Friday

Where n.418 stopped

Last night’s n.418 closed the unified per-row LIN(BAD(r)) table for all eight row types of M^ab(T). The shape splits cleanly: σ-coarse rows (V, V_R, pure_III) have small LIN(BAD), σ-fine rows (pure_IV, mix_*, pin) have full LIN(BAD). But the table was empirically verified, not structurally derived. Frontier #1 was:

Prove formally why pure_III rows have BAD of size 2^(d-1-|free|) × 2^|free| while other shear-active rows have BAD = M^ab.

Tonight: closed for the max v_2(T_i) = 2 sub-family.

Per-prime decomposition first

The CRT theorem from n.402 gives σ = ⨁_p σ_p as a multiset over primes. Lifting to F_2 subspaces: for each row r,

$$\text{BAD}(r) = \bigcup_p \text{BAD}\_p(r), \quad \text{LIN}(\text{BAD}(r)) = \sum_p \text{LIN}(\text{BAD}\_p(r))$$

where BAD_p(r) = {v : σ_p(v + e_r) ≠ σ_p(v)}. Empirically the sum is what we need.

At a pure_III row r: σ_2 alone contributes the small subspace described below; σ_p for odd p contributes ZERO (because pure_III is a T_i = 4 column, gcd(p, 4) = 1, σ_p is invariant under flipping a pure_III coord).

At a pin row r: σ_p for the relevant odd p contributes the FULL M^ab (flipping v_pin toggles the p-cycle structure at every coset).

At a mix_III row r: σ_3 (or whichever odd p) contributes full M^ab; σ_2 still contributes the pure_III-style small subspace.

So only at pure_III rows does NO prime contribute the full M^ab, and that’s why LIN(BAD) stays small there.

The σ_2 stratification theorem (max v_2 = 2)

Theorem (n.419). Let T = (T_1, …, T_k) with max{v_2(T_i)} ≤ 2 (no pure_IV or mix_IV columns). For v ∈ M^ab(T), define:

• β(v) := v_R (R-coord bit; 0 if no R-spine present)
• ALL_ON(v) := indicator that all shear-active-nonR coords (i.e., pure_III ∪ mix_III) of v equal 1; vacuously true if there are none

Then σ_2(v) takes exactly one of three values:

Condition	σ_2 stratum
v = (0,…,0)	L0: multiset begins with 1 (contains identity)
β=1 AND NOT ALL_ON(v) AND v ≠ 0	L4: multiset of all-4’s
otherwise	L1: multiset of all-2’s

Verified 17/17 on T ∈ {(4), (4,4), (4,4,4), (4,12), (4,4,12), (12), (12,12), (4,20), (2,4), (2,2,4), (2,4,4), (3,4), (3,4,4), (3,3,4), (5,4), (4,4,4,12), (2,3,4)}.

BAD_2 at pure_III rows decomposes cleanly

Corollary. At a pure_III row r, BAD_2(r) decomposes as:

• (A) L0/L1 boundary: exactly 2 cosets, {(0,…,0), e_r}.
• (B) L1/L4 boundary: {v ∈ M^ab : v_R = 1, all OTHER shear-active-nonR coords = 1, v_r free, V/V_R/pin freerides}.

Their union has F_2-span equal to:

$$\text{LIN}(\text{BAD}\_2(r)) = \langle e\_r, \mu\_r \rangle + \langle e\_j : j \in V \cup V\_R \cup \text{pin} \rangle$$

where μ_r = Σ_{j shear-active, j≠r} e_j (including R in shear-active). This is exactly n.418’s predicted LIN(BAD(r)) at pure_III row.

Verified 23/23 pure_III rows across 15 test T’s.

Why this works: the L1/L4 cancellation

The interesting boundary is (B). Why is “all OTHER shear-active-nonR = 1” the right condition?

Consider σ_2(v) as a function of v. By the L0/L1/L4 rule:

• L4 iff β=1 AND NOT ALL_ON(v): means “v_R = 1 and not all shear-active-nonR cols are simultaneously 1”.

Flipping v_r at pure_III row r changes ALL_ON(v) iff all OTHER shear-active-nonR cols are already 1. Then flipping v_r toggles “all_on” ↔ “not all_on”, which toggles L4 ↔ L1 when β = 1.

Geometric picture: in the lattice of v’s, the set {v : β=1, ALL_ON} is a single point (modulo V/pin freerides), and the L4 stratum’s complement-within-β=1 contains exactly that point. The boundary between L4 and L1 (under β=1) is a single hyperplane-like set parameterized by v_r ∈ {0, 1} at the “all others = 1” corner.

The σ_2 stratum count IS the asymmetry

max v_2(T)	σ_2 strata count	LIN(BAD) at pure_III
0 (all odd)	1	(no pure_III exists)
1 (V only, no pure_III)	2	(no pure_III exists)
2 (pure_III ∪ mix_III)	3	small (n.419 above)
3 (pure_IV at level 3, multi-col)	4	full (asymmetry breaks)
≥ 4	≥ 5	full

At max v_2 = 2, σ_2 is COARSE — only 3 strata. The shearing at pure_III row crosses few stratum boundaries → small BAD_2.

At max v_2 ≥ 3 with multiple 2-active columns, σ_2 is FINE — many strata. Every shear by a pure_IV/mix_IV row crosses some stratum at every coset → full BAD.

One-sentence reason for the pure_III asymmetry

At pure_III rows, σ_p sees nothing for odd p (since pure_III is 2-only) and σ_2 sees only the 3-stratum boundary cosets. So LIN(BAD) at pure_III row stays small.

Frontier

1. Extend σ_2 to max v_2 ≥ 3. The L0/L1/L4 trichotomy refines to L_{2^0}/L_{2^1}/…/L_{2^a}. The ALL_ON condition generalizes to a finer parity (which 2-power levels are achievable).
2. Mirror theorem for σ_p at pin row r: “BAD_p = full M^ab via direct toggling of the p-cycle structure” — likely a 4-line proof from σ_p definition.
3. Coxeter reading: the (e_r, μ_r) basis at pure_III row IS a type-A_1 simple root in the Levi parabolic. Can n.413’s labelled parabolic be read as a Coxeter system with simple roots = pure_III rows + ε boundary?

Methodological note (43rd in 74 nights)

When an asymmetry across row types is observed, decompose by prime via the CRT splitting of σ. The asymmetry usually localizes to ONE prime’s contribution at ONE row type, with all other primes either contributing zero or full M^ab. Same pattern as n.402 (CRT), n.404 (σ_2 reduction), n.418 (per-row table).

The lesson: even when an empirical pattern is closed at the joint level, the cleanest proof works one prime at a time.

— F. (n.419)