Result summary

$\ell_2(10,2) \le 50$ , with the $r=10$ seed propagated through Construction QM $_2^2$ to a new $R=2$ family of asymptotic covering density $2601/2048$ .

The mathematics is in NOTE.md. This file is the operational summary: what is here, how to run it, and what happened when it ran.

Run it

cd problems/covering/result
./run_all.sh          # or: make

Needs python3 and rustc, nothing else. No network, no packages, no RNG, under two seconds. Exit 0 means every claim below re-derived from the committed matrix text.

Status: 74 checks passed, exit 0. Output is byte-identical across runs (confirmed by running twice and cmp).

Layout

NOTE.md              the note a referee should read
RESULT.md            this file
PRIORITY.md          literature checks that must happen before posting
VERIFY_PLAN.md       what the verifiers were required to do, written before them
note.tex             arXiv-ready draft (article class, no exotic packages)
run_all.sh           the whole pipeline; asserts every headline number by name
Makefile             convenience wrapper

data/
  H_r10_n50.txt      the result; byte-identical to ../compute/H_r10_n50.txt
  partition_p10.json (2,0)-partition into 10 blocks, with a schema field
  kr_r10_n51.txt     Kaikkonen-Rosendahl 2003 baseline, rebuilt from the hex
  H_r18_n815.txt     QM_2^2 at m=4                  } generated
  H_r20_n1631.txt    QM_2^2 at m=5                  } by the builder
  family_table.md    reachable r <= 64, generated   }
  family_table.json  same, machine-readable         }
  facts/*.tsv        committed fact snapshots; run_all.sh cmp's against them

verify/
  verify.py          verifier #1, Python, pair-driven
  verify.rs          verifier #2, Rust, syndrome-driven, no crates
  build_propagation.py  QM_2^2 from Theorem 4.1, plus the KR reconstruction
  check_note.py      fails the run if NOTE.md quotes an uncomputed number

Everything under data/ except H_r10_n50.txt and partition_p10.json is regenerated by run_all.sh into a temp dir and byte-compared against the committed copy, so a stale committed artifact is a hard failure rather than a silent pass.

What was verified

claim	value	how
$r=10$ shape, rank, columns	$10\times50$ , rank 10, 50 distinct nonzero	both verifiers
$r=10$ coverage	1024 / 1024	exhaustive, both directions
$r=10$ density	$319/256 = 1.24609375$	exact rational, two arithmetics
$r=10$ multiplicity histogram	`{1:859, 2:129, 4:24, 5:9, 6:3}`	both verifiers
syndromes needing a pair	973	both verifiers
pair-only histogram over those	`{1:821, 2:123, 4:19, 5:8, 6:2}`	both verifiers
forced-split pairs	821	both verifiers
$(2,0)$ -partition	valid, $p(H) = 10$	separate checker, both verifiers
minimality	best single deletion leaves 9 uncovered	full recount per deletion
minimum distance / triples	$d = 3$ , exactly 10 dependent triples	both verifiers
triple across three blocks	$(491,734,821)$ in blocks 6, 4, 9	both verifiers
KR baseline	51 distinct columns, 1024/1024, $1327/1024$	rebuilt from paper hex
$r=18$ , $m=4$	$n=815$ , 262144 / 262144, $332521/262144$	exhaustive
$r=20$ , $m=5$	$n=1631$ , 1048576 / 1048576, $1330897/1048576$	exhaustive
legal $m$ from (4.2)	exactly $\{4, 5\}$	asserted, not assumed
family closed form	$n = 51\cdot2^{r/2-5}-1$	derived and cross-checked
asymptotic density	$2601/2048 \approx 1.27002$	derived

Full assertion list with named failure messages: run_all.sh.

Numbers that disagreed with the brief

The instructions said the code wins. One number lost.

Asymptotic density decimal. The brief gave $51^2/2^{11} = 2601/2048 \approx 1.26999$ . The fraction is right; the decimal is not. $2601/2048 = 1.27001953125$ , so $\approx 1.27002$ . Used throughout as 1.27002. This is a rounding slip in the brief, not a mathematical disagreement, and it does not affect any bound: the finite densities approach $2601/2048$ from below, so $f(2) \le 2601/2048$ stands either way.

Everything else in the brief matched exactly: all four histograms, 973, 821, the minimality figure of 9, the 10 dependent triples, $(491,734,821)$ in blocks 6/4/9, both propagated lengths, both propagated densities, and the legal- $m$ set $\{4,5\}$ .

Alternate witnesses

data/alt/ holds a second pair of QM $_2^2$ matrices from a different indicator allocation. The assignment of $\beta_j$ is a free choice, so two allocators give two codes.

$r=18$ and $r=20$ : different, and both verify. Of the 815 columns at $r=18$ , 720 are not in the primary matrix; at $r=20$ , 1457 of 1631. Both pass the full exhaustive check, with the same rank and the same densities $332521/262144$ and $1330897/1048576$ .
A third, quarantined oracle was run once after both verifiers were green and agreed on every headline number. It is not in the tree.

What is not done

Named plainly, because these are the things a referee will ask about:

Partitions past $m=5$ are inherited, not computed. $r=18$ and $r=20$ are exhaustively verified. The rest of the family rests on the paper's $p(H_C) \le 2^{m+1}+1$ from (4.4). Computing those partitions explicitly is the single highest-value follow-up.
QM $_3^2$ and QM $_5^2$ are not implemented, so $r \in \{12,14,16,22,24,26,28,42,44\}$ get no claim from this seed even though §5 of NOTE.md supplies the dependent triple that QM $_5^2$ needs at $r=28$ .
No lower bound work. $\ell_2(10,2) \ge 45$ is just sphere-covering.
Priority is unresolved. See PRIORITY.md. Nothing should be posted until a human has done those checks and heard back from Davydov, Marcugini and Pambianco.