For organizations competing across several product or service categories, total AI representation is not the simple average of per-category authority but the prominence-weighted average, where prominence is each category's share of total buyer queries. The paper sets out a quarterly monitoring and reallocation protocol, and treats cross-category spillover as a benefit when categories share vocabulary and buyer type and a liability when they do not.

Abstract and Evidence Status

Abstract (verbatim from the Zenodo record)

This paper states Theorem 3 of The Ontological Dominance Papers, the Portfolio Extension, governing organizations that compete across multiple product or service categories. The theorem establishes that total AI representation across a multi-category portfolio is not the unweighted average of per-category authority but the prominence-weighted average, where prominence is the fraction of total buyer queries falling into each category. The theorem specifies a quarterly monitoring and reallocation protocol and characterizes cross-category spillover as a benefit when categories share vocabulary and buyer type, and a liability when they do not. Building on the mathematical foundation in OD-S and extending OD-1, Theorem 3 was classified VERDICT B upon adversarial review: a formally correct extension of OD-1's scope conditions, not an independent theorem, confirmed by three independent mathematical derivations and contingent on OD-1's empirical field confirmation, which is pending execution.

Evidence status

Classified VERDICT B on adversarial review: a formally correct extension of OD-1's scope conditions, not an independent theorem. Confirmed by three independent mathematical derivations and contingent on OD-1's empirical field confirmation, which is pending.

Concepts and Related Reading

Keywords

Portfolio Extension · category prominence · cross-category spillover · prominence-weighted average · cross-category correlation · Byrum's Law of Ontological Dominance · The Ontological Dominance Papers

Related glossary entries

Practitioner companion article

78% of Competitors Invisible to AI: Map & Own Their Gaps

Cite This Paper

APA

Byrum, J. (2026). The Ontological Dominance Papers Theorem 3: The Portfolio Extension [Working paper]. Zenodo. https://doi.org/10.5281/zenodo.21683247

BibTeX

@techreport{byrum2026_od3,
  author       = {Byrum, Joseph},
  title        = {The Ontological Dominance Papers Theorem 3: The Portfolio Extension},
  year         = {2026},
  month        = jul,
  type         = {Working paper},
  institution  = {Zenodo},
  doi          = {10.5281/zenodo.21683247},
  url          = {https://doi.org/10.5281/zenodo.21683247}
}

DOI: https://doi.org/10.5281/zenodo.21683247 · Zenodo: https://zenodo.org/records/21683247

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