Add exact E6 sheared-frame cost census

research/e6-sheared-frame-seedb401/OVERLAP.json

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+{
+  "schema_version": 1,
+  "claim": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-18938219",
+  "search": {
+    "command": "python3 scripts/find_overlap.py --refresh --task complex-rank-shape-decoder --limit 100 'E6 basis shear' 'six-term gauge' 'basis transvection' 'one-extra rank-one frame' 'full-rank E6 decoder' --json",
+    "searched_at": "2026-10-15T01:58:52.749494+00:00",
+    "status": "POSSIBLE_OVERLAP_NOT_A_NOVELTY_CERTIFICATE",
+    "coverage": {
+      "open_prs": {
+        "fetched_at": "2026-10-15T01:55:22.022247+00:00",
+        "complete_pagination": true,
+        "records": 477
+      },
+      "discussions": {
+        "fetched_at": "2026-10-15T01:58:52.549544+00:00",
+        "complete_pagination": true,
+        "records": 951
+      }
+    },
+    "direct_live_check": "Fresh complete overlap refresh found no other remote match for the basis-transvection terms; the only direct Discussion match was this cycle claim #18938219. PR #2240 remains open and mergeable at head c8a2b37056e766715c457f56e2dd8172f5bdbccd; PR #2241 remains open and mergeable at head 5688d8b17830dbb813fba5a509f52449383903fa. Both had checks queued at the live status query."
+  },
+  "nearest": [
+    {
+      "pr": 1590,
+      "head": "08e134b4a0c7f04869dae703f2a2e688e53c44f0",
+      "state": "merged",
+      "scope": "Root-unit model and Theorem C pricing context."
+    },
+    {
+      "pr": 2228,
+      "head": "ad1757e319225f3a5b86e3d3fa72897bd9f6db40",
+      "state": "open",
+      "scope": "Dense root-code floors; leaves overcomplete E6 m=7 open."
+    },
+    {
+      "pr": 2137,
+      "head": "6a022d50e21bdcf2014a26937a34d44eed3473f0",
+      "state": "open",
+      "scope": "Rank-four E6 B0 coordinate-form family."
+    },
+    {
+      "pr": 2232,
+      "head": "3391f27dd0f226f0e1ead8b2b829eb6d3ec567d3",
+      "state": "open",
+      "scope": "Rank-four projected Gram with adapted pairing."
+    },
+    {
+      "pr": 2240,
+      "head": "c8a2b37056e766715c457f56e2dd8172f5bdbccd",
+      "state": "open",
+      "scope": "Prior Cycle30 family fixes A=I; this census varies A over signed elementary transvections."
+    },
+    {
+      "discussion": "#1167 comment #18936773",
+      "scope": "Active S3 rank-nine claim; different object."
+    },
+    {
+      "pr": 2241,
+      "scope": "Recent Cycle42 sum-anchor helper inventory; different object, just completed."
+    }
+  ],
+  "scope_distinction": "Full-rank six-factor basis is changed by A=I+epsilon E_ij before the rank-one redundant term; fixed E6 Gram and B0; 60 matrices, 365 directions, all exact lambda support cells.",
+  "selected_frontier_changed": false
+}

research/e6-sheared-frame-seedb401/PROOF.md

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+# Exact statement and support-cell proof
+
+## Finite statement
+
+Fix the exact E6 root lines and basis reconstructed from the pinned `ade.E6b()` ordering, their Gram matrix `H`, the coordinate matrix `X`, and the dense pairing `B0` with unit diagonal and off-diagonal ones only on orthogonal root pairs. For every signed elementary transvection
+
+```text
+A = I + epsilon E_ij,   i != j,   epsilon in {-1,+1},
+```
+for every direction `a` in the union of the 36 projective E6 root directions and the 364 projective nonzero ternary vectors in the pinned basis, and for every rational `lambda != 0`, define `c=A^-1 a` and
+
+```text
+p_j = X A e_j,      q_j = X(H A^-T e_j - lambda c_j a),  j=1,...,6,
+p_7 = X a,           q_7 = lambda X a.
+```
+
+The exhaustive support-cell census finds minimum raw cost 37 and minimum active-term Theorem C cost 37. Thus no seven-active decoder in this declared finite family has cost at most 35. The result is not a lower bound for all E6 seven-term decoders.
+
+## Identity
+
+As matrices in the six-dimensional coefficient space, the first six factors have sum
+
+```text
+A (H A^-T)^T - lambda (A c) a^T
+  = A A^-1 H^T - lambda a a^T
+  = H - lambda a a^T,
+```
+
+using symmetry of `H` and `A c=a`. The seventh factor contributes `lambda a a^T`. Multiplying by `X` and `X^T` proves `sum_{j=1}^7 p_j q_j^T = X H X^T = G`. Each root has norm squared 2, and `B0` is one only on the diagonal or orthogonal pairs, so `B0` is admissible. The verifier checks this identity for both `A=I` and `A=I+E_01`, then rejects a reversed correction sign and an omitted seventh term.
+
+The first independent basis rows have root indices `[0,1,2,4,6,20]`; the first nonsingular ambient minor uses columns `[0,3,4,5,6,7]`. The canonical root digest, exact basis Gram and `rank_F2(B0)=7` are pinned in [CERTIFICATE.json](CERTIFICATE.json).
+
+## Exhaustive lambda cells
+
+For each `A,a`, the left supports are fixed. The seventh right support is fixed for nonzero lambda. Each coordinate of the remaining right factors is affine:
+
+```text
+q_j(t) = X_t H A^-T e_j - lambda * (A^-1 a)_j * (X_t a).
+```
+
+A nonzero slope has one exact rational zero. A zero slope gives either an identically zero coordinate or one that never vanishes. The verifier inserts every nonzero zero with `0` as a partition cut, evaluates every nonzero breakpoint itself, and one rational point in every open interval, including both tails. This represents every allowed lambda with exactly the same support. Degenerate terms are counted by their actual active total count; no zero term is charged the floor `w=5`.
+
+Across the 60 signed transvections and 365 directions there are 143,343 distinct nonzero breakpoints. Including the zero cut gives `2*(60*365+143343)=330486` point/interval representatives. The search is split into two reproducible shards of 30 transvections.
+
+## Cost
+
+For each `B0[R_j,A_j]`, binary Gaussian elimination computes `r_j` exactly. The raw cost is `sum r_j`; the full Theorem C charge is `sum max(5,r_j)` over active terms. The complete cost histogram ranges from 37 through 45, with counts in [CERTIFICATE.json](CERTIFICATE.json). The least-cost witness is `A=I+E_23`, `a=(1,-1,1,0,0,1)`, `lambda=1/2`; its ranks are `(6,0,6,6,6,6,7)`, leaving six active totals at cost 37. All seven-active cells also have cost at least 37 because the full histogram contains no cost below 37. The six-term square benchmark 36 is cited from the open request and is not recomputed here.
+
+No source accesses, copied-information realization, frame transitions, hosting, restoration, precision, recursive children or supplier integration are supplied. The result does not change the selected frontier or establish a multiplication bound.

research/e6-sheared-frame-seedb401/README.md

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+# E6 sheared base factorization with one extra frame
+
+**Scoped negative result; no new κ.** For the 60 signed elementary transvections of the pinned six-coordinate E6 Gram factorization, the exact one-extra rank-one frame construction has minimum raw pairing cost **37** and minimum fully charged Theorem C cost **37**. The census covers **330,486** rational support cells across the 60 transvections and the union of 36 root directions with 364 projective ternary directions. No seven-active decoder reaches cost 35.
+
+The least-cost cell is degenerate to six active totals: its rank profile is `(6,0,6,6,6,6,7)`, at `A=I+E_{2,3}`, `a=(1,-1,1,0,0,1)`, `lambda=1/2`. Its charged cost 37 is still above the square-decoder benchmark 36 in the open request. The finite result does not settle arbitrary rational basis changes, other seven-term identities, sparse pairings, or the unrestricted E6 question.
+
+## Construction
+
+Let `X` be the exact 36-by-6 E6 root-coordinate matrix and `H` the Gram matrix of its pinned basis, so `G=X H X^T`. For a signed transvection `A=I+epsilon E_ij`, define `c=A^{-1}a`. For each nonzero projective direction `a` and `lambda != 0`, use
+
+```text
+p_j = X A e_j
+q_j = X(H A^{-T} e_j - lambda c_j a),  j=1,...,6
+p_7 = X a
+q_7 = lambda X a.
+```
+
+Since `A c=a`, the six correction terms sum to `-lambda X a a^T X^T`, which the seventh term restores. Hence the decoder is exactly `G`. The dense pairing `B0` stays fixed: diagonal one and off-diagonal one exactly on orthogonal root pairs.
+
+The direction set is the union of all 36 E6 root-line directions and all 364 projective vectors in `{-1,0,1}^6` in the pinned basis. For each transvection and direction, every rational breakpoint where a coordinate of `q_1,...,q_6` becomes zero is included, as is one exact rational representative from each complementary interval. Zero lambda is excluded. The census records degenerate terms and charges only active totals.
+
+The full cost histogram is `37:6, 38:12, 39:18, 40:131, 41:369, 42:1397, 43:29259, 44:117297, 45:181997`. There are 26 cells with six active totals and 330,460 with seven. The exact per-transvection records, witness, support-cell counts and root pins are in [CERTIFICATE.json](CERTIFICATE.json); the two shards are in [SHARD-0.json](SHARD-0.json) and [SHARD-1.json](SHARD-1.json).
+
+A hypothetical seven-active win has optimistic floor `7*5=35`, one below the square benchmark 36. The tested family’s minimum is 37 even when degenerate cells are included. This is only an algebraic root-unit cost. No supplier implementation, per-call paid profile, source/copy/frame/hosting/restoration/precision charge, or κ improvement is established.
+
+## Reproduce
+
+From repository root, with Python 3.12 and only the standard library:
+
+```sh
+python3 scripts/research_package.py --package research/e6-sheared-frame-seedb401 --check-only
+python3 scripts/research_package.py --package research/e6-sheared-frame-seedb401
+```
+
+The registered replay exhausts two shards of 30 transvections each and checks their saved results plus the combined certificate. The exact verifier can also run a quick identity-control smoke check with `python3 -B research/e6-sheared-frame-seedb401/verify.py --smoke`.
+
+## Overlap and attribution
+
+The current open request is [Discussion #1166, comment #18936173](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-18936173). The completed Cycle30 [PR #2240](https://github.com/CrocSwap/integer-mult-bounds/pull/2240), head `c8a2b37056e766715c457f56e2dd8172f5bdbccd`, fixes `A=I`; this contribution changes the six-term base factorization by signed elementary transvections. Merged [PR #1590](https://github.com/CrocSwap/integer-mult-bounds/pull/1590), head `08e134b4a0c7f04869dae703f2a2e688e53c44f0`, supplies the root-unit model and Theorem C context. Open [PR #2228](https://github.com/CrocSwap/integer-mult-bounds/pull/2228) leaves overcomplete E6 at seven terms open. Open [PR #2137](https://github.com/CrocSwap/integer-mult-bounds/pull/2137) and [PR #2232](https://github.com/CrocSwap/integer-mult-bounds/pull/2232) are different rank-four families.
+
+The exact root ordering follows the pinned `research/root-unit-decoder-lemmas/lib/ade.py` input in [SOURCES.json](SOURCES.json). Credit the authors of PR #1590 and the cited E6 decoder work for the model and predecessors. Prepared by OpenAI Codex with substantial GPT-6 assistance in research, exact derivation, implementation, verification and writing. No independent human review is claimed. Apache-2.0.
+
+## Open next questions
+
+Can a seven-term E6 decoder outside the elementary-transvection family attain fully charged cost 35? A next step is to test one structurally different basis family, such as a two-coordinate or two-shear composition, after screening its optimistic charge. Arbitrary `GL_6(Q)` changes are unbounded, and this package supplies no physical supplier mapping.