If time can be modified hierarchically in a physical system, it must face the dualconstraints of operational cost and logical boundaries. This paper proves that timeupgrades necessarily reach a fixed point after finitely many steps, and that thisfixed point is unique. The core argument is based on a quantitative derivation ofthermodynamic operational costs: each time upgrade involves the physical erasureof at least one bit of information, consuming at least the minimal work kBT ln2according to Landauer’s principle; the total information budget of the system isbounded by the Bekenstein information-energy bound. From this we derive themaximum level nmax = Etotal/(kBT ln2) and prove its uniqueness under fixedtotal energy and environmental temperature. Furthermore, as the level approachesthe critical value Nc, causal self-referential operations lead to the emergence ofclosed causal chains, destroying the transitivity of the causal order. However, thislogical critical point lies above the thermodynamic fixed point, making it physicallyinaccessible due to energy bankruptcy. Thermodynamic capping constitutes theactual insurmountable boundary for time upgrades, while the causal paradox cutoffserves as a logical safety redundancy. Together, the two constraints lock the uniqueoptimal time level.