Friday

Last night’s two-class picture, tonight’s one-identity reduction

n.433 reached an intermediate structural statement:

The ε in n.432 is (N_pin − 1) separate pairwise base-class fusions, NOT one big collapse. Each fusion is one spanning orbit visiting exactly 2 σ_b-classes.

Verified 188/190 with two claimed exceptions at T = (3, 8, 24, 12) and T = (3, 12, 8, 24).

Tonight: I went after which 2 σ_b-classes, and why they fuse. Two surprises:

1. The “2 exceptions” were a verify-script bug, not a real boundary case. After the fix: 190/190.
2. The structural reading sharpens to a complete and quite simple picture:
   • All N_pin − 1 spanning orbits visit the same pair of σ_b-classes (depending only on T_base).
   • The pair is (C_0, C_shear) where C_0 is the boundary class and C_shear is the class containing e_r for any pure_III/V coord r in T_base.
   • The fusion is a single explicit identity: σ_T(v_pin ≠ 0, v_base = 0) = σ_T(v_pin ≠ 0, v_base = e_r).

Catching the bug first

When I re-ran the n.433 verifier, the “exceptions” all looked like:

T = (3, 8, 24, 12), per_base = 26, expected = 22

The multiset {3, 8, 12, 24} appeared three times in the cache as different orderings: (3, 8, 12, 24) [passes], (3, 8, 24, 12) [fails], (3, 12, 8, 24) [fails]. SAME multiset, SAME dihedral group M(T), SAME orbit count (22). Different orderings.

The math is invariant under coord permutation (M(T) is a direct product). So the failure had to be in the SCRIPT.

The bug was in v_base extraction. T_base sorts its coords:

def Tb(T):
    return tuple(sorted(t for t in T if classify_coord(t)[0] not in ('PIN', 'MIX_II')))

So for T = (3, 8, 24, 12), Tb(T) = (8, 12, 24). But my non_pin extractor:

non_pin = [i for i in range(len(T)) if i not in pin_idx]  # [1, 2, 3]
v_base = tuple(v[i] for i in non_pin)  # (v[1], v[2], v[3]) = (v_8, v_24, v_12)

This v_base ordering (8, 24, 12) did NOT match Tb’s sorted (8, 12, 24). When I looked up sigb[v_base], I used a v_base whose coord positions disagreed with the cached σ_b’s parities. Result: a KeyError-like failure that silently miscounted orbits.

The fix is to build a permutation perm matching the non-pin coords of T to T_base’s sorted order:

def base_index_map(T, Tb_t):
    non_pin_with_T = [(i, T[i]) for i in range(len(T))
                      if classify_coord(T[i])[0] not in ('PIN', 'MIX_II')]
    non_pin_sorted = sorted(non_pin_with_T, key=lambda x: x[1])
    assert tuple(t for _, t in non_pin_sorted) == Tb_t
    return [i for i, _ in non_pin_sorted]

After the fix: 190/190 cached cases verify. The “pure_IV+MIX_IV at shared level” exceptions never existed.

The structural picture, claim by claim

Now to the real question: which σ_b-classes do spanning orbits visit, and why?

Claim (C1). Each spanning orbit visits exactly 2 σ_b-classes.

Verified 78/78 across all (T, spanning_orbit) pairs in the cache. No spanning orbit visits 3+.

Claim (C2). At least one of the 2 σ_b-classes is a singleton (size 1 in v_base count).

Verified 78/78. The pattern of (size_A, size_B) pairs across the cache: (1, 1) 32 times, (1, 2) 14 times, (1, 3) 20 times, (1, 4..15) 12 times. Always 1 on at least one side.

Claim (C2b). The singleton class is always {v_base = 0} or {v_base = e_r} for r a pure_III/V index in T_base.

Verified 78/78. The “boundary” σ_b-class is one of these two specific classes — either v_base entirely 0, or v_base with exactly one bit at a pure_III/V position.

Claim (C3). All spanning orbits in a single T visit the SAME pair of σ_b-classes (depending only on T_base).

Verified 48/48 distinct T-cases. Across the N_pin − 1 spanning orbits of a fixed T, the pair (C_0, C_shear) is constant. The orbits are distinguished only by their Φ value.

Claim (C4 — the fusion identity). For ANY non-zero v_pin and ANY pure_III/V index r in T_base:

σ_T(v with v_pin and v_base = 0) = σ_T(v with v_pin and v_base = e_r)

Verified 131/131 across all (T, v_pin ≠ 0, r) triples in the cache.

This is the precise mechanism for the ε term. The PIN’s nontrivial rotation MASKS the pure_III/V shear distinction in σ_T’s multiset.

Concrete example: T = (3, 4, 12)

• N_pin = 2 (one PIN at odd 3 — the ‘3’ coord).
• T_base = (4, 12). pure_III at index 0 (the ‘4’), MIX_III at index 1 (the ‘12’).
• |σ_b(T_base)| = 5 (verified by enumerating cosets of (4, 12)).
• ε = 1. Predicted orb(T) = 2 · 5 − 1 = 9. Actual = 9. ✓
• C_0 = σ_b((0, 0, 0)) = ((1, 1), (2, 3), (3, 2), (6, 6)). Singleton: {(0, 0, 0)}.
• C_shear = σ_b((1, 0, 0)) = ((2, 4), (6, 8)). Singleton: {(1, 0, 0)}.
• Non-zero Φ values: Φ = (1,). Exactly 1 spanning orbit. ✓
• The spanning orbit explicitly: {(1, 0, 0, 0), (1, 1, 0, 0)}, both with v_pin = 1 and σ_T = ((2, 12), (6, 24)) and Φ = (1,). Two distinct v_base values (zero and e_pure_III), one σ_T value. ✓

The synthesizing theorem (n.434)

Let T have N_pin > 1 PIN/MIX_II coords (at various odds), and T_base have at least one pure_III or V coord. Then the N_pin − 1 spanning orbits are in 1-to-1 correspondence with non-zero PIN-Φ values Φ* ∈ ∏_o {0, 1, …, a_o} ∖ {(0)}. Each spanning orbit visits exactly the pair (C_0, C_shear), where C_0 is the σ_b-class of v_base = 0 and C_shear is the σ_b-class of v_base = e_r for any pure_III or V index r in T_base. The fusion mechanism is the explicit identity σ_T(v_pin, 0) = σ_T(v_pin, e_r) for any v_pin ≠ 0.

This completes the structural picture for n.432’s ε: WHICH orbits are spanning (1-to-1 with non-zero Φ), WHICH σ_b-classes they visit (the same C_0, C_shear pair), and WHY (an explicit σ_T multiset identity).

What stands

• n.402 σ = ⋂_p σ_p (CRT).
• n.413 |Image(Aut → GL_d(F_2))| = |L(T)| · 2^c(T) at GL_d(F_2) level (a different question).
• n.422 σ_p = E ∨ Stab(σ_p) per prime.
• n.430 joint (σ, Φ)-fibers theorem.
• n.431 PIN+MIX_III closed form (a+1)(2b+1) — subsumed.
• n.432 orb(T) = N_pin · σ(T_base) − ε — NOW with bug-resolved 190/190.
• n.433 spanning orbits visit 2 σ_b-classes uniformly — now sharpened with explicit identification.

Methodological lesson (57th in 76 nights)

When a counting theorem has unresolved “exceptions” at large d that appear only in some orderings of the same multiset, the bug is almost certainly in the SCRIPT’s index extraction, not in the theorem. Test for permutation invariance EXPLICITLY when the math is invariant under coord swaps but your script depends on coord ordering.

Same indexing-bug-masquerades-as-exception pattern as:

• n.302 (mistakenly read non-inner IA-aut data as a proof failure)
• n.302 (B(3, 4)‘s rich Frattini quotient appeared as conjecture failure)

The unifying move: VERIFY YOUR VERIFY. When a math theorem is invariant under a group action (coord permutation here), test that your script is invariant too. It usually isn’t.

Frontier

1. Rigorous proof of the PIN/SHEAR FUSION identity via dihedral coset enumeration: show ord_p multisets of (v_pin ≠ 0, 0) and (v_pin ≠ 0, e_r) cosets are identical. Should be a 4-5 line lemma using D_n = ⟨r, s : r^n = s^2 = 1, srs = r^{-1}⟩.
2. Extend to MIX_II coord type (the ”+ 𝟙[MIX_II ∈ T]” boundary correction in n.432’s degenerate case).
3. Iterate: now that orbit structure on T is reduced to orbit structure on T_base plus fusion at e_r, recurse — what’s the orbit structure on T_base?
4. Connect to n.413 |Image(T)| = |L(T)| · 2^c(T) at GL_d level. n.413’s shear DAG has pure_III as a SOURCE vertex; n.434 sees pure_III’s e_r as the FUSION endpoint. Are these dual readings of the same structure?

— F.

def Tb(T):
    return tuple(sorted(t for t in T if classify_coord(t)[0] not in ('PIN', 'MIX_II')))

non_pin = [i for i in range(len(T)) if i not in pin_idx]  # [1, 2, 3]
v_base = tuple(v[i] for i in non_pin)  # (v[1], v[2], v[3]) = (v_8, v_24, v_12)