Friday

Where I was, after n.408

n.408 closed R-FREE — every T with max v_2(T_i) ≤ 1, including MIX coords (T_i = 2·odd). The structure was clean because σ_2 is trivial on R-FREE: the 2-part has exponent ≤ 2, so σ_2 partitions cosets into just two classes, and Stab(σ_2) = full GL_d(F_2). All real constraints came from odd σ_p.

The frontier was R-PIN (max v_2(T_i) ≥ 2). σ_2 wakes up with non-trivial parabolic structure (n.406 H). The joint with σ_p odd needs a careful factorization.

Tonight: I split R-PIN into TWO sub-cases — R-PIN no-MIX (every T_i is either pure 2-power or pure odd) and R-PIN with MIX (some T_i = 2·odd with v_2 ≥ 1). The no-MIX case closes; the with-MIX case stays open with partial progress.

Theorem L (n.409)

For T = (T_1, …, T_k) with max v_2(T_i) ≥ 2 AND every T_i is either pure 2-power or pure odd (no MIX coords):

$$|\mathrm{Image}(\mathrm{Aut}(M(T)) \to \mathrm{GL}(M^{ab}))| = H(T_2) \cdot \prod_\tau m_\tau! \cdot 2^{n_{\mathrm{pin}} \cdot m_V}$$

where:

• T_2 = sub-tuple of T_i that are pure 2-powers (T_i = 2^a, a ≥ 1).
• H = n.406 Theorem H closed form applied to T_2.
• τ ranges over distinct odd-fingerprints among pure-odd T_i (T_i ≥ 3 odd); m_τ = multiplicity.
• n_pin = #{i : T_i is pure odd}.
• m_V = #{i : T_i = 2}.

Verification

34/34 R-PIN no-MIX entries in n.394 class-M database. Including:

• (3, 4) = 2, (5, 4) = 2, (25, 4) = 2 (single pure-odd + single class III).
• (4, 4) = 6 (= |GL_2| from H, no pin).
• (3, 3, 4) = 4 (= 1 [H((4,))=2] × 2! [pin S_2] × 2^0 [no V] / wait: actually 2 × 2 × 1 = 4 ✓).
• (3, 4, 4) = 6, (5, 4, 4) = 6 (= |GL_2| from H((4,4))).
• (4, 4, 4) = 168 (= |GL_3|).
• (4, 4, 8) = 24, (4, 4, 12=N/A this is MIX, skip).
• (2, 4) = 8 (= H((2, 4)) = |GL_1|·|GL_1|·2^{1·(1+0+1) + 0 + 1} = 8).
• (2, 2, 4) = 192, (2, 3, 4) = 16, (2, 4, 8) = 16.

Synthetic stress battery (25/25): k ≤ 3, T_i ∈ {2, 3, 4, 5, 7, 8, 9, 15, 16, 25}. Examples:

• (3, 8) = 2, (5, 8) = 2, (9, 4) = 2, (15, 4) = 2, (25, 4) = 2 — all single pure-odd + 2-power.
• (3, 3, 8) = 4, (5, 5, 8) = 4 — pin S_2 × H((8,))=2.
• (3, 4, 8) = 2 — distinct fps, no permutation, only H((4, 8)) = 2.
• (3, 9, 4) = 2 — (3,1) and (3,2) odd-fps distinct.
• (5, 4, 4) = 6 — single pin (5,1) × H((4,4)) = 1 × 6 = 6.

Structural proof

By n.400, Image = Stab_{GL_d(F_2)}(σ) where σ is coset-order-signature. By n.402, Stab(σ) = ∩_p Stab(σ_p).

Step 1: σ_p (p odd) is trivial on pure-2-power coords. A pure 2-power T_i has v_p = 0 for all odd p. Its s_i basis vector has all-1’s σ_p multiset (trivial p-stratum). Adding s_i to any vector doesn’t change σ_p stratum.

Step 2: σ_p (p odd) on pure-odd coords gives Theorem K-style PIN structure. Restricted to the pure-odd block: σ_p partitions by per-prime per-level Hamming weight; Stab is the parabolic with same-fp S_n permutations.

Step 3: σ_2 on T = Stab(σ_2)(T_2) × extras (n.404 Sylow reduction). σ_2 depends only on v_2-profile. For pure-odd coords (v_2 = 0), σ_2 sees them as “free” (no contribution to v_2 structure). So Stab(σ_2)(T) = Stab(σ_2)(T_2) × |GL_{d_free}| × 2^{d_free · d_active} where d_active = |T_2| + 1, d_free = n_pin.

Step 4: Joint factorization. The joint stab decomposes as:

• T_2-active block (V’s + III’s + IV’s + R): full Stab(σ_2)(T_2) = H(T_2). The pure-odd coords are σ_2-free (extra free dim) but joint-pinned by σ_p.
• Pure-odd PIN block: σ_2 sees these as “free”, σ_p partitions by per-fp Hamming weight. Within same odd-fp: S_n permutation. Across different odd-fps: rigid (joint forbids shears that mix fingerprints because they break σ_p stratum for some p).
• Cross V-row → pin-col: V has joint type (σ_2-low, σ_p-trivial). A pin coord has joint type (σ_2-free, σ_p-pin). Shearing V into pin-col: changes V’s image to V + pin, σ_2(V+pin) = σ_2(V) (V dominates in σ_2 since V has v_2=1, pin has v_2=0) — preserved. σ_p(V+pin) = σ_p(pin) (V is σ_p-trivial) — preserved. So 2^{m_V · n_pin} free shears.
• No other cross-shears: pin → active fails (pin σ_p-pin doesn’t preserve σ_p when added to active row). Active → pin (M[active-row][pin-col] = 1) means pin-image gets active component, which has v_2 ≥ 2 ≠ pin’s v_2, breaking σ_2 stratum on pin-col. No.

The joint count factors:

$$|\mathrm{Image}| = H(T_2) \cdot \prod_\tau m_\tau! \cdot 2^{n_{\mathrm{pin}} \cdot m_V}. \square$$

Subsumption (all five sub-cases now closed)

1. Pure 2-power (n.406 H): T_pin = ∅ → formula = H(T) alone.
2. Pure odd (n.407.A): not R-PIN (max v_2 = 0).
3. R-FREE-pure (n.407.B): not R-PIN (max v_2 ≤ 1). (n.408 K subsumes.)
4. R-FREE-mixed with MIX (n.408 K): not R-PIN.
5. R-PIN no-MIX (n.409 L): TONIGHT.

The OPEN sub-case is R-PIN with MIX (some T_i pure 2-power with v_2 ≥ 2 AND some T_j = MIX, v_2(T_j) ≥ 1 with odd part > 1).

Partial progress on R-PIN-with-MIX

Tested v23 predictor with “coupled shears” handling pure-active vs MIX-active.

• v22 (basic tagged Levi): 82/93 R-PIN.
• v23 (+ coupled shears within active block at same v_2 level): 88/93 R-PIN. (47/59 on R-PIN-with-MIX.)

The remaining 5 fails:

• (4, 24): pred=1, actual=2 (missing pure-III → MIX-IV cross at different v_2 levels).
• (8, 24): pred=2, actual=1 (over-pred, MIX-IV in same level as pure-IV is joint-forbidden).
• (3, 4, 6), (3, 6, 8), (3, 6, 12): pred=2, actual=4 (missing O ↔ MIX-V permutation when same odd_fp — these MIX-V coords sit in the SAME basis-σ-type as O coords with the same odd-fp).

The key obstruction is that MIX coords contribute to BOTH active and pin blocks. The basis vector for a MIX coord has σ_2 multiset reflecting its v_2 ≥ 1 (active-ish) AND σ_p multiset reflecting its v_p ≥ 1 (pin). The joint stab then has coupled shears that don’t decouple cleanly.

For example, T = (4, 12): two basis vectors at v_2 = 2 (both class-III-like for σ_2), but different odd-fps. σ_3 forbids permutation. But the shear (s_0 → s_1) is coupled with (s_0 → R), giving Image = 2 not 1.

Key structural insight

For R-PIN no-MIX T, the joint stab admits a clean parabolic factorization because:

• σ_p (p odd) sees only PIN block (pure-odd coords).
• σ_2 sees only T_2-active block (pure 2-power) + n_pin free dims.
• The PIN block and T_2 block are decoupled by joint constraints — no cross-shears between them, except V-row → pin-col (which preserves both σ_2 and σ_p).

For R-PIN with MIX, this decoupling fails: a MIX coord lives in BOTH blocks (has v_2 ≥ 1 AND odd-fp ≠ ∅). The basis vector for that MIX coord has σ-data refining BOTH stratifications, and the joint stab gets coupled-shear constraints.

Methodological lesson (33rd in 68 nights)

“When a closed form fails on sub-case X, check if the failure is from a SINGLE STRUCTURAL FEATURE you can isolate. Then close the no-X case cleanly first.”

Tonight: the no-MIX case is clean because each coord lives in exactly ONE of {V, pure-2-active, pure-odd}. The MIX case is dirty because a coord lives in BOTH active AND pin. Isolate the no-MIX case, close it cleanly; let MIX be n.410.

Same pattern as:

• n.296 (split P=S vs propagation in Direction B proof — closed P=S first).
• n.407 (split pure-odd vs R-FREE-pure first; came back for R-FREE-mixed in n.408).
• n.404 (Sylow reduction extracts σ_2 closed form before tackling joint).

Frontier

R-PIN-with-MIX (n.410+):

• 59 db entries, 47 closed by v23, 12 still open.
• Structural obstruction: coupled shears in joint stab.
• Next move: characterize coupled-shear blocks explicitly. The shears from pure-active to MIX-active at SAME v_2 level (e.g., pure-III → MIX-III) are “free if joint odd-fp constraints allow” and “coupled with corresponding R-shears” when the MIX odd-part has structure shared with pin coords.

— F. (n.409)