How a Fifth-Power Record Did Not Move

0. What was actually missing

The capacity number $\Theta(C_7)$ already moved in July 2026, but only through gadgets in the 10th and 200th strong powers. The fifth-power independence number is a different finite object: a subset of $(\mathbb Z/7\mathbb Z)^5$ in which every pair has circular distance $>1$ in some coordinate. The published maximum is still 367. The missing degree of freedom is one extra vertex, or a certificate that none exists.

A 368-set is already a new bound. Its fifth root is about 3.2596, which would also beat the Lean-verified 3.258805, but that comparison is not required to count the finite set.

1. Named false starts

Direct homomorphism. If some cyclic orbit of size $n\ge 368$ has min-max circular distance $k\ge 2n/7$, the floor map lands in $C_7$ and we are done. Exhaustive geometric search on $n=300..600$, and random generators on $n=368..400$, never cleared the threshold. The famous $382/108$ ratio is 3.537, still above 3.5. The closest miss was $n=317$, $q=31$, $k=90$ versus need 91 — and 317 is too small anyway.

Linear codes. A good 3-dimensional $\mathbb F_7$-subspace has size 343. Thirty random good codes had empty residual: you cannot add even one extra point. The 24-point gap from 343 to 367 is nonlinear. Eight cosets of a good $2$-dimensional code would have been $392$. Every good $2$-dimensional code is now decided: the $343$-vertex quotient has $\alpha\le 7$. Seven cosets are $343$ and leave no original-graph residual.

Maximum independent set of the folded orbit. Polak–Schrijver throw away every non-isolated image vertex (55 of them). Keeping an MIS of that conflict graph instead produces a larger core (357) but a residual of only 4, total 361. The 40-vertex leftover they kept is worth more than those 30 conflict vertices.

Small swaps. The 367-set is maximal (0 free vertices). Exhaustive 1-out, 2-out, and 3-out searches have gain 0. That matches the paper's 3-out/4-in report and extends it to a complete 3-out census (8,171,255 triples).

Few-flip SAT (stolen from W(2,7)). Cadical with library CardEnc.atmost refuted every odd Hamming distance $\le 9$ from the seed, so 4-out/5-in is also empty. Hamming $11$ and Hamming $13$ are now finished as well: every high-blocker $5$-out/$6$-in or $6$-out/$7$-in frees at most $14$ candidates with $\alpha\le 4$, and the leftover SAT instances on $\le 3$-blocker candidates are both UNSAT. That is still a ball around one seed, not $\alpha\le 367$.

One-dimensional codes. A good $1$-dimensional subspace has $7$-point cosets. Fifty-three independent cosets would be $371$. The $2680$ good generators collapse to $1156$ Cayley graphs on $\mathbb F_7^4$. Hoffman sits around $366$–$516$ and does not kill $53$. Local search reached $46$ cosets ($322$ vertices) with empty residual. Not a $368$-set.

Cyclic $5$-orbits. The coordinate cycle splits the graph into $3360$ orbits of size $5$. Seventy-four independent orbits would be $370$. Greedy packing stopped at $52$ orbits ($260$).

Four-letter local search. Growing the $367$-set inside a $4$-letter slice reached $309$. That number is irrelevant: a $4$-support set cannot reach $368$ at all.

Product template. $10\times 33=330$ is already a maximal independent set. CEGAR cannot grow it. A template can be leftover-free and still 38 short.

Missing a letter. Baumert et al. give $\alpha(C_7^{\boxtimes 4})\le 115$. An adjacent pair of letters in one coordinate induces $K_2\boxtimes C_7^{\boxtimes 4}$ and contributes at most $115$, not $230$, because the two fibers project to one independent set in the fourth power. Every $1$--$6$ letter support therefore has size at most $345$. A $368$-set uses all seven letters in every coordinate.

Two translates of the 367-set. For every nonzero $v\in(\mathbb Z/7\mathbb Z)^5$, the bipartite matching on $A\setminus(A+v)$ versus $(A+v)\setminus A$ is perfect, so $\alpha(A\cup(A+v))=367$. That shape is closed.

A 5-fiber of the published 6th- or 10th-power sets. Itty's 1120-set in dimension 6 has max 5-fiber 165. The 10th-power gadget built from the 367-set has max 5-fiber 367.

2. The useful failure

The isolate/residual split is a real tradeoff, not a sloppy deletion. The 327 isolates leave a 71-vertex, 85-edge residual whose independence number is 40. Eating the conflict vertices first spends that leftover. Any attempt to beat 367 has to change the 327-core, not just re-pack the 71. Puncturing one isolate and re-solving the residual always returned $\alpha=41$, total 367: the extra leftover vertex exactly replaces the one that was deleted.

Orbit ratios also taught a negative: among geometric progressions, $n=382$, $q=7$ is still the closest approach to $7/2$ in the range that could matter. Folding the nearer-ratio $n=317$ orbit into seven symbols explodes the residual and yields totals around 250.

Hoffman on the $8$-coset Cayley graphs was the other useful miss. The graphs are regular of degree $122$–$228$, and the smallest eigenvalue is negative enough that the Hoffman number sits around $33$. A spectral bound that looks like it should kill $\alpha<8$ does not even touch $8$. The leftovers were a node cap, not a hard family.

3. The click

The pair-fiber projection. Two adjacent letters look like they should give $115+115$, but $(0,x)$ is adjacent to every $(1,y)$ in the closed neighbourhood of $x$, so the two fibers cannot both be large. They fold to a single independent set in $C_7^{\boxtimes 4}$. Once that is visible, missing even one letter in one coordinate is fatal for $368$. The published translation $(40,123,40,123,40)$ before the fold $\lfloor 2i/109\rfloor$ remains the best completed construction. Other translations of the same orbit scored 357–360.

4. The argument, in the order it was found

Seed first. The Itty ancillary R367.txt is the Polak–Schrijver appendix. A pairwise verifier checks circular distance $>1$ in some coordinate; it accepts the 367-set. Reconstruction from the circular orbit independently reproduces $|M|=327$, residual $71/85$, $\alpha=40$.

Then the homomorphism search, the S' MIS, the swap census, the linear residuals, and the fold/shift reruns. None produced 368.

The pair-fiber bound came later, while trying to finish the $4$-support search that had been left as a local grow. Hamming $11$ was the other leftover from the few-flip ball; Hamming $13$ was the leftover after that. The $8$-coset census looked unfinished because $1280$ graphs hit an $80000$-node cap. Deduplicating the connection sets dropped $97240$ codes to $9584$ Cayley graphs; a complete search on those unique graphs found no $8$-pack. The $1$-dimensional quotients are larger ($2401$ vertices) and the Hoffman number never approaches $53$.

5. Computer search

• compute/R367.txt — published 367-set, verifier OK
• compute/R_reconstructed.txt — pipeline 367-set, verifier OK, differs in 2 vertices
• compute/R361_sprime.txt — S' MIS construction, size 361, verifier OK
• compute/verify_set.py — pairwise checker
• logs: orbit_search_wide.txt, three_out_log.txt, fold_near.txt, linear_probe.txt, local_search_log.txt, fewflip_sat_log.txt, minconflicts_cage.txt, cegar_product_log.txt
• compute/q1/ — six-shape search: translate matching (complete, $\alpha=367$ for every $v$), private-pair 2-out plateau (best_gain=0 on 108 seeds), isolate-core $k=1$ census (327/327 at total 367), 4-support local best 309, verified 120 good 2-dimensional codes with at most 7 cosets, higher-power fiber replay (165 and 367)
• compute/q2/bound_support.py — $126$ supports of size $1$--$6$, all $\le 345$; pair projections of the $367$-set independent
• compute/q2/hamming11_log.txt, hamming11_sat_log.txt — Hamming $11$ empty
• compute/q2/coset_exact_log.txt — $140050$ subspaces, $97240$ good, no $8$-pack; $1280$ leftover graphs after a second pass (coset_unknown_c_log.txt)
• compute/q3/coset_finish_log.txt — $9584$ unique connection sets, clique cover $359$, exact search $9225$, leftover $0$
• compute/q3/unique_sample_log.txt — Python RREF count and Cadical sample of $159$ unique graphs, all UNSAT
• compute/q3/hamming13_log.txt — Hamming $13$ cases A+B empty (best_mis=4)
• compute/q4/hamming13_four_log.txt, hamming13_sat_log.txt — Hamming $13$ case C empty, leftover SAT UNSAT
• compute/q4/onedim_log.txt — $2680$ good $1$-dimensional codes, $1156$ unique, best pack $43$ then $46$
• compute/q4/cyclic_log.txt — cyclic $5$-orbit greedy $\le 52$
• compute/q4/hamming15_log.txt — Hamming $15$ cases A+B empty (best_mis=5)

6. What is proved vs still open

The 367-set is independent. $\alpha(C_7^{\boxtimes 5})\ge 367$ is old. This search did not find 368 and did not prove 367 is maximum. An independent set that misses a letter in any coordinate has size at most $345$. Hamming distance $13$ from the published $367$-set contains no $368$-set. The union of the published set with any translate has independence number 367. No good $2$-dimensional $\mathbb F_7$-code has eight independent cosets. A $1$-dimensional good-code pack was not found at size $53$; Hoffman does not forbid it. Cyclic $5$-orbit SAT for $74$ orbits and negation-pair SAT for $184$ pairs are unfinished. Hamming $15$ is empty when an added vertex has $6$ or $7$ blockers. Lovász still gives $\alpha\le 401$. $\Theta(C_7)$ is not claimed from this folder.