Witness a characteristic-3 rank-36 pattern on P104.

research/p104-char3-rank36-window/NOTICE

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+Characteristic-3 rank-36 witness for the P104 port window.
+Prepared by unpaidmoon with substantial Cursor Grok assistance.
+The port list is the P104 set published in PR #6933.
+The rank-38 degree-2 pattern without a constant term is the predecessor
+finding in discussion comment 19492602.
+New exposition and checks are Apache-2.0.

research/p104-char3-rank36-window/PROOF.md

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+# Characteristic 3 admits a rank-36 pattern on P104
+
+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 below is the block-mask window of PR #6933. A physical decoder
+for `k > 1` with exactly this mask is not derived, there or here.
+
+## 0. Definitions
+
+`P120` is the 120 vectors in `F_2^9` of weight congruent to 3 mod 4 (weights
+3 and 7). Two distinct vectors are adjacent when their `F_2` dot product is
+1 and orthogonal when it is 0. `P104` is the 104-vector subset in
+`data/ports_p104.json`, byte-identical to the port list of #6933 at
+`a967c521074c2c7450255ae237fda9c439ee1b35` (sha256
+`691c7b7d212112115b1bfc76db89f90d304ec448dc8c4a0c43d9a5954317146c`). Its
+complement in `P120` is 16 vectors and exactly two cliques of size 8. Each
+of those cliques, added to `P104`, is a 112-vector set `P112`.
+
+A **1-block pattern** on a vertex set `S` is a matrix, indexed by `S`, with
+nonzero diagonal and with a zero in both off-diagonal positions of every
+orthogonal pair. Entries on adjacent pairs are free, and the matrix need not
+be symmetric. Write `w_1(S; F)` for the minimum rank of a 1-block pattern
+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)`.
+
+The predecessor note (fork PR 151, discussion comment 19492602) proves
+`w(S; F) ≤ 36` for every nonempty `S ⊆ P120` and every field of
+characteristic not 3, by the matrix `3I − A`, and proves `w(S; F_3) ≤ 38` by
+a degree-at-most-2 factorisation with no constant term. Characteristic 2 is
+already settled by #6933 and is not reopened. 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.
+
+## 1. The adjacency square vanishes
+
+**Lemma 1.** Let `A` be the adjacency matrix of the odd graph on `P120`.
+Over every field of characteristic 3, `A^2 = 0`.
+
+*Proof.* For an undirected simple graph, `(A^2)_ij` is the number of walks of
+length 2 from `i` to `j`, including the degree on the diagonal. `verify.py`
+enumerates all 120 vertices and all `C(120, 2) + 120` unordered pairs
+`{i, j}` with `i ≤ j`, counts the common 1-bits of the two neighbor masks,
+and checks that every count is divisible by 3. The domain is finite, every
+pair is visited once, and the walk interpretation is the definition of `A^2`.
+The same integer counts remain 0 in every field of characteristic 3. ∎
+
+The predecessor proves that the graph is strongly regular with parameters
+`(120, 63, 30, 36)`, which gives the integer identity
+`A^2 = 27I − 6A + 36J`. All three coefficients are divisible by 3, so that
+identity specialises to Lemma 1. The parameter census is not repeated here.
+Lemma 1 does not depend on it.
+
+## 2. The class-constant ansatz cannot have rank 36
+
+Call a matrix **class-constant** when its diagonal entries are equal, its
+adjacent entries are equal, and its orthogonal entries are equal. Every such
+matrix has the form
+
+```
+α I + β A + γ (J − I − A).
+```
+
+**Lemma 2.** Over every field of characteristic 3, every class-constant
+1-block pattern on `P120` has rank 120. In particular, no class-constant
+pattern has rank 36.
+
+*Proof.* An orthogonal pair has `A`-entry 0 and `(J − I − A)`-entry 1, so the
+matrix entry there is `γ`. A pattern forces `γ = 0`. The diagonal entry is
+then `α`. If `α = 0`, the diagonal vanishes and the matrix is not a pattern.
+The reduction of `3I − A` is this illegal case: in characteristic 3 it equals
+`−A`. If `α ≠ 0`, the matrix is `α(I + c A)` with `c = β α^{-1}`. Lemma 1
+gives
+
+```
+(I + c A)(I − c A) = I − c^2 A^2 = I.
+```
+
+So the matrix is invertible and has rank 120. ∎
+
+This is a proof for one restricted ansatz: matrices constant on the three
+relations of the odd graph. It is not a proof that an arbitrary pattern has
+rank greater than 36. The legal class-constant patterns are exactly the
+invertible ones. `verify.py` checks the three monic cases `I + β A` over
+`F_3`, including the inverse identity, and checks that `3I − A` has zero
+diagonal modulo 3.
+
+The same lemma is why the predecessor's rank-36 matrix does not reduce into
+characteristic 3. Closing the gap needs a matrix that is not class-constant.
+
+## 3. The degree-at-most-2 witness
+
+Order the 46 monomials of degree at most 2 on `F_2^9` 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`. This is a `120 × 46` matrix `Φ`.
+
+For each vertex `v`, let `C_v` be the matrix whose rows are the rows of `Φ`
+indexed by `v` and by the vertices orthogonal to `v`. The system
+`C_v y = e`, with right-hand side 1 on the `v` row and 0 on the orthogonal
+rows, is consistent. Its reduced row echelon form has a unique set of pivot
+columns. Take the unique solution `y_v` supported on those pivot columns
+(free coordinates equal to 0). Define `M_uv = Φ_u · y_v`.
+
+**Theorem 1.** `M` is a 1-block pattern over `F_3` of rank 37. Its principal
+submatrix on `P104` has rank 36. Extending scalars, the same matrix is a
+1-block pattern of those same ranks over every field of characteristic 3.
+Consequently, for every such field `F`,
+
+```
+w(P104; F) ≤ w_1(P104; F) ≤ 36,
+w(P120; F) ≤ w_1(P120; F) ≤ 37.
+```
+
+Of the two `P112` extensions, the one that adds the clique
+
+```
+(7, 25, 42, 52, 76, 82, 97, 127)
+```
+
+has principal rank 36, and the one that adds
+
+```
+(67, 104, 137, 162, 266, 289, 448, 491)
+```
+
+has principal rank 37. So one 112-port extension meets the same ceiling 36,
+and the other only meets 37.
+
+*Proof.* `verify.py` rebuilds `Φ` and every `y_v`, substitutes the solutions
+back into the equations, and checks every diagonal entry and both
+off-diagonal entries of every orthogonal pair. Rank is the number of pivots
+in Gaussian elimination over `F_3`. A minor is zero over `F_3` if and only
+if it is zero over an extension, and a nonzero value in `F_3` stays nonzero,
+so the rank is unchanged by scalar extension. A principal submatrix of a
+pattern is a pattern. The sha256 of the `104 × 104` principal submatrix, in
+the port-file order, is
+
+```
+473541e0dcc7e2c22e879d37ebd3cd3e1f88c1de5c0306fd42fa38ff134c6e6a
+```
+
+with entries written as bytes in `{0, 1, 2}` row by row. ∎
+
+The same solver with the constant monomial omitted reproduces the
+predecessor's pattern: rank 38 on `P120`, and rank 37 on the `P104` principal
+submatrix. The predecessor recorded only the upper bound 38 for every subset.
+The constant monomial is what drops the `P104` principal rank from 37 to 36
+in this construction. `M` is not class-constant: it is asymmetric, and the
+adjacent entries take more than one value. Lemma 2 does not apply to it.
+
+## 4. What the witness does not minimise
+
+`Φ` has rank 45. Its one-dimensional kernel is contained in the homogeneous
+solution space of every vertex. The coefficient space of solutions with
+diagonal normalised to 1 therefore has dimension 4 at each weight-3 vertex
+and dimension 9 at each weight-7 vertex, while the space of legal columns
+`Φ y` has dimension 3 and 8 respectively. `verify.py` checks these dimensions
+for every vertex.
+
+Theorem 1 uses one point of that affine family, the point whose free
+coordinates are 0. It is not an exhaustive search of the family, and a
+different point might have smaller rank on `P120`. No such point is claimed
+here. The pairs-only and linear-only feature maps are inconsistent; that
+rules out those smaller maps and does not rule out another 36-dimensional
+model.
+
+So three statements stay distinct:
+
+1. **Proof, restricted ansatz.** Every class-constant pattern has rank 120
+   in characteristic 3 (Lemma 2).
+2. **Proof, one witness.** The explicit matrix of Theorem 1 has rank 36 on
+   `P104` and rank 37 on `P120`.
+3. **Not settled.** Whether every 1-block pattern on `P120` has rank at
+   least 37, and whether some other point of the degree-at-most-2 family
+   has rank 36 on `P120`. A search that did not find such a point 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 | 36 |
+| P112, clique `(7, 25, 42, 52, 76, 82, 97, 127)` | 14 | 36 |
+| P112, the other deleted clique | 14 | 37 |
+| P120 | 15 | 37 |
+
+The lower bounds are not claimed for arbitrary subsets. Characteristic 2
+stays inside #6933's interval `[53/6, 9]` on `P104` and the value 9 on
+`P112` and `P120`.
+
+## 5. What this does not say
+
+A rank-36 pattern is not a multiplication algorithm, not a paid profile, and
+not a change to the selected frontier. It does not show that 36 is the
+minimum on `P104`: the cited lower bound is 13, and the star floor `53/6`
+of #6933 is weaker. It does not show that `P120` admits rank 36. It does
+not touch the `F_5` rank-13 factorisation. No human has reviewed this note.

research/p104-char3-rank36-window/README.md

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+# Characteristic-3 rank-36 pattern on the P104 window (no new κ)
+
+**No new κ.** The selected record stays `781451/250000000`. This package
+exhibits one 1-block pattern of rank 36 on `P104` over every field of
+characteristic 3. 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, vendored byte-for-byte. A 1-block
+pattern has nonzero diagonal and zeros on both off-diagonal positions of
+every orthogonal pair.
+
+1. **Class-constant ansatz, proved impossible at rank 36.** In
+   characteristic 3 the adjacency matrix satisfies `A^2 = 0`. Every matrix
+   that is constant on the diagonal, on edges, and on non-edges is then
+   either illegal (zero diagonal) or invertible of rank 120. The matrix
+   `3I − A` is the illegal case. This is not an unrestricted nonexistence
+   theorem.
+2. **Witness, outside that ansatz.** The 46 monomials of degree at most 2,
+   including the constant, give a 1-block pattern over `F_3` of rank 37 on
+   `P120` and rank 36 on `P104`. Scalar extension preserves both ranks, so
+   `w(P104; F) ≤ 36` and `w(P120; F) ≤ 37` for every field `F` of
+   characteristic 3. One of the two `P112` extensions has rank 36 and the
+   other has rank 37.
+3. **The witness is one point of a positive-dimensional family.** With the
+   diagonal normalised to 1, the coefficient spaces have dimensions 4 and 9,
+   and the legal columns have dimensions 3 and 8. Free coordinates were set
+   to 0. That choice was not minimised over the family.
+
+The same solver without the constant term reproduces the predecessor's rank
+38 on `P120` and gives rank 37 on `P104`. The predecessor recorded only the
+bound 38. The constant term is the step from 37 to 36 on `P104`.
+
+Accepting the lower bounds in #6933, Theorem 4, without replaying them, the
+characteristic-3 window now satisfies `w(P104; F) ∈ [13, 36]`. The matching
+upper bound 36 was previously known only off characteristic 3. No equality
+is claimed.
+
+## 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
+characteristic-3 half of the question left open by the predecessor finding
+on discussion comment 19492602: whether characteristic 3 admits a rank-36
+pattern on the P104 window. Base commit
+`7c49fe32c55de02c2c873655c804b16e4c3fef0a`, priority revision 8.
+
+The `F_5` rank-13 factorisation is a different question and is not
+investigated. Characteristic 2 is not reopened.
+
+## Reproduce
+
+From the repository root, CPython 3.12, standard library only:
+
+```sh
+python3 -B research/p104-char3-rank36-window/verify.py
+python3 -I -O -B research/p104-char3-rank36-window/verify.py
+python3 -B scripts/research_package.py --package research/p104-char3-rank36-window
+```
+
+Expected final line: `PASS_CHAR3_RANK36 p104=36 p120=37 class_constant=120`,
+then `PASS_REGISTERED_RESEARCH_PACKAGE` with outcome `partial`.
+
+The port file is an input copied from #6933 because that commit is not in
+this repository's history. Its sha256 is pinned in `verify.py`. 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 36 in characteristic 3? The
+   witness here has rank 37, and one of the two `P112` extensions also stops
+   at 37. A first step is one explicit `120 × 36` factorisation over `F_3`,
+   or a lower bound of 37 with the hypotheses written down. Nothing in this
+   package shows that every pattern has rank at least 37.
+2. Does any other point of this degree-at-most-2 family, with the recorded
+   nullities 4 and 9, have rank 36 on `P120`? The free-coordinate-zero point
+   has rank 37. The family was not searched. A partial search would not
+   decide the question.
+
+The rank-13 factorisation over `F_5` is open in the predecessor note and is
+intentionally not a next step of this package.
+
+## Credit
+
+DreamingOfClouds, PR #6933, for the window, the port list, and Theorem 4.
+The predecessor finding (discussion comment 19492602, fork PR 151) proved the
+rank-36 pattern off characteristic 3 and the rank-38 degree-2 pattern in
+characteristic 3. Prepared by unpaidmoon with substantial Cursor Grok
+assistance. New work is Apache-2.0.

research/p104-char3-rank36-window/SOURCES.json

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