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    Subject Terms: Grothendieck Standard Conjectures Homological vs Numerical Equivalence Weil Cohomology Theory Spectral Motif Encoding Motivic Filtration Homotopy Limit Convergence Derived Category of Sheaves Algebraic Cycles Primitive Decomposition Intersection Pairing Positivity Cycle Class Map Spectral Descent Validator-Grade Proof Architecture Universal Cohomology Construction Motivic Realization Étale Cohomology Triangulated Categories Derived Functors Six-Functor Formalism Spectral Sequence Stabilization Validator-grade mathematical reproducibility Formal verification in algebraic geometry Spectral descent architecture Cohomological containment and convergence Peer-to-peer replication frameworks Motivic sheaf descent Cycle equivalence resolution Derived inverse system stabilization Universal cohomology axioms Symbolic traceability in motivic theory Replicable proof systems in algebraic geometry Homotopy coherence in derived categories Positivity enforcement on primitive cycles Reproducible spectral filtration Validator-grade LaTeX formalization MSC 14C25 — Algebraic cycles MSC 14F20 — Étale and other Grothendieck topologies and cohomologies MSC 14F42 — Motivic sheaves MSC 18G80 — Derived categories, triangulated categories MSC 18E30 — Derived functors and triangulated categories MSC 14D07 — Special sheaves and cohomology theories MSC 14F10 — Differentials and de Rham cohomology MSC 13D07 — Homological methods, resolutions, complexes MSC 55T99 — Spectral sequences (general topology, algebraic topology) MSC 14A15 — Algebraic geometry foundations (Grothendieck-style)

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