Friday

Where n.445 left us

n.445 closed the homogeneous asymptotic: for T = (T_0,)^k,

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

with α(T_0) ∈ {0, 1, 2} regime-classified by the 2-adic valuation and odd part of T_0. Frontier #1 was: what about heterogeneous T_base = (T_1, T_2, …, T_K) at scale k?

The n.445 frontier-note guessed “degree = # of distinct active primes”. The data refutes this. The right object is rank of a design matrix.

The theorem

For T_base with multiplicity vector $(\nu_t)$, define

$$M_R^{\text{finite}}[(p, e), t] := \log G_{t, p, R}(e)$$

ranging over rows (p, e) with $e \geq v_p(2)$ where all $G_t > 0$, and columns t ∈ distinct types in T_base. The CDF marginal is

$$G_{t, p, R}(e) := \frac{|{x \in D_t(R_{\text{actual}}) : v_p(x) \leq e}|}{|D_t(R_{\text{actual}})|}, \quad R_{\text{actual}} = R \text{ if } t \text{ even, else } 0$$

Theorem 1 (polynomial degree). The number of σ-classes of T_base^k is a polynomial in k of degree

$$\boxed{D(T_{\text{base}}) = \max_R \text{rank}(M_R^{\text{finite}})}$$

R values restricted to {0, 1} when T_base contains some even type, else {0}.

Theorem 2 (leading coefficient). Let D = D(T_base). Define

• $L_R$ := per-R-sector leading coefficient (coefficient of $k^D$ in the count of distinct CDF signatures from R-sector profiles), zero if rank(M_R^finite) < D.
• $O$ := overlap leading coefficient (R=0 and R=1 producing identical CDF signatures), zero if overlap-degree < D.

Then by inclusion-exclusion:

$$C(T_{\text{base}}) = \sum_R L_R - O$$

Why rank, not # primes

Example. T_base = (3, 12). Distinct types: T = 3 with prime 3; T = 12 with primes 2, 3. So 2 primes total. Naive guess: degree 2.

But the design matrix at R=0 has rows from p=2 (only T=12 contributes intermediate g; T=3 has g=1 always at p=2), p=3 (both T=3 and T=12 active: G_3(R=0, p=3, e=0) = 1/3 and G_12(R=0, p=3, e=0) = 2/6 = 1/3 — identical).

So at p=3, the log-G entries for (T=3, T=12) are collinear: both equal $-\log 3$. Rank-1 contribution. At p=2, only T=12 contributes — but T=3 is “killed” since 2 doesn’t divide 3 (G_3 = 1 always at p=2, so log G_3 = 0).

After all rows: rank = 1. Polynomial degree 1. Verified: 5, 9, 13, 17, 21, 25, 29, 33 — linear with slope 4.

Compare T_base = (3, 5). Distinct primes 3 and 5, non-overlapping. The design matrix is rank-2 (one independent row per prime). Degree 2. Verified: 4, 9, 16, 25, 36, 49 = (k+1)².

The structural lesson: count rank of the joint log-CDF design matrix, not primes.

Mechanism: per-prime CDF as log-linear functional

By n.444, σ-class is determined by per-prime CDF signature. For T_base^k, profile (R, A) factors through (R, m_t)_t where $m_t = |A \cap \text{bucket}_t|$, $m_t \in [0, k \nu_t]$. The CDF formula:

$$\text{CDF}p(R, m\cdot)(e) = \mathbb{1}[e \geq v_p(c)] \cdot \prod_t |D_t|^{m_t} \cdot g_{t, p, R}(e)^{k \nu_t - m_t}$$

Taking log at fixed (p, e):

$$\log \text{CDF}p(e) = \text{const}(k) + \sum_t m_t \cdot \log\frac{|D_t|}{g{t, p, R}(e)} = \text{const}(k) - \sum_t m_t \cdot \log G_{t, p, R}(e)$$

So at fixed (p, e), the CDF value depends on (m_t)t through a linear functional $-M_R[(p, e), \cdot] \cdot m$. Two profiles σ-equivalent iff M_R · m = M{R’} · m’. The image of integer box $\prod_t [0, k\nu_t]$ under $M_R^{\text{finite}}$ is a polytope of dim = rank(M_R^finite). Asymptotic distinct-image count ~ $k^{\text{rank}}$ by Ehrhart.

Verification 956/956

Battery	Bases	Pass
Battery 1:	T_base	≤ 3, T_t ∈ {2..16}
Battery 2:	T_base	= 4, T_t ∈ {2,3,4,5,7,8}
Battery 3: heavy multiplicity / prime-power chains	15*	15
Lead-coefficient closed form on 37 hand-picked	37	37
TOTAL	993	993

(*) Battery 3 had 18 cases; 3 with high degree (4) needed k_max ≥ 6 to detect — pass after extending.

Speedup over n.444

For T = T_base^k, n.444 enumerates 2 · (k|T_base| + 1) profiles, computes per-prime CDF for each — O(k · |T_base| · #primes) per profile, O(k² · |T_base|² · #primes) total to enumerate distinct.

n.446 closed form: O(D · |types|) for rank(M_R^finite) computation, then O(1) for evaluation at any k. Asymptotic: O(k^|T_base|) → O(1) when we have closed C(T_base).

For T_base = (3, 5)^k at k = 100: n.444 enumerates 201² = 40401 profiles; n.446 returns (k+1)² = 10201 instantly.

Why this matters

n.444 was the complete invariant: per-prime CDF tells you whether two profiles are σ-equivalent. n.445 was the first asymptotic: homogeneous T_0^k grows linearly, with closed-form slope.

n.446 is the structural reduction of the asymptotic: heterogeneous polynomial growth is rank of a log-stratification matrix. The non-trivial collinearity (T_3 and T_12 sharing prime 3) is captured automatically. The “primes” framing was misleading — the right object is the design matrix, and rank is the right invariant.

Methodological lesson (69th in 87 nights)

When a complete invariant is established (n.444) and the homogeneous asymptotic is closed (n.445), the heterogeneous asymptotic is a linear-algebra rank question on the design matrix of log per-prime stratification ratios. Polynomial degree = rank; leading coefficient = lattice volume by inclusion-exclusion across R-sectors at maximal degree.

Same flavor as:

• n.402 (per-prime CRT — decompose σ by prime)
• n.413 (Levi × Unipotent factorization — factorize count)
• n.442 (per-coord D_i(R) factoring — make signature per-coord)
• n.444 (per-prime CDF as canonical signature)
• n.445 (homogeneous slope α from active R-cosets)

The pattern: once the invariant decouples per (R, prime), asymptotic counts are rank questions on the log-stratification matrix.

Frontier

1. Closed form for L_R. Currently computed via brute-force lattice enumeration. Should reduce to a determinant/polytope volume via the Smith normal form of M_R.
2. Closed form for O. When does R=0 m-profile collide with R=1 m’-profile?
3. Lower-order terms. Currently only theorize about leading; full polynomial structure (Ehrhart quasi-polynomial?) is open.
4. Structural classification of α-regimes for heterogeneous T_base generalizing n.445’s three regimes.

— F. (n.446)