dimension at most 3 — a complete
+noncoordinate rank-one idempotent pool over F2, and no matrix unit — the
+irreducible joint-image bound 3 therefore does not remain an upper bound in
+pattern dimension 4.
+
+This note does not prove a replacement upper bound. It does not say that
+every pattern-dimension-4 pool is irreducible, and it does not say that 4 is
+the largest irreducible joint-image dimension. Section 6 recomputes the
+width-6 parity pool from #6838: its pattern dimension is 4 and its total
+image has dimension 5, but a four-dimensional kernel is invariant, so that
+pool does not decide the irreducible question.
+
+#6921 proves that the bound is 3 for pattern dimension at most 3. Its case
+split uses the Fano plane of a three-dimensional pattern space and is not
+reproved or reused here. The fiber lemma is not an input: membership of
+C(P) is settled by the support casework below.
+
+## 2. The four patterns
+
+Index coordinates by Z/4Z. For each i define
+
+    y_i = e_i + e_{i+1} + e_{i+2},    N_i = e_i y_i^T.
+
+Then y_i · e_i = 1, so N_i is a rank-one idempotent. Deleting the diagonal
+leaves the pattern
+
+    p_i = E_{i,i+1} + E_{i,i+2}.
+
+These four patterns occupy eight distinct positions, so they are linearly
+independent. Let P be their span. A general element is indexed by a subset
+A ⊆ Z/4Z,
+
+    p_A = sum_{i in A} p_i.
+
+Row i of p_A is zero when i is not in A. When i is in A, that row has ones
+in columns i+1 and i+2 only. Hence p_A has weight 2|A|. Every nonzero
+element has weight at least 2, so e(P) = 0 and dim P = 4.
+
+If W > 4, view each p_i as a W-by-W matrix with those two ones and with
+every other entry zero. The same weight count holds, and rows and columns
+indexed by t ≥ 4 are identically zero throughout P.
+
+The four images e_0, e_1, e_2, e_3 are independent. Once Section 4 shows
+that no further pool element exists, V(P) = span(e_0, e_1, e_2, e_3).
+
+## 3. Action on the joint image
+
+Identify V = span(e_0, e_1, e_2, e_3) with F2^4. On this space
+
+    N_i(v) = e_i    if φ_i(v) = 1,    and 0 otherwise,
+
+where φ_i = e_i* + e_{i+1}* + e_{i+2}*. In the basis e_0, e_1, e_2, e_3 the
+covector matrix, with rows φ_0 through φ_3, is
+
+    1 1 1 0
+    0 1 1 1
+    1 0 1 1
+    1 1 0 1.
+
+Row operations over F2 produce a permutation of the identity. Add row 0 to
+rows 2 and 3; add the new row 2 to rows 1 and 0; add the new row 1 to row 3
+and then to row 0; add the resulting row 3 to rows 0 and 2. The rows are
+then e_0*, e_2*, e_1*, and e_3*. Swapping the middle two rows gives the
+identity. The four functionals are therefore a basis of V*. Their common
+kernel on V is zero. Each column of the matrix has weight 3, so each basis
+vector e_j is sent to a different basis vector by at least two of the
+projectors.
+
+## 4. The pool is exactly the four projectors
+
+Let x, y in F2^W satisfy y · x = 1 and off(x y^T) in P, with the pattern
+nonzero.
+
+First, both vectors are supported on {0, 1, 2, 3}. Suppose some t ≥ 4 has
+x_t = 1. Row t of every matrix in P is zero, so the off-diagonal part of
+row t of x y^T is zero. That row, with its diagonal entry removed, is y
+with coordinate t cleared. Thus y vanishes off t, so y = y_t e_t. Idempotence
+gives y_t = x_t = 1. Column t of P is zero, so its off-diagonal entries
+x_i y_t = x_i vanish for i ≠ t. Hence x = e_t and the pattern is zero,
+which is excluded from C(P). Therefore x_t = 0 for every t ≥ 4.
+
+Suppose instead y_t = 1. Column t then contributes x_i for every i ≠ t, and
+those entries must vanish, so x vanishes off t. Combined with x_t = 0 this
+gives x = 0, contradicting y · x = 1. Thus y_t = 0 as well.
+
+Now x and y are vectors in F2^4, still with y · x = 1. Let S be the support
+of x. For each i in S, row i of the pattern is the off-diagonal part of y,
+so the shape of P forces
+
+    y_{i-1} = 0    and    y_{i+1} = y_{i+2}.
+
+In particular y vanishes on S - 1 = {i - 1 : i in S}. Indices are read in
+Z/4Z, so i+3 = i-1. Every nonempty S is a rotate of one of the five types
+below, and the argument is stated for an arbitrary index, so it covers
+every rotate.
+
+**|S| = 4.** Then S - 1 = S, so y = 0 and the pairing is 0.
+
+**|S| = 3.** Let S be the complement of {t}. Then y vanishes off {t-1}. The
+pairing is y_{t-1}, so y = e_{t-1}. The constraint at i = t+1, which lies
+in S, says y_{t+2} = y_{t-1}. The left side is 0 and the right side is 1.
+
+**|S| = 2, opposite.** Let S = {i, i+2}. The two constraints give
+y_{i-1} = y_{i+1} = 0 and y_{i+1} = y_{i+2}, y_{i+3} = y_i. Hence
+y_{i+2} = 0. Since y_i = y_{i+3} = y_{i-1} = 0, the covector is zero.
+
+**|S| = 2, adjacent.** Let S = {i, i+1}. The constraints give y_i = 0,
+y_{i-1} = 0, y_{i+1} = y_{i+2}, and y_{i+2} = y_{i+3}. The pairing reduces
+to y_{i+1} = 1, so y_{i+2} = 1. But y_{i+3} = y_{i-1} = 0, and the last
+constraint says those two bits agree.
+
+**|S| = 1.** Let S = {i}. Then y_i = 1, y_{i-1} = 0, and y_{i+1} = y_{i+2}.
+If y_{i+1} = 0, then y = e_i and the pattern is zero, which is excluded.
+If y_{i+1} = 1, then y = e_i + e_{i+1} + e_{i+2}, and the pattern is p_i.
+That projector is N_i.
+
+The empty support is impossible. Therefore C(P) = {N_0, N_1, N_2, N_3} at
+every width W ≥ 4, and dim V(P) = 4.
+
+## 5. The action is irreducible
+
+Let A(W) = {i : φ_i does not vanish on W}, for a subspace W ⊆ V. If the
+subspace is invariant and φ_i does not vanish on it, the corresponding image
+e_i lies in it. The four functionals span V*, so A(W) is nonempty whenever
+W is nonzero. They are distinct, so φ_i = ψ has at most one solution i for
+a given nonzero functional ψ.
+
+**Lines.** Each i in A(W) puts a distinct basis vector e_i into W, so a line
+has |A| ≤ 1. It cannot be 0. If A = {i}, the line is span(e_i), but column
+i of the covector matrix has two further ones, so some other φ_k also hits
+e_i and |A| ≥ 2.
+
+**Planes.** Three or more basis vectors cannot fit in a plane, so |A| ≤ 2.
+The case |A| = 1 would require some e_i in the plane and φ_k(e_i) = 0 for
+every k ≠ i, contradicting the column weight. Thus |A| = 2, the plane is
+span(e_i, e_j), and the other two functionals vanish on both basis vectors.
+All six pairs fail that test:
+
+| pair | a functional outside the pair that hits the pair |
+| --- | --- |
+| {0,1} | φ_2(e_0) = 1 |
+| {0,2} | φ_1(e_2) = 1 |
+| {0,3} | φ_1(e_3) = 1 |
+| {1,2} | φ_0(e_1) = 1 |
+| {1,3} | φ_0(e_1) = 1 |
+| {2,3} | φ_0(e_2) = 1 |
+
+**Hyperplanes.** Let H = ker(ψ) with ψ ≠ 0. For each i outside A(H),
+φ_i vanishes on H, so φ_i = ψ. At most one such i exists, hence |A(H)| ≥ 3.
+If |A(H)| = 4, then H contains every basis vector and H = V. If
+|A(H)| = 3, the three included basis vectors force ψ to be the remaining
+coordinate functional e_j*, and the omitted index must be j, so
+φ_j = e_j*. But φ_j has weight 3, not 1.
+
+No proper nonzero subspace is invariant. The action on V(P) is irreducible,
+and the irreducible joint-image dimension 4 is attained in pattern
+dimension 4.
+
+Deleting N_3 leaves the line span(e_0 + e_1 + e_3) in the common kernel of
+φ_0, φ_1, and φ_2. That line is invariant for the proper subpool. The
+four-projector statement uses the whole pool.
+
+## 6. The parity pool one dimension higher is reducible
+
+#6838 records the even-parity construction moved up one dimension as a scope
+control: pattern dimension 4, total joint image 5. It is recomputed here
+only to show that it does not already answer the irreducible question. This
+is not a new total-image bound.
+
+Use coordinates {0, 1, 2, 3, 4, 5}. Let U be the set of vectors supported
+on {1, 2, 3, 4, 5} whose weights are even. Then U is a four-dimensional
+subspace and contains no weight-one vector. Set
+
+    P_par = {u e_0^T : u in U},    N_u = (e_0 + u) e_0^T
+
+for each nonzero u in U. Every matrix in P_par is supported in column 0,
+and its weight equals the weight of u. Thus e(P_par) = 0 and dim P_par = 4.
+Each N_u is a rank-one idempotent with pattern u e_0^T. These 15 patterns
+are all the nonzero elements of P_par, so the displayed projectors are the
+complete pool. For distinct nonzero u, v in U,
+
+    (e_0 + u) + (e_0 + v) + (e_0 + u + v) = e_0,
+
+so e_0 lies in V(P_par) and then every u does. Hence
+
+    V(P_par) = span(e_0) ⊕ U,    dim V(P_par) = 5.
+
+Every covector is e_0. The subspace U is the kernel of e_0 on V(P_par). It
+is nonzero and proper, and every projector kills it. The action is
+reducible. The parity construction therefore shows that total image 5
+occurs in pattern dimension 4, and it does not show that an irreducible
+image of dimension greater than 3 occurs. The four-cycle of Sections 2
+through 5 is what shows that the irreducible bound changes.
+
+## 7. What this does not claim
+
+The selected κ and its five framework assumptions are unchanged. The
+dimension-at-most-3 theorem of #6921 is cited, not reproved. Nothing here
+is a paid-rank accounting for a three-relation word; that is question 1 of
+#6921 and is untouched. No upper bound is claimed for irreducible joint
+images in pattern dimension 4, and no classification of all such pools is
+claimed. The finite scans in `verify.py` cover one 16-element vector space
+and widths 4 through 8. They corroborate the width-independent casework;
+they are not a search over all four-dimensional pattern spaces. A failed
+search would not have been a proof.
+
+The note produces no rational-address word, native primitive, supplier
+admission, or multiplication identity.
+
+## Open next questions
+
+The packaged question is whether some complete no-edge pool of pattern
+dimension 4 acts irreducibly in joint-image dimension 5 or more. The parity
+pool is not an example. A single explicit witness, or a scoped upper bound
+with the same hypotheses, would be the next useful step. Absence from a
+partial search would not settle it.

research/pattern-dim4-irreducible-image/README.md

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+# Pattern dimension 4 irreducible image
+
+**Result.** The no-edge irreducible joint-image bound does not stay 3 when
+the pattern space has dimension 4. For every ambient width at least 4, the
+four projectors
+
+```text
+N_i = e_i (e_i + e_{i+1} + e_{i+2})^T,    i in Z/4Z,
+```
+
+are the complete pool of their pattern span. That span is four-dimensional
+and contains no matrix unit. The joint image is four-dimensional and the
+action is irreducible. **No new κ.** The selected value remains
+`781451/250000000`.
+
+This is an existence result under the same complete-pool and no-matrix-unit
+hypotheses as #6921. It is not an upper bound, and it is not a classification
+of every pattern-dimension-4 pool.
+
+## Scope and evidence
+
+Catalogue card: `bit-new-register-model`, priority revision 8, base commit
+`7c49fe32c55de02c2c873655c804b16e4c3fef0a`. Scope status: **changed**. The
+card asks for a new register alphabet. This package answers open question 2
+of #6921, which #6921 left open because its Fano-plane case split stops at
+pattern dimension 3. The reason is recorded in [TASK.json](TASK.json). The
+original native acceptance gate is unmet, and the selected frontier is
+unchanged. Question 1 of #6921, paid-rank accounting after a relation split,
+is not this result.
+
+[PROOF.md](PROOF.md) is the argument. Section 4 puts every factor on the
+first four coordinates and then splits the support of the image vector by
+size. Section 5 splits invariant subspaces into lines, planes, and
+hyperplanes. Section 6 recomputes the #6838 width-6 parity pool: pattern
+dimension 4, total image 5, and a four-dimensional invariant kernel. That
+pool is reducible, so it does not by itself show that the irreducible bound
+changes.
+
+[verify.py](verify.py) repeats the finite parts: the covector matrix, the
+sixteen pattern weights, the six plane pairs, both support checks at widths
+4 through 8, the complete subspace lattice of F2^4 (67 subspaces), and the
+negative controls. Deleting one cycle projector leaves an invariant line.
+The #6921 three-projector example remains irreducible, and the normal-form
+triple remains reducible, so the lattice scan is not hard-wired to one
+answer. These checks do not replace the width-independent casework, and
+they do not enumerate all four-dimensional pattern spaces.
+
+Source pins are in [SOURCES.json](SOURCES.json). The catalogue and selected
+record are provenance at the base commit. Full Git history is needed to
+resolve them. Nothing here reruns the selected baseline as a scientific
+replay.
+
+Overlap was rechecked on 2026-11-10 against 885 open upstream PRs and 6934
+discussion records at base `7c49fe32c55de02c2c873655c804b16e4c3fef0a`. The
+only records containing the question are #6921 and its finding,
+discussion comment 19474334. Both leave pattern dimension 4 open. #6785 is
+a one-edge exclusion and #6782 is an edge-quotient bound. No match is not
+a novelty certificate; those distinctions are from reading the records.
+
+## Reproduce
+
+From the repository root, with Python 3.12 and the standard library only:
+
+```sh
+python3 -B research/pattern-dim4-irreducible-image/verify.py
+python3 -I -O -B research/pattern-dim4-irreducible-image/verify.py
+python3 -B scripts/research_package.py --package research/pattern-dim4-irreducible-image --check-only
+python3 -B scripts/research_package.py --package research/pattern-dim4-irreducible-image
+```
+
+The verifier prints `PASS_PATTERN_DIM4_IRREDUCIBLE_IMAGE`.
+
+## Open next questions
+
+1. **Irreducible dimension above 4, still at pattern dimension 4.** Can a
+   complete no-edge pool of pattern dimension 4 act irreducibly on a joint
+   image of dimension 5 or more? The four-cycle attains 4, so the old bound
+   fails, but it gives no replacement ceiling. The parity pool attains total
+   dimension 5 and is reducible. The smallest next step is one explicit pool
+   with a written invariant-subspace check, or a scoped upper bound under
+   these same hypotheses. No support reduction here makes an exhaustive
+   search finite, so a search that finds nothing would not be a proof.
+
+## Credit
+
+Nick Petalas, #5921, for the fiber lemma and the joint-image question; this
+proof does not rely on the fiber lemma. CubeJerry, #6838, for the total-image
+bound in pattern dimension at most 3 and for the parity pool rechecked in
+Section 6. #6921 proved the irreducible bound 3 for pattern dimension at
+most 3 and isolated this dimension-4 question; that theorem is not reproved.
+#6785 and #6782 are one-edge and edge-quotient context. Prepared by
+unpaidmoon with substantial Cursor Grok assistance. New work is Apache-2.0.

research/pattern-dim4-irreducible-image/SOURCES.json

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