Method and checks

The lambda algebra

The conventions are those of S. O. Ivanov, R. Mikhailov and J. Wu, arXiv:1506.00952, §2, at $p=3$: $\deg\lambda_i=4i-1$ ($i\ge1$), $\deg\mu_j=4j$ ($j\ge0$), $a(k,j)=(-1)^{j+1}\binom{2(k-j)-1}{j}$, $b(k,j)=(-1)^j\binom{2(k-j)}{j}$. A word is admissible if the index following $\lambda_i$ is $\le 3i-1$ and the index following $\mu_i$ is $\le 3i$. $\Lambda(1)$ consists of the admissible words with first index $\le1$.

The paper has $N(k)=k-[\tfrac{k+1}{p}]$. We read the brackets as the ceiling: with the floor, the relation for $k=1$ would contain its own left-hand side. Beyond the ceiling all the binomial coefficients vanish anyway. $N'(k)=k-\lceil k/p\rceil$.

The differential is a derivation with the Koszul sign, $\partial(xy)=\partial x\cdot y+(-1)^{|x|}x\,\partial y$. Inadmissible products are rewritten in the admissible basis by the relations, recursively.

The metalibrary

The admissible words in $\lambda_i,\mu_i$ ($i\ge1$) that begin with $\lambda_1$ or $\mu_1$ and end with a $\lambda$. These are the words of $\Lambda(1)$ without $\mu_0$ that end with a $\lambda$. The words containing $\mu_0$ are exactly the words ending in a run of $\mu_0$'s; they form the subcomplex $\Lambda(1)\mu_0$. The words ending in a $\lambda$ form a subcomplex of the quotient $\Lambda(1)/\Lambda(1)\mu_0$; this was checked on every word. The cell of a word: $n-1=2+4\sum(\text{indices})-\#\lambda$; the colour is the number of $\mu$'s.

Checks

• $\partial^2=0$ and $\partial\Lambda(1)\subseteq\Lambda(1)$ on all 2.2 million admissible words up to stem 43.
• $H(\Lambda(1))$ agrees with the unstable $\operatorname{Ext}$ of the sphere $S^3$ at $p=3$ computed by the independent program sseq/ext (resolve_unstable, module S_3[3]) in every cell (stem, $s$) up to stem 43.
• For $n>3$ the homology of the metalibrary equals $H(\Lambda(1))$ cell by cell in $(n,s,\text{colour})$ (up to $n=46$) and equals $\operatorname{Ext}$ from sseq in $(n,s)$ (up to $n=75$). Discarding $\mu_0$ removes only the tower over $\pi_3=\mathbb Z$.
• Two independent implementations of the enumeration, in Python and in Rust, give byte-for-byte the same result up to $n=56$.

The colour is the number of Bocksteins

For odd $p$ the Steenrod algebra carries an extra grading: the number of $Q$'s in an element $Q(E)P(R)$ of the Milnor basis. All the relations preserve it, and the minimal resolution in sseq is automatically homogeneous with respect to it, so $\operatorname{Ext}$ acquires a third grading $e$ (the $Q$-weight). By degree, the letters correspond as $\lambda_i\leftrightarrow P^i$ (weight 0), $\mu_i\leftrightarrow\beta P^i$ (weight 1).

• Colour $=e$ in all 281 cells $(n,s,\text{colour})$ with $4\le n\le75$, where the colours are known from a direct computation in the lambda algebra.
• Beyond $\pi_{75}$: in all 180 cells (67 of them nonzero) up to $\pi_{85}$ computed by enumerating the blocks of the metalibrary.
• Therefore the coloured $E^2$ on this site up to $n=150$ is taken from the $Q$-weight (2 s of computation); beyond $\pi_{85}$ the equality “colour $=e$” has not been checked independently, but it fails nowhere it has been checked.

The $Q$-weight is computed by the example qgrade_dims in the fork ispacex/sseq (branch math-54-qgrade).

Limits

• The rapacious rule is final up to $n=149$; Toda's 2003 table goes up to $n=82$, so the comparison with it goes up to $n=82$.
• Toda 2003 $\le E_2$ for all $n\le82$.

The code is in our working repository; we will send it on request.