Infinitely many emirps

For an integer with exactly $L$ decimal digits, let $R_L(n)$ reverse those digits. Define the directed count

The target is

For example, 13 and 31 contribute two integers, not one unordered pair. Nonvanishing at arbitrarily large even lengths is sufficient but stronger than the unrestricted target. A route using arbitrary lengths must remove palindromic primes. No positive-density estimate is required, and the source reports no proof of infinitude.

Quantities used in the proposed routes

The sufficient gates below use even digit lengths $L\ge4$, with $X=10^L$. Put

Thus the admissible integers have $L$ digits and both end digits are nonzero. Reversal $R_L$ uses exactly $L$ digits, including zero padding when applied outside this admissible set. All counts are directed, so a number and its different reversal are counted separately.

For $11\le z<X/10$, write $P^-(n)$ for the least prime factor of $n$. The three roughness counts are

Here $\mathcal A_L$ is the admissible set and $\mathsf A_L(z)$ is a count, not the same object. The reported finite triples $(A,B,D)$ abbreviate these three counts at $z=\lfloor X^{1/3}\rfloor$. Composite survivors at this cutoff have two prime factors counted with multiplicity; prime squares are allowed. The source's exact identity at these even lengths is

Consequently, $2\mathsf A_L(z)>\mathsf B_L(z)$ at arbitrarily large even lengths would suffice for infinitude. It is not necessary, because the nonnegative composite–composite correction can also give positivity. The finite computations remain source-reported and unreproduced here.

For the minor-arc route, all logarithms are natural and $e(t)=\exp(2\pi i t)$. The prime-only sums are

These weights count primes, not proper prime powers. Fix a constant $B>0$, set $Q=(\log X)^B$, and let $\mathfrak M$ be the union, on $\mathbb R/\mathbb Z$, of arcs of radius $Q/X$ around reduced fractions $a/q$ with $q\le Q$, including $0/1$. For sufficiently large $X$, the source uses $2Q^3<X$; let $\mathfrak m$ be the complementary minor arcs. Its normalized residual is

The source reports $C_L=(33/20)X+\mathcal U_L+o(X)$, where $C_L=\sum_{p\in\mathcal A_L,\ p\text{ prime},\ R_L(p)\text{ prime}}(\log p)(\log R_L(p))$. Its unproved sufficient gate asks for $\limsup_{L\to\infty,\ L\text{ even}}\mathcal U_L/X>-33/20$, with $B$ fixed.

Finally, the Möbius checkpoint sums over primes $p$ in the same $\mathcal A_L$. The weight $\mu(n)$ is zero when a prime square divides $n$, and otherwise is $(-1)^k$ for $k$ distinct prime factors. A bound for that checkpoint alone is not the full sufficient gate: the additional divisor and truncation controls remain open.