Erdős 699: the finite range at index fourteen

P699 research · Erdős problem 699

Finite-range theorem. There is no admissible bad triple with $i=14$ and $n\le10^{15\,170\,264}$, for the original threshold $p\ge14$. The numerical tail now bounds every possible bad row by $10^{8516269}$, inside this range. Thus index fourteen is completely excluded. The combined checker binds both parts of this proof.

The driver and output retain the calculation. This file supplies the finite-range half of the proof; it does not itself assert a numerical infinite tail.

The retained index-fifteen prime-power strip certificate applies to every pair among $2,3,5,7,11,13$, to differences at most fourteen, and to heights through $10^{15\,170\,264}$. It therefore applies to the index-fourteen numerator factors. Above its explicitly recorded exceptional heights, each edge gives a cofactor exponent $1/3$.

Use the weighted ramp with $k=9$, degree $90$, and row multiplicities $m_t=18-t$ for $0\le t<14$. The exact capacity calculation covers all cofactor witnesses and all collisions of selected prime-power positions. Weakening the finite exponent to $.3333$ gives minimum gain

Its explicit coefficient comparison gives $n<10^{41}$. The elementary ramp threshold and every exceptional height in the first strip certificate are below $10^{41}$. Hence any bad triple in the stated finite range must already lie below this smaller cap.

The new six-prime strip certificate proves exponent $2/5$ on every pair through $10^{41}$, with difference bound thirteen. Every exceptional prime-pair solution is kept, and the interval engine disables its edge on precisely the corresponding exceptional row windows.

The terminal-prime lemma covers the prefix below row $523$. The remaining range is partitioned into 400,818 certified intervals, with no residual singleton rows and no exact column checks. There are $801,635$ visited nodes, $127$ active cofactor graphs, and $289$ exceptional row windows. The smallest certified logarithmic margin is greater than $0.00000309997964345$.

All interval decisions use directed bounds. The output records the interval partition hash and, as its own run record, the names of both finite prime-pair inputs. Reproduction uses the pinned dependencies already used by the other complete small-index calculations:

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_14_finite.py