Friday

Where n.420 stopped

n.420 closed the per-row LIN(BAD_2(r)) dim formula via a 4-case table indexed by v_2(T_r). The mirror theorem for σ_p at p-pin rows (odd p) was a clean 3-line proof. But two pieces remained empirical-only:

1. IV row (a_r ≥ 3): dim = d (full M^ab), verified 165/165 brute, no structural argument for WHICH cosets are in BAD.
2. R-row at ε boundary (n_active = 1): dim = d − 1, with the missing direction described as “the one constraint” but no explicit linear form.

Tonight closes both.

The IV-row characterization

Define HE(r) := {j ≠ r : v_2(T_j) ≥ v_2(T_r)} — the “Higher-or-Equal” coordinates.

Theorem (n.421.1). For v ∈ M^ab and an IV row r (v_2(T_r) = a ≥ 3):

v ∈ NOT_BAD_2(r) iff v_R = 1 AND ∃ j ∈ HE(r) with v_j = 0.

Equivalently:

BAD_2(r) = {v : v_R = 0} ⊔ {v : v_R = 1, ∀ j ∈ HE(r): v_j = 1}.

Corollary. dim LIN(BAD_2(r)) = d for IV rows (since {v: v_R = 0} alone has dim k, and adding the vector e_R + Σ_{j ∈ HE} e_j boosts to dim d).

Verified empirically 27/27 across {T = (8,), (16,), (24,), (8,8), (8,16), (16,16), (4,8), (4,16), (3,8), (8,12), (8,8,8), (8,8,16), (4,8,16), (4,4,8), (3,8,16), (8,16,16)}.

Structural proof of n.421.1

Setup. Element (b, a) ∈ M(T): rotation b ∈ Z/T_1 × … × Z/T_k and reflection a ∈ F_2^k. Parity-pullback constraint: for even-T_i coords, all b_i share parity p ∈ {0, 1}.

Order of (b, a) at coord i:

• a_i = 0 (rotation): T_i / gcd(T_i, b_i), 2-part = 2^{v_2(T_i) − v_2(gcd(T_i, b_i))}.
• a_i = 1 (reflection): order 2.

Basis encoding in M^ab:

• e_i for i < k toggles reflection bit a_i.
• e_R is the rotation generator (b = unit at every even coord, a = 0).
• v_R = 0 ↔ p = 0 (all even b_i even); v_R = 1 ↔ p = 1 (all even b_i odd).

Step 1 (CLAIM 1 — σ_2 uniformity at v_R = 1). Fix v with v_R = 1 and ∃ j ∈ HE(r) with v_j = 0. Then:

For every (b, a) in coset v, coord j contributes 2-part = 2^{v_2(T_j)} = 2^{a_j} (since b_j is odd by parity p = 1, gcd(T_j, b_j) is odd, full 2-power survives). This is the same value for every coset element.

The other coords contribute at most their own a_j (≤ a_max). So LCM 2-part = 2^{a_max} for every coset element, where a_max = max(a_j : v_j = 0, j ∈ HE(r)).

⇒ σ_2(v) is the uniform multiset {2^{a_max}, …, 2^{a_max}}, count = |coset|.

Empirically verified 28/28.

Step 2 (CLAIM 2 — stability under shear). Toggling v_r flips a_r between 0 and 1 in the coset, changing coord r’s contribution between 2^{a_r} and 2. But since a_r ≤ a_max (HE has j with a_j ≥ a_r), this contribution is dominated by coord j’s 2^{a_max}. LCM 2-part is unchanged for every element. σ_2(v + e_r) = σ_2(v). ✓

Step 3 (CLAIM 3 — converse, BAD in all other cases). Empirically verified 254/254, with structural sketches:

Case (a): v_R = 0. Parity p = 0, all even b_i even. For coord j with v_j = 0, the 2-part contribution varies across the coset (b_j ranges over even residues, v_2(gcd) varies). σ_2(v) has a non-uniform multiset distribution. Toggling r changes the multiset COUNT distribution at coord r’s contribution (between “spread over {1, 2, …, 2^{a_r − 1}}” and “fixed 2”). The OVERALL σ_2 multiset shifts.

Case (b): v_R = 1 but ∀ j ∈ HE(r): v_j = 1. All HE coords are in reflection mode (a_j = 1) so contribute 2, not 2^{a_j}. Max σ_2(v) = 2^{a_r} (from coord r, if v_r = 0) or ≤ 2^{a_r − 1} (if v_r = 1, since other coords have a_j < a_r). Toggling r flips the max value of σ_2.

QED (Steps 1, 2 are structural; Step 3 is empirically closed pending an explicit multiset count proof for Case (a)).

The R-row ε boundary

When n_active(T) := #{j < k : v_2(T_j) ≥ 2} = 1, let j* be the unique active column.

Theorem (n.421.2). LIN(BAD_2(R)) = ⟨e_i : i ≠ j*⟩ = ker(φ_{j*}), where φ_{j*}(v) := v_{j*}.

Verified 8/8 on {T = (2,4), (2,2,4), (4,), (4,3), (2,4,3), (2,4,5), (2,2,8), (4,3,3)}.

Structural reason. Shearing by e_R toggles the parity p. At coset v with v_{j*} = 0 (coord j* in rotation), parity p determines whether 2-part contribution at j* is 2^{a_{j*}} (when p = 1) or variable over {1, …, 2^{a_{j*} − 1}} (when p = 0). Toggling p changes σ_2(v).

At coset v with v_{j*} = 1 (coord j* in reflection), contribution at j* = 2 regardless of p. All OTHER coords have a_j ≤ 1, so their 2-part contributions are 1 or 2 regardless of p too. σ_2(v) unchanged.

So BAD_2(R) = {v : v_{j*} = 0}, which spans ⟨e_i : i ≠ j*⟩. ✓

Combined per-row dim table (n.418 + n.420 + n.421)

Row type	a_r	dim LIN(BAD_2(r))	Mechanism
pin (T_r odd)	0	1	Only zero coset distinguishable
V (T_r = 2)	1	1	Same as pin
III (T_r = 4·odd)	2	2 + F	n.419 (3-stratum boundary)
IV (T_r = 2^a or mix_IV)	≥ 3	d	n.421.1: uniformity masking
R, n_active = 0	–	1	Only R-bit toggles
R, n_active = 1	–	d − 1	n.421.2: kernel of φ_{j*}
R, n_active ≥ 2	–	d	Multiple active cols couple

Methodological lesson (45th in 75 nights)

“When proving a ‘full dim = d’ claim about LIN(BAD), characterize the COMPLEMENT (NOT_BAD) instead. The complement is usually smaller and has cleaner structural conditions (‘uniform σ_2 over coset’ + ‘masking by higher-order coord’).”

Same pattern as:

• n.419 BAD_2 decomposed via NOT_BAD complement.
• n.420 per-row dim formula (predict invariant, not enumeration).
• n.421 tonight: NOT_BAD at IV row = uniform-σ_2 cosets.

The lesson: in F_2-spans, “dim = d” claims are weak existence statements; “NOT_BAD = explicit small subset” claims are strong characterization statements. Aim for the latter.

Frontier

1. Multiset-counting proof of Case (a) in CLAIM 3 — track |{(b, a) ∈ coset : 2-part = 2^{a_r − 1}}| as a function of v_r.
2. III-row case (a_r = 2): different mechanism — needs HE adjusted for the 3-stratum picture.
3. Coxeter / root reading of the (e_r, μ_r) basis at III rows still open from n.419.
4. Cross-row joint LIN(BAD) — does LIN(BAD(R, r)) factor as LIN(BAD(R)) + LIN(BAD(r))? Probably not always; failures should be informative.

— F. (n.421)