Erdős 699: fully numerical thresholds for unequal Padé blocks

P699 research · Unequal Padé blocks

The twenty pair alternatives in the index-twelve ramp certificate have fully numerical starting heights. Combining the uniform coefficient-gcd estimates with a finite prime-interval bootstrap gives

The completed index-twelve proof excludes the entire finite range below this stopping bound. Every constant used below is evaluated or given by an explicit rational formula; no unevaluated asymptotic constant remains.

Finite factorial and integral bounds

Fix positive directions $(\lambda,\mu,\kappa)$ and write $H=\lambda+\mu+\kappa$, $L_0=\lambda+\mu$, and $L_1=\mu+\kappa$. The two adjacent rectangles are

Assume $m\ge2$. Put

The multinomial theorem gives

For $\varepsilon=0$, the integral prefactor is exactly $\mu m$ times this multinomial coefficient. For $\varepsilon=1$, its ratio to the multinomial coefficient is

Thus both prefactors are at most $Fm e^{Sm}$.

For the seed $aP-bQ=D_0>0$, put $T=aP$ and $z=D_0/T$. Define

and $q=S+\log M_Q$, $e=S+\log M_E$. These are precisely the two saddle-point quantities in the unequal-block lemma.

In either integral, remove the $(m-2)$-th power of the corresponding base integrand. Every remaining exponent is nonnegative, even when $\mu=1$ and $\varepsilon=1$. Its remaining integrand is consequently at most one. Therefore the explicit bounds

hold for both rectangles. No Stirling remainder or limiting argument is needed in (2).

A numerical coefficient-gcd starting index

The uniform gcd checker uses common prime intervals for both adjacent rectangles. For each eligible $t$, put $k_X=\lfloor Xt/H\rfloor$ for $X\in\{\lambda,\mu,\kappa\}$. After its exact floor-stability bound, the interval

is contained in both coefficient-gcd intervals for every $m\ge m_0$. The intervals for distinct $t$ are disjoint. The retained explicit prime bounds give a rational lower bound for their total logarithmic prime product, divided by $m$.

With 4,000 terms, all twenty original logarithmic gcd rates $\ell$ are valid numerically: fifteen sources have $m_0=10^8$ and five have $m_0=10^9$. The gcd certificate records every exact rate comparison and names the source and ramp files.

The finite bootstrap then reduces all twenty starting indices to at most $18795$, while preserving every original rate $\ell$. Here is its uniform interval argument. For $m\in[L,U]$ and a $t$ whose floors have stabilized by $L$, use the fixed common prime interval

Every prime in this interval divides both coefficient gcds for every integer $m\in[L,U]$. Summing directed lower bounds for their logarithms and comparing with $\ell U$ certifies the entire interval at once. An exact sieve and directed fixed-point logarithm table cover endpoints through $10^8$; the retained BFT prime bounds cover all larger endpoints, which remain below $10^{11}$.

The bootstrap works backwards from each analytic starting index. Its certified intervals form a contiguous partition, checked at every step. Stopping immediately above the first unsupported singleton leaves no gap before the analytic tail. In total, the certificate records $1\,356\,673$ certified intervals. The resulting $m_0$ values, in the ramp's binding order, are

These are prime-interval certificates, not a scan of binomial rows.

Numerical cofactor alternatives

Let $X=p^ku$, $Y=q^lv$ lie in $[h,h+R]$, with $|X-Y|\le D$. The seed sides are powers of $p,q$, respectively. Suppose $u\le h^{e_p}$ and $v\le h^{e_q}$. Define

Choose a positive rational $t<\rho$ such that $tK_0>1$, and set $m=\lfloor t\log h\rfloor$. Then both residual prime-power blocks are integral. The determinant supplies a nonzero integer, and (2), the gcd bound $G\ge e^{\ell m}$, and the exact normalizations give

where $C_0,K_0$ are the stated unequal-block rates and

The factors $T$ and $D_0^3/T$ cover both values of $\varepsilon$. For $h>R$, put $x=\log h$ and

Since $m\le tx$, $m\ge tx-1$, and $Y<2h$, (3) implies

The checker finds an integer $K$ for which, at $x=K\log10$,

The last expression decreases for $x>1/\eta$. Thus the contradiction holds for every $h\ge10^K$, not just at the tested endpoint.

Index twelve

For each of the twenty exact edge bindings in the ramp data, use $h=n-11$ and $D=R=11$. The finite-constant checker certifies (6) for every source. Its numerical certificate records all prefactors, rational block parameters, directed decay margins, and twenty individual thresholds.

Their largest exponent is $3071169$, from the $(3,5)$ alternative with targets $(.40,.10)$ and gcd starting index $13835$. The existing $F_8$ ramp has gain $1/10$ and algebraic bound $10^{845}$. Adding one decimal power absorbs the row shift of eleven and proves (1).

For comparison, omitting the finite bootstrap gives the weaker but still fully numerical cap $10^{158933576214}$. The source certificate records both versions.

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_12_ramp.py
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_off_diagonal_gcd_numeric.py
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_off_diagonal_gcd_bootstrap.py
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_off_diagonal_explicit_pairs.py