No new κ: covers of E7 by hyperplane flats are exactly the D4-frame triples; near-covers are the A3s (checked)

.gitattributes

@@ -119,3 +119,4 @@ research/tintagel-source-lookup/** -text -whitespace
 research/trace-curvature-preload/** -text -whitespace
 research/witness-spec-tools/** -text -whitespace
 research/community-family588-audit/** -text -whitespace
+research/e7-flat-cover-classification/** -text -whitespace

.github/workflows/e7-flat-cover-classification.yml

@@ -0,0 +1,37 @@
+name: Check the E7 flat-cover classification
+
+on:
+  push:
+    paths:
+      - 'research/e7-flat-cover-classification/**'
+      - 'research/format-limits/scripts/d1_cover_numbers.py'
+      - '.github/workflows/e7-flat-cover-classification.yml'
+  pull_request:
+    paths:
+      - 'research/e7-flat-cover-classification/**'
+      - 'research/format-limits/scripts/d1_cover_numbers.py'
+      - '.github/workflows/e7-flat-cover-classification.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  check:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 30
+    strategy:
+      fail-fast: false
+      matrix:
+        python-version: ['3.11', '3.14']
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97 # v7
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Classify covers and near-covers of E7 by hyperplane flats
+        run: python3 -B research/e7-flat-cover-classification/cover_check.py
+      - name: Require committed files to remain unchanged
+        run: git diff --exit-code

research/e7-flat-cover-classification/LICENSE

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research/e7-flat-cover-classification/NOTICE.md

@@ -0,0 +1,8 @@
+# Notice
+
+This note and `cover_check.py` were prepared by DaysSky, with assistance from Claude. They are released under the
+Apache-2.0 license in `LICENSE`.
+
+The checker reads `research/format-limits/scripts/d1_cover_numbers.py` (#595) as text to compare pinned values, without
+modifying or importing it. That file keeps its own authors, notices and license; see the repository's `NOTICE`,
+`CONTRIBUTORS.md` and the `format-limits` package.

research/e7-flat-cover-classification/README.md

@@ -0,0 +1,88 @@
+# Covers of E7 by hyperplane flats: the D4-frame triples are the only covers
+
+**No new κ is claimed.** Main's design-free cover numbers (`research/format-limits` §D,
+`scripts/d1_cover_numbers.py`) establish that the 63 readers of a bit source need three hyperplane sections. The
+readers are the lines of E7 = t⊥. Two flats cover at most 52 lines, and some three D6 flats cover all 63. This note
+classifies the covers exactly.
+
+1. **The covering triples are exactly the D4 frames through t.**
+   - Three hyperplane flats of E7 cover all 63 lines if and only if they are E7 ∩ a⊥, E7 ∩ b⊥, E7 ∩ c⊥, where
+     {t, a, b, c} is an orthogonal frame spanning a D4 subsystem.
+   - There are 315 such triples, one per D4 through t.
+   - The other 3780 orthogonal frames through t span only 4A1. Each of their triples leaves at least 8 lines
+     uncovered.
+2. **The near-covers are exactly the A3 subsystems through t.**
+   - A near-cover is a triple that covers exactly 62 of the 63 lines.
+   - Every near-cover is E6(A) + E6(B) + D6(m) for an A3 through t. Here A and B are the A3's two A2 subsystems
+     through t, m is its one line orthogonal to t, and m is the missed line.
+   - There are 378 = 63 × 6 of them, and no other type occurs.
+3. **A cover count alone does not force 3 registers per D6 at rank 5.**
+   - In a covering triple, each D6 has 16 private lines, covered by it alone. They span the D6.
+   - Two rank-5 subflats never cover a whole D6: at most 28 of its 30 lines.
+   - But two of them do cover the 16 private lines (75 such pairs per D6), and six rank-5 flats, two per D6, cover
+     all of E7.
+   - So "two 5-dim subflats cover ≤ 28 of 30" does not by itself force a third register in each D6 pool. A
+     3-per-pool count needs a further argument about the shared lines.
+
+**Consequences for bit skeletons:**
+
+- The D4 skeletons of #493, #510 and #543 read each target through the three D6 frames {t, e}⊥, where e runs over
+  the other lines of t's 4-frame. By (1), every top layer made of three hyperplane sections has this form.
+- The only near-alternative is (2), with one line read separately.
+- (3) corrects a counting step that has been proposed for a floor of 540 = 180 × 3 bit registers in this paradigm.
+
+```sh
+python3 -B research/e7-flat-cover-classification/cover_check.py
+```
+
+The check uses the standard library only and takes about 6 minutes. It ends with
+`PASS_E7_FLAT_COVER_CLASSIFICATION (21 checks, ...)`.
+
+## Method
+
+- **Lines and flats.**
+  - The E8 lines are rebuilt in Construction-A coordinates from the [8, 4, 4] Hamming code.
+  - t is one fixed line. All lines are equivalent under W(E8), so one choice suffices.
+  - The closed flats of E7 are enumerated by rank. The rank-(r+1) flats over a rank-r flat F are the classes of the
+    remaining lines, grouped by direction modulo span(F).
+- **Exact ranks.** All ranks are computed mod p = 2³¹ − 1. By Hadamard's inequality, a minor of at most 8 rows of
+  squared norm 4 has |det| ≤ 2⁸, so ranks mod p equal rational ranks. No floating point is used.
+- **Exhaustive searches.** Every search in the table below is exhaustive.
+
+| Step | Check | Result |
+|---|---|---|
+| Flats | flats of E7 by rank 1–6 | 63 / 1281 / 10395 / 33411 / 36435 / 8821 |
+| Hyperplane flats | by size | 5040 × 10, 2016 × 13, 1008 × 16, 666 × 21, 63 × 30 (D6), 28 × 36 (E6) |
+| Agreement with main | `d1_cover_numbers.py` pins 8821 flats, the size histogram, best pair 52 and cover number 3 | identical |
+| Identification | D6 flats = E7 ∩ a⊥ (63); E6 flats = E7 ∩ A⊥ for the 28 A2 subsystems through t | exact set equality |
+| Pairs | best union of two hyperplane flats | 52; a larger union needs two flats of size ≥ 21, and all 757 such flats are tried |
+| Covering triples | all triples, members in size order | 315, all D6 + D6 + D6 |
+| Frames | orthogonal triples {a, b, c} ⊂ E7: span of {t, a, b, c} | 315 span a D4 (12 lines), 3780 span 4A1 (4 lines) |
+| Classification | covering triples = D4-frame triples; each 4A1 triple misses ≥ 8 lines; each D4 through t has one frame through t | exact |
+| Near-covers | for each of the 63 lines m, all triples covering E7 − {m} and avoiding m | 378 = 63 × 6, all E6 + E6 + D6 |
+| A3 | the 378 A3 subsystems through t and their E6(A) + E6(B) + D6(m) triples | equal to the near-covers, with the same missed line |
+| Inside a cover | private lines per D6; shared lines | 16 per D6, rank 6; 15 shared, 12 of them in all three |
+| Rank-5 subflats of a D6 | count and sizes | 268: 120 × 8, 80 × 9, 30 × 13, 32 × 15, 6 × 20 |
+| Pool step | best pair of rank-5 subflats on a D6; pairs covering the 16 private lines; a 2 + 2 + 2 cover of E7 | 28 of 30; 75 per D6; exists |
+
+## Scope
+
+- These are statements about covers by flats, which is the model behind main's §D cover numbers.
+- (1) and (2) say which covers exist. They do not price a skeleton.
+- (3) shows only that one counting step does not follow from coverage. It does not show that a design with 2
+  registers per D6 pool exists: frame nesting, the readers' frames and restoration are not modelled.
+
+## Related work
+
+- Main: `research/format-limits/README.md` §D and `scripts/d1_cover_numbers.py` (#595, DreamingOfClouds). These give
+  the census of 8821 flats, the best pair 52, and the existence of a three-D6 cover. The classification of all covers
+  and near-covers is new here.
+- #463 (DreamingOfClouds): the carrier bound, that two hyperplane sections cannot cover the 4-design s⊥.
+- #493, #510, #543 (eumemic, DreamingOfClouds): the D4-skeleton bit words that use the D6 triples.
+- Duplicates were checked against main at `f567884a` and against the titles and bodies of all hub PRs through #883.
+
+## Credits
+
+- DreamingOfClouds, for #595's cover-number census and #463's carrier bound.
+- eumemic, for the D4 skeleton of #493 and #543.
+- Prepared by DaysSky, with assistance from Claude.