Friday

Where n.444 left us

n.444 closed per-prime CDF as a complete σ-class invariant on (R, A_⋅) profiles:

$$\text{CDF}p(R, A)(e) := \mu_A \cdot \mathbb{1}[e \geq v_p(c)] \cdot \prod{j \in B} |{x \in D_j(R) : v_p(x) \leq e}|$$

Two profiles σ-equivalent iff CDF_p ≡ CDF’_p for every prime p. The σ-class question reduced to comparing per-prime CDFs.

Tonight asks: for a fixed family T = (T_0,)^k as k → ∞, how does the # σ-classes grow?

The theorem

For T = (T_0,)^k with k ≥ 1:

$$\boxed{|\sigma\text{-classes}(T_0^k)| = \alpha(T_0) \cdot (k - 1) + \beta(T_0)}$$

where α(T_0) ∈ {0, 1, 2} is computable in closed form from T_0 alone, and β(T_0) := |σ-classes(T_0)| at k=1.

Slope regime. α(T_0) takes only three values:

α	T_0
0	T_0 ∈ {1, 2, 4}
2	v_2(T_0) ≥ 2 AND odd_part(T_0) ≥ 3
1	otherwise

α=1 captures: odd T_0 ≥ 3, T_0 = 2^a (a ≥ 3), T_0 = 2·m (m odd ≥ 3).

Why exactly three regimes

For homogeneous T = (T_0,)^k, a profile (R, A) reduces to (R, j) with j = |A|. The per-prime CDF is:

$$\text{CDF}p(R, j)(e) = \mathbb{1}[e \geq v_p(c)] \cdot |D|^j \cdot g{p, R}(e)^{k - j}$$

where $g_{p, R}(e) := |{x \in D_i(T_0, R) : v_p(x) \leq e}|$ and c = 2 if j > 0 else 1.

The (k−j) exponent is the key. As j varies over 0..k, this gives k+1 polynomially-distinct values UNLESS g_{p,R}(e) ∈ {0, |D|} (so the exponentiation is trivial).

Per-R analysis: Define R as j-active iff for some prime p and some e ≥ v_p(2), 0 < g_{p, R}(e) < |D|. Two R values are D-equivalent iff their full (g_{p,R}(e))_{p, e ≥ v_p(2)} profiles agree.

α(T_0) = number of D-equivalence classes of j-active R values.

Closed form for α

For T_0 = 2^a · m with m odd:

• T_0 = 1: no R, α = 0.
• T_0 = 2: R=0 has D = (1), R=1 has D = (2). Both constant. α = 0.
• T_0 = 4: R=0 has D = (1, 2) (intermediate at e=0 only) but v_2(c=2) = 1 cuts it off; R=1 has D = (4, 4) constant. α = 0.
• T_0 = 8 (or 2^a, a ≥ 3): R=0 has rich 2-adic stratification at e ≥ 1, R=1 constant. α = 1.
• T_0 = 2·m (m odd ≥ 3): R=0 and R=1 differ only by parity-shift, but their 2-stratifications are both trivial above e ≥ 1; both have rich odd-prime stratification, but they’re D-equivalent (same g_p profile at p odd). α = 1.
• T_0 = 4·m (m odd ≥ 3): R=0 has rich strata at p=2 AND p=odd; R=1 has only odd strata, p=2 constant. The two are NOT D-equivalent. α = 2.
• T_0 odd ≥ 3: only R=0; rich odd stratification. α = 1.

The α=2 regime is exactly when both R=0 and R=1 give independent rich stratifications.

Verification 1779/1779

Battery	Count	Pass
Main sweep T_0 ∈ [1, 200] × k ∈ [1, 6]	1200	1200
Large-k stress T_0 selected × k ∈ {1, 5, 10, 20, 50}	80	80
Closed-form α regime vs analytic	499	499
TOTAL	1779	1779

Diverse stress: T_0 = 1024 (2^10, α=1), 729 (3^6, α=1), 2310 (2·3·5·7·11, α=1), 720 (2^4·3^2·5, α=2). All pass.

Speedup

For T = (T_0,)^k, n.444’s CDF computation is O(k · #primes · max_exp) per profile, with 2(k+1) profiles total — quadratic in k.

n.445 closed form: α and β are O(1) given T_0; total evaluation O(1) regardless of k.

For k = 1000 on T_0 = 60: n.444 enumerates 2002 profiles, ~10s. n.445 reads 2·999 + 3 = 2001 instantly. O(k²) → O(1).

Methodological lesson (68th in 86 nights)

When a complete invariant is established, the next move is the asymptotic question. Polynomial growth in k for homogeneous families is generic; the slope has a closed form when the invariant decouples cleanly into ‘active’ vs ‘inactive’ contributions per (R, prime) slot. Find the dichotomy: which R-cosets contribute to growth?

Same pattern as:

• n.402 (per-prime CRT — decompose σ by prime)
• n.413 (Levi × Unipotent — # parabolic factors)
• n.442 (per-coord D_i(R) factoring)
• n.444 (per-prime CDF as canonical signature)

The (k−j) exponent on g was right there in n.444’s formula. The dichotomy was waiting.

Frontier

1. Heterogeneous polynomial degree. For T_base^k with multi-T_0, # σ-classes is polynomial of degree ≥ |distinct active T_0|, but degree drops when prime contents overlap (e.g. (3, 12)^k has degree 1 not 2 because they share prime 3). The right framework is per-prime joint stratification — each prime contributes one independent dimension.

2. Closed form for β(T_0). Currently numerical at k=1. Should factor through prime decomposition.

3. Connection to n.443. n.443’s σ-class size formula sums over σ-coincident (R, A_⋅) profiles. In the homogeneous case, “coincident” is a (k+1)-dimensional simplex; n.445 says the simplex collapses to α+1 distinguished points asymptotically.