Scope
A record persists when histories over increasing observation horizons admit a coherent continuation. A label replicates when separate designated carriers in the causal future of its origin admit individual decoders. This submission specifies both concepts for a history sheaf of spaces and a finite causal implementation with explicitly declared observable states, laws, and thermodynamic resources.
The accompanying PDF is the complete manuscript. It includes a higher diagram atlas and the proofs summarized below. The editable source archive includes the TeX, the vector diagrams, the references, and a short independent reproduction of finite probability and free-energy examples.
Semantic results
Let $\mathcal E=\operatorname{Sh}_\infty(\mathcal C_M)$ be the topos of sheaves of spaces on a spacetime observation site, let $o:H\to B$ be a history observation, and let $b:U\to B$ be prescribed. The space of reconstructions is $\operatorname{Map}_{\mathcal E_{/B}}((U,b),(H,o))$. Equivalently it is the homotopy fiber of $H(U)\to B(U)$ at $b$. This space satisfies relative Čech descent by mapping out of the colimit of the cover nerve in the slice. The proof extends to hyperdescent under the stated hypotheses. The manuscript gives concrete examples with nontrivial higher automorphisms and an extension criterion for Eilenberg–Mac Lane-valued histories using the long exact sequence of a pair.
Persistence over increasing horizons is the inverse limit of reconstruction spaces. Nonempty compact Hausdorff levels in the set-valued case have a nonempty limit; compact component spaces likewise suffice for nonemptiness of the homotopy limit of a tower. A tower of doubling maps on $K(\mathbb Z,1)$ shows how choices of comparison paths can distinguish coherent continuations even when every level has one component. An affine tower shows that arbitrarily long finite continuations alone do not imply an infinite continuation. For effective finite presentations, the resulting existence question is $\Pi^0_1$-complete.
Causal computation and entropy
For a finite directed acyclic network with a causal placement, let $C$ be a tuple of signals through a past cut, $W$ declared side information, and $X$ an $m$-ary source. With conditional independence of additional outside inputs, the downstream law factors through $(C,W)$. A decoder of error $e$ requires $$ H(X\mid W)-h_{\rm b}(e)-e\log(m-1)\ \le\ I(X;C\mid W)\ \le\ H(C\mid W). $$ A common trunk can supply several receivers without sending the source entropy several times across that trunk. An interactive version includes the receiver's private randomness, and a zero-error interface with receiver-only side information is characterized by the chromatic number of a finite confusability graph.
For a joint law of the source and $r$ carrier states $Y=(Y_1,\ldots,Y_r)$, the exact identity is $$ \operatorname{TC}(Y\mid W) =(r-1)H(X\mid W)-\sum_i H(X\mid Y_i,W) +H(X\mid Y,W)+\operatorname{TC}(Y\mid X,W). $$ It bounds the required correlation for independently readable instances of a common label. For tree-shaped families of compatible marginals, the separator law $q$ is the unique entropy maximizer. Every other coupling $p$ has deficit $H(q)-H(p)=D(p\|q)$. A cyclic family of pairwise laws can obstruct any joint coupling, even when all local supports are full.
Free energy and persistence
For a finite system with energy $E$, temperature $T$, and Gibbs law $\pi_E$, the nonequilibrium free energy is $F_E(p)=\mathbb E_p E-k_{\rm B}T H(p)$. A protocol alternating quenches with Gibbs-stationary finite stochastic steps obeys $$ W_{\rm in}-\Delta F=k_{\rm B}T\sum_j \left[D(p_{j-1}\|\pi_{E_j})-D(p_j\|\pi_{E_j})\right]\ge0 . $$ For additive endpoint energies the joint free energy splits into marginal terms and $k_{\rm B}T$ times total correlation. This yields a work bound for creating replicas with specified errors, initial correlations, and blank states. Pure blanks can supply the free energy for reversible classical copying. For a fair retained reference and $r$ initially thermal flat bits, the conditional equilibration infimum for independent symmetric crossover $e>0$ is $r k_{\rm B}T(\log 2-h_{\rm b}(e))$. Exact writing from full support is treated through its limiting cost in this finite-energy class.
If passive noise degrades a binary record from crossover $e_0$ to $e_-$ with the reference retained, restoring it has work infimum $k_{\rm B}T[h_{\rm b}(e_-)-h_{\rm b}(e_0)]$ under controlled equilibration. A finite-activity jump protocol has an additional positive lower bound on dissipation. For nonuniform target energies and a fixed decoder, a conditional Gibbs tilt gives the exact minimum endpoint free energy for an average accuracy target. The tilt depends on the Gibbs weight of each correct decoder fiber. The manuscript also decomposes the cost of local modules that destroy correlation with inaccessible spectators.
Evidence and authorship
The manuscript contains mathematical proofs under stated hypotheses. Worked examples include shared upstream noise, erasure archives, and branching histories. The calculations describe declared finite models rather than experimental measurements. The byline is anonymous. Generative AI assisted with drafting, typesetting, and checking selected finite examples.
Selected references
- Jacob Lurie, Higher Topos Theory. https://arxiv.org/abs/math/0608040
- Emily Riehl and Dominic Verity, Elements of Infinity-Category Theory. https://www.cambridge.org/core/books/elements-of-infinity-category-theory/DAC48C449AB8C2C1B1E528A49D27FC6D
- Antonio N. Bernal and Miguel Sánchez, Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. https://arxiv.org/abs/gr-qc/0401112
- Tobias Fritz, A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. https://arxiv.org/abs/1908.07021
- Udo Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines. https://arxiv.org/abs/1205.4176
- Juan M. R. Parrondo, Jordan M. Horowitz, and Takahiro Sagawa, Thermodynamics of information. https://www.nature.com/articles/nphys3230