Théorème

T4 — Spectral convergence

$\alpha_k \to 1/2$ as the sieve depth $k \to \infty$.

Statement

Consider the deep sieve at $k$ stages (eliminating multiples of $p_1, \ldots, p_k$ ). Define the mixed-transition fraction:

\alpha_k = \frac{n_{12}(k)}{n_{12}(k) + n_{21}(k)},

where $n_{ij}(k)$ counts transitions from class $i$ to class $j$ among survivors at depth $k$ . Then:

\boxed{\lim_{k \to \infty} \alpha_k = s = \frac{1}{2}.}

The proof rests on three pillars:

Spectral annihilation $r_2(0) = 0$ — the antisymmetric eigenvector vanishes at site 0 (structural result).
Mertens compactness — asymptotic bounds on survivors $\prod (1 - 1/p)$ .
Gordin decomposition — the sequence $\{\alpha_k\}$ is a quasi-martingale plus a bounded remainder.

Théorème

Plain reading. The deeper we refine the sieve (the more multiples we remove), the closer the fraction of “1 → 2” transitions vs. “2 → 1” gets to 50/50. At infinite depth, it’s exactly 50/50. That $1/2$ is the stationary value defining $s$ . T4 proves that the sieve “finishes the job”: no residual bias remains.

Why it matters

T4 is the bridge between the finite sieve (at depth $k$ ) and the ideal sieve (at infinite depth). T1, T2, T3 are exact statements about the ideal transfer matrix; T4 guarantees that the empirical matrix computed at finite depth converges to that ideal matrix.

Without T4, $|\alpha_k - 1/2| > 0$ would persist, meaning a new constant (different from $s = 1/2$ ) would emerge in the limit. PT would then be ill-defined. T4 closes this objection.

Proof — outline

Reduce the convergence of $\alpha_k$ to convergence of a Markov chain on the transition state.
Decompose $\alpha_k$ into quasi-martingale plus bounded remainder (Gordin).
Bound the remainder via Mertens compactness.
Annihilate the antisymmetric mode at site 0 by the $r_2(0) = 0$ argument.
Conclude: only the symmetric mode survives, eigenvalue $\lambda_1 = 1$ , uniform stationary distribution — hence $\alpha_\infty = 1/2$ .

Detailed proof

Step 1 — Exact recurrence

At each depth $k$ , the deviation $D(k) = \alpha_k - 1/2$ satisfies an exact recurrence (Recurrence Theorem, ch. 7):

D(k + 1) = (p_k - 3) D(k) + \Delta_k,

where $\Delta_k$ is a remainder depending on correlations between primes eliminated at step $k+1$ .

This recurrence is verified for $k = 3, \ldots, 11$ by direct computation on survivors.

Step 2 — Spectral factorisation

The characteristic polynomial of the recurrence operator factors as:

f(1) = 4 (\alpha - 1/2)^2 (\alpha^2 + (p - 3)\alpha + 1).

The first factor $(\alpha - 1/2)^2$ has a double zero at $\alpha = 1/2$ , forcing the asymptotic fixed point. The second factor has complex roots for $|p - 3| < 2$ and real roots in $(-1, 0)$ for $|p - 3| > 2$ — in all cases, $|R_\text{spec}| < 1$ .

Step 3 — Spectral bound

The global spectral bound is:

R_\text{spec} \approx 0.287 \ll 1.

Strictly less than 1, ensuring geometric convergence of $D(k) \to 0$ .

Step 4 — Annihilation $r_2(0) = 0$

To rigorously close the convergence, the remainder $\Delta_k$ must not contribute to the dominant mode $\lambda_2^2 \approx 0.36$ . Here the spectral annihilation intervenes:

The antisymmetric eigenvector $r_2$ associated with $\lambda_2$ has an exactly zero value at state 0, hence:

\langle r_2, \delta_0 \rangle = r_2(0) = 0.

This result is structural (cf. ch. 7, Spectral Closure Theorem) and unconditional on PNT. Consequence: only the mode $\lambda_1^2 \approx 0.012$ survives in correlations starting from state 0, and convergence is exponential with strongly damped exponent.

Step 5 — Gordin decomposition

The sequence $\alpha_k$ writes:

\alpha_k = M_k + R_k,

with $M_k$ a martingale relative to the survivor filtration, and $R_k$ a remainder bounded by Mertens compactness:

|R_k| \leq C \cdot \prod_{i \leq k} \left(1 - \frac{1}{p_i}\right) \to 0.

By the martingale convergence theorem and $R_k \to 0$ , $\alpha_k$ converges a.s. to a limit. Identification of that limit with $1/2$ is forced by steps 1–4.

Consequence: closing the hierarchy problem

The result $r_2(0) = 0$ has an unexpected consequence: it closes the PT hierarchy problem. The mode $\lambda_2^2 \approx 0.36$ would have contributed a hierarchical phase shift between scales; its annihilation at site 0 ensures no new scale spontaneously emerges. This is what allows T5 to fix a unique fixed point $\mu^* = 15$ without hierarchical corrections.

For the complete derivation and auxiliary proofs (recurrence, factorisation, spectral bound, annihilation), see chapter 7 of the monograph.