Erdős 699: finite exclusion at index nine through 10^241547629

P699 research · Erdős problem 699 · The companion index-seven calculation

Theorem. There is no admissible bad triple with $i=9$ and $n\le10^{241\,547\,629}$, for the original threshold $p\ge9$.

This is an unconditional finite-range result. A numerical infinite-tail theorem at or below this exponent would complete index nine; that theorem is a separate input, not asserted by this file.

The combination checker and its certificate use two finite cofactor calculations:

Both ramps have $k=6$, degree $42$, and row multiplicities $12-t$ for $0\le t<9$. Their minimum gains, including every possible collision of selected prime-power positions, are respectively $3/10$ and $3/2$. The first pair certificate's exceptional heights are below $10^{122}$. The second certificate's largest exceptional threshold is below $10^{26}$. Thus every bad triple in the stated finite range would have to satisfy $n<10^{26}$.

The first clique was certified by the logarithmic strip generator. It uses 128, 123, 123, 120, 120, and 117 anchors for the six pairs. The earlier certificate through $10^{27\,027\,659}$ is preserved. The extension adds respectively 13, 14, 13, 13, 13, and 14 anchors, with no new exceptions. The generator verifies that every retained boundary check was untruncated, then continues from its final exponent orthant. Each continued certificate names its prefix file and reproduces the prefix's anchors, boundary checks and height cap, as in the index-seven continuation proof. The second clique uses the same exact CRT argument at the smaller cap. Both allow every factor difference through eight.

The final interval driver also uses the older complete $2/5$ clique through $10^{1732}$ as a fallback where a stronger edge is unavailable. Each edge is disabled on its precise exceptional row windows.

The terminal-prime lemma covers the prefix below row $113$. The remaining range is excluded by 607,975 certified intervals and thirteen exact rows, all at most $5,832$. The exact part checks 6,762 columns and finds a common prime at least nine in every case. The smallest directed logarithmic margin is greater than $0.00000218172979727$.

The interval output records the partition hash, the source file names, every residual row, and the count of exact column gcd checks. No modular polynomial ranks are recomputed for this finite argument. Reproduce the interval and combination steps with the pinned dependencies installed in the repository's .venv/:

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_9_finite_intervals.py
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_9_finite.py

Complete result. The combined proof excludes every admissible bad triple at nine. The estimates below are retained as components of that proof.