Erdős 699: a numerical stopping bound for index fourteen

P699 research · The interpolation family · Finite Padé constants · Finite-range certificate

Theorem. Every bad triple $(n,14,j)$ for the original threshold $p\ge14$ satisfies

All source thresholds in this statement are numerical. The bound lies below the $10^{15\,170\,264}$ upper limit of the separately retained finite-range certificate. Together they exclude every bad triple at fourteen and complete the cutoff-ten theorem.

Sixty-four sources suffice

Keep the polynomial family from the existing index-fourteen proof:

Its residual interpolation degree is $56$. The symmetric jet map has rank $831$ on $841$ monomials, as verified modulo $65521$. The nonvanishing family has residual degree at most $57$, hence total degree $375$ after multiplication by the coordinate factors.

The numerical proof needs only $64$ pair alternatives: $61$ from the retained off-diagonal table and the following three additional sources. Here the targets are in the displayed prime order, and $\ell$ is the logarithmic coefficient-gcd rate per Padé index $m$.

These replace three earlier alternatives by slightly weaker targets, which admit more practical numerical constants. The additional-source checker verifies their finite gcd sums, positive decay rates, and strict endpoint margins with directed intervals. Its source records contain the complete parameters.

The selected alternatives have $543$ componentwise minimal witnessing vectors. The checker regenerates all of them, then all $13002$ distinct capacity lists, including every collision pattern of the selected positions. Since the multiplicities are nonincreasing, the same rearrangement argument covers arbitrary positions. Exact arithmetic gives

The worst capacity list, in decreasing order, is $(1,.95,.9274,.83,.72)$. The complete list of selected edges and all minimal vectors are retained in the numerical-tail data.

Uniform gcd bounds with a finite bootstrap

The uniform prime-interval argument first supplies analytic numerical starting indices for the coefficient gcds. The finite bootstrap then checks every intervening integer index down to a smaller $m_0$ by certifying entire intervals of indices at once. Both adjacent Padé rectangles are included throughout.

The $61$ selected sources use the selected analytic inputs and bootstrap records. The other three use their analytic inputs and bootstrap records. Together the $64$ records certify $5\,197\,837$ intervals of Padé indices. Every requested rate is unchanged across its bootstrap, and every retained minimum margin is positive.

The exact sieve covers prime endpoints through $10^8$. Between that cutoff and $10^{11}$, the checker uses the retained BFT bounds. For larger endpoints it uses Dusart's $|\theta(x)-x|<.01x/\log^2x$ estimate. An integer lower bound for $\log x$, obtained from $2^k\le x$ and $\log2>69/100$, keeps these last comparisons rational. Thus the last source's long analytic tail does not introduce an unevaluated prime-number estimate.

Smaller explicit integral constants

The finite-constant argument used for index twelve remains valid here, with two improvements. For directions $(\lambda,\mu,\kappa)$, split the base integrand at the real power

The remaining exponents are nonnegative: the only potentially negative one is $\mu\gamma-1-\varepsilon=1-\varepsilon$, where $\varepsilon\in\{0,1\}$ selects the adjacent rectangle. Consequently

where $F,M_Q,M_E,q,e$ have the definitions in the finite-constant note. This gives smaller prefactors than splitting off two full powers of the base integrand.

Also retain the two decay rates separately. With $x=\log h$, $m=\lfloor tx\rfloor$, the nonzero-integer estimate is

where

Choose a verified rational $t$ just below the available integral-block limit. The checker bounds each displayed term by $1/2$ separately, using directed logarithmic comparisons. Both comparisons decrease after an explicitly checked starting point, so the resulting bound holds at every larger height. These improvements are implemented as options of the explicit pair checker; the earlier index-twelve calculations retain their original parameters.

Every source is evaluated with $h=n-13$, factor gap at most $13$, and factor heights in $[h,h+13]$. The largest source exponent is $8516268$, coming from the last row of the table above.

A sharper coefficient bound for the same polynomial family

No new interpolation rank calculation is needed. A Hasse jet of one monomial of degree at most $56$, at nonnegative coordinates at most $13$, has absolute value at most $26^{56}$. A symmetric basis element contains at most two monomials, so every matrix entry is at most $2\cdot26^{56}$. The exact integer inequality

bounds every row norm of an $831\times831$ minor by $10^{81}$. Thus Hadamard's determinant bound uses $831\cdot81$ in place of the earlier general-purpose $831\cdot118$ estimate.

Keeping the same generator, coordinate-factor, and bounded-shift constants as in the original proof gives the explicit logarithmic bound

By (2), the algebraic part of the stopping bound is therefore

Taking the maximum with the numerical pair thresholds, and adding one decimal power to absorb the row shift of thirteen, proves (1).

Reproduction

The whole numerical-tail checker regenerates the cofactor vectors, collision capacities, gain, coefficient bound, and all $64$ numerical source thresholds. It also checks every source and bootstrap binding. With the pinned interpolation dependencies:

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_14_numerical_tail.py

The source and finite-prime calculations can be rebuilt separately:

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_14_gcd_selected.py
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_14_gcd_bootstrap.py
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_14_additional_pairs.py
mkdir -p tmp/erdos699
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_off_diagonal_gcd_bootstrap.py --input wiki/research/erdos_699/computations/erdos_699_index_14_additional_gcd.json --output tmp/erdos699/erdos_699_index_14_additional_bootstrap.json

The last command writes under the ignored repository tmp/; compare its output with the retained bootstrap file on its mathematical fields, since elapsed times differ between runs. Updating the retained record in place is a separate step: after replacing the bootstrap files deliberately, erdos_699_index_14_numerical_tail.py --write repeats the full numerical proof and then rewrites the record, including the names of the rebuilt files as its own run record.