Theoretical Foundations of Manifestation Finiteness (P4): From the No Signalling Theorem to the Inevitable Barrier of the Basal Structure

Manifestation Finiteness (P4) is one of the four core properties of the ITC framework. It asserts that the manifestation of basal information in emergent spacetime requires a local time τ, and the outcome of a single manifestation is random. This property seemingly contradicts P2 (Instantaneous Connectivity), but in fact it is the crucial barrier safeguarding causality and ensuring compatibility with relativity. This paper systematically demonstrates the necessity of P4 from six levels. First, at the experimental evidence level, the experimental confirmation of the no‑signalling theorem unequivocally shows that quantum correlations cannot be used for superluminal communication, providing the empirical cornerstone for P4. Second, from the first principle, the existence of absolute time requires a definite order of events, and the local manifestation time τ in P4 guarantees the causal order in emergent spacetime. Third, an analysis of spacelike separation and compatibility with relativity shows that even if τ is extremely small, leading to spacelike separated events, the randomness of single outcomes in P4 prevents causal paradoxes; moreover, the information extraction event itself has a definite time point, ensuring that the final information extraction is causally later than the sending event. Fourth, the feasibility of statistical detection—demonstrated by Pan's team's 2026 DI‑QKD experiment—proves that weak signals can be extracted from random data through large samples, providing a methodological basis for overcoming P4. Fifth, the complementary relationship with P2 shows that P2 guarantees instantaneous connectivity at the basal level, while P4 prevents this connectivity from being used for single‑shot controllable information transmission; together they constitute the complete picture of the two‑layer reality. Sixth, at the level of quantum mechanical foundations, the randomness of measurement collapse and the projection postulate are deeply compatible with P4. Through these six levels of argumentation, we conclude that Manifestation Finiteness is an inherent property of the basal structure, a core element for the logical self‑consistency of the ITC framework, and the fundamental obstacle that statistical detection schemes must face and overcome.