Erdős 699: the initial index-seven cofactor bounds from a numerical height

P699 research · Choosing the Padé index

All five initial cofactor alternatives in the relaxed index-seven proof hold for

Three alternatives already have smaller numerical thresholds. The two remaining alternatives are

Here $p^{v_p(n-r_p)}u_p=n-r_p$, with the selected numerator factors in $[h,h+6]$. The checker and full witness chains prove (2) throughout the interval from (1) to the previously certified numerical tails. Those tails complete the assertion for every larger height.

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

The exact target box

Assume both cofactor upper bounds in one alternative hold, and write $x=\log h$. For a fixed integer Padé index $m$, the denominator cost is bounded by

where $\kappa_p=L_{s(p)}a_p\log p$, $a_p$ is the seed exponent, $b_p$ is the product multiplicity, and $e_p$ is the explicit target. There is no requirement that $m$ be at most the source's original block parameter times $x$.

The first term of the nonzero-integer estimate decreases with $x$. For the second term, $d x+D(x)$ is the maximum of the affine functions

indexed by subsets $J$ of the prime components. There are four such functions in each case here. Thus two rational endpoint checks for each affine function certify the entire interval. The boxed sparse-witness helper rounds the $\kappa_p$ upwards and the intrinsic decay rates downwards, then performs the interval checks using integer and rational arithmetic.

The Padé index is chosen near the balanced value or a point at which one of the denominator contributions vanishes. These choices are only proposals. Each accepted index has an independently checked pointwise coefficient-gcd lower bound and strict interval inequalities.

Complete interval chains

The prime-log prefix through $10^9$ is shared with the other source calculations and loaded read-only. Proper prime powers through $10^{14}$ give additional certified logarithmic weight. Each coefficient-gcd calculation uses 8000 disjoint BFT intervals and applies to both adjacent Padé rectangles. No uniform estimate over untested intermediate indices is assumed: a single tested integer $m$ proves an entire interval of real logarithmic heights.

Both chains start at the integer 230258508, which is strictly below $100000000\log10$. Each right endpoint is exactly the next interval's left endpoint. Their final endpoints exceed the logarithms of the two displayed numerical tails. The complete 122868-step certificate records every Padé index, interval endpoint, coefficient-gcd lower bound, and number of prime intervals used.

An independent replay recomputed every retained pointwise gcd bound and every rational interval inequality. It used the existing prime cache; no second prime sieve or binomial-row search was performed.

Verification

The ordinary checker command replays the full certificate, using the shared prime cache:

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_7_initial_height_cover.py

For a combined proof wrapper, check(data, replay_gcd=False) verifies all source bindings, every interval inequality, and complete coverage. Setting replay_gcd=True additionally recomputes the pointwise gcd bounds. The returned summary also checks that the other three initial alternatives have thresholds below (1).