A continuous free product system not generated by its tensor core

We construct an associative pointed free product system of separable complex Hilbert spaces whose maximal ordered tensor subsystem does not generate the whole free system. Its tensor core is the time-ordered finite-word Fock system, and the closed free hull of that core is the full finite-word Fock system. The ambient system has additional nonzero orthogonal sectors of ordinal words of length below omega squared. A recoverable sigma-finite Gaussian mixture and coordinate-retaining finite-prefix absorption yield countable maximal-run charts and pointed free-product unitaries. The paper proves full associativity and an explicit continuous-field structure including time zero, positive-time local triviality, and continuity of the structural maps and their inverses. A quantitative finite-positive-measure grade-omega witness proves properness for every positive time. This gives a negative answer in the scalar specialization to the universal tensor-core generation question posed by Michael Skeide in Oberwolfach Reports 6 (2009), 528-530, DOI10.4171/OWR/2009/09. It does not resolve adapted-cocycle, forward tensor-recovery or general classification questions. The stated explicit continuity convention is part of the theorem; arbitrary stronger regularity axioms are not asserted. Classical Fock constructions, standard Gaussian/Radon-Nikodym tools and nonclassical tensor-noise context are credited. The known forward construction from any spatial tensor input has generating tensor core by a direct hull-monotonicity argument, so it cannot itself give this proper-core-hull obstruction. The source and verification ZIP preserves all eleven original files and three sealed scoped AI review families, primary-source URLs/hashes and exact reading scopes, new explicitly guarded integrity/reproduction tools, and meaningful normal/optimized negative controls. The bounded source investigation authenticated no earlier exact negative construction in the inspected passages. The named in-preparation manuscript remains unrecovered; unrecorded talks, unpublished/unindexed work and differently formulated results are not excluded. No worldwide-firstness or still-open-in-2026 claim is made. Historical assertion-based scripts are preserved unchanged and replayed only with assertions enabled by the new wrapper. Finite computations supplement, and do not certify, the analytic infinite-dimensional proof. AI tools were extensively involved in research planning, proof development/reconstruction, computational checks, literature search/comparison, manuscript writing and adversarial auditing. This is an unrefereed preprint and has received no formal independent human peer review or proof-assistant certification. Inherited-context AI scientific reviews are explicitly distinguished from blind or human review.