A continuous free product system not generated by its tensor core
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Abstract
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 subsystem generated by that core is precisely the full finite-word Fock system. The ambient system has, in addition, nonzero orthogonal sectors of words of ordinal length below ω². A recoverable Gaussian mixture supplies a sigma-finite measure class invariant under finite-prefix absorption. Finite monochromatic-run charts turn concatenation into the free product structure. We prove separability, an explicit continuous Hilbert-field structure including time zero, and continuity of the full structural maps and their inverses. This gives a negative answer, in the scalar specialization, to the universal tensor-core generation question stated by Skeide in 2009. It does not answer the separate adapted-cocycle or forward tensor-recovery questions. The argument is analytic; accompanying finite computations do not certify its infinite-dimensional assertions. This draft is unrefereed and extensively AI-assisted.
The idea in plain English
A product system describes how mathematical spaces for shorter time intervals combine into a space for a longer interval. There are different ways to combine them. Tensor products impose one kind of ordering; free products allow a richer collection of alternating combinations.
The question is whether the largest tensor subsystem inside a free product system always contains enough information to rebuild the whole free system. This paper constructs an example where it does not. Its tensor core generates all the finite-word sectors, but the full system also contains orthogonal sectors made from infinite words. Taking limits of the finite-word sectors cannot reach those extra sectors.
The construction retains separable Hilbert spaces and proves a specified continuous dependence on time, including at time zero. The obstruction therefore persists beyond a purely algebraic example.
The precise result
There exists an associative pointed free product system E of separable complex Hilbert spaces with the following structure at every positive time t. Its maximal ordered tensor subsystem F is the time-ordered finite-word Fock system. The closed free subsystem generated by F is the full finite-word Fock system K, and K_t is a proper subspace of E_t. A nonzero grade-ω sector remains orthogonal to K_t.
The system has a continuous vacuum and norm-continuous support-subspace sections, is locally trivial at positive time, and has jointly continuous structural maps and inverses, including at time zero.
This gives a negative answer in the scalar case to the universal tensor-core generation question stated by Michael Skeide in 2009.
Why the result holds
A recoverable Gaussian mixture gives a sigma-finite measure class on convergent infinite sequences that is preserved under finite-prefix absorption.
Words of ordinal length below ω² split at a time cut into finitely many monochromatic runs, producing associative pointed free-product unitaries and the specified continuity structure.
Intersecting ordered tensor ranges leaves decreasing finite words. Their free hull contains exactly finite words, while a positive finite-measure grade-ω set supplies a nonzero orthogonal remainder.
Scope and limitations
The continuity claim uses the explicit Hilbert-field convention stated in the paper. Stronger regularity requirements would need to be checked separately.
The result concerns tensor-core generation. The adapted-cocycle and forward tensor-recovery questions remain separate, as does a classification of free product systems.
Status and AI use
This is an unrefereed preprint. AI tools were used extensively in the research, proof development, computational checks, literature review, and writing. The supporting package contains the published source and verification materials.
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Collaborate on this result
I welcome collaboration on an independent mathematical review of the construction, particularly its measure-class factorization and continuity arguments, or on developing a journal submission.
Authorship and responsibilities would be discussed in relation to the substantive contributions made.
Cite this paper
Alec Kriebel. (2026). A continuous free product system not generated by its tensor core (Version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23539283
Background and prior work
- Michael Skeide, Free Product Systems (2009): Introduces the free-product-system framework and asks whether the maximal tensor subsystem generates the whole free system. The present construction addresses this universal converse in the scalar case.
- Michael Skeide, The index of (white) noises and their product systems (2006): Section 10.2 develops the earlier free-product-system perspective and the construction from a supplied spatial tensor system.
- Michael Skeide, Spatial Markov semigroups admit Hudson–Parthasarathy dilations (2022): Provides context on noises and cites the forward spatial-tensor-to-free construction, a distinct direction from the counterexample here.