Sustainability Constraint

The Sustainability Constraint is the ceiling on recursion: the data capacity after n completed pulses, ℜ(n) · ℨᵇ, may not exceed the maximum capacity of the substrate, I▱⥣. Part 1.5 sets it beside the Recursive Growth Law. Because capacity doubles at every pulse, ℜ(n) = 2ⁿ, the doubling is exact and unrelenting, and it is the finite substrate that ends each stage: when the system holds all the substrate can hold it hits a critical threshold and must reorganize, as a computer restarts when it runs out of memory. The ceiling is reached at a definite depth, the Phase Transition Criterion ℜ(n⨶) = T▱, n⨶ = log₂(T▱), where the system reorganizes into the next level of structure; the ceiling is finite (Bekenstein 1981; Bousso 2002), so no capacity approaches infinity. Together the two properties give the full law of recursive limits. Whether I▱⥣ and T▱ name one ceiling, and whether n⨶ is the Rupture Rule's L, a universe rupturing at 2^L of its own pulses, is an open question the book frames. The book writes ℜ(n) · ℨᵇ ≤ I▱⥣, where ℜ(n) is the total data capacity after n completed pulses [∅], ℨᵇ the Zinf bit [1ᵇ] and I▱⥣ the maximum substrate capacity, the absolute computational limit the substrate architecture imposes [1ᵇ]: [∅] × [1ᵇ] ≤ [1ᵇ]. It connects to The Recursive Growth Law, Phase Transition Criterion, Rupture Rule, Two-Sided Growth, Substrate Threshold and Synchronization Cost.