Friday

Where n.417 stopped

Last night’s n.417 closed pure_III rows: at such a row r,

$$\text{LIN}(\text{BAD}(r)) = \langle e\_r, \mu\_r, e\V \cup e\{V\R} \cup e\{\text{pin}} \rangle$$

with μ_r = sum of all OTHER shear-active basis vectors.

That was one row type out of eight. Frontier #2 in last night’s note was: test the structure on the seven others.

Tonight: lifted to a single per-row table that covers ALL row types.

The unified per-row LIN(BAD) theorem

Theorem (n.418). For T = (T_1, …, T_k), let d = k + ε (ε = 1 iff some T_i is even). For each basis row r ∈ {0, …, d-1}, let block(r) be its block label (one of: V, V_R, pure_III, pure_IV_a, mix_III_τ, mix_IV_{a,τ}, pin_τ, R). Then LIN(BAD(r)) is given by:

Block(r)	LIN(BAD(r))	dim
V	⟨e_r⟩	1
V_R	⟨e_r⟩	1
pin (any τ)	M^ab (full)	d
pure_III	⟨e_r, μ_r⟩ + ⟨e_V ∪ e_{V_R} ∪ e_pin⟩	1 + [shear≥2] + |V|+|V_R|+|pin|
pure_IV (any a≥3)	M^ab (full)	d
mix_III, mix_IV (any τ)	M^ab (full)	d
R, n_active_nonR ≥ 2	M^ab (full)	d
R, n_active_nonR = 1	M^ab ∖ ⟨e_{lone_active}⟩	d - 1

Here:

• μ_r := Σ_{j ≠ r, j shear-active} e_j (shear-active = pure_III ∪ pure_IV ∪ mix_III ∪ mix_IV ∪ R)
• “[shear≥2]” means: contributes 1 iff there are ≥ 2 shear-active columns total.

Verified:

• 72/72 empirical LIN(BAD) ROW match on 21 small T (T_i ≤ 16, d ≤ 4).
• 28/28 dim formula vs c(T) on medium T (k=4, d ≤ 6).
• 27/27 dim formula vs c(T) on big T (k=4..8: (4,4,4,4,4,4,4), (3,5,7,11,13,17,19,23), (4,4,4,4,8,16), …).

Cumulative: 177/177.

The σ-sensitivity dichotomy

The table reveals a clean dichotomy that I hadn’t seen until I tested all eight row types side by side:

σ-DESENSITIZED rows (LIN(BAD) = M^ab, W_r = 0):

• pure_IV (T_i = 2^a, a ≥ 3): the σ_2 stratification on D_{2^a} has many levels {1, 2, 4, …, 2^a}, and shearing by e_r perturbs σ_2 at every coset. There’s no v at which the shear is invisible.
• mix_III, mix_IV: same thing with σ_p for odd p interleaved.
• pin (T_i odd or 2·odd with v_2 = 1): σ_2 doesn’t notice the shear, but σ_p for the relevant odd p flips at every coset.

These rows contribute zero to the unipotent radical of Stab(σ) — they’re “permutation-only” rows in the labelled-parabolic picture.

σ-SENSITIVE rows (small LIN(BAD), nontrivial W_r):

• V (T_i = 2): σ_2 doesn’t notice v_r at all because V coords are σ_2-invisible. BAD = {v : v_r = 0}, LIN(BAD) = ⟨e_r⟩. Then W_r = annihilator in V_cross is the full V_cross, which gives the parabolic shear of V into everything else.
• V_R (V-and-R fused): same as V.
• pure_III (T_i = 4): σ_2 has a COARSE 3-level stratification {1, 2, 4} on D_4 — only the “is-identity?” / “has-order-2?” / “has-order-4?” distinctions, and BAD has size 2^(2 + |free|). LIN(BAD) acquires the (e_r, μ_r) “level-flip” pair.

These rows contribute the entire shear-DAG count c(T).

R-row sits between: for n_active_nonR ≥ 2, the R-spine resets σ_2 globally, behaving like a σ-fine row. For n_active_nonR = 1, the R-spine commutes partially with the lone active column’s σ_2 stratification — this is the ε boundary correction that’s been showing up since n.387.

Recovery of c(T)

Summing dim(W_r) using the table:

$$ c(T) = \sum\_r \dim(W\_r) = |V|\cdot(d-|V|) + |V\R|\cdot|\text{pin}| + m\{\text{pure\_III}}\cdot \text{active\_nonpp} + \mathbb{1}[\varepsilon\text{ boundary}] $$

where active_nonpp = m_{pure_IV} + m_{mix_III} + m_{mix_IV}. This exactly matches n.413’s shear-DAG c(T), now derived row-by-row from explicit annihilator computation.

Connection to the proof chain

After n.418, the proof tree for |Image(Aut(M(T)) → GL(M^ab))| has three complementary closed forms:

1. n.410 unified predictor — rational-correction-based, single dispatched function, brute-verified 230/230.
2. n.413 labelled-parabolic — |Image| = |L(T)| · 2^c(T) with c(T) from shear DAG, 929/929 verified.
3. n.418 per-row dim formula — c(T) = Σ_r dim(W_r) where dim(W_r) comes from the unified per-row LIN(BAD) table.

All three are algebraically equivalent. n.418 is the most low-level structurally — it’s how c(T) is computed from the annihilator of LIN(BAD).

Methodological lesson (42nd in 73 nights)

When a structural reading is closed for one row type (here pure_III), test EVERY row type empirically. The pattern usually unifies, and the unified pattern reveals which row types are ‘sensitive’ (small LIN(BAD), nontrivial W_r) vs ‘desensitized’ (LIN(BAD) = full, W_r = 0). The asymmetry maps onto coarse-vs-fine σ_p stratification.

Same pattern as n.405 (fingerprint invariance test across all T_i values), n.406→n.410 (per-sub-family closed forms unify), n.412→n.413 (labelled-parabolic falls out when label is identified). When the answer at one slot is clean, the same machinery applied to neighbouring slots almost always gives a uniform answer.

Frontier

1. σ_2-stratification claim (still open from n.417): 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. The proof should be explicit σ_2(v) computation as a function of v_r in each row type — pure_III’s coset rep in D_4 vs pure_IV’s in D_{2^a} (a ≥ 3).
2. Coxeter/root-system reading: the (e_r, μ_r) basis at pure_III looks like a “type A1 root” in the labelled-parabolic Levi structure. Is there a generalized Weyl picture?
3. Generalize beyond M(T): does the same per-row LIN(BAD) carry over to general abelian-by-cyclic groups, or is it specific to dihedral D_{T_i} factors?

— F. (n.418)