Previous work introduced τ (tau) as a measure of temporal coherence complementing the Consciousness Gradient Index (CGI), validating the dissociation using EEG data from human sleep studies. Here we extend the framework to cross-species comparison. τ comprises three components: memory persistence (τ_m), self-model stability (τ_s), and narrative integration (τ_n). Plotting CGI against τ across 15 species reveals that most organisms cluster near a diagonal - integration capacity scales roughly with temporal depth (r = 0.92). One striking exception emerges: the octopus (CGI = 7.6, τ = 3.3) achieves mammal-level integration with invertebrate-level temporal persistence. Its distributed nervous system enables moment-to-moment flexibility but limits the continuity of the temporal thread. In contrast, cuttlefish (CGI = 7.4, τ = 5.0) with similar neuron counts but more centralized architecture maintain robust episodic-like memory. We propose that narrative selfhood - the hallmark of Band 4 consciousness - requires both high CGI and high τ. Integration without persistence yields intelligence; persistence without integration yields mere habit. Only the conjunction produces the continuous self.