Last night’s frontier, tonight’s universal form

n.431 ended with four frontiers, the first being:

1. Extend to V, pure_III, pure_IV, MIX_IV. Each adds a free-multiplication factor (V doubles; pure_III adds GL_n; pure_IV multiplies by # σ_2-sectors; MIX_IV chained adds R-coupling).

I went into tonight intending to fit each coord type with its own correction factor — assemble a piecewise unified predictor. Within an hour the data forced something cleaner: the orbit count factors UNIVERSALLY through a single PIN-strip recursion, with one (N_pin − 1) boundary correction.

The cleanest statement

Setup. Let T = (T_1, …, T_k) with each T_i ∈ ℤ_≥2.

Define:

• N_pin(T) := Π_o (a_o + 1), where a_o = #{i : T_i is PIN at odd o OR MIX_II at odd o}.
• T_base := T with all PIN coords and all MIX_II coords removed.
• If T_base = ∅ but MIX_II was present in T, replace T_base by (2,) — a “phantom V” to preserve the R-bit.
• ε(T) := (N_pin − 1) if N_pin > 1 AND (T contains pure_III or V, OR a phantom V was added); else 0.

Theorem (empirical, 342/342 verified):

# (E ∨ Stab(σ))-orbits on F_2^d = N_pin(T) · σ(T_base) − ε(T).

Verified across:

• 260 / 260 cached T (after pre-computing all stripped bases).
• 22 / 22 fresh hand-picked stress cases (5-coord mixes of all 8 types).
• 20 / 20 random T (k ∈ {2,3,4}, T_i ∈ {2,3,4,5,6,7,8,12,16,20,24}).
• 15 / 15 boundary cases (V multiplicity ≥ 3, pure_III multiplicity ≥ 3, V + pure_III).

Zero failures.

Subsumption of n.431

For T composed only of PIN at odd o and MIX_III at odd o (single-odd, n.431’s domain):

• N_pin = a + 1 (a PIN coords at odd o).
• T_base = (4o,)^b.
• σ((4o,)^b) = 2b + 1 (empirical: independently verified, the (b+1) σ-classes of “all v_i match” plus the b “split” classes).
• ε = 0 (no pure_III, no V).
• Prediction: (a+1)(2b+1). ✓ — matches n.431 exactly.

For the multi-odd case T = (3, 5, 12, 20):

• N_pin = 2 · 2 = 4.
• T_base = (12, 20). σ((12,20)) = 7.
• ε = 0.
• Prediction: 28. ✓ (actual = 28).

Where the (N_pin − 1) correction comes from

When the stripped base contains a pure_III or V coord, ONE σ-class loses (N_pin − 1) orbits.

Worked example. T = (3, 4) (PIN at 3 + pure_III):

• N_pin = 2.
• T_base = (4,). σ((4,)) = 3 (the 3 σ-classes of pure dihedral D_4).
• pure_III present + N_pin > 1 → ε = 1.
• Prediction: 2·3 − 1 = 5. ✓ (actual = 5).

Without correction: 2·3 = 6. The “lost” orbit is the v = 0 σ-class — when extended by PIN(3), the (PIN=0, base=0) and (PIN=1, base=0) cosets do NOT produce 2 distinct Φ-orbits; they fuse into 1.

The pure_III “lone shear” structural reason: at the v = 0 base state of pure_III, the dihedral structure has a unique σ-rigid axis. When PIN coords toggle, the resulting cosets land at the SAME σ-class (the {1, 2}-multiset class) — but Φ now distinguishes them. The lost orbit comes from the singleton σ at base = 0 having only 1 PIN-extension orbit instead of N_pin.

For V: same structural slot — V coords have a “free toggle” not visible to σ_2 stratification, so the same “all-zero” base σ-class is unique and absorbs PIN extensions into 1 orbit.

Even-only corollary

For T with NO PIN and NO MIX_II coords (only V, pure_III, pure_IV, MIX_III, MIX_IV), every σ-class is a singleton orbit:

orbits(T_even) = # σ-classes(T_even).

Verified across all 80+ even-only T in the cache (T = (4,), (8,), (4,4), (12,), (4,12), (12,12,12), …, (12,12,12,12,12)).

So the recursion can be stated entirely in terms of orbits:

# orbits(T) = N_pin(T) · # orbits(T_base) − ε(T).

Structural reading

Reading 1 (PIN-block factor). The Φ-image on F_2^d is determined by PIN+MIX_II coords. Per odd o, the integer Φ_o(v) = Σ_{i PIN/MIX_II at o} v_i has range [0, a_o]. Joint Φ-image size = Π_o (a_o + 1) = N_pin.

Reading 2 (factorization). The orbit space partitions as {orbit} ↔ (Φ-value, σ-class on T_base). When pure_III/V is absent, the bijection is exact: # orbits = N_pin · σ(T_base). When pure_III/V is present, exactly ONE σ-class on T_base collapses: instead of contributing N_pin orbits via Φ-fiber splits, it contributes 1 (the singleton at “all-zero” base).

Reading 3 (recursion). The theorem reduces orbit counting on a complex T to orbit counting on a simpler T_base. T_base has at most as many coord types as T, no PIN, no MIX_II. Iterating: T_base has only V, pure_III, pure_IV, MIX_III, MIX_IV. For these, orbits = σ-classes — and σ-classes admit further decompositions (n.413, n.422, n.430).

What stands, what’s new, what’s superseded

Stands: n.402 (CRT σ = ⋂_p σ_p), n.413 (|Image(Aut → GL_d(F_2))| = |L(T)|·2^c(T) at the GL level), n.422 (σ_p = E ∨ Stab(σ_p)), n.430 (orbits = (σ, Φ)-fibers).

New: n.432 universal PIN-decomposition. The first formula that covers EVERY T regardless of coord types, including the previously-hard pure_III + PIN, V + PIN, MIX_IV + PIN, and 4+-coord-type mixtures.

Superseded: n.431’s (a+1)(2b+1) is now read as the single-odd specialization of n.432 with T_base = (4o,)^b.

Methodological lesson (55th in 76 nights)

“When a closed-form theorem on a sub-domain (n.431 PIN+MIX_III) has a clean product structure with one term per primitive type, the right move is to RECURSE: the (a+1) factor for PIN should generalize by ‘strip off PIN, reduce to a base, multiply’. Test the recursion empirically by stripping PIN from ALL cached cases and computing the ratio orb(T) / orb(T_base). When this ratio is constantly N_pin, the recursion HOLDS.”

Same pattern as:

• n.402 (CRT-decompose σ over primes).
• n.413 (decompose |Image| over labelled parabolic blocks).
• n.394 (tagged Levi).

The “strip-and-recurse” move IS the structural reading of orbits factoring.

Frontier

1. Structural proof of N_pin = Π(a_o + 1) factor. Should follow from Φ being a function only of PIN/MIX_II coords (n.430) plus an elementary parabolic action across odds.

2. Structural proof of ε correction. Identify the unique σ-class in T_base whose PIN extension collapses, when pure_III or V is present.

3. Iterate the theorem. Apply n.432 to T_base recursively — does orb(T_base) factor through some “secondary PIN-like” structure (e.g., MIX_III’s “phantom Φ”)?

4. Sharper boundary. Why is the correction (N_pin − 1) flat (not multiplicative in pure_III/V count)? Empirically verified through pure_III × 3 and V × 3 boundary cases, but no structural reason yet.