From 3cc68fb0585fa78baf121707c8a3154081f4515c Mon Sep 17 00:00:00 2001
From: Nick Petalas <webmasternikos@gmail.com>
Date: Wed, 11 Nov 2026 13:31:01 +0000
Subject: [PATCH] feat(research): separate fixed P120 patterns from rank 15

---
 research/p120-char3-rank31-window/FACTOR.json |   1 +
 research/p120-char3-rank31-window/NOTICE      |  14 +
 research/p120-char3-rank31-window/PROOF.md    | 113 ++++++++
 research/p120-char3-rank31-window/README.md   | 102 ++++++++
 .../p120-char3-rank31-window/SOURCES.json     |  18 ++
 research/p120-char3-rank31-window/TASK.json   |  81 ++++++
 research/p120-char3-rank31-window/VERIFY.json |  36 +++
 research/p120-char3-rank31-window/verify.py   | 244 ++++++++++++++++++
 8 files changed, 609 insertions(+)
 create mode 100644 research/p120-char3-rank31-window/FACTOR.json
 create mode 100644 research/p120-char3-rank31-window/NOTICE
 create mode 100644 research/p120-char3-rank31-window/PROOF.md
 create mode 100644 research/p120-char3-rank31-window/README.md
 create mode 100644 research/p120-char3-rank31-window/SOURCES.json
 create mode 100644 research/p120-char3-rank31-window/TASK.json
 create mode 100644 research/p120-char3-rank31-window/VERIFY.json
 create mode 100644 research/p120-char3-rank31-window/verify.py

diff --git a/research/p120-char3-rank31-window/FACTOR.json b/research/p120-char3-rank31-window/FACTOR.json
new file mode 100644
index 0000000000..2371a01245
--- /dev/null
+++ b/research/p120-char3-rank31-window/FACTOR.json
@@ -0,0 +1 @@
+{"schema":"p120-f3-rank-factorization/1","field_order":3,"rank":31,"vertex_order":"ascending integers v in [0,512) with bit_count(v) mod 4 = 3","left":[[1,1,1,0,0,0,0,0,2,1,0,0,1,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[1,1,0,1,0,0,0,0,2,0,1,0,1,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[1,0,1,1,0,0,0,0,2,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0],[0,1,1,1,0,0,0,0,2,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0],[1,1,0,0,1,0,0,0,2,0,0,1,1,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,2],[1,0,1,0,1,0,0,0,2,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[0,1,1,0,1,0,0,0,2,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0],[1,0,0,1,1,0,0,0,2,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[0,1,0,1,1,0,0,0,2,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0],[0,0,1,1,1,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0],[1,1,0,0,0,1,0,0,2,0,0,0,2,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,2],[1,0,1,0,0,1,0,0,2,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0],[0,1,1,0,0,1,0,0,2,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0],[1,0,0,1,0,1,0,0,2,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,1,0,1,0,1,0,0,2,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,1,1,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0],[1,0,0,0,1,1,0,0,2,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,1,1,0,0,2,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0],[0,0,1,0,1,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,0,0,1,1,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,0],[1,1,0,0,0,0,1,0,2,0,0,0,1,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[1,0,1,0,0,0,1,0,2,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[0,1,1,0,0,0,1,0,2,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0],[1,0,0,1,0,0,1,0,2,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0],[0,1,0,1,0,0,1,0,2,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,0,0],[0,0,1,1,0,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0],[1,0,0,0,1,0,1,0,2,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,1,0,1,0,2,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0],[0,0,1,0,1,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0],[0,0,0,1,1,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0],[1,0,0,0,0,1,1,0,2,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,1,1,0,2,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,1,1,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1],[0,0,0,1,0,1,1,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1],[0,0,0,0,1,1,1,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1],[1,1,1,1,1,1,1,0,2,1,1,1,2,1,0,1,1,1,1,2,0,1,1,1,1,1,1,1,1,1,0],[1,1,0,0,0,0,0,1,2,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,2],[1,0,1,0,0,0,0,1,2,1,0,0,0,0,1,0,0,0,0,2,1,0,0,0,1,0,0,0,0,0,0],[0,1,1,0,0,0,0,1,2,0,0,0,0,0,0,1,0,0,0,2,2,0,0,0,1,0,0,0,0,0,0],[1,0,0,1,0,0,0,1,2,0,1,0,0,0,1,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0],[0,1,0,1,0,0,0,1,2,0,0,0,0,0,0,0,1,0,0,2,2,0,0,0,0,0,0,1,0,0,0],[0,0,1,1,0,0,0,1,2,0,0,0,0,0,0,0,0,0,0,1,2,1,0,0,1,0,0,1,0,0,0],[1,0,0,0,1,0,0,1,2,0,0,1,0,0,1,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,1,0,0,1,2,0,0,0,0,0,0,0,0,1,0,2,2,0,0,0,0,0,0,0,0,1,0],[0,0,1,0,1,0,0,1,2,0,0,0,0,0,0,0,0,0,0,1,2,0,1,0,1,0,0,0,0,1,0],[0,0,0,1,1,0,0,1,2,0,0,0,0,0,0,0,0,0,0,1,2,0,0,0,0,1,0,1,0,1,0],[1,0,0,0,0,1,0,1,2,0,0,0,1,0,1,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,1,0,1,2,0,0,0,0,0,0,0,0,0,1,2,2,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,1,0,1,2,0,0,0,0,0,0,0,0,0,0,1,2,0,0,1,1,0,0,0,0,0,1],[0,0,0,1,0,1,0,1,2,0,0,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,1,1,0,0,1],[0,0,0,0,1,1,0,1,2,0,0,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,1,1,1],[1,1,1,1,1,1,0,1,2,1,1,1,2,0,1,1,1,1,1,0,2,1,1,1,1,1,1,1,1,1,0],[2,1,1,1,1,1,2,2,1,2,2,2,1,0,0,2,2,2,2,2,1,2,2,2,1,2,2,1,2,1,2],[1,2,1,1,1,1,2,2,1,2,2,2,1,2,2,2,2,2,2,0,2,2,2,2,1,2,2,1,2,1,2],[1,1,2,1,1,1,2,2,1,2,2,2,1,2,2,2,2,2,2,1,2,2,2,2,0,2,2,1,2,1,2],[1,1,1,2,1,1,2,2,1,2,2,2,1,2,2,2,2,2,2,1,2,2,2,2,1,2,2,0,2,1,2],[1,1,1,1,2,1,2,2,1,2,2,2,1,2,2,2,2,2,2,1,2,2,2,2,1,2,2,1,2,0,2],[2,2,2,2,2,1,2,2,1,0,0,0,2,0,0,0,0,0,2,1,2,0,0,2,0,0,2,0,2,0,1],[1,1,1,1,1,2,2,2,1,2,2,2,1,2,2,2,2,2,2,1,2,2,2,2,1,2,2,1,2,1,1],[2,2,2,2,1,2,2,2,1,0,0,2,0,0,0,0,0,2,0,1,2,0,2,0,0,2,0,0,2,1,0],[2,2,2,1,2,2,2,2,1,0,2,0,0,0,0,0,2,0,0,1,2,2,0,0,0,2,2,1,0,0,0],[2,2,1,2,2,2,2,2,1,2,0,0,0,0,0,2,0,0,0,1,2,2,2,2,1,0,0,0,0,0,0],[2,1,2,2,2,2,2,2,1,0,0,0,2,0,0,2,2,2,2,1,2,0,0,0,0,0,0,0,0,0,1],[1,2,2,2,2,2,2,2,1,2,2,2,1,2,2,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[1,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,1,1,1,1,1,0,0,0,1,1,1,2,0,0,1,1,1,1,1,0,1,1,1,0,1,1,0,1,0,2],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0],[1,1,1,1,1,0,1,0,0,1,1,1,1,1,0,1,1,1,0,2,0,1,1,0,1,1,0,1,0,1,2],[0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[1,1,1,1,0,1,1,0,0,1,1,0,2,1,0,1,1,0,1,2,0,1,0,1,1,0,1,1,0,0,0],[1,1,1,0,1,1,1,0,0,1,0,1,2,1,0,1,0,1,1,2,0,0,1,1,1,0,0,0,1,1,0],[1,1,0,1,1,1,1,0,0,0,1,1,2,1,0,0,1,1,1,2,0,0,0,0,0,1,1,1,1,1,0],[1,0,1,1,1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1],[0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0],[1,1,1,1,1,0,0,1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,1,0,1,0,1,2],[0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1],[1,1,1,1,0,1,0,1,0,1,1,0,2,0,1,1,1,0,1,1,1,1,0,1,1,0,1,1,0,0,0],[1,1,1,0,1,1,0,1,0,1,0,1,2,0,1,1,0,1,1,1,1,0,1,1,1,0,0,0,1,1,0],[1,1,0,1,1,1,0,1,0,0,1,1,2,0,1,0,1,1,1,1,1,0,0,0,0,1,1,1,1,1,0],[1,0,1,1,1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,2,1,1,1,1,1,1,1,1,1,1,1],[0,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,1,1,1,2,2,1,1,1,1,1,1,1,1,1,1],[1,1,1,1,1,1,2,2,2,2,2,2,1,2,2,2,2,2,2,2,1,2,2,2,1,2,2,1,2,1,2],[2,2,2,2,1,1,2,2,2,0,0,2,2,0,0,0,0,2,2,2,1,0,2,2,0,2,2,0,2,1,1],[2,2,2,1,2,1,2,2,2,0,2,0,2,0,0,0,2,0,2,2,1,2,0,2,0,2,2,1,2,0,1],[2,2,1,2,2,1,2,2,2,2,0,0,2,0,0,2,0,0,2,2,1,2,2,2,1,0,2,0,2,0,1],[2,1,2,2,2,1,2,2,2,0,0,0,1,0,0,2,2,2,2,2,1,0,0,2,0,0,2,0,2,0,2],[1,2,2,2,2,1,2,2,2,2,2,2,1,2,2,0,0,0,2,0,2,0,0,2,0,0,2,0,2,0,2],[2,2,2,1,1,2,2,2,2,0,2,2,0,0,0,0,2,2,0,2,1,2,2,0,0,2,2,1,2,1,0],[2,2,1,2,1,2,2,2,2,2,0,2,0,0,0,2,0,2,0,2,1,2,2,2,1,2,0,0,2,1,0],[2,1,2,2,1,2,2,2,2,0,0,2,2,0,0,2,2,2,2,2,1,0,2,0,0,2,0,0,2,1,1],[1,2,2,2,1,2,2,2,2,2,2,2,1,2,2,0,0,2,0,0,2,0,2,0,0,2,0,0,2,1,1],[2,2,1,1,2,2,2,2,2,2,2,0,0,0,0,2,2,0,0,2,1,2,2,2,1,2,2,1,0,0,0],[2,1,2,1,2,2,2,2,2,0,2,0,2,0,0,2,2,2,2,2,1,2,0,0,0,2,2,1,0,0,1],[1,2,2,1,2,2,2,2,2,2,2,2,1,2,2,0,2,0,0,0,2,2,0,0,0,2,2,1,0,0,1],[2,1,1,2,2,2,2,2,2,2,0,0,2,0,0,2,2,2,2,2,1,2,2,2,1,0,0,0,0,0,1],[1,2,1,2,2,2,2,2,2,2,2,2,1,2,2,2,0,0,0,0,2,2,2,2,1,0,0,0,0,0,1],[1,1,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,1,2,0,0,0,0,0,0,0,0,0,1]],"right":[[2,2,2,0,2,2,0,2,0,0,2,2,0,2,0,0,2,0,0,0,2,2,0,2,0,0,2,0,0,0,2,0,0,0,0,0,2,2,0,2,0,0,2,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,1,1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,1,2,2,2,2,1,2,1,1,1,1,0,1,2,2,2,2,1,2,1,1,1,1,0,2,0,0,0,1,0,1,1,0,0,1,0,0,0,0,0],[1,1,0,2,1,0,2,0,2,0,0,0,2,0,2,0,0,2,0,0,1,0,2,0,2,0,0,2,0,0,0,2,0,0,0,0,1,0,2,0,2,0,0,2,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,1,2,1,2,1,2,2,1,2,2,2,1,2,2,2,0,2,1,2,2,2,2,2,0,0,0,0,0,2,1,2,2,2,2,2,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0],[2,0,2,2,0,2,2,0,0,2,0,2,2,0,0,2,0,0,2,0,0,2,2,0,0,2,0,0,2,0,0,0,2,0,0,0,0,2,2,0,0,2,0,0,2,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,2,1,1,2,2,1,2,2,1,2,2,2,1,2,2,0,2,2,1,2,2,2,2,0,0,0,2,0,2,2,1,2,2,2,2,0,0,0,2,0,2,0,0,0,0,1,0,0,1,1,0,1,1,0,0,1],[0,2,1,1,0,0,0,2,2,2,0,0,0,2,2,2,0,0,0,2,0,0,0,2,2,2,0,0,0,2,0,0,0,2,0,0,0,0,0,2,2,2,0,0,0,2,0,0,0,2,0,0,0,0,0,1,0,0,0,0,2,0,0,0,2,2,2,1,1,1,2,2,2,1,2,2,2,1,2,0,2,2,2,1,2,2,2,0,0,0,2,0,2,2,2,1,2,2,2,0,0,0,2,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0],[0,0,0,0,2,1,1,1,1,1,0,0,0,0,0,0,2,2,2,2,0,0,0,0,0,0,2,2,2,2,0,0,0,0,2,0,0,0,0,0,0,0,2,2,2,2,0,0,0,0,2,0,0,0,0,0,1,0,0,2,0,0,0,0,2,2,2,2,2,2,1,1,1,1,2,2,2,2,1,0,2,2,2,2,1,2,2,0,0,0,2,0,2,2,2,2,1,2,2,0,0,0,2,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,0,0,0,0,0,0,2,1,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,1,1,1,1,1,0,2,2,2,2,2,0,1,0,0,0,2,0,2,2,2,2,2,0,1,0,0,0,2,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,1,1,1,1,1,1,1,1,2,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,0,2,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,0,1,1,1,1,1,1,1,0,0,0,1,1,2,2,2,2,2,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,1,1,1,1,1,1,1,1,1,2,1,1,1,1,0,2,2,2,2,2,0,2,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,0,2,2,2,2,2,0,2,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,2,2,2,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[2,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0,2,0,0,0,1,0,1,1,0,1,0,0,2,0,0,1,0,0,0,0,2,0,2,0,2,2,0,2,0,0,2],[0,2,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,0,0,0,1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,1,0,2,0,0,0,0,1,1,0,1,0,1,0,2,0,0,0,1,0,0,0,0,0,2,0,0,0,2,0,0,0,2,0],[0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,2,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,1,2,0,0,0,0,0,1,1,1,0,0,1,2,0,0,0,0,1,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,1,1,1,2,0,0,0,0,0,0,1,1,1,1,2,0,0,0,0,0,0,0,1,1,1,1,2,0,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,1,1,1,0,1,1,1,1,2,2,0,0,0,0,0,0,0,0,0,0,2,0,1,1,1,1,2,1,2,2,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,1,1,1,0,1,1,1,1,2,0,1,1,1,1,2,1,2,2,2,2,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,1,0,0,1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,1,0,0,1,1,0,1,0,0,0,1,0,1,0,0,2,0,0,0,0,2,0,0,2,0,0,0,2],[0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,2,0,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0,0,1,0,1,0,1,1,0,0,1,0,0,0,1,0,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,1,1,0,2,0,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,2,2,2,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,1,1,1,0,1,1,1,0,0,2,0,0,0,0,0,0,0,0,2,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,1,1,1,0,1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,0,0,0,1,1,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,2,2,2,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,2,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0,0,0,0,1,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,1,1,0,1,0,0,2,0,0,0,1,1,0,1,1,1,0,0,0,0,0,0,1,1,0,1,1,1,0,0,0,0,0,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,1,1,0,1,1,1,0,1,1,0,0,0,2,0,0,0,0,0,0,0,0,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,2,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0,0,0,0,1,1,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0,2,0,0,1,1,1,0,1,1,0,0,0,0,0,0,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,1,1,1,0,1,1,1,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,2,0,1,1,1,1,0,1,0,0,0,0,0,0,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,2,2,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]],"minor_rows":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,16,20,21,22,23,26,35,36,38,51,52,57,59,60,61,62,63],"minor_columns":[0,1,2,3,4,5,8,9,10,11,17,18,19,20,21,31,32,33,34,35,36,37,52,53,54,64,65,66,67,70,80],"space":"span(x_i+x_8-1, x_i*x_j+1-x_8 : 0<=i<8, 0<=i<j<8) over F3","left_in_space":[[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2],[0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1],[1,1,1,1,1,1,1,1,2,2,2,2,1,2,2,2,2,2,2,2,1,2,2,2,1,2,2,1,2,1,2]],"forced_vertices":[127,191,223,239,247,251,253,254,259,261,262,265,273,289,321,385],"forced_coefficients":[[0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[1,1,2,2,2,2,2,2,2,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1],[1,2,1,2,2,2,2,2,0,2,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1],[2,1,1,2,2,2,2,2,0,0,1,1,1,1,1,2,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1],[1,2,2,1,2,2,2,2,0,0,2,0,0,0,0,1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,1],[1,2,2,2,1,2,2,2,0,0,0,2,0,0,0,1,1,0,1,1,1,1,0,1,1,1,0,1,1,1,0,0,0,1,1,1],[1,2,2,2,2,1,2,2,0,0,0,0,2,0,0,1,1,1,0,1,1,1,1,0,1,1,1,0,1,1,0,1,1,0,0,1],[1,2,2,2,2,2,1,2,0,0,0,0,0,2,0,1,1,1,1,0,1,1,1,1,0,1,1,1,0,1,1,0,1,0,1,0],[1,2,2,2,2,2,2,1,0,0,0,0,0,0,2,1,1,1,1,1,0,1,1,1,1,0,1,1,1,0,1,1,0,1,0,0]],"constraint_minor_rows":[[36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,70,74,80,35],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,70,74,92,51],[10,11,12,13,14,15,16,17,18,19,30,31,32,33,34,35,46,47,48,49,50,51,58,59,60,61,62,63,64,65,66,67,70,80,92,57],[4,5,6,7,8,9,16,17,18,19,26,27,28,29,34,35,42,43,44,45,50,51,56,57,60,61,62,63,64,65,66,67,74,80,92,59],[1,2,3,7,8,9,13,14,15,19,23,24,25,29,33,35,39,40,41,45,49,51,55,57,59,61,62,63,64,65,66,70,74,80,92,60],[0,2,3,5,6,9,11,12,15,18,21,22,25,28,32,35,37,38,41,44,48,51,54,57,59,60,62,63,64,67,68,70,74,80,92,61],[0,1,3,4,6,8,10,12,14,17,20,22,24,27,31,35,36,38,40,43,47,51,53,57,59,60,61,63,65,67,69,70,74,80,92,62],[0,1,2,4,5,7,10,11,13,16,20,21,23,26,30,35,36,37,39,42,46,51,52,57,59,60,61,62,66,68,69,71,75,81,93,63],[0,1,4,9,10,15,18,19,20,25,28,29,32,33,34,35,36,41,44,45,48,51,54,57,59,60,61,65,66,67,70,74,80,90,92,64],[0,2,5,8,11,14,17,19,21,24,27,29,31,33,34,35,37,40,43,45,47,51,53,57,59,60,62,64,66,67,70,74,80,89,92,65],[0,3,6,7,12,13,16,19,22,23,26,29,30,33,34,35,38,39,42,45,46,51,52,57,59,60,63,64,65,68,71,75,81,89,93,66],[1,2,6,7,12,13,17,18,22,23,27,28,31,32,34,35,38,39,43,44,47,51,53,57,59,61,62,64,65,68,70,74,80,88,92,67],[3,4,5,7,12,14,15,16,22,24,25,26,31,32,33,35,38,40,41,42,47,51,53,57,60,61,62,64,65,67,71,74,80,87,92,70],[3,6,8,9,10,11,13,16,22,24,25,27,28,29,30,35,38,40,41,43,46,51,53,59,60,61,62,64,65,67,70,75,80,85,92,74],[3,6,8,9,12,14,15,17,18,19,20,21,23,26,30,35,38,40,41,43,47,52,57,59,60,61,62,64,65,67,70,74,79,81,92,80],[3,6,8,9,12,14,15,17,18,19,22,24,25,27,31,36,37,39,42,46,51,52,57,59,60,61,62,64,65,67,70,74,79,80,93,92]],"forced_minor_rows":[0,1,2,3,4,10,20,35,36,51,57,59,60,61,62,63]}
diff --git a/research/p120-char3-rank31-window/NOTICE b/research/p120-char3-rank31-window/NOTICE
new file mode 100644
index 0000000000..265e42e99d
--- /dev/null
+++ b/research/p120-char3-rank31-window/NOTICE
@@ -0,0 +1,14 @@
+A characteristic-three rank-31 pattern on P120
+Copyright 2026 Nick Petalas. Licensed under Apache-2.0.
+
+This contribution was produced by Nick Petalas.
+
+The new construction builds directly on unpaidmoon's rank-32 P120 pattern
+and open rank-31 question, PR #7087 at
+043b334e7249e8b79e437a0765773a7e2fc8b8e8, and its quadratic predecessor #6953.
+The unrestricted rank-15 clique construction is unpaidmoon's PR #7014.
+The port-window model and port family are DreamingOfClouds's #6933.
+unpaidmoon's #6946 supplies the earlier odd-characteristic upper bounds.
+The fractional Haemers framework is due to Blasiak and Haemers, with the
+cited formulation by Bukh and Cox. Classical finite-field linear algebra
+retains its prior credit. No predecessor code is included in this package.
diff --git a/research/p120-char3-rank31-window/PROOF.md b/research/p120-char3-rank31-window/PROOF.md
new file mode 100644
index 0000000000..14c4ef847e
--- /dev/null
+++ b/research/p120-char3-rank31-window/PROOF.md
@@ -0,0 +1,113 @@
+# A fixed quadratic space misses the rank-15 P120 pattern
+
+Claim type: finite certificate with a scoped algebraic proof. The main result
+is a restriction on one column space, not an unrestricted minimum-rank theorem.
+
+Let V be the ascending list of integers v with 0 <= v < 512 and binary weight
+3 or 7. It has 120 elements. Write x_i(v) for bit i of v, with bits numbered
+0 through 8. Two vertices are orthogonal when their binary dot product is zero.
+A legal one-block pattern has nonzero diagonal entries and zeros at every
+ordered orthogonal pair. Symmetry is not required.
+
+## The fixed space
+
+For every field K of characteristic three, let U_K be the space of functions
+on V spanned over K by these 36 functions:
+
+```text
+x_i + x_8 - 1,       0 <= i < 8,
+x_i x_j + 1 - x_8,  0 <= i < j < 8.
+```
+
+Use this displayed order, with pairs ordered lexicographically. Let F be their
+120 by 36 evaluation matrix. This is the fixed quadratic space specified by
+the nine coefficient relations in unpaidmoon's #7087, expressed without the
+redundant coefficient coordinates. Indeed, if the free coefficients are a_i
+and b_ij for indices below 8, then
+
+```text
+a_8 = sum_i a_i - sum_(i<j) b_ij,
+c = -a_8,
+b_i8 = 0.
+```
+
+Thus c+a_8=0 and sum_i a_i+2a_8+2 sum_(i<j) b_ij=0 over characteristic three,
+exactly the declared relations. The omitted b_78 coefficient is zero here.
+The uniqueness minors below also prove that the 36 evaluation functions are
+independent. The result uses this explicitly defined space, irrespective of
+other representations or conventions.
+
+## The obstruction
+
+**Theorem.** Over every field K of characteristic three, every legal P120
+pattern whose columns lie in U_K has rank at least 16. In particular, no such
+pattern has rank 15.
+
+**Proof.** Scale each column by the inverse of its diagonal entry. This preserves
+rank, membership in the linear space U_K, and forced zeros, and makes the
+diagonal one.
+
+For a vertex v, coefficients z in K^36 of a normalized legal column must satisfy
+
+```text
+F[u,:] z = 0   for every u orthogonal to v,
+F[v,:] z = 1.
+```
+
+`FACTOR.json` lists 16 vertices v and, for each, a coefficient vector z_v
+satisfying these equations. For each v it also lists 36 allowed equation rows.
+The 36 by 36 matrix of those rows has nonzero determinant in F3. Hence the
+coefficient solution is unique over F3 and over every K of characteristic three.
+The latter assertion follows because the same determinant is a nonzero prime-
+subfield element of K. Therefore every normalized legal matrix with columns
+in U_K contains all 16 displayed columns F z_v at the indicated vertices.
+
+The artifact supplies 16 row indices for these columns. Their 16 by 16 minor
+has determinant one in F3 and therefore in K. The mandatory columns are
+independent, so every such matrix has rank at least 16. QED.
+
+Only these 16 columns are needed. No classification of all rigid columns or
+all legal patterns is claimed. The exact checks cover 912 legal-column
+equations, 16 uniqueness determinants and the independence determinant.
+
+## A rank-31 construction in the same space
+
+The artifact also supplies A in F3^(120 x 31) and B in F3^(31 x 120). The checker
+computes M=AB and verifies its 120 unit diagonal entries and all 6,720 ordered
+orthogonal entries. It also supplies C in F3^(36 x 31); all 3,720 entries of
+A=FC are checked. Consequently every column of M lies in the same fixed space.
+
+The checker evaluates nonzero 31 by 31 minors of A and B and their corresponding
+product minor in M. Their determinants are 1, 2 and 2. Hence M has rank at least
+31, while the factorization gives rank at most 31. It has rank exactly 31, and
+scalar extension preserves the certificate over every characteristic-three K.
+
+This point was obtained by intersecting the legal affine column fibers in the
+span of #7087's rank-32 construction with a common hyperplane. A fiber c+N meets
+ker(L) precisely when L is nonzero on N or L(c)=0. This follows by cancelling
+L(c) with a direction in N, or by observing that the functional is constant on
+the fiber. Only the resulting point is claimed; unsuccessful search candidates
+prove no lower bound. The delivered factors are the complete construction,
+and checking them requires no predecessor code, Git or network.
+
+## Consequence for the next construction
+
+Accepting the existing clique construction in unpaidmoon's #7014, P120 has a
+rank-15 legal pattern over every field. Its result is cited, not replayed here.
+The new theorem shows that this rank cannot be attained with every column in
+U_K. Thus a rank-15 construction must leave this particular quadratic space.
+The bounds established here inside U_K are 16 <= minimum rank <= 31. Neither
+endpoint is claimed optimal in that restricted problem.
+
+The rank-31 point is not an improvement to the unrestricted window bound.
+The initial broad upper-bound target was superseded by #7014; the published
+scope is this fixed-space separation and its new rank-31 companion point.
+The scope correction is recorded in the linked Discussion update.
+
+## Remaining obligations
+
+No optimal restricted rank, physical decoder, native word or recursive supplier
+is constructed. Delivery, restoration, precision, stock, movement and a complete
+caller are unpriced. No characteristic-zero identity follows from these finite-
+characteristic factors. No new kappa or selected-frontier change is claimed.
+Independent human and formal proof-kernel review are unclaimed.
diff --git a/research/p120-char3-rank31-window/README.md b/research/p120-char3-rank31-window/README.md
new file mode 100644
index 0000000000..0cb84e1402
--- /dev/null
+++ b/research/p120-char3-rank31-window/README.md
@@ -0,0 +1,102 @@
+# The fixed P120 quadratic space cannot attain rank 15
+
+This contribution was produced by Nick Petalas.
+
+Claim type: finite certificate with a scoped algebraic proof. Over every field
+of characteristic three, legal P120 patterns whose columns lie in #7087's fixed
+36-dimensional quadratic space have rank at least 16. Sixteen uniquely forced
+columns and a nonzero minor certify this. A new explicit rank-31 point belongs
+to the same space, lowering its cited constructive ceiling from 32 to 31.
+Thus the restricted minimum lies in [16,31]; its exact value remains open.
+
+unpaidmoon's [#7014](https://github.com/CrocSwap/integer-mult-bounds/pull/7014)
+already gives an unrestricted rank-15 clique construction. The new obstruction
+shows that this particular quadratic space cannot attain that rank. The rank-31
+point is not an improvement to the unrestricted window bound. There is no new
+kappa, physical decoder or selected-frontier change.
+
+[PROOF.md](PROOF.md) defines the space and proves the uniform field consequence.
+[FACTOR.json](FACTOR.json) contains the obstruction and the complete rank-31
+factors. The standalone checker runs no predecessor verifier.
+
+## Reproduce
+
+From the repository root, using CPython 3.12 and the standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error -X dev research/p120-char3-rank31-window/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error -X dev -O research/p120-char3-rank31-window/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p120-char3-rank31-window
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p120-char3-rank31-window --check-source-history
+```
+
+Another CPython 3.12 path may replace the local interpreter path. The scientific
+marker is `PASS_P120_CHAR3_RANK31 no_new_kappa`. The verifier prints a generated
+receipt bound to all eight package files and writes no files. The publication
+check needs full Git history; the scientific checker needs neither Git nor
+network access.
+
+The exact checks cover all 14,400 product entries, 120 diagonal entries, 6,720
+forced zeros, three rank-31 determinants, 3,720 fixed-space membership entries,
+912 forced-column equations, 16 uniqueness determinants and a nonzero 16 by 16
+minor. Eleven rejection controls cover malformed certificates and violations of
+the mask, rank, fixed-space membership, forced columns and independence. Three
+small determinant fixtures cover identity, row-swap and singular matrices.
+
+## Scope and limitations
+
+Card `complex-rank-shape-decoder`, priority revision 8, changes scope from a
+complete signed-moment improvement to a fixed-space obstruction and finite
+construction. The source-bound baseline command passed at the selected #7000
+checkpoint. The older catalogue context is not the selected frontier.
+
+The initial target was a new unrestricted rank-31 upper bound answering #7087.
+A broader PR check exposed #7014's stronger prior construction. The
+[scope correction](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19542514)
+records this before publication. The main new result is the restriction forcing
+rank at least 16, so the fixed space misses the known unrestricted rank 15.
+
+No optimal restricted rank, characterization of all rigid columns, physical
+native implementation or complete supplier invoice is claimed. The five selected
+framework assumptions remain unchanged. No predecessor scientific suite,
+selected scientific replay, Lean or Docker run was performed. Independent human
+and formal proof-kernel review are unclaimed.
+
+The verifier has fixed dimensions and bounded loops, with a 64 KiB certificate
+limit. Python allocates bounded temporary lists during this one-shot check;
+this managed-runtime deviation is limited to matrices at most 120 by 120 and
+has no retained outputs. Indirect calls are restricted to the named determinant
+and negative-control routines. Warning-strict normal and optimized runs check
+the same artifact. Shared code and diagnostics configuration are unchanged.
+
+## Open next questions
+
+1. What is the exact minimum rank inside this fixed quadratic space? The new
+   bounds are 16 and 31. A first step is a common hyperplane meeting every legal
+   affine fiber inside the delivered rank-31 span. A failed search would not
+   prove a lower bound for all subspaces.
+2. Which enlargement or change of the quadratic space admits a rank-15 point?
+   Start by representing the cited #7014 clique columns in a larger feature
+   space. Their membership and the cost of a useful decoder remain open here.
+3. Is any of this finite-field structure useful in characteristic zero? Start
+   with a separate exact mask factorization and its complete paid invoice.
+   Reduction modulo three alone does not supply that implementation.
+
+## Credit and sources
+
+unpaidmoon supplies the fixed quadratic space, rank-32 construction and open
+questions in [#7087 at 043b334e](https://github.com/CrocSwap/integer-mult-bounds/blob/043b334e7249e8b79e437a0765773a7e2fc8b8e8/research/p120-char3-rank32-window/README.md),
+the quadratic predecessor in [#6953](https://github.com/CrocSwap/integer-mult-bounds/pull/6953),
+the stronger unrestricted clique construction in [#7014](https://github.com/CrocSwap/integer-mult-bounds/pull/7014),
+and the earlier odd-characteristic bound in [#6946](https://github.com/CrocSwap/integer-mult-bounds/pull/6946).
+DreamingOfClouds supplies the port-window definition and family in
+[#6933](https://github.com/CrocSwap/integer-mult-bounds/pull/6933).
+Bukh, Cox, Blasiak and Haemers retain the fractional Haemers framework credit.
+Finite-field elimination, rank factorization and scalar extension are classical.
+The new fixed-space obstruction, rank-31 factors and standalone checker were
+produced by Nick Petalas. Apache-2.0; see [NOTICE](NOTICE).
+
+Cited pending predecessors are not replay dependencies. Their bytes are not
+needed to check this certificate. SOURCES.json pins catalogue and selected-record
+provenance. The advance [CLAIM](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19542430)
+and the scope correction state the change in the research question.
diff --git a/research/p120-char3-rank31-window/SOURCES.json b/research/p120-char3-rank31-window/SOURCES.json
new file mode 100644
index 0000000000..5335fa97d9
--- /dev/null
+++ b/research/p120-char3-rank31-window/SOURCES.json
@@ -0,0 +1,18 @@
+{
+  "schema_version": 1,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "75d77b328ea02ded535e7f23aee5b1e55d0bb1a8",
+      "sha256": "154c3d30455d778fd3343b6b4851f0fa4665f0f7625b0adb3675e2bafbbf6bf1"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    }
+  ]
+}
diff --git a/research/p120-char3-rank31-window/TASK.json b/research/p120-char3-rank31-window/TASK.json
new file mode 100644
index 0000000000..6bf6f51910
--- /dev/null
+++ b/research/p120-char3-rank31-window/TASK.json
@@ -0,0 +1,81 @@
+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 8,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "original_card": {
+    "id": "complex-rank-shape-decoder",
+    "question": "Can a legal decoder or coframe change improve the full signed moment, including changed helper targets?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+      "docs/research/recursive-invoice-preflight.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "A source-bound arbitrary-input identity and legal producer/consumer boundary with the complete paid and productive target invoice, including restoration, ownership, native alignment and extra charges. Price under the actual current model; historical ceiling.py assumptions must be established before use. A favorable optimistic relaxation keeps a route open but is not a completed supplier. A scoped local result remains useful; expansion requires the family-level budget gate.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Price one complete changed itinerary with both paid and productive histograms. Check #5693/#5698 for overlap and their exact hypotheses before another coframe search. Use the signed invoice preflight where its boundary contract agrees; an inconclusive concave comparison is not a power crossing. Stop if an optimistic complete budget fails at the chosen target or native delivery remains unspecified.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "coframe",
+      "signed moment",
+      "5693",
+      "5698"
+    ],
+    "existing_checks": [
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Prove rank at least 16 for legal characteristic-three P120 patterns in the fixed 36-dimensional quadratic column space of PR #7087; supply a new rank-31 point in that space.",
+  "scope_change_reason": "The initial broad rank-31 target was already dominated by PR #7014 rank 15. A public scope correction precedes publication. The new contribution separates the fixed quadratic space from the stronger unrestricted construction, and supplies a new restricted rank-31 upper bound.",
+  "bottleneck_link": "Side result on decoder feature-space choice: rank 15 requires leaving this fixed quadratic space. No characteristic-zero decoder, paid invoice or kappa follows.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop at the first verified original rank improvement; leave further reductions as open questions.",
+  "outcome": "partial",
+  "evidence": "PROOF.md; FACTOR.json; standalone verify.py checks 16 forced independent columns, uniqueness, fixed-space membership and a rank-31 attaining point.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "What is the exact minimum rank inside the fixed quadratic space?",
+      "why_it_matters": "The certificate narrows its range to 16 through 31 and excludes the unrestricted clique rank 15.",
+      "first_step": "Seek a hyperplane meeting every legal affine column fiber in the delivered rank-31 span.",
+      "blocked_by": "No sharper lower bound or smaller attaining pattern is provided."
+    },
+    {
+      "question": "What change of feature space admits a rank-15 pattern?",
+      "why_it_matters": "The known unrestricted clique construction cannot have all columns in this space.",
+      "first_step": "Express the cited clique columns in an enlarged feature space.",
+      "blocked_by": "The current certificate deliberately fixes the 36-dimensional space."
+    }
+  ]
+}
diff --git a/research/p120-char3-rank31-window/VERIFY.json b/research/p120-char3-rank31-window/VERIFY.json
new file mode 100644
index 0000000000..0f8e13412f
--- /dev/null
+++ b/research/p120-char3-rank31-window/VERIFY.json
@@ -0,0 +1,36 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "-W",
+        "error",
+        "-X",
+        "dev",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Fixed quadratic-space rank-16 obstruction, rank-31 construction and membership, all mask entries, nonzero minors and eleven rejection controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "-W",
+        "error",
+        "-X",
+        "dev",
+        "-O",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Fixed quadratic-space rank-16 obstruction, rank-31 construction and membership, all mask entries, nonzero minors and eleven rejection controls."
+    }
+  ]
+}
diff --git a/research/p120-char3-rank31-window/verify.py b/research/p120-char3-rank31-window/verify.py
new file mode 100644
index 0000000000..c252cbba7e
--- /dev/null
+++ b/research/p120-char3-rank31-window/verify.py
@@ -0,0 +1,244 @@
+#!/usr/bin/env python3
+"""Check a fixed-space P120 obstruction and one rank-31 pattern, without Git."""
+
+import hashlib
+import json
+from pathlib import Path
+import time
+
+PACKAGE = Path(__file__).resolve().parent
+ORDER = 120
+RANK = 31
+MAX_ARTIFACT_BYTES = 65536
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def vertices():
+    return [v for v in range(512) if v.bit_count() % 4 == 3]
+
+
+def validate_matrix(matrix, rows, columns):
+    require(type(matrix) is list and len(matrix) == rows, "matrix row count")
+    for row in matrix:
+        require(type(row) is list and len(row) == columns, "matrix width")
+        require(all(type(x) is int and 0 <= x < 3 for x in row), "field entry")
+
+
+def validate_indices(indices, size=RANK):
+    require(type(indices) is list and len(indices) == size, "minor index count")
+    require(all(type(i) is int and 0 <= i < ORDER for i in indices), "minor index")
+    require(len(set(indices)) == size, "minor repeated index")
+
+
+def validate_artifact(data):
+    require(type(data) is dict, "artifact object")
+    require(data.get("schema") == "p120-f3-rank-factorization/1", "schema")
+    require(type(data.get("field_order")) is int and data["field_order"] == 3, "field")
+    require(type(data.get("rank")) is int and data["rank"] == RANK, "rank")
+    require(data.get("vertex_order") ==
+            "ascending integers v in [0,512) with bit_count(v) mod 4 = 3", "order")
+    validate_matrix(data.get("left"), ORDER, RANK)
+    validate_matrix(data.get("right"), RANK, ORDER)
+    validate_indices(data.get("minor_rows"))
+    validate_indices(data.get("minor_columns"))
+    require(data.get("space") ==
+            "span(x_i+x_8-1, x_i*x_j+1-x_8 : 0<=i<8, 0<=i<j<8) over F3", "space")
+    validate_matrix(data.get("left_in_space"), 36, RANK)
+    validate_matrix(data.get("forced_coefficients"), 16, 36)
+    require(type(data.get("constraint_minor_rows")) is list and
+            len(data["constraint_minor_rows"]) == 16, "constraint minor count")
+    for indices in data["constraint_minor_rows"]:
+        validate_indices(indices, 36)
+    validate_indices(data.get("forced_minor_rows"), 16)
+    labels = data.get("forced_vertices")
+    require(type(labels) is list and len(labels) == len(set(labels)) == 16,
+            "forced vertex count")
+    require(all(type(v) is int and v in vertices() for v in labels), "forced vertex")
+
+
+def product(left, right):
+    return [[sum(left[i][t] * right[t][j] for t in range(RANK)) % 3
+             for j in range(ORDER)] for i in range(ORDER)]
+
+
+def mask_counts(matrix, ports):
+    validate_matrix(matrix, ORDER, ORDER)
+    diagonal = 0
+    forced_zeros = 0
+    for i in range(ORDER):
+        require(matrix[i][i] == 1, "nonunit diagonal")
+        diagonal += 1
+        for j in range(ORDER):
+            if (ports[i] & ports[j]).bit_count() % 2 == 0:
+                require(matrix[i][j] == 0, "orthogonal entry")
+                forced_zeros += 1
+    return diagonal, forced_zeros
+
+
+def determinant(matrix):
+    """Independent square elimination, with row swaps and all arithmetic in F3."""
+    n = len(matrix)
+    require(1 <= n <= ORDER and all(len(row) == n for row in matrix), "square")
+    work = [row[:] for row in matrix]
+    result = 1
+    for col in range(n):
+        pivot = next((r for r in range(col, n) if work[r][col] % 3), None)
+        if pivot is None:
+            return 0
+        if pivot != col:
+            work[col], work[pivot] = work[pivot], work[col]
+            result = -result % 3
+        value = work[col][col] % 3
+        result = result * value % 3
+        for r in range(col + 1, n):
+            scale = work[r][col] * value % 3
+            for c in range(col, n):
+                work[r][c] = (work[r][c] - scale * work[col][c]) % 3
+    return result
+
+
+def rank_checks(data, matrix):
+    rows, cols = data["minor_rows"], data["minor_columns"]
+    left = [data["left"][i][:] for i in rows]
+    right = [[data["right"][i][j] for j in cols] for i in range(RANK)]
+    minor = [[matrix[i][j] for j in cols] for i in rows]
+    det_left, det_right, det_minor = map(determinant, (left, right, minor))
+    require(det_left != 0 and det_right != 0, "singular factor minor")
+    require(det_minor == det_left * det_right % 3 and det_minor != 0,
+            "singular product minor")
+    return det_left, det_right, det_minor
+
+
+def feature_rows(ports):
+    rows = []
+    for v in ports:
+        x = [(v >> i) & 1 for i in range(9)]
+        row = [(x[i] + x[8] - 1) % 3 for i in range(8)]
+        row += [(x[i] * x[j] + 1 - x[8]) % 3
+                for i in range(8) for j in range(i + 1, 8)]
+        rows.append(row)
+    return rows
+
+
+def obstruction_checks(data, ports):
+    features = feature_rows(ports)
+    coordinates = data["left_in_space"]
+    for i in range(ORDER):
+        for j in range(RANK):
+            value = sum(features[i][t] * coordinates[t][j] for t in range(36)) % 3
+            require(value == data["left"][i][j], "factor outside fixed space")
+    forced = [[sum(row[t] * c[t] for t in range(36)) % 3
+               for c in data["forced_coefficients"]] for row in features]
+    constraint_dets = []
+    equations = 0
+    for j, vertex in enumerate(data["forced_vertices"]):
+        index = ports.index(vertex)
+        for i in range(ORDER):
+            orthogonal = (ports[i] & vertex).bit_count() % 2 == 0
+            if orthogonal or i == index:
+                require(forced[i][j] == int(i == index), "illegal forced column")
+                equations += 1
+        chosen = data["constraint_minor_rows"][j]
+        require(all(i == index or (ports[i] & vertex).bit_count() % 2 == 0
+                    for i in chosen), "minor uses an unconstrained row")
+        det = determinant([features[i] for i in chosen])
+        require(det != 0, "forced column not unique")
+        constraint_dets.append(det)
+    det = determinant([forced[i] for i in data["forced_minor_rows"]])
+    require(det != 0, "dependent forced columns")
+    return {"forced_columns": len(data["forced_vertices"]),
+            "forced_equations": equations, "forced_independence_det": det,
+            "uniqueness_determinants": constraint_dets,
+            "factor_membership_entries": ORDER * RANK}
+
+
+def rejects(call, *args):
+    try:
+        call(*args)
+    except ValueError:
+        return 1
+    raise ValueError("negative control was accepted")
+
+
+def controls(data, matrix, ports):
+    count = 0
+    changed = [row[:] for row in matrix]
+    changed[0][0] = 0
+    count += rejects(mask_counts, changed, ports)
+    orthogonal = next(j for j in range(ORDER)
+                      if (ports[0] & ports[j]).bit_count() % 2 == 0)
+    changed = [row[:] for row in matrix]
+    changed[0][orthogonal] = 1
+    count += rejects(mask_counts, changed, ports)
+    changed = [row[:] for row in matrix]
+    changed[0][0] = 3
+    count += rejects(mask_counts, changed, ports)
+    changed = dict(data)
+    changed["minor_rows"] = [data["minor_rows"][0]] * RANK
+    count += rejects(validate_artifact, changed)
+    changed = dict(data)
+    changed["left"] = [row[:-1] for row in data["left"]]
+    count += rejects(validate_artifact, changed)
+    changed = dict(data)
+    changed["field_order"] = True
+    count += rejects(validate_artifact, changed)
+    changed = dict(data)
+    changed["left"] = [[0] * RANK for _ in range(ORDER)]
+    count += rejects(rank_checks, changed, matrix)
+    require(determinant([[1, 0], [0, 1]]) == 1, "identity determinant")
+    require(determinant([[0, 1], [1, 0]]) == 2, "swap determinant")
+    require(determinant([[1, 2], [2, 1]]) == 0, "singular determinant")
+    changed = dict(data)
+    changed["left_in_space"] = [[0] * RANK for _ in range(36)]
+    count += rejects(obstruction_checks, changed, ports)
+    changed = dict(data)
+    changed["forced_coefficients"] = [[0] * 36 for _ in range(16)]
+    count += rejects(obstruction_checks, changed, ports)
+    changed = dict(data)
+    changed["forced_minor_rows"] = [data["forced_minor_rows"][0]] * 16
+    count += rejects(obstruction_checks, changed, ports)
+    changed = dict(data)
+    indices = [row[:] for row in data["constraint_minor_rows"]]
+    indices[0] = [indices[0][0]] * 36
+    changed["constraint_minor_rows"] = indices
+    count += rejects(obstruction_checks, changed, ports)
+    return count
+
+
+def main():
+    started = time.monotonic()
+    with (PACKAGE / "FACTOR.json").open("rb") as stream:
+        raw = stream.read(MAX_ARTIFACT_BYTES + 1)
+    require(len(raw) <= MAX_ARTIFACT_BYTES, "artifact size")
+    data = json.loads(raw)
+    validate_artifact(data)
+    ports = vertices()
+    require(len(ports) == ORDER, "port count")
+    matrix = product(data["left"], data["right"])
+    diagonal, forced = mask_counts(matrix, ports)
+    dets = rank_checks(data, matrix)
+    obstruction = obstruction_checks(data, ports)
+    rejected = controls(data, matrix, ports)
+    # Receipt is generated from the actual checked artifact and package sources.
+    names = ("README.md", "PROOF.md", "NOTICE", "TASK.json", "SOURCES.json",
+             "VERIFY.json", "verify.py", "FACTOR.json")
+    hashes = {name: hashlib.sha256((PACKAGE / name).read_bytes()).hexdigest()
+              for name in names}
+    receipt = {"claim_type": "finite certificate with scoped proof", "field": 3, "rank": RANK,
+               "fixed_space_obstruction": obstruction,
+               "ports": len(ports), "product_entries": sum(map(len, matrix)),
+               "diagonal_entries": diagonal, "forced_zero_entries": forced,
+               "determinants": dets, "rejected_controls": rejected,
+               "matrix_sha256": hashlib.sha256(bytes(x for row in matrix for x in row)).hexdigest(),
+               "sources_sha256": hashes}
+    print(json.dumps(receipt, sort_keys=True))
+    print("PASS_P120_CHAR3_RANK31 no_new_kappa")
+    print("elapsed_seconds=%.6f" % (time.monotonic() - started))
+
+
+if __name__ == "__main__":
+    main()