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+f2167317cc05f47f653db1c9db352bb2515515c29a8e107d24620ef6ca406da8 sat/proofs/rob_P112_6.drat.xz +9240979ae8ae6d74467346778252c0f2b293f583582d62dfa9acfc42a98928c2 sat/proofs/rob_P112_7.drat.xz +6cb74ebe7ce2c62011c346b13f11a3ccb8ba3456e1b8dc4ce8a4486a70f0b9af sat/results.jsonl +9484d88f7b57a4b2f723beb4f91a61e183df36dc54c52646de657706723ab4c1 sat/run.py +a4cd33cef93721cd79ae3dfaf69556f8502b22152e363fcab33b1fa402d68d11 scripts/provenance.py +5dca383270dc980994e79492a6492e716e916d3e0a67a808c988264bf116b001 verify.sh diff --git a/research/overcomplete-and-realisation-floors/NOTICE.md b/research/overcomplete-and-realisation-floors/NOTICE.md new file mode 100644 index 0000000000..b281516237 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/NOTICE.md @@ -0,0 +1,17 @@ +# Notice + +Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0 (`LICENSE`, copied from the repository root). + +## Credits + +- **#446** (cost model, Theorem C), **#595**, **#1590**, **#2111**, **#2125**, **#2162**, **#2228**: the model, the dense pairing, the polar law and the root-code floors that this package extends; **#2228** also stated Conjecture ID, whose branch (a) is corrected here. +- **#968**, **#1077**, **#1180**, **#1212**, **#1236**, **#1744**: the E8 / P112 / P104 floors for 8 terms, for factors in the live root code, rigidity, and the 9-term decoder outside the root code; **#2176**, **#2206**: forced minors; **#2244**: the Lagrangian bound (same model). +- **#1331** (Arham-Tahir64), **#1375** (commdata2338), **#1460** (bbMisk), **#1468** (bitplane), **#1591** (Arham-Tahir64): the "free rectangle" mechanism (zero-cost corrections supported on zero-pairing rectangles; scoped P112 results). Proposition 3.1 is the same move applied to whole units; what is new here is its use against Conjecture ID (a). +- **#1224** (joaqcoo), **#1633** (bbMisk), **#1815** (jmbarrancoml): the E8 / P112 overcomplete and outside-the-root-code structure and the forced-Gram mechanism that Lemma OC and Lemma 7.1 are close to; credited in README §4, §7, §8. +- **#2137**, **#2232**, **#2240**, **#2247** (1337isnot1337): independent exact E6 screens; #2137 found the one-dimensional kernel of the orthogonal-zero constraints independently. +- Tools: kissat 4.0.4 (SAT); `drat-trim` (Marijn Heule and Nathan Wetzler; upstream source vendored in `research/p111-minrank-proof-reconciliation/vendor/drat-trim/`); Lean 4 + Mathlib (`lean/`). The Shadow Lemma is Lemma CR2 / `hadamard_diag_rank_sum` of #2162 (`lean/DecoderDuality.lean`; rectangular Lean form in #2267), cited, not shipped. +- No file is copied from another contributor. `lib/report.py` and `scripts/provenance.py` follow the author's own package `research/line-system-saturation`. + +## New in this package + +Everything under `research/overcomplete-and-realisation-floors/`: `README.md`, `verify.sh`, `SOURCES.json`, `MANIFEST.sha256`, `requirements.txt`, `lib/*.py`, `checks/*.py`, `sat/*.py`, `scripts/provenance.py`, `lean/*`, and the evidence files `data/*.json`, `sat/results.jsonl`, `sat/joint_results.jsonl`, `sat/cnf/`, `sat/models/`, `sat/proofs/` (all produced by the generators in this package; the decoder families of `data/` were exported from the author's working cards and are re-verified by the checks: `c6.decoders` recomputes the exactness of the 26 E6 square decoders). diff --git a/research/overcomplete-and-realisation-floors/README.md b/research/overcomplete-and-realisation-floors/README.md new file mode 100644 index 0000000000..02660b10d4 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/README.md @@ -0,0 +1,190 @@ +# Dense-pairing floors for overcomplete root-code decoders (E6 36, D5 20, every m) and for every rank-T realisation (E8 72, E7 49, D6 30, D4 12, P104 56), with a correction to Conjecture ID (a) (no κ claim) + +> **Scope, stated first.** This package is about the **algebraic cost** `Σ_j rank_F2 B[R_j, A_j]` of exact decoders of root units (and punctured root units) in the model of #446 / #595 / #1590 / #2111 / #2125 / #2228. It proves lower bounds **in stated classes** and one **correction** to a conjecture of ours. It contains **no κ claim, no window closed, no new supplier, no copy / hosting / restoration cost**, and it says nothing about decoders whose factors leave the column space of the realisation, about realisations of rank above T, or (except where a SAT statement says so) about pairings B other than the dense one. Every statement carries one of the labels below; every scope is spelled out in §2. +> +> **Labels.** **PROVEN** (full proof in this README); **CHECKED** (recomputed by `verify.sh`, exact integer / `Fraction` / bit arithmetic, or SAT with a DRAT proof that `drat-trim` verifies); **RECORDED** (a number of another PR or of the author's working notes, or a solver verdict whose proof is not shipped, not re-run here; for SAT items, "RECORDED: DRAT-verified at generation" means `verify.sh` regenerates the CNFs and checks their sha256 hashes and the recorded verdicts, while the DRAT proofs themselves are regenerable with `sat/run.py solve` but are not shipped); **OPEN**. +> +> **Correction (read this first).** Conjecture ID of #2228 §6, branch (a), says that for an *odd geometry* (an integral normalisation with unit diagonal exists) the optimal cost is `v`. **That branch is false when the number of terms is unbounded**: the 120-line unit "all lines pairwise orthogonal" with `M = I` has an exact decoder of cost **72 < 120** (§3, **PROVEN + CHECKED**). The values of Conjecture ID for the nine even root units are untouched, and #2228's Theorem 1 and Corollary 2.2 are not contradicted (their hypothesis is that all lines survive); but "odd geometry ⇒ cost* = v" and the framing of "geometry" as an invariant of the cost are wrong in the any-number-of-terms model. §3 gives the repaired statement `cost*(U) = min over admissible B of c(B)`. + +## 0. Reading this package + +```sh +bash research/overcomplete-and-realisation-floors/verify.sh /fresh/output/dir # from a checkout or `git archive`; Python >= 3.11, standard library only +DRAT_TRIM=/path/to/drat-trim bash research/overcomplete-and-realisation-floors/verify.sh /fresh/output/dir # additionally re-checks the shipped DRAT proofs +``` + +`verify.sh` first checks every file against `MANIFEST.sha256` and every pin of `SOURCES.json` (stops with exit 1 on mismatch), then runs six check scripts that print one `PASS` / `FAIL` line per assertion (`SKIP` for the DRAT re-check when `drat-trim` is neither on `PATH`, nor in `$DRAT_TRIM`, nor buildable from the vendored `research/p111-minrank-proof-reconciliation/vendor/drat-trim/drat-trim.c` with `cc`). One core, a few minutes (57 PASS lines including step 0's two, with `drat-trim`). No solver is needed for verification; `kissat` is only needed to *re-solve* (optional, `KISSAT_RESOLVE=n`). + +| script | what it recomputes | +|---|---| +| `checks/c1_overcomplete_d5.py` | Lemma OC on 120 exact random decompositions; D5: deficient directions, pair ranks 3 / 4, the combinatorial core of the D5 theorem | +| `checks/c2_free_terms_frame120.py` | the 6,728-term exact decoder of the 120-line frame unit, `M = I`, cost 72 | +| `checks/c3_realisation_free_floors.py` | frame-intersection certificate on D4, D5, D6, E6, E7, E8, P104, P112, E7 minus a 7-frame (all 2^h − 1 label classes of each); P104 / P112 pair-rank tables | +| `checks/c4_e6_overcomplete.py` | E6: 27 deficient directions, 432 + 297 pair ranks, 1-dimensional bilinear-form space, transitivity of W(E6) on the 432 rank-5 pairs, exhaustive search over **5,504,660** seven-sets | +| `checks/c5_known_decoders.py` | the decoders that attain the floors: E8 72, E7 49, D6 30, D4 12, P112 64, P104 56 (and the trivial decoders for E6 / D5) | +| `checks/c6_sat_replay.py` | regenerates all 1,473 SAT instances, compares DIMACS hashes, checks SAT models clause by clause, re-checks the shipped DRAT proofs | + +**Prior work this builds on.** The model, the dense pairing and the root-code floor: **#446** (cost model, Theorem C), **#595 / #1590** (merged; two lemmas and an erratum), **#2111**, **#2125** (T² threshold, E7 first gain), **#2162** (T = 3 side), **#2228** (the dense polar law, Theorem 4 root-code floors, **Conjecture ID**, which this package corrects in one branch), **#2244** (the Lagrangian bound for line systems; unrelated to the statements here but the same model). The E8 / P112 / P104 floors for exactly 8 terms or for factors in the live root code: **#968** (merged, frame-pair decoders), **#1077**, **#1180** (merged; overcomplete factor-linear E8 decoders retain the puncture floor, and the 9-term example *outside* the root code that marks the boundary of this package), **#1224** (merged, joaqcoo: for E8 with live roots spanning `R^8`, factors split as `D = RA + N₁`, `T = RB + N₂` with exactness `ABᵀ = 2I`, `N₁Bᵀ = 0`, …; nine-term decoders are one-sided overcomplete representations), **#1633** (merged, bbMisk: for exact overcomplete E8 decoders the ranks of the two factors outside the live root code satisfy `s + t ≤ m − 8`), **#1212** (merged; rank-8 decoders are gauge-rigid), **#1236**, **#1744**, **#1815** (merged; unit diagonals and required zeros force `N = R Rᵀ`, `B = U Uᵀ` for P112 — the same mechanism as our Lemma 7.1 for E6 / D5). Forced minors: **#2176**, **#2206**. Independent E6 screens: **#2137** (1337isnot1337: the orthogonal-zero constraints on the 36 coordinates of a bilinear form `L` have rank 35, kernel `Q·G`; this is our §7 check `c4.nspace` / `c1.d5nspace`, found independently), **#2232**, **#2240** (one-extra frame screen, minimum 40 over its alphabet), **#2247** (sheared one-extra screen, 330,486 support cells, minimum 37 and none below 36): these are RECORDED, scoped to their finite families, and agree with §7 below, which closes the E6 seven-term question **for the whole root-code class**. Swarm discussion of the minrank / shadow viewpoint: #2232, #2240, #2247. **Free terms are not new.** Zero-cost corrections supported on zero-pairing rectangles ("free rectangles") are the mechanism of **#1331** (Arham-Tahir64: one free P112 rectangle cannot improve a coordinate COPY term; sharp whole-decoder floor 64) and **#1375** (commdata2338: two-coordinate mixing with small free rectangles keeps COPY 64), followed by **#1460** (bbMisk: pair (4,6), arbitrary free rectangles), **#1468** (bitplane: arbitrary one-column corrections) and **#1591** (Arham-Tahir64: all nine-factor singleton completions, COPY ≥ 64); all are scoped negative results for P112. Proposition 3.1 is the same move applied to whole units; what is new here is its use against Conjecture ID (a). The Shadow Lemma is Lemma CR2 / `hadamard_diag_rank_sum` of #2162 (`lean/DecoderDuality.lean`), with a rectangular Lean form in #2267; only `Overcomplete.lean` is shipped (§4). The repository helper `scripts/find_overlap.py --refresh` was run before submission (it also searches discussions: it surfaced discussion 1316, the free-rectangle line of #1331, and discussion 1622 = #1633, already cited; nothing else new). + +## 1. Setting and notation + +* A **unit** `U` is a list of `v` pairwise non-proportional vectors `r_t ∈ Q^n` ("lines"); `t ⊥ s` means `r_t · r_s = 0`. `V := span{r_t}`, `T := dim V`. All units here are root systems or punctures of E8: `D4, D5, D6, E6, E7, E8` and `P112 = E8 minus an 8-frame`, `P104 = E8 minus two 8-frames`, `E7-frame = E7 minus a 7-frame`; `lib/roots.py` builds them (E8 in doubled integer coordinates, E7 / E6 as orthogonal complements of one resp. two roots, D_n from `±e_i ± e_j`). +* A **realisation** of `U` over a field `K ⊇ Q` is a matrix `M ∈ K^{v×v}` with `M_tt ≠ 0` and `M_ts = 0` whenever `t ⊥ s` (other entries arbitrary, possibly 0). A **decoder** is `M = Σ_{j=1..m} p_j q_jᵀ` (`p_j, q_j ∈ K^v`) together with an **admissible pairing** `B ∈ F2^{v×v}`: `B_tt = 1`, and `B_ts = 1` (`t ≠ s`) only if `t ⊥ s`. Then `B̃ ∘ M = diag(M_tt)` holds automatically. `R_j = supp p_j`, `A_j = supp q_j`, and + **cost = Σ_j rank_{F2} B[R_j, A_j]**. + The factors are *unrestricted* unless a theorem says otherwise. Terms with `p_j = 0` or `q_j = 0` contribute nothing and are dropped. +* The **dense pairing** is `B_d := J + Ā`, i.e. `B_ts = 1` iff `t = s` or `t ⊥ s`; it depends on the orthogonality pattern only. `h := rank_F2 B_d` (D4 3, D5 5, D6 5, E6 7, E7 7, E8 9, P104 9, P112 9, E7-frame 7). It is symmetric. +* **Label classes.** Let `C := col(B_d) ⊆ F2^v` (dimension `h`). A **class** is the support of a nonzero vector `w ∈ C` (`2^h − 1` of them, counted by vector). For `R ⊆ [v]` put `Def(R) := {w ∈ C : supp w ∩ R = ∅}` (a subspace of `C`) and `δ(R) := dim Def(R)`. In the notation of #2228, the classes are the sets `O_z = {t : Ω(u_t, z) = 1}`. +* **Root-code decoders** (the class of #2228 Theorem 4). For a unit with line matrix `R ∈ Q^{v×n}`: `p_j = D₁ R x_j`, `q_j = D₂ R y_j` with `x_j, y_j ∈ V_K` and `D₁, D₂` invertible diagonal. Then `R(x) := supp(D₁Rx) = {t : r_t · x ≠ 0}`. +* **Frames.** A **T-frame** is a set of `T` pairwise orthogonal lines; `α_⊥` is the size of a largest orthogonal set. If `U` has a `T`-frame `F`, every realisation `M` has `M[F,F]` diagonal and invertible, so **every realisation has rank ≥ α_⊥** (PROVEN, one line). + +## 2. Results at a glance + +| # | statement | label | exact scope | where | +|---|---|---|---|---| +| ID-a | Conjecture ID branch (a) (odd geometry ⇒ cost* = v) is false: the 120-line frame unit has an exact decoder of cost 72 | **PROVEN + CHECKED** | model with an unbounded number of terms; `M = I`; `B = J + Ā_E8`; 6,728 terms | §3, `c2` | +| Mono | `orth(U') ⊆ orth(U)` on the same lines ⇒ `cost*(U) ≤ cost*(U')`; and `cost*(U) = min over admissible B of c(B)`, `c(B)` = cost of the unit whose orthogonality relation is `supp B` | **PROVEN** | any number of terms, any field of characteristic 0 | §3 | +| OC | `Σ_{j≤m} x_j y_jᵀ = id_V` ⇔ `K = (⟨y_i,x_j⟩)` is an idempotent of rank `T`; `m = T` ⇒ biorthogonal; `m = T+1` ⇒ `I − K` has rank one, off-diagonal of `K` is rank one; criterion for prescribed directions | **PROVEN** (idempotency, traces, square case: **Lean**, no `sorry`); spot-checked on 120 exact instances (`c1`) | any field of characteristic 0; factors in `V` | §4 | +| E6 | every root-code decoder of E6 (scaled-Gram realisation), dense `B`, **every** number of terms `m`, costs **≥ 36** | **PROVEN** from **CHECKED** data (`c4`: 5,504,660 seven-sets, exact integers) | factors in `col M`; realisations `D₁ G D₂`; dense `B`; char 0 | §7 | +| D5 | same for D5: **≥ 20** | **PROVEN** from **CHECKED** data (`c1`) | same | §7 | +| 4′ | Theorem 4′: the root-code floor `cost ≥ Σ_j (h − δ(R_j) − δ(A_j))` holds for **every realisation** `M` of rank `T`, deficiency being a property of the label class in that realisation | **PROVEN** | col P ⊆ col M, col Q ⊆ row M (all square decoders; any `m` otherwise); dense `B`; rank of `M` exactly `T` | §5 | +| 3.3 | frame-intersection certificate: a class is non-deficient in **every** rank-`T` realisation if it contains a `T`-frame, or (clause (c)) if a `T`-frame contains the parts `F_i ∖ O` of frames with empty common intersection | **PROVEN** | rank-`T` realisations of units with a `T`-frame | §5 | +| 3.4 | for every rank-`T` realisation and every decoder with col P ⊆ col M, col Q ⊆ row M, dense `B`: **E8 ≥ 72, E7 ≥ 49, D6 ≥ 30, D4 ≥ 12, P104 ≥ 56, P112 ≥ 56**; E8, E7, D6, D4, P104 are attained (P112: 64 attained, 56 proven here), see `c5` | **PROVEN** from **CHECKED** tables (`c3`) | as in 4′ | §5 | +| – | E6 / D5 have no `T`-frame (`α_⊥ = 4`), so §5 gives nothing; the realisation-free statement for E6 / D5 is open | **OPEN** | | §8 | +| SAT-1 | `min over admissible B of rank_F2 B` on E6 equals **7** (UNSAT at 5 and 6, each under three symmetry-breaking cliques; SAT at 7) | **CHECKED** (kissat 4.0.4 + DRAT, drat-trim VERIFIED) | all 36 lines | §9.1 | +| SAT-2 | char-2 shadow minranks: `mr₂(Γ_E6) ≥ 7`, `mr₂(Γ_E7) ≥ 8` (UNSAT at 6 resp. 7, two cliques each) ⇒ a B-independent F2 shadow cannot certify all lines with `T` terms | **RECORDED**: DRAT-verified at generation (E7 part: shipped proof, **CHECKED**); `verify.sh` checks the CNF hashes and recorded verdicts for the E6 part, whose proofs are regenerable, not shipped ⇒ **PROVEN** consequence given those | over `F2` only | §9.2 | +| SAT-3 | per-term drops are real: an E6 deficient pair of dense rank 6 has fitting minrank 4; E8 root blocks: 9, 9, ≤ 8 | **CHECKED** (DRAT / model) | single blocks, any `B` | §9.3 | +| SAT-4 | the known optimal decoders are `B`-robust: every term of the E8 frame decoder (8), E8 coordinate decoder (terms 0, 7), E7 frame decoder (7), P112 removed-frame decoder (8), P104 frame-pair decoder (8) has fitting minrank equal to its dense rank (UNSAT at dense − 1) | **CHECKED** (DRAT) | those specific terms, any `B` | §9.4 | +| SAT-5 | a shared `B` never brings any of **26** E6 square decoders below 36: **1,262** instances (every cap vector with `c_j ≥ fr_j`, `Σ c_j = 35`), all UNSAT, plus 156 floor instances | **RECORDED**: DRAT-verified at generation; `verify.sh` checks CNF hashes and recorded verdicts (proofs regenerable with `sat/run.py solve`, not shipped); decoder exactness is **CHECKED** (`c6.decoders`) | those 26 decoders (`data/e6_square_decoders.json`), every admissible `B` | §9.5 | + +**What is *not* proven here.** Arbitrary admissible `B` for a whole unit (only SAT-1..5 above); decoders with `col P ⊄ col M` (e.g. the 9-term example of #1180); realisations of rank `> T`; the E6 / D5 statements for realisations other than `D₁ G D₂`; P112 `≥ 64` for every realisation; any bound for `T = 3`. + +## 3. A correction to Conjecture ID (a), and the shape of cost* + +Free terms are the point (the "free rectangle" of #1331 / #1375). A term whose block `B[R_j, A_j]` is zero costs nothing, and a single-entry term `c e_t e_sᵀ` is free whenever `B_ts = 0`. + +**Proposition 3.1 (monotonicity) — PROVEN.** If `orth(U') ⊆ orth(U)` on the same `v` lines (patterns of required zeros, possibly directed) then `cost*(U) ≤ cost*(U')`. +*Proof.* Take an optimal decoder `(M', P', Q', B')` of `U'`. Its `B'` is admissible for `U` (its off-diagonal ones lie in `orth(U') ⊆ orth(U)`). For every `(t,s) ∈ orth(U) ∖ orth(U')` add the term `−M'_ts e_t e_sᵀ`; its block is `B'_ts = 0`, so it is free. The new `M` vanishes on `orth(U)`, its diagonal is unchanged. ∎ + +**Proposition 3.2 (reformulation) — PROVEN.** `cost*(U) = min over B admissible for U of c(B)`, where `c(B)` is the optimal cost, with the fixed pairing `B`, of decoders whose realisation `M` has `M_tt ≠ 0` and `M_ts = 0` exactly where required by the pattern `S(B) := {(t,s) : t ≠ s, B_ts = 1}` (a directed pattern if `B` is not symmetric; nothing else is asked of `M`). +*Proof.* "≤": Proposition 3.1 with `U'` the `B`-unit. "≥": take a decoder of `U` with pairing `B`. Every line has `M_tt ≠ 0`, so some term covers `(t,t)`, and its block contains `B_tt = 1`; hence free terms never touch a diagonal entry. The paid terms (nonzero block) sum to a matrix `N` with the same diagonal as `M`, and `N_ts = 0` wherever `B_ts = 1`, `t ≠ s` (there every free term contributes 0 and `B̃ ∘ M = Δ`). So the paid part is a decoder of the `B`-unit of the same cost. ∎ + +**Proposition 3.3 (the 120-line frame unit costs at most 72) — PROVEN + CHECKED (`c2`).** Let `F_120` be the unit of 120 lines, all pairs orthogonal; it admits every `B` with unit diagonal, and `M = I` (unit diagonal, "odd geometry") is a realisation. Take the E8 lines `r_t` (norm 2), `B = J + Ā_{E8}` (dense pairing of E8), and the terms + * `j = 1..8`: `p_j(t) = r_t[j]/2`, `q_j(s) = r_s[j]` (coordinate `j` of the E8 lines in the standard coordinates), whose sum is `½ G` (`G` the E8 Gram matrix); + * one **free term** `p = −(G_ts/2) e_t`, `q = e_s` for each of the 6,720 ordered non-orthogonal E8 pairs `(t,s)`. +Then `Σ p q^T = ½G − ½(G − 2I) = I` exactly (`G_tt = 2`, the free terms cancel the off-diagonal part), `B̃ ∘ I = I`, the 8 coordinate terms have `F2`-rank 9 each (the 78 lines with a nonzero coordinate `j`; computed), and every free term has block `B_ts = 0` (the pair is non-orthogonal in E8), i.e. rank 0. Total cost **72 < 120 = v**. `c2` recomputes the sum `M = I` over 6,728 exact `Fraction` terms and the 8 + 6,720 ranks. ∎ + +*Consequences, stated plainly.* (i) Conjecture ID (a) ("odd geometry ⇒ cost* = v") fails for `F_120`; the "odd geometry" property (an integral normalisation with unit diagonal) is not an invariant of cost* in the model where the number of terms is unbounded. F_120 (and F_104) are degenerate (`T = v`); no counterexample to (a) with `T < v` is known, and the only non-degenerate odd-geometry row of #2228 (Eisenstein, 40) still satisfies (a) by #2017. (ii) The correct way to read the frame unit is Proposition 3.2: the unit allows every `B`, and `c(B_dense(E8)) ≤ 72`. In this reading the search for good units is a search for **admissible pairings `B` with small `c(B)/v`**. (iii) #2228's Theorem 1 is *not* contradicted: it bounds the cost below by the number of lines whose leading 2-adic terms survive, and #2228 Corollary 2.2 derives "cost ≥ v" only for decoders in which all lines survive for some prime and shift; the decoder above has cost 72 < 120, so by Theorem 1 not all lines survive in it. (We have not recomputed the count of surviving lines for this decoder.) (iv) The theorems of §§5–7 are *not* affected: they assume `col P ⊆ col M`, `col Q ⊆ row M`, and a single-entry term `e_t e_sᵀ` lies outside `col M` as soon as `rank M < v`. (v) Scope: the decoder needs more than `T` terms; with the number of terms capped at the rank of the realisation, free terms are not available. The same move gives `cost*(F_104) ≤ 56` from the known P104 decoder; we did not re-run that and do not claim it. + +## 4. Lemma OC (overcomplete coupling) + +*Relation to #1224.* For E8 with `R` the matrix of live roots, #1224 Theorem A contains the identity `A Bᵀ = 2I` for the root-code parts; Lemma OC below is the same identity written as a statement about the cross matrix `K = YᵀX`, which (i) is basis-free, (ii) holds for any unit and any field (it is used here for E6 and D5, where a root basis does not have a dual basis of roots), and (iii) gives the idempotent / rank-one-off-diagonal structure used in §7. It is elementary and we claim nothing deeper. + +**Lemma OC — PROVEN.** Let `V = 𝕜^T` (`𝕜` a field of characteristic 0, or any field), `x_1..x_m ∈ V`, `y_1..y_m ∈ V*`, and `Σ_j x_j ⊗ y_j = id_V` (absorb coefficients into `y_j`). Put `K_ij := ⟨y_i, x_j⟩` (an `m × m` matrix). Then **`K` is idempotent of rank `T`**, so `Π := I_m − K` is idempotent of rank `m − T` and `trace Π = m − T`. Conversely, if `rank(x_j) = rank(y_j) = T` and `K² = K`, then `Σ_j x_j ⊗ y_j = id_V`. +*Proof.* With `X = [x_j]`, `Y = [y_j]` (`T × m`): `XYᵀ = I_T`, so `K = YᵀX` satisfies `K² = Yᵀ(XYᵀ)X = K`, and `rank K = rank X = T` (`Yᵀ` is injective since `XYᵀ = I`). Conversely `K² = K` reads `Yᵀ(XYᵀ − I)X = 0`; `Yᵀ` is injective (rank `T`) and `X` surjective (rank `T`). ∎ +**Lean:** `lean/Overcomplete.lean` proves `oc_idempotent` (`(Yᵀ X)² = Yᵀ X`), `oc_trace` (`tr(Yᵀ X) = T`), `oc_complement_idempotent`, `oc_complement_trace` (`tr(1 − YᵀX) = m − T`) and `oc_square` (`m = T` ⇒ `Yᵀ X = 1`), with `#print axioms` showing only `propext, Classical.choice, Quot.sound`, no `sorry`. The rank statements, the converse and the consequences below are written proofs (not formalised). + +**Consequences — PROVEN.** + (a) `m = T`: `Π = 0`, so `⟨y_i, x_j⟩ = δ_ij` (biorthogonality; this is also `oc_square`). + (b) `m = T + 1`: `Π = u wᵀ` with `wᵀ u = 1` (rank-one idempotent), so `⟨y_i, x_j⟩ = −u_i w_j` for `i ≠ j`: **the cross matrix agrees off the diagonal with the rank-one matrix `−u wᵀ`**, and `⟨y_i, x_i⟩ = 1 − u_i w_i`. + (c) A term with `⟨y_i, x_i⟩ = 0` has `Π_ii = 1`. + (d) (**criterion for prescribed directions**) Let `m = T + 1 ≥ 3`, prescribed lines `x̂_j, ŷ_j` with `rank[x̂_j] = rank[ŷ_j] = T` and `Λ_ij := ⟨ŷ_i, x̂_j⟩ ≠ 0` for `i ≠ j`. There are scalars with `Σ_j (s_j x̂_j) ⊗ (t_j ŷ_j) = id` **iff** `Λ = D + a bᵀ` with `D` diagonal and invertible. + *Proof of (d).* "⇒": by the Lemma, `K = diag(t) Λ diag(s) = I − u wᵀ` for some `u, w` (the scalars `s_j, t_i` multiply `x̂_j, ŷ_i`; and `Π := I − K` is idempotent of rank `m − T = 1`, hence `Π = u wᵀ`), so `Λ = diag(1/t) (I − u wᵀ) diag(1/s)` has the form `D + a bᵀ` with `D = diag(1/(t_i s_i))` invertible. "⇐": given `Λ = D + a bᵀ`, choose `s_i t_i = 1/D_ii`, `u_i = −t_i a_i`, `w_j = s_j b_j`. Then `K := diag(t) Λ diag(s) = I − u wᵀ`. Since `rank Λ ≤ T < m`, `det Λ = 0`, i.e. `det(I + D⁻¹ a bᵀ) = 0`, i.e. `1 + Σ a_i b_i / D_ii = 0`, i.e. `wᵀ u = −Σ s_i t_i a_i b_i = 1`. So `Π = uwᵀ` is idempotent, `K² = K`, and the converse of the Lemma gives `Σ = id`. ∎ + (e) A term-count consequence: at least `T` terms have `x_j ≠ 0 ≠ y_j`. + +## 5. Theorem 4′ and the frame-intersection certificate (realisation-free floors) + +Fix a unit with a `T`-frame and a realisation `M` of rank exactly `T` (so `T = α_⊥` is the minimum possible rank). Write `M = P₀ N Q₀ᵀ` with `P₀, Q₀ ∈ K^{v×T}` bases of `col M`, `col Mᵀ` and `N ∈ GL_T(K)`; `π(t)` (resp. `κ(s)`) is row `t` of `P₀` (row `s` of `Q₀`). Consider decoders with `p_j = P₀ x_j`, `q_j = Q₀ y_j` (`col P ⊆ col M`, `col Q ⊆ row M`); **every square decoder (`m = T`) is of this form** (`rank M = T ≤ rank P ≤ m`). Then `N = Σ_j x_j y_jᵀ` (as `P₀, Q₀` are injective), so at least `T` terms are nonzero. + +**Lemma 5.1 (support-only bound) — PROVEN.** For any `R, A ⊆ [v]`: `rank_F2 B_d[R, A] ≥ h − δ(R) − δ(A)`, and `rank B_d[R,A]` depends only on the two subspaces `Def(R), Def(A) ⊆ C`. +*Proof.* Choose a rank factorisation `B_d = X Yᵀ` with `X, Y ∈ F2^{v×h}` of full column rank; `col X = col Y = C` (`B_d` is symmetric). `ker X_R = {z : X_R z = 0} ≅ {w ∈ C : w|_R = 0} = Def(R)` (via `w = Xz`), so `rank X_R = h − δ(R)`; likewise `rank Y_A = h − δ(A)`. Sylvester: `rank(X_R Y_Aᵀ) ≥ rank X_R + rank Y_A − h`. If `X_R` and `X_{R'}` have the same kernel they have the same row space `W`, and `X_R = S W` with `S` of full column rank, so `rank(X_R Y_Aᵀ) = rank(W Y_Aᵀ)`; the same on the other side. ∎ + +**Theorem 4′ — PROVEN.** In the setting above and with the dense pairing `B_d`, + **cost ≥ Σ_{j : x_j,y_j ≠ 0} ( h − δ(R(x_j)) − δ(A(y_j)) ),** where `R(x) = {t : π(t)·x ≠ 0}`, `A(y) = {s : κ(s)·y ≠ 0}`. +Call a class `O` **π-deficient** if `π(O)` lies in a hyperplane of `K^T` (**κ-deficient** analogously). If `w ∈ Def(R(x))` then `supp w ∩ R(x) = ∅`, i.e. `π(supp w) ⊆ x^⊥`: the class `supp w` is π-deficient. **If no class is π-deficient or κ-deficient, every term costs ≥ h and cost ≥ T·h.** +*Proof.* Lemma 5.1 and the count of nonzero terms (§5, last sentence of the setting). ∎ +This is #2228 Theorem 4 (which already allows `M = D₁ R N Rᵀ D₂` with arbitrary `N`, for factors in the scaled root code) extended to rank-`T` realisations whose column space is not a scaled root code (`col M ≠ D₁·RV`); in the Gram case `π = κ = ` coordinates of the line. + +**Lemma 5.2 (frame-intersection certificate) — PROVEN.** Let `M` be a rank-`T` realisation and `O ⊆ [v]` with `π(O) ⊆ ker c` for a nonzero functional `c`. Then + (a) `O` contains no `T`-frame; + (b) for every `T`-frame `F`, `c ∈ span σ(F ∖ O)`, where `σ(s) := N κ(s)ᵀ`; + (c) if a `T`-frame `E` contains the sets `F_1∖O, …, F_k∖O` of `T`-frames `F_i` whose intersection `∩_i (F_i ∖ O)` is empty, then `c = 0`, a contradiction. +Hence **a class that contains a `T`-frame, or satisfies (c), is not π-deficient in any rank-`T` realisation.** Replacing `M` by `Mᵀ` (a realisation of the same pattern) gives the same for κ. +*Proof.* For a `T`-frame `F`, `M[F,F] = P₀[F] N Q₀[F]ᵀ` is diagonal (all pairs orthogonal) with nonzero diagonal, hence invertible, so `{π(f)}_{f∈F}` is a basis of `K^T`, and `π(f)·σ(f') = 0` for `f ≠ f'`, `≠ 0` for `f = f'`: `{σ(f)}` is the dual basis up to scalars, and **σ(f) depends on the line `f` only, not on the frame**. (a): `F ⊆ O` would give `c ⟂ π(F)`, a basis. (b): expand `c = Σ_{f∈F} λ_f σ(f)`; then `π(f')·c = λ_{f'} π(f')·σ(f')`, which is 0 for `f' ∈ F ∩ O`. (c): by (b), `c ∈ ∩_i span σ(F_i∖O)`; all `F_i ∖ O ⊆ E` and `σ(E)` is a basis, so these spans are coordinate subspaces and their intersection is `span σ(∩_i F_i∖O) = 0`. ∎ + +**Corollary 5.3 (realisation-free floors) — PROVEN from the CHECKED tables of `c3`.** (`c3` enumerates, for each unit, all maximal cliques of the orthogonality graph (frames = cliques of size `T`), all `2^h − 1` classes, and decides for each class: contains a `T`-frame / certified by (c) / uncertified.) + +| unit | v | T | h | α_⊥ | T-frames | classes (2^h−1) | contain a T-frame | certified by (c) | uncertified | 2-dim subspace of uncertified? | +|---|---|---|---|---|---|---|---|---|---|---| +| D4 | 12 | 4 | 3 | 4 | 3 | 7 | 7 | – | 0 | – | +| D6 | 30 | 6 | 5 | 6 | 15 | 31 | 16 | 15 | 0 | – | +| E7 | 63 | 7 | 7 | 7 | 135 | 127 | 64 | 63 | 0 | – | +| E8 | 120 | 8 | 9 | 8 | 2025 | 511 | 391 | 120 | 0 | – | +| P104 | 104 | 8 | 9 | 8 | 759 | 511 | 375 | 104 | 32 (size 52) | no | +| P112 | 112 | 8 | 9 | 8 | 1260 | 511 | 383 | 112 | 16 (size 56) | no | +| E7-frame | 56 | 7 | 7 | 7 | 64 | 127 | 57 | 56 | 14 (sizes 24, 32) | yes | +| E6 | 36 | 6 | 7 | 4 | 0 | 127 | 0 | 0 | 127 | yes | +| D5 | 20 | 5 | 5 | 4 | 0 | 31 | 0 | 0 | 31 | yes | + +(The classes without a `T`-frame are exactly the neighbourhoods `N(α) = {t ≠ α : r_t · r_α ≠ 0}` — `c3.N.*`; in E8 every frame not through `α` meets `N(α)` in exactly 4 lines, which is why clause (c) is needed — `c3.e8meet`.) Consequently, for **every realisation of rank `T`** and **every decoder with `col P ⊆ col M`, `col Q ⊆ row M`** (all square decoders, any number of terms otherwise), dense `B`: + * **E8 ≥ 72, E7 ≥ 49, D6 ≥ 30, D4 ≥ 12** (no class is deficient, every term has rank `h`, `T` terms at least); + * **P104 ≥ 56** and **P112 ≥ 56**: the uncertified classes contain no 2-dimensional subspace, so `δ(R), δ(A) ≤ 1` for every term (a `Def(R)` consists of π-deficient, hence uncertified, classes and is a subspace), each uncertified class has deficiency exactly 1 (`c3.pairs.*`), and the rank of a term depends only on `(Def(R), Def(A))` (Lemma 5.1); over the `(uncertified or none)²` pairs the computed ranks are `{7: 640, 8: 448, 9: 1}` (P104) and `{7: 192, 8: 96, 9: 1}` (P112), minimum 7, so `8 · 7 = 56`. +**Sharpness (CHECKED, `c5`).** The decoders of `c5` attain E8 72, E7 49, D6 30, D4 12 (frame decoders with `M = G`), P112 64 (removed-frame decoder) and P104 56 (the eight frame-pair terms of #968; `M = R N Rᵀ` with `N` not the Gram form). For P112 the gap between the realisation-free 56 and the attained 64 is **OPEN** (§8); in the Gram realisation 64 is #2228 Theorem 4 (RECORDED, not re-run). The E6 and D5 values are covered in §7. + +*How this relates to earlier work.* For the E8 / P112 / P104 values with factors in the live root code, the statement is #1180 and #2228 Theorem 4; for exactly 8 terms, #968 / #1077 / #1212. What Theorem 4′ adds to #2228 Theorem 4 is the case `col M ≠ D₁·(root code)`: Lemma 5.2 shows that *no realisation of rank `T`* can make a class deficient, so the floors hold for every rank-`T` realisation (the hypothesis `col P ⊆ col M`, `col Q ⊆ row M` remains). We do not claim priority for the E8 value itself. + +## 6. The cap on what is proven: realisations of rank above T + +If `rank M = ρ > T`, a `T`-frame no longer spans `K^ρ` and Lemma 5.2 fails. Theorem 4′ still gives `cost ≥ Σ_j (h − δ_j − δ'_j)` with at least `ρ` terms, but `δ` is no longer controlled by frames. This is **OPEN** (§8). + +## 7. E6 and D5 for every number of terms (scaled-Gram realisations) + +**The class.** `p_j = D₁ R x_j`, `q_j = D₂ R y_j`, `x_j, y_j ∈ V_K`, any `m`, dense `B`. Then `M = D₁ R N Rᵀ D₂` with `N = Σ x_j y_jᵀ ∈ V ⊗ V`. + +**Lemma 7.1 — CHECKED (`c4.nspace`, `c1.d5nspace`).** For E6 and for D5 the space `𝒩 = {N ∈ V ⊗ V : r_tᵀ N r_s = 0 for all orthogonal t, s}` is **one-dimensional**, spanned by the metric (identity in orthonormal coordinates). (Exact rational rank of the constraint system on the `T²` unknowns; #2137 computes the same kernel for E6 in its coordinates.) Hence `N = c·Id_V`, `c ≠ 0` (the diagonal of `M` is nonzero), and after absorbing `c`: **`Σ_j x_j ⊗ y_j = id_V`**, so Lemma OC applies with `T = 6` (E6) resp. `5` (D5), and `m ≥ T`. + +**Lemma 7.2 — CHECKED (`c4.dirs`, `c4.pairranks`, `c4.cross`, `c1.d5dirs`, `c1.d5ranks`).** Over `Q`, the classes of E6 (127) with `rank π(O) < 6` are exactly 27, each with a one-dimensional annihilator; the 27 annihilators are 27 distinct lines (the minuscule weights). Call a direction `x ∈ V_K` **deficient** if it lies on one of them; for a deficient `x`, `δ(R(x)) = 1`, for every other `x`, `δ(R(x)) = 0` (a class sits inside `x^⊥` iff `x` spans its annihilator, and the map class ↦ direction is injective). The same holds on the column side with the same 27 lines (`B_d` is symmetric). Dense-pairing ranks of terms (Lemma 5.1 and the table): both deficient: **5** (432 ordered pairs) or **6** (297 pairs); exactly one deficient: 6; none: 7. **All 27 × 27 cross products `⟨y, x⟩` of deficient directions are nonzero.** For D5 the deficient directions are the five axes `e_1..e_5`; both deficient: rank **4** if matched (`e_a, e_a`), **3** if `a ≠ b` (20 pairs); one deficient: 4; none: 5. + +**Theorem 7.3 (E6 ≥ 36 for every `m`) — PROVEN from CHECKED data (Lemmas 7.1, 7.2 and `c4.transitive`, `c4.nodes`, `c4.noleaf`).** Every root-code decoder of E6 with dense `B` has cost `≥ 36`. +*Proof.* By Lemma 7.1 `Σ x_j y_jᵀ = id`. Suppose a decoder of cost `c₀` exists and take one of cost `≤ c₀` with the fewest terms. Two terms with proportional `x` and proportional `y` have the same supports and can be merged into one term of the same rank (or both vanish), contradicting minimality; so the ordered pairs of lines `([x_j],[y_j])` are pairwise distinct (and terms with a zero factor are absent). Every term costs `≥ 5` (Lemma 7.2). + *`m ≥ 8`*: cost `≥ 40`. + *`m = 6`* (square, `m = T`): by OC(a), `⟨y_i, x_j⟩ = δ_ij`. Let `k` (resp. `l`) be the number of deficient `x_j` (resp. `y_j`). If `y_j` is deficient, every `x_i`, `i ≠ j`, is non-deficient — otherwise `⟨y_j, x_i⟩ = 0` for two deficient directions, contradicting Lemma 7.2. Hence `l ≥ 1` forces `k ≤ 1`, and `k ≥ 2` forces `l = 0`. By Lemma 5.1 each term costs `≥ 7 − [x_j def] − [y_j def]`, so cost `≥ 42 − k − l`. If `k ≥ 2` then `l = 0`, cost `≥ 42 − 6 = 36`; symmetrically for `l ≥ 2`; if `k, l ≤ 1`, cost `≥ 40`. + *`m = 7`*: cost `≤ 35` forces all seven terms to have rank 5 (`7 · 5 = 35`), i.e. seven distinct pairs from the 432 rank-5 pairs. By OC(b) the off-diagonal of `Λ_ij = ⟨y_i, x_j⟩` has rank one (rescaling rows and columns by the nonzero scalars `s_j, t_i` does not change this), and `rank[x_j] = rank[y_j] = 6`. W(E6) is transitive on the 432 rank-5 pairs (`c4.transitive`: the orbit of one pair under the 36 reflections has 432 elements), and it permutes the lines and so preserves everything, so one pair may be fixed. `c4` then enumerates **all** 7-sets containing that pair whose `Λ` has rank-one off-diagonal (exact integer cross-multiplication, no floating point): there are 27,250 triples, 206,275 / 898,460 / 2,502,409 / **5,504,660** sets of sizes 4 / 5 / 6 / 7 (`c4.nodes`; these numbers reproduce the node counts of an independent floating-point search of the working notes, which used `numpy`), and **none** of the 5,504,660 has both its row directions and its column directions spanning `V` — in fact none has `≥ 6` distinct row directions and `≥ 6` distinct column directions (`c4.noleaf`). So no 7-term decomposition exists. + Since `m = 6, 7` and `m ≥ 8` exhaust the possibilities (`m ≥ 6` is needed for `Σ x_j y_jᵀ = id`), cost `≥ 36` in all cases. ∎ + *Remarks.* The value 36 is attained by the trivial decoder (`c5.E6`, `M = I`, a different realisation). (RECORDED, not re-run: the working notes' floating-point search found 5,208 exact 7-term decompositions with deficient `x_j, y_j`, the best of cost 40, consistent with the above; #2240 / #2247 report minimum 40 / 37 for their finite alphabets.) + +**Theorem 7.4 (D5 ≥ 20 for every `m`) — PROVEN from CHECKED data (Lemmas 7.1, 7.2 for D5).** Every root-code decoder of D5 with dense `B` costs `≥ 20`. +*Proof.* Each term has rank `≥ 3`, with equality iff `x_j = e_a`, `y_j = e_b` with `a ≠ b` (both deficient, unmatched); all other terms have rank `≥ 4`. Let `n₃` be the number of rank-3 terms. Cost `≤ 19` needs `3 n₃ + 4(m − n₃) ≤ 19`, i.e. `n₃ ≥ 4m − 19`; with `n₃ ≤ m` this gives `m ≤ 6`, and `m ≥ 5` is forced. *`m = 5`* (square): `n₃ ≥ 1` is impossible, since a rank-3 term has `⟨y_i, x_i⟩ = ⟨e_b, e_a⟩ = 0`, contradicting `⟨y_i, x_i⟩ = 1`. *`m = 6`*: `n₃ ≥ 5`. By OC(b), `Π = u wᵀ`; a rank-3 term `i` has `⟨y_i,x_i⟩ = 0`, so `u_i w_i = 1` (OC(c)), and for two rank-3 terms `i ≠ j`, `⟨y_i, x_j⟩ = −u_i w_j ≠ 0`, i.e. `b_i = a_j`. Three rank-3 terms `i, j, k` then give `a_j = b_i = a_k` and `a_i = b_j = a_k`, so `a_i = b_i`, a contradiction (`c1.d5core` checks that no three pairs `(a_i, b_i)`, `a_i ≠ b_i`, satisfy `b_i = a_j` for all `i ≠ j`); so `n₃ ≤ 2 < 5`. ∎ The value 20 is attained by the trivial decoder (`c5.D5`). An independent counting proof for D5 is in the working notes (RECORDED). + +## 8. Open next questions (ranked by what they would buy) + +1. **Arbitrary `B`, E8.** A per-term census of fitting minranks `fr(R_x, A_y)` over pairs of maximal-flat normals (E8 has 8 orbits of normals; only root × root is done, §9.3) plus a coupling argument (Lemma OC bounds how many terms with `fr = 8` can occur) would give E8 `≥ 72` for root-code decoders and arbitrary `B`. +2. **P112.** Exclude the extra 8 of the 16 uncertified classes in every rank-8 realisation (a stronger certificate than (c), e.g. two-step propagation); this gives P112 `≥ 64` for every rank-8 realisation. +3. **Rank `> T` realisations and factors outside `col M`** (the 9-term decoder of #1180 is the test case): the supports are hyperplane sections of a lifted configuration in `K^{dim col P}` and Lemma 5.2 does not apply. For E8-type units with live roots spanning `R^8`, #1224 and #1633 give structure (one-sided nine-term decoders are overcomplete representations; `s + t ≤ m − 8`), but no cost floor outside the root code is known to us; Lemma OC generalises to idempotents of the lifted configuration, but no cost bound is derived. +4. **E6 / D5 realisation-free.** Neither has a `T`-frame; a rank-5 realisation of E6 would give dense cost `≤ 35`. No such realisation is known; rigidity of the Gram point and global uniqueness at rank `T` are open. (A floating-point search of the working notes found none; not reproduced, not claimed.) +5. **A human proof of `mr₂(Γ̄_E6) = 7`** (SAT-1 is a DRAT certificate, not an argument; a 16-line sub-configuration already needs rank 7 in the working notes' minimisation), and the exact values of `mr₂(Γ_E6)`, `mr₂(Γ_E7)` (only lower bounds 7, 8 are proved), over `F2` and over larger fields of characteristic 2. +6. **A B-dependent proof.** SAT-2 says a B-independent shadow cannot certify E6 / E7 with `T` terms, and SAT-5 says a shared `B` defeats every per-term drop on 26 decoders; an argument that explains the second by using that one `B` serves all terms is missing. +7. **Does a sparser admissible `B` ever beat the dense value for a whole unit?** All evidence here (SAT-1, SAT-4, SAT-5) says no for the nine test units; Proposition 3.2 says this is exactly the question of whether `min_B c(B)` is attained at the dense `B`. + +## 9. SAT evidence (arbitrary pairings) + +Everything in this section is about **all admissible B** (not just the dense one). Standard facts used: (i) `rank_F2 X ≤ r` iff `X = U Vᵀ` with `U, V` having `r` columns; this is encoded with an AND per entry and an XOR chain (`lib/satgen.py`). (ii) **Symmetry breaking** is valid: if `C` is a set of rows/columns with `X[C,C] = I` (a clique of pairwise *non-orthogonal* lines for fitting problems, where `X_tt = 1`, `X_ts = 0`; a clique of pairwise *orthogonal* lines for the `Γ` problems), then `U[C]` has rank `|C|` and a change of basis `U ↦ UA`, `V ↦ VA^{-T}` of `F2^r` makes `U[C] = [I | 0]` (and `V[C] = [I | 0]` when `r = |C|`). Any clique may be chosen; we use several. (iii) Every UNSAT is a **DRAT proof** checked by `drat-trim` (built from the upstream source vendored in `research/p111-minrank-proof-reconciliation/vendor/drat-trim/`); every SAT answer is a **model** checked clause by clause. (iv) **Controls** (SAT at the known feasible rank, so the encoding is not vacuously unsatisfiable): E6 at rank 7, E8 / P104 / E7 frame blocks at their dense ranks, three E6 shared-B instances at the dense ranks (`jointctrl_d00/07/18`). +Solver: kissat 4.0.4 only. Every instance is regenerated by `sat/instances.py` (deterministic); `sat/results.jsonl` and `sat/joint_results.jsonl` record the sha256 of its DIMACS text, the status, and the DRAT verdict; `c6` regenerates all 1,473 and compares. Proof files: the 44 DRAT proofs of the primary instances other than the two E6 `Γ` ones are shipped (`sat/proofs/*.drat.xz`, 2 MB compressed), with all 55 primary CNFs (`sat/cnf/*.cnf.xz`) and the 9 SAT models (`sat/models/`); the two E6 `Γ` proofs (about 90 MB each) and the 1,418 shared-B / floor proofs (about 600 MB in total, each ≤ 2.5 MB) are **not** shipped: regenerate with `python3 sat/run.py solve OUT NAME...` (`kissat` and `drat-trim` on `PATH`; sub-second per shared-B instance) and verify with `drat-trim OUT/NAME.cnf OUT/NAME.drat`. All of them were solved and DRAT-verified by the author (`seconds` and `proof_bytes` in the results files). + +### 9.1 SAT-1: the smallest rank of an admissible B on E6 is 7 — CHECKED +`e6glob_r5_c{0,1,2}`, `e6glob_r6_c{0,1,2}`: UNSAT (three different 5-cliques of pairwise non-orthogonal lines); `e6glob_r7_c0`: SAT (model checked). So **every** admissible `B` on the 36 E6 lines has rank `≥ 7` (the 5-clique already forces `≥ 5`), and the dense `B` attains 7. Note `ω(Γ) = 5 < 7`, so this is not a clique bound. Meaning: a decoder cannot lower its cost below `m · 7` by using a globally low-rank `B` (the bound "cost ≤ m · rank B" is useless below 42 for `m = 6`). + +### 9.2 SAT-2: B-independent shadows cannot certify E6 or E7 — E7 CHECKED (shipped DRAT), E6 RECORDED (DRAT-verified at generation), consequence PROVEN +**Shadow Lemma — PROVEN** (this is Lemma CR2 / `hadamard_diag_rank_sum` of #2162, `lean/DecoderDuality.lean`; rectangular Lean form in #2267; cited, not shipped). For any field `L ⊇ F2`, `x_j, y_j ∈ L^v` with `supp x_j ⊆ R_j`, `supp y_j ⊆ A_j`, and `N := Σ_j x_j y_jᵀ`: for every admissible `B`, **cost ≥ rank_L(B ∘ N)**. *Proof:* `B ∘ N = Σ_j D(x_j) B D(y_j)` and each summand factors through `B[R_j, A_j]`, so has rank `≤ rank_F2 B[R_j,A_j]` (rank is invariant under field extension); rank is subadditive. If moreover `N` vanishes on every orthogonal pair, `B ∘ N` is diagonal (a non-orthogonal pair has `B_ts = 0`), so **cost ≥ #{t : N_tt ≠ 0} for every B**. +A shadow `N` certifying all `v` lines with `m` terms is a matrix of rank `≤ m` over `L` with nonzero diagonal vanishing on orthogonal pairs. Over `L = F2` this is a matrix of rank `≤ m` with unit diagonal and zeros on orthogonal pairs: its minimum rank is `mr₂(Γ)`. **E6 RECORDED** (DRAT-verified at generation; the two ~90 MB proofs are not shipped, `verify.sh` checks the CNF hashes and verdicts) **and E7 CHECKED:** `gamma_E6_r6_c{0,1}` UNSAT (so `mr₂(Γ_E6) ≥ 7 > 6 = T`) and `gamma_E7_r7_c{0,1}` UNSAT (`mr₂(Γ_E7) ≥ 8 > 7 = T`); symmetry-breaking cliques of pairwise orthogonal lines (E6 has `α_⊥ = 4`). **PROVEN consequence:** for square decoders (`m = T`) no **F2-valued, B-independent** Shadow-Lemma certificate certifies all lines of E6 or of E7. Larger fields of characteristic 2 are not covered (OPEN, §8.5). The E-type floors for arbitrary `B`, if true, need an argument using that one `B` is shared. + +### 9.3 SAT-3: per-term drops — CHECKED +`e6term_r4` (SAT) and `e6term_r3` (UNSAT): a pair of deficient directions with dense rank 6 and `|R ∩ A| = 8` has fitting minrank exactly 4 over all admissible `B`. E8 root blocks (`e8blk_*`, any `B`): `fr(R_α, R_α) ≥ 9` and `fr(R_α, R_β) ≥ 9` for non-orthogonal `α, β` (UNSAT at 8); `fr(R_α, R_β) ≤ 8` for orthogonal `α, β` (SAT). So per-term monotonicity in `B` fails (a sparser `B` can lower a *single* term) — which is what makes SAT-5 informative. + +### 9.4 SAT-4: the known optimal decoders are B-robust — CHECKED +For each of the following terms the F2-fitting minrank over all admissible `B` equals the dense rank (UNSAT at dense − 1; SAT at the dense rank for the controls): E8 frame-normal decoder, 8 terms (dense 9); E8 coordinate decoder, terms 0 and 7 (78 lines each; dense 9; the other coordinates are equivalent under coordinate permutations of E8); E7 frame decoder, 7 terms (dense 7); P112 removed-frame decoder, 8 terms (dense 8); P104 frame-pair decoder, 8 terms (dense 7). So **no admissible `B` lowers the cost of these decoders** (72, 49, 64, 56), i.e. the sharp dense values are also robust to the choice of `B` *for these decoders*. (This is a statement about those decoders, not a lower bound for the unit.) + +### 9.5 SAT-5: a shared B does not rescue per-term drops on E6 — RECORDED (DRAT-verified at generation; CNF hashes and verdicts checked by verify.sh) +`data/e6_square_decoders.json` lists 26 square (`m = 6`) root-code decoders of E6 (found by hill climbing over bases of normals of maximal flats; dense costs 38 to 41). For each decoder `d` and term `j` let `lo_j` be the F2-fitting minrank of `B[R_j, A_j]` over all admissible `B` (computed with kissat from the dense rank downwards; each `lo_j > 1` is backed by an UNSAT instance `floor_dXX_tY` at rank `lo_j − 1`, DRAT-verified; 156 of them). Every admissible `B` has `rank B[R_j,A_j] ≥ lo_j`, and `Σ_j lo_j ≤ 35` for all 26 decoders. If some `B` had total cost `≤ 35`, then with `c_j := rank B[R_j,A_j] ≥ lo_j` and `Σ c_j ≤ 35` we may enlarge the `c_j` to reach `Σ = 35`; `B` then satisfies `rank B[R_j,A_j] ≤ c_j` for **a** cap vector `c ≥ lo` with `Σ c = 35`. So it suffices to refute each such cap vector with the shared variables (`B` entries on orthogonal pairs shared by all six blocks): **1,262 instances** (`joint_dXX_c1-…-c6`; 1, 6, 21, 56, 126 or 252 per decoder), **all UNSAT, all DRAT-verified**. Controls: with the dense ranks as caps the instances are SAT (`jointctrl_*`). So **no admissible `B` brings any of these 26 decoders to cost `≤ 35`**, although the sums of their per-term floors are as low as 30. Scope: those 26 decoders only. + +## 10. What was *not* reproduced, and discrepancies +`verify.sh` reproduces every CHECKED number above. Items quoted from other PRs or from the working notes are marked RECORDED in place and are not claimed. The earlier helper `mrsub.py` of the working notes reports solver timeouts as SAT and was **not** used; all SAT / UNSAT verdicts here come from `sat/run.py`, which records `UNKNOWN` / `TIMEOUT` separately (there are none). diff --git a/research/overcomplete-and-realisation-floors/SOURCES.json b/research/overcomplete-and-realisation-floors/SOURCES.json new file mode 100644 index 0000000000..597e573ff7 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/SOURCES.json @@ -0,0 +1,95 @@ +{ + "description": "Pins for research/overcomplete-and-realisation-floors. files = library, generator and check code (own work); evidence = data exported from the author's working cards (data/e6_square_decoders.json: 26 E6 square decoders, sources pinned inside; data/p104_frames.json: the two E8 frames and frame pairs of the P104 construction of #968; data/e6_joint_plan.json: per-term fitting-minrank floors computed by sat/mkplan.py with kissat 4.0.4), the recorded SAT / DRAT results, and the Lean file. Tools used (not vendored): kissat 4.0.4; drat-trim from research/p111-minrank-proof-reconciliation/vendor/drat-trim/drat-trim.c (sha256 d834b649f437e091597f5347f259b9f681087f89ca0844d0cee250a1a1a0c2ee). scripts/provenance.py checks the pins and MANIFEST.sha256.", + "files": [ + { + "file": "checks/c1_overcomplete_d5.py", + "sha256": "820b6711ea1b5a6800579043705cc425784c91790ca7b1c8a09158c08be0b145" + }, + { + "file": "checks/c2_free_terms_frame120.py", + "sha256": "80b3a542d4a2b9ee6f827e02b90369246e5070be2c17c1f1f90510098787594a" + }, + { + "file": "checks/c3_realisation_free_floors.py", + "sha256": "94c1d41db0395c96e4d5aa44b98c76d412c3fb2130fcf700210b4b469ef4a761" + }, + { + "file": "checks/c4_e6_overcomplete.py", + "sha256": "ae6e331b03db9c695dd093022a840578119a7b80b6ad7420c0df422bc3daebb0" + }, + { + "file": "checks/c5_known_decoders.py", + "sha256": "30e26f3c9b5c6420d43bbf14465cc6786b640158c2096db2220acbaed80dcbbc" + }, + { + "file": "checks/c6_sat_replay.py", + "sha256": "279967868f9519bd65fdbf1ee4fb17a4ce70d661dbe0401cee44ac3e73b52a97" + }, + { + "file": "lib/dirs.py", + "sha256": "b56a474db62ef365cfc7458fe2815218b1afb171bfd83cf9a9dd6ab2e944e775" + }, + { + "file": "lib/frames.py", + "sha256": "b5e9f53dbdf15cb7667fd1e7d6db92f61fc773822746e895a56c2d21b6fa4612" + }, + { + "file": "lib/report.py", + "sha256": "fc4bb6836cb34d310ca7b3275ef15ef2d5d03e511063ddb06d0eb60e7072a640" + }, + { + "file": "lib/roots.py", + "sha256": "b2bdaa63376ab0c718b256e879d669f1465f4d3407b64ecc876ca0f3e6a32acd" + }, + { + "file": "lib/satgen.py", + "sha256": "567df3a6f3277179f88cf17dffbe69dd40b1f6d8277baab3cb79e7332361077f" + }, + { + "file": "sat/instances.py", + "sha256": "37523be9effa35d1de7f6f71a77ae5b78d87f0d525701bdc74de22f3b120caf1" + }, + { + "file": "sat/mkplan.py", + "sha256": "5456a3485b1b60181e4c80faf8590a863a4a2ff1f35c3e04e30cdab703a91a2c" + }, + { + "file": "sat/run.py", + "sha256": "9484d88f7b57a4b2f723beb4f91a61e183df36dc54c52646de657706723ab4c1" + } + ], + "evidence": [ + { + "file": "data/e6_joint_plan.json", + "sha256": "fd63c3a40911865d9622d8c0e88e1fa7b5a310810ace99bf16e1e2a92144af59" + }, + { + "file": "data/e6_square_decoders.json", + "sha256": "ef327e475750e9129bb9fd7aef27061cb06bcda63c376d66041e68feb8466df6" + }, + { + "file": "data/p104_frames.json", + "sha256": "52e1a2b0e963311eb74c1ec2bbb110497da57b606e132f784f9000603d10a306" + }, + { + "file": "lean/Overcomplete.lean", + "sha256": "06ed7a0c208e328bfdbc34609ed851085dd8832ee9b38e8e5b5790dfae96f0cd" + }, + { + "file": "lean/lake-manifest.json", + "sha256": "1f62920310807cf74bb08e834265e7e2281eac76650ea6026cdc676e516652b7" + }, + { + "file": "lean/lakefile.toml", + "sha256": "6d1584fba35d26bb56eb4b4e756b41caa4299d544b69bc937dc00fd8bc8791b1" + }, + { + "file": "sat/joint_results.jsonl", + "sha256": "b5dc67f3b8972011e1fee9bb48fd16276935c662f2453852f7ea922e3bdd7b9c" + }, + { + "file": "sat/results.jsonl", + "sha256": "6cb74ebe7ce2c62011c346b13f11a3ccb8ba3456e1b8dc4ce8a4486a70f0b9af" + } + ] +} \ No newline at end of file diff --git a/research/overcomplete-and-realisation-floors/checks/c1_overcomplete_d5.py b/research/overcomplete-and-realisation-floors/checks/c1_overcomplete_d5.py new file mode 100644 index 0000000000..38e02a4314 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/checks/c1_overcomplete_d5.py @@ -0,0 +1,75 @@ +"""Lemma OC (exact numerical instances), the D5 floor for every number of terms, and the finite facts used by its proof. +Standard library only; Fractions.""" +import sys, os, random, itertools +sys.path.insert(0, os.path.join(os.path.dirname(os.path.abspath(__file__)), '..', 'lib')) +from report import check, finish +from fractions import Fraction as Fr +from dirs import * + +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 tr(A): return [list(r) for r in zip(*A)] +def inv(A): + n = len(A); M = [[Fr(x) for x in r] + [Fr(int(i == j)) for j in range(n)] for i, r in enumerate(A)] + for c in range(n): + p = next(r for r in range(c, n) if M[r][c] != 0); M[c], M[p] = M[p], M[c] + pv = M[c][c]; M[c] = [x / pv for x in M[c]] + for r in range(n): + if r != c and M[r][c] != 0: + f = M[r][c]; M[r] = [x - f * y for x, y in zip(M[r], M[c])] + return [r[n:] for r in M] +def ident(n): return [[Fr(int(i == j)) for j in range(n)] for i in range(n)] +def sub(A, B): return [[a - b for a, b in zip(r, s)] for r, s in zip(A, B)] + +random.seed(20261014) +def random_decomposition(T, m): + """X, Y in Q^{T x m} with X Y^T = I_T: Y^T = X^T (X X^T)^{-1} + (I - X^T (X X^T)^{-1} X) W""" + while True: + X = [[Fr(random.randint(-5, 5)) for _ in range(m)] for _ in range(T)] + try: G = inv(matmul(X, tr(X))) + except StopIteration: continue + Y0 = matmul(tr(X), G) # m x T, X Y0 = I + W = [[Fr(random.randint(-5, 5)) for _ in range(T)] for _ in range(m)] + Pm = sub(ident(m), matmul(Y0, X)) # projector onto ker X + Yt = [[a + b for a, b in zip(r, s)] for r, s in zip(Y0, matmul(Pm, W))] # Y^T + return X, Yt + +ok_idem = ok_trace = ok_sq = ok_rank1 = True; ninst = 0 +for (T, m) in [(2, 2), (3, 3), (4, 4), (3, 4), (4, 5), (5, 6), (6, 7), (5, 7), (6, 8), (4, 7)]: + for _ in range(12): + X, Yt = random_decomposition(T, m); ninst += 1 + assert matmul(X, Yt) == ident(T) # X Y^T = I (Yt = Y^T) + K = matmul(Yt, X) # K_ij = + Pi = sub(ident(m), K) + ok_idem &= matmul(K, K) == K and matmul(Pi, Pi) == Pi + ok_trace &= sum(K[i][i] for i in range(m)) == T and sum(Pi[i][i] for i in range(m)) == m - T + ok_rank1 &= rank_q(K) == T and rank_q(Pi) == m - T + if m == T: ok_sq &= K == ident(m) + if m == T + 1: + # Pi = u w^T, w^T u = 1, K_ij = -u_i w_j (i != j): off-diagonal of K is rank one + ok_sq &= rank_q(Pi) == 1 and all(K[i][j] * K[k][l] == K[i][l] * K[k][j] for i in range(m) for j in range(m) for k in range(m) for l in range(m) if len({i, k}) == 2 and len({j, l}) == 2 and i != j and k != l and i != l and k != j) +check('c1.oc_idem', ok_idem, 'Lemma OC: K = Y^T X and I - K are idempotent on %d random exact instances (T,m up to 6,8)' % ninst) +check('c1.oc_trace', ok_trace and ok_rank1, 'Lemma OC: trace K = T, trace(I-K) = m-T, rank K = T, rank(I-K) = m-T on the same instances') +check('c1.oc_cases', ok_sq, 'Lemma OC: m = T gives K = I; m = T+1 gives I - K of rank one and a rank-one off-diagonal cross matrix') + +# D5 +L, T = UNITS['D5']; Br = b0rows(L) +dm = deficient_directions(L, T); X5 = sorted(set(dm.values())) +unit = sorted(tuple(int(i == j) for j in range(5)) for i in range(5)) +check('c1.d5dirs', X5 == unit and len(dm) == 5, 'D5: the deficient directions are exactly the five coordinate axes e_1..e_5, one per deficient class') +check('c1.d5nspace', bilinear_space_dim(L, T) == 1, 'D5: the space of bilinear forms vanishing on orthogonal pairs is 1-dimensional') +rkt = {(a, b): blk(Br, support(L, unit[a]), support(L, unit[b])) for a in range(5) for b in range(5)} +check('c1.d5ranks', all(rkt[a, b] == (4 if a == b else 3) for a in range(5) for b in range(5)), 'D5: pair rank = 4 for matched (e_a, e_a), 3 for (e_a, e_b), a != b') +# rank of a pair with a non-deficient side is >= 4: via the F2 rank of B on random supports with one non-deficient direction (spot check) and Lemma 3.1 in the README +rng = random.Random(5); bad = 0 +for _ in range(300): + x = tuple(rng.choice([-2, -1, 0, 1, 2]) for _ in range(5)); y = tuple(rng.choice([0, 1, -1]) for _ in range(5)) + if not any(x) or not any(y): continue + if x in unit or tuple(-c for c in x) in unit: continue # x non-deficient (not +-e_a up to scale) -- scalings of e_a are caught below + if sum(1 for c in x if c) == 1: continue + if blk(Br, support(L, x), support(L, y)) < 4: bad += 1 +check('c1.d5nondef', bad == 0, 'D5: 300 random pairs with a non-deficient row direction all have rank >= 4 (spot check of Lemma 3.1; proof in README)') +# the combinatorial core of the m = 6 argument: three terms (a_i, b_i) with a_i != b_i and b_i = a_j for all i != j do not exist +trip = [(a, b) for a in range(5) for b in range(5) if a != b] +found = [t for t in itertools.permutations(trip, 3) if all(t[i][1] == t[j][0] for i in range(3) for j in range(3) if i != j)] +check('c1.d5core', not found, 'D5: no three rank-3 terms (e_a_i, e_b_i), a_i != b_i, satisfy != 0 for all i != j (checked over all %d ordered triples)' % (len(trip) * (len(trip) - 1) * (len(trip) - 2))) +finish() diff --git a/research/overcomplete-and-realisation-floors/checks/c2_free_terms_frame120.py b/research/overcomplete-and-realisation-floors/checks/c2_free_terms_frame120.py new file mode 100644 index 0000000000..aff0b24db3 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/checks/c2_free_terms_frame120.py @@ -0,0 +1,36 @@ +"""Free terms and the 120-line frame unit. A decoder of the unit 'all 120 lines pairwise orthogonal' (so M = I is allowed and every +off-diagonal entry of B may be 1) with the E8 dense pairing B = J + Abar_E8 has exact cost 72 (8 coordinate terms of rank 9, 6720 free +single-entry terms of rank 0). Exact Fractions; standard library only.""" +import sys, os +sys.path.insert(0, os.path.join(os.path.dirname(os.path.abspath(__file__)), '..', 'lib')) +from report import check, finish +from fractions import Fraction as Fr +from roots import * + +L = E8; v = len(L); Gd = gram(L) # doubled coordinates: real inner product = Gd/4, real coordinates = L/2 +Br = b0rows(L) +real = lambda t, j: Fr(L[t][j], 2) +Greal = lambda t, s: Fr(Gd[t][s], 4) +terms = [] # (p, q) as dicts +for j in range(8): + p = {t: real(t, j) / 2 for t in range(v) if L[t][j] != 0} + q = {t: real(t, j) for t in range(v) if L[t][j] != 0} + terms.append((p, q)) +nfree = 0 +for t in range(v): + for s in range(v): + if t != s and Gd[t][s] != 0: + terms.append(({t: -Greal(t, s) / 2}, {s: Fr(1)})); nfree += 1 +M = {} +for p, q in terms: + for t, a in p.items(): + for s, b in q.items(): M[t, s] = M.get((t, s), 0) + a * b +isI = all(M.get((t, s), 0) == int(t == s) for t in range(v) for s in range(v)) +check('c2.M_is_I', isI, 'E8 coordinate terms (p_j = R_j/2, q_j = R_j) plus %d free single-entry terms sum to M = I_120 exactly (%d terms)' % (nfree, len(terms))) +ranks = [blk(Br, sorted(p), sorted(q)) for p, q in terms] +paid = [r for r in ranks if r] +check('c2.cost72', sum(ranks) == 72 and paid == [9] * 8 and nfree == 6720 and all(r == 0 for r in ranks[8:]), + 'B = J + Abar_E8: the 8 coordinate terms have F2-rank 9 each, the %d free terms have rank 0; total cost %d < 120 = v (M = I, unit diagonal)' % (nfree, sum(ranks))) +# B fits the frame unit: B_tt = 1 and B_ts = 1 only where allowed -- in the frame unit every pair is orthogonal, so every B with unit diagonal fits +check('c2.fits', all(Br[t] >> t & 1 for t in range(v)), 'B has unit diagonal; the frame unit allows any off-diagonal B (all 120 lines pairwise orthogonal)') +finish() diff --git a/research/overcomplete-and-realisation-floors/checks/c3_realisation_free_floors.py b/research/overcomplete-and-realisation-floors/checks/c3_realisation_free_floors.py new file mode 100644 index 0000000000..4d10cbfc8f --- /dev/null +++ b/research/overcomplete-and-realisation-floors/checks/c3_realisation_free_floors.py @@ -0,0 +1,51 @@ +"""Frame-intersection certificate (README Lemma 3.3) and the realisation-free floors of Corollary 3.4. Exact bitmask arithmetic; standard library only. +For each unit: all maximal cliques of the orthogonality graph (frames = cliques of size T), all 2^h - 1 label classes (supports of the +nonzero vectors of col B, B = dense pairing), and for each class: does it contain a T-frame, or is it certified by clause (c)?""" +import sys, os, time +sys.path.insert(0, os.path.join(os.path.dirname(os.path.abspath(__file__)), '..', 'lib')) +from report import check, finish +from collections import Counter +from frames import * + +U = units(); R = {} +t0 = time.time() +for n, (L, T) in U.items(): R[n] = analyse(L, T) +print(' (analysis %.0f s)' % (time.time() - t0)) +exp = { # name: (v, T, h, alpha_orth, #T-frames, classes with a T-frame, certified by (c), uncertified, uncertified sizes, contains 2-dim subspace) + 'D4': (12, 4, 3, 4, 3, 7, 0, 0, [], False), 'D6': (30, 6, 5, 6, 15, 16, 15, 0, [], False), + 'E7': (63, 7, 7, 7, 135, 64, 63, 0, [], False), 'E8': (120, 8, 9, 8, 2025, 391, 120, 0, [], False), + 'P104': (104, 8, 9, 8, 759, 375, 104, 32, [52], False), 'P112': (112, 8, 9, 8, 1260, 383, 112, 16, [56], False), + 'E7-frame': (56, 7, 7, 7, 64, 57, 56, 14, [24, 32], True), + 'E6': (36, 6, 7, 4, 0, 0, 0, 127, [16, 20, 36], True), 'D5': (20, 5, 5, 4, 0, 0, 0, 31, [8, 12, 20], True)} +for n, e in exp.items(): + a = R[n] + got = (a['v'], a['T'], a['h'], a['alpha'], a['nframes'], len(a['has_frame']), len(a['cert_c']), len(a['unc']), + sorted(set(popcount(w) for _, w in a['unc'])), a['sub2']) + check('c3.table.' + n, got == e, '%s: v,T,h,alpha_orth,#T-frames,#classes with T-frame,#certified by (c),#uncertified,sizes,2-dim subspace of uncertified = %s' % (n, got)) +# the classes without a T-frame are exactly the neighbourhoods N(alpha) (lines non-orthogonal to alpha, alpha excluded) +for n in ('D6', 'E7', 'E8', 'P112', 'P104', 'E7-frame'): + L, T = U[n]; a = R[n]; G = gram(L); v = len(L) + N = set(sum(1 << s for s in range(v) if s != x and G[x][s] != 0) for x in range(v)) + ws = set(w for _, w in a['cert_c']) + check('c3.N.' + n, ws == N, '%s: the %d classes without a T-frame (all certified by clause (c)) are exactly the %d neighbourhoods N(alpha)' % (n, len(ws), len(N))) +# E8: every frame not through alpha meets N(alpha) in exactly 4 lines +L, T = U['E8']; G = gram(L); a = R['E8'] +meets = Counter() +for al in range(0, 120, 7): + Na = sum(1 << s for s in range(120) if s != al and G[al][s] != 0) + for f in a['frames']: + if not f >> al & 1: meets[popcount(f & Na)] += 1 +check('c3.e8meet', set(meets) == {4}, 'E8: every 8-frame not through alpha meets N(alpha) in exactly 4 lines (alpha ranging over 18 lines; W(E8) is transitive): %s' % dict(meets)) +# P104 and P112: with delta <= 1 on each side the pair ranks of the uncertified-or-full supports give the floor +for n, want in (('P104', 56), ('P112', 56)): + L, T = U[n]; a = R[n]; v = len(L); Br = a['Br']; full = (1 << v) - 1; basis = a['basis']; h = a['h'] + def rows_rank(S): # dim of span of the F2 row-coordinates of the lines of S in col(B) (delta = h - this) + # coordinates of line t w.r.t. 'basis': c_t(i) = bit t of ... use the row space description instead: rank of the matrix B[S, :] + return rank_f2([Br[t] for t in bits(S)]) + comp = [full & ~w for _, w in a['unc']] + [full] + okd = all(h - rows_rank(c) == 1 for c in comp[:-1]) and rows_rank(full) == h + cnt = Counter(blk(Br, bits(R_), bits(A_)) for R_ in comp for A_ in comp) + check('c3.pairs.' + n, okd and min(cnt) * T == want, '%s: (uncertified-or-full) x (uncertified-or-full) pair ranks %s; each uncertified class has deficiency exactly 1; floor T*min = %d' % (n, dict(sorted(cnt.items())), T * min(cnt))) +# D5 / E6: no T-frame, every class uncertified (nothing claimed beyond the Gram realisation) +check('c3.noframes', R['E6']['nframes'] == 0 and R['D5']['nframes'] == 0 and R['E6']['alpha'] == 4 and R['D5']['alpha'] == 4, 'E6 and D5: alpha_orth = 4 < T, no T-frame (realisation-free statement not available)') +finish() diff --git a/research/overcomplete-and-realisation-floors/checks/c4_e6_overcomplete.py b/research/overcomplete-and-realisation-floors/checks/c4_e6_overcomplete.py new file mode 100644 index 0000000000..177f83fb07 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/checks/c4_e6_overcomplete.py @@ -0,0 +1,98 @@ +"""E6, dense B, root-code decoders (factors in the scaled-Gram realisation's column/row space), any number of terms m. +Exact integer arithmetic, standard library only. + (i) deficient directions (27), pair-rank table (432 pairs of rank 5, 297 of rank 6), all 27 x 27 cross pairings nonzero, + one deficient direction per deficient class, the bilinear-form space is 1-dimensional; + (ii) W(E6) is transitive on the 432 rank-5 pairs (orbit of one pair under the 36 reflections), so one pair may be fixed; + (iii) exhaustive search for 7 rank-5 pairs containing pair 0 whose cross matrix Lambda_pq = has rank one off the diagonal + (the necessary condition of Lemma OC for m = T+1): node counts per size, and the rank test of the direction sets on every leaf. +Input arithmetic is integers only: rank-one is tested by cross-multiplication, never by floats.""" +import sys, os, itertools, time +sys.path.insert(0, os.path.join(os.path.dirname(os.path.abspath(__file__)), '..', 'lib')) +from report import check, finish +from dirs import * + +L, T = UNITS['E6']; Br = b0rows(L) +dmap = deficient_directions(L, T) +X = sorted(set(dmap.values())) +check('c4.dirs', len(dmap) == 27 and len(X) == 27, 'E6: 27 deficient classes, 27 distinct deficient directions, one direction per class (so delta = 1 for each)') +check('c4.nspace', bilinear_space_dim(L, T) == 1, 'E6: the space of bilinear forms N on V with r_t N r_s = 0 for all orthogonal t, s is 1-dimensional (spanned by the metric)') +supp = {x: support(L, x) for x in X} +rk = {(i, j): blk(Br, supp[X[i]], supp[X[j]]) for i in range(27) for j in range(27)} +from collections import Counter +cnt = Counter(rk.values()) +check('c4.pairranks', dict(cnt) == {5: 432, 6: 297}, 'E6 deficient x deficient pair ranks (dense B): %s' % dict(sorted(cnt.items()))) +nz = all(dot(X[i], X[j]) != 0 for i in range(27) for j in range(27)) +check('c4.cross', nz, 'E6: != 0 for all 27 x 27 pairs of deficient directions (the column directions are the same 27 lines as the row directions)') +P = sorted(k for k, r in rk.items() if r == 5); n = len(P) +check('c4.rank5', n == 432, 'E6: 432 rank-5 pairs') + +# (ii) transitivity of W(E6) on the rank-5 pairs +idx = {x: i for i, x in enumerate(X)} +def act(r, i): return idx[reflect(X[i], r)] +seen = {P[0]}; fr = [P[0]] +while fr: + nf = [] + for (i, j) in fr: + for r in L: + q = (act(r, i), act(r, j)) + if q not in seen: seen.add(q); nf.append(q) + fr = nf +check('c4.transitive', seen == set(P), 'E6: the 36 reflections generate a single orbit on the 432 rank-5 pairs (so pair 0 may be fixed WLOG); orbit size %d' % len(seen)) + +# (iii) search. Lam[p][q] = with y_p = X[P[p][1]], x_q = X[P[q][0]] +Lam = [[dot(X[P[p][1]], X[P[q][0]]) for q in range(n)] for p in range(n)] +t0 = time.time() +def run(): + depth = Counter(); leaves = [] + for p in range(1, n): + L0p = Lam[0][p]; Lp0 = Lam[p][0] + for q in range(p + 1, n): + L0q = Lam[0][q]; Lq0 = Lam[q][0]; Lpq = Lam[p][q]; Lqp = Lam[q][p] + if L0p * Lpq * Lq0 != L0q * Lqp * Lp0: continue + depth[3] += 1 + # a = (1, Lpq/L0q, Lqp/L0p), b = (Lp0 L0q/Lpq, L0p, L0q); candidate c > q must have a_c = L_ci/b_i (i in {0,p,q}) and b_c = L_ic/a_i + cand = [] + for c in range(q + 1, n): + Lc0 = Lam[c][0]; Lcp = Lam[c][p]; Lcq = Lam[c][q] + if Lcp * L0q != Lcq * L0p: continue + if Lc0 * Lpq * L0p != Lcp * Lp0 * L0q: continue + L0c = Lam[0][c] + if Lam[p][c] * L0q != L0c * Lpq: continue + if Lam[q][c] * L0p != L0c * Lqp: continue + cand.append(c) + m = len(cand) + if not m: continue + # compatibility of candidates c < d: L_dc L_0p == L_dp L_0c and L_cd L_0p == L_cp L_0d + adj = [0] * m + for u in range(m): + c = cand[u] + for w in range(u + 1, m): + d = cand[w] + if Lam[d][c] * L0p == Lam[d][p] * Lam[0][c] and Lam[c][d] * L0p == Lam[c][p] * Lam[0][d]: + adj[u] |= 1 << w; adj[w] |= 1 << u + def ext(S, size, cm): + depth[size] += 1 + if size == 7: leaves.append(tuple(S)); return + while cm: + low = cm & -cm; u = low.bit_length() - 1; cm ^= low + ext(S + [cand[u]], size + 1, cm & adj[u]) + # cliques of the compatibility graph, ordered increasingly + for u in range(m): + ext([0, p, q, cand[u]], 4, adj[u] & ~((1 << (u + 1)) - 1)) + return depth, leaves +depth, leaves = run() +exp = {3: 27250, 4: 206275, 5: 898460, 6: 2502409, 7: 5504660} +check('c4.nodes', dict(depth) == exp, 'E6 rank-one-off-diagonal sets containing pair 0, by size: %s (%.0f s)' % (dict(sorted(depth.items())), time.time() - t0)) +rcache = {} +def rk6(vecs): + key = tuple(sorted(set(vecs))) + if key not in rcache: rcache[key] = rank_q([list(v) for v in key]) if len(key) >= 6 else len(key) + return rcache[key] +ok = 0; dis = 0 +for S in leaves: + xs = [P[s][0] for s in S]; ys = [P[s][1] for s in S] + if len(set(xs)) < 6 or len(set(ys)) < 6: continue + dis += 1 + if rk6([X[i] for i in xs]) == 6 and rk6([X[j] for j in ys]) == 6: ok += 1 +check('c4.noleaf', ok == 0 and dis == 0, 'E6: of %d candidate 7-sets, %d have >= 6 distinct row AND >= 6 distinct column directions, and %d have both direction sets of rank 6 (none: no 7-term decomposition from seven rank-5 pairs)' % (len(leaves), dis, ok)) +finish() diff --git a/research/overcomplete-and-realisation-floors/checks/c5_known_decoders.py b/research/overcomplete-and-realisation-floors/checks/c5_known_decoders.py new file mode 100644 index 0000000000..7a5269b3f7 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/checks/c5_known_decoders.py @@ -0,0 +1,48 @@ +"""The floors of Corollary 3.4 are attained: explicit decoders and their exact dense-B costs (E8 72, E7 49, D6 30, D4 12, P112 64, P104 56), +and the trivial decoder (cost v) for E6 and D5. Exact Fractions; standard library only. +Frame decoder of a unit with an orthogonal frame f_1..f_T spanning its space: p_j(t) = , q_j(s) = /; then +M = sum_j p_j q_j^T is the Gram matrix of the live lines. (P112 uses the frame of the removed lines.)""" +import sys, os, json +sys.path.insert(0, os.path.join(os.path.dirname(os.path.abspath(__file__)), '..', 'lib')) +from report import check, finish +from fractions import Fraction as Fr +from roots import * +from frames import units + +HERE = os.path.dirname(os.path.abspath(__file__)) +D = json.load(open(os.path.join(HERE, '..', 'data', 'p104_frames.json'))); F0, F1, pairs = D['F0'], D['F1'], D['pairs'] + +def frame_decoder(live, frame_vecs): + v = len(live); Br = b0rows(live); G = gram(live); cost = 0 + M = [[Fr(0)] * v for _ in range(v)] + for f in frame_vecs: + nf = dot(f, f); p = [dot(r, f) for r in live] + for t in range(v): + if p[t]: + for s in range(v): + if p[s]: M[t][s] += Fr(p[t] * p[s], nf) + R = [t for t in range(v) if p[t] != 0]; cost += blk(Br, R, R) + return M == [[Fr(G[t][s]) for s in range(v)] for t in range(v)], cost + +U = units(); L8 = UNITS['E8'][0] +for name, want in (('D4', 12), ('D6', 30), ('E7', 49), ('E8', 72)): + L, T = U[name]; fr = bits(next(c for c in maximal_cliques(orth_graph(L)) if popcount(c) == T)) + ok, cost = frame_decoder(L, [L[i] for i in fr]) + check('c5.' + name, ok and cost == want, '%s: frame decoder with %d terms has M = Gram (zero on orthogonal pairs, nonzero diagonal) and dense-B cost %d' % (name, T, cost)) +live112 = [L8[t] for t in range(120) if t not in set(F0)] +ok, cost = frame_decoder(live112, [L8[i] for i in F0]) +check('c5.P112', ok and cost == 64, 'P112 (E8 minus a frame): removed-frame decoder has M = Gram and dense cost %d' % cost) +# P104: E8 minus two frames; terms (a_j, b_j) with a_j in F0, b_j in F1 (pairs from the #968 construction as recorded in data/p104_frames.json): +live104i = [t for t in range(120) if t not in set(F0) | set(F1)]; live104 = [L8[t] for t in live104i]; v = len(live104); Br = b0rows(live104); G = gram(live104) +M = [[sum(dot(live104[t], L8[x]) * dot(live104[s], L8[y]) for x, y in pairs) for s in range(v)] for t in range(v)] +okM = all(M[t][t] != 0 for t in range(v)) and all(M[t][s] == 0 for t in range(v) for s in range(v) if t != s and G[t][s] == 0) +cost = 0 +for x, y in pairs: + R = [i for i, t in enumerate(live104i) if dot(L8[t], L8[x]) != 0]; A = [i for i, t in enumerate(live104i) if dot(L8[t], L8[y]) != 0] + cost += blk(Br, R, A) +check('c5.P104', okM and cost == 56, 'P104: the 8 frame-pair terms give a realisation M (zero on orthogonal pairs, nonzero diagonal; not the Gram matrix) of dense cost %d' % cost) +for name, v_ in (('E6', 36), ('D5', 20)): + L, T = U[name]; Br = b0rows(L) + cost = sum(blk(Br, [t], [t]) for t in range(len(L))) + check('c5.' + name, len(L) == v_ and cost == v_, '%s: the trivial decoder (M = I, one single-entry term per line) costs %d = v; this equals the floor proved for the scaled-Gram realisations (M = I is a different realisation)' % (name, cost)) +finish() diff --git a/research/overcomplete-and-realisation-floors/checks/c6_sat_replay.py b/research/overcomplete-and-realisation-floors/checks/c6_sat_replay.py new file mode 100644 index 0000000000..3571f0bd56 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/checks/c6_sat_replay.py @@ -0,0 +1,102 @@ +"""Replay of the SAT/DRAT evidence (standard library only; drat-trim optional). + * every instance of the registry (sat/instances.py) is regenerated from its generator and its DIMACS sha256 must equal the recorded one + (sat/results.jsonl for the 51 primary instances + controls, sat/joint_results.jsonl for the 1,262 shared-B instances and 156 floor instances); + * every recorded status equals the expected status (UNSAT for all lower bounds, SAT for all controls and existence claims); + * every SAT model is checked clause by clause against the regenerated CNF; + * every shipped DRAT proof (sat/proofs/*.drat.xz) is checked with drat-trim when it is available (environment DRAT_TRIM, or `drat-trim` on PATH); + otherwise those lines are reported as SKIP (not PASS). +Set KISSAT_RESOLVE=n to also re-solve the first n joint instances with kissat and check their proofs.""" +import sys, os, json, hashlib, lzma, subprocess, shutil, tempfile, time +HERE = os.path.dirname(os.path.abspath(__file__)); PKG = os.path.join(HERE, '..') +sys.path.insert(0, os.path.join(PKG, 'lib')); sys.path.insert(0, os.path.join(PKG, 'sat')) +from report import check, finish +import instances as I +from run import model_satisfies +from fractions import Fraction +from roots import UNITS, dot, gram, rank_q + +def sha(b): return hashlib.sha256(b).hexdigest() +def rows(fn): return [json.loads(l) for l in open(os.path.join(PKG, 'sat', fn))] +prim = rows('results.jsonl'); joint = rows('joint_results.jsonl') +A = I.all_instances(); J = I.joint_instances(); ALL = dict(A); ALL.update(J) +t0 = time.time() +# registry coverage +check('c6.coverage', set(r['name'] for r in prim) == set(A) | {k for k in J if k.startswith('jointctrl')} and set(r['name'] for r in joint) == {k for k in J if not k.startswith('jointctrl')}, + 'every registered instance has a recorded result: %d primary (incl. controls), %d shared-B and floor instances' % (len(prim), len(joint))) +plan = json.load(open(os.path.join(PKG, 'data', 'e6_joint_plan.json'))) +check('c6.plan', all(sum(p['lo']) <= plan['total'] for p in plan['plan']) and len(plan['plan']) == 26 and sum(1 for k in J if k.startswith('joint_d')) == 1262 and sum(1 for k in J if k.startswith('floor_')) == 156, + 'plan: 26 decoders, floors sum to <= 35 for each (so the cap vectors with c_j >= lo_j, sum 35 are non-empty), 1,262 cap-vector instances, 156 floor instances') +# statuses +bad = [r['name'] for r in prim + joint if r['status'] != r['expect'] or (r['status'] == 'UNSAT' and r.get('drat') != 'VERIFIED') or (r['status'] == 'SAT' and not r.get('model_ok'))] +check('c6.status', not bad, 'recorded statuses: %d UNSAT instances DRAT-VERIFIED when recorded, %d SAT instances with checked models, none unexpected %s' % ( + sum(r['status'] == 'UNSAT' for r in prim + joint), sum(r['status'] == 'SAT' for r in prim + joint), bad[:3])) +# hashes +def regen_hash(name): + c = ALL[name]['build'](); return sha(c.dimacs().encode()), c +badh = []; cnfs = {} +for r in prim: + h, c = regen_hash(r['name']) + if h != r['cnf_sha256']: badh.append(r['name']) + if r['status'] == 'SAT': cnfs[r['name']] = c +check('c6.hash_primary', not badh, 'regenerated DIMACS of all %d primary instances has the recorded sha256 %s' % (len(prim), badh[:3])) +t1 = time.time(); badj = [] +for r in joint: + h, _ = regen_hash(r['name']) + if h != r['cnf_sha256']: badj.append(r['name']) +check('c6.hash_joint', not badj, 'regenerated DIMACS of all %d shared-B/floor instances has the recorded sha256 (%.0f s) %s' % (len(joint), time.time() - t1, badj[:3])) +# shipped CNFs +badc = []; nc = 0 +for f in sorted(os.listdir(os.path.join(PKG, 'sat', 'cnf'))): + name = f[:-len('.cnf.xz')] + if lzma.open(os.path.join(PKG, 'sat', 'cnf', f)).read().decode() != ALL[name]['build']().dimacs(): badc.append(name) + nc += 1 +check('c6.cnf', not badc and nc == len(prim), 'the %d shipped CNFs (sat/cnf/*.cnf.xz) equal the regenerated DIMACS text %s' % (nc, badc[:3])) +# models +nm = 0; okm = True +for r in prim: + if r['status'] != 'SAT': continue + pos = set(map(int, open(os.path.join(PKG, 'sat', 'models', r['name'] + '.model')).read().split())) + okm &= model_satisfies(cnfs[r['name']].dimacs(), pos); nm += 1 +check('c6.models', okm and nm > 0, 'all %d shipped SAT models satisfy the regenerated CNF clause by clause' % nm) +# the 26 E6 square decoders are exact (stdlib exact arithmetic): x_j, y_j span a 6-dim space, != 0 iff i = j (so sum_j x_j y_j^T / = id), +# the realised matrix M_ts = sum_j / equals the Gram matrix of the 36 lines exactly, and the stored dense sums equal the F2 ranks +from roots import b0rows, blk +from dirs import support +_L = UNITS['E6'][0]; _G = gram(_L); _Br = b0rows(_L); _n = len(_L) +_dec = json.load(open(os.path.join(PKG, 'data', 'e6_square_decoders.json')))['decoders']; _bad = [] +for _d, _x in enumerate(_dec): + X, Y = _x['X'], _x['Y']; dd = [dot(Y[j], X[j]) for j in range(6)] + okd = rank_q([[Fraction(v) for v in r] for r in X]) == 6 and rank_q([[Fraction(v) for v in r] for r in Y]) == 6 + okd &= all((dot(Y[i], X[j]) != 0) == (i == j) for i in range(6) for j in range(6)) + okd &= all(sum(Fraction(dot(_L[t], X[j]) * dot(_L[s], Y[j]), dd[j]) for j in range(6)) == _G[t][s] for t in range(_n) for s in range(_n)) + dn = [blk(_Br, support(_L, X[j]), support(_L, Y[j])) for j in range(6)] + okd &= dn == _x['dense'] and sum(dn) == _x['sum'] + if not okd: _bad.append(_d) +check('c6.decoders', len(_dec) == 26 and not _bad, 'the 26 E6 square decoders of data/e6_square_decoders.json are exact: rank 6 factors, != 0 iff i = j, M = Gram of the 36 lines (Fractions), stored dense ranks and sums recomputed %s' % _bad[:3]) +# DRAT +drat = os.environ.get('DRAT_TRIM') or shutil.which('drat-trim') +proofs = sorted(f for f in os.listdir(os.path.join(PKG, 'sat', 'proofs')) if f.endswith('.drat.xz')) +if drat and os.path.exists(drat): + ok = 0; badp = [] + with tempfile.TemporaryDirectory() as td: + for f in proofs: + name = f[:-len('.drat.xz')] + cnf = os.path.join(td, name + '.cnf'); pr = os.path.join(td, name + '.drat') + open(cnf, 'w').write(ALL[name]['build']().dimacs()); open(pr, 'wb').write(lzma.open(os.path.join(PKG, 'sat', 'proofs', f)).read()) + d = subprocess.run([drat, cnf, pr, '-t', '3000'], capture_output=True, text=True) + if 's VERIFIED' in d.stdout: ok += 1 + else: badp.append(name) + os.remove(cnf); os.remove(pr) + check('c6.drat', ok == len(proofs) and ok > 0, 'drat-trim (%s) verifies all %d shipped DRAT proofs against the regenerated CNFs %s' % (os.path.basename(drat), ok, badp[:3])) +else: + print('SKIP c6.drat drat-trim not found (set DRAT_TRIM=/path/to/drat-trim): %d shipped proofs not re-checked' % len(proofs), flush=True) +n = int(os.environ.get('KISSAT_RESOLVE', '0') or 0) +if n and shutil.which('kissat') and drat: + from run import solve + names = [r['name'] for r in joint if r['name'].startswith('joint_')][:n]; okk = 0 + with tempfile.TemporaryDirectory() as td: + for nm_ in names: + res = solve(nm_, ALL[nm_], td, tmo=600); okk += res['status'] == 'UNSAT' and res.get('drat') == 'VERIFIED' + check('c6.resolve', okk == len(names), 're-solved %d shared-B instances with kissat; all UNSAT and DRAT-VERIFIED' % len(names)) +print(' (replay %.0f s)' % (time.time() - t0)) +finish() diff --git a/research/overcomplete-and-realisation-floors/data/e6_joint_plan.json b/research/overcomplete-and-realisation-floors/data/e6_joint_plan.json new file mode 100644 index 0000000000..4d5cc40071 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/data/e6_joint_plan.json @@ -0,0 +1,499 @@ +{ +"total": 35, +"plan": [ +{ +"decoder": 0, +"dense": [ +7, +7, +6, +6, +7, +7 +], +"lo": [ +5, +5, +5, +5, +6, +6 +] +}, +{ +"decoder": 1, +"dense": [ +7, +7, +7, +7, +6, +7 +], +"lo": [ +5, +5, +7, +6, +6, +5 +] +}, +{ +"decoder": 2, +"dense": [ +7, +7, +6, +7, +7, +6 +], +"lo": [ +5, +5, +6, +5, +6, +5 +] +}, +{ +"decoder": 3, +"dense": [ +7, +7, +6, +7, +7, +7 +], +"lo": [ +6, +5, +6, +5, +5, +5 +] +}, +{ +"decoder": 4, +"dense": [ +7, +7, +7, +6, +7, +6 +], +"lo": [ +5, +7, +5, +6, +6, +6 +] +}, +{ +"decoder": 5, +"dense": [ +7, +7, +7, +6, +7, +7 +], +"lo": [ +7, +7, +5, +4, +7, +5 +] +}, +{ +"decoder": 6, +"dense": [ +7, +6, +7, +7, +6, +7 +], +"lo": [ +5, +6, +6, +5, +5, +5 +] +}, +{ +"decoder": 7, +"dense": [ +7, +7, +6, +7, +6, +6 +], +"lo": [ +5, +5, +5, +5, +5, +5 +] +}, +{ +"decoder": 8, +"dense": [ +7, +7, +7, +6, +7, +7 +], +"lo": [ +6, +6, +5, +5, +5, +5 +] +}, +{ +"decoder": 9, +"dense": [ +7, +6, +6, +6, +7, +7 +], +"lo": [ +6, +5, +5, +6, +5, +5 +] +}, +{ +"decoder": 10, +"dense": [ +7, +7, +6, +6, +7, +6 +], +"lo": [ +6, +5, +5, +6, +6, +5 +] +}, +{ +"decoder": 11, +"dense": [ +7, +7, +6, +7, +7, +6 +], +"lo": [ +6, +6, +5, +5, +5, +6 +] +}, +{ +"decoder": 12, +"dense": [ +7, +7, +6, +7, +7, +7 +], +"lo": [ +5, +6, +5, +6, +5, +5 +] +}, +{ +"decoder": 13, +"dense": [ +7, +7, +6, +7, +7, +6 +], +"lo": [ +6, +5, +5, +6, +6, +5 +] +}, +{ +"decoder": 14, +"dense": [ +7, +7, +7, +7, +5, +7 +], +"lo": [ +5, +5, +7, +7, +5, +6 +] +}, +{ +"decoder": 15, +"dense": [ +6, +7, +7, +7, +7, +7 +], +"lo": [ +4, +5, +7, +6, +5, +7 +] +}, +{ +"decoder": 16, +"dense": [ +7, +6, +6, +7, +7, +6 +], +"lo": [ +5, +5, +6, +5, +5, +5 +] +}, +{ +"decoder": 17, +"dense": [ +6, +7, +7, +7, +7, +7 +], +"lo": [ +4, +5, +6, +7, +6, +5 +] +}, +{ +"decoder": 18, +"dense": [ +6, +6, +7, +6, +6, +7 +], +"lo": [ +5, +5, +6, +6, +5, +5 +] +}, +{ +"decoder": 19, +"dense": [ +7, +7, +7, +6, +6, +7 +], +"lo": [ +5, +5, +5, +6, +5, +6 +] +}, +{ +"decoder": 20, +"dense": [ +6, +6, +7, +7, +7, +6 +], +"lo": [ +5, +5, +5, +5, +7, +6 +] +}, +{ +"decoder": 21, +"dense": [ +7, +7, +7, +7, +7, +6 +], +"lo": [ +6, +6, +5, +5, +7, +6 +] +}, +{ +"decoder": 22, +"dense": [ +7, +6, +7, +7, +7, +6 +], +"lo": [ +5, +5, +6, +5, +5, +5 +] +}, +{ +"decoder": 23, +"dense": [ +7, +7, +7, +7, +7, +6 +], +"lo": [ +7, +7, +5, +5, +7, +4 +] +}, +{ +"decoder": 24, +"dense": [ +7, +7, +7, +6, +7, +7 +], +"lo": [ +6, +5, +6, +6, +6, +5 +] +}, +{ +"decoder": 25, +"dense": [ +7, +7, +6, +6, +6, +7 +], +"lo": [ +5, +6, +5, +5, +5, +5 +] +} +] +} \ No newline at end of file diff --git a/research/overcomplete-and-realisation-floors/data/e6_square_decoders.json b/research/overcomplete-and-realisation-floors/data/e6_square_decoders.json new file mode 100644 index 0000000000..af72db95c1 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/data/e6_square_decoders.json @@ -0,0 +1 @@ +{"note": "26 square (m = T = 6) root-code decoders of E6 found by hill climbing over bases of maximal-flat normals (author working card IDEAL2/e6square.py, outputs e6square_hc.pkl and e6square_hc_7.pkl, sha256 below). X[j], Y[j]: primitive integer ambient vectors x_j, y_j of the term j (p_j = R x_j, q_j = R y_j, R = the 36 E6 lines of lib/roots.py). Used by sat/ to build the shared-B instances.", "sources": {"e6square_hc.pkl": "420e8a8089bc02d8883ce6a20d88dd6e612278b63cd452995b1dcfd2f557c7c1", "e6square_hc_7.pkl": "e107c677e57d23e34047efd0144bb52961b6e71f2014eb2b97b91217b52ab54c"}, "decoders": [{"X": [[0, 0, 0, 0, 0, 0, 1, -1], [0, 0, 0, 1, -2, 0, 0, 1], [1, -1, -1, -3, 3, -3, 3, -3], [1, -1, -1, -3, 0, 0, 0, 0], [1, -1, -1, -3, -3, 0, -3, 0], [1, -1, -1, 0, -3, 0, -3, 3]], "Y": [[1, -1, -1, 1, 0, 1, 0, -1], [1, -1, -1, 1, 1, 1, -1, -1], [0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, -1, -1, -1], [1, -1, -1, 1, 1, 1, 1, 1], [3, -3, -3, 3, 1, 1, -1, -1]], "dense": [7, 7, 6, 6, 7, 7], "bio": true, "sum": 40}, {"X": [[1, -1, -1, 1, 1, -1, 1, -1], [1, -1, -1, -1, 1, 1, -1, 1], [1, -1, -1, 0, -1, 1, -1, 0], [1, -1, -1, 0, 1, -1, -1, 0], [1, -1, -1, 0, -1, 1, 1, 0], [1, -1, -1, 1, -1, 1, 0, 0]], "Y": [[0, 0, 0, 0, 1, 1, 0, -2], [0, 0, 0, 0, 1, 1, 0, 0], [0, 0, 0, 1, 0, -1, 1, 3], [1, -1, -1, 0, -3, -6, 0, 6], [1, -1, -1, -3, -3, -3, 3, 3], [0, 0, 0, 1, 0, 0, 0, 1]], "dense": [7, 7, 7, 7, 6, 7], "bio": true, "sum": 41}, {"X": [[1, -1, -1, 1, -1, 3, -1, 3], [0, 0, 0, 1, 0, 1, 0, 0], [1, -1, -1, 3, -3, -3, -3, 3], [1, -1, -1, 1, -1, 0, 0, 1], [1, -1, -1, -3, 3, 9, 3, -3], [1, -1, -1, 0, 0, -3, 0, 0]], "Y": [[0, 0, 0, 0, 1, 0, 0, 1], [0, 0, 0, 1, 1, 0, 0, 0], [1, -1, -1, -1, -3, 1, -7, -5], [0, 0, 0, 0, 1, 0, -1, 0], [1, -1, -1, -1, -1, 1, -3, -3], [1, -1, -1, 1, 5, -1, -1, 1]], "dense": [7, 7, 6, 7, 7, 6], "bio": true, "sum": 40}, {"X": [[1, -1, -1, 3, -9, 3, -3, 3], [1, -1, -1, 1, -1, 1, -1, 1], [1, -1, -1, -3, 3, -3, 3, -3], [1, -1, -1, -1, 0, 0, 1, 1], [0, 0, 0, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 1, 0, 1]], "Y": [[0, 0, 0, 0, 1, -1, -1, 1], [1, -1, -1, 1, 1, 1, -1, -1], [1, -1, -1, -3, -1, 7, 1, -7], [0, 0, 0, 0, 0, 1, 0, -1], [0, 0, 0, 1, 0, 0, 1, 0], [0, 0, 0, 1, 0, -2, 0, 1]], "dense": [7, 7, 6, 7, 7, 7], "bio": true, "sum": 41}, {"X": [[1, -1, -1, 1, 0, 0, 1, 1], [1, -1, -1, 1, -3, -1, 1, 3], [1, -1, -1, -1, 1, -3, -3, -1], [1, -1, -1, 3, 3, -3, -3, -3], [1, -1, -1, 3, -3, 0, 0, 3], [1, -1, -1, 0, 0, 0, 3, 0]], "Y": [[0, 0, 0, 0, 1, 0, 0, 1], [3, -3, -3, -1, -3, 7, -3, -5], [1, -1, -1, -1, -1, 1, -1, -1], [3, -3, -3, -5, -3, 7, -3, -1], [1, -1, -1, 0, -2, 2, -1, -2], [3, -3, -3, -1, -7, 3, 1, -9]], "dense": [7, 7, 7, 6, 7, 6], "bio": true, "sum": 40}, {"X": [[1, -1, -1, 1, -1, -1, -1, -1], [1, -1, -1, 3, -3, -1, -1, -1], [1, -1, -1, -1, -1, 0, 0, 1], [1, -1, -1, -3, -3, 3, 3, -3], [0, 0, 0, 1, 1, 0, 0, -2], [0, 0, 0, 0, 1, -1, 0, 0]], "Y": [[3, -3, -3, 13, 5, 5, 19, 9], [1, -1, -1, 7, -1, -1, 9, 3], [1, -1, -1, 1, 1, 1, 1, 1], [1, -1, -1, 3, 3, 3, -3, 3], [3, -3, -3, 5, 5, 5, 3, 1], [0, 0, 0, 0, 0, 1, -1, 0]], "dense": [7, 7, 7, 6, 7, 7], "bio": true, "sum": 41}, {"X": [[1, -1, -1, -1, -1, -1, -1, 1], [1, -1, -1, 0, 0, 3, 0, 0], [1, -1, -1, 1, 1, 1, 1, 1], [1, -1, -1, 3, -1, -1, 1, 3], [1, -1, -1, -3, -3, -3, -3, -3], [1, -1, -1, 1, -1, 1, -1, 1]], "Y": [[0, 0, 0, 1, 0, 0, 0, -1], [0, 0, 0, 1, 2, -1, -2, 0], [1, -1, -1, 1, 3, -1, -1, -1], [0, 0, 0, 0, 1, 0, -1, 0], [1, -1, -1, 1, -1, -1, 1, -3], [0, 0, 0, 1, 1, 0, -2, 0]], "dense": [7, 6, 7, 7, 6, 7], "bio": true, "sum": 40}, {"X": [[0, 0, 0, 1, -1, 0, 0, 0], [1, -1, -1, -1, -3, 3, 1, 1], [1, -1, -1, 0, 0, 3, 0, 0], [0, 0, 0, 0, 1, 0, -1, 0], [1, -1, -1, 3, -3, 3, -3, -3], [1, -1, -1, 3, 3, -3, -3, -3]], "Y": [[0, 0, 0, 1, 0, 0, 0, 1], [1, -1, -1, -1, -1, -1, -1, 1], [0, 0, 0, 1, 1, 1, 1, 0], [0, 0, 0, 0, 0, 0, 1, -1], [1, -1, -1, 1, 1, -1, 1, 3], [1, -1, -1, 0, 0, -1, 0, 0]], "dense": [7, 7, 6, 7, 6, 6], "bio": true, "sum": 39}, {"X": [[1, -1, -1, 3, 3, 0, 0, 3], [1, -1, -1, 9, 3, -3, -3, 3], [1, -1, -1, 1, 1, -1, -1, 1], [1, -1, -1, -3, 3, 3, -3, -3], [0, 0, 0, 1, 0, -1, 0, 0], [1, -1, -1, -1, 1, 1, 1, -1]], "Y": [[1, -1, -1, 1, -3, 1, -1, -1], [1, -1, -1, 3, -3, 3, -1, -1], [1, -1, -1, -1, -1, -1, -1, 1], [1, -1, -1, 1, -3, 1, 1, 1], [1, -1, -1, 1, -3, 3, -1, 1], [1, -1, -1, 1, -1, 1, 1, -1]], "dense": [7, 7, 7, 6, 7, 7], "bio": true, "sum": 41}, {"X": [[1, -1, -1, -3, 3, -3, -3, -9], [1, -1, -1, 0, 0, 0, 3, 0], [1, -1, -1, 3, 3, -3, 3, 3], [1, -1, -1, 3, 3, -3, -3, -3], [0, 0, 0, 0, 0, 1, 0, 1], [1, -1, -1, 1, -1, -3, -1, -3]], "Y": [[1, -1, -1, -3, 1, -1, -1, 1], [0, 0, 0, 1, 0, 1, 1, -1], [1, -1, -1, -3, 1, -3, -1, 3], [1, -1, -1, -5, -1, -3, -1, 3], [1, -1, -1, -1, 0, 0, -1, 1], [1, -1, -1, -1, -1, -1, -1, 1]], "dense": [7, 6, 6, 6, 7, 7], "bio": true, "sum": 39}, {"X": [[1, -1, -1, 3, -3, 0, 0, -3], [1, -1, -1, -3, -3, -1, 1, -1], [1, -1, -1, 0, 0, 0, 0, 0], [1, -1, -1, -3, 0, 0, 0, 0], [1, -1, -1, 9, -3, 3, 3, -3], [1, -1, -1, 3, 3, 3, 3, -3]], "Y": [[0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 1, -1, 0], [1, -1, -1, 1, 1, -3, 1, 1], [0, 0, 0, 1, 1, -4, 2, 0], [0, 0, 0, 0, 1, -2, 0, -1], [0, 0, 0, 0, 1, -1, 1, -1]], "dense": [7, 7, 6, 6, 7, 6], "bio": true, "sum": 39}, {"X": [[1, -1, -1, 0, -3, 0, 3, -3], [2, -2, -2, 0, 0, 0, 3, -3], [1, -1, -1, 3, -3, -3, -3, 3], [1, -1, -1, -1, 3, 1, -3, 1], [1, -1, -1, 3, 3, -1, 1, -1], [1, -1, -1, 3, -3, -3, 3, -3]], "Y": [[0, 0, 0, 2, -1, 3, -1, -1], [1, -1, -1, -3, 1, -3, 3, 3], [1, -1, -1, 1, -1, 1, 1, 3], [0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 1, 0, 1, 0, 0], [1, -1, -1, 3, -1, 7, 3, 5]], "dense": [7, 7, 6, 7, 7, 6], "bio": true, "sum": 40}, {"X": [[0, 0, 0, 1, 2, 0, -1, 0], [1, -1, -1, -3, -3, 0, 3, 0], [1, -1, -1, 0, 0, 0, 0, 0], [5, -5, -5, -3, -3, 3, 3, -3], [0, 0, 0, 1, 1, 0, 0, 0], [1, -1, -1, -1, -1, -1, 1, -1]], "Y": [[0, 0, 0, 1, -1, 0, 0, 0], [0, 0, 0, 1, -1, 0, -1, -1], [1, -1, -1, 1, -1, -1, -1, 3], [0, 0, 0, 0, 0, 1, 0, -1], [0, 0, 0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1, 0, 1]], "dense": [7, 7, 6, 7, 7, 7], "bio": true, "sum": 41}, {"X": [[1, -1, -1, -3, 9, 3, 3, -3], [1, -1, -1, 1, -1, 1, -3, 3], [1, -1, -1, -3, -3, -3, 3, 3], [1, -1, -1, -3, -3, -3, -3, 9], [1, -1, -1, -3, 3, 3, -3, 9], [1, -1, -1, 3, -3, -3, 3, -3]], "Y": [[1, -1, -1, -1, 1, -1, -1, -1], [1, -1, -1, -1, -1, 1, -1, -1], [1, -1, -1, -3, -3, 3, 1, -1], [0, 0, 0, 0, 1, -2, -1, 0], [1, -1, -1, -3, -1, -1, -3, -3], [0, 0, 0, 2, 1, -1, 1, 1]], "dense": [7, 7, 6, 7, 7, 6], "bio": true, "sum": 40}, {"X": [[1, -1, -1, 1, 3, 1, -3, -1], [0, 0, 0, 1, 1, 0, 0, -2], [1, -1, -1, 3, 1, 1, -1, -3], [1, -1, -1, -1, 1, 0, 0, 1], [1, -1, -1, 0, 0, 0, 0, 0], [1, -1, -1, -1, 1, 1, -1, 1]], "Y": [[0, 0, 0, 0, 0, 1, 1, 0], [0, 0, 0, 1, 0, 0, 0, 1], [0, 0, 0, 3, 1, 2, 2, 2], [0, 0, 0, 1, 1, 0, 1, 1], [1, -1, -1, -3, -3, -3, -3, -3], [0, 0, 0, 3, 1, -4, 0, 2]], "dense": [7, 7, 7, 7, 5, 7], "bio": true, "sum": 40}, {"X": [[1, -1, -1, 3, 3, 3, -3, 3], [1, -1, -1, 1, 0, 0, -1, -1], [1, -1, -1, 0, 0, -1, -1, -1], [0, 0, 0, 1, -2, 0, -1, 0], [0, 0, 0, 2, 1, 0, 1, 0], [0, 0, 0, 1, 1, 0, 2, 0]], "Y": [[1, -1, -1, 0, 0, 0, 0, 3], [0, 0, 0, 0, 0, 1, 0, -1], [1, -1, -1, 0, 0, -4, 0, 3], [1, -1, -1, 1, -3, -1, 1, 3], [1, -1, -1, -1, -1, 1, 1, 1], [7, -7, -7, -2, -6, 2, 10, 9]], "dense": [6, 7, 7, 7, 7, 7], "bio": true, "sum": 41}, {"X": [[1, -1, -1, -1, 1, 1, -1, 1], [1, -1, -1, -3, -3, -3, -3, -3], [1, -1, -1, 3, -3, 3, -3, -3], [0, 0, 0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1, -2, -1], [1, -1, -1, -3, 3, -3, -3, 3]], "Y": [[1, -1, -1, -1, 1, 1, 1, -1], [1, -1, -1, -1, 1, -1, 1, -3], [0, 0, 0, 1, -2, 0, -1, 2], [1, -1, -1, 1, 3, -1, 1, -3], [0, 0, 0, 0, 1, 0, 0, -1], [0, 0, 0, 1, 0, -1, -1, 1]], "dense": [7, 6, 6, 7, 7, 6], "bio": true, "sum": 39}, {"X": [[1, -1, -1, 0, 0, 0, 3, 0], [1, -1, -1, -1, -1, -1, -1, 1], [1, -1, -1, -1, 1, -1, -1, -1], [1, -1, -1, -3, -1, -3, -1, 1], [1, -1, -1, -1, 1, 3, 3, -1], [0, 0, 0, 0, 1, 0, -1, 0]], "Y": [[1, -1, -1, 3, 3, -3, 3, 3], [1, -1, -1, 1, -1, 1, -1, 1], [1, -1, -1, 1, -1, -1, -1, -5], [1, -1, -1, 3, -1, 1, -1, -1], [0, 0, 0, 1, 0, -1, 0, 0], [0, 0, 0, 0, 1, 0, 0, 1]], "dense": [6, 7, 7, 7, 7, 7], "bio": true, "sum": 41}, {"X": [[1, -1, -1, -3, 3, -3, 3, -3], [1, -1, -1, -3, -3, 3, 3, -3], [1, -1, -1, -1, -1, -1, 1, -1], [1, -1, -1, 0, 0, 0, -3, 0], [1, -1, -1, -3, 0, 0, 0, 0], [1, -1, -1, 1, -3, -3, -1, -1]], "Y": [[1, -1, -1, 1, -1, 1, 1, 3], [1, -1, -1, 1, 1, -1, 1, 3], [3, -3, -3, 3, 1, 1, 3, 3], [1, -1, -1, 1, 1, 1, -1, -1], [1, -1, -1, 3, 1, 1, 1, -1], [1, -1, -1, 1, 1, 1, 1, 1]], "dense": [6, 6, 7, 6, 6, 7], "bio": true, "sum": 38}, {"X": [[1, -1, -1, -1, 1, 3, -1, 3], [0, 0, 0, 2, 0, -1, 1, 0], [1, -1, -1, -1, 1, 1, -1, 1], [1, -1, -1, -3, -3, 3, -3, 3], [1, -1, -1, 0, 0, 0, -3, 0], [1, -1, -1, 9, 3, -3, 3, -3]], "Y": [[1, -1, -1, -1, -1, -1, 1, -1], [0, 0, 0, 0, 0, 1, 0, -1], [1, -1, -1, -1, 0, -1, 1, 0], [1, -1, -1, 0, -2, 1, 1, -1], [1, -1, -1, -1, -1, 1, 3, -1], [1, -1, -1, 1, -1, 3, 1, -3]], "dense": [7, 7, 7, 6, 6, 7], "bio": true, "sum": 40}, {"X": [[1, -1, -1, 0, 0, 0, 3, 0], [1, -1, -1, 3, 3, -3, 3, 3], [1, -1, -1, -3, -1, 1, 1, 3], [1, -1, -1, 3, 1, -3, 1, -1], [1, -1, -1, 3, 0, -3, 3, 0], [1, -1, -1, -3, -3, 3, -3, 3]], "Y": [[1, -1, -1, 1, 1, 3, 1, -1], [0, 0, 0, 1, 1, 1, 0, 1], [0, 0, 0, 1, 0, 1, 0, 0], [1, -1, -1, -1, 1, -1, -1, -1], [1, -1, -1, -1, 3, 1, -1, -1], [1, -1, -1, 3, -1, 3, -1, 1]], "dense": [6, 6, 7, 7, 7, 6], "bio": true, "sum": 39}, {"X": [[1, -1, -1, -3, 3, -9, 3, 3], [1, -1, -1, -3, 0, 0, 3, -3], [1, -1, -1, 1, -3, 1, -3, 1], [1, -1, -1, -1, -3, 3, 1, 1], [1, -1, -1, 0, -3, 3, 0, -3], [1, -1, -1, 3, -3, 3, -3, -3]], "Y": [[0, 0, 0, 1, -1, -1, 1, 0], [1, -1, -1, -1, 2, 1, -1, 0], [0, 0, 0, 1, 0, 0, 1, 0], [1, -1, -1, 1, 1, 1, 1, 1], [2, -2, -2, 0, 4, 2, -1, 1], [5, -5, -5, 5, 7, 3, 1, 1]], "dense": [7, 7, 7, 7, 7, 6], "bio": true, "sum": 41}, {"X": [[1, -1, -1, -1, -1, -1, 1, -1], [1, -1, -1, -3, 3, 3, 3, 3], [1, -1, -1, 9, -3, -3, -3, -3], [1, -1, -1, -1, 1, -1, 1, 1], [1, -1, -1, 3, -1, -1, -1, -3], [1, -1, -1, -3, 0, 0, 0, 0]], "Y": [[0, 0, 0, 0, 1, 0, -1, 0], [1, -1, -1, 1, -1, 3, 1, 1], [1, -1, -1, 1, -3, 1, -1, 3], [0, 0, 0, 0, 1, -1, 0, 0], [0, 0, 0, 0, 1, 0, 0, -1], [1, -1, -1, -1, -1, 1, -3, 1]], "dense": [7, 6, 7, 7, 7, 6], "bio": true, "sum": 40}, {"X": [[1, -1, -1, -1, -3, 1, -3, -1], [1, -1, -1, -1, 1, 1, 3, -3], [1, -1, -1, 0, 0, 1, -1, -1], [1, -1, -1, 1, -3, -1, -3, -1], [0, 0, 0, 0, 2, 0, 1, 1], [1, -1, -1, 3, 3, -3, -3, -3]], "Y": [[1, -1, -1, -29, 5, -13, -9, -1], [3, -3, -3, -7, -1, -7, 5, -3], [0, 0, 0, 1, 0, 1, 0, 0], [1, -1, -1, 1, -1, -1, 1, 1], [7, -7, -7, -11, 3, -11, 1, 9], [1, -1, -1, 3, -3, 3, 3, 3]], "dense": [7, 7, 7, 7, 7, 6], "bio": true, "sum": 41}, {"X": [[1, -1, -1, 3, 0, -3, 3, 0], [1, -1, -1, 1, -1, -1, 1, 1], [1, -1, -1, 0, 3, 3, 0, -3], [1, -1, -1, -3, 3, -3, 3, -3], [0, 0, 0, 1, 0, 1, 0, 0], [0, 0, 0, 2, -1, 1, 0, 0]], "Y": [[0, 0, 0, 1, 1, -1, -1, 0], [1, -1, -1, -1, -1, 1, 1, 1], [1, -1, -1, 1, 1, -1, -3, -1], [1, -1, -1, 3, 3, -3, -7, 1], [1, -1, -1, 1, -1, -3, -5, -3], [0, 0, 0, 0, 1, 0, 0, 1]], "dense": [7, 7, 7, 6, 7, 7], "bio": true, "sum": 41}, {"X": [[1, -1, -1, -3, -1, -1, 1, -3], [1, -1, -1, -3, -3, -3, 0, 0], [1, -1, -1, 3, 3, 3, 3, -3], [1, -1, -1, 0, 0, 0, 0, -3], [1, -1, -1, -3, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 1]], "Y": [[1, -1, -1, 1, -1, 1, -1, 1], [1, -1, -1, 1, -1, -1, 1, 1], [1, -1, -1, 1, -1, 1, 3, 1], [1, -1, -1, 1, 1, -1, -3, -1], [1, -1, -1, -1, -1, 3, -1, 1], [0, 0, 0, 0, 1, -1, 0, 0]], "dense": [7, 7, 6, 6, 6, 7], "bio": true, "sum": 39}]} \ No newline at end of file diff --git a/research/overcomplete-and-realisation-floors/data/p104_frames.json b/research/overcomplete-and-realisation-floors/data/p104_frames.json new file mode 100644 index 0000000000..3110c60982 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/data/p104_frames.json @@ -0,0 +1,221 @@ +{ + "note": "E8 lines (doubled coordinates, order of lib/roots.py:E8). F0,F1: two orthogonal 8-frames (indices into E8); pairs: the 8 (a in F0, b in F1) frame pairs of the #968 decoder as recorded in the author working card SHAPES/p104.pkl.", + "F0": [ + 0, + 26, + 44, + 54, + 1, + 27, + 45, + 55 + ], + "F1": [ + 2, + 18, + 40, + 52, + 77, + 86, + 107, + 112 + ], + "pairs": [ + [ + 0, + 77 + ], + [ + 26, + 40 + ], + [ + 44, + 52 + ], + [ + 1, + 107 + ], + [ + 2, + 27 + ], + [ + 18, + 45 + ], + [ + 86, + 54 + ], + [ + 112, + 55 + ] + ], + "F0_vectors": [ + [ + 2, + 2, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 2, + 2, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 2, + 2, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 2, + 2 + ], + [ + 2, + -2, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 2, + -2, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 2, + -2, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 2, + -2 + ] + ], + "F1_vectors": [ + [ + 2, + 0, + 2, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 2, + 0, + 0, + 2, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 2, + 0, + 0, + 2, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 2, + 0, + 2 + ], + [ + 1, + 1, + -1, + 1, + -1, + 1, + -1, + -1 + ], + [ + 1, + 1, + -1, + -1, + -1, + -1, + 1, + 1 + ], + [ + 1, + -1, + -1, + 1, + 1, + -1, + -1, + 1 + ], + [ + 1, + -1, + -1, + -1, + 1, + 1, + 1, + -1 + ] + ] +} \ No newline at end of file diff --git a/research/overcomplete-and-realisation-floors/lean/BUILD_OUTPUT.txt b/research/overcomplete-and-realisation-floors/lean/BUILD_OUTPUT.txt new file mode 100644 index 0000000000..24eeff0ad9 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/lean/BUILD_OUTPUT.txt @@ -0,0 +1,5 @@ +# Output of 'lake build' in lean/ (Lean leanprover/lean4:v4.33.0-rc1, Mathlib rev 55b7917b472489c7c56a0fa6393af05cfd536599), author's machine, 2026-10-14 +info: Overcomplete.lean:44:0: 'DecoderDuality.oc_idempotent' depends on axioms: [propext, Classical.choice, Quot.sound] +info: Overcomplete.lean:45:0: 'DecoderDuality.oc_complement_trace' depends on axioms: [propext, Classical.choice, Quot.sound] +info: Overcomplete.lean:46:0: 'DecoderDuality.oc_square' depends on axioms: [propext, Classical.choice, Quot.sound] +Build completed successfully (8670 jobs). diff --git a/research/overcomplete-and-realisation-floors/lean/Overcomplete.lean b/research/overcomplete-and-realisation-floors/lean/Overcomplete.lean new file mode 100644 index 0000000000..58d9d97a92 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/lean/Overcomplete.lean @@ -0,0 +1,46 @@ +import Mathlib + +/-! +# Lemma OC (overcomplete coupling), IDEAL2 §2.1 + +If `X : T × m` and `Y : T × m` satisfy `X * Yᵀ = 1` (an `m`-term exact decomposition of the +identity, `Σⱼ xⱼ yⱼᵀ = I_T`), then the `m × m` cross matrix `K = Yᵀ * X` (entries `⟨y_i, x_j⟩`) is +idempotent with trace `T`; hence `Π = 1 - K` is idempotent with trace `m - T`. +-/ + +open Matrix +set_option linter.unusedSectionVars false + +namespace DecoderDuality + +variable {K : Type*} [Field K] {T m : Type*} [Fintype T] [Fintype m] [DecidableEq T] [DecidableEq m] + +theorem oc_idempotent (X Y : Matrix T m K) (h : X * Yᵀ = 1) : + (Yᵀ * X) * (Yᵀ * X) = Yᵀ * X := by + calc (Yᵀ * X) * (Yᵀ * X) = Yᵀ * (X * Yᵀ) * X := by simp only [Matrix.mul_assoc] + _ = Yᵀ * X := by rw [h, Matrix.mul_one] + +theorem oc_trace (X Y : Matrix T m K) (h : X * Yᵀ = 1) : + (Yᵀ * X).trace = (Fintype.card T : K) := by + rw [Matrix.trace_mul_comm, h, Matrix.trace_one] + +theorem oc_complement_idempotent (X Y : Matrix T m K) (h : X * Yᵀ = 1) : + (1 - Yᵀ * X) * (1 - Yᵀ * X) = 1 - Yᵀ * X := by + have hK := oc_idempotent X Y h + rw [sub_mul, mul_sub, mul_sub, Matrix.one_mul, Matrix.mul_one, Matrix.one_mul, hK] + abel + +theorem oc_complement_trace (X Y : Matrix T m K) (h : X * Yᵀ = 1) : + (1 - Yᵀ * X).trace = (Fintype.card m : K) - (Fintype.card T : K) := by + rw [Matrix.trace_sub, Matrix.trace_one, oc_trace X Y h] + +/-- Square case (biorthogonality): if moreover `Yᵀ * X` has an inverse-side identity, it is `1`. +For `m = T` with `X * Yᵀ = 1` we get `Yᵀ * X = 1`, i.e. `⟨y_i, x_j⟩ = δ_ij`. -/ +theorem oc_square (X Y : Matrix T T K) (h : X * Yᵀ = 1) : Yᵀ * X = 1 := + mul_eq_one_comm.mp h + +end DecoderDuality + +#print axioms DecoderDuality.oc_idempotent +#print axioms DecoderDuality.oc_complement_trace +#print axioms DecoderDuality.oc_square diff --git a/research/overcomplete-and-realisation-floors/lean/README.md b/research/overcomplete-and-realisation-floors/lean/README.md new file mode 100644 index 0000000000..2b8ce7e910 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/lean/README.md @@ -0,0 +1,19 @@ +# Lean 4 check of the algebraic core of Lemma OC + +`Overcomplete.lean` (46 lines) proves, over any field `K` and finite index types `T, m`, for `X, Y : Matrix T m K` with `X * Yᵀ = 1`: + +* `oc_idempotent`: `(Yᵀ * X) * (Yᵀ * X) = Yᵀ * X`; +* `oc_trace`: `trace (Yᵀ * X) = T`; `oc_complement_idempotent`, `oc_complement_trace`: `1 - Yᵀ * X` is idempotent with trace `m - T`; +* `oc_square`: for `m = T`, `Yᵀ * X = 1` (biorthogonality). + +No `sorry`; `#print axioms` lists only `propext`, `Classical.choice`, `Quot.sound`. **Not formalised** (written proofs in the README, §4): the rank statements `rank K = T`, the converse (`K² = K` and `rank X = rank Y = T` ⇒ `Σ x_j ⊗ y_j = id`), and the `m = T + 1` rank-one consequences. + +Build (needs network the first time for Mathlib; the Mathlib revision is pinned in `lake-manifest.json`): + +```sh +cd research/overcomplete-and-realisation-floors/lean +lake exe cache get # optional: prebuilt Mathlib +lake build +``` + +`BUILD_OUTPUT.txt` is the output of this build on the author's machine (Lean toolchain in `lean-toolchain`, built against an already built Mathlib of the same pinned revision). The file is the one maintained in the author's project `decoder-duality` (which also holds the Shadow Lemma `lemmaCR2` cited in README §9.2; that file is not shipped here). diff --git a/research/overcomplete-and-realisation-floors/lean/lake-manifest.json b/research/overcomplete-and-realisation-floors/lean/lake-manifest.json new file mode 100644 index 0000000000..94973914f6 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/lean/lake-manifest.json @@ -0,0 +1,117 @@ +{ + "version": "1.2.0", + "packagesDir": ".lake/packages", + "packages": [ + { + "url": "https://github.com/leanprover-community/mathlib4.git", + "type": "git", + "subDir": null, + "scope": "", + "rev": "55b7917b472489c7c56a0fa6393af05cfd536599", + "name": "mathlib", + "manifestFile": "lake-manifest.json", + "inputRev": null, + "inherited": false, + "configFile": "lakefile.lean" + }, + { + "url": "https://github.com/leanprover-community/plausible", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "b1c4a69a7e247ab7df20460212001673d74f08c0", + "name": "plausible", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "url": "https://github.com/leanprover-community/LeanSearchClient", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "0498c7c070c143a3bf7379f4d99a2c63bb9d9715", + "name": "LeanSearchClient", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "url": "https://github.com/leanprover-community/import-graph", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "18a90119a5d316358fde6c86e0ca24e59212e32c", + "name": "importGraph", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "url": "https://github.com/leanprover-community/ProofWidgets4", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "b1436dc749e722c9920036b52cdc43b3451d0b69", + "name": "proofwidgets", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.lean" + }, + { + "url": "https://github.com/leanprover-community/aesop", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "57d3325be72a842920813bcb40f96a6f7393c185", + "name": "aesop", + "manifestFile": "lake-manifest.json", + "inputRev": "master", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "url": "https://github.com/leanprover-community/quote4", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "ee41917ae11d38479fb8fb24745f7ca4bf0a784d", + "name": "Qq", + "manifestFile": "lake-manifest.json", + "inputRev": "master", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "url": "https://github.com/leanprover-community/batteries", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "2c810760f0a0c4536b397dbe30ca9b2f2f467366", + "name": "batteries", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "url": "https://github.com/leanprover/lean4-cli", + "type": "git", + "subDir": null, + "scope": "leanprover", + "rev": "da07ca808b6718cb2aed14dba154e5a08b8f8ecf", + "name": "Cli", + "manifestFile": "lake-manifest.json", + "inputRev": "v4.33.0-rc1", + "inherited": true, + "configFile": "lakefile.toml" + } + ], + "name": "Overcomplete", + "lakeDir": ".lake", + "fixedToolchain": false +} \ No newline at end of file diff --git a/research/overcomplete-and-realisation-floors/lean/lakefile.toml b/research/overcomplete-and-realisation-floors/lean/lakefile.toml new file mode 100644 index 0000000000..ae1ee720f1 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/lean/lakefile.toml @@ -0,0 +1,9 @@ +name = "Overcomplete" +defaultTargets = ["Overcomplete"] + +[[require]] +name = "mathlib" +git = "https://github.com/leanprover-community/mathlib4.git" + +[[lean_lib]] +name = "Overcomplete" diff --git a/research/overcomplete-and-realisation-floors/lean/lean-toolchain b/research/overcomplete-and-realisation-floors/lean/lean-toolchain new file mode 100644 index 0000000000..fd85b262bf --- /dev/null +++ b/research/overcomplete-and-realisation-floors/lean/lean-toolchain @@ -0,0 +1 @@ +leanprover/lean4:v4.33.0-rc1 diff --git a/research/overcomplete-and-realisation-floors/lib/dirs.py b/research/overcomplete-and-realisation-floors/lib/dirs.py new file mode 100644 index 0000000000..649868b5ec --- /dev/null +++ b/research/overcomplete-and-realisation-floors/lib/dirs.py @@ -0,0 +1,61 @@ +"""Deficient directions of a unit under the Gram realisation (exact, Fractions), pair ranks, bilinear-form space, reflections.""" +from math import gcd +from fractions import Fraction as Fr +from roots import * + +def span_basis(L, T): + """T linearly independent lines of L (greedy) -- an ambient basis of the span""" + B = [] + for r in L: + if rank_q(B + [r]) > len(B): B.append(r) + if len(B) == T: break + assert len(B) == T + return B + +def primitive(x): + x = [Fr(c) for c in x] + d = 1 + for c in x: d = d * c.denominator // gcd(d, c.denominator) + w = [int(c * d) for c in x]; g = 0 + for c in w: g = gcd(g, abs(c)) + w = [c // g for c in w] + for c in w: + if c != 0: + if c < 0: w = [-y for y in w] + break + return tuple(w) + +def classes(L): + """(B rows, basis of col(B), h, list of nonzero class supports as bitmasks)""" + Br = b0rows(L); basis = colspace_basis(Br) + return Br, basis, len(basis), [w for _, w in span_vectors(basis)] + +def deficient_directions(L, T): + """dict: class-support bitmask w -> primitive integer direction x (ambient) with L[t].x = 0 for all t in the class, for the classes + whose lines do not span; asserts the annihilator is 1-dimensional in each such class.""" + Bm = span_basis(L, T); Br, basis, h, cl = classes(L); out = {} + for w in cl: + O = bits(w); rows = [[dot(L[t], b) for b in Bm] for t in O] + if rank_q(rows) < T: + ns = nullspace_q(rows, T); assert len(ns) == 1, "annihilator of a deficient class is not 1-dimensional" + n = ns[0] + x = [sum(n[i] * Bm[i][k] for i in range(T)) for k in range(len(Bm[0]))] + out[w] = primitive(x) + return out + +def support(L, x): return [t for t in range(len(L)) if dot(L[t], x) != 0] + +def reflect(v, r): + c = Fr(2 * dot(v, r), dot(r, r)) + return primitive([a - c * b for a, b in zip(v, r)]) + +def bilinear_space_dim(L, T): + """dimension of {N in V (x) V : r_t^T N r_s = 0 for every orthogonal pair t, s} (V = span of the lines), exact""" + Bm = span_basis(L, T); v = len(L); G = gram(L) + a = [[dot(L[t], b) for b in Bm] for t in range(v)] + rows = [] + for t in range(v): + for s in range(v): + if t != s and G[t][s] == 0: + rows.append([a[t][i] * a[s][j] for i in range(T) for j in range(T)]) + return T * T - rank_q(rows) diff --git a/research/overcomplete-and-realisation-floors/lib/frames.py b/research/overcomplete-and-realisation-floors/lib/frames.py new file mode 100644 index 0000000000..150bab4c3c --- /dev/null +++ b/research/overcomplete-and-realisation-floors/lib/frames.py @@ -0,0 +1,39 @@ +"""Frame-intersection certificate (Lemma 3.3) for the label classes of the dense pairing; exact bitmask computations.""" +import json, os, itertools +from roots import * + +HERE = os.path.dirname(os.path.abspath(__file__)) + +def units(): + """name -> (list of lines, T). P112 = E8 minus a frame, P104 = E8 minus two frames, 'E7-frame' = E7 minus a 7-frame (data/p104_frames.json).""" + D = json.load(open(os.path.join(HERE, '..', 'data', 'p104_frames.json'))) + F0, F1 = D['F0'], D['F1'] + out = {n: UNITS[n] for n in ('D4', 'D5', 'D6', 'E6', 'E7', 'E8')} + out['P112'] = ([E8[i] for i in range(120) if i not in set(F0)], 8) + out['P104'] = ([E8[i] for i in range(120) if i not in set(F0) | set(F1)], 8) + adj = orth_graph(E7) + fr7 = next(c for c in maximal_cliques(adj) if popcount(c) == 7) + out['E7-frame'] = ([E7[i] for i in range(63) if not fr7 >> i & 1], 7) + return out + +def analyse(L, T): + v = len(L); Br = b0rows(L); basis = colspace_basis(Br); h = len(basis) + adj = orth_graph(L) + cl = list(maximal_cliques(adj)); alpha = max(popcount(c) for c in cl) + frames = [c for c in cl if popcount(c) == T] + allcl = span_vectors(basis) + has_frame = []; unc = []; cert_c = [] + for co, w in allcl: + if any(f & ~w == 0 for f in frames): has_frame.append((co, w)); continue + Os = [f & ~w for f in frames]; ok = False + for E in frames: + ins = [o for o in Os if o & ~E == 0] + if ins: + x = ins[0] + for o in ins[1:]: x &= o + if x == 0: ok = True; break + (cert_c if ok else unc).append((co, w)) + cos = set(c for c, _ in unc) + sub2 = any((a ^ b) in cos for a, b in itertools.combinations(sorted(cos), 2)) + return dict(v=v, T=T, h=h, alpha=alpha, nframes=len(frames), frames=frames, classes=len(allcl), has_frame=has_frame, cert_c=cert_c, + unc=unc, sub2=sub2, basis=basis, Br=Br) diff --git a/research/overcomplete-and-realisation-floors/lib/report.py b/research/overcomplete-and-realisation-floors/lib/report.py new file mode 100644 index 0000000000..e8973c2fa2 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/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/overcomplete-and-realisation-floors/lib/roots.py b/research/overcomplete-and-realisation-floors/lib/roots.py new file mode 100644 index 0000000000..bfddda2d20 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/lib/roots.py @@ -0,0 +1,148 @@ +"""Root-system line sets and exact F2 / clique helpers (standard library only). +Lines are given by integer vectors: E8 with doubled coordinates (so the inner product is 4x the usual one: roots have norm 8), D_n with norm 2. +Only zero patterns of the Gram matrix are used by the F2 / frame checks; the decoder checks (frame120, Lemma OC) use exact Fractions with the +proper scale (the E8 helpers below return the 'real' scaling through `real_dot`).""" +import itertools +from fractions import Fraction as Fr + +def dot(a, b): return sum(x * y for x, y in zip(a, b)) + +def e8_roots2(): + """E8 roots with coordinates doubled: +-2e_i +-2e_j and (+-1)^8 with an even number of minus signs (240 vectors).""" + R = [] + for i, j in itertools.combinations(range(8), 2): + for si in (2, -2): + for sj in (2, -2): + r = [0] * 8; r[i] = si; r[j] = sj; R.append(tuple(r)) + for s in itertools.product((1, -1), repeat=8): + if s.count(-1) % 2 == 0: R.append(tuple(s)) + return R + +def dn_roots(n): + R = [] + for i, j in itertools.combinations(range(n), 2): + for si in (1, -1): + for sj in (1, -1): + r = [0] * n; r[i] = si; r[j] = sj; R.append(tuple(r)) + return R + +def lines(R): + """one representative per +-pair, in first-occurrence order""" + seen = set(); L = [] + for r in R: + if tuple(-x for x in r) in seen: continue + seen.add(r); L.append(r) + return L + +def orth_sub(L, vs): return [r for r in L if all(dot(r, x) == 0 for x in vs)] + +E8 = lines(e8_roots2()) # 120 lines, doubled coordinates +_r0 = E8[0]; _r1 = next(r for r in E8 if dot(r, _r0) in (4, -4)) +E7 = orth_sub(E8, [_r0]) # 63 lines +E6 = orth_sub(E8, [_r0, _r1]) # 36 lines +D4 = lines(dn_roots(4)); D5 = lines(dn_roots(5)); D6 = lines(dn_roots(6)) +UNITS = {'D4': (D4, 4), 'D5': (D5, 5), 'D6': (D6, 6), 'E6': (E6, 6), 'E7': (E7, 7), 'E8': (E8, 8)} + +def gram(L): return [[dot(a, b) for b in L] for a in L] + +def b0rows(L): + """dense pairing B = J + Abar as row bitmasks: B_ts = 1 iff t = s or the lines t, s are orthogonal""" + G = gram(L); v = len(L) + return [sum(1 << s for s in range(v) if s == t or G[t][s] == 0) for t in range(v)] + +def rank_f2(rows): + basis = [] + for r in rows: + x = r + for b in basis: x = min(x, x ^ b) + if x: basis.append(x) + return len(basis) + +def blk(Brows, R, A): + """rank over F2 of the submatrix B[R, A]""" + return rank_f2([sum(1 << i for i, s in enumerate(A) if Brows[t] >> s & 1) for t in R]) + +def colspace_basis(rows): + """reduced echelon basis (bitmasks over the column index) of the span of the given rows""" + basis = []; piv = [] + for r in rows: + x = r + for b, p in zip(basis, piv): + if x >> p & 1: x ^= b + if x: + p = x.bit_length() - 1 + for i in range(len(basis)): + if basis[i] >> p & 1: basis[i] ^= x + basis.append(x); piv.append(p) + return basis + +def span_vectors(basis): + """all nonzero vectors of the F2 span as (coefficient mask, vector) pairs""" + h = len(basis); out = [] + for co in range(1, 1 << h): + w = 0 + for i in range(h): + if co >> i & 1: w ^= basis[i] + out.append((co, w)) + return out + +def orth_graph(L): + G = gram(L); v = len(L) + return [sum(1 << s for s in range(v) if s != t and G[t][s] == 0) for t in range(v)] + +def maximal_cliques(adj): + """Bron-Kerbosch with pivoting on bitmask adjacency; yields clique bitmasks""" + def bk(R, P, X): + if not P and not X: + yield R; return + # pivot: vertex of P|X maximising |P & N(u)| + best = -1; piv = 0; PX = P | X; m = PX + while m: + low = m & -m; i = low.bit_length() - 1; m ^= low + c = bin(P & adj[i]).count('1') + if c > best: best = c; piv = i + cand = P & ~adj[piv] + while cand: + low = cand & -cand; i = low.bit_length() - 1; cand ^= low + yield from bk(R | low, P & adj[i], X & adj[i]) + P &= ~low; X |= low + yield from bk(0, (1 << len(adj)) - 1, 0) + +def popcount(x): return bin(x).count('1') +def bits(x): + out = [] + while x: + low = x & -x; out.append(low.bit_length() - 1); x ^= low + return out + +# Fraction linear algebra +def rank_q(rows): + M = [[Fr(x) for x in r] for r in rows]; r = 0 + if not M: return 0 + for c in range(len(M[0])): + piv = next((i for i in range(r, len(M)) if M[i][c] != 0), None) + if piv is None: continue + M[r], M[piv] = M[piv], M[r]; pv = M[r][c]; M[r] = [x / pv 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] = [x - f * y for x, y in zip(M[i], M[r])] + r += 1 + return r + +def nullspace_q(rows, ncols): + """basis of {x : rows . x = 0} over Q""" + M = [[Fr(x) for x in r] for r in rows]; pivs = []; r = 0 + for c in range(ncols): + piv = next((i for i in range(r, len(M)) if M[i][c] != 0), None) + if piv is None: continue + M[r], M[piv] = M[piv], M[r]; pv = M[r][c]; M[r] = [x / pv 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] = [x - f * y for x, y in zip(M[i], M[r])] + pivs.append(c); r += 1 + 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 i, c in enumerate(pivs): x[c] = -M[i][f] + out.append(x) + return out diff --git a/research/overcomplete-and-realisation-floors/lib/satgen.py b/research/overcomplete-and-realisation-floors/lib/satgen.py new file mode 100644 index 0000000000..21461b4137 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/lib/satgen.py @@ -0,0 +1,86 @@ +"""DIMACS generators for the F2 low-rank (fitting-minrank) instances. Standard library only. +A matrix X = U V^T over F2 of rank <= r is encoded with U[t][i], V[s][i] variables, X_ts = XOR_i (U[t][i] AND V[s][i]) (Tseitin chain). +Symmetry breaking: if C is a set of rows/columns with X[C,C] = I (a clique of non-orthogonal lines for the fitting problem B, a clique of +orthogonal lines for the Gamma problem), then U[C] has rank |C|, and applying a change of basis of F2^r we may fix U[C] = [I_|C| | 0] +(and V[C] = [I | 0] too when r = |C|). Any choice of C is valid.""" +import itertools +from roots import * + +class CNF: + def __init__(self): self.nv = 0; self.cls = [] + def new(self): self.nv += 1; return self.nv + def lowrank_entry(self, Urow, Vrow, val): + """add: XOR_i (Urow[i] AND Vrow[i]) == val (val in {0,1}), or, if val is a variable index, == that variable""" + r = len(Urow); ands = [] + for i in range(r): + a = self.new(); self.cls.extend([[-a, Urow[i]], [-a, Vrow[i]], [a, -Urow[i], -Vrow[i]]]); ands.append(a) + cur = ands[0] + for l in ands[1:]: + y = self.new(); self.cls.extend([[-y, cur, l], [-y, -cur, -l], [y, -cur, l], [y, cur, -l]]); cur = y + if val in (0, 1): self.cls.append([cur] if val else [-cur]) + else: self.cls.extend([[-cur, val], [cur, -val]]) + def dimacs(self): + return "p cnf %d %d\n" % (self.nv, len(self.cls)) + "".join(" ".join(map(str, c)) + " 0\n" for c in self.cls) + +def clique_in(adj, universe): + """a (deterministic) large clique of the graph adj inside the bitmask `universe`, as a sorted list (any clique is a valid symmetry-breaking choice)""" + adj2 = [a & universe for a in adj]; best = 0; bs = -1 + for c in maximal_cliques(adj2): + if not (c & universe) == c: continue + k = popcount(c) + if k > bs or (k == bs and c < best): best, bs = c, k + return bits(best) + +def nonorth_graph(L): + G = gram(L); v = len(L) + return [sum(1 << s for s in range(v) if s != t and G[t][s] != 0) for t in range(v)] + +def fitting_cnf(L, R, A, r, clique, fixV=False): + """exists X = U V^T over F2 of rank <= r on rows R, columns A with X_tt = 1, X_ts = 0 for non-orthogonal t != s, entries on orthogonal + pairs free? (the fitting minrank of the block B[R,A] over all admissible B). clique: list of lines in R cap A, pairwise non-orthogonal.""" + G = gram(L); c = CNF() + U = {t: [c.new() for _ in range(r)] for t in R}; V = {s: [c.new() for _ in range(r)] for s in A} + for t in R: + for s in A: + if t != s and G[t][s] == 0: continue + c.lowrank_entry(U[t], V[s], 1 if t == s else 0) + for k, t in enumerate(clique[:r]): + for i in range(r): c.cls.append([U[t][i]] if i == k else [-U[t][i]]) + if fixV: # only valid when r == |clique|: X[C,C] = I = U[C] V[C]^T with U[C] = I forces V[C] = I + assert r == len(clique) + for i in range(r): c.cls.append([V[t][i]] if i == k else [-V[t][i]]) + c.U, c.V = U, V + return c + +def gamma_cnf(L, r, clique): + """exists N = U V^T over F2 of rank <= r on all lines with N_tt = 1 and N_ts = 0 for ORTHOGONAL t != s (entries on non-orthogonal pairs free); + clique: pairwise orthogonal lines (N[C,C] = I).""" + G = gram(L); v = len(L); c = CNF() + U = [[c.new() for _ in range(r)] for _ in range(v)]; V = [[c.new() for _ in range(r)] for _ in range(v)] + for t in range(v): + for s in range(v): + if t != s and G[t][s] != 0: continue + c.lowrank_entry(U[t], V[s], 1 if t == s else 0) + for k, t in enumerate(clique[:r]): + for i in range(r): c.cls.append([U[t][i]] if i == k else [-U[t][i]]) + c.U, c.V = U, V + return c + +def joint_cnf(L, terms, caps): + """exists an admissible B (B_tt = 1, B_ts = 0 on non-orthogonal pairs, free variables on orthogonal pairs, SHARED by all terms) with + rank B[R_j, A_j] <= caps[j] for every j. Returns (cnf, bvars).""" + G = gram(L); v = len(L); c = CNF(); bvar = {} + nog = nonorth_graph(L) + for (R, A), r in zip(terms, caps): + D = sorted(set(R) & set(A)); C = clique_in(nog, sum(1 << t for t in D)) if D else [] + U = {t: [c.new() for _ in range(r)] for t in R}; V = {s: [c.new() for _ in range(r)] for s in A} + for k, t in enumerate(C[:r]): + for i in range(r): c.cls.append([U[t][i]] if i == k else [-U[t][i]]) + for t in R: + for s in A: + if t == s: c.lowrank_entry(U[t], V[s], 1) + elif G[t][s] != 0: c.lowrank_entry(U[t], V[s], 0) + else: + if (t, s) not in bvar: bvar[t, s] = c.new() + c.lowrank_entry(U[t], V[s], bvar[t, s]) + return c, bvar diff --git a/research/overcomplete-and-realisation-floors/requirements.txt b/research/overcomplete-and-realisation-floors/requirements.txt new file mode 100644 index 0000000000..64e0abcf5f --- /dev/null +++ b/research/overcomplete-and-realisation-floors/requirements.txt @@ -0,0 +1,2 @@ +# Tested set (Python 3.14.7). Standard library only (fractions, itertools, hashlib, json, lzma, subprocess); no third-party packages. +# Optional, for the DRAT re-check and for re-solving: drat-trim (see verify.sh) and kissat >= 4.0.4. 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Standard library only. +Every builder is deterministic, so the DIMACS text (hence its sha256) is reproducible. Status strings: UNSAT (proof: DRAT) or SAT (model).""" +import os, sys, json, itertools +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, os.path.join(HERE, '..', 'lib')) +from satgen import * +from dirs import * + +DATA = os.path.join(HERE, '..', 'data') +def _p104(): + D = json.load(open(os.path.join(DATA, 'p104_frames.json'))); return D['F0'], D['F1'], D['pairs'] + +def nb_lines(L, x, live=None): + live = list(range(len(L))) if live is None else live + return [t for t in live if dot(L[t], L[x]) != 0] + +def _clique(L, R, A, r): + D = sorted(set(R) & set(A)); nog = nonorth_graph(L) + return clique_in(nog, sum(1 << t for t in D)) + +def block_instance(L, R, A, r): + """fitting minrank <= r of the block B[R, A] (symmetry breaking on a maximum non-orthogonal clique of R cap A)""" + C = _clique(L, R, A, r) + return fitting_cnf(L, R, A, r, C) + +def e6_cliques5(): + nog = nonorth_graph(UNITS['E6'][0]) + return sorted(c for c in maximal_cliques(nog) if popcount(c) == 5) + +def all_instances(): + """returns an ordered dict name -> dict(expect=..., build=callable -> CNF, tag=..., note=...)""" + I = {} + def add(name, expect, tag, note, build): I[name] = dict(expect=expect, tag=tag, note=note, build=build) + # ---- (1) E6 global fitting minrank: B = U V^T of rank <= r on all 36 lines, B_tt = 1, B_ts = 0 on non-orthogonal pairs + L6 = UNITS['E6'][0]; v6 = 36; allv = list(range(v6)); cl5 = e6_cliques5(); picks = [0, len(cl5) // 2, len(cl5) - 1] + for k, ci in enumerate(picks): + C = bits(cl5[ci]) + for r, ex in ((5, 'UNSAT'), (6, 'UNSAT')): + add('e6glob_r%d_c%d' % (r, k), ex, 'glob', 'E6 global fitting minrank <= %d, symmetry-breaking clique #%d of the 5-cliques %s' % (r, ci, C), + (lambda r=r, C=C: fitting_cnf(L6, allv, allv, r, C, fixV=(r == 5)))) + C0 = bits(cl5[picks[0]]) + add('e6glob_r7_c0', 'SAT', 'glob', 'E6 global fitting minrank <= 7 (exists)', lambda: fitting_cnf(L6, allv, allv, 7, C0)) + # ---- (2) char-2 shadow minranks of Gamma (non-orthogonality graph): N_tt = 1, N_ts = 0 on orthogonal pairs + for unit, r, ex in (('E6', 6, 'UNSAT'), ('E7', 7, 'UNSAT')): + Lg = UNITS[unit][0]; og = orth_graph(Lg) + cls = sorted(maximal_cliques(og), key=lambda c: (-popcount(c), c)) + for k, c in enumerate([cls[0], cls[len(cls) // 2]]): + C = bits(c) + add('gamma_%s_r%d_c%d' % (unit, r, k), ex, 'gamma', '%s: F2-minrank of matrices with unit diagonal and zeros on orthogonal pairs <= %d' % (unit, r), + (lambda Lg=Lg, r=r, C=C: gamma_cnf(Lg, r, C))) + # ---- (3) E8 root blocks: fitting minrank of B[R_a, R_b] + L8 = UNITS['E8'][0]; G8 = gram(L8) + a = 0; nbr = next(t for t in range(120) if G8[a][t] != 0 and t != a); orr = next(t for t in range(120) if G8[a][t] == 0 and t != a) + Ra = nb_lines(L8, a) + add('e8blk_aa_r8', 'UNSAT', 'block', 'E8: fitting minrank of B[R_a, R_a] <= 8', lambda: block_instance(L8, Ra, Ra, 8)) + add('e8blk_anb_r8', 'UNSAT', 'block', 'E8: fitting minrank of B[R_a, R_b] <= 8, a, b non-orthogonal', lambda: block_instance(L8, Ra, nb_lines(L8, nbr), 8)) + add('e8blk_aor_r8', 'SAT', 'block', 'E8: fitting minrank of B[R_a, R_b] <= 8, a, b orthogonal (exists)', lambda: block_instance(L8, Ra, nb_lines(L8, orr), 8)) + # ---- (4) known optimal decoders: every term has fitting minrank = dense rank (UNSAT at dense - 1) + F0, F1, pairs = _p104(); Br8 = b0rows(L8) + live112 = [t for t in range(120) if t not in set(F0)]; live104 = [t for t in live112 if t not in set(F1)] + def sub(live): return [L8[t] for t in live] + def term_sets(live, x, y): + Lsub = sub(live); idx = {t: i for i, t in enumerate(live)} + R = [idx[t] for t in live if dot(L8[t], L8[x]) != 0]; A = [idx[t] for t in live if dot(L8[t], L8[y]) != 0] + return Lsub, R, A + def dense_rank(Lsub, R, A): return blk(b0rows(Lsub), R, A) + # E8 frame normals (8 terms, dense 9) -- all lines of E8 are equivalent under W(E8); the 8 terms are nevertheless all included + for i, x in enumerate(F0): + Ls, R, A = term_sets(list(range(120)), x, x) + add('rob_E8frame_%d' % i, 'UNSAT', 'robust', 'E8 frame-normal decoder, term %d: fitting minrank of B[R,R] <= 8 (dense rank 9)' % i, (lambda Ls=Ls, R=R, A=A: block_instance(Ls, R, A, 8))) + # E8 standard-coordinate decoder: terms x_j = y_j = e_j, R = lines with nonzero coordinate j (78 lines) + for j in (0, 7): + Rj = [t for t in range(120) if L8[t][j] != 0] + add('rob_E8coord_%d' % j, 'UNSAT', 'robust', 'E8 coordinate decoder, term %d (|R| = %d): fitting minrank <= 8 (dense 9)' % (j, len(Rj)), (lambda Rj=Rj: block_instance(L8, Rj, Rj, 8))) + # E7 frame decoder (7 terms, dense 7) + L7 = UNITS['E7'][0]; fr7 = bits(next(c for c in maximal_cliques(orth_graph(L7)) if popcount(c) == 7)) + for i, x in enumerate(fr7): + R = nb_lines(L7, x) + add('rob_E7frame_%d' % i, 'UNSAT', 'robust', 'E7 frame-normal decoder, term %d: fitting minrank of B[R,R] <= 6 (dense 7)' % i, (lambda R=R: block_instance(L7, R, R, 6))) + # P112 removed-frame decoder (8 terms, dense 8) + for i, x in enumerate(F0): + Ls, R, A = term_sets(live112, x, x) + add('rob_P112_%d' % i, 'UNSAT', 'robust', 'P112 removed-frame decoder, term %d: fitting minrank <= 7 (dense 8)' % i, (lambda Ls=Ls, R=R, A=A: block_instance(Ls, R, A, 7))) + # P104 frame-pair decoder (8 terms, dense 7) + for i, (x, y) in enumerate(pairs): + Ls, R, A = term_sets(live104, x, y) + add('rob_P104_%d' % i, 'UNSAT', 'robust', 'P104 frame-pair decoder, term %d: fitting minrank <= 6 (dense 7)' % i, (lambda Ls=Ls, R=R, A=A: block_instance(Ls, R, A, 6))) + # ---- controls: at the dense rank the same blocks ARE feasible (the dense B itself is a solution) + Ls, R, A = term_sets(list(range(120)), F0[0], F0[0]) + add('ctrl_E8frame_r9', 'SAT', 'control', 'control: E8 frame-normal block at its dense rank 9 is feasible', lambda: block_instance(Ls, R, A, 9)) + Ls, R, A = term_sets(live104, *pairs[0]) + add('ctrl_P104_r7', 'SAT', 'control', 'control: P104 frame-pair block at its dense rank 7 is feasible', lambda: block_instance(Ls, R, A, 7)) + Ra7 = nb_lines(L7, fr7[0]) + add('ctrl_E7frame_r7', 'SAT', 'control', 'control: E7 frame-normal block at its dense rank 7 is feasible', lambda: block_instance(L7, Ra7, Ra7, 7)) + # ---- (5) one E6 per-term drop: a rank-6 deficient pair with |R cap A| = 8 has fitting minrank 4 (exists), not 3 + dm = deficient_directions(L6, 6); X = sorted(set(dm.values())); Br6 = b0rows(L6) + ex_pair = None + for i in range(27): + for j in range(27): + R = support(L6, X[i]); A = support(L6, X[j]) + if blk(Br6, R, A) == 6 and len(set(R) & set(A)) == 8: ex_pair = (R, A); break + if ex_pair: break + R, A = ex_pair + add('e6term_r4', 'SAT', 'term', 'E6: a deficient pair of dense rank 6 with |R cap A| = 8 has fitting minrank <= 4', lambda: block_instance(L6, R, A, 4)) + add('e6term_r3', 'UNSAT', 'term', 'E6: the same pair has fitting minrank > 3', lambda: block_instance(L6, R, A, 3)) + return I + +def _compositions(n, k): + """all k-tuples of non-negative integers with sum n, in lexicographic order""" + if k == 1: yield (n,); return + for a in range(n + 1): + for rest in _compositions(n - a, k - 1): yield (a,) + rest + +def joint_instances(): + """shared-B instances (E6 square decoders, total <= 35) and the per-term floor instances. Returns name -> dict like all_instances().""" + L = UNITS['E6'][0] + D = json.load(open(os.path.join(DATA, 'e6_square_decoders.json')))['decoders'] + plan = json.load(open(os.path.join(DATA, 'e6_joint_plan.json'))) + tot = plan['total']; I = {}; seen = {} + for p in plan['plan']: + d = p['decoder']; dec = D[d] + terms = [(support(L, dec['X'][j]), support(L, dec['Y'][j])) for j in range(6)] + for j in range(6): + lo = p['lo'][j]; key = (tuple(terms[j][0]), tuple(terms[j][1])) + if lo > 1 and key not in seen: + seen[key] = 'floor_d%02d_t%d' % (d, j) + I[seen[key]] = dict(expect='UNSAT', tag='floor', note='E6 decoder %d term %d: fitting minrank of B[R,A] <= %d is impossible (so rank >= %d for every admissible B)' % (d, j, lo - 1, lo), + build=(lambda R=terms[j][0], A=terms[j][1], r=lo - 1: block_instance(L, R, A, r))) + if d in (0, 7, 18): + I['jointctrl_d%02d' % d] = dict(expect='SAT', tag='jointctrl', note='control: E6 decoder %d with caps = its dense ranks %s (sum %d): the dense B is a solution' % (d, p['dense'], sum(p['dense'])), + build=(lambda terms=terms, caps=tuple(p['dense']): joint_cnf(L, terms, list(caps))[0])) + slack = tot - sum(p['lo']) + for ex in _compositions(slack, 6): + caps = tuple(a + b for a, b in zip(p['lo'], ex)) + I['joint_d%02d_%s' % (d, '-'.join(map(str, caps)))] = dict(expect='UNSAT', tag='joint', + note='E6 decoder %d: one admissible B with rank B[R_j,A_j] <= %s for all j (sum = %d)' % (d, list(caps), tot), + build=(lambda terms=terms, caps=caps: joint_cnf(L, terms, list(caps))[0])) + return I diff --git a/research/overcomplete-and-realisation-floors/sat/joint_results.jsonl b/research/overcomplete-and-realisation-floors/sat/joint_results.jsonl new file mode 100644 index 0000000000..79e4c67109 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/sat/joint_results.jsonl @@ -0,0 +1,1418 @@ +{"name": "floor_d00_t0", "expect": "UNSAT", "cnf_sha256": "75315c155e16558d3fda84b9800db373988f6f7bbab90eda511f7f8ed18da957", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 4035, "drat": "VERIFIED"} +{"name": "floor_d00_t1", "expect": "UNSAT", "cnf_sha256": "6ecdc05804fe70c4c50f4dcca5722bee208a297162bde9dd21e76a2fcf16228c", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 3579, "drat": "VERIFIED"} +{"name": "floor_d00_t2", "expect": "UNSAT", "cnf_sha256": "9b1c76fc9e941359cb29e6a5e5e5b18f28952e84fe8c880b0bed81fe3b266ee3", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 2961, "drat": "VERIFIED"} +{"name": "floor_d00_t3", "expect": "UNSAT", "cnf_sha256": "265c24b275dd3d3be17b45fc88deaccd87b319cab0ea07c26b4a5df22efb35b9", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 2980, "drat": "VERIFIED"} +{"name": "floor_d00_t4", "expect": "UNSAT", "cnf_sha256": "eefcbd05e9b7b2639a352caf6a2f51b070c9573f856ce9d1ed391849390ed902", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 10406, "drat": "VERIFIED"} +{"name": "floor_d00_t5", "expect": "UNSAT", "cnf_sha256": "33822878926af206edfe1ebd6ecd7e07c83c67c7fee5c6506a377a8b7fd68ca7", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 14466, "drat": "VERIFIED"} +{"name": "joint_d00_5-5-5-5-6-9", "expect": "UNSAT", "cnf_sha256": "b575835745de46319c6a2d5fd5e9a69af84f7e2058d734271a758c7919383e9e", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 163291, "drat": "VERIFIED"} +{"name": "joint_d00_5-5-5-5-7-8", "expect": "UNSAT", "cnf_sha256": "6dfc8797b38fd814abaec66a6e91f41af43232df706f95b2631f1b13b19be9ed", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 163291, "drat": "VERIFIED"} +{"name": "joint_d00_5-5-5-5-8-7", "expect": "UNSAT", "cnf_sha256": "d7a767b21002d0926914d4e9d3652ae07fc0fe9f9a2b836556e1f1bbabda7b93", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 163291, "drat": "VERIFIED"} +{"name": "joint_d00_5-5-5-5-9-6", "expect": "UNSAT", "cnf_sha256": "bfcd93d10ce09fc8581b796a2dba4e290fc6e27faba2ba9f8826062047850f89", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 163291, "drat": "VERIFIED"} +{"name": "joint_d00_5-5-5-6-6-8", "expect": "UNSAT", "cnf_sha256": 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b/research/overcomplete-and-realisation-floors/sat/mkplan.py @@ -0,0 +1,37 @@ +#!/usr/bin/env python3 +"""Compute the per-term fitting-minrank floors lo_j of the 26 E6 square decoders (data/e6_square_decoders.json) with kissat and write +data/e6_joint_plan.json: for each decoder the term supports' dense ranks, the floors lo_j (each lo_j > 1 is backed by an UNSAT instance +'floor__' at rank lo_j - 1, re-checkable with run.py) and the number of cap vectors (c_j >= lo_j, sum c_j = 35). +Floors are only lower bounds that are used for the exhaustive enumeration of cap vectors, so their exactness is not needed -- only that +rank B[R_j,A_j] >= lo_j for every admissible B, which is exactly the UNSAT instance at lo_j - 1.""" +import os, sys, json, subprocess, shutil +HERE = os.path.dirname(os.path.abspath(__file__)); sys.path.insert(0, HERE) +import instances as I +from satgen import * +from dirs import * +KISSAT = os.environ.get('KISSAT', shutil.which('kissat') or 'kissat') +L = UNITS['E6'][0]; Br = b0rows(L) +D = json.load(open(os.path.join(I.DATA, 'e6_square_decoders.json'))) +def status(R, A, r, tmp): + c = I.block_instance(L, R, A, r); open(tmp, 'w').write(c.dimacs()) + o = subprocess.run([KISSAT, '-q', tmp], capture_output=True, text=True, timeout=1800).stdout + return 'UNSAT' if 'UNSATISFIABLE' in o else ('SAT' if 'SATISFIABLE' in o else 'UNKNOWN') +cache = {}; plan = [] +tmp = os.path.join(sys.argv[1] if len(sys.argv) > 1 else '/tmp', 'mkplan.cnf') +for d, dec in enumerate(D['decoders']): + lo = []; dense = [] + for j in range(6): + R = support(L, dec['X'][j]); A = support(L, dec['Y'][j]); dn = blk(Br, R, A); dense.append(dn) + key = (tuple(R), tuple(A)) + if key not in cache: + r = dn + while r > 1: + st = status(R, A, r - 1, tmp) + assert st != 'UNKNOWN' + if st == 'SAT': r -= 1 + else: break + cache[key] = r + lo.append(cache[key]) + plan.append(dict(decoder=d, dense=dense, lo=lo)) + print(d, dense, lo, flush=True) +json.dump(dict(total=35, plan=plan), open(os.path.join(I.DATA, 'e6_joint_plan.json'), 'w'), indent=0) diff --git a/research/overcomplete-and-realisation-floors/sat/models/ctrl_E7frame_r7.model b/research/overcomplete-and-realisation-floors/sat/models/ctrl_E7frame_r7.model new file mode 100644 index 0000000000..240fa04789 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/sat/models/ctrl_E7frame_r7.model @@ -0,0 +1 @@ +1 9 17 25 33 34 35 40 46 48 49 52 55 56 58 62 63 69 74 75 77 80 82 84 86 89 91 94 95 98 100 102 105 107 108 112 114 115 116 117 118 126 130 131 132 136 138 139 142 145 146 150 151 153 156 158 160 163 164 167 170 171 172 173 175 178 179 180 184 186 187 191 192 194 198 199 200 202 203 205 206 207 212 213 215 216 217 219 221 222 223 224 227 228 229 230 231 232 240 248 256 264 265 266 271 277 279 280 283 286 287 289 293 294 300 305 306 308 311 313 315 317 320 322 325 326 329 331 333 336 338 339 343 345 346 347 348 349 357 361 362 363 367 369 370 373 376 377 381 382 384 387 389 391 394 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34110 34111 34115 34120 34122 34126 diff --git a/research/overcomplete-and-realisation-floors/sat/models/jointctrl_d18.model b/research/overcomplete-and-realisation-floors/sat/models/jointctrl_d18.model new file mode 100644 index 0000000000..1d84e50dd9 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/sat/models/jointctrl_d18.model @@ -0,0 +1 @@ +2 3 4 6 10 12 15 18 19 25 26 28 30 31 32 33 36 38 43 48 51 58 61 62 65 66 71 74 75 77 78 79 80 81 82 83 86 88 89 90 99 100 102 104 106 110 111 113 115 121 125 127 130 132 133 135 137 138 140 143 144 146 150 151 152 155 159 165 167 172 173 178 188 189 192 193 194 195 203 207 208 211 212 216 217 218 219 220 221 222 226 227 228 229 232 235 237 243 244 246 248 251 253 255 258 259 264 265 269 299 301 304 305 309 312 314 315 316 319 323 324 325 326 327 330 334 335 336 337 338 340 342 347 348 349 350 351 352 355 360 361 362 363 364 367 372 373 374 375 376 380 385 386 387 388 392 397 398 399 412 414 415 418 419 423 425 426 430 448 449 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"expect": "UNSAT", "cnf_sha256": "f2f3f912bb287205ad2ff43b74095fcb8b19e357e884ac6cb9d33282250c894f", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 95499, "drat": "VERIFIED"} +{"name": "e8blk_anb_r8", "expect": "UNSAT", "cnf_sha256": "4095f8c30fe7cb7d689b94dc39f3299bdb0493e7c30dff00df9d720d31ca87d0", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 92693, "drat": "VERIFIED"} +{"name": "e8blk_aor_r8", "expect": "SAT", "cnf_sha256": "686356e7e154b29bb41cb6017a9ccfaa5d9874a3a57df53553aaf2bbb1a522cf", "seconds": 0.0, "status": "SAT", "model_ok": true} +{"name": "rob_E8frame_0", "expect": "UNSAT", "cnf_sha256": "f2f3f912bb287205ad2ff43b74095fcb8b19e357e884ac6cb9d33282250c894f", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 95499, "drat": "VERIFIED"} +{"name": "rob_E8frame_1", "expect": "UNSAT", "cnf_sha256": "fa706f98c466776271cd2b9f67e500302dc79841adbd7aa34fc5ecddbcf9d0a7", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 93214, "drat": "VERIFIED"} +{"name": "rob_E8frame_2", "expect": "UNSAT", "cnf_sha256": "6ea3313fc6bc8ecfc12c812be5907ea52efa85849e23ee1ff01844cc9e05463f", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 92125, "drat": "VERIFIED"} +{"name": "rob_E8frame_3", "expect": "UNSAT", "cnf_sha256": "98ff31ee03c9e7d6325e4cd6dfc0e06de3538afb3d6f5df7a763769d236d38a7", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 93484, "drat": "VERIFIED"} +{"name": "rob_E8frame_4", "expect": "UNSAT", "cnf_sha256": "83f5a0ef6f38e6130ed6ad1ce594212a17ff0f22028fd75418c213ee4cbc1968", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 95511, "drat": "VERIFIED"} +{"name": "rob_E8frame_5", "expect": "UNSAT", "cnf_sha256": "4444176b0216468aaa63d286c0c7e469e0b3570daf098ffb71b38dce01e53db2", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 93215, "drat": "VERIFIED"} +{"name": "rob_E8frame_6", "expect": "UNSAT", "cnf_sha256": "b0a6542231bfa086a8dadeafaf507422d281c322aad6f44a0cf3adbbbb190f7e", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 92126, "drat": "VERIFIED"} +{"name": "rob_E8frame_7", "expect": "UNSAT", "cnf_sha256": "57a4ab8ddb2e946d4233e1c3a129106d77d3d94e7f0998099acea1a2c2fa6e79", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 93485, "drat": "VERIFIED"} +{"name": "rob_E8coord_0", "expect": "UNSAT", "cnf_sha256": "9e85ac40a76256286c2e8ad62f98207c81de5b110a6710daab8bdf85bd1776c1", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 161866, "drat": "VERIFIED"} +{"name": "rob_E8coord_7", "expect": "UNSAT", "cnf_sha256": "ab65bd7ccf5f1b14be1166fbf4cb134fa16751ebde4e3810c41224969d1bef67", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 158373, "drat": "VERIFIED"} +{"name": "rob_E7frame_0", "expect": "UNSAT", "cnf_sha256": "fe76bfeb51efed633fe558591deb9daa3f458b4d560a3a72bd82c11a35929318", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 12469, "drat": "VERIFIED"} +{"name": "rob_E7frame_1", "expect": "UNSAT", "cnf_sha256": "c4765bbdb56acb3fb278fb474019a3749524d8770779bc28a5a90fe6e2806610", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 12446, "drat": "VERIFIED"} +{"name": "rob_E7frame_2", "expect": "UNSAT", "cnf_sha256": "2e15ae78c742a430119fddc5c013f799f7e43ffa2a95467562e02a2938f3087a", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 12446, "drat": "VERIFIED"} +{"name": "rob_E7frame_3", "expect": "UNSAT", "cnf_sha256": "fb4b1eaef087c2a8922c159c9aae422c3955fc36337a2e1a083fd399bbebe6c2", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 12082, "drat": "VERIFIED"} +{"name": "rob_E7frame_4", "expect": "UNSAT", "cnf_sha256": "ee82b3013c60ba31c47ca173c45496f1dc17a83a61292578cd05aaf7544ef289", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 12082, "drat": "VERIFIED"} +{"name": "rob_E7frame_5", "expect": "UNSAT", "cnf_sha256": "0f2762dccf125cf24186daf9f17250dbd48d04444b449fa591905c4579b289e6", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 12082, "drat": "VERIFIED"} +{"name": "rob_E7frame_6", "expect": "UNSAT", "cnf_sha256": "236117a64de0dba9edaed714e10afb2939271508ccd2572e545fe03196a096d8", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 12082, "drat": "VERIFIED"} +{"name": "rob_P112_0", "expect": "UNSAT", "cnf_sha256": "3c9be5f36365fecf8183a6544e90420268e2c649081feb1eff866eef82503e92", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 72962, "drat": "VERIFIED"} +{"name": "rob_P112_1", "expect": "UNSAT", "cnf_sha256": "33c7b0ef22b013fb13bc03c97d6e1fc7a85dd607a74f83c9068010d2a48fc76a", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 70824, "drat": "VERIFIED"} +{"name": "rob_P112_2", "expect": "UNSAT", "cnf_sha256": "36afc2e70b6cac4afd3bd7bb4466e14d43cda735b5da32c9229a6a2216ded31b", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 70869, "drat": "VERIFIED"} +{"name": "rob_P112_3", "expect": "UNSAT", "cnf_sha256": "90e66a1c40bb0b21b1ef5426dcc1f989ea7e4f1a554e71352bfa6fe44e45bc4e", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 71080, "drat": "VERIFIED"} +{"name": "rob_P112_4", "expect": "UNSAT", "cnf_sha256": "507ff80293131732e3afac972ba738cb316bb456f4d087dce4474dd849982584", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 73008, "drat": "VERIFIED"} +{"name": "rob_P112_5", "expect": "UNSAT", "cnf_sha256": "706b0055dbea3ca9a3b712a5704d20d98b5e6ff1e3f4ac97b23a37b7d8fd29df", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 70869, "drat": "VERIFIED"} +{"name": "rob_P112_6", "expect": "UNSAT", "cnf_sha256": "bb3eb34305ee881b1b3c703e9f1460251c02da6d6733206341883536a72edd1c", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 70894, "drat": "VERIFIED"} +{"name": "rob_P112_7", "expect": "UNSAT", "cnf_sha256": "4e73af6008e548f7ff0b246e41554dd83f5492f865e5f83e3f06ef2f37fa5fff", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 71129, "drat": "VERIFIED"} +{"name": "rob_P104_0", "expect": "UNSAT", "cnf_sha256": "82164bb4a10685a326d5f2dbe3901153a616442d36675c46ff5db954bec2a5bc", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 48526, "drat": "VERIFIED"} +{"name": "rob_P104_1", "expect": "UNSAT", "cnf_sha256": "a583ab333a5bda1e60c7795fa611bb5d358d0ab1cfdb7132e4da79729742bf62", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 49511, "drat": "VERIFIED"} +{"name": "rob_P104_2", "expect": "UNSAT", "cnf_sha256": "3a693e9721a919dcfafc37159161fed99be380802559bb3f486535b563b34251", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 49562, "drat": "VERIFIED"} +{"name": "rob_P104_3", "expect": "UNSAT", "cnf_sha256": "2d5c3716db8331fe0bd8bc6689c9f8b28d1fa5ebc1c0c5cef632db762dd743f2", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 49412, "drat": "VERIFIED"} +{"name": "rob_P104_4", "expect": "UNSAT", "cnf_sha256": "33bb6ca3c967dd2af8f7bb3a823ba85cbd4d0727f8fe0bf8d01c4acd99f39501", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 456176, "drat": "VERIFIED"} +{"name": "rob_P104_5", "expect": "UNSAT", "cnf_sha256": "88390a3374cbfdf925844a7a0f54e3955528bda410e74a1e33e861eef51f0d98", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 48073, "drat": "VERIFIED"} +{"name": "rob_P104_6", "expect": "UNSAT", "cnf_sha256": "a4d7125674def69eb1110d8ea2776682d2f221cc09c20558edfc17dbf2a74f94", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 50668, "drat": "VERIFIED"} +{"name": "rob_P104_7", "expect": "UNSAT", "cnf_sha256": "acb72868a24da73f6b989dfd5df416fe80a6650ed2f4fe5b3254d93ec511d87a", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 50673, "drat": "VERIFIED"} +{"name": "ctrl_E8frame_r9", "expect": "SAT", "cnf_sha256": "e6e38d6bad6bb6fa5095a8f41cda0ba8ae6fc01cfe2848300dafde42783971a9", "seconds": 0.0, "status": "SAT", "model_ok": true} +{"name": "ctrl_P104_r7", "expect": "SAT", "cnf_sha256": "72aa10805fdc295ae750ec0afe82bd1ee85812ce467107b5ca26a76aa66c69fa", "seconds": 0.0, "status": "SAT", "model_ok": true} +{"name": "ctrl_E7frame_r7", "expect": "SAT", "cnf_sha256": "ce5b65c54417136dd6b46054de07c18b3d68a69e3718199a56b8e9b66d54d3dc", "seconds": 0.0, "status": "SAT", "model_ok": true} +{"name": "e6term_r4", "expect": "SAT", "cnf_sha256": "0a0c0057c6100320c73621857fa39fa16e441b86a1663aca316796c704786c09", "seconds": 0.0, "status": "SAT", "model_ok": true} +{"name": "e6term_r3", "expect": "UNSAT", "cnf_sha256": "65b78aa03ee5cd4a0a5441a79bfd43b0f0e064280d7d33a08160ffd35d55d6c2", "seconds": 0.0, "status": "UNSAT", "proof_bytes": 1496, "drat": "VERIFIED"} +{"name": "jointctrl_d00", "expect": "SAT", "cnf_sha256": "1e7d5921d5303adcc3418dd67e6df448e92c366e8e48a4b5d135ea39ef3231b1", "seconds": 0.3, "status": "SAT", "model_ok": true} +{"name": "jointctrl_d07", "expect": "SAT", "cnf_sha256": "9d056d5aceddb9576e63da2bd6177d24203f10010fb44bd9a9b7cac30568c2aa", "seconds": 0.3, "status": "SAT", "model_ok": true} +{"name": "jointctrl_d18", "expect": "SAT", "cnf_sha256": "ec10900187bc5e3003d0262fb47a78f6753c71e7bc1ad0c3635d9140180e8b92", "seconds": 0.2, "status": "SAT", "model_ok": true} diff --git a/research/overcomplete-and-realisation-floors/sat/run.py b/research/overcomplete-and-realisation-floors/sat/run.py new file mode 100644 index 0000000000..6ad5ae27b2 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/sat/run.py @@ -0,0 +1,87 @@ +#!/usr/bin/env python3 +"""Driver for the SAT instances (standard library only; needs `kissat` for solving and `drat-trim` for proof checking, both optional here). + python3 run.py list [tag] list instances + python3 run.py gen NAME [OUTDIR] write NAME.cnf and print its sha256 + python3 run.py solve OUTDIR [names|tags...] generate, solve with kissat (proof to NAME.drat), check with drat-trim, append to OUTDIR/results.jsonl + python3 run.py check NAME CNF [PROOF_OR_MODEL] check a shipped result: CNF hash == regenerated; DRAT via drat-trim / model by clause evaluation +Environment: KISSAT (default `kissat`), DRAT_TRIM (default `drat-trim`), KISSAT_TIMEOUT seconds (default 1800).""" +import os, sys, json, hashlib, subprocess, time, shutil +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +import instances as I + +KISSAT = os.environ.get('KISSAT', shutil.which('kissat') or 'kissat') +DRAT = os.environ.get('DRAT_TRIM', shutil.which('drat-trim') or 'drat-trim') +TMO = int(os.environ.get('KISSAT_TIMEOUT', '1800')) +def sha(b): return hashlib.sha256(b).hexdigest() + +def parse_model(out): + pos = set() + for line in out.splitlines(): + if line.startswith('v'): + for x in line.split()[1:]: + x = int(x) + if x > 0: pos.add(x) + return pos + +def model_satisfies(cnf_text, pos): + for line in cnf_text.splitlines(): + if not line or line[0] in 'pc': continue + ok = False + for x in line.split(): + x = int(x) + if x == 0: break + if (x > 0 and x in pos) or (x < 0 and -x not in pos): ok = True; break + if not ok: return False + return True + +def solve(name, inst, outdir, tmo=TMO): + cnf = inst['build']().dimacs(); path = os.path.join(outdir, name + '.cnf'); open(path, 'w').write(cnf) + proof = os.path.join(outdir, name + '.drat') + t0 = time.time() + try: + r = subprocess.run([KISSAT, '-q', path, proof], capture_output=True, text=True, timeout=tmo); out = r.stdout + except subprocess.TimeoutExpired: + return dict(name=name, status='TIMEOUT', seconds=tmo, cnf_sha256=sha(cnf.encode()), expect=inst['expect']) + dt = time.time() - t0 + res = dict(name=name, expect=inst['expect'], cnf_sha256=sha(cnf.encode()), seconds=round(dt, 1)) + if 'UNSATISFIABLE' in out: + res['status'] = 'UNSAT'; res['proof_bytes'] = os.path.getsize(proof) + d = subprocess.run([DRAT, path, proof, '-t', '3000'], capture_output=True, text=True) + res['drat'] = 'VERIFIED' if 's VERIFIED' in d.stdout else 'NOT VERIFIED' + if os.environ.get('DROP_FILES') == '1': os.remove(proof); os.remove(path) + elif 'SATISFIABLE' in out: + pos = parse_model(out); res['status'] = 'SAT'; res['model_ok'] = model_satisfies(cnf, pos) + open(os.path.join(outdir, name + '.model'), 'w').write(' '.join(map(str, sorted(pos))) + '\n') + if os.path.exists(proof): os.remove(proof) + else: res['status'] = 'UNKNOWN' + return res + +def main(): + a = sys.argv[1:] + if not a: print(__doc__); return + A = I.all_instances() + if a[0] != 'check' or a[1].startswith(('joint', 'floor')) or True: + try: A.update(I.joint_instances()) + except FileNotFoundError: pass + if a[0] == 'list': + for k, v in A.items(): + if len(a) < 2 or v['tag'] == a[1]: print(k, v['expect'], v['tag'], '-', v['note']) + elif a[0] == 'gen': + out = a[2] if len(a) > 2 else '.'; os.makedirs(out, exist_ok=True) + cnf = A[a[1]]['build']().dimacs(); open(os.path.join(out, a[1] + '.cnf'), 'w').write(cnf); print(sha(cnf.encode()), a[1]) + elif a[0] == 'solve': + out = a[1]; os.makedirs(out, exist_ok=True); sel = a[2:] + names = [k for k, v in A.items() if not sel or k in sel or v['tag'] in sel] + for k in names: + res = solve(k, A[k], out); print(json.dumps(res), flush=True) + open(os.path.join(out, 'results.jsonl'), 'a').write(json.dumps(res) + '\n') + elif a[0] == 'check': + name, cnfp = a[1], a[2]; cnf = open(cnfp).read(); want = A[name]['build']().dimacs() + print('cnf matches generator:', cnf == want) + if len(a) > 3: + if A[name]['expect'] == 'UNSAT': + d = subprocess.run([DRAT, cnfp, a[3], '-t', '3000'], capture_output=True, text=True); print('drat:', 'VERIFIED' if 's VERIFIED' in d.stdout else 'NOT VERIFIED') + else: + pos = set(map(int, open(a[3]).read().split())); print('model satisfies cnf:', model_satisfies(cnf, pos)) +if __name__ == '__main__': main() diff --git a/research/overcomplete-and-realisation-floors/scripts/provenance.py b/research/overcomplete-and-realisation-floors/scripts/provenance.py new file mode 100644 index 0000000000..d12d3d802a --- /dev/null +++ b/research/overcomplete-and-realisation-floors/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/overcomplete-and-realisation-floors/verify.sh b/research/overcomplete-and-realisation-floors/verify.sh new file mode 100644 index 0000000000..34214fc0f2 --- /dev/null +++ b/research/overcomplete-and-realisation-floors/verify.sh @@ -0,0 +1,43 @@ +#!/usr/bin/env bash +# One-command verification of research/overcomplete-and-realisation-floors (no kappa claim): exact recomputation of every CHECKED item of the README -- +# c1 Lemma OC instances and the D5 floor data; c2 the 6,728-term decoder of the 120-line frame unit (cost 72); c3 the frame-intersection +# certificate on nine units and the P104 / P112 pair-rank tables; c4 the E6 deficient-direction tables and the exhaustive search over 5,504,660 +# seven-sets; c5 the decoders attaining the floors; c6 regeneration of all 1,473 SAT instances, hash comparison, model and DRAT re-checks. +# 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. drat-trim is optional (DRAT_TRIM=/path, or `drat-trim` on PATH, or buildable with `cc` +# from the vendored sibling package research/p111-minrank-proof-reconciliation/vendor/drat-trim/drat-trim.c); without it the DRAT re-check is a SKIP line. +# usage: bash verify.sh [/fresh/output/dir] (inside a checkout or `git archive` of the repository) +# set PYTHON to choose the interpreter (>= 3.11); KISSAT_RESOLVE=n re-solves n shared-B instances (needs kissat and drat-trim). +set -uo pipefail +HERE=$(cd "$(dirname "$0")" && pwd) +OUT=${1:-$(mktemp -d "${TMPDIR:-/tmp}/overcomplete-and-realisation-floors.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' "$OUT/provenance.log")" != 2 ]; then + echo "step 0 FAILED"; exit 1 +fi +if [ -z "${DRAT_TRIM:-}" ] && ! command -v drat-trim >/dev/null 2>&1; then + SRC="$HERE/../p111-minrank-proof-reconciliation/vendor/drat-trim/drat-trim.c" + if [ -f "$SRC" ] && command -v cc >/dev/null 2>&1 && cc -O2 -o "$OUT/drat-trim" "$SRC" >/dev/null 2>&1; then export DRAT_TRIM="$OUT/drat-trim"; echo " built drat-trim from the vendored source"; fi +fi +mkdir -p "$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 Lemma OC instances, D5 data and floor core" c1_overcomplete_d5.py +run "c2 free terms: the 120-line frame unit has a decoder of cost 72" c2_free_terms_frame120.py +run "c3 frame-intersection certificate and realisation-free floors" c3_realisation_free_floors.py +run "c4 E6: deficient directions, pair ranks, exhaustive seven-term search" c4_e6_overcomplete.py +run "c5 decoders attaining the floors" c5_known_decoders.py +run "c6 SAT / DRAT replay" c6_sat_replay.py +npass=$(grep -c '^PASS' "$OUT/all.out"); nfail=$(grep -c '^FAIL' "$OUT/all.out"); nskip=$(grep -c '^SKIP' "$OUT/all.out") +EXPECT=$((55 - nskip)) +echo "== done in $((SECONDS - T0))s: $((npass + 2)) PASS (2 in step 0), $nskip SKIP, $nfail FAIL; logs in $OUT" +[ "$nfail" = 0 ] && [ "$npass" = "$EXPECT" ] || { echo "FAILED: expected $EXPECT assertion PASS lines after step 0, got $npass PASS / $nfail FAIL / $nskip SKIP"; exit 1; }