← Legendre's Conjecture: The Base-Space Prime-Fitting Program / Phase 32 · The end of the series

Legendre's Conjecture: The Base-Space Prime-Fitting Program Phase 32 · The end of the series Author line in the source: Neo.K × AI collaboration (no AI named)

The Fully Nondegenerate Toric Core: the Opposite-Pair Three-GAP Normal Form, Alternating Product Conservation, Opposite-GCD Defect and Multiplicative Slope Coupling

Reduces the fully nondegenerate core to a torically coupled three-GAP energy problem (TTG3): alternating-product conservation A₁A₃A₅ = A₂A₄A₆, d₁d₂d₃ | Δ², slope coupling u₁u₃v₂ = u₂v₁v₃, pair-resonance thresholds; TTG₃ ≪ R^{4+ε}K³ is OPEN, as are full CRE6, the BA6 exact-zero provider and Legendre's conjecture; a Phase 33 is requested.

The end of the series: the fully nondegenerate core is reduced to a torically coupled three-GAP energy problem; full CRE6, the BA6 exact-zero provider and Legendre's conjecture all remain OPEN. No code or data ships (a bare md), so its 'finite audits: COMPLETE' cannot be checked; the pre-publication read-through re-verified the invariants only on the explicit example of Phase 28 (Δ = 336, A₁A₃A₅ = A₂A₄A₆, Σ h_i A_i = 0). Phase 33 is requested but is not in the package; there is no final verdict, synthesis or updated program.

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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.

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