The Thermodynamic Cost of a Final Boundary Condition: A Resource Theory of Informational Pruning and a Rank-Indexed Contextuality Conjecture

Abstract

In prior work I modelled the macroscopic development of a closed quantum system as a directed graph of coarse-grained histories on which a final boundary condition induces an admissible subgraph, and I graded terminal conditions into a trivial, structured, and computational typology. That programme left two structural gaps: it imported its formal backbones rather than proving results of its own, and it never joined its pruning half to its agency-and-contextuality half or assigned pruning a thermodynamic cost. This paper addresses both. I construct a resource theory whose objects are two-boundary-conditioned decoherent-history ensembles, whose free objects are those in which the terminal condition leaves the history distribution unchanged, and whose free operations are boundary-compatible coarse-grainings. Theorem 1 establishes that the relative entropy of the conditioned against the unconditioned history distribution is a faithful monotone under every such coarse-graining, jointly convex in the pair of history distributions; I also exhibit a counterexample showing that the more obvious counting-based deficit is not a monotone, which is why the relative-entropy form is the correct object. Theorem 2 establishes that a register satisfying the Reeb-Wolf hypotheses, carried from the unconditioned to the conditioned distribution, deposits heat at least kBT ln 2 times the entropy difference, and, combined with the Craig rank-product bound, at least kBT ln[n0 / (rαrω)] for equidistributed history weights. Lower terminal rank therefore raises the derived cost floor under the stated assumptions. I am explicit that this is a floor and not a total ordering: rank does not by itself order terminal conditions by resource content, and full rank does not imply a trivial condition. I then advance, explicitly as a conjecture rather than a theorem, a bridge to the contextuality half of the programme: along a nested family of normalized projectors terminating at the identity, within the paradoxical class, and against a noncontextual baseline, the Contextuality-by-Default measure of pre- and post-selected cyclic statistics should be non-increasing in the family index. I do not claim that the universe physically erases histories; the more limited claim is that a register implementing admissibility filtering pays a rank-graded cost. Conjecture C is, in the first instance, a mathematical conjecture about quantum statistics, settleable by computation, not a proposed departure from

Keywords