Tolman's Own Ledger: Cycle Growth, the Reversibility Split, and the Entropy per Baryon, Priced

Richard Tolman's cyclic universe of 1931–34 is usually quoted as a slogan — entropy rises, so cycles grow — and my own earlier paper on Tolman's toll priced the general wall every bounce must face. This paper prices what that paper did not: Tolman's own construction, in his own terms. Three numbers carry the result. First, the growth law: for a closed, radiation-filled universe, the largest size of a cycle and its period both grow as the two-thirds power of the entropy, so a cycle that doubles its entropy comes back 59 percent larger and 59 percent longer. Second, the past is finite in a precise sense: run the film backward and the shrinking periods add up to a finite total — with a one-percent entropy rise per cycle, all of the past fits inside about 150 present cycles, no matter how many cycles you count. Third, the ledger itself: the universe today carries about ten billion units of entropy per baryon, and the black holes Tolman never lived to price carry about a hundred thousand trillion times the entropy of all its light. A bounce that carries that ledger pays for it; a bounce that does not carry it owes a named mechanism for where it went. Tolman supplied neither a carrying mechanism nor a shedding one. He supplied the bill. Grade: the growth law and the finite-past sum are derived here in the open; the assumed step — the bounce's handling of the ledger — stands priced, still un-derived.