Witness a characteristic-3 rank-32 pattern on P120.

research/p120-char3-rank32-window/NOTICE

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+Characteristic-3 rank-32 witness for the P120 port window.
+Prepared by unpaidmoon with substantial Cursor Grok assistance.
+The port list is the P104 set published in PR #6933 and vendored by #6953.
+The rank-37 degree-at-most-2 pattern is PR #6953.
+New exposition and checks are Apache-2.0.

research/p120-char3-rank32-window/PROOF.md

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+# Characteristic 3 admits a rank-32 pattern on P120
+
+No new κ. The selected record stays `781451/250000000`. This note does not
+build a word, a decoder, or a paid profile, and it does not change the
+selected frontier.
+
+The window is the block-mask window of PR #6933. A physical decoder for
+`k > 1` is not derived.
+
+## 0. Definitions
+
+`P120` is the 120 vectors in `F_2^9` of weight congruent to 3 mod 4. Two
+distinct vectors are adjacent when their `F_2` dot product is 1 and orthogonal
+when it is 0. `P104` is the port list of #6933, the same file vendored by
+#6953, sha256
+`691c7b7d212112115b1bfc76db89f90d304ec448dc8c4a0c43d9a5954317146c`. Its
+complement is the two 8-cliques recorded in #6953.
+
+A **1-block pattern** has nonzero diagonal and a zero in both off-diagonal
+positions of every orthogonal pair. Adjacent entries are free, and the matrix
+need not be symmetric. Write `w_1(S; F)` for the minimum rank of a 1-block
+pattern on `S` over `F`, and `w(S; F)` for the infimum of `rank(B)/k` over
+k-block patterns in the sense of #6933. Then `w(S; F) ≤ w_1(S; F)`.
+
+#6953 proves `w_1(P120; F_3) ≤ 37` and `w_1(P104; F_3) ≤ 36` by one
+degree-at-most-2 point, and proves that every class-constant pattern on
+`P120` has rank 120 in characteristic 3. The lower bounds `w(P104) ≥ 13`,
+`w(P112) ≥ 14` and `w(P120) ≥ 15` in odd characteristic are #6933, Theorem 4.
+They are cited and not replayed. This note does not extend those lower bounds.
+
+## 1. The degree-at-most-2 chart
+
+Order monomials as the constant `1`, the coordinates `x_0, …, x_8`, and the
+products `x_i x_j` for `0 ≤ i < j ≤ 8`. Evaluate them at a vector as integers
+in `{0, 1}` and read the values in `F_3`.
+
+**Lemma 1.** On `P120`, `sum_{i<j} x_i x_j = 0` as a function to `F_3`. The
+remaining 45 monomials, with `x_7 x_8` omitted, have evaluation rank 45.
+
+*Proof.* For a 0-1 vector the pair sum equals `binom(wt(v), 2)`. That binomial
+coefficient is 3 at weight 3 and 21 at weight 7, both divisible by 3.
+`verify.py` evaluates the sum at all 120 vectors and row-reduces the 45-column
+evaluation matrix. The domain is finite and every vector is visited once. ∎
+
+Every degree-at-most-2 function therefore has a unique coefficient vector in
+this 45-coordinate chart. Call that space of functions `E`.
+
+## 2. The witness span of the previous point
+
+For each vertex `v`, let the legal equations require value 1 at `v` and value
+0 at every vertex orthogonal to `v`. Gaussian elimination over `F_3`, with
+free variables set to 0, produces a solution `q_v` in the 45-chart. This is
+the point constructed in #6953. The matrix `M^0_{uv} = φ(u) · q_v` is a
+1-block pattern of rank 37. The vectors `q_v` span a 37-dimensional subspace
+`S ⊆ E`. Both statements are the row counts of the reduced forms in
+`verify.py`.
+
+The reduced row echelon form is unique: elimination runs left to right, each
+pivot is scaled to 1, and the pivot column is cleared above and below. The
+kernel basis used below is the standard one for that unique form. For each
+free column, in increasing order, the basis vector has a 1 in that free
+column, 0 in the other free columns, and pivot entries equal to the negation
+of the reduced coefficient. It does not depend on row order.
+
+## 3. Eight rigid columns
+
+**Lemma 2.** Inside `S`, the eight weight-7 vectors that do not contain
+coordinate 8 have no legal direction. Each has a unique legal column in `S`,
+namely its witness column `q_v`, and those eight columns are linearly
+independent.
+
+The eight vectors are
+
+```
+127, 191, 223, 239, 247, 251, 253, 254.
+```
+
+*Proof.* A direction in `S` is a linear combination of an RREF basis of `S`.
+`verify.py` imposes vanishing on `{v} ∪ orth(v)` and counts the kernel. The
+kernel is zero at exactly these eight vertices, and the eight witness
+coordinate vectors have rank 8. The vertex list is the full set of weight-7
+vectors with bit 8 clear: a weight-7 vector with bit 8 clear has its single
+other zero among the first eight coordinates, so there are eight of them. ∎
+
+This uniqueness is inside `S`. It is not a uniqueness statement in `E`, and
+not a uniqueness statement among all patterns.
+
+## 4. One hyperplane of the witness span
+
+Let `U` be the span of those eight rigid columns, and let `π_0, …, π_28` be
+the canonical kernel basis of `U` in the 37 coordinates of `S`.
+
+**Lemma 3.** The hyperplane `ker(π_1)` is the unique hyperplane in this
+29-element basis that still meets every legal affine flat of `S`. The space
+`W = ker(π_1) ∩ S` has dimension 36.
+
+*Proof.* `verify.py` tests each of the 29 basis vectors. For a form `π`, it
+solves the legal equations inside `S` together with `π · c = 0`. The system
+is consistent for every vertex if and only if `i = 1`. The two nonzero scales
+of `π_1` determine the same kernel. Consistency is checked at all 120
+vertices. This is an exhaustive census of this basis, not of every hyperplane
+of `S`. ∎
+
+The same space `W` is the set of coefficient vectors satisfying the nine
+independent equations
+
+```
+c + a_8 = 0,
+sum_{i=0}^{7} a_i + 2 a_8 + 2 sum_{0≤i<j≤7} b_{ij} = 0,
+b_{i8} = 0 for i = 0, …, 6,
+```
+
+with `b_78 = 0` already built into the chart. `verify.py` checks that these
+nine rows span the annihilator of `W`.
+
+## 5. The rank-32 pattern
+
+Let `ψ_1, …, ψ_36` be the canonical kernel basis of `π_1` inside `S`, lifted
+to the 45-chart, and let `Φ` be their `120 × 36` evaluation matrix. For each
+vertex `v`, let `y_v` be the solution of the legal equations `Φ y = e_v` with
+free coordinates set to 0. Define `M_uv = Φ_u · y_v`.
+
+**Theorem 1.** `M` is a 1-block pattern over `F_3` of rank 32. Its pivot
+columns and the coordinates of the remaining columns are a `120 × 32`
+factorisation of `M` over `F_3`. Extending scalars does not change the rank,
+so for every field `F` of characteristic 3,
+
+```
+w(P120; F) ≤ w_1(P120; F) ≤ 32.
+```
+
+The same matrix has principal rank 32 on `P104` and on both `P112`
+extensions. A principal submatrix of a pattern is a pattern, so the same
+upper bound 32 holds for those three sets.
+
+*Proof.* `verify.py` rebuilds `Φ` and every `y_v`, checks every diagonal
+entry and both off-diagonal entries of every orthogonal pair, and counts
+pivots of `M`, of the column-reversed matrix, and of the transpose. All three
+counts are 32. The sha256 of `M`, entries in `{0, 1, 2}` in increasing vertex
+order, row by row, is
+
+```
+81e209bb5be3c3e9b9a5d593190fd9b19540656bb0162164f4df7a1f0bf4a8c4
+```
+
+The 32 pivot columns are linearly independent, and every column of `M` reduces
+into their span, which is the factorisation. A minor vanishes over `F_3` if
+and only if it vanishes over an extension, and a nonzero value stays nonzero,
+so the rank is unchanged by scalar extension. The port-file principal
+submatrix and both 112-vertex principals are checked in the same way. ∎
+
+`M` is not class-constant: it is asymmetric, and the adjacent entries take
+more than one value. The class-constant matrices `I + β A` still have rank
+120, as in #6953. The pure quadratic monomials remain inconsistent, so this
+is not that smaller map.
+
+## 6. What the rank is attached to
+
+`W` is 36-dimensional, but the free-zero rule depends on the basis. In the
+canonical kernel basis of the nine relations of section 4, the same legal
+systems have a solution, and that other point is a 1-block pattern of rank
+36. `verify.py` rebuilds it. Rank 32 belongs to the basis carried from `S`
+in Theorem 1. It is not an invariant of the set of all patterns with columns
+in `W`.
+
+Inside that basis the legal solution spaces have affine dimensions 0 at 36
+vertices, 2 at 56 vertices, and 5 at 28 vertices. `verify.py` counts these
+nullities. The witness uses one point of each affine space, the free-zero
+point. It was not minimized over those spaces. The checked rank-36 point of
+the previous paragraph already shows that `W` does not by itself force rank
+32.
+
+So three statements stay distinct:
+
+1. **Proof, one witness.** The matrix of Theorem 1 has rank 32. The checks
+   that identify it are exhaustive on a finite domain: every vertex, every
+   orthogonal pair, one reduced form, and all 29 vectors of one kernel basis.
+2. **Proof, one other point of the same space.** Another explicit basis of
+   `W` gives a legal pattern of rank 36.
+3. **Not settled.** Whether some pattern, inside `W` or outside `E`, has
+   rank at most 31, and whether every 1-block pattern on `P120` has rank at
+   least 32. A search that did not find a smaller pattern would not decide
+   either question. This note does not report one.
+
+Accepting #6933, Theorem 4, without replaying it, the characteristic-3
+intervals are now
+
+| set | lower bound, cited | upper bound proved here |
+|---|---|---|
+| P104 | 13 | 32 |
+| either P112 extension | 14 | 32 |
+| P120 | 15 | 32 |
+
+#6953 had upper bounds 36, 36, 37 and 37 on these four sets. The upper end
+32 is strictly smaller than the rank of `3I − A` off characteristic 3, which
+is 36. No equality is claimed. Characteristic 2 stays inside #6933.
+
+## 7. What this does not say
+
+A rank-32 pattern is not a multiplication algorithm, not a paid profile, and
+not a change to the selected frontier. It does not show that 32 is the
+minimum on `P120` or on `P104`. It does not touch the `F_5` rank-13
+factorisation. The construction uses the coordinate chart of `F_2^9` and is
+not claimed to be invariant under a larger automorphism group. No human has
+reviewed this note.

research/p120-char3-rank32-window/README.md

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+# Characteristic-3 rank-32 pattern on P120 (no new κ)
+
+**No new κ.** The selected record stays `781451/250000000`. This package
+exhibits one 1-block pattern of rank 32 on `P120` over every field of
+characteristic 3, together with a `120 × 32` factorisation over `F_3`. The
+same matrix has principal rank 32 on `P104` and on both copies of `P112`.
+It does not build a word or a paid profile.
+
+The argument is in [PROOF.md](PROOF.md). The exact checks are in
+[verify.py](verify.py).
+
+## Result
+
+`P120` is the odd graph on the 120 vectors in `F_2^9` of weight 3 or 7.
+`P104` is the port list of PR #6933, the file already vendored by #6953.
+A 1-block pattern has nonzero diagonal and zeros on both off-diagonal
+positions of every orthogonal pair.
+
+1. **The previous point still has rank 37.** The free-zero degree-at-most-2
+   pattern of #6953 is rebuilt and still has rank 37 on `P120`. Its columns
+   span a 37-dimensional space `S`.
+2. **Eight columns are rigid inside that span.** The weight-7 vectors that
+   omit coordinate 8 have a unique legal column in `S`. Those eight columns
+   are independent.
+3. **One canonical hyperplane of `S` has a rank-32 pattern.** In the kernel
+   basis of those eight columns, the second basis vector cuts `S` to a
+   36-dimensional space that still meets every legal flat. The free-zero
+   point in the basis carried from `S` is a 1-block pattern of rank 32.
+   Scalar extension preserves the rank, so `w(P120; F) ≤ 32` for every field
+   `F` of characteristic 3. The principal submatrices on `P104` and both
+   `P112` extensions also have rank 32.
+4. **The same space has another point of rank 36.** Free-zero coordinates in
+   the kernel basis of the nine coefficient relations give a different legal
+   pattern, of rank 36. Rank 32 is attached to the basis in item 3, not to
+   every pattern with columns in the space.
+
+Accepting the lower bounds in #6933, Theorem 4, without replaying them, the
+characteristic-3 window now satisfies `w(P120; F) ∈ [15, 32]`, and the same
+upper bound 32 holds for `P104` and both copies of `P112`. #6953 had upper
+bounds 37, 36, 36 and 37 on these sets. No equality is claimed. No κ changes.
+
+The census of the 29 kernel-basis vectors is exhaustive for that basis. It
+is not a census of all hyperplanes, and a failed search for a smaller
+pattern is not part of the claim.
+
+## Scope
+
+Original card: `complex-rank-shape-decoder`. Scope status: **changed**. The
+card asks whether a decoder or coframe change improves the signed moment.
+This package does not change a word or a moment. It answers the P120 half of
+the question left open by #6953: whether characteristic 3 admits a 1-block
+pattern of rank 36 on `P120`. The witness here has rank 32, which is stronger
+than that requested ceiling. Base commit
+`17469fff78c22941280f04d39dc5a1faee09c840`, priority revision 8. The selected
+κ in `certificates/selected-result.json` is `781451/250000000`; the older
+decimal quoted in `docs/research/current-priorities.md` is not the live
+frontier.
+
+The `F_5` rank-13 factorisation is open in #7014 and is not investigated.
+The degree-at-most-2 family is not searched beyond the two explicit points.
+Characteristic 2 is not reopened.
+
+## Reproduce
+
+From the repository root, CPython 3.12, standard library only:
+
+```sh
+python3 -B research/p120-char3-rank32-window/verify.py
+python3 -I -O -B research/p120-char3-rank32-window/verify.py
+python3 -B scripts/research_package.py --package research/p120-char3-rank32-window
+```
+
+Expected final line: `PASS_P120_CHAR3_RANK32 rank=32 p104=32 p112=[32, 32]`,
+then `PASS_REGISTERED_RESEARCH_PACKAGE` with outcome `partial`.
+
+The port file is a live input, the bytes already pinned by #6953. The
+catalogue and the selected record are provenance pins at the base commit.
+
+## Open next questions
+
+1. Does `P120` admit a 1-block pattern of rank at most 31 in characteristic 3?
+   The witness is one free-zero point of one hyperplane of the #6953 column
+   span. Inside that 36-dimensional space the legal fibers have affine
+   dimensions 0, 2 and 5, and this point was not minimized over them. Another
+   basis of the same space has a legal point of rank 36. A first step is one
+   other explicit pattern of rank at most 31, inside this space or outside
+   degree at most 2. A failed search is not a lower bound.
+2. Does every 1-block pattern on `P120` in characteristic 3 have rank at
+   least 32? The cited lower bound is 15, and the class-constant ansatz has
+   rank 120. The rank-36 point in the same space shows that this column space
+   does not by itself force rank 32. A first step is one linear condition
+   satisfied by every legal column, strong enough to force rank at least 32,
+   with the hypotheses written down.
+
+## Credit
+
+DreamingOfClouds, PR #6933, for the window, the port list, and Theorem 4.
+The predecessor pattern of rank 37 on `P120` and rank 36 on `P104` is
+#6953. Prepared by unpaidmoon with substantial Cursor Grok assistance.
+New work is Apache-2.0.

research/p120-char3-rank32-window/SOURCES.json

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