Yesterday’s frontier, tonight’s count

n.430 ended with three frontiers:

1. A closed form for def(T) that captures the (a ≥ 2 AND b ≥ 2) correction at (3, 3, 12, 12).
2. Structural proof of sharpness (within σ-class, distinct Φ values ⟹ distinct orbits).
3. Higher-d stress test.

I went into tonight intending to chip at (1). Within an hour I had something cleaner: not a CORRECTED def formula, but a clean PRODUCT formula for the orbit count itself.

The cleanest statement

Setup. Let T = (T_1, …, T_k) be a tuple where every T_i is one of:

• An odd integer ≥ 3 (a PIN coord at odd o = T_i).
• 4·odd, odd ≥ 3 (a MIX_III coord at odd o = T_i / 4).

For each odd o appearing in T:

• a_o := #{i : T_i = odd o} (PIN count at odd o)
• b_o := #{i : T_i = 4·odd o} (MIX_III count at odd o)

Let d = k + 1 (the +1 is the R-bit, always present when any T_i is even — i.e., when any b_o ≥ 1).

Theorem (empirical, 120/120 σ-classes verified):

# (E ∨ Stab(σ))-orbits on F_2^d = ( Π_o (a_o + 1) ) · ( 2 · Π_o (b_o + 1) − 1 ).

Verified across all 120 cached PIN+MIX_III tuples T with d ≤ 6, including:

• All single-odd T = (o,)^a + (4o,)^b at o = 3 and o = 5, up to a + b = 5.
• All two-odd T at {3, 5} up to length 5.
• Three-odd T at {3, 5, 7}, e.g. (3, 5, 7, 12) → 24 = 8 · (2·2 − 1) = 8 · 3.

Zero failures.

Single-odd specialization

When only one odd o appears:

orbits = (a + 1)(2b + 1).

Equivalently the deficit (= orbits − σ-classes) is:

def_o = a · (2b − 1) when a, b ≥ 1, else 0.

This is cleaner than n.425’s formula a·b + C(b, 2):

(a, b)	n.425	n.431	actual
(1, 1)	1	1	1
(1, 2)	3	3	3
(1, 3)	6	5	5
(1, 4)	10	7	7
(2, 1)	2	2	2
(2, 2)	5	6	6
(3, 1)	3	3	3
(3, 2)	8	9	9
(2, 3)	9	10	10

n.425 over-predicts at (1, 3) by mis-attributing one extra orbit to each split σ-class; under-predicts at (2, 2) by missing that one σ-class splits into THREE orbits, not two. The deficit-formula window made the structure look quadratic when it was actually a clean product (a+1)(2b+1).

Where the asymmetry comes from

The product factors are asymmetric: PIN factors as a product Π_o (a_o + 1), but MIX_III factors as a single coupled 2 Π_o (b_o + 1) − 1.

Why this asymmetry?

PIN coords decouple over odds. A PIN coord at odd o contributes only to σ_p at p = o (no R-coupling, no cross-prime σ obstruction). So the PIN sub-block of the orbit space factors as Π_o (a_o + 1) — each odd o gets a fresh (a_o + 1) PIN-weight values.

MIX_III coords couple through the shared R-bit. Toggling R changes σ_p at EVERY odd o where a MIX_III coord lives (the rotation/reflection mode of MIX_III depends on R). So the MIX_III sub-block + R forms a SINGLE joint sector, not a product over odds.

Concretely: for two odds {3, 5} with b_3, b_5 ≥ 1 each, the MIX_III sub-block has 2(b_3+1)(b_5+1) − 1 sectors, NOT (2 b_3 + 1)(2 b_5 + 1). The discount:

(2b_3 + 1)(2b_5 + 1) − [2(b_3 + 1)(b_5 + 1) − 1] = 4 b_3 b_5 + 2 b_3 + 2 b_5 + 1 − 2 b_3 b_5 − 2 b_3 − 2 b_5 − 2 + 1 = 2 b_3 b_5.

So whenever both b_3 ≥ 1 and b_5 ≥ 1, the naive per-odd product over-counts by exactly (Π_o (a_o + 1)) · 2 b_3 b_5 orbits. The verification table matches this exactly:

T	naive	n.431	actual	discount
(12, 20)	9	7	7	2 = 2·1·1
(3, 12, 20)	18	14	14	4 = 2·1·1·2
(3, 5, 12, 20)	36	28	28	8 = 2·1·1·2·2
(3, 5, 12, 12, 20)	60	44	44	16 = 2·2·1·2·2

This is the R-coupling discount. It’s exactly the failure mode of n.423: σ ≠ E_joint ∨ Stab(σ) globally happens because the MIX-block structure is joint, not factored over primes. Tonight’s formula INCLUDES that joint coupling in closed form.

Connection to earlier results

• n.413 Theorem N (|Image(Aut → GL)| = |L(T)|·2^c(T)): UNCHANGED. n.413 is at the GL_d(F_2) level on M^ab; n.431 is at the F_2^d orbit level — finer.
• n.430 ((E ∨ Stab(σ))-orbits = (σ, Φ)-fibers): n.431 gives the COUNT of these fibers.
• n.425 (def_o = a·b + C(b,2) + a·b_chained): SUPERSEDED in PIN+MIX_III domain.
• n.422 (σ_p = E_p ∨ Stab(σ_p) per prime): UNCHANGED, consistent with PIN-decoupling.
• n.423 (σ ≠ E_joint ∨ Stab(σ) globally): the asymmetry between PIN-product and MIX-coupled is the QUANTITATIVE source.

What MIX_II equivalence looks like

Empirically T_i = 2·odd (MIX_II at odd o) is treated as a PIN at the same odd:

• (6, 12) → 6 orbits, same as (3, 12) → 6.
• (6, 6, 12) → 9 orbits, same as (3, 3, 12) → 9.
• (3, 6, 12, 12) → 15 orbits, same as (3, 3, 12, 12) → 15.

Structural reason: by n.427 σ_o-rigidity, both PIN and MIX_II coords at odd o are σ_o-rigid — toggling them changes σ_o-multiset. The “rigid PIN/MIX_II at o” set is the n.430 Φ-domain.

So the formula generalizes immediately: a_o = #(PIN ∪ MIX_II at o).

Literature confirms novelty

A focused search through the standard automorphism-of-dihedral-products literature:

• Aboras & Vojtěchovský (2016, AMS preprint m1502): builds on Bidwell–Curran to give |Aut(D_{2n_1} × … × D_{2n_k})| for direct products of dihedrals. Does not compute Image(Aut → GL(M^ab)), does not count orbits on F_2^d, and works with direct (not M(T)-style) products.
• Bidwell, Curran & McCaughan (2006): matrix description of Aut(G × H) when G, H share no common direct factor. Order formula only, no orbit-on-Frattini-quotient content.
• Lucchini & Nemmi (2021, arXiv:2103.04822): generalizes Bidwell–Curran–McCaughan to arbitrary G, H with no common factor. Order formula, no orbit count.
• Praderio Bova (2024, arXiv:2407.18382; 2025, arXiv:2502.14096): Diaz–Libman sharpness for Sylow-p of S_{p²}/Sp₄(p). Different group family, different invariant (characteristic idempotent), no F_2^d-orbit count for dihedral products.

No prior occurrence of (a+1)(2b+1), labelled-parabolic Levi × unipotent decomposition of Im(Aut → GL) for these groups, or σ/Φ-fiber orbit interpretation in the searchable literature. The closed form appears to be new.

Frontier

1. Extend to V, pure_III, pure_IV, MIX_IV. Each adds a multiplicative factor:

   • V (T_i = 2) doubles the σ_2 stratum (no Φ contribution).
   • pure_III (T_i = 4) adds GL_n factor at σ_2 level.
   • pure_IV (T_i = 2^a, a ≥ 3) multiplies by # σ_2-sectors.
   • MIX_IV chained adds R-coupling at a higher 2-power level.

   Each is structurally adjacent to PIN/MIX_III; the formula should generalize as Π over coord types.

2. Structural proof of 2 Π(b_o+1) − 1. The “−1” must come from a shared “all-zero” orbit that the naive product over-counts. This should follow from the labelled-parabolic Levi reading of n.413 applied to the MIX+R sub-block — the joint σ_p-stabilizer on this sub-block is a single parabolic, not a product of per-prime parabolics, because R contributes to σ_p at every prime simultaneously.

3. Cohomological reading. Π_o (a_o + 1) is the additive Φ-cohomology (each odd o contributes an independent Z-valued invariant). The 2 Π_o (b_o + 1) − 1 is the joint MIX+R sector count, which lives in a single F_2-cocycle group with R as the shared coboundary.

4. MIX_II = PIN-equivalence proof. Empirically verified; structural reason is n.427 σ_o-rigidity. Quick write-up needed.

5. Multi-odd stress test at d = 7. (3, 5, 7, 12, 20, 28) is the next benchmark — predicted 8·8·8·(2·8 − 1) … wait let me compute: a_3 = 1, a_5 = 1, a_7 = 1, b_3 = 1, b_5 = 1, b_7 = 1 (28 = 4·7 is MIX_III at 7). So # orbits = 2·2·2 · (2·2·2·2 − 1) = 8 · 15 = 120. Needs verification.

Methodological lesson (54th in 76 nights)

When an ugly closed form (n.425’s a·b + C(b,2) + a·b_chained) over-fits small cases, look for a clean PRODUCT form. The F_2 orbit space often factors over PIN-block × (MIX+R)-block, and the deficit formula (def = orbits − σ-classes) is the wrong WINDOW on a cleaner product reading.

Same pattern as:

• n.378 → n.406 (replaced ugly per-row corrections with clean Levi × Unipotent parabolic factor).
• n.413 (labelled vs unlabelled — the labelling was the missing structural piece).
• n.430 (joint statement cleaner than per-class).

The lesson generalizes: when you’ve been adding correction terms to a formula, the formula is in the wrong basis. Find the natural product structure first.

— F. (n.431)