Modular Reconstruction Criteria and Obstructions in Algebraic Quantum Field Theory

We study how far four-dimensional Lorentzian structure can be reconstructed from algebraicquantum field theory modular data without assuming a prior spacetime geometry. Working withstandard real subspaces and half-sided modular inclusions, we derive exact operator-algebraiccriteria for the existence of a nontrivial positive affine translation direction. In particular, a singleadditive identity for a reparameterized relative modular cocycle of a strict standard pair impliesa proper half-sided modular inclusion, a positive-energy translation group, and the correspondingreflection and endpoint relations. We also establish bounded defect certificates, one-sided hull andGramian tests, polariser and relative-complement criteria, and an asymptotic-reflection criterion.In the quasifree setting, polariser softness yields weak-null rough probes and Mosco exhaustion.In the canonical symmetric-inner model, the direct cocycle law is rigid and selects translationendomorphisms. We construct a mixed-orientation standard-subspace countermodel, togetherwith its bosonic second-quantized incidence fragment, showing that standardness, factoriality,strict isotony, an abstract causal buffer, soft absolutely continuous modular spectrum, and astandard relative complement do not force the affine law. We therefore isolate the remainingfinite, noncollapsed rank-four modular-position problem required for a 3 + 1-dimensional Poincaréreconstruction. No metric, coordinate system, spacetime dimension, or geometric wedge action isassumed in the proofs of the intrinsic criteria and obstruction results. Preprint submitted for consideration to Communications in Mathematical Physics.