From RKHS to Exact Reproduction: Why Non-Projective Closure Matters | Zenodo Skip to main Communities My dashboard Log in Sign up Theoretical Physics There is a newer version of the record available. Published April 23, 2026 | Version v1 Preprint Open From RKHS to Exact Reproduction: Why Non-Projective Closure Matters Authors/Creators Kreder III, Karl J (Contact person) Description This note is an explanatory comparison between standard reproducing-kernel Hilbert-space (RKHS) theory and the Distinguished Reproducing Kernel (DRK) viewpoint. The aim is not to present a new proof, but to clarify a change in organizing principle. In standard RKHS theory, the kernel represents evaluation in a pre-existing Hilbert space of functions. In the DRK viewpoint, the kernel is treated as primary, and exact self-reproduction is taken as the decisive condition. The central distinction is between projective closure, T_K(Σ)=cK, and non-projective closure, T_K(Σ)=K. The note argues that if closure/evaluation is interpreted as a real physical reproduction law, then projective closure is not merely a harmless rescaling: unless c=1, repeated evaluation changes the physical scale of the object, so the theory is not intrinsically stable. From this perspective, non-projective closure is not a cosmetic preference but the natural condition for exact, physically sensible self-consistency. The note is intended as a conceptual bridge between familiar RKHS language and the stronger fixed-point demand of a Distinguished Reproducing Kernel. Files Files (10.9 kB) Name Size Download all rkhs_vs_drk_nonprojective_closure_note_with_citations.tex md5:32941d41d0496c409d444cd839ead8b0 10.9 kB Download 108 Views 192 Downloads Show more details All versions This version Views Total views 108 56 Downloads Total downloads 192 18 Data volume Total data volume 33.1 MB 195.7 kB More info on how stats are collected.... Versions External resources Indexed in OpenAIRE Communities Details DOI DOI Badge DOI 10.5281/zenodo.19712957 Markdown [](https://doi.org/10.5281/zenodo.19712957) reStructuredText .. image:: https://zenodo.org/badge/DOI/10.5281/zenodo.19712957.svg :target: https://doi.org/10.5281/zenodo.19712957 HTML <a href="https://doi.org/10.5281/zenodo.19712957"><img src="https://zenodo.org/badge/DOI/10.5281/zenodo.19712957.svg" alt="DOI"></a> Image URL https://zenodo.org/badge/DOI/10.5281/zenodo.19712957.svg Target URL https://doi.org/10.5281/zenodo.19712957 Resource type Preprint Publisher Zenodo Rights License Creative Commons Attribution 4.0 International The Creative Commons Attribution license allows re-distribution and re-use of a licensed work on the condition that the creator is appropriately credited. Read more Citation Export Technical metadata Created April 23, 2026 Modified April 23, 2026 Jump up About About Policies Infrastructure Principles Projects Roadmap Contact Blog Blog Support Help FAQ Developers REST API OAI-PMH Contribute GitHub Donate Funded by Powered by CERN Data Centre & InvenioRDM Status Privacy policy Cookie policy Terms of Use This site uses cookies. Find out more on how we use cookies Accept all cookies Accept only essential cookies