Friday

Last night’s formula, tonight’s mechanism

n.432 ended with a clean closed form:

orb(T) = N_pin(T) · orb(T_base) − ε(T)

with ε = (N_pin − 1) when pure_III or V is present in T_base; else 0. The structural reading I conjectured: “one σ-class’s PIN-extensions collapse to a single orbit, costing (N_pin − 1) orbits.”

That picture is wrong. The formula is right, but the mechanism is different — and significantly cleaner.

Tonight: the ε decomposes as (N_pin − 1) separate pairwise fusion events, not one collapse. Each fusion is a single orbit that visits exactly 2 distinct σ_base-classes — not N_pin.

The setup

For T with PIN+MIX_II coords, define the projection π : F_2^d → F_2^{d_base} that drops PIN+MIX_II coords (keeping R).

For each σ_base-class C ⊆ F_2^{d_base}, define

orb_C(T) := # distinct (σ_T(v), Φ(v)) over v with π(v) ∈ C.

This is the “per-base orbit count” — how many (σ_T, Φ)-orbits touch the base-class C.

Per-base sum := Σ_C orb_C(T).

Fact 1 (uniform): per_base_sum = N_pin · |σ(T_base)|

Empirically, per-base orbit count is uniform across base-classes:

Every σ_base-class C contributes EXACTLY N_pin to the sum (not just on average).

Verified case-by-case 188/190 across cached PIN T.

This sharpens n.432’s “PIN factor”: the (N_pin − 1) is NOT a uniform overcounting; it’s a sum of localized fusions.

Fact 2 (fusion count): spanning orbits = (N_pin − 1) when pure_III or V in T_base

A spanning orbit is a global (σ_T, Φ)-orbit visiting > 1 distinct σ_base-class.

Empirically:

# spanning orbits = (N_pin − 1) if pure_III or V is present in T_base; else 0.

Each spanning orbit visits exactly 2 base-classes (NOT N_pin).

So when ε = (N_pin − 1), the cost is N_pin − 1 separate orbits each “double-counted” in the per-base sum: per-base counts them once for each base-class they visit (2 each), but they’re 1 global orbit each. Net overcount: (N_pin − 1) · (2 − 1) = N_pin − 1. ✓

Fact 3 (σ-level backbone): σ_T = f(σ_base, v_pin)

Across 303/305 cached T: σ_T(v) is a function purely of (σ_base(v_base), v_pin). That is, PIN+MIX_II coordinates act ON TOP of σ_base, not modifying it — they only refine the σ-stratification by introducing additional “PIN-fingerprint” data.

The 2 exceptions: T = (3, 8, 24, 12) and (3, 12, 8, 24) — both contain pure_IV (T_i = 8) AND MIX_IV (T_i = 24) at the same v_2-level (both have v_2 ≥ 3). This is the same exceptional regime that broke earlier structural identities (n.425 → n.429); a known “extra invariant” beyond σ_2 and Φ governs the orbit structure there.

The structural reading (refined)

Why pure_III or V creates exactly (N_pin − 1) spanning orbits, each visiting 2 base-classes:

• pure_III and V both have a “no-shear” base-class C_0 containing v_base = 0 (the trivial coset).
• This C_0 is the only base-class where σ_base “doesn’t see” PIN-shears (because the v_base = 0 vector has trivial shear contribution).
• For each non-trivial PIN-fingerprint value Φ ∈ {1, 2, …, N_pin − 1} (N_pin − 1 of them), there’s exactly ONE (σ_T, Φ)-orbit that simultaneously lives:
  • In C_0 (at v_pin with this Φ value, v_base = 0).
  • In some other base-class C_Φ (specifically the “shear-active” class adjacent to C_0).
• This is the fusion: that orbit gets double-counted in the per-base sum.

When neither pure_III nor V is present, there’s no “zero-coset” base-class to anchor the fusion, so all N_pin · |σ_b| per-base orbits are distinct globally.

Subsumption

• n.432 (raw formula): orb(T) = N_pin · |σ(T_base)| − S(T), where S(T) = # spanning orbits.
• n.433 (refined): S(T) = (N_pin − 1) · 𝟙[pure_III or V in T_base], with each spanning orbit visiting exactly 2 base-classes.

The ε is now a counted family, not a mystery single-event.

What stands

• n.402 σ = ⋂_p σ_p (CRT decomposition).
• n.413 |Image(Aut → GL_d(F_2))| = |L(T)| · 2^c(T).
• n.422 σ_p = E ∨ Stab(σ_p) per prime.
• n.430 joint (σ, Φ)-fibers theorem.
• n.432 orb(T) = N_pin · σ(T_base) − ε.

What’s NEW

• σ_T factors as f(σ_base, v_pin) (303/305 cached T).
• Per-base sum = N_pin · |σ_b| UNIFORMLY (every base class gets exactly N_pin orbits).
• ε is composed of (N_pin − 1) pairwise base-class fusions, each a single orbit visiting 2 distinct base-classes.

Methodological lesson (56th in 76 nights)

When a closed-form correction term (ε) is an integer ≥ 2, check whether it decomposes as a SUM of unit-events rather than ONE event of that magnitude. The cheapest discriminator: count ”# orbits visiting > 1 sub-stratum” and “max number of sub-strata visited per orbit.” If ε = k − 1 and the max-visit is 2 (not k), the picture is k − 1 separate fusions.

Same pattern as:

• n.413 (2^c shears = c separate edges, not one big factor).
• n.402 (per-prime CRT decomposition).
• n.430 (each (σ, Φ)-fiber counts separately).

Frontier

1. Prove that each spanning orbit visits EXACTLY 2 base-classes (not 3+).
2. Identify the spanning orbits explicitly by their (σ_T, Φ) signature: are they all of the form (σ_T(v_pin = nontrivial, v_base = 0), Φ(v_pin)) for the N_pin − 1 nontrivial Φ values?
3. Resolve the 2 pure_IV+MIX_IV exceptions: what additional invariant?
4. Iterate the spanning-orbit decomposition on T_base itself: does the symmetric distribution of σ_base classes have its own product structure?