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

[DaysSky]

No new κ. Main's cover numbers (research/format-limits §D, #595) establish that the 63 readers of a bit source (E7 = t⊥) need three hyperplane sections: two flats cover at most 52 lines, and some three D6 flats cover all 63. This package classifies the covers exactly.

Change and mathematical scope

One self-contained package, research/e7-flat-cover-classification/, plus one path-filtered workflow and one .gitattributes line. No existing file changes.

1. The covering triples are exactly the D4 frames through t.
   • Three hyperplane flats cover E7 if and only if they are E7 ∩ a⊥, E7 ∩ b⊥, E7 ∩ c⊥ for an orthogonal frame {t, a, b, c} spanning a D4.
   • There are 315 such triples, one per D4 through t.
   • The other 3780 orthogonal frames span only 4A1, and their triples miss at least 8 lines.
2. The near-covers are exactly the A3 subsystems through t. A near-cover is a triple covering exactly 62 of the 63 lines.
   • Each near-cover is E6(A) + E6(B) + D6(m). Here A and B are the two A2s through t of the A3, and m is the A3's line orthogonal to t, which is the missed line.
   • There are 378 = 63 × 6, and no other type occurs.
3. The cover count does not force 3 registers per D6 pool.
   • In a covering triple, each D6 has 16 private lines, which span it.
   • Two rank-5 subflats never cover a whole D6 (at most 28 of 30). But they do cover its 16 private lines (75 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 alone give the proposed 180 × 3 = 540 count.

Consequence for bit skeletons.

• The D4 skeletons of Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493, Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510 and Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543 read each target through the D6 triple of its 4-frame. By (1), every three-section top layer has this form.
• The only near-alternative is (2), with one line read separately.

Scope.

• These are statements about covers by flats, the model of §D. They do not price a skeleton.
• (3) does not construct a 2-per-pool design: frame nesting, the readers' frames and restoration are not modelled.

Duplicates.

• Main has the flat census, the best pair and the existence of a D6 cover. It does not classify covers or near-covers.
• Conditional κ = 0.001283479303751122: E8 bit word R715 (carrier lower bound + F2 overlap cancellation) with #458's complex supplier (+0.711% over #460) #463 has the two-section carrier bound.
• Checked against main at f567884a and against the titles and bodies of every hub PR through Research construction: four-primitive X4 preconditions save 784 per orientation (no new kappa) #883.

Validation

python3 -B research/e7-flat-cover-classification/cover_check.py uses the standard library only.

• Run. It prints PASS_E7_FLAT_COVER_CLASSIFICATION (21 checks) in about 6 minutes. It passed on Linux aarch64 with Python 3.12.3 and 3.11.16, against main at f567884a. CI runs Python 3.11 and 3.14.
• Exact arithmetic. Ranks are computed mod 2³¹ − 1. By Hadamard, minors of at most 8 norm-4 rows are at most 256 in absolute value, so these equal the rational ranks.
• Exhaustive searches. Every search is exhaustive.
• Agreement with main. The checker confirms main's pinned D1 values (8821 flats, the size histogram, the best pair 52, cover number 3) both from its own census and from the text of d1_cover_numbers.py.

Credits and AI disclosure

• DreamingOfClouds: Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems #595's cover-number census and Conditional κ = 0.001283479303751122: E8 bit word R715 (carrier lower bound + F2 overlap cancellation) with #458's complex supplier (+0.711% over #460) #463's carrier bound.
• eumemic: the D4 skeleton of Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493 and Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543.
• Prepared by DaysSky with assistance from Claude.

[Mister-Meeseeks]

100-PR batch review at 12c4789c61ac843dcd00e2c71e5388d362298b1a.

This remains open after scoped triage. Next review step: After repairing a missing historical format-limits input, the full E7 classification replay reached 180 seconds. Finish the cover/near-cover domain and remaining controls; the initial missing-file error was our fixture setup.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

Batch review, attribution and evidence. Thank you for the contribution.