Attack log — C7 fifth power

2026-08-27 — q4, new shapes, Hamming 13 finished

Replay: python3 compute/verify_set.py compute/R367.txt --min-size 367 → OK. Opened Polak–Schrijver arXiv:1808.07438v2 (Table 1 still $367$–$401$), Itty et al. 2607.21517v2 (fifth-power record still 367), Gao 2607.27869 (“it remains the largest currently known such set”), Buys–Polak–Zuiddam 2607.29681 (profile $(367,8,367,322)$). No 368 in those papers.

q3 closed the $8$-coset leftover. This pass hunted shapes that are not an $8$-coset pack, not a missing letter, and not Hamming $11$ from the seed.

Hamming $13$ is finished. q3 already killed $5$- and $6$-blocker adds (best_mis=4). Case C: every $4$-blocker plus two extra removals, $199{,}671{,}417$ pairs, max_freed=14, no $\alpha\ge 7$. Leftover SAT on the $1767$ vertices with $\le 3$ blockers ($153967$ clauses): Cadical UNSAT in $548$s. Combined with q1's odd ball through $9$ and q2's Hamming $11$, the odd Hamming ball through $13$ around this seed is empty. Logs: compute/q4/hamming13_four_log.txt, hamming13_sat_log.txt.

$1$-dimensional good codes. $(7^5-1)/6=2801$ subspaces, $2680$ good, $1156$ unique connection sets on $\mathbb F_7^4$. Degrees $188$–$242$. Hoffman never dropped below $53$ (best float $366.01$). Random greedy plus a short improve reached $43$ cosets ($301$ vertices); a harder ejection on five generators reached $46$ ($322$). Those packs had empty original-graph residual. No $371$-set. Cadical on a $53$-pack is unfinished (residue). compute/q4/onedim_log.txt, grow_1dim_log.txt.

Cyclic-coordinate $5$-orbits. $3360$ orbits, $3262$ internally independent. Greedy $\le 52$ orbits ($260$) or $53$ plus $\{00000,22222,44444\}$ ($268$). Cadical for $74$ orbits ($370$) is unfinished (residue). compute/q4/cyclic_log.txt.

Negation pairs. $8282$ internally independent pairs $\{v,-v\}$. Cadical for $184$ pairs ($368$) is unfinished (residue).

Hamming $15$ high-blocker. $2842$ vertices with $7$ blockers have freed $\alpha\le 4$; $1{,}406{,}817$ extensions of a $6$-blocker by one extra have freed $\alpha\le 5$. Residue: add only $\le 5$-blocker vertices. compute/q4/hamming15_log.txt.

Ejection from $343$. $80$ random good $3$-dimensional codes, $25000$ delete-and-repack steps each, best=343. The linear sets did not move. Residue. From-scratch GRASP, $40$ restarts: best $297$. Wider fold grid recovered $361$ by greedy residual (the published $367$ needs the exact $40$-set). Mixing the published $367$-set with the reconstructed pipeline set (symmetric difference $4$) has independence number $367$.

No $368$-set. No $\Theta(C_7)$ claim. $367^{1/5}\approx 3.25787<3.258805$. This is not a dent.

2026-08-27 — q3, 8-coset leftover finished

Replay: python3 compute/verify_set.py compute/R367.txt --min-size 367 → OK. Opened Polak–Schrijver arXiv:1808.07438 (Table 1 still $367$–$401$), Itty et al. 2607.21517v2 (fifth-power record still 367; v2 only adds a $C_{15}$ capacity bound), Gao 2607.27869 (“it remains the largest currently known such set” in the fifth power), Buys–Polak–Zuiddam 2607.29681 (profile $(367,8,367,322)$). No 368 in those papers.

q2 left $1280$ quotient graphs after a node-capped search. Those were not $1280$ distinct problems. The $97240$ good $2$-dimensional codes collapse to $9584$ unique connection sets on $\mathbb F_7^3$ (degrees $122$–$228$). A Hoffman bound on the Cayley eigenvalues never dropped below $8$ (best float $33.12$). A multi-start clique cover killed $359$ graphs. An exact $8$-set search with clique-cover pruning and a $2\cdot 10^6$ node cap finished the other $9225$: yes=0 leftover=0. Every $7$-pack still has empty residual in the original graph (q2). No $392$-set of this shape.

Python replay of the RREF count matches $140050$ subspaces and $97240$ good codes. Cadical on a deterministic sample of $159$ unique graphs is UNSAT. The leftover file is empty, so the SAT queue is empty.

Hamming $13$ (6-out/7-in) around the published $367$-set, high-blocker split: $3897$ vertices with $6$ blockers have freed $\alpha\le 4$; $1{,}343{,}744$ extensions of a $5$-blocker vertex by one extra removal have freed $\alpha\le 4$. Residue: adding only $\le 4$-blocker vertices (Cadical on $4806$ candidates, $697012$ clauses, not finished).

No $368$-set. No $\Theta(C_7)$ claim. $367^{1/5}\approx 3.25787<3.258805$. This is not a dent.

2026-08-27 — q2 start, 4-support finished

Replay: python3 compute/verify_set.py compute/R367.txt --min-size 367 → OK. Opened Polak–Schrijver arXiv:1808.07438 (Table 1 still $367$–$401$), Itty et al. 2607.21517, Gao 2607.27869 (“367 remains the largest currently known” in the fifth power), Buys–Polak–Zuiddam 2607.29681 (profile $(367,8,367,322)$). No 368 in those papers.

q1 left 4-support as a local-search residue (best grown $309$). That shape is impossible, and so are 5- and 6-support.

An adjacent pair of letters in one coordinate induces $K_2\boxtimes C_7^{\boxtimes 4}$. The two fibers project to a single independent set in $C_7^{\boxtimes 4}$, so they contribute at most $\alpha(C_7^{\boxtimes 4})\le 115$ (Baumert via Polak–Schrijver Table 1), not $230$. Every $k$-subset of $C_7$ with $k\le 6$ has matching number large enough that

All $126$ such subsets were enumerated (compute/q2/bound_support.py). Toy checks: $\alpha(K_2\boxtimes C_7)=3$, $\alpha(C_7^{\boxtimes 2})=10=\alpha(K_2\boxtimes C_7^{\boxtimes 2})$. The published $367$-set is $7$-surjective on every coordinate; its $35$ adjacent pair-fibers have size $100$–$107$ and project to independent $4$-tuples.

A $368$-set uses all seven letters in every coordinate.

Hamming $11$ (5-out/6-in) around the published $367$-set is finished. Split on the added vertices' blocker counts:

• 3712 vertices with $5$ blockers: each determines the removal $5$-set. Freed graphs have size $\le 11$, $\alpha\le 3$.
• $1{,}103{,}157$ extensions of a $4$-blocker vertex by one extra removal: freed size $\le 11$, $\alpha\le 4$.
• Leftover: add only $\le 3$-blocker vertices. Cadical, $1767$ candidates, $153218$ clauses: UNSAT in $102$s.

No $368$-set at Hamming distance $11$ from this seed. Combined with q1's Hamming $\le 9$, the odd ball through $11$ is empty. Logs: compute/q2/hamming11_log.txt, compute/q2/hamming11_sat_log.txt.

Ejection sample: $16$ restarts, $250000$ one-outs, $16672$ four-outs (best_gain=0), $8336$ five-outs (best_gain=1, size stayed $367$). Residue. compute/q2/ejection_log.txt.

Exact $8$-coset census of every $2$-dimensional subspace of $\mathbb F_7^5$: $140050$ subspaces (Gaussian $\binom{5}{2}_7$), $97240$ good ($V\cap\{-1,0,1\}^5=\{0\}$). Greedy and the capped exact search never found $8$ independent cosets. Every materialised $7$-pack had empty residual in the original graph (total $343$). First pass left $7040$ graphs at the node cap; a second Cayley search proved $5760$ of those have $\alpha<8$ and left $1280$ unknown. Residue, not a proof that no $8$-coset pack exists. No $392$-set. Logs: compute/q2/coset_exact_log.txt, compute/q2/coset_unknown_c_log.txt.

No $368$-set. No $\Theta(C_7)$ claim. $367^{1/5}\approx 3.25787<3.258805$.

2026-08-23 — replay

• Checkout grok/c7-shannon. Replayed python3 compute/verify_set.py compute/R367.txt --min-size 367 → size=367 unique=367, OK: independent in C7^{box5}. Same verifier on R_reconstructed.txt (367) and R361_sprime.txt (361) also OK. This is the Polak–Schrijver record, not a dent.

2026-08-23 — six shapes for a 368-set (q1)

Work backwards from the shape, not from another 3-out census of the same seed. Baumert–McEliece–Rodemich–Rumsey–Stanley–Taylor give $\alpha(C_7^{\boxtimes 4})\le 115$ (their lemma $\alpha(C_n^d)\le \alpha(C_n^{d-1})n/2$). Polak–Schrijver Table 1 already records that 108–115 interval.

The published 367-set meets every letter in every coordinate, with fiber sizes 50–55. It differs from the reconstructed pipeline set in two vertices (34035,64340 versus 24635,64246), which are two of Gao's eight private pairs. The eight single-blocker vertices of the 367-set are exactly the eight private neighbours $q_j$.

1. Three mutually non-adjacent slices. Support in one coordinate on a 3-set of circular distance $\ge 2$ (e.g. $\{0,2,4\}$). The three fibers do not interact, so the size is at most $3\alpha(C_7^{\boxtimes 4})\le 345<368$. This shape is impossible. Every 368-set uses at least four letters in every coordinate.

2. Four-letter support. Average fiber 92, each $\le 115$. A 4-subset of $C_7$ always contains an edge; the split $\{0,1,3,5\}$ has a single adjacent pair and two free fibers.

3. Punctured isolate core. Polak–Schrijver keep $|M|=327$ isolates of the folded 382-orbit and add $\alpha=40$ from a 71-vertex residual. Deleting $k$ isolates and re-solving the whole residual (not a 3-out of the finished 367-set) aims at $327-k+\alpha(\text{new residual})\ge 368$. The S' MIS extreme ($357+4=361$) already showed that eating conflict vertices spends the leftover; this shape is the intermediate punctures.

4. Eight cosets of a good 2-dimensional $\mathbb F_7$-code. A 2-dimensional $V$ with $V\cap\{-1,0,1\}^5=\{0\}$ has 49-point cosets. Eight independent points of the 343-vertex quotient give 392 vertices. Seven cosets are only 343.

5. A 5-dimensional fiber of a published higher-power set. Itty–Rosin–Carstensen–Reichman $\alpha(C_7^{\boxtimes 6})\ge 1120$: after dropping one coordinate, the largest 5-fiber has size 165 (replayed from their CC_6_7_1120.txt, 1120 unique, independent). Their 10th-power gadget $I=(B\times B)\cup A_H\cup A_V$ of size 134753 has largest 5-fiber 367 (left or right), never 368. Those two certificates do not hide a 368-set in a fiber.

6. A 367-set on the private-pair / translate plateau, plus one vertex. Few-flip SAT only killed odd Hamming distance $\le 9$ around one seed. The even 1-out-1-in moves are the eight private-pair swaps; 108 of the 256 subsets remain independent 367-sets, including the reconstructed list. None of those 108 has a 1-out-2-in with newly-free set of size $\le 16$. Separately, the union $A\cup(A+v)$ is bipartite aside from the intersection. Among all 16806 nonzero translations, 8422 are disjoint from $A$; none has an isolated vertex in the bipartite conflict graph, so there is no cheap "$A$ plus one point of $A+v$". An imperfect maximum matching would still give 368; that is a C check, not the old 3-out SAT.

2026-08-23 — q1 results, no 368

Worked the six shapes backwards. No independent set of size 368. Timeouts and samples are residue, not a lower bound. The 3-support obstruction and the two-translate matching census are complete for those shapes.

1. 3-support. Impossible: $3\cdot 115=345<368$. Recorded.

2. 4-support. search_4support.py: every 4-subset of letters, slice the 367-set and grow. Best grown size 309 on $\{0,2,4,5\}$. Greedy fibers of $\{3,5\}$ plus a $\{0,1\}$-slice: $81+81+81=243$. Not a 368-set. Residue: not an exact $\alpha$ on the 9604-vertex induced subgraphs.

3. Punctured isolate core. search_core_puncture.py. $k=0$ recovers $327+40=367$. All 327 single punctures: residual 72 or 73, $\alpha=41$, total 367. Deleting one isolate frees exactly one extra leftover vertex. $k=2$ was stopped after 19 trials (residue). Log: compute/q1/core_puncture_log.txt.

4. Eight cosets. First C quotient encoding was wrong (claimed 13 cosets, $13\cdot 49=637>401$, impossible). Replaced by neighbourhood-true coset ids. C enumeration of RREF generators: $\ge 25000$ good codes, greedy $\le 7$ (total 343). Verified Python sample of 120 random good 2-dimensional codes: best 7, leftover of every 7-pack empty, so those packs are maximal in the quotient. Not a proof that no 8-coset pack exists. Residue. Log: compute/q1/coset_sample_log.txt.

5. Higher-power fibers. Replayed. Itty 1120-set in dimension 6 is independent; max 5-fiber 165. Itty/Gao 10th-power gadget from R367: max 5-fiber 367. Those two certificates do not contain a 368-set as a fiber. compute/q1/profile_log.txt.

6. Plateau / translates / ejection. search_translates.c: all 16806 nonzero $v$, $8422$ disjoint and $8384$ intersecting, maximum matching always perfect, $\alpha(A\cup(A+v))=367$ for every $v$. No 368-set of this shape. search_plateau.c: 108 private-pair 367-sets, $7253388$ two-out packs, best_gain=0. search_ejection.c: 6 restarts, 250000 one-out steps each, 7500 sampled four-outs, global_best=367, four_out_best_gain=0. The translate census is complete for that shape; ejection is a sample (residue).

No $\Theta(C_7)$ claim. $367^{1/5}\approx 3.25787<3.258805$. A 368-set is still the finite bound that would move $\alpha(C_7^{\boxtimes 5})$.

2026-08-16

• Folder created. Grok 4.6 cloud agent launched.

2026-08-16 — reconstruction

• Read Polak–Schrijver IPL 2019 / arXiv:1808.07438. The 367-set is an adapted Reed–Solomon orbit in $C_{108,382}^{\boxtimes 5}$, not a linear $\mathbb F_7$-code (those max out at $7^3=343$).
• July 2026 papers (Itty et al. 2607.21517, Gao 2607.27869, Buys–Polak–Zuiddam 2607.29681) improve $\Theta(C_7)$ in dimensions 10 and 200. All three still treat 367 as the fifth-power record.
• Copied the published 367 words from Itty c7/R367.txt into compute/R367.txt.
• Verifier python compute/verify_set.py compute/R367.txt --min-size 367 → OK, 367 unique, independent.
• reconstruct_polak.py replays §3: fold unique 382, $|M|=327$, residual 71 verts / 85 edges, $\alpha=40$, $|R|=367$. Matches the paper. Reconstructed set differs from the published list in 2 vertices (two different 40-sets in the same residual). Both verify.

2026-08-16 — searches, no 368

• Direct geometric orbits. search_orbits.py over $n=300..600$: no pair with $k(n,5,q)\ge 2n/7$. Closest misses: $317/90\approx 3.522$ ($q=31$, one short of the homomorphism) and the published $382/108\approx 3.537$. Random non-geometric generators $t\cdot(1,a,b,c,d)$ for $n=368..400$, 80 samples each: no hit.
• S' MIS instead of isolates. Conflict component of the folded 382-set has 55 verts, 35 edges, $\alpha=30$. Isolates + that MIS gives $M=357$, residual 4, total 361 (R361_sprime.txt, verifies). Greedy on $S'$ shrinks the leftover that was worth 40.
• Local swaps from the 367-set. Free vertices: 0 (maximal). Exhaustive 1-out: none. Exhaustive 2-out: 67161 trials, none. Exhaustive 3-out: 8171255 trials, best_gain=0 in 419s. Sampled 4-out (3000) and two anneal restarts: stayed at 367.
• Linear 3-dim codes. 30 random good $\mathbb F_7^3$ subspaces: every residual empty, total stuck at 343. $V+\{-1,0,1\}^5=(\mathbb Z/7\mathbb Z)^5$.
• Fold-and-repair on near-miss orbits (317, 382, 309, 303, 301, 339, 367, 362). Best is the published shift on $n=382$, $q=7$, den=109: 327+40=367. Other 382-translations scored 357–360. Smaller $n$ collapsed to $M\sim 30$ and totals $\sim 250$.
• Shifts. Published $(40,123,40,123,40)$ recovered as the best completed score. Broader translation grid not finished (exact MIS on 70-vertex leftovers is slow). No 368 in the completed prefix.

2026-08-16 — steal from W(2,7)

• Sibling finished with no dent (best verified colouring length 3703). Methods stolen: CardEnc.atmost only; few-flip SAT around the published seed; treat a one-violation min-conflicts stall as a cage; CEGAR on a product template and stop if leftovers plateau.
• Scripts: search_fewflip_sat.py, search_minconflicts_cage.py, search_cegar_product.py. A 367-set is not a dent. No $\Theta(C_7)$ claim.
• Few-flip SAT. Cadical195 + CardEnc.atmost (kmtotalizer; at-least via negated at-most). Free vertices 0. Hamming $\le 9$ is UNSAT, including 4-out/5-in (4806 candidates, 67s). Hamming 11 with 5 removals timed out. Local cage, not a bound.
• Min-conflicts cage. 367 plus one extra, 8 trials: stalled at 787–846 adjacent pairs. Not one leftover; still a cage, not optimality.
• Product CEGAR. SAT built a 10-set in $C_7^{\boxtimes 2}$ and a 33-set in $C_7^{\boxtimes 3}$. Product has size 330 and empty residual. Incremental constraints have nothing to add; leftover count is already 0 and the size is 38 short. Stopped.

Result

No independent set of size $\ge 368$. A 367-set is not a dent. No certificate that 367 is maximum: Lovász still allows 401; few-flip SAT only kills a Hamming ball of radius 9 around one seed. Do not claim a new $\Theta(C_7)$. $367^{1/5}\approx 3.25787 < 3.258805$.