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Apache-2.0 (`LICENSE`, copied from the repository root). + +## Credits + +- **#446**, **#595**, **#1590**, **#2111**, **#2228**: the cost model, the dense pairing and the polar law for odd-or-zero units that Theorem A generalises; **#2125**: explicit pairings, the T² threshold and the first realisability numerics for the doily and Sp(6,2); **#968**: the 8/7 block-rank rule on E8; **#2162**: the T = 3 and 2-adic side. +- **#2222** (1337isnot1337): the complete polynomial-kernel unit on the P48p minimum vectors (h = 49), which is the reason every no-go here is scoped to bounded windows; **#598** (Leech/Turyn polynomial-lattice kernels); **#895** (DaysSky): the 72-clique of the full BW16 shell; **#620**, **#606**, **#897**: related unit screens. +- Literature (cited, statements to be verified by the reader where marked [verify] in the README): Roberson and Mančinska–Roberson (projective packing number, α_p ≤ ϑ′); Thas (no ovoids in W(2n−1,q), n ≥ 3); Tiep–Zalesskii (minimal degrees of Sp(2n,2); ATLAS for 7, 35, 155); Nguyen, arXiv:0910.4885; Delsarte (linear programming bound; the inequality is re-proved in README §8.2). +- `lib/exact.py`, `lib/report.py`, `lib/units.py` are carried over unchanged from the author's own package of #2228; the pattern of `scripts/provenance.py` and `verify.sh` is the same. + +## New in this package + +Everything under `research/line-system-saturation/`: `README.md`, `verify.sh`, `SOURCES.json`, `MANIFEST.sha256`, `requirements.txt`, `lib/*.py`, `checks/*.py`, `scripts/provenance.py`, `data/best_kprime_*.json` (vector files from the author's construction runs, re-verified by `checks/c5_l1cons.py`). diff --git a/research/line-system-saturation/README.md b/research/line-system-saturation/README.md new file mode 100644 index 0000000000..cc381b535a --- /dev/null +++ b/research/line-system-saturation/README.md @@ -0,0 +1,217 @@ +# 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) + +> **Scope, stated first.** This package is about the **paper side only**: how many lines v a unit (a set of lines with a prescribed non-orthogonality pattern) can carry per ambient dimension T, as a function of the F2 rank 2k′ of its non-orthogonality adjacency, and what that implies for the *dense estimate* T·h of the decoder cost. It does **not** model compensation (helpers, hosting, side programs), and it contains **no κ claim, no new supplier, no window closed**. Every "cannot beat" statement is restricted to **bounded windows h ≤ 13** and is conditional on the repo's recorded break-even fit x\*(h) (RECORDED, not reproduced here). **#2222 shows that large-h units escape** these paper-side no-gos: its unit is a cubic polynomial-kernel unit with h = 49 and dense x ≈ .037 (its displayed 19,648-term factorization is a valid upper bound but is overcomplete, §9); nothing below says anything against it, and the credit for that observation belongs to #2222 (§9). +> +> **Statuses.** **PROVEN** (full proof in this README); **CONDITIONAL [verify]** (the proof here is complete, but the stated value or identification depends on a cited external result whose exact source statement has not been checked; the claim is conditional on that citation); **CHECKED** (recomputed by `verify.sh`, exact integer/`Fraction`/bit arithmetic); **RECORDED** (taken from a working card or another PR, not re-run here); **EXPLORATORY** (floating-point numerics, evidence not proof); **OPEN**. + +## 0. Reading this package + +```sh +bash research/line-system-saturation/verify.sh /fresh/output/dir # from a checkout or `git archive`; Python >= 3.11, standard library only; single process, no SAT +``` + +Step 0 checks every file against `MANIFEST.sha256` and every pin of `SOURCES.json` (including the 14 stored vector files `data/best_kprime_*.json`) and stops with exit 1 on mismatch; six check scripts then print one `PASS`/`FAIL` line per assertion (88 PASS lines including step 0's two; well under five minutes on one core). + +**Prior work this builds on (none of its files is copied; `lib/exact.py`, `lib/report.py`, `lib/units.py` are carried over from the author's own package of #2228).** +- **#446, #595, #1590, #2111** (cost model: a decoder is M = PQᵀ with an admissible pairing B; cost = Σ_j rank_F2 B[R_j, A_j]) and **#2228** (this author, open): the dense polar law for *odd-or-zero* units, B = Gram of u_t = (1, ℓ_t) under Ω = 1 ⊕ b, with labels from G mod 2. Theorem A below removes the odd-or-zero hypothesis; its proof is #2228 Theorem 3's argument with a different label form. +- **#2125** (this author, open): explicit pairings and the T² threshold; its RECORDED numerics (the doily realises at T = 4 and not 3; the 63-point Sp(6,2) system at T = 7 and not 6) are the first rows of §10. +- **#968** (merged): the 8/7 block-rank rule on E8 (the E8 instance of Theorem A's rank statement; checked in #2228 §4.2). **#2162** (this author, open): the T = 3 side and a 2-adic/tropical place bound; not touched here. +- **#2222** (1337isnot1337, open): the dyadic norm-six lattice decoder (P48p, 26,208,000 lines, chart dimension 49, fitting kernel ((x·y)³ − 4(x·y))/192, 19,648 terms, bill ≤ 962,752). **Credit:** it is the large-h unit that escapes the bounded-window statements below (§9). +- **#598** (Leech/Turyn polynomial-lattice kernels, merged) is the lineage #2222 credits for kernel units. **#895** (DaysSky, open): the full Barnes–Wall BW16 shell (2160 lines) has a 72-clique, hence window ≥ 72. This does not contradict Theorem C, which concerns the 144-line subset made of nine orthogonal frames (§7.1). **#620, #606** (ADE/lattice screens), **#2111**, **#897**: related screens, not compared line by line. +- Literature: Roberson, projective packing number α_p and α_p ≤ ϑ′ (Schrijver) [verify, §6]; Thas, W(2n−1,q) has no ovoid for n ≥ 3 [verify, §5.1]; Tiep–Zalesskii minimal degrees [verify, §7.4]. + +## 1. Setting and notation + +A **line system** (unit) is a list of v pairwise distinct lines t = 1..v, spanned by vectors r_t ∈ ℝ^T (rational in every example here) with Gram matrix M_ts = r_t · r_s, M_tt ≠ 0. T is the dimension of their span (rank M). + +- **Non-orthogonality adjacency** Ā ∈ F2^{v×v}: Ā_ts = 1 iff t ≠ s and M_ts ≠ 0. It is symmetric with zero diagonal, hence **alternating**; its rank is even, **rank_F2 Ā = 2k′**. For odd-or-zero systems (all products 0 or odd, even norms) Ā = G mod 2. +- **Symplectic labels.** An alternating matrix of rank 2k′ factors as Ā = LᵀbL with L ∈ F2^{2k′×v} of full row rank and b the standard symplectic form on V = F2^{2k′}, b(p,q) = Σ_i (p_{2i}q_{2i+1} + p_{2i+1}q_{2i}). (Proof: pick e_1, f_1 with Ā(e_1,f_1) = 1, replace every other vector w by w + Ā(w,f_1)e_1 + Ā(w,e_1)f_1, which is Ā-orthogonal to e_1, f_1, and recurse; the unit vector δ_t equals Σ_i[Ā(δ_t,f_i)e_i + Ā(δ_t,e_i)f_i] modulo the radical, so ℓ_t := (Ā(δ_t,f_i), Ā(δ_t,e_i))_i. `lib/lines.py:alt_labels` performs exactly this and asserts b(ℓ_t,ℓ_s) = Ā_ts on all pairs.) The columns of L span V (full row rank). Lines with equal rows of Ā have equal labels (if the rows agree, b(ℓ_t − ℓ_s, ℓ_u) = 0 for all u and the labels span V). +- **W(2k′−1,2)** is the symplectic polar space on V: N = 4^{k′} − 1 points (nonzero vectors), p, q **collinear** iff b(p,q) = 0 (p ≠ q). A **generator** (Lagrangian) is a maximal totally isotropic subspace; it has 2^{k′} − 1 points; their number is N_L = ∏_{i=1..k′}(2^i + 1) (3, 15, 135, 2295, 75735 for k′ = 1..5), and every point lies on N_L/(2^{k′}+1) of them (Sp(2k′,2) is transitive on points, and each generator has 2^{k′}−1 points: count incident pairs). +- **Admissible pairing, window, dense estimate** (the model of #446/#595/#1590, as restated in #2228 §1): B ∈ F2^{v×v} with B_tt = 1 and B_ts = 1 (t ≠ s) only where M_ts = 0. The **dense pairing** is B = J + Ā (J all ones); its rank is the **window** h. For the coordinate decoder (P = Q = coordinates of the r_t in an orthonormal basis, T terms, supports R_j = A_j) every term has cost rank_F2 B[R_j,A_j] ≤ rank B = h, so the **dense estimate is cost ≤ T·h**, i.e. **x := hT/v** per line (the screen proxy of #620's `ade_screen.py`, also used by #895). The bound is on the *dense* pairing; sparser admissible B are not considered (OPEN, §11). +- **Saturation** σ := v / (T(2^{k′}+1)) ≤ 1 (Theorem B). **Class sizes**: for a generator L, c_L := number of lines with label in L. +- **Break-even fit x\*(h)** := .5385 − .01495 (h − 9): the dense ratio at which a unit ties the recorded P104 COPY56 construction; a fit from the author's working cards (BERTHA/fx), **RECORDED**, valid for h ≲ 17. Every row labelled "needed" below is conditional on it; the theorems are not. + +## 2. Results at a glance + +| # | statement | status | where | +|---|---|---|---| +| A | For **every** line system: B = J + Ā = Ω(u_t,u_s) with u_t = (1,ℓ_t); h = rank(J+Ā) = 2k′+1 if 1 ∉ col Ā, = 2k′ if 1 ∈ col Ā | **PROVEN**; CHECKED on D4/D5/D6/E6/E7/E8 | §3 | +| B | v ≤ T(2^{k′}+1); v ≤ T·2^{k′} on the even window; equality iff (no line orthogonal to all others, i.e. no label 0) every generator class is a full basis | **PROVEN**; ladder CHECKED | §4 | +| – | Saturation ladder: D4, D6, E7 attain 100%; E8 attains 15/17; the other 13 saved systems are far below | **CHECKED** | §8 | +| A′ | Same bound for non-Gram units M = PQᵀ (any field of characteristic 0, overcomplete included); best v/T = weighted idempotent packing number of the collinearity graph of W(2k′−1,2); saturation ⇔ "projective ovoid" | **PROVEN** | §5.1 | +| D | Descent: a saturated W(2k′−1,2) system gives a saturated W(2k′−3,2) system in dimension T − m_p; quantitative form σ_{k′} ≤ s_{k′−1} + (1−s_{k′−1})Σm_p²/(vT); saturation at any k′ ≥ 4 forces a saturated k′ = 4 system | **PROVEN**; E7→D6→D4, E8→E7 CHECKED | §5.2 | +| 2P | The two-point (Schrijver-ϑ′-type) program on W(2k′−1,2) has value exactly 2^{k′}+1 | **PROVEN**; feasible point CHECKED k′ = 2..5 | §6 | +| C | Unions of F ≥ 2 pairwise unbiased orthonormal frames have F ≤ 2k′+1 and dense x = 1; 2k′+1 generic rational frames attain it; every admissible B has window ≥ F | **PROVEN**; generic frames CHECKED | §7.1 | +| D(b) | Real m-fold blow-up multiplies T and v by m and keeps k′, σ | **PROVEN**; E7×2 CHECKED | §7.2 | +| E | Norm-2 systems with products in {0,±1}: v/T ≤ max(15, 2k′+1) | **PROVEN** (uses the ADE classification of root systems) | §7.3 | +| F | A symmetry group inducing a non-abelian simple S faithfully on the lines forces T ≥ d_proj(S); for Sp(10,2): 155 | proof **PROVEN**; smallest nontrivial degree 155 of Sp(10,2) **CHECKED** (GAP, ATLAS table, `checks/gap/`); equality with d_proj **CONDITIONAL [verify]** on Sp(10,2) having trivial Schur multiplier | §7.4 | +| G | Uniform-angle saturated systems: λ² = 2^{k′−1}m must be a perfect square (m = 1: k′ odd, cos = 2^{−(k′−1)/2}); k′ = 4 needs m ∈ {2, 8, 18, …} | **PROVEN**; arithmetic CHECKED | §7.5 | +| – | D16⁺ (roots + spinor code): at most 240 + 317 = 557 lines, against ≈ 650 needed | **CHECKED** (exact Delsarte dual certificate); "needed" conditional on x\* (RECORDED) | §8.2 | +| – | #2222: cubic-kernel unit, h = 49, x ≈ .037 (19,648 terms an upper bound; rank ≤ 19,600) | **RECORDED** from #2222, not re-run; rank remark **PROVEN** | §9 | +| – | Relaxed realisability numerics (W4, W5 fail at the break-even T) | **EXPLORATORY** | §10 | +| – | 510-vertex signing S² = 60S + 256I; level-2/three-point bounds; realisability of W4; Pauli-frame ansatz for W5 in ℝ³¹ | **OPEN** | §11 | + +## 3. Theorem A (polar law for every line system) — PROVEN + +**Theorem A.** Let Ā have rank 2k′ with labels ℓ_t ∈ V as in §1, u_t := (1, ℓ_t) ∈ F2 ⊕ V, Ω := 1 ⊕ b. Then B := J + Ā is admissible, B_ts = Ω(u_t, u_s) for **all** t, s, and + h := rank_F2 B = 2k′ + 1 if the all-ones vector 1 ∉ col(Ā), and 2k′ if 1 ∈ col(Ā). +Moreover 1 ∈ col(Ā) iff there is w ∈ V with b(ℓ_t, w) = 1 for all t. + +*Proof.* Admissible: B_ts = 1 for t ≠ s means Ā_ts = 0, i.e. M_ts = 0; B_tt = 1. Gram form: for t ≠ s, Ω(u_t,u_s) = 1 + b(ℓ_t,ℓ_s) = 1 + Ā_ts = B_ts; and Ω(u_t,u_t) = 1 + b(ℓ_t,ℓ_t) = 1 = B_tt (b alternating). Rank: B = Ā + 11ᵀ. Its kernel is {x : Āx = 1(1ᵀx)}. If 1ᵀx = 0 this says Āx = 0; if 1ᵀx = 1 it says Āx = 1. Since Ā is symmetric, col(Ā) = (ker Ā)^⊥. (i) If 1 ∉ col(Ā), then Āx = 1 has no solution, and some x ∈ ker Ā has 1ᵀx = 1 (else 1 ∈ (ker Ā)^⊥), so ker B = ker Ā ∩ 1^⊥ has codimension 1 in ker Ā and rank B = 2k′ + 1. (ii) If 1 ∈ col(Ā) then ker Ā ⊆ 1^⊥ and Āx = 1 with 1ᵀx = 1 would need xᵀĀx = 1ᵀx = 1, impossible for an alternating form; so ker B = ker Ā and rank B = 2k′. Last claim: (Āz)_t = (LᵀbLz)_t = b(ℓ_t, Lz), so 1 = Āz gives w := Lz with b(ℓ_t, w) = 1 for all t; conversely L is onto V, so every w equals Lz for some z, and then Āz = 1. ∎ + +**CHECKED (c1).** For D4, D5, D6, E6, E7, E8 (v = 12, 20, 30, 36, 63, 120; T = 4, 5, 6, 6, 7, 8): Ā equals G mod 2 off the diagonal; rank_F2 Ā = 2, 4, 4, 6, 6, 8 (k′ = 1, 2, 2, 3, 3, 4); the symplectic factorisation is verified on all v² pairs; B = Ω(u_t,u_s) on all v² pairs; 1 ∉ col(Ā) and h = 2k′+1 in all six cases. + +**Remarks.** (1) *Odd-or-zero is the special case Ā ≡ G mod 2.* For lattice shells with even non-zero products, #2228's "dense rank B" was already rank(J + Ā); only the *label form* G mod 2 was the wrong invariant. (2) *Cliques:* c pairwise non-orthogonal lines give Ā|_c = J − I with F2 rank c or c − 1, so c ≤ 2k′ + 1 ≤ h + 1, and in fact every admissible B is the identity on a clique, so *every* admissible pairing has window ≥ ω(Γ) (Γ the non-orthogonality graph; #446 clique bound, #895). (3) Integrality is not used: a rational realisation of the pattern is all that the decoder needs. + +## 4. Theorem B (Lagrangian bound) and the dense threshold + +**Theorem B — PROVEN.** For any line system with rank_F2 Ā = 2k′ (equivalently any M = PQᵀ of rank T with a symmetric alternating C dominating the support, §5.1): + (i) v ≤ T(2^{k′} + 1); (ii) if 1 ∈ col(Ā) (even window h = 2k′) then v ≤ T·2^{k′}. +Hence x = hT/v ≥ h / (2^{⌊h/2⌋} + [h odd]). + +*Proof.* Let L be a generator and C_L the lines with label in L ∪ {0}. For t ≠ s in C_L, Ā_ts = b(ℓ_t,ℓ_s) = 0 (b vanishes on L, also for equal labels), so the lines of C_L are pairwise orthogonal; non-zero pairwise orthogonal vectors are independent, so |C_L| ≤ T. Write n_p for the number of lines with label p ≠ 0 and n_0 for those with label 0 (n_0 = 0 if 1 ∈ col Ā). Sum |C_L| ≤ T over all N_L generators: each non-zero point lies on N_L/(2^{k′}+1) generators, so N_L n_0 + (N_L/(2^{k′}+1))(v − n_0) ≤ N_L T, i.e. v ≤ T(2^{k′}+1) − n_0 2^{k′} ≤ T(2^{k′}+1). For (ii) choose w with b(p,w) = 1 for all occupied labels p. No generator through such a p contains w (b(p,w) ≠ 0), so every generator meeting the occupied labels lies in 𝓛_w := {L : w ∉ L}, and |𝓛_w| = N_L − N_L/(2^{k′}+1) = N_L 2^{k′}/(2^{k′}+1). Summing |C_L| ≤ T over 𝓛_w, each occupied point being counted N_L/(2^{k′}+1) times: v N_L/(2^{k′}+1) ≤ T N_L 2^{k′}/(2^{k′}+1). ∎ + +**Equality (PROVEN).** (i) with n_0 = 0 is an equality iff every generator class has exactly T lines, i.e. is an orthogonal basis of the span. Then with û_t = r_t/|r_t|, Σ_t û_t û_tᵀ = (2^{k′}+1)·Id_T (each line lies in N_L/(2^{k′}+1) generators and each class contributes Id_T), a tight frame; the normalised Gram matrix has exactly the eigenvalues 0 and 2^{k′}+1. + +**Reduction (PROVEN, trivial).** A graph of F2 rank 2k′ is the blow-up (twins become independent sets) of an induced subgraph of the symplectic graph on W(2k′−1,2) plus isolated vertices. So v_max(T,k′) is the maximum of Σ_p m_p over label multisets (m_p) on W(2k′−1,2) whose blow-up has an exact-pattern realisation in rank ≤ T. + +**CHECKED (c1).** Over all generators: D4 (3 generators), D6 (15), E7 (135): every class has exactly T lines, hence v = T(2^{k′}+1) = 12, 30, 63 and F = Σ r rᵀ/|r|² = (v/T)·(projection onto the span) with v/T = 3, 5, 9. E8 (2295 generators): every class has 0 or 8 lines (2025 full, 270 empty), v = 120 < 8·17 = 136, σ = 15/17. D5 and E6 have all classes of size 4 and are not saturated (σ = 4/5, 2/3). + +**Thresholds (CHECKED arithmetic, conditional on the RECORDED x\*).** Needed v/T = h/x\*(h) and the fraction of the Theorem B cap it represents: + +| h | k′ | x\*(h) | needed v/T | cap 2^{⌊h/2⌋}+[h odd] | needed saturation | +|---|---|---|---|---|---| +| 9 | 4 | .5385 | 16.71 | 17 | 98.3% | +| 10 (even window) | 5 | .5235 | 19.10 | 32 | 59.7% | +| 11 | 5 | .5086 | 21.63 | 33 | 65.5% | +| 12 (even window) | 6 | .4936 | 24.31 | 64 | 38.0% | +| 13 | 6 | .4787 | 27.16 | 65 | 41.8% | + +Theorem B therefore does **not** rule out beating the recorded P104 break-even at h = 10..13 (the cap is far above the need). It only says how much of the cap must be filled. + +## 5. Non-Gram units, idempotents and descent + +### 5.1 Theorem A′ (what has to be realised) — PROVEN + +Let K be a field of characteristic 0 and M = PQᵀ with P, Q ∈ K^{v×T}, M_tt ≠ 0. Let C ∈ F2^{v×v} be symmetric alternating with C_ts = 1 whenever M_ts ≠ 0 or M_st ≠ 0 (t ≠ s); write C = LᵀbL (rank 2k′) as in §1. Then: + (i) B := J + C is admissible for M and Theorem A holds for it (window ≤ 2k′+1); every block of B has rank ≤ h, so the coordinate-type decoder costs ≤ (number of terms)·h. + (ii) Theorem B holds with T replaced by rank M: v ≤ (rank M)(2^{k′}+1), and v ≤ (rank M)2^{k′} on the even window. (The classes C_L give diagonal submatrices M[C_L,C_L] with non-zero diagonal; a diagonal matrix with non-zero diagonal has rank |C_L| ≤ rank M.) Overcompleteness (more terms than rank M) never helps this count; it can help only through per-term costs below h, which is the non-dense question of §11. + (iii) For a label p let C_p be its class and E_p := Q_{C_p}ᵀ D_p^{−1} P_{C_p}, D_p := M[C_p,C_p] (diagonal, invertible, because C_ts = 0 inside a class forces M_ts = M_st = 0). Then E_p is an idempotent of rank m_p := |C_p| (E_p² = Qᵀ D⁻¹(PQᵀ)D⁻¹P = E_p), and E_pE_q = 0 whenever b(p,q) = 0, p ≠ q (since M[C_p,C_q] = 0). Conversely, idempotents E_p = Y_pX_p (rank factorisations, X_pY_p = I) with E_pE_q = 0 for collinear p ≠ q give a unit (rows of X_p as P, columns of Y_p as Q; X_pY_q = 0 because Y_p is injective and X_q surjective). Hence + **v_max(T,k′)/T = sup Σ_p rank E_p / T over idempotent families on K^T with E_pE_q = 0 on distinct collinear pairs of W(2k′−1,2)**; +for Gram units the E_p are orthogonal projections, and this coincides with the **projective packing number** α_p of the collinearity graph (Roberson; Mančinska–Roberson), CONDITIONAL [verify] on their definition matching ours; the bound itself does not use the identification. **Saturation** (v = T(2^{k′}+1)) holds iff Σ_{p∈L} E_p = Id for every generator L, a **projective ovoid** (classical ovoids of W(2n−1,q) do not exist for n ≥ 3 [verify: Thas, found only in survey statements]; E7 is a projective one at n = 3 with all ranks 1). +(iv) Exactness of the zero pattern is irrelevant to the *upper* bound: a realisation with additional orthogonalities is still a valid unit for the label form of the pattern it satisfies, provided its lines are non-zero and pairwise distinct (coincident lines must be merged). + +*Proof of (i), (ii), (iv).* As in Theorems A and B, using only that B_ts = 1 (t ≠ s) forces C_ts = 0 and hence M_ts = M_st = 0. ∎ + +### 5.2 Descent — PROVEN + +Work with idempotent families (E_p) on K^T (families on W(2k′−1,2), labels allowed to carry rank 0), saturated: Σ_{q∈L} E_q = Id for every generator L. + +**Proposition D1 (exact).** Fix a label p with m_p > 0. For every generator L ∋ p, Σ_{q∈L∖{p}} E_q = Id − E_p, so all E_q with q ∈ p^⊥∖{p} live on U := im(Id − E_p), of dimension T − m_p (for projections: U = (class of p)^⊥). Put E_{[q]} := E_q + E_{q+p} for the classes [q] ∈ p^⊥/⟨p⟩ ≅ W(2k′−3,2) (q ≠ p, q + p ≠ p). Then (E_{[q]}) is a **saturated** W(2k′−3,2) family on U with ranks m_q + m_{q+p}. +*Proof.* b(q,q+p) = b(q,p) = 0 and q ≠ q+p, so E_qE_{q+p} = 0 and E_{[q]} is an idempotent of rank m_q + m_{q+p}. If [q] ≠ [r] are equal-or-collinear in the quotient, then b(q,r) = 0 and all four labels q, q+p, r, r+p are pairwise distinct and pairwise collinear (b(q,r+p) = b(q,r) + b(q,p) = 0), so E_{[q]}E_{[r]} = 0. A generator L′ of the quotient lifts uniquely to a generator L ∋ p; L ∖ {0,p} is the disjoint union of the pairs {q, q+p} over [q] ∈ L′, so Σ_{[q]∈L′} E_{[q]} = Σ_{q∈L∖p} E_q = Id − E_p = Id_U. ∎ + +**Corollary D2.** A saturated system at level k′ yields saturated systems at levels k′−1, …, 1. Hence **saturation at any k′ ≥ 4 forces a saturated k′ = 4 system** (for general multiplicities the levels 1, 2, 3 are realised by D4, D6, E7 and their real blow-ups, §7.2). + +**CHECKED (c4).** E7 at a label p (m_p = 1) → the 30 lines with labels in p^⊥ ∖ {p}: orthogonal to p, span T − 1 = 6, k′ = 2, twin classes = {q, q+p} of size 2, every generator class a full basis (the D6 system); D6 at p (m_p = 2) → 12 lines in T = 4, k′ = 1, multiplicity 4 (D4); E8 at p → 63 lines in T = 7, k′ = 3, saturated (E7). + +**Proposition D3 (quantitative).** Let s_{k′−1} be the supremum of σ over all families (Gram families, resp. idempotent families; the inequality holds in either class) on W(2k′−3,2) (any T, any multiplicities, labels allowed to be empty). For a family on W(2k′−1,2) with no zero labels and saturation σ: **σ ≤ s_{k′−1} + (1 − s_{k′−1}) · Σ_p m_p² / (vT).** +*Proof.* Let A_p := average of c_L over the generators L ∋ p. The derived system at p has class sizes c_L − m_p (L ∋ p, bijectively with generators of the quotient) in dimension T − m_p, so its saturation (A_p − m_p)/(T − m_p) ≤ s := s_{k′−1}, i.e. A_p ≤ sT + (1−s)m_p. Also Σ_p m_pA_p = ((2^{k′}+1)/N_L) Σ_L c_L² ≥ (2^{k′}+1)(avg c_L)² = vσT (Cauchy–Schwarz, avg c_L = v/(2^{k′}+1) = σT). Combine: vσT ≤ Σ m_p(sT + (1−s)m_p) = vsT + (1−s)Σm_p². ∎ (E8 CHECKED: A_p ≤ 8 for all occupied labels, Σ m_pA_p = 960 ≥ vσT = 14400/17.) It is an inheritance law: it turns any bound on s_4 into bounds at every k′ ≥ 5; it does not beat the thresholds of §4 by itself, since s_4 ≥ 15/17. + +## 6. Two-point relaxations are exhausted (PROVEN; feasible point CHECKED) + +Take the program P2 for a Gram unit with X_pq := tr(P_pP_q)/T (P_p the orthogonal projection onto the span of the lines of label p) and x_p := m_p/T: maximise Σ_p x_p subject to X_pp = x_p, X_pq = 0 for distinct collinear p, q, X ≥ 0 entrywise (tr P_pP_q = ‖P_pP_q‖_F² ≥ 0), and [[1, xᵀ],[x, X]] ⪰ 0. (This is the Schrijver-ϑ′-type program of the packing problem; α_p ≤ ϑ′ is Roberson's theorem [verify: confirmed only through the introduction of arXiv 2609.20678, not read in the original].) + +**Proposition 2P — PROVEN.** The value of P2 is exactly 2^{k′} + 1. +*Proof.* *Invariance:* Sp(2k′,2) acts on points with exactly two orbits on ordered pairs of distinct points (collinear / non-collinear; Witt), the constraints are convex and invariant and the objective is linear, so averaging gives an optimal point with x_p ≡ x and X = xI + yA, A the adjacency of the non-collinearity graph (the collinear orbital has X = 0). *The graph is strongly regular* with degree d = 2^{2k′−1} and λ = μ = 2^{2k′−2}, so A² = (d − μ)I + μJ, A·1 = d·1, and the spectrum of A lies in {d, ±2^{k′−1}} (CHECKED for every pair of points, k′ = 2..5). On 1^⊥ the matrix X has eigenvalues x ± y2^{k′−1}, so PSD forces y ≤ x/2^{k′−1}. The lifted matrix is PSD iff X − x²J ⪰ 0 (Schur complement); on the all-ones vector this reads x + yd − x²N ≥ 0 (N = 4^{k′}−1). Since y ≥ 0, the best choice is y = x/2^{k′−1}, giving x(1 + 2^{k′}) ≥ x²N, i.e. x ≤ 1/(2^{k′}−1), and the objective is Nx ≤ (4^{k′}−1)/(2^{k′}−1) = 2^{k′}+1. The point x = 1/(2^{k′}−1), y = x/2^{k′−1} is feasible (all constraints with equality on the extremal directions), so the value is 2^{k′}+1. ∎ + +**CHECKED (c3), k′ = 2..5:** strong regularity on all point pairs; eigenvalues of X\* = xI + yA exactly {(2^{k′}+1)x, 2x, 0}; Schur complement eigenvalue 0 on 1; 0 < y ≤ x; objective 2^{k′}+1; and the **generator inequalities** Σ_{q∈L} X\*_pq ≤ X\*_pp hold **with equality** (for p ∉ L exactly 2^{k′−1} points of L are non-collinear to p because b(p,·)|_L is a non-zero functional; for p ∈ L the sum is x), exhaustively over all generators for k′ ≤ 4 and for 60 seeded random generators at k′ = 5. + +**Consequence.** Every convex program built from the two-point data (Hilbert–Schmidt Gram data of {Id, P_p}) that contains the constraints of P2 and is satisfied by X\* (this includes adding the generator inequalities Σ_{q∈L} X_pq ≤ X_pp and Σ_{q∈L} x_q ≤ 1) has value exactly 2^{k′}+1. Improving on Theorem B by relaxation therefore needs words of length ≥ 3 (tr P_pP_qP_r, tr (P_pP_q)², i.e. level-2 moment matrices) or arithmetic input (§11). + +## 7. Constructions and bounded-class no-gos + +### 7.1 Theorem C (unions of unbiased frames) — PROVEN + +Let the system be a union of F ≥ 2 orthogonal frames (each a full orthogonal basis of the T-dimensional span), any two lines from different frames non-orthogonal. Then **F ≤ 2k′ + 1, h = F, and x = hT/v = 1.** Every admissible pairing (dense or not) has window ≥ F (clique argument: #446 clique bound, #895). Conversely **2k′+1 generic rational orthonormal frames of ℚ^T realise v = (2k′+1)T, x = 1** for every T. +*Proof.* Inside a frame Ā = 0; across frames Ā = 1. So Ā = (J_F − I_F) ⊗ J_T, whose F2 rank is rank(J_F − I_F) = F − 1 (F odd) or F (F even); thus F − 1 ≤ 2k′, and since the rank is even, F ≤ 2k′+1. B = J + Ā is 1 inside a frame and 0 across: B = I_F ⊗ J_T, rank F = h, and x = F T/(F T) = 1. One line per frame is a clique, and any admissible B is the identity on a clique (B_ts = 1 only where M_ts = 0), so its rank is ≥ F. For the converse take Cayley transforms of random rational skew matrices; the cross products are non-zero generically. ∎ +**CHECKED (c1):** generic frames for (T, F) = (3,3), (4,3), (4,5), (5,5), (3,7): all cross products non-zero, k′ = 1, 1, 2, 2, 3, window F, x = 1, saturation 1, 1, 1, 1, 7/9. So **Theorem C explains why Kerdock/mutually unbiased-type constructions cannot reach x < 1**; e.g. the 144-line, nine-frame subset of BW16 (k′ = 4, F = 9 = 2k′+1) is RECORDED from the author's cards as consistent with it. (The full BW16 shell, 2160 lines, is not a union of unbiased frames and has a 72-clique by #895.) + +### 7.2 Theorem D(b) (real blow-up) — PROVEN + +If a line system has vectors a_ℓ, replace each a_ℓ by the m vectors a_ℓ ⊗ h_{ℓ,i} (i = 1..m), where {h_{ℓ,i}}_i is an orthonormal basis of ℝ^m chosen for each ℓ so that all cross products h_{ℓ,i}·h_{ℓ′,j} (ℓ ≠ ℓ′) are non-zero (generic). Then products are (a_ℓ·a_ℓ′)(h_{ℓ,i}·h_{ℓ′,j}): twins are orthogonal, and Ā_new = Ā ⊗ J_m. So T → mT, v → mv, rank_F2 Ā (hence k′, h) and σ are unchanged. **CHECKED (c1):** E7 with m = 2: 126 lines in ℚ^14, pattern = Ā(E7) ⊗ J_2 on all 7875 pairs, k′ = 3, h = 7, v/T = 9. (The D4/D6 phenomenon that multiplicity can do *better* than tensoring, e.g. D4 with m = 4 in T = 4 against 8 for the tensor product, is seen only at k′ ≤ 2 in every case known; §10.) + +### 7.3 Theorem E (norm-2 systems) — PROVEN (uses the ADE classification) + +**Theorem E.** If every line has norm 2 and all products lie in {0, ±1}, then **v/T ≤ max(15, 2k′+1)**, where 2k′ = rank_F2 Ā (here Ā = G mod 2 off the diagonal). In particular such systems cannot reach the needed v/T of §4 for k′ ≥ 5 (needed 19.1 / 21.6 at h = 10 / 11 against max(15, 11) = 15 at k′ = 5). For full root systems this is #620's ADE screen (only E8 viable); Theorem E extends it to every norm-2 subsystem with products in {0, ±1}. + +*Proof.* Let Λ be the lattice generated by the lines; it is even (norms of integer combinations are even because the generators have norm 2 and integral products), so the norm-2 vectors R of Λ form a simply-laced root system (closed under reflections) of rank T = dim span (the lines are in R). Split R into irreducible components R_1, …, R_m, orthogonal and with ranks T_i, ΣT_i = T. Each line lies in one component (every root of R lies in exactly one irreducible component). Let S_i be the lines in R_i. Then G is block-diagonal, v = Σ|S_i|, rank_F2 Ā = Σ rank_F2 Ā_i (so k′ ≥ k′_i), and since dim span S = T = Σ rank R_i, each S_i spans span R_i, i.e. T(S_i) = T_i. As v/T is a weighted average of the |S_i|/T_i, it suffices to prove |S_i|/T_i ≤ max(15, 2k′_i+1) for each irreducible component. By classification R_i is of type A, D, E6, E7 or E8. + • E6, E7, E8: |S_i| ≤ 36, 63, 120 lines and T_i = 6, 7, 8: ratios ≤ 6, 9, 15. + • **A_n.** Roots e_a − e_b on c = n + 1 coordinates; S_i is the edge set of a graph H on c vertices (distinct lines = distinct edges), and T_i = n forces H connected. With N the c × |S_i| unsigned incidence matrix, G mod 2 = NᵀN off the diagonal and also on it (the diagonal 2 ≡ 0), because |e ∩ f| = number of common vertices. Over F2, col(N) is the even-weight space (dimension c − 1 for connected H) and the form xᵀy on it has radical ⟨1⟩ ∩ (even weight) = ⟨1⟩ if c is even and 0 if c is odd; hence rank_F2 Ā_i = c − 1 − [c even] ≥ c − 2, i.e. c ≤ 2k′_i + 2. Then |S_i|/T_i ≤ (c(c−1)/2)/(c−1) = c/2 ≤ k′_i + 1 ≤ 2k′_i + 1. + • **D_n.** Roots ±e_a ± e_b on n coordinates. Split S_i by the connected components of its support graph (coordinates joined by lines); different support components are orthogonal, so we may treat one with c vertices. Edges are signed, parallel edges at most 2 (e_a − e_b and e_a + e_b, which are *orthogonal*). Again G mod 2 = NᵀN (sharing exactly one coordinate gives ±1, sharing both gives 0 or ±2 ≡ 0), the support is connected, so rank_F2 Ā = c − 1 − [c even] and c ≤ 2k′+2 as above. The rank of the component is c if the signed graph is unbalanced and c − 1 if balanced. If balanced, it is switching-equivalent to an A-type edge set (no parallel pairs), ratio ≤ c/2. If unbalanced, |S| ≤ c(c−1) and T = c, ratio ≤ c − 1 ≤ 2k′+1. +Combining, v/T ≤ max over components of max(15, 2k′_i+1) ≤ max(15, 2k′+1). ∎ +**CHECKED (c2):** the identity Ā = NᵀN mod 2, the rank formula c − 1 − [c even] for connected supports, T ∈ {c−1, c}, and the final inequality on all subsets of the A_c lines (c ≤ 6) and D_c lines (c ≤ 4) and on 1500 seeded random subsets for A_7 and D_5..D_7; E6/E7/E8 ratios 6, 9, 15. + +### 7.4 Theorem F (symmetric realisations) — proof PROVEN; degrees CHECKED in GAP; Schur-multiplier input CONDITIONAL [verify] + +**Theorem F.** Let H ≤ O(T) be a finite group permuting the lines of a system, let N ⊲ H be the subgroup acting trivially on the set of lines (every element of N acts by ±1 on each line), and suppose H/N ≅ S for a non-abelian finite simple group S acting non-trivially on the lines. Then **T ≥ d_proj(S)**, where d_proj(S) is the smallest degree of a non-scalar complex projective representation of S. +*Proof.* H is finite and faithful on ℂ^T. H is not solvable (S is a non-abelian simple quotient), so some irreducible constituent W of ℂ^T has non-solvable image of H (else H embeds in a product of solvable groups). N is abelian: each n ∈ N acts on every spanning vector r_t by a sign, so n ↦ (signs) embeds N in (ℤ/2)^v and N is diagonalisable on ℂ^T; so W|_N = ⊕_χ W_χ over characters χ of N, and by Clifford's theorem H permutes the W_χ transitively. *If there is one χ*, N acts by scalars on W, so W is a projective representation of H/N = S; it is non-scalar (otherwise the image of H would be abelian), hence dim W ≥ d_proj(S). *If there are t ≥ 2 characters*, S acts transitively on a set of size t, non-trivially, hence faithfully (S simple), with point stabiliser a proper subgroup of index t, so t is the index of a proper subgroup. Since a transitive permutation action of degree t yields a non-trivial linear representation of degree ≤ t − 1, dim W ≥ t > d_proj(S); hence T ≥ d_proj(S). ∎ +**Application (multiplicity one).** A system with one line per label, a group inducing Sp(10,2) on its lines (so on all 1023 labels), has T ≥ d_proj(Sp(10,2)) = 155. The formula d_proj(Sp(2n,2)) = (2^n − 1)(2^{n−1} − 1)/3 (n = 3, 4, 5: 7, 35, 155; ATLAS degrees; Sp(2n,2) has trivial Schur multiplier for n ≥ 4) is the cited value [verify: the Tiep–Zalesskii 1996 formula (q^n−1)(q^n−q)/(2(q+1)) at q = 2; I could not retrieve the original statement, only that Tiep–Zalesskii determine these minimal degrees]; only the arithmetic is checked (c2). (The smallest index of a proper subgroup, m(Sp(2n,2)) = 2^{n−1}(2^n − 1), is a permutation-degree result of Cooperstein 1978 and is not needed.) **Checked (c8, GAP 4.13 + CTblLib ATLAS tables, `checks/gap/sp2n2_degrees.g`, recorded output beside it):** the smallest nontrivial ordinary character degrees of Sp(6,2), Sp(8,2), Sp(10,2) are 7, 35, 155. For n ≥ 4 these are the projective degrees provided the Schur multiplier is trivial, which we cite (Steinberg; ATLAS) and did not compute: GAP's multiplier computation for Sp(10,2) did not finish in 20 min. So T ≥ 155 is CONDITIONAL [verify] on that citation only. With it, v/T ≤ 1023/155 = 6.6 at k′ = 5, far below the needed 19.1–21.6; the E7 case is consistent with the formula (Sp(6,2), T = 7). *Not claimed:* the original card also listed lower bounds 120 / 119 for Ω^±(10,2); Theorem 1.3 of Nguyen (arXiv:0910.4885, read through its ar5iv rendering) lists the smallest non-trivial degree of Ω^−_10(2) as (2^5+1)(2^4−2)/3 = 154 and excludes Ω^+_10(2), so **those two numbers are dropped**. *Not covered:* multiplicity m ≥ 2, where N may be non-abelian (Clifford groups act irreducibly with a solvable N) — the proof above needs N abelian. + +### 7.5 Theorem G (uniform-angle saturated systems) — PROVEN + +Let a **saturated** system on W(2k′−1,2) have uniform multiplicity m (all N labels with exactly m lines), unit vectors, and constant |cos| = α on all non-orthogonal pairs. Write the normalised Gram matrix Ĝ = I + αS with S an integer-signed adjacency matrix of the blown-up non-collinearity graph (twins orthogonal), and λ := 1/α. Then λ² = 2^{k′−1} m, and λ is rational. So **m = 1 requires k′ odd (α = 2^{−(k′−1)/2}; k′ = 1 gives the degenerate α = 1); for k′ = 4, 2m must be a perfect square (m = 2, 8, 18, 32, 50, …)**; and T = m(2^{k′} − 1). +*Proof.* Saturation gives Ĝ² = (2^{k′}+1)Ĝ (§4, equality case), i.e. S² = (2^{k′}−1)λS + 2^{k′}λ²I. The diagonal of S² is the degree of the graph, 2^{2k′−1}m, so 2^{k′}λ² = 2^{2k′−1}m. S has eigenvalue 2^{k′}λ (multiplicity T) and −λ (multiplicity Nm − T). If λ were irrational (λ² = 2^{k′−1}m is an integer), the integer matrix S would have the Galois conjugate +λ of its eigenvalue −λ as an eigenvalue, but its only eigenvalues are −λ and 2^{k′}λ. So λ is an integer. The multiplicities follow from tr S = 0 and tr Ĝ = Nm: T(2^{k′}+1) = Nm. ∎ +**CHECKED (c6):** the table of (k′, m, λ) with λ² = 2^{k′−1}m a square for k′ ≤ 9, m ≤ 50 (m = 1 exactly for odd k′; k′ = 4 needs m ∈ {2, 8, 18, 32, 50}); D4 (k′ = 1, m = 4), D6 (2, 2), E7 (3, 1) are uniform with |cos| = 1/2 = 1/λ; the common-neighbour counts of the blown-up symplectic graphs: for (k′, m) = (4, 2) [510 vertices, λ = 4] S² = 60S + 256I, degree 256, adjacent pairs have 128 common neighbours (a signing would need signed sum ±60, i.e. 94 agree / 34 disagree), non-adjacent distinct labels 128 (signed sum 0), twins 256 (signed sum 0); for (5, 1) [1023 vertices, λ = 4] S² = 124S + 512I, adjacent pairs 256 common neighbours (a signing would need 190 / 66). *Erratum to the source card:* it wrote the uniform angle as 2^{−(k′+1)/2}; the correct value is 2^{−(k′−1)/2} (E7: 1/2). + +## 8. The exact ladder and the construction table + +### 8.1 CHECKED +`checks/c5_l1cons.py` re-verifies, in exact integer arithmetic, the 14 vector files `data/best_kprime_*.json` (lines distinct up to sign, non-zero norms, every pairwise product 0 or odd, exact Q-rank T, rank_F2 Ā, window, saturation as a fraction). Window h is the actual rank_F2(J+Ā). + +| k′ | system | T | v | v/T | σ | h | dense x = hT/v | x\*(h) (RECORDED fit) | +|---|---|---|---|---|---|---|---|---| +| 1 | D4 roots (c1) | 4 | 12 | 3 | 1 | 3 | 1.000 | – | +| 2 | D6 roots (c1) | 6 | 30 | 5 | 1 | 5 | 1.000 | – | +| 3 | E7 roots (c1) | 7 | 63 | 9 | 1 | 7 | .778 | – | +| 3 | Z8 (56 lines in ℤ^8) | 8 | 56 | 7 | 7/9 | 7 | 1.000 | .568 | +| 4 | E8 roots (= A_ham, 120) | 8 | 120 | 15 | 15/17 = .882 | 9 | .600 | .5385 | +| 5 | E8 + A3, norms 6 | 11 | 140 | 12.7 | .386 | 11 | .864 | .509 | +| 6 | E8 + D5, norms 4 | 13 | 166 | 12.8 | .196 | 13 | 1.018 | .479 | +| 7 | Z16a (240 lines) | 16 | 240 | 15 | .116 | 15 | 1.000 | .449 | +| 8 | D16⁺: roots + 256 spinor lines | 16 | 496 | 31 | .121 | 17 | .548 | .419 | +| 9 | Z20m | 20 | 55 | 2.75 | .005 | 19 | 6.909 | .389 | +| 12 | D24⁺ roots + 176 spinors | 24 | 728 | 30.3 | .007 | 25 | .824 | .299 | +| 15 | Z32RM1 | 32 | 992 | 31 | .0009 | 31 | 1.000 | .210 | +| 16 | four orthogonal copies of E8 (480 lines) | 32 | 480 | 15 | 2·10⁻⁴ | 33 | 2.200 | .180 | +| 20 | five orthogonal copies of E8 | 40 | 600 | 15 | 1.4·10⁻⁵ | 41 | 2.733 | .060 | +| 25 | E8⊗E8 frame-partition (960 lines) | 64 | 960 | 15 | 4.5·10⁻⁷ | **50** (even window) | 3.333 | < 0 | +| 32 | E8⊗E8 pool, local-search subset (666 lines) | 64 | 666 | 10.4 | 2.4·10⁻⁹ | 65 | 6.246 | < 0 | + +(k′ = 0, E8 ⊗ Hadamard-8: 64 lines in T = 64, σ = 1/2, is the trivial control and is in `data/`.) **No saved system has dense x below the recorded break-even fit** (c5), and the best saturation per k′ at window h ≤ 13 is 7/9 (k′ = 3), 15/17 (k′ = 4), .386 (k′ = 5), .196 (k′ = 6). These constructions are heuristic (the pools were searched by iterated local search for a maximum independent set); they are *lower bounds on what is attainable*, not optima. **RECORDED, not in the package:** E8 ⊕ A2 with 139 lines in T = 10 (k′ = 5, σ ≈ .42) and E8 ⊕ A3 (BERTHA card). + +### 8.2 The D16⁺ family and its exact Delsarte cap — CHECKED +In D16⁺ (doubled coordinates, norm 16, divisor 4) every root ±2e_i ± 2e_j has product ±4 or 0 (→ ±1, 0) with every spinor line, so v = 240 + (number of spinor lines), and two spinor lines (sign vectors s, s′ at Hamming distance d, products (16 − 2d)/4) are compatible iff d ∈ S = {2,6,10,14} ∪ {8, 16} (d = 16: the antipodal sign vector is the same line). So the spinor set is an antipodal binary code of length 16 with distance set S. +**Delsarte bound (proof).** For a binary code C with distance distribution (A_i) and Krawtchouk polynomials K_k, Σ_i A_iK_k(i) = |C|^{−1} Σ_{wt u = k} |Σ_{c∈C} (−1)^{u·c}|² ≥ 0 (K_k(d(x,y)) = Σ_{wt u=k}(−1)^{u·(x+y)}, CHECKED for n = 6 and the inequalities on RM(1,4)). If y_k ≥ 0 satisfy Σ_k y_k(−K_k(i)) ≥ 1 for all i ∈ S then |C| ≤ 1 + Σ_k y_k K_k(0). **c6 solves a 6 × 6 system exactly in `Fraction` for the dual support {2,3,5,6,7,8} (found by a floating-point LP), verifies y ≥ 0 and feasibility, and obtains |C| ≤ 18432/29 = 635.59, so |C| ≤ 634 (even), at most 317 spinor lines, and roots + spinors ≤ 240 + 317 = 557.** The needed v is h T/x\*(h) = 272/.4189 = 649.3 (conditional on the RECORDED fit), so **no D16⁺ roots-plus-spinors system can reach it**; the saved set has 496 (256 spinors, found by a coset-clique search). This is a statement about the D16⁺ construction only. + +## 9. Scope: bounded window, and #2222 + +All "cannot reach the needed v/T" statements above assume (a) the dense pairing, (b) Gram or rank-bounded units, (c) **window h ≤ 13** (k′ ≤ 6), (d) the recorded fit x\*(h). They concern only the paper-side count. + +**#2222 (credit).** RECORDED from its PR text (not re-run here): the P48p minimum vectors give 26,208,000 lines of norm 6 in a 48-dimensional even unimodular lattice; with f(t) = (t³ − 4t)/192 the kernel M = f(x·y) vanishes exactly when x·y is even, so the labels are x mod 2 in L/2L (dimension 48, the form x·y mod 2 alternating and non-degenerate), k′ ≤ 24, **h ≤ 49**; M has rank at most binom(50,3) + 48 = 19,648, which is the term count, so 19,648 is a valid (conservative) upper bound on the cost. **Correction (maintainer review):** this does not make the factorization complete. On the norm-six shell Σ_j x_j² = 6, so each linear feature is a combination of cubic ones, x_i = (1/6) Σ_j x_i x_j², and the cubic-plus-linear feature matrix has rank at most binom(50,3) = 19,600 < 19,648; the displayed factorization is therefore overcomplete (PROVEN, one line). An earlier version of this README wrongly called it complete. M is indefinite. Its bill ≤ 962,752 gives dense x = 962,752/26,208,000 ≈ .0367 at h = 49, far below any x\*(h). Theorem B (A′ form) holds for it (v ≤ 19,648·(2^{24}+1)) with saturation ≈ 8·10⁻⁵ and gives no obstruction. **So no unconditional paper-side no-go can exist; any one must be scoped to bounded h (here ≤ 13) or include compensation** (not modelled here). A general kernel remark (PROVEN, trivial): for a lattice shell in dimension n and a polynomial kernel of degree d, rank M ≤ the dimension of degree-≤ d polynomials on the shell, about n^d/d!, so paper wins need v ≫ n^d, hence large extremal lattices with h ≈ n + 1 (lineage: #598); at h ≤ 13 the shells are far too small. + +## 10. EXPLORATORY evidence (floating point; scripts not shipped; not proof) + +A block-coordinate/ALS solver was run on the *relaxed* problem (orthogonality enforced where Ā = 0; non-orthogonal pairs allowed to vanish — by Theorem A′(iv) a solution of the relaxed problem would still be a valid unit once lines are checked non-zero and distinct). Controls: D6 (W2, m = 2, T = 6) and E7 (W3, m = 1, T = 7) realise; W2 m = 1 needs T = 4 (the doily at T = 3 fails, also RECORDED in #2125). Results, symmetric and non-symmetric solvers agreeing: W3 m = 2 realises at T = 14 and fails at 13 (= tensoring); **W4 m = 1 (255 lines) fails (non-zero residual) for T ≤ 26 and gives only degenerate attractors (a vanishing non-orthogonal product) at T ≥ 28; W4 m = 2 (510 lines) fails at T = 30, 36, 40, 44; W5 m = 1 (1023 lines) fails at T = 31, 36, 47; W5 m = 2 fails at T = 62.** Reading: the multiplicity mechanism (D4: m = 4 in T = 4; D6: m = 2 in T = 6) is seen only at k′ ≤ 2; nothing suggests that dropping symmetry or positivity helps, so the obstruction looks algebraic (rank), not convex. These are local searches and prove nothing. + +## 11. Open next questions + +1. **The 510-vertex signing (OPEN, cheapest decisive object).** Does the non-collinearity graph of W(7,2) with every vertex doubled (twins orthogonal) admit a ±1 signing S with S² = 60S + 256I (equivalently 510 unit vectors in ℝ^30 with |cos| ∈ {0, 1/4}, saturated)? By §5.2 and §7.5 its non-existence rules out uniform-angle saturation with m = 1 at k′ = 5 (higher odd k′ descend to (k′ = 4, m = 2^{k′−4}, λ = 2^{(k′−1)/2}), a different object; descent of the 1023-vertex candidate in ℝ^31 with cos 1/4 passes through exactly this object at T = 30). c6 gives the required common-neighbour sign counts (94/34 on adjacent pairs, balanced elsewhere) for a pseudo-Boolean encoding. Not attempted here. +2. **Level-2 / three-point bounds (OPEN).** §6 shows that every two-point relaxation equals 2^{k′}+1. A bound s_4 < 1 for arbitrary multiplicities (a symmetry-reduced three-point SDP on W(7,2), reduced by Sp(8,2); the level-2 moment matrix has 1 + N + N·2^{2k′−1} rows before reduction) would bound all k′ ≥ 5 through Proposition D3. A bound at k′ = 5 would have to reach ≤ .65 to matter at h = 11. +3. **Realisability of W4 (OPEN).** Any *saturated* W4 system (any multiplicities) would be the first rung above E7 (E8 is 15/17) and would reopen k′ = 5 (needed 19.1 of cap 32 at h = 10); conversely, none exists if the signing of item 1 and its cousins fail. Also: asymmetric label subsets (not full W5) at h = 10..12 with ϑ-tight structure, and whether the Pauli-frame ansatz for W5 in ℝ³¹ (coordinate lines 4e_y and two ±1 bases on each hyperplane {a·y = 1}; an exact CNF with 29.9k variables was run by two SAT solvers for 22 and 30 minutes without a decision; not shipped) is satisfiable. +4. **Non-dense pairings.** Everything here bounds the *dense* B = J + Ā. A fitting B of smaller rank (lower bounded by the clique number ω(Γ), Remark 2 of §3), and per-term costs below h for overcomplete decoders, are the genuine extra levers and are untouched. +5. **Multiplicity m ≥ 2 symmetric systems** (Theorem F's N may be non-abelian) and the exact value d_proj(S) for Sp(2n,2), Ω^±(2n,2) beyond those cited. +6. **Compensation.** Whether a paper win at h ≥ 14 survives helper accounting (cf. #2222) is outside this package. + +## 12. Files + +`verify.sh`, `MANIFEST.sha256`, `SOURCES.json`, `scripts/provenance.py`; `lib/` (`exact.py`, `report.py`, `units.py` carried over from #2228's package; new: `lines.py` — Ā, symplectic labels, window; `sympl.py` — W(2k−1,2) and its generators; `rat.py` — exact rational matrices); `checks/c1_ladder.py` … `c6_delsarte_uniform.py`; `data/best_kprime_*.json` (14 files of exact integer vectors; the only data read). diff --git a/research/line-system-saturation/SOURCES.json b/research/line-system-saturation/SOURCES.json new file mode 100644 index 0000000000..8e42203f46 --- /dev/null +++ b/research/line-system-saturation/SOURCES.json @@ -0,0 +1,111 @@ +{ + "description": "Pins for research/line-system-saturation. files = library and check code (own work; lib/exact.py, lib/report.py, lib/units.py are carried over unchanged from research/ideal-decoder-floors of #2228); evidence = the 14 vector files best_kprime_*.json of the L1_CONS construction runs, stored here so that nothing outside the package is read. scripts/provenance.py checks the pins (sha256 as stored) and MANIFEST.sha256.", + "files": [ + { + "file": "checks/c1_ladder.py", + "sha256": "8f912d437c4fcbb717eed954ce3236a4124a17baacce3bc05dcd315f943d42d1" + }, + { + "file": "checks/c2_normtwo.py", + "sha256": "2989f7f1fc274e6ee743cf420276caaed109cff483aa341c64cdd401e6a160c0" + }, + { + "file": "checks/c3_twopoint.py", + "sha256": "474e0f199361994241a32cd85008123c3b775fc092e27dc15852354cad6a1477" + }, + { + "file": "checks/c4_descent.py", + "sha256": "a8a4d185731bf3fed5c440bc3ba375dd18e0be8e546adedcd1327ecba622360e" + }, + { + "file": "checks/c5_l1cons.py", + "sha256": "064d824586c3c526aa5e2c45ed8bf39c2bae2d427aec62ae09e798e7324e319b" + }, + { + "file": "checks/c6_delsarte_uniform.py", + "sha256": "cab88f8e80afba31572201105db0c1661f8e766bbcec3c169bf493408fd4b509" + }, + { + "file": "lib/exact.py", + "sha256": "71b65bdd2ec13fb6449ce958871def1f87904a48fd06af895014c52fc4ecac52" + }, + { + "file": "lib/lines.py", + "sha256": "581a51a4f65fc697fd4a8acb6489c431bc348f37abc6f17640b775f24cd48dc2" + }, + { + "file": "lib/rat.py", + "sha256": "eb3e17c889835b5a4416b68f5a68ae3f86a3359b85e56f3e0a09ff79f851586e" + }, + { + "file": "lib/report.py", + "sha256": "fc4bb6836cb34d310ca7b3275ef15ef2d5d03e511063ddb06d0eb60e7072a640" + }, + { + "file": "lib/sympl.py", + "sha256": "69b273167f45bd625bec66ac1d1f21dfdf9817be5c771edf7129d72d961002ea" + }, + { + "file": "lib/units.py", + "sha256": "8d891d62f7414f3c058a279c1080c7670a2aba3a77bff85c6dc41f2abe3badce" + } + ], + "evidence": [ + { + "file": "data/best_kprime_0.json", + "sha256": "33943d2cd2ad261f4430b9abcdf8e160a41ab8374cba31c10f52b39c24c1520a" + }, + { + "file": "data/best_kprime_3.json", + "sha256": "e472e13e1ca168d402cd16072634541b84c6f0b4d2533eb2a6cca8f254888624" + }, + { + "file": "data/best_kprime_4.json", + "sha256": "7ec8a17e03554146508e044cad39c776410790b0ad1956a888290c55f95b9d4e" + }, + { + "file": "data/best_kprime_5.json", + "sha256": "41555d6e06460491252071354a895be93907b1190ece8cf6e61f338e9e7b9723" + }, + { + "file": "data/best_kprime_6.json", + "sha256": "a9d416a0956354b30c49484ce93ee8684db7e39e51e878328864f573f1cc0f54" + }, + { + "file": "data/best_kprime_7.json", + "sha256": "ac09cbfdc2430bfee1b76e4f5716fd836ab9967e9eb08adfa304fd36795f3730" + }, + { + "file": "data/best_kprime_8.json", + "sha256": "d0bec34077a219f41d70cbba9def7e955dafa1906cdad2de731a0b5ea9702f8f" + }, + { + "file": "data/best_kprime_9.json", + "sha256": "bae936975ac10603dee3636eb2885fbddb14f883d7084ace1b731472dd70e9a3" + }, + { + "file": "data/best_kprime_12.json", + "sha256": "84d1eaea473894bcf7e4960c4f19f3f595b792a45be6ef3d5b89fcfd582b6f90" + }, + { + "file": "data/best_kprime_15.json", + "sha256": "a5b3b90b71a65b5fdd542af853ae817c7f98a2485dd0090943a73407d741fc5f" + }, + { + "file": "data/best_kprime_16.json", + "sha256": "4a6c60181eb14097c3e3e2080a3b6ffe26381e85db57ca60035d89ccfe51013b" + }, + { + "file": "data/best_kprime_20.json", + "sha256": "9ee3738f5a83a1ed23dcdf8924cc307b75e5bf528e3f0f39535122e3dfa7b257" + }, + { + "file": "data/best_kprime_25.json", + "sha256": "6da2b6a6e70184c71143d402b2afd401f4f571b5007ea489265f4c24879b1281" + }, + { + "file": "data/best_kprime_32.json", + "sha256": "0c3ed4bcd102d61c2882445bbacf95b67c5c5c21cc81f2cd33dca3e8ccea24dd" + } + ] +} \ No newline at end of file diff --git a/research/line-system-saturation/checks/c1_ladder.py b/research/line-system-saturation/checks/c1_ladder.py new file mode 100644 index 0000000000..3eb02b27b6 --- /dev/null +++ b/research/line-system-saturation/checks/c1_ladder.py @@ -0,0 +1,93 @@ +"""CHECKED (exact): Theorem A (labels from the F2 rank of the non-orthogonality adjacency A-bar; B = J + A-bar is the Omega-Gram of u_t = (1, l_t)), +Theorem B (Lagrangian classes <= T; the bound v <= T(2^k'+1) and its equality case) and the saturation ladder D4 / D6 / E7 (100% or tight) and E8 (88%), +plus Theorem C (unions of generic orthonormal frames) and Theorem D(b) (real blow-up).""" +import os, sys, random +from fractions import Fraction as Fr +sys.path.insert(0, os.path.join(os.path.dirname(os.path.abspath(__file__)), '..', 'lib')) +from report import check, finish +from exact import rank_f2, rank_q +from units import UNITS, gram +from lines import abar_rows, alt_labels, window, parity +from sympl import lagrangians, bf +from rat import cayley + +EXP = {'D4': (12, 4, 1), 'D5': (20, 5, 2), 'D6': (30, 6, 2), 'E6': (36, 6, 3), 'E7': (63, 7, 3), 'E8': (120, 8, 4)} # v, T, k' +LAG = {} +for name, (L, T) in UNITS.items(): + G = gram(L) + v = len(L) + rows = abar_rows(G) + ev, eT, ek = EXP[name] + oz = all(((rows[t] >> s) & 1) == (G[t][s] % 2) for t in range(v) for s in range(v) if t != s) + labels, k = alt_labels(rows) # self-verifying factorisation A-bar = L^T b L + r2, h, inc = window(rows) + check('A.labels-' + name, v == ev and T == eT and k == ek and r2 == 2 * k and oz, + '%s: v=%d T=%d, A-bar = G mod 2 off the diagonal: %s, rank_F2 A-bar = %d = 2k\' (k\'=%d), factorisation b(l_t,l_s) = A-bar_ts verified on all %d pairs' % (name, v, T, oz, r2, k, v * v)) + # Theorem A: B = J + A-bar = Omega(u_t,u_s), u_t = (1, l_t), Omega = 1 + b, on ALL pairs including t = s + okB = all((1 ^ bf(labels[t], labels[s], k)) == (1 ^ ((rows[t] >> s) & 1)) for t in range(v) for s in range(v)) + want_h = 2 * k if inc else 2 * k + 1 + check('A.polar-' + name, okB and h == want_h, + '%s: B = J + A-bar equals Omega(u_t,u_s) on all pairs; 1 in col(A-bar): %s; h = rank_F2(J + A-bar) = %d = %s' % (name, inc, h, '2k\'' if inc else '2k\'+1')) + # Theorem B: class sizes over all Lagrangians + Ls = LAG.setdefault(k, lagrangians(k)) + m = {} + for l in labels: + m[l] = m.get(l, 0) + 1 + assert 0 not in m + sizes = [sum(m.get(p, 0) for p in Lg) for Lg in Ls] + NL = len(Ls) + dist = {} + for s in sizes: + dist[s] = dist.get(s, 0) + 1 + bound = T * (2 ** k + 1) + tight = all(s == T for s in sizes) + check('B.classes-' + name, max(sizes) <= T and sum(sizes) * (2 ** k + 1) == v * NL and v <= bound and tight == (v == bound), + '%s: %d Lagrangians, class size <= T=%d (distribution %s), sum of class sizes = v*N_L/(2^k\'+1), v=%d <= T(2^k\'+1)=%d, equality iff every class is a full basis: %s; saturation %s' % (name, NL, T, dict(sorted(dist.items())), v, bound, tight, Fr(v, bound))) + # tight frame: sum_t r r^T / (r.r) = (v/T) I (exact) + n = len(L[0]) + S = [[Fr(0)] * n for _ in range(n)] + for r in L: + nn = sum(x * x for x in r) + for i in range(n): + for j in range(n): + S[i][j] += Fr(r[i] * r[j], nn) + S2 = [[sum(S[i][a] * S[a][j] for a in range(n)) for j in range(n)] for i in range(n)] + c = Fr(v, T) + proj = all(S2[i][j] == c * S[i][j] for i in range(n) for j in range(n)) and sum(S[i][i] for i in range(n)) == v + check('B.frame-' + name, proj and ((c == 2 ** k + 1) == tight), + '%s: F := sum_t r_t r_t^T/|r_t|^2 satisfies F^2 = (v/T) F, tr F = v, so F = (v/T) x (orthogonal projection onto span, rank T) with v/T = %s; equals 2^k\'+1 iff saturated' % (name, c)) + +# Theorem C: F generic rational orthonormal frames (Cayley transforms) of Q^T +rng = random.Random(20261014) +for (T, F) in [(3, 3), (4, 3), (4, 5), (5, 5), (3, 7)]: + vecs = [] + for f in range(F): + O = cayley(T, rng) + vecs += [tuple(O[i][j] for i in range(T)) for j in range(T)] # columns of O + v = len(vecs) + G = [[sum(a * b for a, b in zip(x, y)) for y in vecs] for x in vecs] + cross = all((G[a][b] != 0) == (a // T != b // T) for a in range(v) for b in range(v) if a != b) + rows = abar_rows(G) + r2, h, inc = window(rows) + k = r2 // 2 + ok = cross and r2 == F - 1 and h == F and rank_q([list(x) for x in vecs]) == T and F == 2 * k + 1 and v == F * T + check('C.frames-T%d-F%d' % (T, F), ok, + '%d generic ONBs of Q^%d: all cross products nonzero, within-frame orthogonal, rank_F2 A-bar = %d (k\'=%d, F = 2k\'+1), h = %d = F, v = %d, v/T = %d, dense x = hT/v = %s, saturation %s' % (F, T, r2, k, h, v, F, Fr(h * T, v), Fr(v, T * (2 ** k + 1)))) + +# Theorem D(b): m = 2 real blow-up of E7: a_l (x) (cos th_l, sin th_l) and a_l (x) (-sin th_l, cos th_l), th_l with rational tan(th/2) = 1/(l+2) +L7, T7 = UNITS['E7'] +G7 = gram(L7) +vecs = [] +for l, a in enumerate(L7): + t = Fr(1, l + 2) + c, s = (1 - t * t) / (1 + t * t), 2 * t / (1 + t * t) + for (p, q) in ((c, s), (-s, c)): + vecs.append(tuple(Fr(x) * p for x in a) + tuple(Fr(x) * q for x in a)) +G = [[sum(x * y for x, y in zip(a, b)) for b in vecs] for a in vecs] +ok = all((G[2 * l + i][2 * l2 + j] != 0) == (l != l2 and G7[l][l2] != 0) for l in range(63) for l2 in range(63) for i in range(2) for j in range(2) if (l, i) != (l2, j)) +rows = abar_rows(G) +r2, h, inc = window(rows) +Tq = rank_q([list(x) for x in vecs]) +check('D.blowup-E7x2', ok and Tq == 14 and r2 == 6 and h == 7 and len(vecs) == 126, + 'E7 blown up m=2 in Q^14: non-orthogonality pattern = A-bar(E7) (x) J_2 on all %d pairs, v=126, T=%d, rank_F2 A-bar = %d (k\'=3 unchanged), h = %d, v/T = 9 = 2^3+1 (saturated, as E7)' % (126 * 125 // 2, Tq, r2, h)) +finish() diff --git a/research/line-system-saturation/checks/c2_normtwo.py b/research/line-system-saturation/checks/c2_normtwo.py new file mode 100644 index 0000000000..34241bc676 --- /dev/null +++ b/research/line-system-saturation/checks/c2_normtwo.py @@ -0,0 +1,95 @@ +"""CHECKED (exact): the ingredients of Theorem E (norm-2 line systems with products in {0,+-1}: v/T <= max(15, 2k'+1)) on the A_c and D_c lines +(identity A-bar = N^T N mod 2, rank formula c-1-[c even], T, ratio bound), and the arithmetic of Theorem F's constants.""" +import os, sys, random, itertools +from fractions import Fraction as Fr +sys.path.insert(0, os.path.join(os.path.dirname(os.path.abspath(__file__)), '..', 'lib')) +from report import check, finish +from exact import rank_f2, rank_q +from lines import abar_rows + + +def lines_A(c): + out = [] + for i, j in itertools.combinations(range(c), 2): + v = [0] * c; v[i], v[j] = 1, -1; out.append(tuple(v)) + return out + + +def lines_D(c): + out = lines_A(c) + for i, j in itertools.combinations(range(c), 2): + v = [0] * c; v[i], v[j] = 1, 1; out.append(tuple(v)) + return out + + +def connected(S, c): + par = list(range(c)) + def f(x): + while par[x] != x: + par[x] = par[par[x]]; x = par[x] + return x + for r in S: + a, b = [i for i, x in enumerate(r) if x] + par[f(a)] = f(b) + return len({f(i) for i in range(c)}) == 1 + + +def test(c, pool, subsets): + n = 0 + worst = Fr(0) + for S in subsets: + S = [pool[i] for i in S] + if not S: + continue + G = [[sum(x * y for x, y in zip(a, b)) for b in S] for a in S] + assert all(G[t][t] == 2 and abs(G[t][s]) <= 1 for t in range(len(S)) for s in range(len(S)) if t != s) + rows = abar_rows(G) + # A-bar = N^T N mod 2 (N = |incidence|): off-diagonal parity of the number of shared coordinates + for t in range(len(S)): + for s in range(len(S)): + if t != s: + sh = sum(1 for x, y in zip(S[t], S[s]) if x and y) + assert ((rows[t] >> s) & 1) == (sh % 2), "A-bar != N^T N mod 2" + T = rank_q([list(r) for r in S]) + r2 = rank_f2(rows) + k = r2 // 2 + if connected(S, c): + assert r2 == c - 1 - (c % 2 == 0), (c, r2) + assert T in (c - 1, c) + ratio = Fr(len(S), T) + assert ratio <= max(15, 2 * k + 1), (c, len(S), T, k) + worst = max(worst, Fr(len(S), T) / (2 * k + 1)) + n += 1 + return n, worst + + +rng = random.Random(7) +for typ, gen in (('A', lines_A), ('D', lines_D)): + tot = 0 + for c in range(2, 8): + pool = gen(c) + m = len(pool) + if m <= 15: + subs = (tuple(i for i in range(m) if (mask >> i) & 1) for mask in range(1, 1 << m)) + mode = 'exhaustive' + else: + subs = (tuple(i for i in range(m) if rng.random() < rng.choice((0.3, 0.6, 0.9))) for _ in range(1500)) + mode = 'seeded random' + n, worst = test(c, pool, subs) + tot += n + check('E.%s%d' % (typ, c), True, '%s_%d lines (%d lines in the pool), %s: %d subsets, A-bar = N^T N mod 2, connected support => rank_F2 A-bar = c-1-[c even], v/T <= max(15, 2k\'+1) (max v/(T(2k\'+1)) seen: %s)' % (typ, c, m, mode, n, worst)) +# exceptional types: v/T of full root systems +check('E.exceptional', all(Fr(a, b) <= 15 for a, b in ((36, 6), (63, 7), (120, 8))), 'E6, E7, E8 lines: v/T = 6, 9, 15 (counts 36, 63, 120 are CHECKED in c1)') + + +# Theorem F constants (formula arithmetic only; the group-theoretic values themselves are citations) +vals = [] +for n in (3, 4, 5): + d = Fr((2 ** n - 1) * (2 ** (n - 1) - 1), 3) + m = 2 ** (n - 1) * (2 ** n - 1) + assert d.denominator == 1 + vals.append((n, int(d), m, 4 ** n - 1)) +check('F.constants', [(n, d, m) for n, d, m, _ in vals] == [(3, 7, 28), (4, 35, 120), (5, 155, 496)], + "formulas (2^n-1)(2^(n-1)-1)/3 and 2^(n-1)(2^n-1) for Sp(2n,2): n=3,4,5 -> degrees 7, 35, 155 and indices 28, 120, 496 [values are citations, only the arithmetic is checked]; " + "a multiplicity-1 system on all 4^n-1 labels would then have v/T <= %s" % ', '.join('%d/%d=%.2f' % (N, min(d, m), N / min(d, m)) for n, d, m, N in vals)) +finish() diff --git a/research/line-system-saturation/checks/c3_twopoint.py b/research/line-system-saturation/checks/c3_twopoint.py new file mode 100644 index 0000000000..71c9626bed --- /dev/null +++ b/research/line-system-saturation/checks/c3_twopoint.py @@ -0,0 +1,58 @@ +"""CHECKED (exact; exhaustive where stated): the Sp-invariant two-point point X* = x I + y A of the packing relaxation on W(2k-1,2), k = 2..5. + (1) the non-collinearity graph A is strongly regular with lambda = mu = 2^(2k-2), degree 2^(2k-1) (all point pairs), so A^2 = (d-mu) I + mu J and + spec A is contained in {2^(2k-1), +-2^(k-1)}; (2) X* is PSD, its lifted moment matrix is PSD (Schur complement), entries lie in [0, min(x_p,x_q)], + objective sum_p X_pp = 2^k+1; (3) every generator inequality sum_{q in L} X_pq <= X_pp holds with equality (all generators for k <= 4, 60 seeded random for k = 5), + and every point lies on N_L/(2^k+1) generators (all generators, k <= 4).""" +import os, sys, random +from fractions import Fraction as Fr +sys.path.insert(0, os.path.join(os.path.dirname(os.path.abspath(__file__)), '..', 'lib')) +from report import check, finish +from sympl import bf, lagrangians, random_lagrangian + +rng = random.Random(5) +for k in (2, 3, 4, 5): + N = 4 ** k - 1 + nb = [0] * (N + 1) + for p in range(1, N + 1): + m = 0 + for q in range(1, N + 1): + if bf(p, q, k): + m |= 1 << (q - 1) + nb[p] = m + d, lam = 2 ** (2 * k - 1), 2 ** (2 * k - 2) + deg_ok = all(nb[p].bit_count() == d for p in range(1, N + 1)) + srg = all((nb[p] & nb[q]).bit_count() == lam for p in range(1, N + 1) for q in range(p + 1, N + 1)) + check('T.srg-k%d' % k, deg_ok and srg, "k'=%d: %d points, non-collinearity graph regular of degree %d and every pair (adjacent or not) has exactly %d common neighbours, so A^2 = %d I + %d J and spec A within {%d, +-%d}" + % (k, N, d, lam, d - lam, lam, d, 2 ** (k - 1))) + x = Fr(1, 2 ** k - 1) + y = x / 2 ** (k - 1) + eig = [x + y * d, x + y * 2 ** (k - 1), x - y * 2 ** (k - 1)] + schur = eig[0] - x * x * N # X* - x^2 J on the all-ones direction (other eigenvalues unchanged since J kills them) + check('T.psd-k%d' % k, min(eig) == 0 and eig[1] == 2 * x and schur == 0 and 0 < y <= x and N * x == 2 ** k + 1, + "k'=%d: X* = x I + y A, x = 1/(2^k'-1), y = x/2^(k'-1): eigenvalues %s >= 0 (min 0); moment matrix [[1,x1^T],[x1,X*]]: Schur complement X*-x^2 J has eigenvalue %s on the all-ones vector; 0 < y <= x; objective N x = %s = 2^k'+1" + % (k, [str(e) for e in eig], schur, N * x)) + if k <= 4: + Ls = lagrangians(k) + cnt = {} + for L in Ls: + for p in L: + cnt[p] = cnt.get(p, 0) + 1 + thru = set(cnt.values()) == {len(Ls) // (2 ** k + 1)} and len(cnt) == N + mode = 'all %d generators' % len(Ls) + else: + Ls = [random_lagrangian(k, rng) for _ in range(60)] + thru = True + mode = '60 seeded random generators' + ok = True + for L in Ls: + lm = 0 + for q in L: + lm |= 1 << (q - 1) + for p in range(1, N + 1): + c = (nb[p] & lm).bit_count() + ok &= (c == 0) if p in L else (c == 2 ** (k - 1)) + s = (x if p in L else c * y) # sum_{q in L} X*_pq (q = p contributes x, collinear q != p contribute 0) + ok &= s == x + check('T.clique-k%d' % k, ok and thru, "k'=%d: %s x all %d points p: #{q in L : b(p,q)=1} = 0 (p in L) or 2^(k'-1), so sum_{q in L} X*_pq = x = X*_pp (equality); generators through a point: N_L/(2^k'+1)%s" + % (k, mode, N, ' (exhaustive)' if k <= 4 else ' (not enumerated at k\'=5; follows from transitivity, proved in README)')) +finish() diff --git a/research/line-system-saturation/checks/c4_descent.py b/research/line-system-saturation/checks/c4_descent.py new file mode 100644 index 0000000000..1ebd73e82f --- /dev/null +++ b/research/line-system-saturation/checks/c4_descent.py @@ -0,0 +1,79 @@ +"""CHECKED (exact): the descent construction of Section 5 (saturation passes from W(2k-1,2) to W(2k-3,2)) on E7 -> D6 -> D4 and E8 -> E7, and the inequalities +used in the quantitative version on E8.""" +import os, sys +from fractions import Fraction as Fr +sys.path.insert(0, os.path.join(os.path.dirname(os.path.abspath(__file__)), '..', 'lib')) +from report import check, finish +from exact import rank_q +from units import UNITS, gram +from lines import abar_rows, alt_labels, window +from sympl import bf, lagrangians + + +def system(L): + G = gram(L) + rows = abar_rows(G) + labels, k = alt_labels(rows) + return G, rows, labels, k + + +def classes(L, labels, k): + Ls = lagrangians(k) + m = {} + for l in labels: + m[l] = m.get(l, 0) + 1 + return m, [sum(m.get(p, 0) for p in Lg) for Lg in Ls], Ls + + +def descend(name, L, p_index): + G, rows, labels, k = system(L) + T = rank_q([list(r) for r in L]) + m, sizes, _ = classes(L, labels, k) + p = labels[p_index] + mp = m[p] + D = [L[t] for t in range(len(L)) if labels[t] != p and bf(labels[t], p, k) == 0] + inp = [L[t] for t in range(len(L)) if labels[t] == p] + orth = all(sum(a * b for a, b in zip(x, y)) == 0 for x in D for y in inp) + T2 = rank_q([list(r) for r in D]) + G2, rows2, labels2, k2 = system(D) + m2, sizes2, _ = classes(D, labels2, k2) + # derived multiplicities: twin classes of the derived system must be the unions {q, q+p} + pair = {} + for t in range(len(L)): + q = labels[t] + if q != p and bf(q, p, k) == 0: + key = min(q, q ^ p) + pair[key] = pair.get(key, 0) + 1 + mult_ok = sorted(pair.values()) == sorted(m2.values()) and len(pair) == len(m2) + sat = Fr(len(D), T2 * (2 ** k2 + 1)) + return dict(k=k, T=T, mp=mp, v2=len(D), T2=T2, k2=k2, orth=orth, mult_ok=mult_ok, sat2=sat, full=all(s == T2 for s in sizes2), sizes=sizes, lab=labels, D=D) + + +L7, _ = UNITS['E7'] +L6, _ = UNITS['D6'] +r1 = descend('E7', L7, 0) +check('S.E7-D6', r1['orth'] and r1['T2'] == r1['T'] - r1['mp'] == 6 and r1['v2'] == 30 and r1['k2'] == 2 and r1['mult_ok'] and r1['full'] and r1['sat2'] == 1, + 'E7 (k\'=3, T=7, saturated) at a label p with m_p=1: the %d lines with labels in p^perp \\ {p} are orthogonal to the class of p, span T-m_p=%d, form a system with k\'=%d, twin classes = {q,q+p} (multiplicity 2), every Lagrangian class is a full basis (saturated; this is the D6 system)' % (r1['v2'], r1['T2'], r1['k2'])) +r2 = descend('D6', r1['D'], 0) +check('S.D6-D4', r2['orth'] and r2['T2'] == 4 and r2['v2'] == 12 and r2['k2'] == 1 and r2['mult_ok'] and r2['full'] and r2['sat2'] == 1 and r2['mp'] == 2, + 'D6 (k\'=2, T=6, m_p=2) descends to %d lines in T=%d, k\'=%d, multiplicity 4, saturated (the D4 system)' % (r2['v2'], r2['T2'], r2['k2'])) +L8, _ = UNITS['E8'] +r3 = descend('E8', L8, 0) +check('S.E8-E7', r3['orth'] and r3['T2'] == 7 and r3['v2'] == 63 and r3['k2'] == 3 and r3['full'] and r3['sat2'] == 1, + 'E8 (k\'=4, saturation 15/17) descends to %d lines in T=%d, k\'=%d, saturated (E7); the level-below saturation is 1, so the inequality of Section 5.3 is slack for E8' % (r3['v2'], r3['T2'], r3['k2'])) +# quantitative ingredients on E8: A_p = average class size over generators through p; A_p <= sT + (1-s)m_p with s = 1; sum_p m_p A_p >= v sigma T +G, rows, labels, k = system(L8) +m, sizes, Ls = classes(L8, labels, k) +T = 8 +v = len(L8) +sigma = Fr(v, T * (2 ** k + 1)) +ok1 = True +tot = Fr(0) +for p, mp in m.items(): + cs = [sizes[i] for i, Lg in enumerate(Ls) if p in Lg] + Ap = Fr(sum(cs), len(cs)) + ok1 &= Ap <= T + tot += mp * Ap +check('S.quant-E8', ok1 and tot >= v * sigma * T and sigma == Fr(15, 17), + 'E8: A_p <= T for every occupied label, sum_p m_p A_p = %s >= v sigma T = %s (Cauchy-Schwarz step); sigma = 15/17' % (tot, v * sigma * T)) +finish() diff --git a/research/line-system-saturation/checks/c5_l1cons.py b/research/line-system-saturation/checks/c5_l1cons.py new file mode 100644 index 0000000000..3cc5fc4fb3 --- /dev/null +++ b/research/line-system-saturation/checks/c5_l1cons.py @@ -0,0 +1,86 @@ +"""CHECKED (exact integer arithmetic): the 14 best systems of the L1_CONS construction table (data/best_kprime_*.json). For each: lines distinct up to sign, nonzero norms, +every pairwise product 0 or odd (after division by 'div' or through the stored lattice Gram), T = exact Q-rank, A-bar and its F2 rank 2k', window h = rank_F2(J + A-bar), +saturation v / (T (2^k'+1)) as a Fraction. The comparison with the break-even fit x*(h) is relative to a RECORDED repo fit (not proved here).""" +import os, sys, json, glob +from fractions import Fraction as Fr +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, os.path.join(HERE, '..', 'lib')) +from report import check, finish +from exact import rank_f2, rank_q +from lines import abar_rows, window + + +def pos_def(M): + n = len(M) + A = [[Fr(x) for x in r] for r in M] + for i in range(n): + if A[i][i] <= 0: + return False + for j in range(i + 1, n): + f = A[j][i] / A[i][i] + A[j] = [a - f * b for a, b in zip(A[j], A[i])] + return True + + +def xstar(h): + return Fr(5385, 10000) - Fr(1495, 100000) * (h - 9) + + +files = sorted(glob.glob(os.path.join(HERE, '..', 'data', 'best_kprime_*.json')), key=lambda f: int(f.split('_')[-1][:-5])) +assert len(files) == 14 +EXPECT = {0: (64, 64), 3: (56, 8), 4: (120, 8), 5: (140, 11), 6: (166, 13), 7: (240, 16), 8: (496, 16), 9: (55, 20), 12: (728, 24), + 15: (992, 32), 16: (480, 32), 20: (600, 40), 25: (960, 64), 32: (666, 64)} +rowsout = [] +for f in files: + d = json.load(open(f)) + V = [tuple(int(x) for x in r) for r in d['vectors']] + v = len(V) + n = len(V[0]) + canon = {tuple(x if next(y for y in r if y) > 0 else -x for x in r) for r in V} + distinct = len(canon) == v + if 'gram' in d: + G0 = d['gram'] + assert len(G0) == n and rank_q(G0) == n and pos_def(G0) + W = [[sum(r[i] * G0[i][j] for i in range(n)) for j in range(n)] for r in V] + G = [[sum(a * b for a, b in zip(W[s], V[t])) for t in range(v)] for s in range(v)] + else: + div = d['div'] + G = [[0] * v for _ in range(v)] + for s in range(v): + for t in range(s, v): + x = sum(a * b for a, b in zip(V[s], V[t])) + assert x % div == 0 + G[s][t] = G[t][s] = x // div + norms = all(G[t][t] > 0 for t in range(v)) + oz = all(G[s][t] == 0 or G[s][t] % 2 != 0 for s in range(v) for t in range(v) if s != t) + T = rank_q([list(r) for r in V]) + rows = abar_rows(G) + r2, h, inc = window(rows) + k = r2 // 2 + sat = Fr(v, T * (2 ** k + 1)) + ex = EXPECT[d['kprime']] + ok = distinct and norms and oz and (v, T) == ex and k == d['kprime'] and r2 == d['rank_mod2_gram'] and abs(float(sat) - d['saturation']) < 1e-12 and d['v'] == v and d['T'] == T + x = Fr(h * T, v) + xs = xstar(h) + rowsout.append((k, d['name'], v, T, h, inc, sat, x, xs)) + check('L.%s' % d['name'], ok, + "%s (k'=%d): v=%d distinct lines, T=%d, all products 0/odd, rank_F2 A-bar=%d, window h=%d%s, v/T=%.2f, saturation %s=%.4f; dense x=hT/v=%.3f vs break-even fit x*(h)=%.3f [RECORDED fit]%s" + % (d['name'], k, v, T, r2, h, ' (even window, 1 in col A-bar)' if inc else '', v / T, sat, float(sat), float(x), float(xs), '' if xs <= 0 or x >= xs else ' BELOW x*')) +below = [r for r in rowsout if r[8] > 0 and r[7] < r[8]] +check('L.none-below', not below, 'no system in the table has dense x below the recorded break-even fit x*(h) (rows with x*(h) <= 0 are outside the fit)') +inscope = [r for r in rowsout if r[4] <= 13] +best = {} +for r in inscope: + best[r[0]] = max(best.get(r[0], Fr(0)), r[6]) +check('L.best-scope', best == {0: Fr(1, 2), 3: Fr(7, 9), 4: Fr(15, 17), 5: Fr(140, 11 * 33), 6: Fr(166, 13 * 65)}, + 'best saturation per k\' for window h <= 13 in the saved table: %s' % ', '.join('k\'=%d: %.3f' % (k, float(s)) for k, s in sorted(best.items()))) +# thresholds: required v/T and saturation to tie the recorded break-even fit x*(h) at window h (conditional on the RECORDED fit) +trow = [] +for h in range(9, 14): + k = h // 2 + need = Fr(h) / xstar(h) + cap = Fr(2 ** k + (h % 2)) + trow.append((h, k, float(xstar(h)), float(need), int(cap), float(need / cap))) +check('L.thresholds', [round(r[5] * 100) for r in trow] == [98, 60, 66, 38, 42] and [r[4] for r in trow] == [17, 32, 33, 64, 65], + 'needed v/T = h/x*(h) and saturation vs the Lagrangian cap (2^(h//2) + [h odd]): ' + '; '.join('h=%d (k\'=%d): x*=%.4f, need v/T=%.2f of cap %d = %.1f%%' % (r[0], r[1], r[2], r[3], r[4], 100 * r[5]) for r in trow)) +finish() diff --git a/research/line-system-saturation/checks/c6_delsarte_uniform.py b/research/line-system-saturation/checks/c6_delsarte_uniform.py new file mode 100644 index 0000000000..ab6f1d2f16 --- /dev/null +++ b/research/line-system-saturation/checks/c6_delsarte_uniform.py @@ -0,0 +1,125 @@ +"""CHECKED (exact): (1) Delsarte linear-programming cap for the D16+ construction: an exact rational DUAL certificate (derived by solving a 6x6 system in Fractions, then verified +feasible) bounds the antipodal binary codes of length 16 with distances in S = {2,6,10,14} u {8,16}: at most 317 spinor lines, so roots + spinors <= 240 + 317 = 557. +(2) Theorem G arithmetic: which (k', m) admit a uniform-angle saturated system (lambda^2 = 2^(k'-1) m a perfect square), the known cases D4 / D6 / E7, and the common-neighbour counts +of the 510-vertex and 1023-vertex signed graphs.""" +import os, sys, itertools +from math import comb, isqrt +from fractions import Fraction as Fr +sys.path.insert(0, os.path.join(os.path.dirname(os.path.abspath(__file__)), '..', 'lib')) +from report import check, finish +from units import UNITS, gram +from sympl import bf + + +def K(n, k, i): + return sum((-1) ** j * comb(i, j) * comb(n - i, k - j) for j in range(k + 1)) + + +# --- Krawtchouk identity K_k(d(x,y)) = sum_{wt u = k} (-1)^{u.(x+y)} (n = 6, all pairs), and Delsarte inequalities on a real code, RM(1,4) +n6 = 6 +ok = all(K(n6, k, bin(z).count('1')) == sum((-1) ** (bin(u & z).count('1')) for u in range(1 << n6) if bin(u).count('1') == k) for z in range(1 << n6) for k in range(n6 + 1)) +g = [0xFFFF, 0xFF00, 0xF0F0, 0xCCCC, 0xAAAA] +rm = {0} +for gen in g: + rm |= {x ^ gen for x in rm} +dd = {} +for a in rm: + for b in rm: + d = bin(a ^ b).count('1') + dd[d] = dd.get(d, 0) + 1 +A = {d: Fr(c, len(rm)) for d, c in dd.items()} +delsarte_ok = all(sum(A[d] * K(16, k, d) for d in A) >= 0 for k in range(17)) +check('D.krawtchouk', ok and len(rm) == 32 and set(A) == {0, 8, 16} and delsarte_ok, + 'Krawtchouk identity verified for n=6 on all (word, k); Delsarte inequalities sum_i A_i K_k(i) >= 0 hold for all k=0..16 on the real code RM(1,4) (32 words, distance distribution %s)' % dict(sorted(A.items()))) + +# --- exact dual certificate +n = 16 +S = [2, 6, 8, 10, 14, 16] +supp = [2, 3, 5, 6, 7, 8] # dual support found from a floating-point LP; the multipliers below are recomputed exactly +M = [[Fr(-K(n, k, i)) for k in supp] + [Fr(1)] for i in S] +for c in range(6): + p = next(r for r in range(c, 6) if M[r][c] != 0) + M[c], M[p] = M[p], M[c] + iv = 1 / M[c][c] + M[c] = [x * iv for x in M[c]] + for r in range(6): + if r != c and M[r][c] != 0: + f = M[r][c] + M[r] = [a - f * b for a, b in zip(M[r], M[c])] +y = {k: M[i][6] for i, k in enumerate(supp)} +feas = all(sum(y[k] * -K(n, k, i) for k in y) >= 1 for i in S) and all(v >= 0 for v in y.values()) +bound = 1 + sum(y[k] * K(n, k, 0) for k in y) +# proof of the bound: sum_{i in S\0}A_i <= sum_i A_i (-sum_k y_k K_k(i)) = -sum_k y_k sum_i A_i K_k(i) <= sum_k y_k K_k(0) +words = int(bound) # floor +lines = (words - words % 2) // 2 +check('D.dual', feas and bound == Fr(18432, 29) and lines == 317, + 'exact dual y>=0 on k=%s: y=%s; for every i in S, sum_k y_k(-K_k(i)) >= 1; so |C| <= 1 + sum_k y_k K_k(0) = %s = %.4f; C antipodal (even size) => |C| <= %d => at most %d spinor lines' + % (supp, {k: str(v) for k, v in y.items()}, bound, float(bound), words - words % 2, lines)) +need = Fr(17 * 16) / (Fr(5385, 10000) - Fr(1495, 100000) * 8) +check('D.cap', 240 + lines == 557 and 240 + lines < need and 16 * 15 == 240, + "D16+ family: 16*15 = 240 root lines + <= %d spinor lines = %d < %.1f = h T / x*(h) at h=17, T=16 (x* is the RECORDED repo fit); this cap is for roots+spinors of D16+ only" % (lines, 240 + lines, float(need))) + +# --- Theorem G arithmetic +sols = [] +for k in range(1, 10): + for m in range(1, 51): + l2 = 2 ** (k - 1) * m + if isqrt(l2) ** 2 == l2: + sols.append((k, m, isqrt(l2))) +m1 = [(k, l) for k, m, l in sols if m == 1] +check('G.table', [k for k, _ in m1] == [1, 3, 5, 7, 9] and (4, 2, 4) in sols and (2, 2, 2) in sols and (1, 4, 2) in sols and (3, 1, 2) in sols and (5, 1, 4) in sols + and not any(k == 4 and m == 1 for k, m, _ in sols), + "lambda^2 = 2^(k'-1) m a perfect square (k'<=9, m<=50): m=1 exactly for odd k' = 1,3,5,7,9 (lambda = %s); k'=4 needs m in %s; D4 (k'=1,m=4), D6 (2,2), E7 (3,1) satisfy it with lambda = 2" + % ([l for _, l in m1], [m for k, m, _ in sols if k == 4])) +for nm, (k, m) in (('D4', (1, 4)), ('D6', (2, 2)), ('E7', (3, 1))): + L, T = UNITS[nm] + G = gram(L) + unif = all(G[t][t] == 2 and G[t][s] in (0, 1, -1) for t in range(len(L)) for s in range(len(L)) if t != s) + lam = 2 + N = 4 ** k - 1 + trace0 = lam * 2 ** k * (m * (2 ** k - 1)) - lam * (N * m - m * (2 ** k - 1)) == 0 + check('G.known-' + nm, unif and len(L) == N * m and T == m * (2 ** k - 1) and trace0 and 2 ** k * lam ** 2 == 2 ** (2 * k - 1) * m, + '%s: all non-orthogonal pairs have |cos| = 1/2 = 1/lambda, v = m(4^k\'-1) = %d, T = m(2^k\'-1) = %d, tr S = 0 for spectrum {2^k\' lambda (x T), -lambda (x v-T)}, degree 2^(2k\'-1) m = 2^k\' lambda^2' % (nm, len(L), T)) + + +# --- 510-vertex (k'=4, m=2) and 1023-vertex (k'=5, m=1) signed-graph counts +def counts(k, m): + N = 4 ** k - 1 + nb = [] + for p in range(1, N + 1): + mm = 0 + for q in range(1, N + 1): + if bf(p, q, k): + mm |= 1 << (q - 1) + nb.append(mm) + # doubled graph: vertex (p, j); neighbours = all copies of points q with b(p,q)=1 + big = [] + for p in range(N): + row = 0 + for q in range(N): + if (nb[p] >> q) & 1: + for j in range(m): + row |= 1 << (q * m + j) + big.append(row) + seen = {} + V = N * m + for u in range(V): + for w in range(u + 1, V): + typ = 'twin' if u // m == w // m else ('adj' if (nb[u // m] >> (w // m)) & 1 else 'nonadj') + seen.setdefault(typ, set()).add((big[u // m] & big[w // m]).bit_count()) + return seen, big[0].bit_count() + + +for (k, m, lam, label) in ((4, 2, 4, '510-vertex'), (5, 1, 4, '1023-vertex')): + seen, deg = counts(k, m) + a = (2 ** k - 1) * lam # S^2 = a S + b I + b = 2 ** k * lam * lam + adj = seen['adj'].pop() + non = seen['nonadj'].pop() + tw = seen.get('twin', {None}).pop() + agree, dis = Fr(adj + a, 2), Fr(adj - a, 2) + ok = deg == b and len(seen['adj']) == len(seen['nonadj']) == 0 and agree.denominator == 1 and dis >= 0 + check('G.%s' % label, ok, + "%s signed graph (k'=%d, m=%d, lambda=%d): S^2 = %d S + %d I, degree %d (= b); common neighbours: adjacent %d (signed sum +-%d: %d agree / %d disagree), non-adjacent distinct %d (signed sum 0), twins %s (signed sum 0)" + % (label, k, m, lam, a, b, deg, adj, a, agree, dis, non, tw if tw is not None else 'none (m=1)')) +finish() diff --git a/research/line-system-saturation/checks/gap/sp2n2_degrees.g b/research/line-system-saturation/checks/gap/sp2n2_degrees.g new file mode 100644 index 0000000000..85dd1a5a83 --- /dev/null +++ b/research/line-system-saturation/checks/gap/sp2n2_degrees.g @@ -0,0 +1,15 @@ +# Smallest nontrivial ordinary character degrees of Sp(2n,2), n = 3,4,5, from the +# ATLAS character tables in the GAP Character Table Library (CTblLib). +# Run: gap -q -b sp2n2_degrees.g < /dev/null (GAP >= 4.12 with package ctbllib) +OnBreak := function() Print("GAP_ERROR\n"); FORCE_QUIT_GAP(1); end; +if LoadPackage("ctbllib") <> true then Print("CTBLLIB_MISSING\n"); FORCE_QUIT_GAP(2); fi; +for pair in [["S6(2)",7],["S8(2)",35],["S10(2)",155]] do + t := CharacterTable(pair[1]); + degs := Set(List(Irr(t), chi -> chi[1])); + m := Filtered(degs, d -> d > 1)[1]; + Print(pair[1], " order=", Size(t), " smallest_nontrivial_degree=", m, + " expected=", pair[2], " ", (m = pair[2]), "\n"); + if m <> pair[2] then FORCE_QUIT_GAP(3); fi; +od; +Print("GAP_SP2N2_PASS\n"); +QUIT; diff --git a/research/line-system-saturation/checks/gap/sp2n2_degrees.out.txt b/research/line-system-saturation/checks/gap/sp2n2_degrees.out.txt new file mode 100644 index 0000000000..a45725c4c9 --- /dev/null +++ b/research/line-system-saturation/checks/gap/sp2n2_degrees.out.txt @@ -0,0 +1,5 @@ +S6(2) order=1451520 smallest_nontrivial_degree=7 expected=7 true +S8(2) order=47377612800 smallest_nontrivial_degree=35 expected=35 true +S10(2) order=24815256521932800 smallest_nontrivial_degree=155 expected= +155 true +GAP_SP2N2_PASS diff --git a/research/line-system-saturation/checks/pr2222_rank.py b/research/line-system-saturation/checks/pr2222_rank.py new file mode 100644 index 0000000000..8cfaba81c5 --- /dev/null +++ b/research/line-system-saturation/checks/pr2222_rank.py @@ -0,0 +1,47 @@ +"""Maintainer correction (PR2222 classification): on the norm-six shell the 48 linear +features are combinations of the cubic ones, x_i = (1/6) * sum_j x_i * x_j**2, so the +cubic-plus-linear feature matrix has rank <= binom(50, 3) = 19,600 < 19,648 terms. + +Exact checks (stdlib only): + 1. binom(50,3) = 19,600 and binom(50,3) + 48 = 19,648. + 2. The identity holds symbolically: sum_j x_i x_j^2 = x_i * N(x), with N(x) = sum_j x_j^2 = 6 + on the shell (checked as a polynomial identity on random integer vectors of every norm, + and as the stated identity on vectors of norm 6 in Z^48 and in (1/2)Z^48 of the D-type form). + 3. Consequence: each linear feature lies in the span of the cubic monomial features + x_i x_j^2 (j = 1..48), which are among the binom(50,3) degree-3 monomials in 48 variables. +""" +import math, random +from fractions import Fraction as F + +assert math.comb(50, 3) == 19600 and math.comb(50, 3) + 48 == 19648 +assert math.comb(48 + 2, 3) == 19600 # degree-3 monomials in 48 variables + +rng = random.Random(2222) +for _ in range(2000): # polynomial identity sum_j x_i x_j^2 = x_i N(x) + x = [rng.randint(-5, 5) for _ in range(48)] + N = sum(v * v for v in x) + for i in range(48): + assert sum(x[i] * v * v for v in x) == x[i] * N + +def norm6_vectors(k): + """Random vectors of squared norm 6 in Z^48 (patterns 2,1,1 / 1^6) and half-integral (±1/2)^24.""" + out = [] + for _ in range(k): + x = [F(0)] * 48 + kind = rng.randrange(3) + if kind == 0: + idx = rng.sample(range(48), 3); x[idx[0]] = F(rng.choice([-2, 2])) + for j in idx[1:]: x[j] = F(rng.choice([-1, 1])) + elif kind == 1: + for j in rng.sample(range(48), 6): x[j] = F(rng.choice([-1, 1])) + else: + for j in rng.sample(range(48), 24): x[j] = F(rng.choice([-1, 1]), 2) + assert sum(v * v for v in x) == 6 + out.append(x) + return out + +for x in norm6_vectors(3000): + for i in range(48): + assert x[i] == F(1, 6) * sum(x[i] * v * v for v in x) + +print("PASS pr2222_rank: binom(50,3)=19600 < 19648; x_i = (1/6) sum_j x_i x_j^2 on norm-6 vectors (exact)") diff --git a/research/line-system-saturation/data/best_kprime_0.json b/research/line-system-saturation/data/best_kprime_0.json new file mode 100644 index 0000000000..e346b4dc8c --- /dev/null +++ b/research/line-system-saturation/data/best_kprime_0.json @@ -0,0 +1 @@ +{"div": 4, "vectors": [[0, 1, -1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, 1, -1, 0], [1, 0, 0, -1, -1, 0, 0, -1, 1, 0, 0, -1, -1, 0, 0, -1, 1, 0, 0, -1, -1, 0, 0, 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0, 0, 0, 0, 0, 0, 0, -1, -1], [1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, -1], [0, 0, 0, 0, 1, 1, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0], [0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0], [0, 1, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1], [1, 1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0], [1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, -1, 0, 1], [0, 0, 0, 1, 0, 1, -1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0], [1, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, -1, 0], [1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 1, 0, 0, 0, 1, -1, 0, -1, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0]], "name": "Z20m", "T": 20, "v": 55, "kprime": 9, "rank_mod2_gram": 18, "saturation": 0.005360623781676413, "verified": "all pairwise products 0 or odd, nonzero norms, distinct lines; exact integer arithmetic"} \ No newline at end of file diff --git a/research/line-system-saturation/lib/exact.py b/research/line-system-saturation/lib/exact.py new file mode 100644 index 0000000000..bac92b72c1 --- /dev/null +++ b/research/line-system-saturation/lib/exact.py @@ -0,0 +1,92 @@ +"""Exact arithmetic helpers (standard library only): rank / row-reduction over Q (Fractions), ranks over F2 (ints as bit rows), +p-adic valuation at 2, deterministic output. Used by every check.""" +from fractions import Fraction as Fr + + +def rank_f2(rows): + """Rank over F2 of a list of ints (each int = a bit row).""" + basis = {} + r = 0 + for x in rows: + while x: + h = x.bit_length() - 1 + if h in basis: + x ^= basis[h] + else: + basis[h] = x + r += 1 + break + return r + + +def rref(rows): + """Reduced row echelon form over Q of a list of vectors; returns a tuple of tuples of Fractions (canonical).""" + M = [[Fr(x) for x in r] for r in rows] + piv = [] + r = 0 + ncol = len(M[0]) if M else 0 + for c in range(ncol): + p = next((i for i in range(r, len(M)) if M[i][c] != 0), None) + if p is None: + continue + M[r], M[p] = M[p], M[r] + inv = 1 / M[r][c] + M[r] = [x * inv for x in M[r]] + for i in range(len(M)): + if i != r and M[i][c] != 0: + f = M[i][c] + M[i] = [a - f * b for a, b in zip(M[i], M[r])] + piv.append(c) + r += 1 + if r == len(M): + break + return tuple(tuple(row) for row in M[:r]) + + +def rank_q(rows): + return len(rref(rows)) if rows else 0 + + +def in_span(basis_rref, v): + """Is v in the row space of the (reduced) basis? basis_rref from rref().""" + v = [Fr(x) for x in v] + for row in basis_rref: + c = next(i for i, x in enumerate(row) if x != 0) + if v[c] != 0: + f = v[c] + v = [a - f * b for a, b in zip(v, row)] + return all(x == 0 for x in v) + + +def nullspace(rows, ncols): + """Basis (list of tuples of Fractions) of {x : rows . x = 0} over Q.""" + R = rref(rows) if rows else () + pivs = [next(i for i, x in enumerate(r) if x != 0) for r in R] + free = [c for c in range(ncols) if c not in pivs] + out = [] + for f in free: + x = [Fr(0)] * ncols + x[f] = Fr(1) + for r, p in zip(R, pivs): + x[p] = -r[f] + out.append(tuple(x)) + return out + + +def v2(x): + """2-adic valuation of a nonzero rational (None for 0).""" + x = Fr(x) + if x == 0: + return None + n, d, k = x.numerator, x.denominator, 0 + while n % 2 == 0: + n //= 2 + k += 1 + while d % 2 == 0: + d //= 2 + k -= 1 + return k + + +def dot(a, b): + return sum(x * y for x, y in zip(a, b)) diff --git a/research/line-system-saturation/lib/lines.py b/research/line-system-saturation/lib/lines.py new file mode 100644 index 0000000000..409b2593cc --- /dev/null +++ b/research/line-system-saturation/lib/lines.py @@ -0,0 +1,106 @@ +"""Line systems over the integers/rationals: the non-orthogonality adjacency A-bar over F2 (bit rows), its symplectic labels, the dense window. +Standard library only. Vectors are tuples of ints/Fractions; a 'line system' is a list of vectors (one per line, distinct up to sign).""" +from exact import rank_f2 + + +def parity(x): + return x.bit_count() & 1 + + +def abar_rows(G): + """A-bar_ts = 1 iff t != s and G_ts != 0, as a list of int bit rows (bit s of row t).""" + n = len(G) + return [sum((1 << s) for s in range(n) if s != t and G[t][s] != 0) for t in range(n)] + + +def apply(rows, x): + """A x over F2 for a symmetric matrix given by bit rows, x an int bit vector (bit s = coordinate s).""" + out = 0 + s = 0 + while x: + if x & 1: + out ^= rows[s] + x >>= 1 + s += 1 + return out + + +def alt_labels(rows): + """Symplectic factorisation A-bar = L^T b L (b standard). Returns (labels, k) with labels[t] an int whose bit 2i is a_i(t), bit 2i+1 is c_i(t). + Self-verifying: raises AssertionError unless b(l_t, l_s) = A-bar_ts for ALL pairs and rank = 2k.""" + v = len(rows) + for t in range(v): + assert not (rows[t] >> t) & 1, "diagonal must be zero (alternating)" + for s in range(t): + assert ((rows[t] >> s) & 1) == ((rows[s] >> t) & 1), "must be symmetric" + W = [1 << t for t in range(v)] + es, fs = [], [] + while True: + found = None + for xi, x in enumerate(W): + Ax = apply(rows, x) + for yi, y in enumerate(W): + if yi != xi and parity(y & Ax): + found = (xi, yi, Ax) + break + if found: + break + if not found: + break + xi, yi, Ax = found + x, y = W[xi], W[yi] + Ay = apply(rows, y) + new = [] + for wi, w in enumerate(W): + if wi in (xi, yi): + continue + w2 = w ^ (x if parity(w & Ay) else 0) ^ (y if parity(w & Ax) else 0) + new.append(w2) + es.append(x) + fs.append(y) + W = new + k = len(es) + # unit vector delta_t = sum_i [A(t,f_i) e_i + A(t,e_i) f_i] + radical, so a_i(t) = (A f_i)_t, c_i(t) = (A e_i)_t + a = [apply(rows, f) for f in fs] + c = [apply(rows, e) for e in es] + labels = [0] * v + for t in range(v): + l = 0 + for i in range(k): + l |= ((a[i] >> t) & 1) << (2 * i) + l |= ((c[i] >> t) & 1) << (2 * i + 1) + labels[t] = l + # verify A-bar = sum_i (a_i c_i^T + c_i a_i^T) as bit rows + for t in range(v): + r = 0 + for i in range(k): + if (a[i] >> t) & 1: + r ^= c[i] + if (c[i] >> t) & 1: + r ^= a[i] + assert r == rows[t], "symplectic factorisation failed" + assert rank_f2(rows) == 2 * k + return labels, k + + +def window(rows): + """(rank_F2(A-bar), rank_F2(J + A-bar), whether the all-ones vector lies in the column space of A-bar).""" + v = len(rows) + ones = (1 << v) - 1 + r = rank_f2(rows) + h = rank_f2([ones ^ x for x in rows]) + inc = rank_f2(rows + [ones]) == r + return r, h, inc + + +def gram_int(V, div=1): + """Integer Gram matrix (exact) of int vectors, divided by div (asserted exact); upper triangle computed once.""" + n = len(V) + G = [[0] * n for _ in range(n)] + for a in range(n): + va = V[a] + for b in range(a, n): + d = sum(x * y for x, y in zip(va, V[b])) + assert d % div == 0 + G[a][b] = G[b][a] = d // div + return G diff --git a/research/line-system-saturation/lib/rat.py b/research/line-system-saturation/lib/rat.py new file mode 100644 index 0000000000..363eea6a47 --- /dev/null +++ b/research/line-system-saturation/lib/rat.py @@ -0,0 +1,38 @@ +"""Exact rational matrices (Fractions) for the generic-frame constructions.""" +from fractions import Fraction as Fr + + +def matmul(A, B): + return [[sum(a * b for a, b in zip(r, c)) for c in zip(*B)] for r in A] + + +def inv(M): + n = len(M) + A = [[Fr(x) for x in r] + [Fr(int(i == j)) for j in range(n)] for i, r in enumerate(M)] + for c in range(n): + p = next(i for i in range(c, n) if A[i][c] != 0) + A[c], A[p] = A[p], A[c] + iv = 1 / A[c][c] + A[c] = [x * iv for x in A[c]] + for i in range(n): + if i != c and A[i][c] != 0: + f = A[i][c] + A[i] = [a - f * b for a, b in zip(A[i], A[c])] + return [r[n:] for r in A] + + +def cayley(T, rng, den=7): + """A rational orthogonal matrix (I - S)(I + S)^(-1), S skew with small random rational entries.""" + S = [[Fr(0)] * T for _ in range(T)] + for i in range(T): + for j in range(i + 1, T): + x = Fr(rng.randint(-den, den), rng.randint(1, den)) + S[i][j], S[j][i] = x, -x + I = [[Fr(int(i == j)) for j in range(T)] for i in range(T)] + A = [[I[i][j] - S[i][j] for j in range(T)] for i in range(T)] + B = [[I[i][j] + S[i][j] for j in range(T)] for i in range(T)] + O = matmul(A, inv(B)) + Ot = [list(r) for r in zip(*O)] + P = matmul(Ot, O) + assert P == I, "not orthogonal" + return O diff --git a/research/line-system-saturation/lib/report.py b/research/line-system-saturation/lib/report.py new file mode 100644 index 0000000000..e8973c2fa2 --- /dev/null +++ b/research/line-system-saturation/lib/report.py @@ -0,0 +1,16 @@ +"""One PASS/FAIL line per assertion, so verify.sh can count them. A check script exits 1 if any of its lines failed.""" +import sys +_fails = 0 +_n = 0 + + +def check(ident, cond, msg): + global _fails, _n + _n += 1 + if not cond: + _fails += 1 + print('%s %s %s' % ('PASS' if cond else 'FAIL', ident, msg), flush=True) + + +def finish(): + sys.exit(1 if _fails else 0) diff --git a/research/line-system-saturation/lib/sympl.py b/research/line-system-saturation/lib/sympl.py new file mode 100644 index 0000000000..15b9129930 --- /dev/null +++ b/research/line-system-saturation/lib/sympl.py @@ -0,0 +1,38 @@ +"""The symplectic polar space W(2k-1,2) on F2^{2k}: points are nonzero ints below 4^k, b(p,q) = sum_i (p_{2i} q_{2i+1} + p_{2i+1} q_{2i}). +Generators (Lagrangians = maximal totally isotropic subspaces) are frozensets of their 2^k - 1 nonzero points.""" +import random + + +def bf(p, q, k): + m = int('01' * k, 2) + return (((p & m) << 1) & q).bit_count() & 1 ^ (((p & (m << 1)) >> 1) & q).bit_count() & 1 + + +def perp_points(basis, k): + return [q for q in range(1, 4 ** k) if all(bf(q, s, k) == 0 for s in basis)] + + +def lagrangians(k): + """All generators of W(2k-1,2) (count prod_{i=1..k} (2^i + 1)).""" + level = {frozenset([p]) for p in range(1, 4 ** k)} + for _ in range(k - 1): + nxt = set() + for S in level: + for p in perp_points(list(S), k): + if p in S: + continue + nxt.add(frozenset(S | {p} | {p ^ s for s in S})) + level = nxt + return level + + +def random_lagrangian(k, rng): + S = set() + basis = [] + for _ in range(k): + cands = [q for q in perp_points(basis, k) if q not in S] + p = rng.choice(cands) + S |= {p} | {p ^ s for s in S} + basis.append(p) + assert len(S) == 2 ** k - 1 + return frozenset(S) diff --git a/research/line-system-saturation/lib/units.py b/research/line-system-saturation/lib/units.py new file mode 100644 index 0000000000..25b05dea6a --- /dev/null +++ b/research/line-system-saturation/lib/units.py @@ -0,0 +1,155 @@ +"""Root units as exact integer data (coordinates scaled by 2, so every root has norm 8 and the Gram entry of two roots is dot/4), +the dense pairing B (B_ts = 1 iff t = s or r_t _|_ r_s), exact coordinate/frame decoders and their F2 cost. Standard library only.""" +import itertools +from fractions import Fraction as Fr +from exact import rank_f2, dot + + +def e8_lines(): + """The 120 lines of E8 (one root per +- pair), coordinates doubled: +-2e_i +-2e_j (i, p(t) = Gram(basis)^-1 q(t); then P Q^T = Gram exactly. Returns P, Q (lists of rows of Fractions).""" + T = len(basis) + Gb = [[dot(a, b) for b in basis] for a in basis] + Gi = inv_matrix(Gb) + P, Q = [], [] + for r2 in live_roots: + r = [Fr(x, 2) for x in r2] + q = [dot(r, a) for a in basis] + p = [sum(Gi[j][k] * q[k] for k in range(T)) for j in range(T)] + assert all(sum(p[j] * basis[j][i] for j in range(T)) == r[i] for i in range(len(r))), 'root not in span of the basis' + P.append(p) + Q.append(q) + return P, Q + + +def gram_exact(live_roots, P, Q): + """P Q^T equals the Gram matrix of the (unit-norm-2) roots, entry by entry.""" + n, T = len(P), len(P[0]) + for t in range(n): + for s in range(n): + if sum(P[t][j] * Q[s][j] for j in range(T)) != Fr(dot(live_roots[t], live_roots[s]), 4): + return False + return True + + +def decoder_cost(live_roots, basis): + """(exact?, cost, per-term ranks) of the decoder of `live_roots` in `basis` with the dense pairing B of the unit.""" + P, Q = decoder(live_roots, basis) + ok = gram_exact(live_roots, P, Q) + G = gram(live_roots) + Br = bmasks(G) + n, T = len(P), len(P[0]) + ranks = [] + for j in range(T): + R = [t for t in range(n) if P[t][j] != 0] + A = [t for t in range(n) if Q[t][j] != 0] + ranks.append(block_rank(Br, R, A)) + return ok, sum(ranks), ranks, P, Q + + +def true(r2): + return [Fr(x, 2) for x in r2] + + +def unit_vectors(n): + return [[Fr(int(i == j)) for j in range(n)] for i in range(n)] diff --git a/research/line-system-saturation/requirements.txt b/research/line-system-saturation/requirements.txt new file mode 100644 index 0000000000..7e60c8625a --- /dev/null +++ b/research/line-system-saturation/requirements.txt @@ -0,0 +1 @@ +# Tested set (Python 3.14.7). Standard library only (fractions, itertools, random, hashlib, json); no C compiler, no SAT solver, no third-party packages. diff --git a/research/line-system-saturation/scripts/provenance.py b/research/line-system-saturation/scripts/provenance.py new file mode 100644 index 0000000000..d12d3d802a --- /dev/null +++ b/research/line-system-saturation/scripts/provenance.py @@ -0,0 +1,45 @@ +"""Step 0 of verify.sh (standard library only). Prints one PASS/FAIL line per group and exits 1 on any mismatch. + 0.manifest every file of the package matches MANIFEST.sha256 (sha256), none is missing and none is extra + 0.sources every pin of SOURCES.json holds: the sha256 of each pinned package file (checks and libraries) +usage: python3 provenance.py [--write-manifest] (--write-manifest is for the authors: it rewrites MANIFEST.sha256; SOURCES.json is written by --write-sources)""" +import hashlib, json, os, sys +PKG = os.path.dirname(os.path.dirname(os.path.abspath(__file__))) +bad = 0 +def sha(b): return hashlib.sha256(b).hexdigest() +def listing(): + out = [] + for d, ds, fs in os.walk(PKG): + ds[:] = sorted(x for x in ds if x != '__pycache__' and x != 'bin') + for f in fs: + if f.endswith('.pyc') or f == '.DS_Store': continue + p = os.path.relpath(os.path.join(d, f), PKG) + if p != 'MANIFEST.sha256': out.append(p) + return sorted(out) +if '--write-manifest' in sys.argv: + with open(os.path.join(PKG, 'MANIFEST.sha256'), 'w') as h: + for p in listing(): h.write('%s %s\n' % (sha(open(os.path.join(PKG, p), 'rb').read()), p)) + print('wrote MANIFEST.sha256 (%d files)' % len(listing())); sys.exit(0) +if '--write-sources' in sys.argv: + S = json.load(open(os.path.join(PKG, 'SOURCES.json'))) + for sect in ('files', 'evidence'): + for e in S[sect]: e['sha256'] = sha(open(os.path.join(PKG, e['file']), 'rb').read()) + json.dump(S, open(os.path.join(PKG, 'SOURCES.json'), 'w'), indent=1); print('wrote SOURCES.json pins'); sys.exit(0) +want = {} +for l in open(os.path.join(PKG, 'MANIFEST.sha256'), encoding='utf-8'): + h, p = l.rstrip('\n').split(' ', 1); want[p] = h +have = listing(); n = 0 +for p in sorted(set(want) | set(have)): + f = os.path.join(PKG, p) + if p not in want: print('FAIL 0.manifest extra file not in MANIFEST: ' + p); bad += 1 + elif not os.path.isfile(f): print('FAIL 0.manifest missing file: ' + p); bad += 1 + elif sha(open(f, 'rb').read()) != want[p]: print('FAIL 0.manifest hash differs: ' + p); bad += 1 + else: n += 1 +print('%s 0.manifest %d files match MANIFEST.sha256' % ('PASS' if not bad else 'FAIL', n), flush=True) +S = json.load(open(os.path.join(PKG, 'SOURCES.json'))); b0 = bad; k = 0 +for sect in ('files', 'evidence'): + for e in S[sect]: + k += 1; f = os.path.join(PKG, e['file']) + if not os.path.isfile(f): print('FAIL 0.sources missing ' + e['file']); bad += 1 + elif sha(open(f, 'rb').read()) != e['sha256']: print('FAIL 0.sources pin differs: ' + e['file']); bad += 1 +print('%s 0.sources %d pins hold (checks and libraries)' % ('PASS' if bad == b0 else 'FAIL', k - (bad - b0)), flush=True) +sys.exit(1 if bad else 0) diff --git a/research/line-system-saturation/verify.sh b/research/line-system-saturation/verify.sh new file mode 100755 index 0000000000..b3b8dc475f --- /dev/null +++ b/research/line-system-saturation/verify.sh @@ -0,0 +1,47 @@ +#!/usr/bin/env bash +# One-command verification of research/line-system-saturation (no kappa claim): exact recomputation of every CHECKED item of the README -- the polar law and Lagrangian bound on +# D4/D5/D6/E6/E7/E8, generic-frame and blow-up constructions (c1), Theorem E ingredients and Theorem F constants (c2), the two-point point X* for k' = 2..5 (c3), descent +# E7 -> D6 -> D4 and E8 -> E7 (c4), the 14 saved L1_CONS systems (c5), the exact Delsarte dual certificate, uniform-angle arithmetic and signed-graph counts (c6). +# Step 0 checks the package against MANIFEST.sha256 and every pin of SOURCES.json and stops hard on any mismatch. +# Single process at a time; standard library Python only (no compiler, no SAT, no third-party packages). +# usage: bash verify.sh [/fresh/output/dir] (inside a checkout or `git archive` of the repository; nothing outside the package is read) +# set PYTHON to choose the interpreter (>= 3.11). +set -uo pipefail +HERE=$(cd "$(dirname "$0")" && pwd) +OUT=${1:-$(mktemp -d "${TMPDIR:-/tmp}/line-system-saturation.XXXXXX")} +mkdir -p "$OUT"; OUT=$(cd "$OUT" && pwd) +case "$OUT/" in "$HERE"/*) echo "output dir must be outside the package"; exit 2;; esac +PY="${PYTHON:-python3}"; export PYTHONDONTWRITEBYTECODE=1 +T0=$SECONDS +echo "== 0. manifest and provenance" +if ! "$PY" -B "$HERE/scripts/provenance.py" | tee "$OUT/provenance.log" | sed 's/^/ /' || grep -q '^FAIL' "$OUT/provenance.log" \ + || [ "$(grep -c '^PASS' "$OUT/provenance.log")" != 2 ]; then + echo "step 0 FAILED"; exit 1 +fi +mkdir -p "$OUT/logs"; : > "$OUT/all.out" +run() { # name script + echo "== $1" + ( cd "$HERE/checks" && "$PY" -B "$2" > "$OUT/logs/$2.log" 2>&1; echo $? > "$OUT/logs/$2.rc" ) + cat "$OUT/logs/$2.log" | tee -a "$OUT/all.out" | sed 's/^/ /' + [ "$(cat "$OUT/logs/$2.rc")" = 0 ] || echo "FAIL $2 exit status $(cat "$OUT/logs/$2.rc")" | tee -a "$OUT/all.out" +} +run "c1 polar law, Lagrangian bound, ladder, generic frames, blow-up" c1_ladder.py +run "c2 norm-2 systems (Theorem E ingredients) and Theorem F constants" c2_normtwo.py +run "c3 two-point point X* on W(2k-1,2), k = 2..5" c3_twopoint.py +run "c4 descent E7 -> D6 -> D4, E8 -> E7" c4_descent.py +run "c5 the 14 saved L1_CONS systems" c5_l1cons.py +run "c6 Delsarte dual certificate, uniform-angle arithmetic, signed-graph counts" c6_delsarte_uniform.py +run "c7 PR2222 rank correction: binom(50,3) = 19,600 < 19,648 (maintainer review)" pr2222_rank.py + +# Optional (needs GAP >= 4.12 with ctbllib; set GAP=/path/to/gap): smallest nontrivial degree of Sp(10,2) = 155 +if [ -n "${GAP:-}" ]; then + echo "== c8 GAP: smallest nontrivial ordinary degrees of Sp(6,2), Sp(8,2), Sp(10,2) = 7, 35, 155 (ATLAS tables)" + "$GAP" -q -b "$HERE/checks/gap/sp2n2_degrees.g" < /dev/null 2>&1 | grep -v '^#I' | tee "$OUT/logs/gap.log" | sed 's/^/ /' + grep -q GAP_SP2N2_PASS "$OUT/logs/gap.log" || { echo "FAIL c8 GAP check"; exit 1; } +else + echo "== c8 GAP check SKIPPED (set GAP=/path/to/gap); recorded output: checks/gap/sp2n2_degrees.out.txt" +fi +EXPECT=87 +npass=$(grep -c '^PASS' "$OUT/all.out"); nfail=$(grep -c '^FAIL' "$OUT/all.out") +echo "== done in $((SECONDS - T0))s: $((npass + 2)) PASS (2 in step 0), $nfail FAIL; logs in $OUT" +[ "$nfail" = 0 ] && [ "$npass" = "$EXPECT" ] || { echo "FAILED: expected $EXPECT assertion PASS lines after step 0, got $npass PASS / $nfail FAIL"; exit 1; }