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Updated Sep 22

Erdős 699: source margins for a numerical index-seven proof

P699 research · The effective proof · Sharper coefficient bounds · Arithmetic nonvanishing

The saved rational source inequalities were rounded extremely close to their limiting analytic boundaries. They can be weakened slightly while preserving a complete exact proof. This gives room to use smaller coefficient-gcd rates and more practical numerical starting indices.

The polynomial part of the modified proof holds for

h=n-6\ge10^{99\,990\,923}. \tag{1}

Numerical starting heights for the analytic sources must still be included separately. The polynomial family and its verified modular ranks are unchanged.

The modified inequalities

Lower both endpoints of each of the five initial pair alternatives by $1/10000$ . They become

(2,3):(.2920,.2920),\ (.3295,.2915),\qquad (2,5):(.2675,.2710),\ (.2678,.2678),\qquad (3,5):(.2561,.3773).

In each of the $258$ joint inequalities

\sum_p(g_p+w_pe_p)_+\ge R,

keep every $g_p$ and $w_p$ , and replace $R$ by $R-\eta$ . Usually $\eta=1/10000$ ; the five exceptions are

Source ID	$\eta$
`source_069`	$1/100000$
`source_078`	$1/100000$
`source_263`	$97863/10^9$
`source_326`	$1/20000$
`source_361`	$6367/10^9$

Every modified $R$ remains positive. For a fixed source block parameter $t$ , lowering its logarithmic gcd rate from $\ell$ to $\ell'$ changes both analytic margins by $t(\ell'-\ell)$ . Thus a choice satisfying

\ell'\ge\ell-\frac{3\eta}{4t}

retains an additional margin of at least $\eta/4$ in both terms of the integer estimate. This observation guides the numerical source bounds; the coverage argument itself uses only exact rational inequalities.

Complete rational coverage

Uniformly subtracting $1/10000$ from the positive coordinates of the old minimal witnessing vectors gives exactly the new minimal vectors. The original position-zero restriction and collision certificate still apply. Thus the same $540$ position-and-vector cases remain exhaustive.

For a polynomial certificate with coefficient exponent $E_F$ , use its actual finite-height inequality

\sum_p m_{r_p}e_p \le D-\sum_{r\notin\{r_2,r_3,r_5\}}m_r+ \frac{E_F}{K},\qquad h\ge10^K.

At $K=10^8$ , almost all old Farkas witnesses still apply. Ten additional branches, using existing sources, complete the remaining cases. The retained forest has $4336$ nodes, $948$ branches, and $3388$ leaves. Every child of every branch is present, and every leaf has an exact rational Farkas certificate.

For each leaf, let $\alpha$ be the negative weighted right side before the polynomial errors $E_F/K$ are added, and put $C=\sum_F\lambda_FE_F$ . The leaf is impossible whenever $K>C/\alpha$ . The largest exact ratio is

\frac{40482625000000}{404863}<99990923.

This proves (1). The collision bound is smaller, and the separate arithmetic nonvanishing certificate eliminates the former common-zero height restriction.

The checker and data name the original proof and coefficient-bound files by basename. They check the new minimal vectors, every source hypothesis, all exact Farkas leaves, and the final weighted height. No new interpolation matrix or binomial row is tested.

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_7_relaxed_sources.py

Linked from (1)

Supporting proofs for the completed indices

Graph