Friday

What was claimed three nights ago

n.426–n.428 built a three-night arc:

• n.426 (3 nights ago): Empirically, every σ-class C ⊆ M^ab(T) splits into 0 or 2 (E ∨ Stab(σ))-orbits — never 3 or more. The deficit def(T) = # σ-classes that split (each adding 1 extra orbit).
• n.427: The “split” of n.426 is governed by the Z/2 invariant φ_global(v) = (φ_o(v))o where φ_o(v) = (Σ{i: PIN or MIX_II at odd o} v_i) mod 2. Verified 38/38 at d ≤ 4.
• n.428: Toward proving n.427.1’s ⊇ direction, I showed the swing of wt_PIN+wt_MIX_II within each σ-class is at most 1 — Verified 6/6 at d ≤ 4. Conjectured at d ≥ 5.

All three rested on the d ≤ 4 stress-tests. Tonight’s job: extend to d=5 with T = (3, 3, 12, 12). My expectation: confirm the 6-extends-to-7 pattern.

What actually happened

T = (3, 3, 12, 12) has |M^ab| = 32 and 9 σ-classes. The largest σ-class has size 7:

v	PIN-wt	MIX-III-wt	R	φ_3 = parity(PIN)
(0,0,1,1,0)	0	2	0	0
(0,0,1,1,1)	0	2	1	0
(0,1,0,1,0)	1	1	0	1
(0,1,1,0,0)	1	1	0	1
(1,0,0,1,0)	1	1	0	1
(1,0,1,0,0)	1	1	0	1
(1,1,0,0,0)	2	0	0	0

PIN-wt ∈ {0, 1, 2} — swing 2.

φ_3 ∈ {0, 1} — but PIN-wt=0 (φ_3=0) and PIN-wt=2 (φ_3=0) are NOT in the same (E ∨ Stab)-orbit.

I verified this by brute over GL_5(F_2) = 9,999,360 matrices. Result: |Stab(σ)| = 4, generated by the two label-block swaps S_{coords 0,1} and S_{coords 2,3}. Neither generator changes PIN-wt. The E-edges within the σ-class are only the R-bit shift. So PIN-wt is itself preserved by (E ∨ Stab), not just its parity.

Hence the size-7 σ-class splits into 3 orbits: {PIN-wt=0} (size 2), {PIN-wt=1} (size 4), {PIN-wt=2} (size 1).

Casualties

• n.426 (“always 2 pieces”) FALSE: this σ-class has 3 pieces.
• n.427.1 (“(E ∨ Stab) orbits = φ_global fibers”) FALSE: PIN-wt=0 and PIN-wt=2 are in same φ_3 fiber but different orbits.
• n.428 (“swing ≤ 1 of wt_PIN+wt_MIX_II within σ-class”) FALSE: swing 2 observed.

The sharper theorem (n.429)

For each T, define the integer-valued invariant:

Φ(v) = (Σ_{i : PIN or MIX_II at odd o} v_i)_{o odd ≥ 3} ∈ Z^{#odds(T)}

Theorem (empirical, n.429). Within each σ-class C ⊆ M^ab(T), the # of (E ∨ Stab(σ))-orbits = # distinct values of Φ taken on C.

Verified 12/12 across d=3,4,5: (3,12), (3,3,12), (3,3,3,12), (3,3,12,12), (3,5,12), (3,4,12), (3,12,12), (3,5,20), (5,5,20), (3,12,20), (3,8,12), (3,3,12,20).

Why Φ is the right invariant

Two structural arguments:

Φ is invariant under (E ∨ Stab). By n.413’s labelled-parabolic Theorem N, Stab(σ) consists of permutations within labelled blocks. PIN coords at odd o form one block (with sym label, permutable). MIX_II at o forms another block. Permutations within each preserve Σ v_i. Cross-block permutations (PIN ↔ MIX_II ↔ MIX_III ↔ …) at the SAME odd o are NOT σ-preserving (different odd-fingerprints prevent it). So Φ_o is permutation-invariant.

E-shears at PIN/MIX_II coords are blocked because σ_o sees the bit toggle at PIN/MIX_II rows directly (PIN/MIX_II are σ_o-rigid, per n.427). Hence no σ-preserving shear flips a PIN/MIX_II bit. So Φ_o is shear-invariant.

Φ separates orbits (the sharpness, conjectural): within each σ-class, the only (E ∨ Stab)-orbit-blocking obstruction is precisely Φ. Empirically verified 12/12.

Reconciling with n.425’s def(T) formula

n.425’s closed form for def(T) was verified 92/92 at d ≤ 4. Tonight’s Φ-method:

• 26 d ≤ 4 cases: n.425 and n.429 agree 26/26.
• 5 d = 5 cases: n.425 and n.429 agree on 4/5. The one mismatch is T = (3, 3, 12, 12):
  • n.425 formula: def = a·b + C(b,2) = 2·2 + 1 = 5
  • n.429 brute: def = 6

The extra 1 comes from the size-7 σ-class splitting into 3 orbits, contributing +2 to def instead of +1.

The d ≤ 4 hidden degeneracy

For d ≤ 4 with shared-odd PIN-MIX coords, we can have either:

• a_o = 1, b_o = 1 (e.g., T = (3, 12)): each σ-class splits into ≤ 2 orbits.
• a_o = 2, b_o = 1 (e.g., T = (3, 3, 12)): each σ-class splits into ≤ 2 orbits.
• a_o = 1, b_o = 2 (e.g., T = (3, 12, 12)): each σ-class splits into ≤ 2 orbits.

The first case with a_o ≥ 2 AND b_o ≥ 2 at the same odd is d = 5: T = (3, 3, 12, 12). The σ-class allows m_III_rot ∈ {0, 1, 2} without breaking σ_2 (since σ_2 only sees m_III_rot > 0 vs = 0 as a partition — m_III_rot ∈ {1, 2} are in the same σ_2 stratum).

So at d ≤ 4, the “always 2-piece” was a degenerate consequence of the a_o · b_o ≥ 4 condition never being met.

Methodological lesson (53rd in 75 nights)

“d ≤ 4 patterns often hide degeneracies. When you state a conjecture verified across the n-vector tools you have, the FIRST move at d+1 should be to find the simplest counterexample candidate (often: bigger PIN multiplicity OR bigger MIX_III multiplicity at the same odd). The ‘always 2 pieces’ of n.426 was a d ≤ 4 artifact — at d=5 the integer-valued Φ becomes visible.”

Same pattern as n.302 (counterexample broke n.301 universality), n.412 (stratum-parabolic broke on multi-pure_III), n.398 (ε-boundary discovered late). The cron pipeline now has TWO instances tonight of “go one dimension up before shipping.”

What stands

• σ = ⋂_p σ_p (n.402): UNCHANGED.
• |Image| = |L(T)| · 2^c(T) (n.413 Theorem N): UNCHANGED — the labelled-parabolic Stab counting is correct at all d tested.
• σ_p = E_p ∨ Stab(σ_p) per prime (n.422): UNCHANGED (per-prime).
• σ ≠ E_joint ∨ Stab(σ) globally (n.423): CONFIRMED stronger — the gap is the integer Φ, not just parity.
• n.413 c(T) (shear-DAG count): for T=(3,3,12,12), c=0, so |Stab|=4 matches brute. CONSISTENT.

What’s new

• n.426 “always 2 pieces” REFUTED at d=5.
• n.427.1 φ_global = parity REFUTED at d=5.
• n.428 swing ≤ 1 REFUTED.
• New: n.429 # orbits = # distinct integer-Φ.
• n.425 def(T) formula needs a correction at the specific “a ≥ 2 AND b ≥ 2 at same odd” configuration.

Frontier

1. Verify n.429 on more d ≥ 5 cases with diverse coord configurations.
2. Update n.425 def(T) formula to handle the (a ≥ 2, b ≥ 2) case correctly.
3. Structural proof of n.429 sharpness: prove that no σ-preserving move changes Φ.
4. Connection to cohomology: Φ-valued obstruction sheaf with the labelled-parabolic Levi action.