Module steenrod 

The Steenrod algebra acting on H*(BZ/2; Z/2) = Z/2[x] — the deepest layer of cohomology operations, and a place where homotopy theory turns out to be number theory.

The total Steenrod square Sq = Σᵢ Sqⁱ is the ring endomorphism of Z/2[x] determined by Sq(x) = x + x² (Sq⁰x = x, Sq¹x = x² since |x| = 1, Sqⁱx = 0 for i > 1) extended multiplicatively by the Cartan formula Sq(ab) = Sq(a)Sq(b). Computing it is then pure polynomial arithmetic over Z/2, and the answer is striking:

  Sqⁱ(xᵏ) = coefficient of x^{k+i} in (x + x²)ᵏ = C(k, i) mod 2

— the binomial coefficients mod 2, i.e. Lucas’ theorem (C(k,i) ≡ 1 ⟺ i AND k = i bit-wise). The top square Sqᵏ(xᵏ) = x^{2k} agrees with the cup-square (crate::postnikov::cup_square); the whole infinite family of operations is governed by one binomial law.

Enums

AdamsFact

One machine-derived fact about the Adams E₂ chart of the sphere. Every variant is computed from the minimal resolution and its chain-level diagonal — none is asserted.

Functions

act_monomial

The action of a Steenrod monomial on a class in H*(BZ/2) = Z/2[x] (apply the squares right-to-left).

adams_one_line

dim Ext^{1,t}_𝒜(Z/2, Z/2) — the first line of the Adams E₂ page = the indecomposables of the Steenrod algebra in degree t. Equals dim 𝒜_t − dim(𝒜⁺·𝒜⁺)_t. By Milnor it is 1 exactly when t is a power of 2 (the h_i = Sq^{2ⁱ}), and 0 otherwise.

adams_two_line

dim Ext^{2,t}_𝒜(Z/2, Z/2) — the second line of the Adams E₂ page: the minimal generators of ker(d₁) in degree t, i.e. dim ker(d₁)_t − dim(𝒜⁺·ker(d₁))_t. The relations among the Sq^{2ⁱ}.

adem_reduce

Reduce a Steenrod monomial to its Z/2 sum of admissible monomials via the Adem relations — the canonical form in the Steenrod algebra. A HashSet models the Z/2 linear combination (present = coefficient 1; re-insertion cancels).

admissibles_of_degree

The admissible monomials of total degree t — a basis of the Steenrod algebra 𝒜_t.

apply_sq

Apply a single Sqⁱ to a class in H*(BZ/2; Z/2) = Z/2[x]: Sqⁱ(Σ aₖ xᵏ) = Σ aₖ C(k,i) x^{k+i}.

ext

dim Ext^{s,t}_𝒜(Z/2, Z/2) — the full Adams E₂ page from the minimal resolution.

harvest_adams_facts

The self-igniting fact engine: from a single minimal resolution it auto-collects everything the symmetry breaking nets — the chart dimensions, the ring products (multiplying named classes by the indecomposables to a fixpoint, so ring relations surface automatically when two product-paths collide on one generator), the Sq⁰ doubling across the chart, and the exact 2-local stable stems from the h₀-towers. Returns the deduplicated, sorted catalog. Bounded by (max_s, max_t); the doubling reaches through filtration 2 (cup-homotopies Δ₀..Δ₂).

harvest_secondary_facts

Auto-collect the primary chart from a resolution by pure symmetry-breaking — no proving, no chain-level diagonal, so it is cheap and runs to a wide range. Gathers chart dimensions, ring products and the relations that surface by collision, the un-auto-able (secondary/Massey) generators the product sweep cannot name, and the exact 2-local stable stems from the h₀-towers. The sorted/deduped catalog.

is_admissible

Is a monomial Sq^{i₁}…Sq^{iₖ} admissible? (i_j ≥ 2 i_{j+1} for all adjacent pairs.)

is_decomposable_sq

Is Sqⁿ decomposable — a Z/2 combination of products SqᵃSqᵇ (a, b ≥ 1) — in the Steenrod algebra? Decided by reducing every such product to the admissible basis (Adem) and testing whether Sqⁿ lies in their span.

monomial

The monomial xᵏ.

sq

Sqⁱ(xᵏ) on H*(BZ/2; Z/2) — the coefficient of x^{k+i} in Sq(xᵏ) = (x + x²)ᵏ.

total_square

The total Steenrod square Sq, as the ring endomorphism of Z/2[x] sending x ↦ x + x² (Cartan formula). Returns Sq(p) as a polynomial.

yoneda_product

The Yoneda product Ext^{a_s,t_a} ⊗ Ext^{b_s,t_b} → Ext^{a_s+b_s, t_a+t_b} on the cohomology of the Steenrod algebra, computed from the resolution itself. The cocycle dual to generator (a_s, a_gi) is lifted to a comparison chain map φ_•: C_{a_s+•} → C_• (ε·φ_0 = dual, ∂φ_k = φ_{k-1}∂, each step a Z/2 boundary solve), and the product is dual_b ∘ φ_{b_s} — read off as the (b_gi, identity) coefficient. The comparison map is the diagonal X → X×X dualized onto the resolution: the geometric symmetry made algebra. Returns the product’s Z/2 support among the C_{a_s+b_s} generators.