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Topological Derivation of Twin Primes Based on the FNT Asymptotic Continuity Criterion

Author: Zhang Suhang, Luoyang, Henan

Preliminary Known Propositions

There are infinitely many primes (Euclid's classical result).

Let the prime fraction be \rho_p=\dfrac1p, where p is prime. As p\to\infty, \rho_p\to 0^+.

Lemma (First Proof by Contradiction: Establishing the FNT Asymptotic Continuity Criterion)

Proposition: If the local density of the set of prime fractions in the neighborhood of 0^+ tends to infinity, then there is no essential topological cut in this region, and it possesses asymptotic continuity as defined by FNT.

Proof (by reductio ad absurdum):

1. Assumption: The local density of prime fractions in the neighborhood of zero tends to infinity, but there exists an essential cut—that is, there exists an irremovable hard gap such that the scale spacing between points has a fixed lower bound.

2. If such an essential cut existed, then the spacing between points could not be reduced arbitrarily; the local distribution could not then achieve a density tending to infinity.

3. Contradiction: It is known that as primes increase, the scale spacing between adjacent prime fractions can be arbitrarily small, and the local density tends to infinity.

4. Therefore the original assumption does not hold.

Lemma conclusion: In the neighborhood of 0^+, the prime fractions have no essential cut and satisfy the asymptotic continuity of FNT scale topology.

Note: Here asymptotic continuity is defined within the FNT paradigm and is distinct from the continuum of classical real number theory.

Main Derivation (Second Proof by Contradiction: Topological Constraint on Twin Primes)

Derivation goal: Based on the preceding lemma, examine the existence of twin primes within the FNT framework.

Proof (by reductio ad absurdum):

1. Assumption: There are only finitely many pairs of twin primes.

   Then there exists a sufficiently large constant N such that when p>N, p and p+2 are no longer both prime.

2. Map this condition into prime-fraction topological space:

   Beyond the corresponding threshold, point pairs of the specific scale spacing

\Delta\rho=\frac1p-\frac1{p+2}=\frac{2}{p(p+2)}

will completely disappear.

The disappearance of this specific scale spacing destroys the completeness of the topological space. In the neighborhood of zero already proved to be asymptotically continuous and without essential cuts, a global severance of such structural pairings occurs, forming a structural topological vacancy.

3. Contradiction: According to the lemma, the neighborhood of 0^+ is an asymptotically continuous scale space in the FNT sense, and does not permit such structural complete rupture.

4. Therefore the assumption "there are finitely many twin primes" cannot hold within the FNT scale-topological system.

Derivation conclusion (within the FNT paradigm): There are infinitely many pairs of twin primes.

Conclusion

Thus, the Twin Prime Conjecture is proved within the pure mathematical topological system of FNT. Classical number theory approaches infinity by analytic estimates, whereas FNT locks down infinity by topological continuity.

All derivations in this paper are established on the FNT (fractal continued fraction geometry) scale-topological paradigm. The definitions of asymptotic continuity and essential cut are stipulated by this system; its logical self-consistency holds within this framework and is not equivalent to a complete proof under classical analytic number theory or standard point-set topology. This scheme provides a new approach to understanding the Twin Prime Conjecture from topological structure.