Within Lacanian psychoanalytic theory, the "end of analysis" resists closure. Lacan outlined several endpoints—revelation of subjective truth, traversal of fantasy, identification with the symptom, and identification with the sinthome—while stressing that analysis promises neither cure nor happiness. This study reframes these endpoints through a mathematical analogy, treating the analytic process as a function over signifiers and its ending as a matter of limits rather than final values.


The subject's discourse is modeled as a function F mapping signifiers to meanings. Traumatic points appear where F is undefined, yet limits may still exist nearby. An analytic "end" is thus conceptualized as convergence: the limit of F(x) as x approaches a, where the process circles around objet a without reaching a final value. The sinthome, in this view, names a re-knotting that stabilizes this orbit of jouissance, permitting consistency without closure.


To explore this model clinically, the study draws on a single-case design (Ms. S., 20 sessions). Clinical material, anonymized for confidentiality, showed her repetitive positioning as the object of the Other's demand—"serving"—and her occupational shift from engineering to flight attendant work ("hosting"). Findings indicate a shift from symptomatic repetition toward a reconfiguration consistent with Lacan's later teaching: rather than abolishing the symptom, Ms. S. moved toward an identification that stabilized jouissance. In functional terms, the signifier "hostess/host" rewrote the mapping so that trajectories once divergent circulate within a bounded neighborhood; the limit exists even if the value is not taken.


These results clarify why analytic endings are not cures or definitive closures but convergences toward a workable regime of desire and jouissance, often through identification with the sinthome. Thus, the end of analysis is best conceived not as a last word but as a limit.