No new κ: BW16 as a unit — window c ≥ 72 puts its ceiling at 0.649× E8's (checked)

[DaysSky]

No new κ. This screens the Barnes–Wall lattice BW16 as a unit, using the screen main already applies to the ADE root systems, K12, Leech and the Niemeier lattices (research/e8-decoder-floors-and-cold14-notes/ade_screen.py, #620; #598; #606). BW16 has not been screened before. Its ceiling comes out 35% below E8's.

Change and mathematical scope

One self-contained package, research/bw16-unit-screen/, plus one path-filtered workflow and one .gitattributes line. No existing file changes.

• Lattice. BW16 = RM(1,4) + 2·RM(3,4) + 4ℤ¹⁶ (Construction D). It is rebuilt from code generators. All 4320 minimal vectors are enumerated: the shapes (±2²) and (±1⁸) are proved to be the only norm-8 vectors. They give 2160 lines.
  • Every line has the same |⟨x, y⟩| profile: {0: 855, 2: 1024, 4: 280}.
• Window.
  • data/clique72.json holds 72 pairwise non-orthogonal lines. A deterministic greedy reproduces it exactly.
  • At the decoder pattern of format-limits §C.6 (Format obstructions (no new κ): exact bounds on window width, copy cost and label dimension #446 Theorem A), the clique forces B[K, K] = I, so the window is c ≥ 72. For comparison, E8 has c = 9, and its largest non-orthogonal set is exactly 8, which is also checked.
  • The pattern's natural decoder B₀ has F₂ rank 191, so 72 ≤ c_min ≤ 191.
• Screen.
  • The screen uses ade_screen.py's proxy x = c·h/v = 8/15 and Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems #595's certified skel.X, imported unchanged.
  • Result: X(s, 16, 1, 8/15) ≤ 3.8923·10⁻³ at every s = 3…39. That is 0.649× E8's bit ceiling (6.0005·10⁻³, reproduced) and below the full-E8 ceiling (5.1598·10⁻³).
• Break-even. BW16 loses at every s if and only if c ≥ 63. So the conclusion does not depend on finding the largest clique.
• Cause of the loss. It is entirely the copy charge. With free copies, X(3, 16, 1, 0) = 1.96·10⁻², more than 3× E8's ceiling.

Scope.

• This is a ranking screen of the same kind as main's ADE, Leech and Niemeier screens. It is not a κ bound.
• Only decoders on the §C.6 pattern are covered. k = 1, and helper cost is not priced (Research notes (no κ claim): E8 decoder floors are tight; extension-ring, triple and Lorentzian units lose after helpers; cold-end, dispatch and fusion-search negatives #620 §4). BW16 has 855 orthogonal neighbours per line against E8's 63, which points the same way.

Duplicates. Checked against main at f567884a and against the titles and bodies of every hub PR through #883: none screens BW16. The Gaussian E8 clique ceilings (#738, #770, #868) concern a different unit.

Validation

python3 -B research/bw16-unit-screen/bw16_check.py uses the standard library only. It prints PASS_BW16_UNIT_SCREEN (21 checks) in about 2 min 40 s. It passed on Linux aarch64 with Python 3.12.3 and 3.11.16, against main at f567884a. CI runs it on Python 3.11 and 3.14.

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 certified ceiling and the format-limits decoder-pattern results.
• Joel Pulikkan (GamingPuzzled): Research notes (no κ claim): E8 decoder floors are tight; extension-ring, triple and Lorentzian units lose after helpers; cold-end, dispatch and fusion-search negatives #620's unit screen.
• The authors of Format obstructions (no new κ): exact bounds on window width, copy cost and label dimension #446, Leech/Turyn research (no κ claim): scoped fitting bounds and cost screens #598 and Niemeier research (no κ claim): operator factors and paid spinor circuits #606.
• BW16 and Construction D are classical (Barnes and Wall 1959; Conway and Sloane, SPLAG, ch. 4 and 8).
• Prepared by DaysSky with assistance from Claude.

[xiayicheng3-code]

Note (no new κ): sub-units of BW16 with span rank 15 or 16 do not beat E8's bit ceiling
A small extension of this PR's screen from the full BW16 unit to its sub-units, using the orbit-averaged clique method of #703 / #738.
Setup. BW16 minimal vectors from Construction D on RM(1,4): 4320 vectors, 2160 lines. Let K be a 72-line pairwise non-orthogonal set: 8 lines of shape (±2,±2,0¹⁴) on a coordinate star, plus 64 lines (±1)⁸ on weight-8 RM(1,4) words. All pairwise |⟨x,y⟩| ∈ {2,4}. Let G be generated by the AGL(4,2) coordinate permutations and the RM(2,4) sign flips. G preserves the minimal vectors and has two line orbits, D (240) and H (1920). K meets each in exactly 1/30.
Lemma. For every set S of BW16 lines, some g ∈ G has |gK ∩ S| ≥ ⌈|S|/30⌉. Proof: for uniform g, gk is uniform on the orbit of k, so E|gK ∩ S| = Σ_{k∈K} |S∩O(k)|/|O(k)| = |S|/30. gK stays pairwise non-orthogonal, and the forced identity block on gK∩S gives window c(S) ≥ ⌈|S|/30⌉.
Consequence. With x = c·h/v and h = rank span(S):

• h = 16 gives x ≥ 16/30 = 0.533, against a loss threshold of 0.4613 (binding at s = 3).
• h = 15 gives x ≥ 0.500, against a threshold of 0.4745.
  So every BW16 sub-unit whose span has rank 15 or 16 has ceiling ≤ X(s,16,1,8/15) ≤ 3.89e-3, below E8's bit ceiling 6.0005e-3, for s = 3..39. For s ≥ 41 even x = 0 loses. This uses skel.X from format-limits unchanged, at the §C.6 decoder pattern with k = 1 and helpers unpriced.
  Scope and limits.
• Rank ≤ 14 is not covered: averaging gives only x ≥ h/30, which falls below the threshold. This is unavoidable in rank 8, since the 120 lines on one octad are √2·E8, so every punctured E8 (including P111) is a BW16 sub-unit.
• I also checked whether Scoped negative result: Gaussian E8 copy floor 40 excludes three stages #738-style weighted averaging could lift the full-BW16 floor above 72. Three independent local searches found maximum cliques of exactly 72 (Hoffman gives ω ≤ 135). For copy charge, an explicit flat pair of pairing rank 1, with hit fraction 11/120, caps any uniform G·K certificate at ≈785, short of the ≈997 needed. So this route does not strengthen No new κ: BW16 as a unit — window c ≥ 72 puts its ceiling at 0.649× E8's (checked) #895's c·h premise.
  Verification. Group preservation was checked over all 322,560 × 2048 elements, and the average |S|/30 was checked exactly on 12 test sets. The work was independently re-implemented and checked twice. I'm happy to package the scripts as a research PR if useful. Feedback is welcome, including whether this is too minor to merit one.

[Mister-Meeseeks]

100-PR batch review at 3bf6fccaf17521ffdb9b3bec2740ba4886478556.

This remains open after scoped triage. Next review step: The BW16 primary screen reached 180 seconds before all 21 declared checks completed. Continue its exact geometry/copy-budget replay; no conclusion is inferred from the timeout.

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.