Long description What can “first actuality” mean if the inquiry admits no prior actual event, material substrate, physical time, probability space, external executor, or positive root determination? This bilingual preprint separates ordinary nothing from absolute undefinedness. Ordinary nothing can be stated from outside as a stable absence. Absolute undefinedness, denoted, is more radical: it is not an empty object, empty domain, truth value, initial state, premise, cause, or earlier moment. It functions as a metalanguage boundary for the phrase “连无都无”: even “nothing” is not yet available as a determinate object. The paper then asks a deliberately restricted question. Suppose there is an actual, rooted, self-contained, non-self-exempting first occurrence whose retrospective root status is exhausted by two labels: - : enacted determination;

  • : absence or negation of that enacted determination. Let be the retrospective quotient of full root configurations, and let be a total autonomous full-state update. If every update reverses its quotient status, then the quotient update is well-defined and unique: This total fixed-point-free involution is the paper’s minimal PR (paradoxical self-reference) normal form. The exchange law is not separately assumed. The proof first classifies all total fixed-point-free endomaps of a two-element set and then derives the quotient map from the full-state dynamics. Sharpness is demonstrated by countermodels obtained when binarity, total continuation, or root-level non-fixation is removed. The exact formal result is model-indexed: Accordingly, the word “necessarily” does not mean,, or. Those notations would incorrectly turn the undefinedness boundary into a proposition, a cause/earlier state, or a probability condition. Necessity means truth in every member of the explicitly defined model class. The paper also marks its limits. It does not prove that actuality can be derived from ; that every logic must adopt revision semantics; that PR alone produces chaos; that a concrete universe-generating Rule has been found; or that the resulting structure is empirically identical with our observable universe. The later route is presented only as a conditional research programme, with failure points made explicit. The publication contains complete English and Chinese versions, a dependency ledger, explicit rival models, exact non-claims, and four recorded internal adversarial review rounds. These rounds are internal review, not external peer review. ## Short description A bilingual logical–mathematical study of first actuality at the boundary of absolute undefinedness. Under an explicit actual, rooted, self-contained, binary, continuing, and root-non-fixating model class, the paper proves that the unique quotient dynamics is the period-two PR exchange, while sharply separating this conditional uniqueness theorem from claims about the production of actuality, chaos, matter, or our observable universe. ## Suggested keywords 1. absolute undefinedness
  1. first actuality
  2. PR
  3. paradoxical self-reference
  4. negative self-reference
  5. fixed-point-free dynamics
  6. revision semantics
  7. primordial actualization
  8. foundational ontology
  9. pregeometry
  10. 绝对未定义
  11. 第一实际性
  12. 悖论自指
  13. 无不动点动力学
  14. 本源实际化 ## Suggested Zenodo metadata — author review required | Field | Proposed value | Status | |---|---|---| | Upload type | Publication | Suggested | | Publication type | Preprint | Suggested | | Title | The First Actuality: Why Absolute Undefinedness Necessarily Co-Arises with PR | Frozen | | Subtitle | A Logical and Mathematical Proof Beyond Nothing, Time, and Probabilistic Choice | Frozen | | Creator | Jia, Baolong | Frozen | | Version | 1.0 | Frozen | | Language | English (full Chinese parallel text included) | Suggested | | Manuscript date | July 2026 | Frozen | | Publication date | — | Author must confirm at deposit time | | ORCID | — | Author must confirm; do not invent | | Affiliation | Independent Researcher | Author should confirm wording | | License | — | Author must choose | | Access right | Open access | Suggested, subject to license choice | | DOI | Reserved by Zenodo at deposit | Platform-generated | | Communities | — | Optional; author decides | | Funding | — | Author confirms if applicable | | Related identifiers | — | Author confirms if applicable | | Embargo | None suggested | Author confirms | | External peer review | No | Frozen factual declaration | | Internal review | Four recorded adversarial rounds | Frozen factual declaration |