← Legendre's Conjecture: The Base-Space Prime-Fitting Program
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Phase 20 · The landing-graph moment hierarchy (a conditional theorem)
Legendre's Conjecture: The Base-Space Prime-Fitting Program
Phase 20 · The landing-graph moment hierarchy (a conditional theorem)
Author line in the source: Neo.K × AI collaboration (no AI named)
The Landing-Graph Moment Hierarchy and GCD Stratification: Fourth-Order Bulk Mixing, Large-GCD Closure, the Sixth-Order Closure Gate and the Primitive Correlation Triangle
Landing-graph moments: the diagonal and large-gcd fourth-moment sectors are closed at X^{2/3+ε}; the fourth moment alone is borderline; Theorem 20.1: IF M₆ = Tr(LL*)³ ≪ X^{5/6+ε} THEN the critical factorable remainder is ≪ X^{17/36+ε} = o(H/log X); the all-distinct triangle energy is OPEN; the text adds that this is 'still only analytic provider closure', not Legendre.
A conditional theorem (assuming a sixth-moment bound) plus partial deterministic closures; the paper: 'this is still only analytic provider closure, not the completion of the whole sieve architecture for Legendre'.
The sixth-moment hypothesis is not proved; Phases 27–32 reduce it to BA6 / CRE6 and stop there. Phase 19 corrected the spectral target to only o(X^{1/6}/log X) (X^{1/12} is just the random scale) and this paper builds on that.
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This is a full-text translation of the Chinese source, which remains authoritative for its research claims. Translation is not independent mathematical verification and does not upgrade the original evidence level.