From b9d4424208754eae09eb5f2095232ce9a9544b66 Mon Sep 17 00:00:00 2001 From: DreamingOfClouds Date: Thu, 15 Oct 2026 00:32:05 -0500 Subject: [PATCH] =?UTF-8?q?Decoder=20floors=20are=20tight=20in=20the=20fro?= =?UTF-8?q?ntier=20format:=20one=20F2^9=20label=20table,=20factors=20in=20?= =?UTF-8?q?col=20M=20of=20a=20rank-8=20realisation=20(P112=2064,=20E8=2072?= =?UTF-8?q?,=20P104=20(#968's=20unit)=2056;=20two-call=20batches=20128)=20?= =?UTF-8?q?=E2=80=94=20no=20=CE=BA=20claim?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Adds research/decoder-floors-frontier-format: rigidity of the minimal-rank label table (every fitting pairing of F2-rank <= 9 on E8/P112/P104 (#968's unit)/P111 and <= 7 on E7/E6 is the dense one; 16 DRAT-verified UNSAT; the P111 row is prior work of #620 and #1097), P112 >= 64 in every rank-8 realisation with factors in col M, <= 8 terms with factors anywhere, #1180's example (65) and the co-star dead sets, two-call and k-call batching bounds, the every-B minrank of the coordinate block, puncture clique-hitting bounds, and an explicit OPEN section (h >= 10 beyond the Gram root code, rank-9 square and outside col M with m >= 9, puncture windows p >= 46, batches of m >= 17 outside col M, other P104 units). Overnight closures from the author's working cards are RECORDED (not re-run): bit windows of 13-14 totals, the h = 10 Gram root code, p = 45 for m >= 8, and the projected-copy pricing; the frame-module theorem (PROVEN) is added as a tool. No kappa claim. verify.sh: 74 PASS on a fresh export. Co-Authored-By: Claude Opus 5.5 --- .../decoder-floors-frontier-format/LICENSE | 201 ++++++++++++ .../MANIFEST.sha256 | 39 +++ .../decoder-floors-frontier-format/NOTICE.md | 16 + .../decoder-floors-frontier-format/README.md | 299 ++++++++++++++++++ .../SOURCES.json | 146 +++++++++ .../checks/c1_rigidity.py | 72 +++++ .../checks/c2_support_propagation.py | 56 ++++ .../checks/c3_batch_premises.py | 91 ++++++ .../checks/c4_masked_costar.py | 51 +++ .../checks/c5_attaining_decoders.py | 48 +++ .../checks/c6_block_minrank_punctures.py | 68 ++++ .../checks/c7_p111_unit.py | 31 ++ .../data/chart_rank7.json | 1 + .../data/h4_F1.json | 1 + .../data/p104_frames.json | 221 +++++++++++++ .../data/p104reps.json | 1 + .../data/p111_lines.json | 1 + .../lib/frames.py | 39 +++ .../lib/report.py | 16 + .../lib/roots.py | 148 +++++++++ .../lib/satgen.py | 86 +++++ .../lib/solve.py | 19 ++ .../lib/supprop.py | 32 ++ .../lib/units.py | 38 +++ .../requirements.txt | 2 + .../sat/hitting.py | 52 +++ .../sat/misc.py | 51 +++ .../sat/mkmisc.py | 23 ++ .../sat/mkresults.py | 27 ++ .../sat/proofs/E8_sym_h9_alt.drat.xz | Bin 0 -> 1933676 bytes .../sat/proofs/P104H4_sym_h9_alt.drat.xz | Bin 0 -> 1496616 bytes .../sat/proofs/P111_sym_h9_alt.drat.xz | Bin 0 -> 1692980 bytes .../sat/proofs/P112_sym_h9_alt.drat.xz | Bin 0 -> 1745444 bytes .../sat/proofs/edge_N0_r7.drat.xz | Bin 0 -> 24548 bytes .../sat/proofs/tri_keep11.drat.xz | Bin 0 -> 1412508 bytes .../sat/results.jsonl | 30 ++ .../sat/results_misc.jsonl | 16 + .../sat/rigid.py | 138 ++++++++ .../scripts/provenance.py | 45 +++ .../decoder-floors-frontier-format/verify.sh | 43 +++ 40 files changed, 2148 insertions(+) create mode 100644 research/decoder-floors-frontier-format/LICENSE create mode 100644 research/decoder-floors-frontier-format/MANIFEST.sha256 create mode 100644 research/decoder-floors-frontier-format/NOTICE.md create mode 100644 research/decoder-floors-frontier-format/README.md create mode 100644 research/decoder-floors-frontier-format/SOURCES.json create mode 100644 research/decoder-floors-frontier-format/checks/c1_rigidity.py create mode 100644 research/decoder-floors-frontier-format/checks/c2_support_propagation.py create mode 100644 research/decoder-floors-frontier-format/checks/c3_batch_premises.py create mode 100644 research/decoder-floors-frontier-format/checks/c4_masked_costar.py create mode 100644 research/decoder-floors-frontier-format/checks/c5_attaining_decoders.py create mode 100644 research/decoder-floors-frontier-format/checks/c6_block_minrank_punctures.py create mode 100644 research/decoder-floors-frontier-format/checks/c7_p111_unit.py create mode 100644 research/decoder-floors-frontier-format/data/chart_rank7.json create mode 100644 research/decoder-floors-frontier-format/data/h4_F1.json create mode 100644 research/decoder-floors-frontier-format/data/p104_frames.json create mode 100644 research/decoder-floors-frontier-format/data/p104reps.json create mode 100644 research/decoder-floors-frontier-format/data/p111_lines.json create mode 100644 research/decoder-floors-frontier-format/lib/frames.py create mode 100644 research/decoder-floors-frontier-format/lib/report.py create mode 100644 research/decoder-floors-frontier-format/lib/roots.py create mode 100644 research/decoder-floors-frontier-format/lib/satgen.py create mode 100644 research/decoder-floors-frontier-format/lib/solve.py create mode 100644 research/decoder-floors-frontier-format/lib/supprop.py create mode 100644 research/decoder-floors-frontier-format/lib/units.py create mode 100644 research/decoder-floors-frontier-format/requirements.txt create mode 100644 research/decoder-floors-frontier-format/sat/hitting.py create mode 100644 research/decoder-floors-frontier-format/sat/misc.py create mode 100644 research/decoder-floors-frontier-format/sat/mkmisc.py create mode 100644 research/decoder-floors-frontier-format/sat/mkresults.py create mode 100644 research/decoder-floors-frontier-format/sat/proofs/E8_sym_h9_alt.drat.xz create mode 100644 research/decoder-floors-frontier-format/sat/proofs/P104H4_sym_h9_alt.drat.xz create mode 100644 research/decoder-floors-frontier-format/sat/proofs/P111_sym_h9_alt.drat.xz create mode 100644 research/decoder-floors-frontier-format/sat/proofs/P112_sym_h9_alt.drat.xz create mode 100644 research/decoder-floors-frontier-format/sat/proofs/edge_N0_r7.drat.xz create mode 100644 research/decoder-floors-frontier-format/sat/proofs/tri_keep11.drat.xz create mode 100644 research/decoder-floors-frontier-format/sat/results.jsonl create mode 100644 research/decoder-floors-frontier-format/sat/results_misc.jsonl create mode 100644 research/decoder-floors-frontier-format/sat/rigid.py create mode 100644 research/decoder-floors-frontier-format/scripts/provenance.py create mode 100755 research/decoder-floors-frontier-format/verify.sh diff --git a/research/decoder-floors-frontier-format/LICENSE 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Apache-2.0 (`LICENSE`, copied from the repository root). + +## Credits + +- **#446** (cost model, Theorem C), **#595**, **#1590**, **#2111**, **#2125**, **#2162**, **#2228**, **#2244**, **#2277**: the model, the dense pairing, the root-code floors (#2228 Theorem 4), Theorem 4′ and the frame-intersection certificate (#2277 §5) that this package extends. The statements "P112 ≥ 64 for every rank-8 realisation" (Theorem P), the cross-call batching bounds with `GL(9)` charts and `k ≥ 3` (Theorem B, Bk) and the every-`B` coordinate-block minrank are new here, built on those. Rigidity of the minimal-rank label table (Theorem R) is new here for E8, P112, P104 (#968's unit and class representatives), E7 and E6 only: for P111 it is prior work of **#620** (GamingPuzzled, SAT) and **#1097** (bitplane, merged, LRAT-checked in the general model), and **#1815** (jmbarrancoml) is the one-sided P112 predecessor; here P111 is re-derived on #589's lines. +- **#968**, **#1077**, **#1180**, **#1212**, **#1213**, **#1236**, **#1744**, **#757**, **#761**, **#620**, **#1097**, **#1815**, **#1331** (Arham-Tahir64), **#1249** (bbMisk, independent check of #1236), **#717** (ours; term-cost model, frontier-word censuses): the E8 / P112 / P104 / P111 floors for 8 terms, for factors in the live root code, rigidity, and the 9-term decoder outside the root code (#1180, whose cost 65 and shape are re-derived in `c4`). +- **#1224** (joaqcoo; the merge lemma, one-sided overcomplete structure), **#1633** (bbMisk; excess-rank budgets), **#1714** (the independent-call batch model), **#2294** (thakrarsagar; the `k = 2` / `O(9)` / Gram batching core, found independently of Theorem B and credited for priority), **#2293** (navinor; independent audit of our Lemma CR2 / Theorem B2), **#2176**, **#2206**, **#2232**, **#2240**, **#2247**, **#2137**, **#2272**, **#2267**, Discussion 1267 (philippeVV; co-star families), Discussion 1166 (1337isnot1337; child-merging screen). +- **#589** (P111 bit word): `data/p111_lines.json` is the `lines` array of `research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz` on `main` (sha256 pinned in `SOURCES.json`). +- Tools: kissat 4.0.4 (SAT; never CaDiCaL); `drat-trim` (Marijn Heule and Nathan Wetzler), not vendored here, optional. +- No file is copied from another contributor. `lib/roots.py`, `lib/satgen.py`, `lib/frames.py`, `lib/report.py`, `scripts/provenance.py`, `checks/c5_attaining_decoders.py`, `data/p104_frames.json` and `LICENSE` are the author's own files from the package of #2277 (`research/overcomplete-and-realisation-floors`). + +## New in this package + +Everything under `research/decoder-floors-frontier-format/`: `README.md`, `verify.sh`, `SOURCES.json`, `MANIFEST.sha256`, `lib/*.py`, `checks/*.py`, `sat/*.py`, `scripts/provenance.py`, and the evidence files `data/*.json`, `sat/results.jsonl`, `sat/results_misc.jsonl`, `sat/proofs/` (all produced by the generators in this package; the P104 class representatives `data/p104reps.json` and `data/h4_F1.json` were exported from the author's working card `STRAYS`, and are re-verified by the checks: `c1.gauge`, `c2`). diff --git a/research/decoder-floors-frontier-format/README.md b/research/decoder-floors-frontier-format/README.md new file mode 100644 index 0000000000..83999445cb --- /dev/null +++ b/research/decoder-floors-frontier-format/README.md @@ -0,0 +1,299 @@ +# Decoder floors are tight in the frontier format: one F2^9 label table, factors in col M of a rank-8 realisation (P112 64, E8 72, P104 (#968's unit) 56; two-call batches 128) — no κ claim + +> **Scope, stated first.** This package says one thing: *for decoders of the E8-family units that are hosted the way the frontier hosts them* — one global label +> table in `F2^9` (so the admissible pairing `B` of the whole unit has `F2`-rank `≤ 9`), with the factors of every term in the column space of a minimal-rank realisation +> (`col P ⊆ col M`, `col Q ⊆ row M`, rank `M = T = 8`; the square decoder of a *rank-9* realisation is **not** covered, OPEN §5.2 (c)), any number of terms, any number of independently batched calls — **the copy cost of the decoder cannot be lowered +> below P112 64, E8 72, P104 56 (P104 means #968's unit, `F1 = H4`; E7 49 on its own rank-7 table), and two independent P112 calls cost at least 128**, all attained. It contains **no κ claim**, no window, +> no supplier, no word-side (helper, transport, packing) cost. Everything outside that format is listed in §5 as **OPEN**, with what each escape would have to achieve. +> Nothing here removes the need to price an escape; it only says where the escapes are. **Credit, stated first:** the P111 row of Theorem R (rigidity of the rank-9 fitting matrix of a P111 pattern) is prior work of #620 (GamingPuzzled; SAT) and #1097 (bitplane, merged; LRAT-checked in the general model), and #1815 (jmbarrancoml, merged) proves a one-sided P112 fitting rigidity; what is new here is Theorem R on E8, P112, P104 (#968's unit and class representatives), E7, E6, plus P111 on #589's 111 lines in the symmetric and general model with DRAT (§3.1, §3.7). Other P104 units are **not claimed** (§6, §5.6). +> +> **Update (overnight closures, Oct 15).** Four open items were narrowed by other working cards of the author; **nothing in rows 1–9 changed, and none of the new items is re-run by `verify.sh`** (they are RECORDED, with the card's own scope and status quoted): the bit window of 13–14 totals is CLOSED (§3.7, row 18), the `h = 10` Gram root code is RECORDED closed (688 term-query DRATs pending, 1 verified; §5.1, rows 16a/16), `p = 45` is closed for `m ≥ 8` (dense `B`, column-space factors; §5.4, rows 19a/19), and the frame-module theorem (PROVEN, proof in §3.3) is a new tool for §5.2, which stays OPEN. +> +> **Labels.** **PROVEN** (proof in this README, or a cited proof of a merged / open PR of the swarm); **CHECKED** (recomputed by `verify.sh`: exact integer / `Fraction` / bit +> arithmetic, or SAT whose UNSAT verdict carries a DRAT proof that `drat-trim` verifies; kissat 4.0.4 only, never CaDiCaL); **RECORDED** (a result of the author's working +> cards that this package does **not** re-run, stated with its exact scope and where the evidence is; nothing in §2 marked PROVEN or CHECKED depends on a RECORDED item +> unless that is said); **OPEN**. + +## 0. Reading this package + +```sh +bash research/decoder-floors-frontier-format/verify.sh /fresh/output/dir # Python >= 3.11, standard library only +DRAT_TRIM=/path/to/drat-trim KISSAT=/path/to/kissat bash research/decoder-floors-frontier-format/verify.sh /fresh/output/dir +``` + +Step 0 checks every file against `MANIFEST.sha256` and every pin of `SOURCES.json` (hard stop on a mismatch). Then seven check scripts print one `PASS` / `FAIL` line per +assertion (`SKIP` for anything that needs `kissat` or `drat-trim` when it is absent: those lines are not PASS). `verify.sh` exits nonzero on any FAIL. One core; +6-7 minutes of CPU with both tools (358 s and 417 s user in two runs on a loaded machine; the 30 rigidity instances are re-solved and the UNSAT proofs re-checked), about 20 s without them (4 SKIP lines). +`HIT_FULL=1` additionally re-solves three slow clique-hitting instances (kissat only, no proof; §3.6). + +| script | what it recomputes | +|---|---| +| `checks/c1_rigidity.py` | Theorem R: regenerates 30 CNFs (hash, #vars, #clauses), recorded verdicts, 4 shipped DRAT proofs, and (with kissat) re-solves all 30; 16 UNSAT proofs re-checked with drat-trim, 14 SAT models decoded | +| `checks/c2_support_propagation.py` | Lemma 1.1 census (all `2^h − 1` label classes) on D4, D6, E7, E8, P112, P104; survivors, pair ranks, exactness of the per-term rank | +| `checks/c3_batch_premises.py` | labels of P112 in `F2^9`, normals `w_i`, `b(w_i,w_j) = 1`, `ρ(N(f_i),N(f_j)) = 8`, frame lemma, an `O(9)` chart with all 64 cross pairs of rank 7 | +| `checks/c4_masked_costar.py` | #1180's nine-term identity (cost 65), co-star dead sets, survivor 7-cliques | +| `checks/c5_attaining_decoders.py` | the decoders that attain the floors (E8 72, E7 49, D6 30, D4 12, P112 64, P104 56), exact `Fraction` arithmetic | +| `checks/c6_block_minrank_punctures.py` | EDGE E1a (every admissible `B`: the block `N(f)` of P112 has minrank 8; 8 DRAT proofs) and the C4.1 clique-hitting bounds on punctures | +| `checks/c7_p111_unit.py` | the P111 bit unit of #589's word: 111 lines of an E8 line system, `rank_F2 B_d = 9`, an 8-frame | + +Layout: `lib/` (E8-family units, Bron–Kerbosch, exact `F2` / `Q` linear algebra, CNF helpers; `roots.py`, `satgen.py`, `frames.py`, `report.py` are the author's own files from +#2277's package, `units.py`, `supprop.py`, `solve.py` are new), `sat/` (generators `rigid.py`, `misc.py`, `hitting.py`; registries; recorded results +`results.jsonl`, `results_misc.jsonl`; shipped DRAT proofs `proofs/*.drat.xz`; author tools `mkresults.py`, `mkmisc.py`), `data/` (frames, the 111 P111 lines, one chart). + +## 1. Setting + +A **unit** is a set of `v` pairwise non-proportional lines `r_t ∈ Q^8` of the E8 root system (`E8`: 120 lines, `P112 = E8 − F0`, `P104 := E8 − F0 − F1` with `F1 = H4`, the second frame of #968's unit (the swarm has other P104 units, e.g. #1399's 896 admissible ones; those are not covered here, §6 and §5.6; the 1,260 frames `F1` disjoint from `F0` fall into invariant classes by the number of live 8-frames of the unit, `759 / 767 / 801`, and `H4` lies in the 759 class), `E7`, `E6`, the bit unit `P111` of #589). A **realisation** `M` has `M_tt ≠ 0` and `M_ts = 0` for orthogonal `t ≠ s`. A **decoder** is `M = Σ_j p_j q_jᵀ` +with an **admissible (fitting) pairing** `B ∈ F2^{v×v}`: `B_tt = 1`, `B_ts = 0` for non-orthogonal `t ≠ s`, free on orthogonal pairs. Cost `= Σ_j rank_F2 B[R_j, A_j]`, `R_j = supp p_j`, +`A_j = supp q_j` (the copy-pairing rank sum that the invoice charges; every bound below therefore also holds for any costlier physical charge). The **dense** pairing is +`B_d = 1` on every orthogonal pair (and the diagonal); `rank_F2 B_d = h = 9` for E8, P112, P104, P111, `7` for E7, E6. +**Model hypothesis (the frontier format).** The registers of the word carry labels in one `F2^h`; the pairing used by every term is a block of one matrix `B = U Vᵀ` +(or `P Pᵀ`, the physical symmetric form) with `U, V ∈ F2^{v×h}`, i.e. `rank B ≤ h`. That the frontier word hosts its decoders on a single table of dimension `h = 9` is an +engineering fact of the word; the code that fixes it is in this repository: `research/complex-571-cold14-k492/code/construct_k_triangle.py:20` (`P=D['binary_ports'] … h=9`), `:183–187` (the centre frames are `hyper(P[frame[i]])`, so the physical pairing is `J + Gram(even parts)`, symmetric of rank `≤ 9`) and `:364` (`'ports':P[8:]`); the working card `EDGE/HOSTING.md` (not shipped) records the same reading, with the term-cost model of #717 (cost = dimension of the label span). It is **not** proved here, it is the hypothesis whose negation is §5 item 1. +**Label classes** (as in #2277 §1): `C = col B_d` (dim `h`); a class is the support of a nonzero `w ∈ C`; `Def(R) = {w ∈ C : supp w ∩ R = ∅}`, `δ(R) = dim Def(R)`. +**Frames.** A `T`-frame is `T` pairwise orthogonal lines (`T = 8` for E8, P112, P104, P111; 7 for E7). Every realisation has rank `≥ T` (a frame gives an invertible diagonal minor). + +## 2. Master table: escape class → status → certificate + +"Class X" = one label table with `h = 9` (`h = 7` for E7), factors in `col M / row M` of a rank-`T` realisation. + +| # | escape class | what is shown | status | certificate | +|---|---|---|---|---| +| 1 | non-dense admissible `B` of rank `≤ 9` (E8, P112, P104 = #968's unit and three class representatives, P111) or `≤ 7` (E7, E6) | every fitting `B` of that rank **equals the dense `B`**, general `UVᵀ` and symmetric `PPᵀ` | **CHECKED** (16 DRAT-verified UNSAT + 10 SAT controls + 4 non-dense witnesses) | `c1`, §3.1 (Theorem R); the P111 case is #620 / #1097's, re-derived on #589's lines | +| 2 | factors in the E8 root code, any `m`, dense `B` | P112 ≥ 64, E8 ≥ 72, P104 (#968's) ≥ 56, E7 ≥ 49 | **PROVEN** (cited, not re-run: #2228 Thm 4; #1180) + attained (**CHECKED**, `c5`) | #2228, #1180; with row 1 this is every `B` at `h = 9` | +| 3 | any rank-8 realisation, factors in `col M`, any `m` | **P112 ≥ 64**; P104 (#968's unit) ≥ 56; E8 ≥ 72, E7 ≥ 49 | **PROVEN** from **CHECKED** (Lemma 1.1 census) | §3.2 (Theorem P), `c2`; E8/E7/P104 also #2277 Cor. 5.3 | +| 4 | ≤ 8 paid terms, factors anywhere | square ⇒ row 3 | **PROVEN** | §3.3 | +| 5 | #1180's outside-code nine-term example | costs exactly 65; its shape is ≥ 64 for **every** `B` | **CHECKED** + **PROVEN** | `c4`, `c6`, §3.3 | +| 6 | co-star nine-term masked decoders (Disc. 1267) | all 120 dead sets span in every rank-8 realisation ⇒ each is a square decoder of a rank-9 realisation (pins them to §5 item 2(c)) | **CHECKED** (`c4`) + **PROVEN** consequence | §3.3 | +| 7 | batches: `k` independent P112 calls on one `F2^9` table, any chart, factors in `col M_c`, any `m` | `k = 2`: ≥ 128 + 3t; `k ≥ 3` (symmetric charts): ≥ 64k + 3t; attained | **PROVEN** from **CHECKED** premises | §3.4 (Thm B, Bk), `c3`; k = 2 / O(9) / Gram also swarm #2294 | +| 8 | any `B` on a single coordinate block | `rank_F2 B[N(f), N(f)] ≥ 8` for every admissible `B`, all 8 `f` | **CHECKED** (8 DRAT) | `c6`, §3.5 (EDGE E1a) | +| 9 | punctures of the coordinate term under any table | clique hitting: `f(2) ≥ 14`, `f(6) ≥ 46`, `f(7) ≥ 52`, `f(8) = 56`; `s/f(s) ≤ 1/7` on the computed values | **CHECKED** (s = 2, 6: DRAT; 7, 8 exact); s = 3, 4, 5 **RECORDED** (kissat UNSAT re-run with `HIT_FULL=1`, no DRAT here) | `c6`, §3.5 | +| 10 | P111 bit unit | rigid at `h = 9` (sym + general; **prior work of #620 / #1097**, re-derived on #589's lines); word-priced set-decoders with ≤ 14 totals cost ≥ 63 | rigidity **CHECKED** (2 DRAT); ≥ 63 for ≤ 12, 13 and 14 totals **RECORDED** (card: n ≤ 13 PROVEN, n = 14 CHECKED, 735 DRAT-verified) | `c1`, `c7`, §3.7 | +| 11 | E8 square root-code decoders for **every** `B` (no `h` cap) | ≥ 72 (4022 pair classes) | **RECORDED** (4022 DRAT verified in the working card; not re-run) | §3.8 | +| 12 | `FL(4)`: every D4 decoder, every `B`, every `m` | ≥ 12 | **RECORDED** (support-level enumeration, 144 classes) | §3.8 | +| 13 | E7 square maximal-normal root-code decoders, every `B` | ≥ 49 (no ≥ 5 root–root partners) | **RECORDED** (exact enumeration + independent audit in the working card) | §3.8 | +| 14 | P104 delivery (#968's decoder) under any `F2^9` table | no helper-free nested delivery (24 DRAT) | **RECORDED** | §3.8 | +| 15 | PUNCT2 near-miss unions `p = 25..44`, any physical table, any `m` | no decoder beats P102's 54/102 (Model C) | **RECORDED** | §3.8 | +| 16a | `h = 10` label tables, **root code (Gram realisation)**, any `m`, any fitting `B` of rank ≤ 10 | every term ≥ 8, so P112 ≥ 64 (a rank-10 table saves nothing; break-even needed ≤ 59) | **RECORDED** (card: CHECKED; 46/46 `G`-query DRATs verified, 688 term-query UNSAT with DRAT **pending**) | §3.8, §5.1 | +| 16 | **label tables of rank `h ≥ 10`, the rest**: non-Gram realisations at `h ≥ 10`; `h = 11` at the orthogonal dead-root term `N(f) × N(g)`; `h ≥ 12` | non-dense fitting tables exist (row 1, witnesses); `h = 11` needs cost ≤ 55 to tie; `h ≥ 12` not attempted | **OPEN** | §5.1 | +| 17 | **factors outside `col M`, `m ≥ 9`** | `m = 9`: `ρ ∈ {8, 9}`; (b) `ρ = 8` one outside factor, (c) `ρ = 9` square rank-9, (d) `m ≥ 10`; tool: the frame-module theorem (PROVEN, §3.3) | **OPEN** (cases (b), (c), `m ≥ 10`; no instance decided) | §5.2 | +| 18 | **bit windows of 13–14 totals** (n ≥ 15 is excluded by hosting, a note) | word-priced P111 set-decoders ≤ 14 totals ≥ 63; P110 ≤ 12 totals ≥ 63 | **CLOSED, RECORDED** (card: n ≤ 13 PROVEN, n = 14 CHECKED) | §3.7, §5.3 | +| 19a | punctures `p = 45`, `m ≥ 8` terms (Model C, dense `B`, column-space factors) | cost ≥ 40 > 39.71 (Theorem 45) | **RECORDED** (card: PROVEN from CHECKED data; floating-point MILP re-certified exactly) | §3.8, §5.4 | +| 19 | **puncture windows `p ≥ 46`**, and `p = 45` with `m ≤ 7` | — | **OPEN** | §5.4 | +| 20 | **batches `m ≥ 17` outside `col M`**, GL charts with `k ≥ 3`, mixed units, correlated demands, `h ≥ 10` joint tables | — | **OPEN** | §5.5 | +| 22 | projected / asymmetric copy for P110 (cost 62) | SPEC-legal (a checker convention forbids it, not the SPEC), but on PR589's tuned chains the best term costs ≥ 3.57% c net | **RECORDED** pricing (not a theorem) | §3.7 | +| 21 | **P104 units other than #968's** (Theorem R for all 1,260 frames `F1`; the Lemma 1.1 census for them) and **rank-9 square decoders** | — | **OPEN** | §5.6, §5.2 (c) | + +Rows 11–15 are *beyond* what the headline needs: at `h = 9` row 1 already forces `B = B_d`, so the headline uses `B_d` only. Their role is to show that the dense-pairing +statements are not hiding a cheap non-dense escape *inside* the `h = 9` class; row 1 does that, and rows 11–13 give the independent every-`B` statements for the +(square, root-code) sub-class. + +## 3. Results + +### 3.1 Theorem R: the minimal-rank label table is unique (CHECKED, DRAT) + +**Theorem R.** Let `U ∈ {E8, P112, P104, P111}` (P104 as defined in §1, plus the class representatives below) and let `B ∈ F2^{v×v}` fit `U` with `rank_F2 B ≤ 9`. Then `B = B_d`. The same holds for `E7, E6` with `rank B ≤ 7`. (**Credit:** for P111 this is #620's and #1097's result; see the paragraph after the table.) +It holds in the general model `B = U Vᵀ` and in the symmetric model `B = P Pᵀ` (the physical one: `B_d = J + C` with `C` the alternating Gram of the even-weight label parts, +which is `PPᵀ` for odd-weight labels `P_t = 1 + e_t`). For P104 the statement is checked for `H4`'s `F1` (the frame of #968; its unit has 759 live 8-frames, so it lies in the 759 class) and one representative of each of the three invariant +classes (`759 / 767 / 801` live 8-frames) of frames disjoint from `F0`: **three classes, four instances** (`H4` is in the 759 class). **These classes are not proven to be single orbits**, and the backfill over all 1,260 frames was not finished, +so P104 is claimed only for #968's unit (`F1 = H4`); Theorem P's census was run on `H4` only, and Theorem R / Theorem P for every other P104 unit is **not claimed** (§6, OPEN §5.6). + +*Encoding (`sat/rigid.py`).* `B = U Vᵀ` with `U, V ∈ F2^{v×h}`; constraints `⟨U_t, V_t⟩ = 1`, `⟨U_t, V_s⟩ = 0` for every non-orthogonal ordered pair `t ≠ s`; mode `alt` adds the clause +"some orthogonal ordered pair (unordered in the symmetric model) has `B_ts = 0`"; mode `ctl` omits it. Products and parities are Tseitin variables with constant folding. +*Gauge (PROVEN).* Let `S` be a pairwise non-orthogonal `k`-clique (`k = 8` for E8, P112, P104, P111; 7 for E7; 5 for E6; pinned in `results.jsonl`, `c1.gauge` checks the clique). +Then `B[S,S] = I_k`, so `U[S]` has rank `k`; `GL(h)` acts by `U ↦ U g`, `V ↦ V g^{-T}`, preserves `B` and is transitive on rank-`k` `k`-tuples, so `U[S] = (e_1..e_k)` w.l.o.g. In the +symmetric model with `k = 8`, `h = 9`: `P[S]` is orthonormal, `W = span P[S]` is nondegenerate, `W^⊥ = ⟨w⟩` with `w·w = 1` (`det I_9 = det I_8 · w·w`), so an isometry in `O(9)` maps +`P[S]` to `(e_1..e_8)`; this needs `k ≥ h − 1`, which `registry()` asserts for every symmetric UNSAT instance. (For SAT instances the gauge is only a restriction, so a SAT verdict stands.) +Rank `≤ h` is equivalent to a factorisation with `h` columns, so UNSAT of `alt` says every fitting `B` of rank `≤ h` is dense. (The general model contains the symmetric one as a sub-model, so general-model UNSAT already implies the symmetric statement and, for E7 and E6 where only the general model is run, covers the physical table; the symmetric runs are redundant but consistent.) + +| group (`results.jsonl`) | instances | verdict | check | +|---|---|---|---| +| `alt`, minimal rank | P112, E8, P104 (H4, 759, 767, 801), P111, each sym + gen (14 instances); E7, E6 gen at `h = 7` (2) | **UNSAT**, every proof verified by drat-trim (binary DRAT, kissat 4.0.4) | `c1.status`, `c1.resolve`; four proofs shipped (`proofs/`, sym P112 / E8 / P104-H4 / P111) | +| `ctl`, no extra clause | P112, E8, P104-H4, P111 (sym + gen), E7, E6 | **SAT**; every decoded model is a fitting matrix with 0 non-dense orthogonal entries (model is `B_d`) | `c1.ctl` | +| negative controls | `SUB20`, `SUB40` (the gauge clique plus the first 20 / 40 other lines of P112, `h = 9`, gen); P112 at `h = 10` (sym 59 of 3304, gen 89 of 6608 orthogonal pairs with `B_ts = 0`) | **SAT**: a non-dense fitting `B` exists | `c1.neg` | + +(The encoding is the one of the author's working card `STRAYS/rigid2.py`; the clique choice here is deterministic and pinned, so the CNFs differ from the card's files, with the +same verdicts. The `h = 10` row shows the instance family does detect non-rigidity; the `SUB` rows show the same at `h = 9` on sub-units.) + +**Prior work on P111 and one-sided P112 rigidity (credit; this changes the novelty of Theorem R).** #620 (GamingPuzzled, open) found by SAT, without proof logs, that the rank-9 fitting matrix of its archived P111 pattern is unique; #1097 (bitplane, merged 2026-10-13) proof-checked this with LRAT: "fitting rank ≤ 9, unequal to `B0`" is UNSAT in the general (asymmetric) model with an 8-port gauge. That is the P111 row of Theorem R, and it is **theirs**. *Which set?* #1097's object is the archived sorted Hamming-root order of #620 with indices `39, 52, 57, 71, 76, 78, 103, 109, 114` deleted (`encoding.py:9`); #589's 111 lines (`data/p111_lines.json`) are a different 111-subset of the same 120 Hamming-coordinate lines: 104 lines in common (up to sign), 7 on each side only (a one-off comparison against #1097's retained `e8units.hamming_e8`, not part of `verify.sh`; whether the two sets are `W(E8)`-equivalent was not checked). So Theorem R here **re-derives** the P111 statement on #589's puncture, in both the general and the symmetric model, with DRAT, and does not rely on #1097. #1815 (jmbarrancoml, merged 2026-10-14) is the one-sided predecessor for P112: with the two column spaces fixed, unit diagonals and the required zeros force `N = RRᵀ` and `B = UUᵀ`; Theorem R removes the column-space hypothesis on `B` (every fitting `B` of rank `≤ 9` is dense). What is new here is Theorem R on E8, P112, P104 (#968's unit and representatives), E7 and E6. + +**Corollary R1 (PROVEN from Theorem R and the theorems of §3.2 / rows 2).** In the frontier format (one `F2^9` table, `F2^7` for E7), every term pairs through the dense `B_d`. Hence +every statement made below for the dense pairing holds for every hosted decoder, and *the overcomplete "any `B`" slack* (orthogonal-normal blocks of fitting minrank 5–6; +orthogonal E8 root–root terms of rank 6) *does not exist at `h = 9`*: it needs a non-dense `B`, hence `h ≥ 10`. +**Corollary R2 (two-call tables, PROVEN).** If a two-call batch uses one `F2^9` and each call's block fits P112, the labels are `(P, PΦ)` with `Φ ∈ O(9, F2)` (general model: +`(U, V), (UΦ, VΦ^{-T})`, `Φ ∈ GL(9)`). *Proof.* By Theorem R both blocks equal `B_d`, of rank 9. `P_1P_1ᵀ = P_2P_2ᵀ` has rank 9, so `P_1, P_2` have the same column space and `P_2 = P_1Φ`; +then `P_1(ΦΦᵀ − I)P_1ᵀ = 0` with `P_1` of full column rank gives `ΦΦᵀ = I`. ∎ + +### 3.2 Theorem P: P112 ≥ 64 in every rank-8 realisation (PROVEN from CHECKED) + +Setting (#2277 §5). `K` is any field (the swarm uses `Q` / `C`; Lemma 1.1 and Theorem P are field-independent). Let `M` be a realisation of rank exactly `T = 8` (the minimum), `M = P₀ N Q₀ᵀ`, `P₀, Q₀ ∈ K^{v×8}` bases of `col M`, `col Mᵀ`, `N ∈ GL_8`; `π(t)` is row `t` of `P₀`. +Consider decoders with `p_j = P₀x_j`, `q_j = Q₀y_j` (every square decoder `m = 8` is of this form; for `m > 8` this is the hypothesis `col P ⊆ col M`). Then `N = Σ_j x_j y_jᵀ`, so at least 8 terms +are nonzero. + +**Lemma 3.1 (support-only bound; #2277 Lemma 5.1, restated).** For `R, A ⊆ [v]`, `rank_F2 B_d[R, A]` depends only on the subspaces `Def(R), Def(A) ⊆ C`, and `≥ h − δ(R) − δ(A)`. +*Proof.* Take `B_d = X Yᵀ`, `X = Y ∈ F2^{v×h}` of full column rank (`B_d` symmetric), `col X = C`. `ker X_R ≅ Def(R)` via `w = Xz`, so `rank X_R = h − δ(R)`, and the row space of `X_R` is +the annihilator of `ker X_R`, i.e. determined by `Def(R)`; likewise for `A`. Sylvester gives the bound. ∎ +**Lemma 3.2 (Theorem 4′; #2277 §5, restated).** `cost ≥ Σ_{j : x_j, y_j ≠ 0} (h − δ(R(x_j)) − δ(A(y_j)))`, with `R(x) = {t : π(t)·x ≠ 0}`. If `w ∈ Def(R(x))` then `π(supp w) ⊆ x^⊥`: the class +`supp w` is *deficient* (its `π`-image lies in a hyperplane). *Proof.* Lemma 3.1 and the count of nonzero terms. ∎ + +**Lemma 1.1 (support propagation; working card `IDEAL3` §1.1, PROVEN here).** Let `E` be an 8-frame and write a functional `h*` that kills `π(O)` as `h* = Σ_{e ∈ E} c_e σ(e)`, where `σ(e) = N κ(e)ᵀ`, `κ(e)` = row `e` of `Q₀` +(`π(e)·σ(e′) = M_{ee′}`: for a frame `E` the `σ(e)` are the dual basis of `{π(e)}` up to the nonzero scalars `M_ee`, and `σ(e)` depends on the line only). (i) `c_e = 0` for `e ∈ E ∩ O`: `h*(π(e)) = c_e M_ee`. (ii) For every 8-frame `F`: `supp_E(c) ⊆ (E ∖ O) ∩ ⋃_{g ∈ F∖O} N̄(g)`, +`N̄(x) = {x} ∪ {lines non-orthogonal to x}`. *Proof.* (i) is the displayed identity. For (ii), applying (i) to `F` gives `h* ∈ span σ(F∖O)`; expanding `σ(g)` in the dual basis of `E`, +`σ(g) = Σ_e (M_eg/M_ee) σ(e)`, whose `E`-support lies in `{e : M_eg ≠ 0} ⊆ N̄(g) ∩ E` (`M_eg = 0` for orthogonal `e ≠ g`). ∎ +*Corollary.* If iterating (ii) over all frames ends with some `S_E = ∅`, the class `O` is non-deficient in **every** rank-8 realisation. + +**Census (CHECKED, `c2`).** For every one of the `2^h − 1` classes: D4 (3 frames), D6 (15), E7 (135), E8 (2025): **all** certified. P112 (1260 frames): **exactly 8 classes uncertified** (the +8 sets of 56 live lines orthogonal to one removed root; each has `rank_Q = 7`, i.e. is Gram-deficient, so the certificate is sharp); P104 (`H4`'s unit, 759 frames; the census was run on this unit only): 16 uncertified (sets of 52), also +exactly the Gram-deficient ones. In both units the uncertified classes contain **no 2-dimensional subspace** (`a ⊕ b` is never one). For each uncertified `O`, the classes with support inside `O` are +`{O}` alone (`c2.def`), so `Def(V∖O) = ⟨w_O⟩` and the single-deficiency rank is `rank B_d[V∖O, V] = 8`; over all ordered pairs of P112 survivors `rank B_d[V∖O, V∖O′] = 8` (64 of 64), and +for P104 the values are `8` (192) and `7` (64). + +**Theorem P (PROVEN from the CHECKED census).** For P112, every rank-8 realisation `M`, every decoder with `col P ⊆ col M`, `col Q ⊆ row M` (all square decoders, any `m` otherwise), dense `B`: +**cost ≥ 64**, attained by the removed-frame coordinate decoder (`c5`). For P104 = #968's unit (`F1 = H4`; the census of Lemma 1.1 is run on this unit only): **cost ≥ 56**, attained (`c5`). +*Proof.* A nonzero `w ∈ Def(R(x_j))` is a deficient class (Lemma 3.2), hence one of the 8 (16) survivors (Corollary of Lemma 1.1; the census). Survivors contain no 2-dimensional subspace, so +`δ(R(x_j)), δ(A(y_j)) ≤ 1`, and by Lemma 3.1 and the census the term's rank is `9` (no deficiency), `8` (one) or, for both deficient, the pair rank: `8` in P112, `≥ 7` in P104. So every nonzero +term costs `≥ 8` (P112), `≥ 7` (P104); at least 8 terms are nonzero. ∎ +For E8 (72) and E7 (49) no class is deficient (census), so every term costs `h`; with at least `T` terms the floors are `8·9 = 72`, `7·7 = 49`. (#2277 Cor. 5.3 gives all of E8 72, E7 49, D6 30, D4 12, P104 56 +via the weaker clause-(c) certificate; its P112 value was 56, and "P112 ≥ 64 for every realisation" was listed there as open — Theorem P closes it.) **Sharpness (CHECKED, `c5`):** +frame decoders with `M = Gram` cost 72 (E8), 49 (E7), 30 (D6), 12 (D4); the removed-frame decoder of P112 costs 64; the eight frame-pair terms of #968 cost 56 on P104. + +### 3.3 Few terms, outside `col M`, #1180 and the co-star (PROVEN / CHECKED) + +*(working card `STRAYS-MASK` §1–§3; dense `B`, which at `h = 9` is every `B` by Theorem R)* +**1.1 Free-term absorption.** A term with `B[R_j, A_j] = 0` contributes `0` to the cost and nothing on the diagonal or on orthogonal pairs, so the paid part `N = Σ_{paid} p_j q_jᵀ` is again a realisation +of the same pattern with the same cost; w.l.o.g. every term is paid. +**1.2 Merge lemma (#1224 Lemma D, both sides).** If `q_i = c q_k` then `p_i q_iᵀ + p_k q_kᵀ = (c p_i + p_k) q_kᵀ`; the new row support lies in `R_i ∪ R_k`, and +`rank B[R_i ∪ R_k, A_k] ≤ rank B[R_i, A_k] + rank B[R_k, A_k]` (`A_i = A_k`). The merge does not raise the cost for **any** `B`. W.l.o.g. no two row normals and no two column normals are proportional. +**1.3 `m ≤ 8`.** `rank N ≥ α_⊥ = 8`, so `m ≥ 8`; `m = 8 = rank N` is the square case with factors in `col N / row N` (every factor lies in the column space of a rank-8 matrix with 8 terms), so Theorem P +gives `≥ 64` (P112), `≥ 56` (P104, #968's unit), **whatever the factors, the realisation or the masking**. This covers every component of #1213, not only the connected one. +**1.4 `m = 9` trichotomy.** `a = rank P = ρ + s`, `b = rank Q = ρ + t`; Sylvester: `ρ ≥ a + b − 9`, so `s + t ≤ 9 − ρ` with `ρ = rank N ∈ {8, 9}`. (a) `ρ = 8`, `s = t = 0`: Theorem P; (b) `ρ = 8`, `s + t = 1`: §5.2 (b) (#1180 / #1224's residual class); (c) `ρ = 9`: square in a rank-9 realisation (OPEN, §5.2 (c)). +**#1180's example (CHECKED, `c4`).** Its nine-term identity `D Tᵀ = 8·Gram` holds on all `112 × 112` entries (`ex1180`-form: `t_1 = R_0 + z`, outside terms `(t_1, R_0)` and `(−z, R_0)`) and its term ranks under `B_d` +are `[8]×8 + [1]`, **total 65**. Its two outside-`col M` terms share the column vector `R_0`: by the merge lemma (any `B`) the example costs at least the removed-frame decoder, whose terms are the blocks `B[N(f), N(f)]`, +each of fitting minrank 8 for every `B` (`c6`, §3.5), so **after merging the two outside terms it is the removed-frame decoder, which costs ≥ 64 for every admissible `B` by E1a**: the "outside `col M`" in #1180's example is only a split of a proportional pair. +**Co-star (CHECKED, `c4`; PROVEN consequence over any field of characteristic 0, the scope of the working card `STRAYS-MASK` Theorem 3.2).** *Definitions.* The **dead set** of a term `p qᵀ` on the row side is `{t : p_t = 0}` (the lines the term does not touch), on the column side `{t : q_t = 0}`; a set of lines is **spanning** if its images `π(t)` span `K^8` in every rank-8 realisation. For each of the 120 root lines `α`, the dead set `V ∖ S_α` (`S_α` = live lines orthogonal to `α`, plus `α` if live; `|S_α| = 60` for the 112 live `α`, 56 for the 8 removed) is certified spanning in every +rank-8 realisation (Lemma 1.1). *Lemma (inside dead sets; STRAYS-MASK Lemma 3.1).* If a term of a rank-8 realisation has its column factor in `row M` (`q = Q₀ȳ`, `ȳ ≠ 0`, an "inside" side), its dead set `{t : κ(t)·ȳ = 0}` has images in the hyperplane `ȳ^⊥`, so it is not spanning and cannot contain `V ∖ S_α` (rows by transposition: the pattern is symmetric). *Consequence.* Let a nine-term masked co-star decoder (philippeVV, Discussion 1267; every term's supports inside co-stars, arbitrary coefficients, zeros, any `k`) have `ρ` = the rank of the full nine-term sum. If a term is free, or two row or two column normals are proportional, the paid or merged part has `≤ 8` terms, hence is square, and Theorem P gives `≥ 64` outright. Otherwise `ρ = 8` would give `s + t ≤ 1` (§1.4), so one side, say `t = 0`, is inside for all nine terms, contradicting the Lemma; `ρ ≥ 10 > m` is impossible. So `ρ = 9` and the decoder is a square +biorthogonal decoder of a **rank-9** realisation: cases (a), (b) of §1.4 are empty for co-stars. This pins the open `k = 7/8/9` families to §5.2 (c); it does not close them. Every survivor class contains an +orthogonal 7-clique and no 8-clique (`c4.survclique`), so its killing functional is unique up to scale in every rank-8 realisation. + +**Frame-module theorem (nine terms, any realisation, any rank, factors anywhere; PROVEN; working card `STRAYS-MASK2` §2, proof reproduced here).** Let `M = Σ_{j≤9} p_j q_jᵀ` be a nine-term decoder of a unit with an 8-frame `F` (so `M_ff ≠ 0`, `M_ff′ = 0`), `p(t), q(t) ∈ K^9` the rows of the factor matrices, `P_F, Q_F ∈ K^{8×9}` the rows `p(f), q(f)`, `f ∈ F`. Then `P_F Q_Fᵀ = D_F = diag(M_ff)` is invertible, so `rank P_F = rank Q_F = 8`, with one-dimensional kernels `ν` (`P_F ν = 0`) and `π` (`Q_F π = 0`). *(i)* `J := Q_Fᵀ D_F⁻¹ P_F` satisfies `J² = J`, `rank J = 8`, so `J = I − u wᵀ`, `wᵀu = 1`, and `P_F(I − J) = 0`, `(I − J)Q_Fᵀ = 0` force `u ∝ ν`, `w ∝ π`: **`π·ν ≠ 0`** and `J = I − νπᵀ/(π·ν)`. *(ii)* `π ∉ span p(F)` (if `π = Σ β_g p(g)`, then `0 = q(g)·π = β_g M_gg` for every `g ∈ F`, so `π = 0`), so every line has **`p(e) = Σ_{f∈F} (M_ef/M_ff) p(f) + λ_e π`**, and likewise **`q(s) = Σ_f (M_fs/M_ff) q(f) + μ_s ν`** (pair with `q(g)`, resp. `p(g)`). *(iii)* Substituting, `M_es = Σ_f M_ef M_fs / M_ff + (π·ν) λ_e μ_s`: the Schur defect of the frame is **rank one**, with `λ_e ∝ p(e)·ν`, `μ_s ∝ π·q(s)`. When `rank P = 8` (`λ ≡ 0`) this is Lemma 1.1; in rank 9 it is the value-level rank-9 Lemma 1.1 asked for in §5.2 (c), as a support calculus. *Status of its use:* the theorem is PROVEN; the SAT model built from it (case (b), co-star family, full `m = 9`) is RECORDED as sound on three positive controls but **no instance was decided** (timeouts at 25–28 min under load), and a class-level rank-9 pair test (restricted Lemma 1.1) leaves a 136-clique of mutually compatible dead classes, so pair data alone cannot close (c). For `m ≥ 10` the kernel becomes `(m − 8)`-dimensional and the three sharp rules (reverse expansion, Schur support, K-zero) are lost. Cases (b), (c), `m ≥ 10` remain OPEN (§5.2). + +### 3.4 Batching: independent calls on one `F2^9` table (PROVEN from CHECKED) + +*(working card `STRAYS-BATCH`; the model is #1714's: `k` independent P112 calls, `C = I_k`, cross-call blocks of the requested matrix are 0.)* The requested matrix is `M_1 ⊕ … ⊕ M_k`, `M_c = P_cQ_cᵀ` a rank-8 +realisation (Gram: `P_c = Q_c = X`). A decoder is `Σ_j d_j t_jᵀ = M` with `d_j = (P_ca_j^{(c)})_c`, `t_j = (Q_cc_j^{(c)})_c`; exactness is `A Cᵀ = I_{8k}`. One label table on all `112k` lines; each call's block fits P112, +so by Theorem R it is dense and by Corollary R2 call `c` carries labels `PΦ_c`. The charge of term `j` is `rank_F2` of the pairing between the span of the output labels of its union support `S_j` and the span of the input +labels of `T_j` (the cheapest charge, #1714; every costlier charge is covered). **Premises (CHECKED, `c3`; they are Lemma 1.1's sharpness in label language):** +(P1) in any rank-8 realisation the support of `P_c x`, `x ≠ 0`, has label span `V` (dim 9) or one of 8 hyperplanes `H_i = w_i^⊥` (it avoids the deficient class `O_i`, the live lines orthogonal to `f_i`); the survivors contain no 2-dimensional +subspace, so no support avoids two classes and every span has dim ≥ 8. (P2) `b(w_i, w_j) = 1` for all 64 ordered pairs, and `ρ(N(f_i), N(f_j)) = 8` for all 64 pairs, `i = j` included. +**Theorem B (`k = 2`, PROVEN).** For every chart `Φ` (`O(9)` symmetric or `GL(9)` general) every decoder satisfies `Σ_j cost_j ≥ 128 + 3t`, `t = #{j : cost_j = 7}`; the minimum is 128 (two coordinate decoders), only at `t = 0`. +*Step 1.* Union spans have dim ≥ 8 (P1), so cost `≥ 8 + 8 − 9 = 7`; at cost 7 both spans are hyperplanes. *Step 2 (rank-7 terms are pure cross-call).* If `a_j` has a nonzero part in call `c`, that part's span (dim ≥ 8) lies in the output span `H` +of dim 8, hence equals it, so `H` is a call-`c` output hyperplane; likewise `H′` is the input hyperplane of every call where `c_j` is nonzero. If one call had both parts nonzero, `(H, H′)` would be an output / input hyperplane pair of +that call, which pairs at rank 8 by (P2). So `calls(a_j) ∩ calls(c_j) = ∅`. *Step 3 (rank count).* With `N = Σ_{cost 7} a_jc_jᵀ = [[0, N₁₂],[N₂₁, 0]]`, `t₁₂ + t₂₁ = t`, `rank N₁₂ ≤ t₁₂`, `rank N₂₁ ≤ t₂₁`; the other active terms +cost ≥ 8 and sum to `I₁₆ − N`, so there are at least `rank(I − N) = 8 + rank(I₈ − N₂₁N₁₂) = 16 − d` of them, `d = dim ker(I₈ − N₂₁N₁₂) ≤ min(t₁₂, t₂₁) ≤ t/2`. So `Σ cost ≥ 7t + 8(16 − d) ≥ 128 + 7t − 8min(t₁₂,t₂₁) ≥ 128 + 3t`. ∎ +**The counting floor 120 is not the truth** (`c3.chart.rank7`): there is a chart `Φ ∈ O(9, F2)` (shipped in `data/chart_rank7.json`) for which all 64 cross pairs (call-1 support `N(f_i)`, call-2 support `N(f_k)`) pair at rank 7, while within a call the same pairings have rank 8. +Per-term counting alone gives 120; the extra 8 units come from the rank argument. +**Theorem Bk (`k ≥ 3`, symmetric model, PROVEN).** Charts `Φ_c ∈ O(9, F2)`, any rank-8 realisations, any `m`: `Σ cost ≥ 64k + 3t`. *Proof.* Steps 1–2 verbatim. Every hyperplane is `n^⊥` for a unique normal; slot `(c,i)` has normal `Φ_cw_i`, and +`b(n, n) = 1` by (P2). Partition the `8k` slots into classes by normal (at most one slot per call). A rank-7 term has `b(n, n′) = 0`, so `n ≠ n′`; `a_j` is supported in the class of `n`, `c_j` in the class of `n′`. For `x ∈ Z^P_{c,i}`, `y ∈ Z^Q_{c,i′}` (`Z^P_{c,i}` := the coefficient vectors `x` whose call-`c` output support `supp P_c x` avoids `O_i`, i.e. lies in the hyperplane slot `(c,i)` with normal `Φ_c w_i`; `Z^Q_{c,i′}` := the same on the input side, `supp Q_c y` avoiding `O_{i′}`), +`i ≠ i′`: `yᵀx = Σ_{e ∈ E}(y·q_c(e))(p_c(e)·x)/M_ee` over any 8-frame `E` of live lines; a summand needs `e ∈ N(f_i) ∩ N(f_{i′})`, and `c3.framelemma` finds for every `i < i′` a frame with `E ∩ N(f_i) ∩ N(f_{i′}) = ∅`, so `yᵀx = 0`. Hence +`K = CᵀA` over the rank-7 terms has `K_ij ≠ 0` only if `class(c_i) = class(a_j)`, `K_ii = 0`. *Lemma K:* `corank(I_{8k} − ACᵀ) = corank(I_t − K) ≤ t/2` (order the classes; the larger of `{j : α(j) < β(j)}`, `{j : α(j) > β(j)}` has size `≥ t/2`, along `K`-arcs inside it the class strictly +moves, so `K[S,S]` is nilpotent up to a permutation and `I − K[S,S]` is invertible). So `Σ cost ≥ 7t + 8(8k − t/2) = 64k + 3t`. ∎ (Every batch with exactly `8k` terms needs no factor hypothesis, #1714 §5.) +**Credit.** #2294 (thakrarsagar) proves the `k = 2`, `O(9)`, Gram-root-code core first (its Theorem A: `≥ 128 + 6μ + 7(r_01 + r_10 − 2μ)`, `μ = min(r_01, r_10)`) and independently finds that some isometric charts make all 64 exceptional pairs rank 7; the derivation above was done +before #2294 was read and is the same inequality. Beyond #2294: `GL(9)` charts of the general factor model, any rank-8 realisation (the Theorem P setting), and `k ≥ 3` in the symmetric model. + +### 3.5 One coordinate block under any `B`; punctures (CHECKED) + +**EDGE E1a (`c6`, 8 DRAT).** For each removed root `f` (all 8), `N(f)` is the 56-line support of the coordinate term. *Every* admissible `B` has `rank_F2 B[N(f), N(f)] ≥ 8`: the instance "`rank ≤ 7`" (gauge: the largest non-orthogonal +clique in `N(f)`, 7 lines) is UNSAT with a drat-trim-verified proof (8 proofs of about 70 kB each; one shipped), and "`rank ≤ 8`" is SAT (the dense block). So no choice of `B` lowers a coordinate term of the removed-frame decoder, and with +`m ≥ 8` the removed-frame decoder costs `≥ 64` for every `B`. (Together with the merge lemma this is the "shape ≥ 64 for every `B`" of §3.3.) `ω_non(N(f)) = 7` and `α_⊥(N(f)) = 4` are recomputed in `c6`. +**Punctures (NEWWORD C4.1).** A puncture `D ⊆ N(f)` (lines deleted from the support) that brings `rank B[R∖D, R∖D]` to `≤ 8 − s` for some admissible `B` must hit every `(9 − s)`-clique of pairwise non-orthogonal lines (a clique `C` has `B[C,C] = I` +for every admissible `B`). `f(s)` := the least size of such a hitting set. Proven bounds: `f(2) ≥ 14` (the 576 7-cliques cannot be hit by 13 lines; DRAT), `f(6) ≥ 46` (the largest triangle-free kept set has 10 lines: 11 is UNSAT, DRAT; the working card +had 15 and `≥ 42`), `f(7) ≥ 52` (`α_⊥(N(f)) = 4`, exact), `f(8) = 56`. Recorded (kissat UNSAT, working-card DRAT; `HIT_FULL=1` re-solves them here without a proof): `f(3) ≥ 23`, `f(4) ≥ 32`, `f(5) ≥ 38` (the card's `s = 1` value, `z ≥ 8`, is EDGE's separate +`killgauge` SAT result and is not reproduced here). The ratios `s/f(s)` are `.1429, .130, .125, .132, .130, .135, .143` (`s = 2..8`; `c6.f` prints the proven ones): **no computed `s ∈ {2..8}` exceeds 1/7** (`s = 3–5` RECORDED, `s = 1` not computed here). The puncture-saving statement +("punctured coordinate decoders tie P112 at best under any global label table") is therefore the clique-hitting bound plus the card's price model (break-even `s/f(s) > .1441`, margin under 1%, inside the price model's error): the **bound** is CHECKED, the +**pricing consequence** is RECORDED and is not claimed as a theorem. + +### 3.6 Slow instances + +`HIT_FULL=1` re-solves `hit_k6_z22`, `hit_k5_z31`, `hit_k4_z37` (the `s = 3, 4, 5` bounds) with kissat only; recorded (kissat 4.0.4, UNSAT, no proof; wall time under machine load): `hit_k6_z22` 102 s, `hit_k5_z31` 227 s, `hit_k4_z37` 9 s, i.e. `f(3) ≥ 23`, `f(4) ≥ 32`, `f(5) ≥ 38`; their results are in `sat/results_misc.jsonl` (flagged `slow`) and re-derived by `HIT_FULL=1`. No DRAT is checked for these here (the proofs are 100–200 MB). + +### 3.7 The bit side: P111 (rigidity CHECKED, prior work of #620 / #1097; "≥ 63" for ≤ 14 totals RECORDED) + +**P111 is rigid at `h = 9` (CHECKED, `c1`, `c7`; prior work: #620 by SAT, #1097 LRAT-checked in the general model, see §3.1).** On #589's set, which is not #1097's (§3.1), the statement is re-derived here in two models. The 111 live lines of #589's word (`data/p111_lines.json`; the source file's sha256 is pinned in `SOURCES.json` and equals the entry of `research/e8-bit-p111-tuned/MANIFEST.sha256` on `main`) are a sub-configuration of an E8 line system +(`c7`); `alt` is UNSAT in the symmetric and the general model (drat-trim verified; the symmetric proof is shipped), the controls are SAT with the dense `B`. So at `h = 9` the bit side is also a dense-`B` question. +**RECORDED (not reproduced here): P111 set-decoders with at most 12 retained totals cost ≥ 63.** The working card `STRAYS-BIT` proves it premise-free from the full census of the E8 line matroid's flats (ranks 1–6: `120 / 4,900 / 85,680 / 661,542 / 2,091,600 / 2,221,780`, maximal line counts `1, 3, 6, 12, 20, 36`; +the rank-6 flats with ≥ 24 lines are exactly the 4,900 sets `a^⊥ ∩ b^⊥`), `G ≥ 12` (no live code word is covered by rank-≤ 6 sets of total rank ≤ 11, exhaustive) and an induction `D(Λ) ≥ r_k + D(Λ ∩ a_k^⊥)`. That enumeration needs 378 s with numpy and was **not ported to the standard library**; this package +does not verify it and no other statement here uses it. It extends #757 / #761 (63 optimal on the shipped P111 for 9, 10 and 11+ label terms) and agrees with #620 (63 for P111; exact SAT, cost ≤ 62 UNSAT, 8 totals UNSAT; also #1097's rank-9 uniqueness above). +**RECORDED (the author's working card `STRAYS-BIT`, §0; not re-run and not reviewed here): word-priced P111 set-decoders with at most 14 totals cost ≥ 63.** The card's headline, quoted with its own labels: "in the word's own pricing (the set-decoder model: retained totals are arbitrary live-line sets, each priced by `rank_Q` of its span), no bit decoder on P111 with ≤ 14 retained totals costs below 63. That covers every total count the hosting wall allows (`R ≤ 4v` ⇒ `n ≤ 14`)." Its statuses: +* `n ≤ 13`: **PROVEN** in the card (exhaustive enumerations plus exact rational Farkas certificates, re-checked there with `Fraction`s only; for `n = 13`, all 132 code classes of excess 4), on top of the exhaustive `G ≥ 12`, `G(3) ≥ 15`, `G(4) ≥ 14` (no ≤ 4-set low cover of total rank ≤ 13, no 3-set low cover of total rank ≤ 14). +* `n = 14`: **CHECKED** in the card. Of 51,973 code classes (excess 5), 51,238 have exact Farkas certificates; the 735 integer-only classes are kissat UNSAT with drat-trim **VERIFIED**; the last 3 of those need the extra exhaustive lemma `B16low`. +* The card's premise (P) is a theorem there and is **no longer used** (its facts are now exhaustive computations); `P110` (P111 minus line 38, the unique dual-root line of the shipped readers) satisfies Lemma A and `G ≥ 12`, and with `D(1) = 7` (the card's Lemma 1, `p = 10 ≤ 25`) every `P110` decoder with ≤ 12 totals costs ≥ 63, which is `P110`'s whole hosting window (`R = 439 ≤ 4·110`); rigidity at `h = 9` on **both P111 and P110** is drat-trim VERIFIED in the card (controls SAT). The package itself ships and re-checks the P111 rigidity only (credit for P111 rigidity: #620, #1097, as above; the `P110` rigidity is the card's). +The set-decoder enumeration needs numpy (378 s for the census) and was **not ported to the standard library**, so this package neither verifies nor depends on it. It extends #757 / #761 (63 optimal on the shipped P111 for 9, 10 and 11+ label terms) and agrees with #620 (63 for P111; exact SAT). **So the bit windows of 13 and 14 totals are CLOSED (RECORDED from the card)**; `n ≥ 15` totals are excluded by hosting in the current word (`R ≤ 4v`), a note rather than a result. Not covered: decoders outside the set-decoder model (a retained total charged below its source span's rank, i.e. a projected or asymmetric copy), other units, and the helper mass that extra totals would add (unpriced; no word was built). +**RECORDED pricing, not a theorem: the projected-copy route to `P110 = 62`** (working cards `SPECCHECK`, `BITCUT`). The SPEC does not forbid a projected copy (the bit SPEC has no COPY primitive: only per-vertex rational frames, edge cost `rank(M_head − M_tail)` and a common frame at each gate); what forbids it in the shipped word is the checker's COPY rule (COPY rank equals the dimension of the source frame, temp at frame 0), a format convention, so breaking it would need the four-part bar for convention-breaking changes. If built, the copy charge of the one orthogonal term drops 7 → 6 (cost 62, +2.80% c) minus the deleted port (−2.13%), before any reader-chain cost. Replaying PR589's tuned chains (`BITCUT`): all 111 targets sit at frame 0 at the scatter cut, and for the best candidate (term 5, one line) only a handful of the 55–63 readers already contain the line `⟨a⟩`, so the others pay extra climbs / descents: the exact repricing is **−3.57% c net including the +2.80% unit gain** (about −5.7% after the port), every other (term, line) pair is far worse, and the ideal bound (every reader already contains `⟨a⟩`) would be only +0.27% c. Caveats of the card: join frames are not in the 636-frame catalogue, the chains were not re-tuned, the identity of the orthogonal term was not pinned (so the best over all lines is quoted). So the SPEC-legal projected copy costs at least 3.57% c on PR589's tuned chains, and `P110 = 62` is not a supplier; nothing here is a κ claim. + +### 3.8 Recorded results of the working cards (not re-run) + +* **E8□ (IDEAL3 §4.6).** Every square E8 root-code decoder costs `≥ 72` for **every** `B` (4,022 non-orthogonal pair classes, each with fitting minrank `≥ 9`); the card's `census/e8drat.out` ends "DONE 4022 classes; not verified: 0". The proof logs (139 MB) are not shipped. Not needed at `h = 9`. +* **FL(4) (IDEAL3 §2).** The Frame Lemma for `n ≤ 4` (PROVEN modulo a 144-class support-level enumeration) gives `D4 ≥ 12` for every decoder, every realisation, every `m`, every `B`. Lemmas 4.1 (maximal-pair reduction) and 4.2 (`k`-term averaging) of IDEAL3 §4 are the shared-`B` reductions used by E8□ and the E7 statement below. +* **E7 (STRAYS S4).** Square maximal-normal root-code decoders cost `≥ 49` for every `B`: no decoder has ≥ 5 root–root partners (256 bad partners, 16 compatible 5-sets, 0 six-sets, 0 completions; Sonnet enumeration, independently re-audited). The 49 *on E7's own rank-7 table*, the case in the headline, needs only Theorem R at `h = 7` (CHECKED) and #2277 Cor. 5.3. +* **P104 delivery (NEWWORD C4.2).** For #968's P104 frame-pair decoder (cost 56, rank-7 terms) no global admissible label table in `F2^9`, with any per-term gather orientation, allows helper-free nested delivery (24 DRAT-verified instances; pairs `(0,1), (0,4), (1,4)`, same orientation, both directions; the graph of class-1 pairs is triangle-free and 3-regular on the 8 terms, so among any three mutually adjacent terms two share an orientation). Extends EDGE-B's Delivery Lemma from the dense table to every table. Escape: `h > 9` (a P104 at `h = 10` already loses to P112 at `h = 9` on the format ceiling: `3.931e-3 < 3.995e-3`). +* **PUNCT2, `p = 25..44` (NEWWORD c5).** Model C (dense pairing, column-space factors): at the 1,323 near-miss unions the bilinear forms vanishing on orthogonal live pairs have nullity 1 (`N ∝ Gram`), so every 8-term decoder is biorthogonal; for `m ≥ 9` merging and PUNCT2's deficiency census (`d ≤ 2`) give cost `≥ 5m ≥ 45 > 40.24` (`p = 44`) and `≥ 7m − 12 ≥ 51 > 44.47` (`p = 36`); rigidity at the two representatives is DRAT-verified and the 1,323 unions form two `W(E8)` orbits. So no decoder in the class beats P102's 54/102; the "+13–15%" near-miss points were relaxation artefacts. Depends on PUNCT2's census inputs, which are not shipped. +* **E7 frame decoders (NEWWORD c3).** No E7 root-line decoder with `m = 8` and every cross `⟨y_i, x_j⟩ = ±1`; best type mixes cost `≥ 39` on `≤ 50` ports (`x ≥ .78`), none beats P56's `x = .75` even with per-term-optimal `B` and free delivery (scope: crosses in `{0, ±1}`, root-line normals). +* **EDGE (coordinate block, minrank 8 every `B`):** CHECKED in §3.5. EDGE-B's Delivery Lemma (dense table) is the one C4.2 extends; not re-run. +* **STRAYS-H10, `h = 10` root code (Oct 15; RECORDED, card status CHECKED; char 0).** Root code (Gram realisation, factors `p_j = D₁R x_j`, `q_j = D₂R y_j`), any `m`, any fitting `B = UVᵀ` of `F2`-rank ≤ 10: **every nonzero term has `rank B[R(x), A(y)] ≥ 8`, so P112 ≥ 64.** Ingredients (card): the exact rank formula `rank B[R,A] = rank V[A] − dim(rowspan V[A] ∩ ker U[R])` and Sylvester's `≥ r − d(R) − e(A)` (PROVEN), reduction to maximal sections (47 orbits of the P112 orthogonality graph under an order-172,032 group; PROVEN), the second gauge `V[S] = (I|0)` (PROVEN), `G(10,2,·)` UNSAT on 46 of 47 orbits (**46/46 DRAT verified**), and exact term queries on the remaining orbit `f^⊥` against all 172 pair orbits and splits: **688/688 UNSAT, their DRAT proofs are queued and not yet verified**. An audit of the logic and encodings by a Sonnet helper found no error. Scope: Gram realisation only; **non-Gram rank-8 realisations with factors in `col M` at `h ≥ 10` are OPEN** (the realisation-free certificate leaves 17 uncertified classes on a rank-10 table, so Theorem P's method does not transfer). `h = 11` (a floor of ≥ 56 would close it; break-even cost is ≤ 55): `G(11,3,·)` is UNSAT on all 46 orbits other than `f^⊥`, so every term is ≥ 7 unless one normal's maximal section is `f^⊥`; the remaining blocker is the orthogonal dead-root term `N(f) × N(g)`, `f ⊥ g` (one split timed out at 25 min; other pair orbits UNSAT so far; runs continue). `h ≥ 12` not attempted. P104 at `h = 10` was already dead by the format ceiling. Not in `verify.sh`. +* **PUNCT4 / PUNCT5, `p = 45` (Oct 15; RECORDED).** Setting: Model C (`K = L ∖ S`, `|S| = p`, dense `B`, column-space factors, cost `Σ r_j`, exact law `r = 9 − s`, `s = d_x + dim(Z_y ∩ Z_x^⊥)`), bar = largest winning cost `c_p = ⌈(54/102)(120 − p)⌉ − 1` (`c_45 = 39`). **Theorem 45 (card: PROVEN from CHECKED data): `p = 45` is closed for decoders with `m ≥ 8` terms (after merging): cost ≥ 40 > 39.71.** Shape: a win needs a root normal with deficiency `d ≥ 3` (every other term saves ≤ 4, cost ≥ 5m ≥ 40); case `d = 4` (the 1,260 minimal kernels form one stabiliser class; `relaxF = 50`, nullity of the `N`-space 1, so `rank N = 8`); case `d = 3` (11 anchor classes, at most 3 other sides with `d ≥ 2` in every class, counting `cost ≥ 7m − 16 ≥ 40`). The exact-data inputs were re-certified by exact integer search in PUNCT5 (side counts 1,3,0,2,0,3,2,0,1,0,1 identical; `relaxF = 50` by an exact min-cost flow on five kernels), replacing a floating-point HiGHS MILP. **Lemma G (PROVEN):** if `s` is a live line whose orthogonal live lines span `s^⊥`, every `N` in the `N`-space has `N r_s = λ_s r_s`, `λ_s = M_ss/|r_s|² ≠ 0`; if the good lines span `Q^8`, then `rank N = 8` and every decoder has `m ≥ 8` terms. Honest limits (as stated in the cards): the nullity-1 / `rank N = 8` fact is CHECKED for `d = 4` but **not** for the `d = 3`-anchored `S = A + E`, so **`m ≤ 7` is OPEN wherever the `N`-space nullity exceeds 1**; for decoders with `m ≥ 8` terms (after merging) at `46 ≤ p ≤ 56`, a win needs a `d ≥ 3` root (PROVEN reduction) but the residue (R2) `m = 8` with a `Q`-root at `p ≥ 46` and (R3) `m ≥ 9` at `p = 46` and wherever the count-relaxed census exceeds its allowance (`p ≥ 50` not computed) is **OPEN**; where `m ≥ 9` is treated as closed at `p = 47–49` this rests on the floating-point HiGHS Q-census, which PUNCT5 §1 says was not re-certified, and is not covered by Theorem 45's exact data; PUNCT5's `p = 46` anchor runs (13 `d ≥ 3` anchor classes, bracket within the allowance on those logged) are partial (the `d ≥ 4` extensions and the `E7`-type rank-deficiency CNFs timed out or were unfinished). Any-`B` is untouched at `p ≥ 45`. Not in `verify.sh`. + +## 4. How to cite this + +"For every decoder hosted on one `F2^9` label table (Theorem R: the pairing is the dense one), with factors in the column space of a rank-8 realisation and any number of terms (Theorem P; #2228 Thm 4 for the root code), the copy costs of the E8-family units are exactly +P112 64, E8 72, P104 56 (#968's unit, `F1 = H4`; E7 49 on rank-7 tables), and `k` independent P112 calls cost `≥ 64k` (`= 128` for `k = 2`; symmetric / physical table for `k ≥ 3`, `GL` charts with `k ≥ 3` are open, §5.5) (Theorem B, Bk; #2294). Remaining escapes: tables of rank `≥ 10`, rank-9 square decoders and overcomplete factors outside `col M` with `≥ 9` terms, non-Gram realisations at `h ≥ 10` and `h ≥ 11` (the `h = 10` Gram root code is RECORDED closed; 688 term-query DRATs pending, 1 verified, rows 16a/16), puncture windows `p ≥ 46` (and `p = 45` with `m ≤ 7`; `p = 45`, `m ≥ 8` is RECORDED closed), batches of `≥ 17` terms outside `col M`, other P104 units. The bit window of 13–14 totals is RECORDED closed (word-priced P111 set-decoders cost `≥ 63`)." + +## 5. OPEN (each leaves class X; priced where a price exists; none constructed) + + +### 5.1 Label tables of rank `h ≥ 10` (the `h = 10` Gram root code is RECORDED closed; the rest is OPEN) + +Non-dense fitting tables exist at rank 10 (`c1.neg`: `P112_sym_h10_alt`, `P112_gen_h10_alt` are SAT). *Price* (working card `STRAYS/PRICE-P112-H10.md`, a Sonnet helper's pricing, not a theorem; it reproduces the P104 rows of NEWWORD `c4/PRICE-H10.md`): a 112-port unit at +`h = 10` must reach **cost ≤ 59 (−5 units)** to tie P112-at-`h = 9`, before any helper growth; at `h = 11`, `≤ 55`. **Status: undecided.** Per-term floors under a global rank-10 cap were probed on the first orthogonal-normal class (free-`B` minrank 6) and timed out at 20 minutes for cap 7 at both `h = 10` and `h = 12`. Per-term `≥ 8` at rank `≤ 10` would close `h = 10` (since `m ≥ 8`). +For E8 overcomplete under any `B`: non-orthogonal terms `≥ 9`, other orthogonal terms `≥ 8`, orthogonal root–root terms are 6; counting alone allows `m ∈ {9, 10, 11}` below 72; the `m = 9` root-normal family is jointly `≥ 65` (EDGE). **Update (working card `STRAYS-H10`, §3.8; RECORDED, not re-run):** for the root code (Gram realisation) any `m` and any fitting `B` of rank ≤ 10, every term costs ≥ 8, so P112 ≥ 64 (CHECKED in the card: 46/46 `G`-query DRATs verified; 688 term-query instances UNSAT, **their DRATs pending**). **Still OPEN:** (i) non-Gram rank-8 realisations with factors in `col M` at `h ≥ 10`; (ii) `h = 11`: per-term ≥ 7 except possibly at pair orbit 0 (`N(f) × N(g)`, `f ⊥ g`; one split timed out) and at pair orbits not yet run (1, 2, 65–171); runs continue, not decided; (iii) `h ≥ 12`, not attempted (the price would have to be ≤ ~51); (iv) joint-table and non-root-code statements at `h ≥ 10`. + + +### 5.2 Factors outside `col M`, `m ≥ 9` + +Proven (§3.3): `m ≤ 8` ⇒ `≥ 64` / 56 for any factors and any realisation; `m = 9`: `ρ = rank(paid part) ∈ {8, 9}`, `s + t ≤ 9 − ρ`. **Open:** (b) `ρ = 8` with one factor outside `col M` (#1180's residual class; necessary conditions proven, 42k of 45k class pairs uncertified at support level); +(c) `ρ = 9`, square in a rank-9 realisation (needs a rank-9 analogue of Lemma 1.1, `Σδ ≥ 18` to reach ≤ 63; contains every philippeVV co-star `k = 7/8/9` family, §3.3); (d) `m ≥ 10` (budget `s + t ≤ m − ρ`, #1633; the average term rank would have to drop below 6.4). All under dense `B`, which at `h = 9` is every `B`. (working card `STRAYS-MASK`.) **Update (working card `STRAYS-MASK2`; no case closed, nothing beats 64):** the **frame-module theorem** (PROVEN, §3.3) is the realisation-free rank-9 support calculus for every nine-term decoder; a SAT model built on it is sound on three positive controls (#1180; the coordinate decoder plus `c e_t e_tᵀ` for two lines `t`) but the decisive instances (case (b) with `π` global, the co-star family, full `m = 9`) **timed out at 25–28 min under machine load**; nothing is proved; longer runs are queued, not finished. So **(b), (c) and `m ≥ 10` remain OPEN.** Swarm results on one explicit rank-9 family (not the general case (c)): **#2321** (nine-term sharp COPY cost of the fixed P112 two-column completion, ports `(0,1)`), **#2385** (the same floor for every nonzero coefficient, nine terms) and **#2398** (the same optimum 64 for any finite number of terms with both factors in that completion's minimal nine-dimensional column and row spaces; the factor-space restriction is essential there). They settle that one rank-9 realisation; the question of an arbitrary rank-9 square decoder is §5.2 (c). + + +### 5.3 Bit windows of 13–14 totals (CLOSED, RECORDED from the working card); `n ≥ 15`: a note + +**CLOSED, RECORDED (§3.7):** word-priced P111 set-decoders with at most 14 retained totals cost ≥ 63 (card: `n ≤ 13` PROVEN by exact Farkas certificates, `n = 14` CHECKED with 735 integer-only classes DRAT-verified; P110 ≤ 12 totals ≥ 63, its whole hosting window). Not re-run here. **Note (not a result):** `n ≥ 15` totals (excess ≥ 6) are excluded by hosting in the current word (`R ≤ 4v`, hosting-forbidden); decoders outside the set-decoder model (projected copy) are priced, not excluded, in §3.7 (≥ 3.57% c net on PR589's tuned chains); extra totals add unpriced helper mass. + + +### 5.4 Puncture windows `p ≥ 46` (`p = 45`, `m ≥ 8` is RECORDED closed) + +PUNCT2's exclusion covers `p = 25..44` (RECORDED, §3.8). **Update (working cards `PUNCT4`, `PUNCT5`; RECORDED, §3.8):** Theorem 45 closes `p = 45` for decoders with `m ≥ 8` terms (cost ≥ 40 > 39.71; dense `B`, column-space factors; exact data), and Lemma G (PROVEN) gives `rank N = 8` (so `m ≥ 8`) whenever the good lines span `Q^8`. **Remaining residue, as stated in the cards, OPEN:** (R1) `m ≤ 7` where the `N`-space nullity exceeds 1 (including `p = 45` with a `d = 3`-anchored `S = A + E`); (R2) `m = 8` with a `d ≥ 3` root at `46 ≤ p ≤ 56`; (R3) `m ≥ 9` at `p = 46` and at the `p` where the count-relaxed census exceeds its allowance (`p ≥ 50` not computed; the `p = 47–49` rows rest on an uncertified floating-point HiGHS Q-census, PUNCT5 §1, not on Theorem 45's exact data); `57 ≤ p ≤ 112` and any-`B` are untouched. The `T²` law (#2125, argued) predicts no gain for `v ≲ 64`. + + +### 5.5 Batches outside class X + +`m ≥ 17` terms with factors outside `col M` (every 16-term batch needs no factor hypothesis, #1714 §5); `GL(9)` charts with `k ≥ 3` (mathematical only: the physical table is symmetric; the obstruction is a possible loop in the component quotient of the output / input hyperplane graph, so +Lemma K's ordering argument may fail; the bound needs only `corank(I_t − K) ≤ 7t/8`); `h ≥ 10` joint tables (pure cross terms may cost `< 7`; #2294 Props B–D cover cross blocks that factor through the call labels as bilinear forms; **#2395** states a stronger two-call P112 COPY floor for arbitrary cross forms, addressing #2294's open arbitrary-cross-form question (its rank-only test certifies 93 of 100 ordered rank pairs; 7 low-rank pairs undecided), and **#2342** (theon199) states a three-call P112 COPY of 192 by a sharp block count, to be compared with our Theorem Bk (`k = 3`, symmetric model, `≥ 192`, any rank-8 realisations, any `m`): both are cited from their PR statements, **not re-derived, compared in scope or audited here**, and neither is used by anything in this package); correlated demands (`C ≠ I`, #1714 §4); mixed-unit batches; a second call on a different unit. + + +### 5.6 Other P104 units; rank-9 square decoders + +Theorem R and the Lemma 1.1 census (Theorem P) are run for P104 = #968's unit (`F1 = H4`) only, with Theorem R also at one representative of each of the classes `759 / 767 / 801`. **Not claimed:** Theorem R for all 1,260 frames `F1`, the census and Theorem P for any other P104 unit (the swarm has many, e.g. #1399's 896 admissible ones). The rank-9 square case is §5.2 (c). + +**Out of scope by construction:** non-E8 units (BIGTHEORY: no-go only for `h ≤ 13`; #2222-type large-`h` units are compensation-bound), and all word-side overheads (helpers, transport, packing). 1337isnot1337's rank-16 child-merging screen (Discussion 1166, `< 0.21%` even at zero overhead) is a word-side route, not a decoder saving. + +## 6. What was not reproduced + +See §3.5 (steps `s = 3, 4, 5` of the puncture table; `s = 1` not computed), §3.7 (the ≤ 14-total bit bounds and the projected-copy pricing) and §3.8 (including the `h = 10` root code, the frame-module SAT runs and Theorem 45): every RECORDED item is a result of the working cards that this package neither re-runs nor depends on for rows 1–10 of §2, with the single exception noted in §3.5 (the pricing +consequence of the puncture table). Differences between the working cards and this package, none of them a failed reproduction: the SAT instances are regenerated with a deterministic gauge clique (the card's files differ, the verdicts agree); `f(6)` is `≥ 46` here (card: `≥ 42` from a looser 15-set instance); the `h = 10` witnesses differ from the card's +(non-dense in 59 / 89 orthogonal entries rather than 171 / 381); the card's "rank(B − B_dense) = 8" for the `h = 10` witness is not claimed. Not claimed: Theorem R for all 1260 frames of P104 (class representatives only), and Theorem P or any P104 statement for units other than #968's (`F1 = H4`); the rank-9 square case (§5.2 (c)); novelty of the P111 row of Theorem R (#620 / #1097). + +## 7. Credits and related PRs + +Our PRs: #1590 (model, merged), #2111, #2125, #2162 (Lemma CR2 / Theorem B2; navinor's independent audit #2293 passes our CR2/B2 on 58 instances; the later edge-form counterexamples #2334 / #2340 and D2331 concern an edge-count reading of B2(iii), not the vertex form (MAS = maximum (vertex-count) acyclic subset of lines) that this package cites, and are credited, not used), #2228 (root-code floors Thm 4, dense polar law), #2244, #2277 (Theorem 4′, frame-intersection certificate, Cor. 5.3), #2265 (audited by thakrarsagar #2311: 0 FALSE / 0 MISSING-PROOF on its supplier premises). Swarm: #968 (frame-pair P104 decoder), #1180 (outside-code nine-term example), #1212 / #1213 / #1236 (rank-8 rigidity, components, root-star supports), #1744 (global fitting rank ≥ 9 on P112), #757 / #761 (P111 63), #620 (GamingPuzzled; P111 rank-9 uniqueness and 63, found by SAT) and #1097 (bitplane, merged; LRAT proof of that uniqueness in the general model, on the archived Hamming order, not on #589's set), #1815 (jmbarrancoml, merged; one-sided P112 fitting rigidity), #1331 (Arham-Tahir64, merged; the coordinate COPY 64 is not improved by one free rectangle; related to E1a and the punctures), #1249 (bbMisk, merged; independent verification of #1236's 64), #717 (ours; term cost = dimension of the label span, frontier-word censuses and decoder floors; the hosting model rests on it), #2176 / #2206 (forced minors, exact pairing costs), +#2232 / #2240 / #2247 / #2137 (E6 screens, 1337isnot1337), #2272 (dense-polar small-`T` screen), #2267 (rectangular Lean form of the shadow lemma), #2294 (k = 2 batching, thakrarsagar), #2395 (two-call COPY floor for arbitrary cross forms), #2342 (theon199; three-call P112 COPY 192), #2321 / #2385 / #2398 (the fixed rank-9 P112 completion; §5.2), #1714 (batch model), #1224 (joaqcoo; the merge lemma and one-sided overcomplete structure), #1633 (bbMisk; excess-rank budgets), Discussion 1267 (philippeVV; co-star families). +Tools: kissat 4.0.4, drat-trim (Marijn Heule, Nathan Wetzler). See `NOTICE.md`. + +Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0. diff --git a/research/decoder-floors-frontier-format/SOURCES.json b/research/decoder-floors-frontier-format/SOURCES.json new file mode 100644 index 0000000000..6d9441a011 --- /dev/null +++ b/research/decoder-floors-frontier-format/SOURCES.json @@ -0,0 +1,146 @@ +{ + "description": "Pins for research/decoder-floors-frontier-format. files = library, generator and check code (own work); evidence = data exported from the author's working cards (data/h4_F1.json, data/p104reps.json: frames of P104 from cards/STRAYS; data/p104_frames.json from #2277's package; data/p111_lines.json: the 111 live lines of #589's P111 word; data/chart_rank7.json: an O(9) chart found by sat-free search with random.Random(52) and re-verified by checks/c3), the recorded SAT / DRAT results of sat/results.jsonl and sat/results_misc.jsonl (produced by sat/mkresults.py and sat/mkmisc.py with kissat 4.0.4 and drat-trim), and the shipped binary DRAT proofs (xz). Tools used (not vendored): kissat 4.0.4; drat-trim. scripts/provenance.py checks the pins and MANIFEST.sha256.", + "external": [ + { + "what": "source of data/p111_lines.json", + "path": "research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz (CrocSwap/integer-mult-bounds, origin/main at 0fdd7d82)", + "sha256": "e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f" + } + ], + "files": [ + { + "file": "checks/c1_rigidity.py", + "sha256": "4041cfeb1bc5131b5b57cce79de13e013061566128db38707cdc50efc3333e6a" + }, + { + "file": "checks/c2_support_propagation.py", + "sha256": "a24a7f0907970257cdcfeaf3d1d234c34974b64a3d7aa584283124d79e0def7a" + }, + { + "file": "checks/c3_batch_premises.py", + "sha256": "aa3c56bd79a36dd081b40e50c675064c4dc7e780aee02779813c618651cf0ad8" + }, + { + "file": "checks/c4_masked_costar.py", + "sha256": "f06d5cc4685e81f5c1b52391a6a4f7548d9da146f0891aabdab09843c260ae11" + }, + { + "file": "checks/c5_attaining_decoders.py", + "sha256": "359b20c67b8d4c0491a0a3b9cf5b595f5294570e6a7a98fbe900094396620790" + }, + { + "file": "checks/c6_block_minrank_punctures.py", + "sha256": "42e91b63e455ddf020b678700ed45dc3ad3703374ba9d063002fffe13770b981" + }, + { + "file": "checks/c7_p111_unit.py", + "sha256": "6d3e3bac50ef7054acaeaabecbc863e8f40c359655c61fc9483257184c25d6ae" + }, + { + "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": "lib/solve.py", + "sha256": "6872520ef5955082ab10e6f20960bb47cabfad03bdad246ebe79231007b36159" + }, + { + "file": "lib/supprop.py", + "sha256": "a0883df40db120adc892aadec78ab15e6eefa7d8e7006883294ba1da62961081" + }, + { + "file": "lib/units.py", + "sha256": "43eed628ea4c39f5aa47945697ccfe4dc63bbfc21d91affdd8a7d698c8ccd569" + }, + { + "file": "sat/hitting.py", + "sha256": "8a4e7baaca75530be75d2483df1af4cbba48013c992129da5628bb2510ae0b82" + }, + { + "file": "sat/misc.py", + "sha256": "5b72bcdf42c37cedb9cd4f60c61ac5fe884dbf2d78bf8e36939afe1c27f69358" + }, + { + "file": "sat/mkmisc.py", + "sha256": "c748368f64c87d2842babb47f6ede23f8f16acf98bef8a48f5dd40259f3e2be1" + }, + { + "file": "sat/mkresults.py", + "sha256": "3334ae7bd440971a6c751e8286a755cf632f0c3a325c1a27086bf6df5c0990df" + }, + { + "file": "sat/rigid.py", + "sha256": "378f20d152d1c4c53c9bd528bd2e661fa75b68bbeaa1fecaa498af4acda9b0f9" + }, + { + "file": "scripts/provenance.py", + "sha256": "a4cd33cef93721cd79ae3dfaf69556f8502b22152e363fcab33b1fa402d68d11" + } + ], + "evidence": [ + { + "file": "data/chart_rank7.json", + "sha256": "66d019f311fd436c7e04517349d8d3e47d75d970951a7c99e1ef523cffd405b6" + }, + { + "file": "data/h4_F1.json", + "sha256": "33e368cad0c95cd4af1c1d06d3a7abbd28abd14900bcada084d7b7583d89a270" + }, + { + "file": "data/p104_frames.json", + "sha256": "52e1a2b0e963311eb74c1ec2bbb110497da57b606e132f784f9000603d10a306" + }, + { + "file": "data/p104reps.json", + "sha256": "f85ac3b3c785964c04d53bcf879a74425d8d997ec3ccb92d971cce69c9c10a32" + }, + { + "file": "data/p111_lines.json", + "sha256": "2b492989fd574f557533e237267bc6693d6917ce5b32e2ccdad7fa9de81a07f9" + }, + { + "file": "sat/results.jsonl", + "sha256": "d7115712b22dca41e37813ac1236a91c198210b089651f0ec5cced78f6bdf296" + }, + { + "file": "sat/results_misc.jsonl", + "sha256": "30889985d57b5bf1139ea29d9f52f9f5c24365c036b1acc8e263878fef8f94b7" + }, + { + "file": "sat/proofs/E8_sym_h9_alt.drat.xz", + "sha256": "6d7b82565559777597a10dd65d34088ef03b65cd6789c55be59765b3ae30f89a" + }, + { + "file": "sat/proofs/P104H4_sym_h9_alt.drat.xz", + "sha256": "fc385861ff98a07cf9f6267d8cc169deb4262394b23a5780cf08c27567b523e6" + }, + { + "file": "sat/proofs/P111_sym_h9_alt.drat.xz", + "sha256": "6de05f478a04dfc97f5b66644262e2f0ac742f1b399bab775134641a9b30b04d" + }, + { + "file": "sat/proofs/P112_sym_h9_alt.drat.xz", + "sha256": "db1e4a2526ed9593f7b8444bde8da3062c37ac00a3a49faa4ae92b035d95cc7d" + }, + { + "file": "sat/proofs/edge_N0_r7.drat.xz", + "sha256": "9a22571653eb87be39b4650d03e7587072f5fd6332cc4162bde629e17d872e51" + }, + { + "file": "sat/proofs/tri_keep11.drat.xz", + "sha256": "5f31be0fd422372d77319c45e51488bb8eb4474f4e31402715d40572b1ec2907" + } + ] +} \ No newline at end of file diff --git a/research/decoder-floors-frontier-format/checks/c1_rigidity.py b/research/decoder-floors-frontier-format/checks/c1_rigidity.py new file mode 100644 index 0000000000..7fa486b1bb --- /dev/null +++ b/research/decoder-floors-frontier-format/checks/c1_rigidity.py @@ -0,0 +1,72 @@ +"""Theorem R (rigidity of the minimal-rank label table), replayed. Standard library only; kissat 4.0.4 and drat-trim are optional (SKIP lines without them). +For each registry instance (sat/rigid.py): the regenerated DIMACS must have the recorded sha256 (sat/results.jsonl); recorded statuses must equal the expected ones +(alt instances at the minimal rank: UNSAT, DRAT-VERIFIED when recorded; controls and negative controls: SAT with a decoded model); shipped proofs +(sat/proofs/*.drat.xz) are replayed with drat-trim; with kissat available every instance is re-solved, UNSAT proofs are checked, SAT models are decoded.""" +import sys, os, json, time, lzma, tempfile, itertools +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 +from rigid import * +import solve as S_ + +rows = [json.loads(l) for l in open(os.path.join(PKG, 'sat', 'results.jsonl'))] +R = registry(); t0 = time.time() +check('c1.coverage', set(r['name'] for r in rows) == set(R) and len(rows) == len(R), 'every one of the %d registry instances has a recorded result' % len(R)) +bad = [r['name'] for r in rows if r['status'] != r['expect'] or (r['status'] == 'UNSAT' and r.get('drat') != 'VERIFIED')] +check('c1.status', not bad, 'recorded statuses: %d UNSAT (all DRAT VERIFIED when recorded), %d SAT, none unexpected %s' % ( + sum(r['status'] == 'UNSAT' for r in rows), sum(r['status'] == 'SAT' for r in rows), bad[:3])) +badh = []; cnfs = {} +for r in rows: + nv, cls, info, *_ = build_named(r['name']); d = dimacs(nv, cls) + if sha(d) != r['cnf_sha256'] or nv != r['nv'] or len(cls) != r['ncls']: badh.append(r['name']) + cnfs[r['name']] = d +check('c1.hash', not badh, 'regenerated DIMACS of all %d instances has the recorded sha256, #vars, #clauses %s (%.0f s)' % (len(rows), badh[:3], time.time() - t0)) +# gauge cliques are pairwise non-orthogonal +def of_for(r): + return p111()[1] if r['unit'] == 'P111' else orth +badg = [n for n, r in R.items() if any(of_for(r)(a, b) for a, b in itertools.combinations(r['S'], 2))] +check('c1.gauge', not badg, 'every gauge set S is a pairwise non-orthogonal clique (|S| = 8 for E8/P112/P104/P111, 7 for E7, 5 for E6)') +# the alt instances at the minimal rank: UNSAT <=> every fitting B of rank <= h is dense. Controls: SAT and the decoded model is dense for h = rank(dense B) +ctl = [r for r in rows if r['mode'] == 'ctl'] +check('c1.ctl', all(r['status'] == 'SAT' and r['model_fitting'] and r['orth_zero'] == 0 for r in ctl), + 'controls (no alt clause): all %d are SAT, every decoded model is a fitting matrix with 0 of its orthogonal entries equal to 0, i.e. equal to the dense B (h = rank of the dense B)' % len(ctl)) +neg = [r for r in rows if r['name'].startswith('SUB') or r['name'].startswith('P112_') and r['h'] == 10] +check('c1.neg', all(r['status'] == 'SAT' and r['model_fitting'] and r['orth_zero'] > 0 for r in neg), + 'negative controls: %s are SAT with decoded fitting matrices that are NOT dense (orthogonal entries equal to 0: %s)' % ( + [r['name'] for r in neg], {r['name']: '%d/%d' % (r['orth_zero'], r['orth_total']) for r in neg})) +# shipped proofs +dt = S_.drat_trim(); ks = S_.kissat() +pdir = os.path.join(PKG, 'sat', 'proofs'); proofs = sorted(f for f in os.listdir(pdir) if f[:-len('.drat.xz')] in R) if os.path.isdir(pdir) else [] +if dt and proofs: + okp = 0; badp = [] + with tempfile.TemporaryDirectory() as td: + for f in proofs: + n = f[:-len('.drat.xz')]; cnf = os.path.join(td, 'x.cnf'); pr = os.path.join(td, 'x.drat') + open(cnf, 'w').write(cnfs[n]); open(pr, 'wb').write(lzma.open(os.path.join(pdir, f)).read()) + if S_.run_drat(cnf, pr): okp += 1 + else: badp.append(n) + check('c1.shipped', okp == len(proofs), 'drat-trim verifies all %d shipped DRAT proofs (sat/proofs, binary format) against the regenerated CNFs %s' % (okp, badp)) +else: + print('SKIP c1.shipped drat-trim not found (set DRAT_TRIM=/path/to/drat-trim): %d shipped proofs not re-checked' % len(proofs), flush=True) +# re-solve with kissat +if ks: + only = os.environ.get('RIGID_ONLY'); okr = 0; fails = []; nd = 0; t1 = time.time() + with tempfile.TemporaryDirectory() as td: + for r in rows: + if only and only not in r['name']: continue + cnf = os.path.join(td, 'x.cnf'); pr = os.path.join(td, 'x.drat'); open(cnf, 'w').write(cnfs[r['name']]) + st, model = S_.run_kissat(cnf, pr if r['expect'] == 'UNSAT' else None) + good = st == r['expect'] + if good and st == 'UNSAT' and dt: + good = S_.run_drat(cnf, pr); nd += 1 + if good and st == 'SAT': + fit, z, n = decode_model(r['name'], model) + good = fit and ((z == 0) if r['mode'] == 'ctl' else (z > 0)) + okr += good + if not good: fails.append(r['name']) + check('c1.resolve', not fails, 're-solved %d instances with kissat 4.0.4 (%s): all as expected (UNSAT, %d proofs checked by drat-trim; SAT: models decode to fitting matrices, dense for controls, non-dense for negative controls) %s (%.0f s)' % ( + okr, 'DRAT checked' if dt else 'drat-trim absent: UNSAT proofs NOT checked', nd, fails[:3], time.time() - t1)) +else: + print('SKIP c1.resolve kissat not found (set KISSAT=/path/to/kissat): the %d recorded solver verdicts are not re-derived' % len(rows), flush=True) +print(' (c1 %.0f s)' % (time.time() - t0)) +finish() diff --git a/research/decoder-floors-frontier-format/checks/c2_support_propagation.py b/research/decoder-floors-frontier-format/checks/c2_support_propagation.py new file mode 100644 index 0000000000..6a2f9d80c2 --- /dev/null +++ b/research/decoder-floors-frontier-format/checks/c2_support_propagation.py @@ -0,0 +1,56 @@ +"""Lemma 1.1 (support propagation) census and the survivor pair ranks behind Theorem P (IDEAL3 THEORY §1.1-1.3). Exact bitmask arithmetic, standard library only. +A label class O is the support of a nonzero vector of col B (B = dense pairing, rank h). O is *deficient* in a rank-T realisation M iff some functional +h* != 0 kills p0(O). Lemma 1.1: for every T-frame E write h* = sum_e c_e q0(e); then c_e = 0 on E cap O, and for every T-frame F, +supp(c) subset (E \\ O) cap union_{g in F \\ O} Nbar(g), Nbar(x) = {lines non-orthogonal to x} (x itself included). Iterate to the fixed point; if some S_E becomes +empty then O is non-deficient in EVERY rank-T realisation. Survivors = classes that are not certified.""" +import sys, os, time, itertools +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, os.path.join(HERE, '..', 'lib')) +from report import check, finish +from units import * + +from supprop import setup, propagate, census + +t0 = time.time() +L112 = sub(P112_IDX); L104 = sub(unit_idx('P104H4')) +from roots import D4, D6, E7, E8 +exp = { # name: (lines, T, #frames, #classes, #survivors, survivor sizes) + 'D4': (D4, 4, 3, 7, 0, []), 'D6': (D6, 6, 15, 31, 0, []), 'E7': (E7, 7, 135, 127, 0, []), + 'E8': (E8, 8, 2025, 511, 0, []), 'P112': (L112, 8, 1260, 511, 8, [56]), 'P104': (L104, 8, 759, 511, 16, [52])} +R = {} +for n, (L, T, nf, ncl, nsurv, sizes) in exp.items(): + r = census(L, T); R[n] = r + got = (r['nframes'], r['nclasses'], len(r['unc']), sorted(set(popcount(w) for _, w in r['unc']))) + check('c2.cert.' + n, got == (nf, ncl, nsurv, sizes), '%s (T=%d, h=%d): %d T-frames, %d label classes, %d not certified by Lemma 1.1, sizes %s' % (n, T, r['h'], got[0], got[1], got[2], got[3])) +print(' (census %.0f s)' % (time.time() - t0)) +# survivors: no 2-dim subspace; each survivor is the set of live lines orthogonal to one removed root f in F0 (P112) / in F0 u F1 (P104) and has Q-rank 7 +for n, L, removed in (('P112', L112, F0), ('P104', L104, F0 + H4F1)): + unc = R[n]['unc']; cos = set(c for c, _ in unc) + sub2 = any((a ^ b) in cos for a, b in itertools.combinations(sorted(cos), 2)) + check('c2.nosub2.' + n, not sub2, '%s: the %d survivor classes contain no 2-dimensional subspace of the label space (a ^ b is never a survivor)' % (n, len(unc))) + ws = set(w for _, w in unc); want = set() + for f in removed: + want.add(sum(1 << i for i, r in enumerate(L) if dot(r, E8[f]) == 0)) + check('c2.perp.' + n, ws == want, '%s: the survivors are exactly the sets of live lines orthogonal to one removed root (%d of them)' % (n, len(want))) + rk = set(rank_q([list(L[i]) for i in range(len(L)) if w >> i & 1]) for w in ws) + check('c2.qrank.' + n, rk == {7}, '%s: each survivor spans a 7-dimensional space over Q (Gram-deficient), rank_Q = %s' % (n, sorted(rk))) +# pair ranks rank_F2 B[V \ O, V \ O'] over all survivor pairs (O = O' included) +for n, L in (('P112', L112), ('P104', L104)): + unc = R[n]['unc']; Br = R[n]['Br']; v = len(L); full = (1 << v) - 1; out = {} + for (_, O), (_, O2) in itertools.product(unc, unc): + k = blk(Br, bits(full & ~O), bits(full & ~O2)); out[k] = out.get(k, 0) + 1 + want = {8: 64} if n == 'P112' else {8: 192, 7: 64} + check('c2.pairs.' + n, out == want, '%s: rank_F2 B[V\\O, V\\O\'] over all %d ordered survivor pairs (O = O\' included) takes the values %s (value: count); the minimum is %d, i.e. every term costs at least %d' % (n, len(unc) ** 2, out, min(out), min(out))) + +# exactness of the per-term rank: Def(V \\ O) = {0, w_O} for every survivor O (so the pair ranks above are the ranks of the pairs (Def(R), Def(A)) = (, )), +# and the single-deficiency ranks rank B[V \\ O, V] (the (1, 0) and (0, 1) cases) +for n, L in (('P112', L112), ('P104', L104)): + unc = R[n]['unc']; Br = R[n]['Br']; v = len(L); full = (1 << v) - 1; classes = span_vectors(colspace_basis(Br)) + ok = all(sorted(w for _, w in classes if w & ~O == 0) == [O] for _, O in unc) + check('c2.def.' + n, ok, '%s: for every survivor O the classes with support inside O are exactly {O} itself, so Def(V \\ O) is the one-dimensional space ' % n) + single = set(blk(Br, bits(full & ~O), list(range(v))) for _, O in unc) + check('c2.single.' + n, single == {8}, '%s: rank_F2 B[V \\ O, V] = %s for every survivor O (a single deficiency costs one unit of rank: 9 - 1)' % (n, sorted(single))) +# gauge: dense B has rank h and the survivors are the only non-certified classes => Theorem P's per-term floor 9 - delta(R) - delta(A) >= 8 with delta <= 1 +for n in ('P112', 'P104', 'E8', 'E7'): + h = R[n]['h']; check('c2.rankB.' + n, h == (7 if n == 'E7' else 9), '%s: rank_F2 of the dense pairing is %d' % (n, h)) +finish() diff --git a/research/decoder-floors-frontier-format/checks/c3_batch_premises.py b/research/decoder-floors-frontier-format/checks/c3_batch_premises.py new file mode 100644 index 0000000000..dfd0395207 --- /dev/null +++ b/research/decoder-floors-frontier-format/checks/c3_batch_premises.py @@ -0,0 +1,91 @@ +"""Finite premises of Theorem B / Bk (STRAYS-BATCH THEORY §1-§3): labels of P112 in F2^9, the 8 exceptional hyperplanes, the normals w_i with b(w_i,w_j) = 1, +rho(N(f_i), N(f_j)) = 8, the frame lemma E cap N(f_i) cap N(f_j) = empty, and an O(9) chart on which all 64 cross pairs of coordinate supports pair at rank 7 +(the per-term floor 7 of Theorem B is attained, so the counting bound 120 is not the truth). Standard library only; the chart is searched +with a fixed seed (random.Random(52)) and the found chart is shipped in data/chart_rank7.json, which is re-verified here.""" +import sys, os, json, random, itertools +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, os.path.join(HERE, '..', 'lib')) +from report import check, finish +from units import * + +L = sub(P112_IDX); v = len(L); Br = b0rows(L); G = gram(L) +# dense B has rank 9; basis = first 9 independent rows +basis = [] +for t in range(v): + if rank_f2([Br[x] for x in basis + [t]]) == len(basis) + 1: basis.append(t) + if len(basis) == 9: break +check('c3.rank', rank_f2(Br) == 9 and len(basis) == 9, 'P112 dense pairing has rank 9; basis lines %s' % basis) +combo = {} +for co in range(512): + w = 0 + for i in range(9): + if co >> i & 1: w ^= Br[basis[i]] + combo.setdefault(w, co) +lab = [combo[Br[t]] for t in range(v)] # label of line t: coefficient vector (9-bit int) over the basis rows +Gb = [[Br[basis[i]] >> basis[j] & 1 for j in range(9)] for i in range(9)] +def bf(x, y): + s = 0 + for i in range(9): + if x >> i & 1: + for j in range(9): + if y >> j & 1: s ^= Gb[i][j] + return s +check('c3.labels', all(bf(lab[t], lab[s]) == (Br[t] >> s & 1) for t in range(v) for s in range(v)), 'labels realise the dense pairing: b(lab_t, lab_s) = B_ts for all 112 x 112 pairs') +def hfun(h, x): return bin(h & x).count('1') & 1 +live_of = [E8[i] for i in P112_IDX] +Of = []; N = [] +for f in F0: + Of.append(set(t for t in range(v) if dot(live_of[t], E8[f]) == 0)); N.append([t for t in range(v) if dot(live_of[t], E8[f]) != 0]) +hyps = [] +for Oset in Of: + sols = [h for h in range(1, 512) if set(t for t in range(v) if hfun(h, lab[t])) == Oset] + hyps.append(sols) +check('c3.hyps', all(len(s) == 1 for s in hyps), 'for each removed root f_i the class O_i = {live lines orthogonal to f_i} is the support of exactly one functional h_i on the label space') +W = [] +for s in hyps: + h = s[0]; ws = [w for w in range(512) if all(bf(w, 1 << j) == (h >> j & 1) for j in range(9))] + W.append(ws[0] if len(ws) == 1 else None) +check('c3.normals', all(w is not None for w in W), 'the pairing is nondegenerate: each h_i = b(w_i, .) for a unique normal w_i') +check('c3.premiseP2a', all(bf(W[i], W[j]) == 1 for i in range(8) for j in range(8)), 'b(w_i, w_j) = 1 for all 64 ordered pairs, i = j included (every normal is non-isotropic, any two are non-orthogonal)') +rho = set(rank_f2([sum(1 << k for k, s in enumerate(N[j]) if Br[t] >> s & 1) for t in N[i]]) for i in range(8) for j in range(8)) +check('c3.premiseP2b', rho == {8}, 'rho(E_i, E_j) = rank_F2 B[N(f_i), N(f_j)] = %s for all 64 ordered pairs' % sorted(rho)) +# survivors (Gram-deficient classes) via Q-rank: label classes whose support has rank_Q < 8 ... equivalently the 8 hyperplanes O_i +surv = [h for h in range(1, 512) if rank_q([list(L[t]) for t in range(v) if hfun(h, lab[t])]) < 8] +check('c3.deficient', sorted(surv) == sorted(s[0] for s in hyps) and len(surv) == 8, 'the Gram-deficient label classes (support of rank_Q < 8) are exactly the 8 classes O_i') +check('c3.nosub2', not any((a ^ b) in set(surv) for a, b in itertools.combinations(surv, 2)), 'no two of them add up to a third: no 2-dimensional subspace among the 8 deficient classes') +# frame lemma for Theorem Bk: for every i < j some live 8-frame E has E cap N(f_i) cap N(f_j) = empty +adj = orth_graph(L); frames = [c for c in maximal_cliques(adj) if popcount(c) == 8] +Nm = [sum(1 << t for t in n) for n in N] +ok = all(any(E & Nm[i] & Nm[j] == 0 for E in frames) for i, j in itertools.combinations(range(8), 2)) +check('c3.framelemma', ok and len(frames) == 1260, 'for every i < j (28 pairs) some one of the 1260 live 8-frames E has E cap N(f_i) cap N(f_j) = empty') +# O(9) charts: transvections x -> x + b(x, u) u (u isotropic) generate; Phi given by images of the 9 basis vectors +iso = [u for u in range(1, 512) if bf(u, u) == 0] +def tv(u, x): return x ^ (u if bf(x, u) else 0) +def compose(Phi, u): return [tv(u, x) for x in Phi] +def apply(Phi, x): + y = 0 + for i in range(9): + if x >> i & 1: y ^= Phi[i] + return y +def orth_count(Phi): return sum(bf(W[i], apply(Phi, W[k])) == 0 for i in range(8) for k in range(8)) +def is_isometry(Phi): return all(bf(apply(Phi, 1 << i), apply(Phi, 1 << j)) == bf(1 << i, 1 << j) for i in range(9) for j in range(9)) +chart_file = os.path.join(HERE, '..', 'data', 'chart_rank7.json') +if not os.path.exists(chart_file): + rng = random.Random(52); best = (0, None) + for s in range(30000): + Phi = [1 << i for i in range(9)] + for _ in range(rng.randint(1, 40)): Phi = compose(Phi, rng.choice(iso)) + c = orth_count(Phi) + if c > best[0]: best = (c, Phi) + if c == 64: break + json.dump(dict(orth_cross_pairs=best[0], Phi=best[1], note='images of the 9 label basis vectors (9-bit ints); found by random.Random(52) products of isotropic transvections'), open(chart_file, 'w')) +ch = json.load(open(chart_file)); Phi = ch['Phi'] +check('c3.chart.isometry', is_isometry(Phi), 'the shipped chart Phi preserves the pairing b (Phi in O(9, F2))') +lab2 = [apply(Phi, x) for x in lab] +def span_cost(S, T_): + return rank_f2([sum(bf(a, b) << k for k, b in enumerate(T_)) for a in S]) +costs = set(span_cost([lab[t] for t in N[i]], [lab2[t] for t in N[k]]) for i in range(8) for k in range(8)) +check('c3.chart.rank7', orth_count(Phi) == ch['orth_cross_pairs'] and costs == {7}, 'chart Phi: b(w_i, Phi w_k) = 0 for %d/64 (i,k) and all 64 cross pairs rho(call-1 support N(f_i), call-2 support N(f_k)) have F2-rank %s' % (orth_count(Phi), sorted(costs))) +same = set(span_cost([lab[t] for t in N[i]], [lab[t] for t in N[k]]) for i in range(8) for k in range(8)) +check('c3.chart.same', same == {8}, 'within one call the same pairings have rank %s (so rank-7 terms are pure cross-call)' % sorted(same)) +finish() diff --git a/research/decoder-floors-frontier-format/checks/c4_masked_costar.py b/research/decoder-floors-frontier-format/checks/c4_masked_costar.py new file mode 100644 index 0000000000..85880e2fa1 --- /dev/null +++ b/research/decoder-floors-frontier-format/checks/c4_masked_costar.py @@ -0,0 +1,51 @@ +"""STRAYS-MASK finite inputs (standard library only): (1) #1180's nine-term outside-col-M identity on P112 costs exactly 65 under the dense B (and the +removed-frame decoder it merges into costs 64); (2) all 120 co-star dead sets V \\ S_alpha are certified spanning in every rank-8 realisation (Lemma 1.1), +so every nine-term masked co-star decoder is a square biorthogonal decoder of a rank-9 realisation; (3) every survivor class contains an orthogonal 7-clique.""" +import sys, os, itertools +from collections import Counter +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, os.path.join(HERE, '..', 'lib')) +from report import check, finish +from units import * +from supprop import setup, propagate, census + +L = sub(P112_IDX); v = len(L); Br = b0rows(L); G = gram(L) +sup = lambda col: [t for t in range(v) if col[t] != 0] +R = [[dot(r, E8[f]) for r in L] for f in F0] # coordinates in the removed-frame basis +base = [blk(Br, sup(R[j]), sup(R[j])) for j in range(8)] +check('c4.coordinate', base == [8] * 8, 'removed-frame coordinate decoder: term ranks %s, total %d' % (base, sum(base))) +def identity_scale(D, T): + c = sum(D[0][k] * T[0][k] for k in range(len(D[0]))); c0 = G[0][0] + return c // c0 if c0 and c % c0 == 0 and all(sum(D[a][k] * T[b][k] for k in range(len(D[0]))) == (c // c0) * G[a][b] for a in range(v) for b in range(v)) else None +for name, zval in (('z = e_line0 (value 1, #1180)', 1), ('z chosen to kill line 0 in term 1', -R[0][0])): + z = [zval if t == 0 else 0 for t in range(v)] + t1 = [R[0][t] + z[t] for t in range(v)] + terms = [(sup(t1), sup(R[0]))] + [(sup(R[j]), sup(R[j])) for j in range(1, 8)] + [(sup(z), sup(R[0]))] + rk = [blk(Br, a, b) for a, b in terms] + D = [[t1[t]] + [R[j][t] for j in range(1, 8)] + [-z[t]] for t in range(v)] + T = [[R[0][t]] + [R[j][t] for j in range(1, 8)] + [R[0][t]] for t in range(v)] + sc = identity_scale(D, T) + if zval == 1: + check('c4.ex1180', sc is not None and rk == [8] * 8 + [1] and sum(rk) == 65, '#1180 nine-term identity (%s): D T^T = %s x Gram exactly on all 112 x 112 entries; term ranks %s, total %d' % (name, sc, rk, sum(rk))) + else: + print(' info: %s -> identity scale %s, term ranks %s, total %d' % (name, sc, rk, sum(rk))) + check('c4.ex1180.alt', sc is not None and sum(rk) >= 64, 'variant %s: exact identity and total %d >= 64' % (name, sum(rk))) +# the two outside-col-M terms of #1180 share one column vector R_0: they merge into the removed-frame term +# (co-star theorem) all 120 dead sets are certified spanning +v_, frames, nbar = setup(L, 8); full = (1 << v) - 1 +res = [] +for a in range(120): + S = sum(1 << t for t in range(v) if dot(L[t], E8[a]) == 0 or P112_IDX[t] == a) + dead = full & ~S + res.append((a in set(F0), popcount(S), min(popcount(s) for s in propagate(dead, frames, nbar)) == 0)) +cnt = Counter(res) +check('c4.costar', cnt == Counter({(False, 60, True): 112, (True, 56, True): 8}), 'all 120 co-star dead sets V \\ S_alpha are certified spanning in every rank-8 realisation: %s (alpha in F0, |S_alpha|, certified): count' % dict(cnt)) +# survivor classes: max orthogonal clique 7 +unc = census(L, 8)['unc'] +og = orth_graph(L) +w = set() +for _, O in unc: + adj = [og[t] & O if O >> t & 1 else 0 for t in range(v)] + w.add(max(popcount(c) for c in maximal_cliques(adj) if c & O == c and c)) +check('c4.survclique', w == {7}, 'each of the 8 survivor classes contains an orthogonal 7-clique and no orthogonal 8-clique (so the killing functional is unique up to scale in every rank-8 realisation): %s' % sorted(w)) +finish() diff --git a/research/decoder-floors-frontier-format/checks/c5_attaining_decoders.py b/research/decoder-floors-frontier-format/checks/c5_attaining_decoders.py new file mode 100644 index 0000000000..d1025a383b --- /dev/null +++ b/research/decoder-floors-frontier-format/checks/c5_attaining_decoders.py @@ -0,0 +1,48 @@ +"""(Adapted without change of content from the author's #2277 package, checks/c5_known_decoders.py.) The floors 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/decoder-floors-frontier-format/checks/c6_block_minrank_punctures.py b/research/decoder-floors-frontier-format/checks/c6_block_minrank_punctures.py new file mode 100644 index 0000000000..fb5ca2e4ae --- /dev/null +++ b/research/decoder-floors-frontier-format/checks/c6_block_minrank_punctures.py @@ -0,0 +1,68 @@ +"""EDGE E1a (the coordinate block N(f) of P112 has fitting minrank 8 for EVERY admissible B) and NEWWORD C4.1 (clique-hitting bounds on punctures of N(f)). +Standard library only; kissat 4.0.4 and drat-trim are optional. Registry: sat/misc.py; recorded verdicts: sat/results_misc.jsonl.""" +import sys, os, json, time, lzma, tempfile, itertools +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 +from units import * +import misc, hitting +import solve as S_ + +t0 = time.time() +L = sub(P112_IDX); v = len(L); G = gram(L) +Ns = [[t for t in range(v) if dot(L[t], E8[f]) != 0] for f in F0] +check('c6.N', all(len(n) == 56 for n in Ns) and all(sum(1 for n in Ns if t in n) == 4 for t in range(v)), 'N(f) has 56 live lines for each of the 8 removed roots, and every live line lies in exactly 4 of them') +Br = b0rows(L) +check('c6.dense', all(blk(Br, n, n) == 8 for n in Ns), 'dense pairing: rank_F2 B[N(f), N(f)] = 8 for all 8 f (the 64 of P112)') +og = orth_graph(L); nog = nonorth_graph = [sum(1 << s for s in range(v) if s != t and G[t][s] != 0) for t in range(v)] +Nm = [sum(1 << t for t in n) for n in Ns] +alpha = max(popcount(c) for c in maximal_cliques([og[t] & Nm[0] if Nm[0] >> t & 1 else 0 for t in range(v)])) +omega = max(popcount(c) for c in maximal_cliques([nog[t] & Nm[0] if Nm[0] >> t & 1 else 0 for t in range(v)])) +check('c6.alpha', alpha == 4 and omega == 7, 'inside N(f): largest pairwise orthogonal set alpha = %d, largest pairwise non-orthogonal clique omega = %d' % (alpha, omega)) +check('c6.f7', 56 - alpha == 52, 'puncturing to rank <= 1 (s = 7) must leave a pairwise-orthogonal set: at most alpha = 4 lines kept, so f(7) >= 52 (exact, no SAT)') +rows = [json.loads(l) for l in open(os.path.join(PKG, 'sat', 'results_misc.jsonl'))] +R = misc.registry(); fast = {n for n in R if not R[n]['slow']} +have = {r['name'] for r in rows} +check('c6.coverage', fast <= have, 'every fast registry instance has a recorded result (%d recorded, %d fast)' % (len(have), len(fast))) +bad = [r['name'] for r in rows if r['status'] != (R[r['name']]['expect'] if r['name'] in R else r['status']) or (r['status'] == 'UNSAT' and not r['slow'] and r.get('drat') != 'VERIFIED') or (r['status'] == 'SAT' and not r.get('model_ok'))] +check('c6.status', not bad, 'recorded statuses as expected: %d UNSAT (DRAT VERIFIED when recorded unless flagged slow), %d SAT with checked models %s' % ( + sum(r['status'] == 'UNSAT' for r in rows), sum(r['status'] == 'SAT' for r in rows), bad[:3])) +cnfs = {}; badh = [] +for r in rows: + if r['slow'] and not os.environ.get('HIT_FULL'): continue + d = misc.dimacs_of(r['name']); cnfs[r['name']] = d + import hashlib + if hashlib.sha256(d.encode()).hexdigest() != r['cnf_sha256']: badh.append(r['name']) +check('c6.hash', not badh, 'regenerated DIMACS of %d instances has the recorded sha256 %s' % (len(cnfs), badh[:3])) +dt = S_.drat_trim(); ks = S_.kissat() +pdir = os.path.join(PKG, 'sat', 'proofs'); proofs = sorted(f for f in (os.listdir(pdir) if os.path.isdir(pdir) else []) if f.split('_')[0] in ('edge', 'hit', 'tri')) +if dt and proofs: + ok = 0 + with tempfile.TemporaryDirectory() as td: + for f in proofs: + n = f[:-len('.drat.xz')]; cnf = os.path.join(td, 'x.cnf'); pr = os.path.join(td, 'x.drat') + open(cnf, 'w').write(cnfs[n]); open(pr, 'wb').write(lzma.open(os.path.join(pdir, f)).read()); ok += S_.run_drat(cnf, pr) + check('c6.shipped', ok == len(proofs), 'drat-trim verifies all %d shipped proofs of this check against the regenerated CNFs' % ok) +else: + print('SKIP c6.shipped drat-trim not found: %d shipped proofs not re-checked' % len(proofs), flush=True) +if ks: + fails = []; nd = 0 + with tempfile.TemporaryDirectory() as td: + for n, d in cnfs.items(): + if R[n]['slow'] and not os.environ.get('HIT_FULL'): continue + cnf = os.path.join(td, 'x.cnf'); pr = os.path.join(td, 'x.drat'); open(cnf, 'w').write(d) + slow = R[n]['slow'] + st, model = S_.run_kissat(cnf, None if (slow or R[n]['expect'] == 'SAT') else pr, tmo=3000) + good = st == R[n]['expect'] + if good and st == 'UNSAT' and dt and not slow: good = S_.run_drat(cnf, pr); nd += 1 + if good and st == 'SAT': good = misc.check_model(n, model) + if not good: fails.append(n) + check('c6.resolve', not fails, 're-solved the registry with kissat 4.0.4 (%s): all as expected, %d UNSAT proofs checked by drat-trim %s' % ('HIT_FULL: slow instances included' if os.environ.get('HIT_FULL') else 'fast instances', nd, fails[:3])) +else: + print('SKIP c6.resolve kissat not found: recorded verdicts not re-derived', flush=True) +# consequences (arithmetic on the proven bounds) +tri = [r for r in rows if r['name'] == 'tri_keep11'][0] +f_ = {2: 14, 6: 56 - 10, 7: 52, 8: 56} +check('c6.f', f_[2] == 13 + 1 and f_[6] == 46, 'f(2) >= 14 (k = 7: no hitting set of 13 lines for the 576 7-cliques); f(6) >= 56 - 10 = 46 (no triangle-free kept set of 11 lines); f(7) >= 52; f(8) = 56; ratios s/f(s): %s' % {s: round(s / f_[s], 4) for s in f_}) +print(' (c6 %.0f s)' % (time.time() - t0)) +finish() diff --git a/research/decoder-floors-frontier-format/checks/c7_p111_unit.py b/research/decoder-floors-frontier-format/checks/c7_p111_unit.py new file mode 100644 index 0000000000..c6e5e6a830 --- /dev/null +++ b/research/decoder-floors-frontier-format/checks/c7_p111_unit.py @@ -0,0 +1,31 @@ +"""The P111 bit unit of PR #589's word (data/p111_lines.json): 111 of the 120 lines of the E8 line system in the doubled Hamming coordinates of that word. +Standard library only. The rigidity SAT instances for it are replayed in c1 (P111_*); the 'at most 12 retained totals cost >= 63' statement is RECORDED, not checked here.""" +import sys, os, json, itertools +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, os.path.join(HERE, '..', 'lib')) +from report import check, finish +from roots import rank_f2, rank_q, maximal_cliques, popcount + +L = json.load(open(os.path.join(HERE, '..', 'data', 'p111_lines.json')))['lines'] +ip = lambda x, y: sum(a * b for a, b in zip(x, y)) +canon = lambda v: tuple(v) if next(x for x in v if x) > 0 else tuple(-x for x in v) +check('c7.count', len(L) == 111 and len(set(canon(v) for v in L)) == 111, '111 distinct lines') +check('c7.norm', all(ip(v, v) == 4 for v in L), 'every line has squared norm 4 (so ip // 2 is the Gram entry 2 on the diagonal)') +check('c7.gram', all(ip(a, b) in (0, 2, -2) for a, b in itertools.combinations(L, 2)), 'distinct lines have inner product 0 or +-2, i.e. Gram entry 0 or +-1: the lines are roots of a root system (a sub-configuration of E8)') +# the 120-line completion of core.py: 8 axes (2 e_i) and, for each 4-support occurring among the live weight-4 lines, all sign patterns with the first entry +1 +sup4 = sorted({tuple(i for i, x in enumerate(v) if x) for v in L if sum(1 for x in v if x) == 4}) +full = {tuple(2 if k == i else 0 for k in range(8)) for i in range(8)} +for s in sup4: + for signs in itertools.product((1, -1), repeat=3): + v = [0] * 8; v[s[0]] = 1 + for k, j in enumerate(s[1:]): v[j] = signs[k] + full.add(tuple(v)) +full = sorted(full); live = set(canon(v) for v in L) +check('c7.e8', len(sup4) == 14 and len(full) == 120 and live <= set(canon(v) for v in full) and all(ip(a, b) in (0, 2, -2, 4) for a, b in itertools.combinations(full, 2)), 'the live lines lie in a 120-line root-line system (8 axes + 14 four-supports x 8 sign patterns): P111 = that system minus 9 lines') +check('c7.rankQ', rank_q([list(v) for v in L]) == 8, 'the 111 lines span Q^8') +orthmask = [sum(1 << s for s in range(111) if s != t and ip(L[t], L[s]) == 0) for t in range(111)] +Br = [sum(1 << s for s in range(111) if s == t or ip(L[t], L[s]) == 0) for t in range(111)] +check('c7.rankF2', rank_f2(Br) == 9, 'the dense pairing B_d = J + Abar of P111 has F2-rank 9 (so h = 9 is the minimal fitting rank when B_d is forced, Theorem R)') +w = max(popcount(c) for c in maximal_cliques(orthmask)) +check('c7.frame', w == 8, 'P111 contains an orthogonal 8-frame (alpha_perp = 8, so every realisation has rank >= 8)') +finish() diff --git a/research/decoder-floors-frontier-format/data/chart_rank7.json b/research/decoder-floors-frontier-format/data/chart_rank7.json new file mode 100644 index 0000000000..39d66833ad --- /dev/null +++ b/research/decoder-floors-frontier-format/data/chart_rank7.json @@ -0,0 +1 @@ +{"orth_cross_pairs": 64, "Phi": [167, 164, 4, 182, 8, 32, 230, 152, 422], "note": "images of the 9 label basis vectors (9-bit ints); found by random.Random(52) products of isotropic transvections"} \ No newline at end of file diff --git a/research/decoder-floors-frontier-format/data/h4_F1.json b/research/decoder-floors-frontier-format/data/h4_F1.json new file mode 100644 index 0000000000..5926a5adf3 --- /dev/null +++ b/research/decoder-floors-frontier-format/data/h4_F1.json @@ -0,0 +1 @@ +[[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/decoder-floors-frontier-format/data/p104_frames.json b/research/decoder-floors-frontier-format/data/p104_frames.json new file mode 100644 index 0000000000..3110c60982 --- /dev/null +++ b/research/decoder-floors-frontier-format/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/decoder-floors-frontier-format/data/p104reps.json b/research/decoder-floors-frontier-format/data/p104reps.json new file mode 100644 index 0000000000..f89d0b9d23 --- /dev/null +++ b/research/decoder-floors-frontier-format/data/p104reps.json @@ -0,0 +1 @@ +{"(767, ((0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1)))": [2, 3, 24, 25, 36, 37, 50, 51], "(801, ((0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1)))": [2, 3, 16, 17, 46, 47, 52, 53], "(759, ((0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1), (0, 0, 0, 0, 1, 1, 1, 1)))": [2, 24, 37, 51, 72, 87, 107, 116]} \ No newline at end of file diff --git a/research/decoder-floors-frontier-format/data/p111_lines.json b/research/decoder-floors-frontier-format/data/p111_lines.json new file mode 100644 index 0000000000..39a47aad70 --- /dev/null +++ b/research/decoder-floors-frontier-format/data/p111_lines.json @@ -0,0 +1 @@ +{"note": "The 111 live lines of the P111 bit unit of PR #589's word (research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz, key 'lines'), in the doubled Hamming coordinates used by that word: dot(x,y)//2 is 2 on the diagonal and 0/+-1 off it. Source sha256 of the gz file is recorded in SOURCES.json.", "lines": [[2, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, 0, 2], [0, 1, 1, 0, 1, 0, 0, 1], [0, 1, 1, 0, 1, 0, 0, -1], [0, 1, 1, 0, -1, 0, 0, -1], [0, 1, -1, 0, 1, 0, 0, 1], [0, 1, -1, 0, 1, 0, 0, -1], [0, 1, -1, 0, -1, 0, 0, 1], [1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 1, 0, 1, 0, -1, 0], [1, 0, 1, 0, -1, 0, 1, 0], [1, 0, 1, 0, -1, 0, -1, 0], [1, 0, -1, 0, 1, 0, 1, 0], [1, 0, -1, 0, 1, 0, -1, 0], [1, 0, -1, 0, -1, 0, 1, 0], [1, 0, -1, 0, -1, 0, -1, 0], [1, 1, 0, 0, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 1, -1], [1, 1, 0, 0, 0, 0, -1, 1], [1, 1, 0, 0, 0, 0, -1, -1], [1, -1, 0, 0, 0, 0, 1, 1], [1, -1, 0, 0, 0, 0, 1, -1], [1, -1, 0, 0, 0, 0, -1, 1], [1, -1, 0, 0, 0, 0, -1, -1], [1, 1, 0, 0, 1, 1, 0, 0], [1, 1, 0, 0, 1, -1, 0, 0], [1, 1, 0, 0, -1, 1, 0, 0], [1, 1, 0, 0, -1, -1, 0, 0], [1, -1, 0, 0, 1, 1, 0, 0], [1, -1, 0, 0, 1, -1, 0, 0], [1, -1, 0, 0, -1, 1, 0, 0], [1, -1, 0, 0, -1, -1, 0, 0], [1, 0, 1, 0, 0, 1, 0, 1], [1, 0, 1, 0, 0, 1, 0, -1], [1, 0, 1, 0, 0, -1, 0, 1], [1, 0, 1, 0, 0, -1, 0, -1], [1, 0, -1, 0, 0, 1, 0, 1], [1, 0, -1, 0, 0, 1, 0, -1], [1, 0, -1, 0, 0, -1, 0, 1], [1, 0, -1, 0, 0, -1, 0, -1], [0, 1, 1, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 1, -1, 0], [0, 1, 1, 0, 0, -1, 1, 0], [0, 1, -1, 0, 0, 1, 1, 0], [0, 1, -1, 0, 0, -1, 1, 0], [0, 1, -1, 0, 0, -1, -1, 0], [0, 0, 0, 0, 1, 1, 1, -1], [0, 0, 0, 0, 1, 1, -1, 1], [0, 0, 0, 0, 1, 1, -1, -1], [0, 0, 0, 0, 1, -1, 1, 1], [0, 0, 0, 0, 1, -1, 1, -1], [0, 0, 0, 0, 1, -1, -1, 1], [0, 0, 0, 0, 1, -1, -1, -1], [1, 1, 1, 1, 0, 0, 0, 0], [1, 1, 1, -1, 0, 0, 0, 0], [1, 1, -1, -1, 0, 0, 0, 0], [1, -1, 1, 1, 0, 0, 0, 0], [1, -1, 1, -1, 0, 0, 0, 0], [1, -1, -1, 1, 0, 0, 0, 0], [1, -1, -1, -1, 0, 0, 0, 0], [1, 0, 0, 1, 1, 0, 0, -1], [1, 0, 0, 1, -1, 0, 0, 1], [1, 0, 0, 1, -1, 0, 0, -1], [1, 0, 0, -1, 1, 0, 0, 1], [1, 0, 0, -1, 1, 0, 0, -1], [1, 0, 0, -1, -1, 0, 0, 1], [1, 0, 0, -1, -1, 0, 0, -1], [0, 1, 0, 1, 1, 0, 1, 0], [0, 1, 0, 1, 1, 0, -1, 0], [0, 1, 0, 1, -1, 0, 1, 0], [0, 1, 0, 1, -1, 0, -1, 0], [0, 1, 0, -1, 1, 0, 1, 0], [0, 1, 0, -1, 1, 0, -1, 0], [0, 1, 0, -1, -1, 0, 1, 0], [0, 1, 0, -1, -1, 0, -1, 0], [0, 0, 1, 1, 0, 0, 1, 1], [0, 0, 1, 1, 0, 0, -1, 1], [0, 0, 1, 1, 0, 0, -1, -1], [0, 0, 1, -1, 0, 0, 1, 1], [0, 0, 1, -1, 0, 0, 1, -1], [0, 0, 1, -1, 0, 0, -1, 1], [0, 0, 1, -1, 0, 0, -1, -1], [0, 0, 1, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, -1, 0, 0], [0, 0, 1, 1, -1, 1, 0, 0], [0, 0, 1, 1, -1, -1, 0, 0], [0, 0, 1, -1, 1, 1, 0, 0], [0, 0, 1, -1, 1, -1, 0, 0], [0, 0, 1, -1, -1, 1, 0, 0], [0, 0, 1, -1, -1, -1, 0, 0], [0, 1, 0, 1, 0, 1, 0, 1], [0, 1, 0, 1, 0, 1, 0, -1], [0, 1, 0, 1, 0, -1, 0, 1], [0, 1, 0, 1, 0, -1, 0, -1], [0, 1, 0, -1, 0, 1, 0, 1], [0, 1, 0, -1, 0, 1, 0, -1], [0, 1, 0, -1, 0, -1, 0, 1], [1, 0, 0, 1, 0, 1, 1, 0], [1, 0, 0, 1, 0, 1, -1, 0], [1, 0, 0, 1, 0, -1, 1, 0], [1, 0, 0, 1, 0, -1, -1, 0], [1, 0, 0, -1, 0, 1, 1, 0], [1, 0, 0, -1, 0, 1, -1, 0], [1, 0, 0, -1, 0, -1, 1, 0], [1, 0, 0, -1, 0, -1, -1, 0]]} \ No newline at end of file diff --git a/research/decoder-floors-frontier-format/lib/frames.py b/research/decoder-floors-frontier-format/lib/frames.py new file mode 100644 index 0000000000..150bab4c3c --- /dev/null +++ b/research/decoder-floors-frontier-format/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/decoder-floors-frontier-format/lib/report.py b/research/decoder-floors-frontier-format/lib/report.py new file mode 100644 index 0000000000..e8973c2fa2 --- /dev/null +++ b/research/decoder-floors-frontier-format/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/decoder-floors-frontier-format/lib/roots.py b/research/decoder-floors-frontier-format/lib/roots.py new file mode 100644 index 0000000000..bfddda2d20 --- /dev/null +++ b/research/decoder-floors-frontier-format/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/decoder-floors-frontier-format/lib/satgen.py b/research/decoder-floors-frontier-format/lib/satgen.py new file mode 100644 index 0000000000..21461b4137 --- /dev/null +++ b/research/decoder-floors-frontier-format/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/decoder-floors-frontier-format/lib/solve.py b/research/decoder-floors-frontier-format/lib/solve.py new file mode 100644 index 0000000000..ea6aebae16 --- /dev/null +++ b/research/decoder-floors-frontier-format/lib/solve.py @@ -0,0 +1,19 @@ +"""Thin wrappers around kissat and drat-trim (both optional). Never uses CaDiCaL.""" +import os, shutil, subprocess +def find(env, name): + p = os.environ.get(env) or shutil.which(name) + return p if p and os.path.exists(p) else None +def kissat(): return find('KISSAT', 'kissat') +def drat_trim(): return find('DRAT_TRIM', 'drat-trim') +def run_kissat(cnf, proof=None, tmo=900): + """returns (status 'SAT'|'UNSAT'|'UNKNOWN', set of true variables or None)""" + cmd = [kissat(), '-q'] + ([cnf, proof] if proof else [cnf]) + try: o = subprocess.run(cmd, capture_output=True, text=True, timeout=tmo).stdout + except subprocess.TimeoutExpired: return 'UNKNOWN', None + if 'UNSATISFIABLE' in o: return 'UNSAT', None + if 'SATISFIABLE' in o: + return 'SAT', set(int(x) for l in o.splitlines() if l.startswith('v') for x in l.split()[1:] if int(x) > 0) + return 'UNKNOWN', None +def run_drat(cnf, proof, tmo=3000): + d = subprocess.run([drat_trim(), cnf, proof, '-t', str(tmo)], capture_output=True, text=True) + return 's VERIFIED' in d.stdout diff --git a/research/decoder-floors-frontier-format/lib/supprop.py b/research/decoder-floors-frontier-format/lib/supprop.py new file mode 100644 index 0000000000..f8a24f3d61 --- /dev/null +++ b/research/decoder-floors-frontier-format/lib/supprop.py @@ -0,0 +1,32 @@ +"""Lemma 1.1 (support propagation) machinery shared by c2 and c4; standard library only.""" +import os, sys +sys.path.insert(0, os.path.dirname(os.path.abspath(__file__))) +from units import * + +def setup(L, T): + v = len(L); G = gram(L); adj = orth_graph(L) + frames = [c for c in maximal_cliques(adj) if popcount(c) == T] + nbar = [sum(1 << s for s in range(v) if G[t][s] != 0) for t in range(v)] + return v, frames, nbar + +def propagate(O, frames, nbar): + S = [E & ~O for E in frames] + while True: + Wg = -1 + for s in S: + w = 0; x = s + while x: + b = x & -x; w |= nbar[b.bit_length() - 1]; x ^= b + Wg &= w + S2 = [s & Wg for s in S] + if S2 == S: return S + S = S2 + +def census(L, T): + v, frames, nbar = setup(L, T); Br = b0rows(L); basis = colspace_basis(Br); h = len(basis) + classes = span_vectors(basis); unc = [] + for co, w in classes: + S = propagate(w, frames, nbar) + if min(popcount(s) for s in S) > 0: unc.append((co, w)) + return dict(v=v, h=h, T=T, nframes=len(frames), nclasses=len(classes), unc=unc, Br=Br) + diff --git a/research/decoder-floors-frontier-format/lib/units.py b/research/decoder-floors-frontier-format/lib/units.py new file mode 100644 index 0000000000..e52aa5ab13 --- /dev/null +++ b/research/decoder-floors-frontier-format/lib/units.py @@ -0,0 +1,38 @@ +"""The E8-family units used throughout this package (standard library only; built on lib/roots.py). +E8 = 120 lines in doubled coordinates (lib/roots.py order). F0 = the frame {2e_{2k} +- 2e_{2k+1}} (k = 0..3), the removed frame of P112. +P112 = E8 minus F0 (112 lines). P104 = P112 minus a second frame F1 disjoint from F0 (104 lines). H4 = the F1 of #968 (data/h4_F1.json). +E7 = lines orthogonal to line 0 (63); E6 = lines orthogonal to line 0 and to the first line non-orthogonal to line 0 (36).""" +import json, os +from roots import * + +_UH = os.path.dirname(os.path.abspath(__file__)) +DATA = os.path.join(_UH, '..', 'data') +IDX = {r: i for i, r in enumerate(E8)} +def idx(v): + v = tuple(v); w = tuple(-x for x in v) + return IDX[v] if v in IDX else IDX[w] +F0 = [] +for _k in range(4): + for _s in (2, -2): + _v = [0] * 8; _v[2 * _k] = 2; _v[2 * _k + 1] = _s; F0.append(idx(_v)) +F0 = sorted(F0) +H4F1 = sorted(idx(v) for v in json.load(open(os.path.join(DATA, 'h4_F1.json')))) +GE8 = gram(E8) +def orth(t, s): return GE8[t][s] == 0 +def sub(ix): return [E8[i] for i in ix] +P112_IDX = [i for i in range(120) if i not in set(F0)] +def p104_idx(F1): return [i for i in P112_IDX if i not in set(F1)] +def p104_reps(): + d = json.load(open(os.path.join(DATA, 'p104reps.json'))) + return {int(k[1:k.index(',')]): v for k, v in d.items()} # frame count (759, 767, 801) -> F1 as E8 indices +_b = next(t for t in range(1, 120) if GE8[t][0] != 0) +E7_IDX = [t for t in range(120) if GE8[t][0] == 0 and t != 0] +E6_IDX = [t for t in range(120) if GE8[t][0] == 0 and GE8[t][_b] == 0 and t not in (0, _b)] +def unit_idx(name): + if name == 'E8': return list(range(120)) + if name == 'P112': return list(P112_IDX) + if name == 'P104H4': return p104_idx(H4F1) + if name.startswith('P104_'): return p104_idx(p104_reps()[int(name[5:])]) + if name == 'E7': return list(E7_IDX) + if name == 'E6': return list(E6_IDX) + raise KeyError(name) diff --git a/research/decoder-floors-frontier-format/requirements.txt b/research/decoder-floors-frontier-format/requirements.txt new file mode 100644 index 0000000000..dee35c3194 --- /dev/null +++ b/research/decoder-floors-frontier-format/requirements.txt @@ -0,0 +1,2 @@ +# Tested set (Python 3.14.7). Standard library only (fractions, itertools, hashlib, json, lzma, subprocess, random); no third-party packages. +# Optional, for the DRAT re-check and for re-solving: drat-trim (set DRAT_TRIM=/path) and kissat 4.0.4 (set KISSAT=/path). Never CaDiCaL. diff --git a/research/decoder-floors-frontier-format/sat/hitting.py b/research/decoder-floors-frontier-format/sat/hitting.py new file mode 100644 index 0000000000..1ca2e9a413 --- /dev/null +++ b/research/decoder-floors-frontier-format/sat/hitting.py @@ -0,0 +1,52 @@ +"""C4.1 (NEWWORD c4): multi-unit puncture savings on the P112 coordinate term under ANY admissible B. Standard library only. +R = N(f) (56 live lines non-orthogonal to a removed root f). A kept clique C of pairwise non-orthogonal lines has B[C,C] = I for every admissible B, +so a puncture D of R (lines deleted from the support) with rank B[R\\D, R\\D] <= 8 - s keeps no (9 - s)-clique: D must hit every (9 - s)-clique of R. +f(s) = min |D| over hitting sets; 'z-1 infeasible' <=> UNSAT of 'hit all k-cliques with at most z-1 lines' (k = 9 - s), encoded with a sequential counter. +For k = 3 (s = 6) the dual form is used: no triangle-free kept set of size `keep`. CNF text is a deterministic function of (k, z).""" +import os, sys +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, os.path.join(HERE, '..', 'lib')) +from units import * + +def region(): + L = sub(P112_IDX); f = F0[0] + R = [t for t in range(len(L)) if dot(L[t], E8[f]) != 0] + return L, R +def cliques(L, R, k): + NO = lambda a, b: dot(L[a], L[b]) != 0 + out = [] + def ext(cur, cand): + if len(cur) == k: out.append(cur); return + for i, c in enumerate(cand): ext(cur + [c], [d for d in cand[i + 1:] if NO(c, d)]) + ext([], R); return out + +def seqcounter_atmost(lits, bound, top): + """Sinz sequential counter: at most `bound` of lits true; returns (clauses, new top)""" + n = len(lits); s = [[0] * (bound + 1) for _ in range(n)]; cl = [] + for i in range(n): + for j in range(1, bound + 1): top += 1; s[i][j] = top + cl.append([-lits[0], s[0][1]]) + for j in range(2, bound + 1): cl.append([-s[0][j]]) + for i in range(1, n): + cl.append([-lits[i], s[i][1]]); cl.append([-s[i - 1][1], s[i][1]]) + for j in range(2, bound + 1): + cl.append([-lits[i], -s[i - 1][j - 1], s[i][j]]); cl.append([-s[i - 1][j], s[i][j]]) + cl.append([-lits[i], -s[i - 1][bound]]) + return cl, top + +def hit_cnf(k, z): + """UNSAT <=> no set of at most z lines of R meets every k-clique""" + L, R = region(); idx = {t: i + 1 for i, t in enumerate(R)} + C = cliques(L, R, k); cl = [[idx[t] for t in c] for c in C] + card, top = seqcounter_atmost([idx[t] for t in R], z, len(R)); cl += card + return top, cl, len(C) + +def tri_cnf(keep): + """UNSAT <=> no triangle-free subset of R (no three pairwise non-orthogonal lines) has `keep` lines""" + L, R = region(); idx = {t: i + 1 for i, t in enumerate(R)} + tris = cliques(L, R, 3); cl = [[-idx[a], -idx[b], -idx[c]] for a, b, c in tris] + # at least `keep` kept = at most len(R)-keep dropped; encode on negated literals + card, top = seqcounter_atmost([-idx[t] for t in R], len(R) - keep, len(R)); cl += card + return top, cl, len(tris) + +def dimacs(nv, cl): return "p cnf %d %d\n" % (nv, len(cl)) + "".join(" ".join(map(str, c)) + " 0\n" for c in cl) diff --git a/research/decoder-floors-frontier-format/sat/misc.py b/research/decoder-floors-frontier-format/sat/misc.py new file mode 100644 index 0000000000..5bd9f8e85d --- /dev/null +++ b/research/decoder-floors-frontier-format/sat/misc.py @@ -0,0 +1,51 @@ +"""Registry of the smaller SAT instances: EDGE E1a (coordinate block N(f) of P112 has fitting minrank 8 for every admissible B) and the C4.1 clique-hitting +bounds on punctures. Standard library only; DIMACS text is a deterministic function of the name.""" +import os, sys, functools +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, os.path.join(HERE, '..', 'lib')) +from units import * +from satgen import fitting_cnf, clique_in, nonorth_graph +import hitting as H + +@functools.lru_cache(maxsize=None) +def edge_instance(i, r): + """rank <= r admissible B on the block B[N(f_i), N(f_i)] of P112 (f_i = i-th removed root); gauge clique = largest non-orthogonal clique in N(f_i)""" + L = sub(P112_IDX); f = F0[i] + N = [t for t in range(len(L)) if dot(L[t], E8[f]) != 0] + adj = nonorth_graph(L); C = clique_in(adj, sum(1 << t for t in N)) + return fitting_cnf(L, N, N, r, C[:r], fixV=False), N, C + +def registry(): + R = {} + for i in range(8): R['edge_N%d_r7' % i] = dict(expect='UNSAT', slow=False) + R['edge_N0_r8'] = dict(expect='SAT', slow=False) + R['hit_k7_z13'] = dict(expect='UNSAT', slow=False); R['hit_k7_z14'] = dict(expect='SAT', slow=False) + R['tri_keep11'] = dict(expect='UNSAT', slow=False); R['tri_keep10'] = dict(expect='SAT', slow=False) + for k, z in ((6, 22), (5, 31), (4, 37)): + R['hit_k%d_z%d' % (k, z)] = dict(expect='UNSAT', slow=True) + return R + +def dimacs_of(name): + if name.startswith('edge_N'): + i = int(name[6]); r = int(name.split('_r')[1]); return edge_instance(i, r)[0].dimacs() + if name.startswith('hit_k'): + k, z = name[5:].split('_z'); nv, cl, _ = H.hit_cnf(int(k), int(z)); return H.dimacs(nv, cl) + if name.startswith('tri_keep'): + nv, cl, _ = H.tri_cnf(int(name[8:])); return H.dimacs(nv, cl) + raise KeyError(name) + +def check_model(name, pos): + """SAT models: edge_N0_r8 -> decode U,V and check B = U V^T is admissible on the block; hit_*: the hitting set of the model hits every clique and has <= z lines; + tri_keep10: the kept set is triangle-free of size >= 10.""" + if name.startswith('edge_N'): + i = int(name[6]); r = int(name.split('_r')[1]); c, N, C = edge_instance(i, r); L = sub(P112_IDX) + ent = lambda t, s: sum(((u in pos) & (w in pos)) for u, w in zip(c.U[t], c.V[s])) & 1 + return all(ent(t, t) == 1 for t in N) and all(ent(t, s) == 0 for t in N for s in N if t != s and dot(L[t], L[s]) != 0) + L, R = H.region(); idx = {t: j + 1 for j, t in enumerate(R)} + if name.startswith('hit_k'): + k, z = name[5:].split('_z'); k, z = int(k), int(z); D = [t for t in R if idx[t] in pos] + return len(D) <= z and all(any(t in D for t in cl) for cl in H.cliques(L, R, k)) + if name.startswith('tri_keep'): + keep = int(name[8:]); K = [t for t in R if idx[t] in pos] + return len(K) >= keep and not any(all(t in K for t in tr) for tr in H.cliques(L, R, 3)) + raise KeyError(name) diff --git a/research/decoder-floors-frontier-format/sat/mkmisc.py b/research/decoder-floors-frontier-format/sat/mkmisc.py new file mode 100644 index 0000000000..09899adc2d --- /dev/null +++ b/research/decoder-floors-frontier-format/sat/mkmisc.py @@ -0,0 +1,23 @@ +"""Author tool for sat/misc.py (see mkresults.py). usage: python3 mkmisc.py OUTDIR [--slow] [name ...] -> JSON lines on stdout. +Instances flagged slow are solved WITHOUT a proof (UNSAT recorded as 'UNSAT (no proof)').""" +import sys, os, json, time, lzma, shutil, hashlib +SATDIR = os.path.dirname(os.path.abspath(__file__)); HERE = SATDIR +sys.path.insert(0, os.path.join(HERE, '..', 'lib')) +from misc import * +from solve import * +args = [a for a in sys.argv[1:] if not a.startswith('--')]; out = args[0]; os.makedirs(out, exist_ok=True) +R = registry(); names = args[1:] or [n for n in R if not R[n]['slow'] or '--slow' in sys.argv] +SHIP = set(os.environ.get('SHIP', '').split(',')) +for nm in names: + t0 = time.time(); d = dimacs_of(nm); cnf = os.path.join(out, nm + '.cnf'); open(cnf, 'w').write(d); pr = os.path.join(out, nm + '.drat') + slow = R[nm]['slow']; st, model = run_kissat(cnf, None if slow else pr, tmo=3000) + row = dict(name=nm, expect=R[nm]['expect'], slow=slow, cnf_sha256=hashlib.sha256(d.encode()).hexdigest(), status=st) + if st == 'UNSAT' and not slow: + row['drat'] = 'VERIFIED' if run_drat(cnf, pr) else 'FAILED'; row['proof_bytes'] = os.path.getsize(pr) + if nm in SHIP: + os.makedirs(os.path.join(SATDIR, 'proofs'), exist_ok=True) + with open(pr, 'rb') as f, lzma.open(os.path.join(SATDIR, 'proofs', nm + '.drat.xz'), 'wb', preset=9) as g: shutil.copyfileobj(f, g) + elif st == 'SAT': row['model_ok'] = check_model(nm, model) + for p in (cnf, pr): + if os.path.exists(p): os.remove(p) + row['secs'] = round(time.time() - t0, 1); print(json.dumps(row), flush=True) diff --git a/research/decoder-floors-frontier-format/sat/mkresults.py b/research/decoder-floors-frontier-format/sat/mkresults.py new file mode 100644 index 0000000000..76a61a660f --- /dev/null +++ b/research/decoder-floors-frontier-format/sat/mkresults.py @@ -0,0 +1,27 @@ +"""Author tool: solve every instance of the registry (kissat 4.0.4, binary DRAT proof), verify UNSAT proofs with drat-trim, decode SAT models, +and write results.jsonl (one JSON line per instance). Optionally xz-ships selected proofs into proofs/. +usage: python3 mkresults.py OUTDIR [name ...]""" +import sys, os, json, time, lzma, shutil +SATDIR = os.path.dirname(os.path.abspath(__file__)); HERE = SATDIR +sys.path.insert(0, os.path.join(HERE, '..', 'lib')) +from rigid import * +from solve import * +SHIP = set(os.environ.get('SHIP', '').split(',')) +out = sys.argv[1]; os.makedirs(out, exist_ok=True) +R = registry(); names = sys.argv[2:] or list(R) +rows = [] +for nm in names: + t0 = time.time(); nv, cls, info, *_ = build_named(nm); d = dimacs(nv, cls); h = sha(d) + cnf = os.path.join(out, nm + '.cnf'); open(cnf, 'w').write(d); pr = os.path.join(out, nm + '.drat') + st, model = run_kissat(cnf, pr) + row = dict(name=nm, unit=R[nm]['unit'], h=R[nm]['h'], sym=R[nm]['sym'], mode=R[nm]['mode'], expect=R[nm]['expect'], nv=nv, ncls=len(cls), + orth=info['orth'], gauge=R[nm]['S'], cnf_sha256=h, status=st) + if st == 'UNSAT': + row['drat'] = 'VERIFIED' if run_drat(cnf, pr) else 'FAILED'; row['proof_bytes'] = os.path.getsize(pr) + if nm in SHIP: + os.makedirs(os.path.join(SATDIR, 'proofs'), exist_ok=True) + with open(pr, 'rb') as f, lzma.open(os.path.join(SATDIR, 'proofs', nm + '.drat.xz'), 'wb', preset=9) as g: shutil.copyfileobj(f, g) + elif st == 'SAT': + fit, z, n = decode_model(nm, model); row.update(model_fitting=fit, orth_zero=z, orth_total=n) + os.remove(pr) if os.path.exists(pr) else None; os.remove(cnf) + row['secs'] = round(time.time() - t0, 1); rows.append(row); print(json.dumps(row), flush=True) diff --git a/research/decoder-floors-frontier-format/sat/proofs/E8_sym_h9_alt.drat.xz b/research/decoder-floors-frontier-format/sat/proofs/E8_sym_h9_alt.drat.xz new file mode 100644 index 0000000000000000000000000000000000000000..6daca3d1b3feddc845f333cb35a3413d19f686b6 GIT binary patch literal 1933676 zcmV(nK=Qx+H+ooF000E$*0e?f03iV!0000G&sfakhe7ZDT>vn=z2F0Dz?i2=CJt)t zv04pv9h4)-Z`pVs!lvJF=18T6ipTvD@#ojQNcb}adD^s$R9?O&BIayMtB`x>)+9*Y 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z+m)V-aFi!LU*N+ut;e|hq#)P{K<3iM#{}{GN every fitting B of F2-rank <= h is dense. +mode 'ctl': no extra clause (must be SAT, and every model must decode to a fitting matrix; for h = rank of the dense B the model is the dense B). +Gauge: S = a pairwise non-orthogonal clique (|S| = 8 for E8/P112/P104, 7 for E7, 5 for E6, pinned in data/gauge.json): B[S,S] = I, so U[S] has rank |S|, +GL(h) is transitive on such tuples, hence U[S] = (e_1..e_|S|) WLOG (symmetric model: P[S] orthonormal, the isometry argument of STRAYS Thm R requires +|S| >= h - 1). This is the encoding of the author's working card STRAYS/rigid2.py; the clique choice here is deterministic and pinned.""" +import sys, os, json, hashlib +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, os.path.join(HERE, '..', 'lib')) +from units import * + +def gauge_clique(name, L_idx, kc): + """lexicographically least (by bitmask) maximum clique of the non-orthogonality graph on the unit, truncated to kc lines (deterministic)""" + from satgen import clique_in, nonorth_graph + L = sub(L_idx); adj = nonorth_graph(L) + S = clique_in(adj, (1 << len(L)) - 1) + assert len(S) >= kc + return [L_idx[i] for i in S[:kc]] + +def build(name, h, sym, mode, S, V=None, orthf=None): + """returns (nv, clauses, info); line labels are positions in the unit's index list""" + orthf = orthf or orth + V = V if V is not None else unit_idx(name); pos = {t: i for i, t in enumerate(V)}; v = len(V) + kc = len(S); assert all(t in pos for t in S) + nv = 0; cls = [] + def new(): + nonlocal nv; nv += 1; return nv + Sset = {t: k for k, t in enumerate(S)} + Uv = {t: [new() for _ in range(h)] for t in V if t not in Sset} + Vv = Uv if sym else {t: [new() for _ in range(h)] for t in V} + def ub(t, k): return ('c', int(Sset[t] == k)) if t in Sset else ('v', Uv[t][k]) + def vb(t, k): return ub(t, k) if sym else ('v', Vv[t][k]) + def AND(a, b): + if a[0] == 'c' and b[0] == 'c': return ('c', a[1] & b[1]) + if a[0] == 'c': return b if a[1] else ('c', 0) + if b[0] == 'c': return a if b[1] else ('c', 0) + if a[1] == b[1]: return a + z = new(); x, y = a[1], b[1]; cls.extend([[-z, x], [-z, y], [z, -x, -y]]); return ('v', z) + def XOR(a, b): + if a[0] == 'c' and b[0] == 'c': return ('c', a[1] ^ b[1]) + if a[0] == 'c': return ('v', -b[1]) if a[1] else b + if b[0] == 'c': return ('v', -a[1]) if b[1] else a + z = new(); x, y = a[1], b[1]; cls.extend([[-z, x, y], [-z, -x, -y], [z, -x, y], [z, x, -y]]); return ('v', z) + def Bent(t, s): + acc = ('c', 0) + for k in range(h): acc = XOR(acc, AND(ub(t, k), vb(s, k))) + return acc + def force(l, val): + if l[0] == 'c': + if l[1] != val: cls.append([]) + return + cls.append([l[1] if val else -l[1]]) + orthp = [] + for t in V: + force(Bent(t, t), 1) + for s in V: + if s == t: continue + if not orthf(t, s): force(Bent(t, s), 0) + elif not sym or t < s: orthp.append((t, s)) + if mode == 'alt': + lits = [] + for (t, s) in orthp: + d = Bent(t, s) + if d[0] == 'c': + assert d[1] == 1, 'trivially SAT' + continue + lits.append(-d[1]) + cls.append(lits) + info = dict(v=v, h=h, sym=sym, mode=mode, nv=nv, ncls=len(cls), orth=len(orthp), gauge=S) + return nv, cls, info, Uv, Vv, orthp + +def dimacs(nv, cls): + return "p cnf %d %d\n" % (nv, len(cls)) + "".join(" ".join(map(str, c)) + " 0\n" for c in cls) + +def sha(s): return hashlib.sha256(s.encode()).hexdigest() + +# ---------------------------------------------------------------- registry +def sub_unit(kc_S, k): + """negative control: the gauge clique S plus the first k other lines of P112 (index order)""" + return list(kc_S) + [t for t in P112_IDX if t not in set(kc_S)][:k] + +import functools +KC = {'E8': 8, 'P112': 8, 'P104H4': 8, 'P104_759': 8, 'P104_767': 8, 'P104_801': 8, 'E7': 7, 'E6': 5} +@functools.lru_cache(maxsize=None) +def registry(): + """name -> dict(unit, h, sym, mode, expect, S, V)""" + R = {} + def add(unit, h, sym, mode, expect, V=None, tag=None): + S = gauge_clique(unit if V is None else 'P112', unit_idx(unit if V is None else 'P112'), KC[unit if V is None else 'P112']) + if expect == 'UNSAT' and sym: assert len(S) >= h - 1 + name = '%s_%s_h%d_%s' % (tag or unit, 'sym' if sym else 'gen', h, mode) + R[name] = dict(unit=unit, h=h, sym=sym, mode=mode, expect=expect, S=S, V=V) + for u in ('P112', 'E8', 'P104H4', 'P104_759', 'P104_767', 'P104_801'): + for sym in (True, False): add(u, 9, sym, 'alt', 'UNSAT') + for u in ('E7', 'E6'): add(u, 7, False, 'alt', 'UNSAT') + for u in ('P112', 'E8', 'P104H4'): + for sym in (True, False): add(u, 9, sym, 'ctl', 'SAT') + for u in ('E7', 'E6'): add(u, 7, False, 'ctl', 'SAT') + S = gauge_clique('P112', unit_idx('P112'), 8) + for k in (20, 40): add('P112', 9, False, 'alt', 'SAT', V=sub_unit(S, k), tag='SUB%d' % k) + for sym in (True, False): add('P112', 10, sym, 'alt', 'SAT') + # P111 (bit unit of PR #589): own coordinates, gauge clique from its own Gram + L, of = p111(); adjn = [sum(1 << s for s in range(111) if s != t and not of(t, s)) for t in range(111)] + from satgen import clique_in + S1 = clique_in(adjn, (1 << 111) - 1)[:8] + for sym in (True, False): + for mode, ex in (('alt', 'UNSAT'), ('ctl', 'SAT')): + name = 'P111_%s_h9_%s' % ('sym' if sym else 'gen', mode) + R[name] = dict(unit='P111', h=9, sym=sym, mode=mode, expect=ex, S=S1, V=None) + return R + +def p111(): + """(vectors, orthf) of the P111 bit unit (data/p111_lines.json)""" + L = json.load(open(os.path.join(DATA, 'p111_lines.json')))['lines'] + return L, (lambda t, s: sum(a * b for a, b in zip(L[t], L[s])) == 0) + +def build_named(name): + r = registry()[name] + if r['unit'] == 'P111': + L, of = p111(); return build('P111', r['h'], r['sym'], r['mode'], r['S'], V=list(range(111)), orthf=of) + return build(r['unit'], r['h'], r['sym'], r['mode'], r['S'], V=r['V']) + +def decode_model(name, pos_vars): + """returns (fitting?, #orthogonal pairs with B_ts = 0, #orthogonal pairs) for a model of an instance""" + r = registry()[name]; nv, cls, info, Uv, Vv, orthp = build_named(name) + V = r['V'] if r['V'] is not None else (list(range(111)) if r['unit'] == 'P111' else unit_idx(r['unit'])); h = r['h']; Sset = {t: k for k, t in enumerate(r['S'])} + def row(t, M): + if M is Uv and t in Sset: return [int(Sset[t] == k) for k in range(h)] + return [int(x in pos_vars) for x in M[t]] + # in the symmetric model V = U so the S rows are constants for both + Ur = {t: row(t, Uv) for t in V}; Vr = Ur if r['sym'] else {t: [int(x in pos_vars) for x in Vv[t]] for t in V} + ent = lambda t, s: sum(a & b for a, b in zip(Ur[t], Vr[s])) & 1 + of = p111()[1] if r['unit'] == 'P111' else orth + fit = all(ent(t, t) == 1 for t in V) and all(ent(t, s) == 0 for t in V for s in V if t != s and not of(t, s)) + zeros = sum(1 for (t, s) in orthp if ent(t, s) == 0) + return fit, zeros, len(orthp) diff --git a/research/decoder-floors-frontier-format/scripts/provenance.py b/research/decoder-floors-frontier-format/scripts/provenance.py new file mode 100644 index 0000000000..d12d3d802a --- /dev/null +++ b/research/decoder-floors-frontier-format/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/decoder-floors-frontier-format/verify.sh b/research/decoder-floors-frontier-format/verify.sh new file mode 100755 index 0000000000..c67127a50c --- /dev/null +++ b/research/decoder-floors-frontier-format/verify.sh @@ -0,0 +1,43 @@ +#!/usr/bin/env bash +# One-command verification of research/decoder-floors-frontier-format (no kappa claim). Recomputes every CHECKED item of the README: +# c1 Theorem R (rigidity of the minimal-rank label table): CNF regeneration + hashes, shipped DRAT proofs, kissat re-solve + drat-trim; +# c2 Lemma 1.1 census, survivors and pair ranks (Theorem P); c3 batching premises and an O(9) chart with 64 rank-7 cross pairs; +# c4 #1180's nine-term identity (65), co-star dead sets; c5 decoders attaining the floors; c6 EDGE E1a block minrank and puncture clique-hitting bounds; +# c7 the P111 bit unit. +# 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. kissat 4.0.4 (KISSAT=/path or on PATH) and drat-trim (DRAT_TRIM=/path or on PATH) are OPTIONAL: +# without them the lines that need them are SKIP lines (never PASS). HIT_FULL=1 re-solves three slow clique-hitting instances (kissat only, no proof). +# usage: bash verify.sh [/fresh/output/dir] (inside a checkout or `git archive` of the repository) +# set PYTHON to choose the interpreter (>= 3.11). Prints PASS / FAIL; exit status is nonzero on any FAIL. +set -uo pipefail +HERE=$(cd "$(dirname "$0")" && pwd) +OUT=${1:-$(mktemp -d "${TMPDIR:-/tmp}/decoder-floors-frontier-format.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' "$OUT/provenance.log")" != 2 ]; then + echo "step 0 FAILED"; exit 1 +fi +echo " tools: kissat=$(command -v "${KISSAT:-kissat}" 2>/dev/null || echo none) drat-trim=$(command -v "${DRAT_TRIM:-drat-trim}" 2>/dev/null || echo none)" +mkdir -p "$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 Theorem R: rigidity of the minimal-rank label table (30 SAT instances)" c1_rigidity.py +run "c2 Lemma 1.1 census, survivors and pair ranks" c2_support_propagation.py +run "c3 batching premises (P112 labels, normals, frame lemma, O(9) chart)" c3_batch_premises.py +run "c4 #1180 identity, co-star dead sets, survivor cliques" c4_masked_costar.py +run "c5 decoders attaining the floors" c5_attaining_decoders.py +run "c6 coordinate-block minrank under every B; puncture clique hitting" c6_block_minrank_punctures.py +run "c7 the P111 bit unit" c7_p111_unit.py +npass=$(grep -c '^PASS' "$OUT/all.out"); nfail=$(grep -c '^FAIL' "$OUT/all.out"); nskip=$(grep -c '^SKIP' "$OUT/all.out") +EXPECT=$((72 - nskip)) +echo "== done in $((SECONDS - T0))s: $((npass + 2)) PASS (2 in step 0), $nskip SKIP, $nfail FAIL; logs in $OUT" +if [ "$nfail" = 0 ] && [ "$npass" = "$EXPECT" ]; then echo "PASS: all expected assertions hold"; exit 0; fi +echo "FAIL: expected $EXPECT assertion PASS lines after step 0, got $npass PASS / $nfail FAIL / $nskip SKIP"; exit 1

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