q9 — trajectory of the r=10 record, and exact quotient-block replacement

Everything here is new in quest q9 (2026-08-21). result/ is read only. No search from q2/q4 (SA, guided, lifted, k-swap, invariant orbit DFS) is rerun.

1. The record, recovered and re-verified

build_kr51.py reconstructs the Kaikkonen–Rosendahl 51-column seed from the hexadecimal listing reprinted as display (4.9) of arXiv:2511.02542v1 (their Theorem 4.3, citing Kaikkonen–Rosendahl, New covering codes from an ADS-like construction, IEEE Trans. Inform. Theory 49(7) 1809–1812, 2003, p. 1812). Output H_r10_n51_KR.txt: rank 10, 51 distinct nonzero columns, 1024/1024. RESEARCH.md had recorded this listing as not recovered; it is recovered now.

trajectory.py re-derives the documented upper-bound trajectory at r=10 from the cited formulas and from the explicit matrices in this repository:

The volume bound only gives n ≥ 45, so nothing in the counting excludes 49.

2. What the trajectory actually says

profiles.py computes, for each of the 174251 two-dimensional quotients q : F_2^10 → F_2^2, the block profile (|A|;|B|,|C|,|D|) of a column set, where A = S ∩ ker q and B, C, D are the three nonzero fibres.

• Since 2003 the record has not been a lift. φ(10)=53 is exactly the r=8 record 26 pushed up by the even-r doubling. The 51-set is not: no quotient of the 51-set, and no quotient of the certified 50-set, has its kernel block A covering ker q. (Both do have quotients with |A| ≥ 25, which is large enough by cardinality — they just are not coverings.) The 2003 gain of −2 and the 2026 gain of −1 both left the lift family behind.
• The record is quotient-flat. Profiles concentrate near n/4 in every direction: |A| ranges only over 3..27 (51-set) and 3..26 (50-set) across all 174251 quotients, against a mean of n/4 ≈ 12.5.
• q4 separately exhausted 79 subgroup classes of GL(10,2) at n=49 with zero invariant coverings, including every C_7×C_7 class.

Flat, asymmetric, not a lift, and (q4) provably ≥6-swap-deep at the 7-hole floor. That is a description of an object no small or symmetric move will improve. The move class left over is a large exact rearrangement along a quotient, which is what §3 does.

3. The construction family: block replacement

Fix a quotient with kernel V and coset representatives t00=0, t01, t10, t11 = t01+t10 (this choice makes the twist vanish). Pulling every block back into V, covering radius ≤ 2 is equivalent to four conditions inside V:

(00)  {0} ∪ A ∪ Δ(A) ∪ Δ(B) ∪ Δ(C) ∪ Δ(D) = V        Δ(X) = {x+x' : x ≠ x'}
(01)  (A ∪ {0}) + B  ∪  C + D = V
(10)  (A ∪ {0}) + C  ∪  B + D = V
(11)  (A ∪ {0}) + D  ∪  B + C = V

verify_blocks.py checks this equivalence both ways against a direct syndrome sweep, on the 51-set, the 50-set and both 7-hole residues, matching the exact hole sets and not merely their counts.

Now fix A, B, C and ask for D. Every condition involving D becomes either a hitting-set constraint or a pair constraint:

u ∉ (A⁺+B) ⟹ D ∩ (u+C)  ≠ ∅
u ∉ (A⁺+C) ⟹ D ∩ (u+B)  ≠ ∅
u ∉ (B +C) ⟹ D ∩ (u+A⁺) ≠ ∅
h ∉ {0} ∪ A ∪ Δ(A) ∪ Δ(B) ∪ Δ(C) ⟹ h ∈ Δ(D)

so "is there any block of size ≤ k completing A, B, C to a covering?" is a finite exact question. block_solve.c decides it by constraint-directed DFS with sound sibling exclusion on the hitting branches (where the branches really are "d ∈ D" and so partition the search), non-excluding branching on the pair constraints (where the branches overlap and exclusion would be unsound), and counting prunes on all four families.

Two modes:

• --shrink replace a block of size m by one of size ≤ m−1: n → n−1. Run on the certified 50-set this asks directly for a 49; run on the 51-set it asks for a 50.
• --resolve replace a block of size m by any block of size ≤ m: same n, and on a 7-hole residue it asks for a 49-covering.

This is not a k-swap. A block has up to 20 columns and they are all re-chosen simultaneously and exactly. q4 proved the 7-hole optima admit no swap of ≤ 5 columns; the planted controls below reconstruct a 14-column block from scratch.

4. Controls

• Encoding, positive: --resolve on the certified 50-set returns a valid 50-set on the first instance, re-verified 1024/1024 by an independent flat sweep in the same binary.
• Reduction, both directions: verify_blocks.py, 160 quotient instances over four column sets, exact hole sets agree.
• Search depth, planted: plant.py-style controls replace an entire block of the 50-set by uniformly random elements of V and ask the solver to reconstruct one. 7/7 recovered, block sizes 8, 8, 10, 11, 12, 13, 14, each re-verified 1024/1024. A planted needle at depth 14 is found.
• Independent oracle: --selftest k cross-checks the DFS verdict against brute-force enumeration of every candidate block of size ≤ k, scored only by the flat covering test, sharing no code with the constraint encoding.

Node cap per instance is reported as capped; a capped instance is unknown, not a negative. Blocks larger than --maxblock are reported as skipped. Nothing is silently dropped.

5. Sweep results (2026-08-21)

All numbers over all 174251 quotients x 3 non-kernel blocks = 522753 instances. capped = node cap hit, which is unknown and not a negative.

The last row is the sharp one: narrow blocks are decided 97.2% of the time. The KR row is calibration -- the historical 51 -> 50 step is not a single-block shrink, so this move class does not contain what annealing did in q1.

6. Replay

gcc -O2 -o block_solve block_solve.c
python3 build_kr51.py H_r10_n51_KR.txt
python3 trajectory.py
python3 profiles.py H_r10_n51_KR.txt ../H_r10_n50.txt ../q4/best_sa_config.cols
python3 verify_blocks.py ../H_r10_n50.txt 40
./block_solve --input ../H_r10_n50.txt --shrink --all --maxblock 20 \
              --nodes 2000000 --shard 0/6