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All logarithms are natural. On the following seven points $(a,b)$ use the indicated integer weights, divided by $D=10^6$:

These are Lemm's seven points. Their differences $a-b$ are respectively $-1,0,1,2,3,4,5$. Let $H$ be the entropy of this probability measure, and let $h$ be the largest of the entropies of $a$, $b$ and $a+b$. The marginal weights for $a$ and $b$, in increasing order of their values, are both

$$(250,252406,718938,28406),$$

and those for $a+b$ are $(449000,551000)$. The exact arithmetic certificate gives

$$1.2238605<H<1.2238606,\qquad 0.6879584<h<0.6879585,\qquad \frac hH<\frac{56213}{100000}=:\tau.$$

The logarithm certificate. For a positive rational $t$, reduce it to $t=2^e r$ with $1\le r<2$. For $v=(r-1)/(r+1)$ and any positive integer $n$,

$$0\le\log r-2\sum_{j=0}^{n-1}\frac{v^{2j+1}}{2j+1} \le\frac{2v^{2n+1}}{(2n+1)(1-v^2)}.$$

This follows by integrating the geometric series for $2/(1-v^2)$; the displayed upper bound dominates its positive tail. Use the same series with $v=1/3$ to enclose $\log2$. Since $v<1/3$ for $r<2$, $n=32$ suffices for all the displayed enclosures. Applying these rational bounds to $-\sum p\log p$, with interval directions reversed when needed, proves the strict inequality $h<\tau H$. The retained checker does precisely these rational operations and fails unless that inequality and the marginal identities hold.

Integer cells with injective differences. Replace each source point by $(x,y)=(2b,a+b)$. Its difference $y-x$ equals $a-b$, and its projections $x,y,2y-x$ equal $2b,a+b,2a$. Thus the projection entropies are unchanged. For each integer $m\ge1$ form words of length $Dm$ in these seven cells, using cell $i$ exactly $m w_i$ times, where $w_i$ is its weight. Encode a word as

$$\left(\sum_{j=0}^{Dm-1}x_j11^j, \sum_{j=0}^{Dm-1}y_j11^j\right).$$

The difference digits lie in $[-1,5]\subset[-5,5]$, so balanced base-eleven uniqueness implies that the encoded difference determines the word. The cell set $S_m$ therefore has distinct differences and exactly

$$n_m=\frac{(Dm)!}{\prod_{i=1}^7(mw_i)!}$$

cells. A projected word has fixed multiplicities given by the associated marginal weights. Consequently each projection has at most its corresponding multinomial count. Writing $N_m$ for the largest of these three counts, Stirling's formula gives

$$\log n_m=DmH+O(\log m),\qquad \log N_m=Dmh+O(\log m).$$

These are deductions about exact counts, not numerical extrapolations. In particular $n_m\to\infty$ and $\log N_m/\log n_m\to h/H<\tau$. Choose a fixed number $\sigma$ with $h/H<\sigma<\tau$. For sufficiently large $m$, $N_m\le n_m^\sigma$; increasing a constant covers the remaining $m$.

Every requested number of primes. The sequence $n_m$ is strictly increasing: appending any one fixed word with $w_i$ occurrences of each cell injects the words for $m$ into those for $m+1$, with additional words available. Moreover

$$\frac{n_{m+1}}{n_m} =\frac{\prod_{j=1}^D(Dm+j)} {\prod_{i=1}^7\prod_{j=1}^{w_i}(mw_i+j)} \longrightarrow \frac{D^D}{\prod_i w_i^{w_i}}=e^{DH}.$$

Thus these ratios have a finite bound $R$. Given $u\ge2$, choose the least $m$ with $n_m\ge u$. For $m>1$ we have $n_m<Ru$, while $m=1$ is covered by a fixed larger constant. Select any $u$ cells of $S_m$. They still have injective differences, and their three projection sizes sum to at most $3N_m$. L202 realizes them with exactly this projection sum as the count in a block of $\lfloor\alpha P\rfloor$ consecutive positive integers for every $3\le\alpha<4$. It follows that

$$G_\alpha(u)\le3N_m\le C_0 n_m^\sigma \le C u^\tau.$$

All constants arise from the fixed seven-point measure. They are independent of $\alpha$, which enters only the prime realization.