Erdős 699: additional product constraints at index fourteen

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This is a retained earlier research note. The current reading guide gives the latest status and distinguishes completed proofs from remaining cases.

P699 research · Numerical product thresholds · Forced boundary divisors

The retained index-fourteen interpolation profiles can be strengthened by constant-gap identities involving the primes 11 and 13. This note records nine additional, fully numerical sources. They do not establish effectivity for the whole index: the alternatives still require complete geometric coverage, and some profiles remain compatible with these inequalities.

Choose the maximal small-prime positions $r_p\in\{0,\ldots,13\}$ and put

$$u_p=(n-r_p)/p^{v_p(n-r_p)},\qquad h=n-13.$$

A row in the following table applies when its two position sums are equal and its two unordered position pairs are different. The conclusion is the disjunction $u_p>h^{e_p}$ for at least one of the listed primes. All terminating decimals denote exact rationals.

Every row is valid for

$$h\ge10^{3\,297\,191\,195}.$$

The individual thresholds are smaller and are retained in the directed interval certificate. The checker reconstructs the seed components, certifies their coefficient-gcd estimates, and applies the general numerical product bound. Its only dependency beyond the Python standard library is the retained mpmath requirement.

For example, the third row uses

$$2(3\cdot13)-7\cdot11=1,\qquad s=16/11.$$

If $r_3+r_{13}=r_7+r_{11}$, the products

$$(n-r_3)(n-r_{13}),\qquad(n-r_7)(n-r_{11})$$

have a nonzero constant difference of magnitude at most $13^2$. The quadratic-product Padé estimate gives the stated four-way alternative. It excludes the previous profile

$$(e_2,e_3,e_5,e_7,e_{11},e_{13}) =(.1399,.3453,.2711,.3218,0,.0729)$$

at positions $(r_2,r_3,r_5,r_7,r_{11},r_{13})=(2,5,3,4,1,0)$. This is an exclusion of that limiting cofactor profile. Each resulting larger-cofactor branch still needs its own polynomial inequality before a whole-index conclusion can be drawn.

A broader exploratory catalog found many more possible sources, but is not being promoted as a proof of index fourteen. In particular, simply counting old profile vectors that violate a new alternative is insufficient: new componentwise minimal vectors must be recomputed after adding the alternative. The finite position assignments are also prime-labeled; a sorted capacity comparison cannot transfer a position identity to another assignment of primes.