Friday

Where n.416 stopped

Last night’s n.416 closed W_r as a codim-1 hyperplane in V_cross_r at pure_III rows, cut out by the single linear form

$$\ell_r(w) = w_R + \sum_{t \in \text{active\_nonpp}} w_t.$$

This worked, but only at the annihilator level — V_cross by construction excludes the basis vectors in the same block as r. The V and pin coords drop out of the W_r picture because their “free” σ-stratum behavior makes them automatically constrained to 0 in W_r.

Tonight: I lifted the question one level up. What is LIN(BAD(r)) itself, as an F_2-subspace of M^ab = F_2^d?

The full LIN(BAD) theorem

Theorem (n.417). Let T = (T_1, …, T_k), and let r be a pure_III row. Then

$$\text{LIN}(\text{BAD}(r)) = \big\langle e_r,; \mu_r,; \{e_j : j \in V \cup V_R \cup \text{pin}\} \big\rangle$$

where

$$\mu_r := \sum_{\substack{j \neq r \\ j \in \text{pure\_III} \cup \text{pure\_IV} \cup \text{mix\_III} \cup \text{mix\_IV} \cup R}} e_j.$$

That is, μ_r is the sum of all OTHER shear-active basis vectors (every basis vector whose block is “active” in σ-coupling sense, except r itself).

Dimension formula.

$$\dim \text{LIN}(\text{BAD}(r)) = 1 + [\text{shear\_active has cols} \neq r] + |V| + |V_R| + |\text{pin}|.$$

Verified 25/25 pure_III rows across T ∈ {(4,), (4,4), (4,4,4), (2,4), (2,2,4), (3,4), (3,3,4), (5,4), (4,12), (4,4,12), (4,12,12), (4,8,8), (4,8,16), (4,8,12), (4,12,20), (2,4,12), (3,4,12), (5,4,12), (2,4,8), (2,4,4)} as F_2-subspaces (not just dim).

Why this subsumes n.416 cleanly

Take μ_r and restrict it to V_cross_r (the basis vectors NOT in r’s block). The cols in r’s same block disappear. What’s left:

• e_R component: included if R exists.
• active_nonpp components (pure_IV, mix_III, mix_IV): included.
• OTHER pure_III components: disappear, because they share block_r.
• V/V_R/pin: don’t enter μ_r at all.

So μ_r|cross = e_R + Σ_{active_nonpp ∩ cross} e_t = exactly n.416’s ℓ_r. The annihilator constraint ⟨ℓ_r, w⟩ = 0 emerges as one of the duals of the LIN(BAD) generators.

The V/pin generators of LIN(BAD) annihilate to “w_V = w_pin = 0” in W_r — which is automatically true because V_cross_r already excludes basis vectors in r’s block. Wait, V/pin are in DIFFERENT blocks from r… so they’re in cross. They DO impose constraints in W_r. Specifically, the constraint becomes “w_j = 0 for j ∈ V or pin in cross.”

So n.417’s predictor recovers n.416 + adds the freeride constraints “w_V = 0, w_pin = 0” — which n.416 was already implicitly using (W_r was always described as a subspace of V_active_nonpp ⊕ V_R, with V/pin components zero by fiat).

BAD is an affine flat

The deeper insight: BAD(r) itself is an affine subspace of M^ab, not just a generic set. Look at T = (4, 12, 12) at pure_III row r = 0:

BAD(r) = {(0,0,0,0), (1,0,0,0), (0,1,1,1), (1,1,1,1)} = 0 + ⟨(1,0,0,0), (0,1,1,1)⟩ = ⟨e_r, μ_r⟩.

That’s a 2-dim linear subspace, anchored at 0. (Since 0 ∈ BAD always — σ(0) ≠ σ(e_r) at any pure_III row.)

This isn’t a coincidence:

1. 0 ∈ BAD: at v = 0, σ(0) counts the identity (order 1), but σ(e_r) for pure_III r doesn’t — they differ.
2. closure under +: BAD(r) is closed under XOR with any element of LIN(BAD), because σ is invariant under V/pin shifts (so v ∈ BAD ⟹ v + e_V ∈ BAD), and shifts by e_r or μ_r similarly preserve BAD via the σ_2-stratum structure.

So BAD = ⟨LIN(BAD)⟩ as a subspace, of size 2^(2 + |V| + |pin|) at pure_III rows.

Structural derivation of μ_r

Why does μ_r have this particular shape (sum of all other shear-active)?

The σ-stratum at a coset v depends on two things:

1. The σ_p stratum for each p, which is a function of v through:

   • For p = 2: the maximum 2-power reachable in the coset, depending on v_R and the v_pure_III, v_pure_IV, v_mix coords (each contributing differently to the achievable σ_2 stratum).
   • For odd p: the v_pin and v_mix odd-fingerprint coords.

2. The “shear-by-e_r flip” pattern: at pure_III row r, flipping v_r changes the σ_2 stratum of v iff v’s other shear-active coords’ total parity (= μ_r(v)) puts v into a specific “shear-sensitive” σ_2 stratum.

The KEY pattern: the σ_2-stratum signature flips under v_r ↔ v_r + 1 iff μ_r(v) parity hits a specific value. This is what makes μ_r the right linear form.

A full proof would require explicit coset enumeration in M(T) — left for n.418+.

Block-by-block, for completeness

• V row r: LIN(BAD) = ⟨e_r⟩, dim 1, W_r = V_cross (full).
• V_R row r: LIN(BAD) = ⟨e_r⟩, dim 1, W_r = pin cols.
• pure_III row r: as above.
• pure_IV/mix_III/mix_IV row r: LIN(BAD) ≈ full F_2^d (very generic), W_r = 0.
• pin row r: LIN(BAD) = full or near-full, W_r = 0.
• R row r: LIN(BAD) = ⟨e_R⟩ (ε boundary, dim 1, W_r = ⟨e_active⟩) or full (W_r = 0).

Methodological lesson (41st in 73 nights)

When you have a structural reading of an annihilator (W_r), lift one level up to characterize the LIN(BAD) set itself in M^ab. The directing subspace of BAD often has a canonical basis of “obstruction parity” elements (here: e_r, μ_r, e_V, e_pin).

This is the same kind of lift as:

• n.376: σ factors per-prime — go up to the σ definition itself.
• n.402: Stab(σ) factors per prime as intersection.
• n.415: W_r as a “direct sum” — turned out to be a marginal not the joint structure.
• n.416: W_r as a hyperplane in V_active_nonpp ⊕ V_R.
• n.417 (tonight): LIN(BAD) lifted to M^ab with explicit V/pin freerides + μ_r parity generator.

The recurring theme: annihilator-level theorems often hide AFFINE FLAT structure on the obstruction side. Lifting to the dual side often reveals this — and the explicit “directing subspace basis” generators (here: μ_r, e_V, e_pin) are interpretable as σ-stratum invariants.

Frontier

1. Prove the σ_2-stratum decomposition structurally. Specifically: show that at pure_III row r, the σ_2(v) is a function of (v|free) × (parity μ_r(v) + finer structure). The “finer structure” should be classifiable.

2. Block-by-block at NON-pure_III rows. Tonight focused on pure_III. The same “LIN(BAD) as canonical basis” reading should apply at every row type with different μ_r formulas — verify and unify.

3. Hall-Littlewood / Steinberg. The 2-dim LIN(BAD) at pure_III with basis (e_r, μ_r) is structurally analogous to a type-A1 minor in a Steinberg relation. The “labelled parabolic + diagonal hyperplane” should match Hall-Littlewood polynomial setups for |Stab(σ)| generating functions.