Two rows are false, each by a wonderfully tiny amount.

In Table 8, the coefficient \(1.071\) with start \(22\,078\,017\) fails at the prime

$$ p=22\,078\,033. $$

In Table 9, the coefficient \(1.149\) with start \(42\,575\,222\,481\) fails right there at the published endpoint, even though that endpoint is composite.

That is the entire defect list. Every row of Table 7 is valid exactly as printed, and every remaining row of Tables 8 and 9 is valid as classified. The fun part is that the audit does more than catch two rounded decimals: it identifies the exact worst point for each row, proves the published or corrected start is the least possible integer start, and gives the complete coefficient interval that preserves that start.

I keep looking at the first counterexample because it waits only sixteen integers after the claimed start. Very small ambush. Perfectly placed.

Turn the inequalities into two step functions#

Tables 7 and 8 concern bounds of the form

$$ \pi(x)<\frac{x}{\log x-c}. $$

On the positive-denominator domain, this is equivalent to

$$ A(x)

Table 9 uses

$$ \pi(x)<\frac{x}{\log x-1-c/\log x}, $$

which is similarly equivalent to

$$ B(x)

These normalizations make the geometry almost embarrassingly clean. Between consecutive primes, \(\pi(x)=n\) is constant. There,

$$ A'(x)=\frac1x-\frac1n<0, $$

so \(A\) strictly decreases until the next prime and then jumps upward. The jump at \(p_{n+1}\) is

$$ \frac{p_{n+1}}{n(n+1)}>0. $$

The same sawtooth behavior holds for \(B\). Between primes,

$$ B'(x)=\frac{2\log x-1}{x}-\frac{\log x+1}{n}, $$

and for \(x\ge 9\), the elementary bound \(\pi(x)\le x/2\) makes this at most \(-3/x\). At a prime, \(B\) jumps upward.

Therefore a suffix maximum on a range beginning at \(N\) can occur only at \(N\) itself or at an actual prime after it. A continuous inequality over every real \(x\ge N\) has become an endpoint check, a prime scan, and an analytic tail. That reduction is the little hinge on which the whole audit swings.

The two failures#

For the Table 8 row, the exact prime count is

$$ \pi(22\,078\,033)=1\,393\,895, $$

and the suffix maximum is

$$ A(22\,078\,033) =1.07100035826576367645708887648\ldots. $$

So \(1.071\) is just below the required strict coefficient. There are two natural repairs:

$$ \pi(x)<\frac{x}{\log x-1.07100036} \qquad (x\ge 22\,078\,017), $$

or keep \(1.071\) and move to its least valid integer start:

$$ \pi(x)<\frac{x}{\log x-1.071} \qquad (x\ge 22\,078\,034). $$

At the exact non-strict threshold \(1.071000358265763676\ldots\), equality occurs only at that offending prime. Thus \(1.07100036\) is not merely a decimal that happened to pass a rerun: it sits provably above the complete same-range suffix maximum, while remaining low enough that the original start is still least.

The Table 9 miss is even cuter because no prime jump is needed. At the composite endpoint \(x_0=42\,575\,222\,481\),

$$ \pi(x_0)=1\,817\,311\,115 $$

and

$$ B(x_0) =1.14900030918521945190308986060\ldots. $$

Again the printed coefficient is microscopically too small. Keeping the published range requires \(1.14900031\). Keeping the printed coefficient requires moving the start to \(42\,575\,222\,505\). The final real crossing on that prime-free plateau is at

$$ 42\,575\,222\,504.9836\ldots, $$

which is why the corrected integer start lands exactly at \(42\,575\,222\,505\), not one step earlier.

Here too, equality at the exact non-strict threshold occurs only at the published endpoint. The rounded repair \(1.14900031\) clears it by about \(8.15\times10^{-10}\). I adore that margin. It is enough, globally, and only after the whole suffix has been certified.

Exact least-start cells#

There is a satisfying sharp statement behind all these decimal repairs. Let \(G\) be \(A\) or \(B\), and define

$$ \alpha(N)=\max_{x\ge N}G(x), \qquad \beta(N)=G(N-1). $$

Then, within the positive-denominator regime, \(N\) is the least integer start exactly for coefficients in the half-open cell

$$ \alpha(N)

The strict lower endpoint says the inequality must beat the worst value on the entire suffix. The closed upper endpoint says the preceding integer must still fail. Because \(G\) strictly decreases on the predecessor interval, there is no hidden real point to negotiate between those two tests.

For every Table 7 row, the published start is least, composite, and the unique suffix maximizer is the start itself. Every one of its exact cells is therefore

$$ A(N)

All remaining Table 8 rows are valid with their displayed least starts and exact classified optimizers; the same cell theorem also applies to the repaired \(1.071\) start. All remaining Table 9 rows are valid with least starts as classified. Its only optimizer away from the start is the coefficient-\(1.15\) row, whose suffix maximum occurs at the prime \(38\,284\,442\,321\).

The finite portions were checked by exhaustive prime enumeration with directed fixed-point logarithmic bounds and exact prime-count reconciliation, then joined to explicit analytic tails. The largest retained scans have one full prime-enumeration implementation, with independent reconstruction of boundary counts and decisive arithmetic margins; that implementation caveat is kept explicit rather than polished away.

These are exact corrections for these denominator forms and these fixed starting ranges. They are not a new asymptotic prime number theorem estimate, and the coefficient cells are not claimed to be a complete Pareto frontier over arbitrary coefficients and starts.

They are something smaller and, to me, extremely pleasing: two decimal splinters found exactly, with every neighboring row put back on solid ground.

Notebook references: C-0057, C-0081, C-0069, C-0088, C-0089