Paper-side line counts per dimension: Lagrangian bound v ≤ T(2^k′+1) with exact ladder D4/D6/E7/E8 and no-gos scoped to windows h ≤ 13 (no κ claim)

[DreamingOfClouds]

Paper-side line counts per dimension: Lagrangian bound v ≤ T(2^k′+1) with exact ladder D4/D6/E7/E8 and no-gos scoped to windows h ≤ 13 (no κ claim)

Package research/line-system-saturation. Model, dense pairing and the polar law for odd-or-zero units are those of #446/#595/#1590 and of #2228 (Theorem 3); the T² threshold and first realisability numerics are #2125's; the 8/7 block-rank rule is #968's. This PR is paper-side only: how many lines a unit can carry per dimension T given the F2 rank 2k′ of its non-orthogonality adjacency. Compensation is not modelled.

Polar law for every line system (PROVEN). For any real line system, with labels taken from the F2 factorisation of the non-orthogonality adjacency Ā, J + Ā is admissible and equals the Gram matrix of u_t = (1, ℓ_t) under 1 ⊕ b; its rank is 2k′ + 1, or 2k′ when all labels avoid a hyperplane w^⊥. Odd-or-zero is the case Ā ≡ G mod 2 (CHECKED on D4, D5, D6, E6, E7, E8).

Lagrangian saturation bound (PROVEN). v ≤ T(2^{k′}+1), and v ≤ T·2^{k′} on the even window, with equality iff (no line orthogonal to all others, i.e. no label 0) every generator class of the symplectic polar space is a full orthogonal basis (a tight frame). The same bound holds for non-Gram units M = PQᵀ of rank T, overcomplete ones included, and the best v/T is a weighted idempotent packing number of W(2k′−1,2). CHECKED, exact: D4, D6, E7 attain it (3, 5, 9 lines per dimension), E8 attains 15/17 (2295 generators: 2025 full, 270 empty). A descent theorem (a saturated W(2k′−1,2) system yields a saturated W(2k′−3,2) system in dimension T − m_p; E7 → D6 → D4 and E8 → E7 CHECKED) shows that saturation at any k′ ≥ 4 forces a saturated k′ = 4 system. The Schrijver-ϑ′-type two-point program P2 (and every strengthening satisfied by its symmetric optimum, including the generator inequalities) has value exactly 2^{k′}+1 for Gram units (proof by symmetry reduction; the symmetric point is CHECKED for k′ = 2..5).

Bounded classes (PROVEN). Unions of unbiased frames have dense x = 1, and every admissible pairing has window ≥ F (so Kerdock/MUB-type constructions cannot beat the P104 break-even; consistent with, and not contradicting, #895's 72-clique for the full BW16 shell); norm-2 systems with products in {0, ±1} have v/T ≤ max(15, 2k′+1); symmetric multiplicity-one realisations need T ≥ d_proj(S) (Sp(10,2): 155, PROVEN mod cite [verify] for the group constant d_proj); uniform-angle saturated systems need λ² = 2^{k′−1}m to be a perfect square (m = 1: k′ odd). The 14 saved construction records are re-verified in exact integer arithmetic (best saturation at window h ≤ 13: 15/17 at k′ = 4, .386 at k′ = 5, .196 at k′ = 6), and the D16⁺ roots-plus-spinor family is capped at 240 + 317 = 557 lines by an exact rational Delsarte dual certificate.

Scope and caveats. All "cannot reach" statements are for the dense pairing, bounded windows h ≤ 13, and the repo's recorded break-even fit x*(h) (RECORDED, not reproduced). #2222's complete cubic-kernel unit (h ≤ 49, dense x ≈ .037, numbers RECORDED from its PR) escapes them; no unconditional paper-side no-go exists, and it is credited as the reason for the scoping. EXPLORATORY realisability numerics are in README §10 and are not part of any claim. Tiep–Zalesskii minimal degrees, Roberson's α_p ≤ ϑ′ and Thas' ovoid theorem are cited and marked [verify]. No κ claim, no window closed, no supplier changed.

Open next questions (README §11). A ±1 signing of the 510-vertex doubled W(7,2) graph with S² = 60S + 256I (its non-existence would rule out a uniform-angle saturated system with one line per label at k′ = 5); level-2/three-point bounds on W(7,2); realisability of any saturated W4 system; non-dense pairings; symmetric systems of multiplicity ≥ 2.

Verify. bash research/line-system-saturation/verify.sh /fresh/dir (Python standard library only, single process, no SAT; step 0 checks MANIFEST.sha256 and SOURCES.json and stops hard on mismatch; 88 PASS lines including step 0).

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.
🤖 Generated with Claude Code

[Mister-Meeseeks]

The full declared exact replay passes (including the stored vector systems and Delsarte arithmetic). One correction is needed to the interpretation of PR2222 in the scope box and section 9: a rank upper bound equal to a displayed term count does not prove that the factorization is complete rather than overcomplete. Here the 48 linear features are actually dependent on the cubic features on the norm-six shell: x_i = (1/6) sum_j x_i*x_j^2. Therefore the cubic-plus-linear feature matrix has rank at most binom(50,3)=19,600, strictly below the 19,648 displayed terms. The 19,648-term cost remains a valid conservative upper bound; the claimed “complete (not overcomplete)” conclusion does not follow and is false for that displayed factorization. Please remove or correct that classification without weakening the useful large-window comparison.

Please also keep the group-degree and other [verify] citations explicitly conditional until their precise source statements are checked. The passing arithmetic does not independently certify those external facts or the recorded break-even fit. The broader finite census and scoped line-count arguments are useful; this request concerns the interpretation and proof-status boundaries.

[DreamingOfClouds]

Thank you, and apologies for the lack of clarity in the original classification.

Fixed in place at head 8ef6fa81403973d3cacaf865bdabe9797a6c34a0:

1. PR2222. The scope box, the claims table and §9 no longer call the unit complete. On the norm-six shell Σ_j x_j² = 6, so x_i = (1/6) Σ_j x_i x_j². The cubic-plus-linear feature matrix therefore has rank at most binom(50,3) = 19,600 < 19,648, and the displayed factorization is overcomplete. §9 says the earlier wording was wrong and keeps 19,648 as a valid conservative upper bound. The large-window comparison is unchanged. New exact check: checks/pr2222_rank.py (step c7).
2. [verify] citations. The status "PROVEN mod cite" is now CONDITIONAL [verify], with each dependency named:
   • Theorem F constants. The smallest nontrivial ordinary degrees of Sp(6,2), Sp(8,2) and Sp(10,2) (7, 35, 155) are now checked with GAP 4.13 and the CTblLib ATLAS tables. The script and its recorded output are in checks/gap/, and step c8 runs when GAP= is set. T ≥ 155 stays conditional on one cited fact, that Sp(10,2) has trivial Schur multiplier (Steinberg; ATLAS). We did not compute it: GAP's multiplier computation did not finish within 20 minutes.
   • The α_p identification (Roberson; Mančinska–Roberson) is marked conditional. No bound uses it.
   • The break-even fit x*(h) was already RECORDED and conditional, and it stays that way.

verify.sh from a fresh export passes: 89 PASS, 0 FAIL, with the GAP step included.

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.