Erdős 699: adjusting the Padé block while retaining the cofactor clause

P699 research · Numerical joint constants · Current seven and nine bounds

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The numerical argument need not choose the Padé index as $m=\lfloor t\log h\rfloor$, even when $t$ was used to define the retained cofactor clause. Choosing a smaller index decreases the residual denominator cost. Quantifying this decrease strengthens the numerical estimate without changing the clause or its polynomial proof forest.

The denominator saving

Put $x=\log h$. Write a retained clause as

$$\sum_p(g_p+b_pe_p)_+\ge R,\qquad R>0,$$

where its outward-rounded intercepts satisfy

$$g_p>t\kappa_p-b_p,\qquad \kappa_p=L_{s(p)}a_p\log p>0.$$

Here $a_p$ is the exponent of $p$ in its seed product, $b_p$ is its factor multiplicity, and $L_0=A+B$, $L_1=B+C$ are the two directional block lengths. Choose any positive rational

$$\kappa\le\min_p\kappa_p.$$

At integer Padé index $m\le tx$, the actual logarithmic residual denominator cost is bounded by

$$D_m(x)=\sum_p\bigl(m\kappa_p-b_p(1-e_p)x\bigr)_+.$$

Lemma. If the retained clause fails, then

$$\boxed{D_m(x)\le\bigl(Rx-\kappa(tx-m)\bigr)_+.}\tag{1}$$

Indeed, set $z_p=(g_p+b_pe_p)x$. The expression inside each actual positive part is at most $z_p-\kappa_p(tx-m)$. For arbitrary real $z_p$ and $s_p\ge s\ge0$,

$$\sum_p(z_p-s_p)_+\le\left(\sum_p(z_p)_+-s\right)_+.$$

If no summand survives the subtraction, the assertion is immediate. Otherwise, summing over surviving indices subtracts at least $s$ and uses at most the original sum of positive parts. Apply this with $s=\kappa(tx-m)$ and $\sum_p(z_p)_+<Rx$ to obtain (1).

The subtraction in (1) is made once, not once for every prime. No assumption about the active sign mask is needed.

A fixed index covering a height interval

Use the finite constants $A_1,A_2$ from the linked numerical argument, and let $d$ be the product degree. Write $c,k$ for the two exponential decay rates before adding a coefficient-gcd contribution. If a common divisor of both adjacent Padé rectangles has logarithm at least $G_m$, the nonzero-integer argument gives

$$1\le mA_1\exp\bigl(D_m(x)-cm-G_m\bigr) +2^dmA_2\exp\bigl(dx+D_m(x)-km-G_m\bigr).\tag{2}$$

Let $y=tx\ge m$. The upper bound in (1) becomes

$$D_m(x)\le\max\{0,\kappa m+(R/t-\kappa)y\}.\tag{3}$$

After taking logarithms, twice the first term in (2) is bounded by the maximum of two affine functions of $y$. Twice the second term is bounded by the maximum of two affine functions with slopes

$$d/t,\qquad (R+d)/t-\kappa.$$

These functions are convex. Checking strict negativity at both endpoints therefore certifies every height between them. If $\kappa t\ge R$, the first term is nonincreasing with $y$; only its left endpoint needs checking.

The left endpoint may exceed $m$. Thus, at an already covered height $x_0$, one can choose $m<tx_0$, check (2) there, and use the same index until the right endpoint fails. A sequence of overlapping certified intervals then joins a finite height to an existing uniform analytic tail. The usual condition $h\ge s+1$ ensures that the shifted factors are at most $2h$; it is checked separately from the interval inequalities.

A smaller block for a uniform tail

Choose a rational $0<\sigma\le t$ and set $m=\lfloor\sigma x\rfloor$. Formula (1) implies

$$D_m(x)\le B_\sigma x,\qquad B_\sigma=\max\{0,R-\kappa(t-\sigma)\}.\tag{4}$$

Suppose $\log G_m\ge\ell m$ for every $m\ge m_0$. Put $C=c+\ell$ and $K=k+\ell$. Sufficient positive decay margins are now

$$\boxed{\sigma C>B_\sigma,\qquad \sigma K>B_\sigma+d.}\tag{5}$$

All floor and integral prefactors are handled exactly as in the original numerical argument, with block $\sigma$ and denominator exponent $B_\sigma$. In particular $\sigma x\ge m_0+1$ ensures that the chosen index lies in the uniform gcd range. The two inequalities force $C,K>0$, so replacing $m$ by $\sigma x-1$ introduces the valid factors $e^C,e^K$.

When $K>C$, the value

$$\sigma_* = \frac{d}{K-C}$$

balances the two margins. It is a useful candidate when $\sigma_*\le t$; a nearby rational value is checked with directed bounds. This is a choice to test, not an assertion that it is optimal in every parameter regime. The difference $K-C=k-c$ is independent of the chosen gcd lower rate.

Every whole-index bound must certify every source and join the resulting height to the polynomial and finite bounds. The complete proof at seven supplies these certificates. The complete proof at nine now does so as well.

Initial sources with explicit cofactor targets

For an initial alternative, its negation directly bounds each $e_p\le\alpha_p<1$. One can then use

$$D_m(x)\le\sum_p\bigl(m\kappa_p-b_p(1-\alpha_p)x\bigr)_+.\tag{6}$$

This bound does not require $m\le tx$: the nominal block used to construct a previous clause is irrelevant. In numerical upper bounds for (6), each $\kappa_p$ is rounded upward.

The right side is the maximum, over all subsets $S$ of the component primes, of

$$m\sum_{p\in S}\kappa_p-x\sum_{p\in S}b_p(1-\alpha_p).$$

Every slope is nonpositive, so the first term of (2) need only be checked at the left endpoint. After adding $dx$, the second term is again the maximum of finitely many affine functions. Its endpoint checks certify the entire interval. For a pair there are only four subsets. The balanced choice $m\approx d x_0/(K-C)$ can be tested even when it exceeds the old nominal value $tx_0$.