Erdős 699: a simpler effective proof at index twelve

P699 research · The first effective proof · Off-diagonal source estimates

The new off-diagonal estimates simplify the effective index-twelve proof to one explicit polynomial. Let $X$ be the largest effective threshold for the twenty prime-pair alternatives in the certificate. Then

The number $X$ is effectively computable but has not been numerically evaluated. Thus $10^{845}$ is the algebraic part of the bound, not a claim that the whole finite range has been excluded.

The subsequent numerical proof and finite certificate complete this argument: index twelve has no bad triples.

One polynomial and a positive exponent

Use the explicit coordinate-line product

It has degree $72$, is positive at every admissible $(j,n-j)$, and has multiplicity at least

at every nonnegative grid point on row $x+y=t$. The total multiplicity is $M=126$, so $M-\deg F_8=54$.

The twenty off-diagonal pair alternatives give eighty-two minimal cofactor witnessing vectors in prime order $2,3,5,7,11$. For a singleton selected position, the capacity is $1-v_p$. For a group of two or more primes at the same position, its product of prime powers divides $n-r$ and has capacity at most one.

The exact checker regenerates all witness vectors and the resulting 402 capacity lists. As the multiplicities decrease with $t$, rearrangement puts the largest capacities at the first available rows. In every case,

The exponent gained in the weighted divisibility inequality is therefore at least $54-53.9=1/10$. This covers all positions and all collisions, without an interpolation rank calculation.

The twenty alternatives are weakenings of individual sources in the directed off-diagonal table. The checker revalidates each selected source and its strict endpoint margin. Their two endpoint exponents sum to less than one; if the selected positions coincide, $u_pu_q\ge n-r$ proves the corresponding alternative directly. The zero-gap case is consequently included too.

The algebraic bound

The exact off-maximum valuation constant is

The weighted-ramp inequality, with its bounded row-shift corrections, gives

once the cofactor sources hold. The checker verifies the integer comparison

This proves the stated bound. The earlier twelve-certificate proof remains valid, but its algebraic bound $10^{3\,643\,130\,001}$ is not needed. Both arguments leave the separate task of evaluating source thresholds and excluding the finite range.

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