Thermodynamics of Learning: A Typed Four-Component Accounting of Memory, Fit, and Value

Abstract: What a finite learning device has recorded and what will hold value for it on future tasks are not the same quantity. We develop a typed accounting for finite-state learning devices that separates four components: a training-side fit functional $Φ<em>{\mathrm{fit}}$, the record-correlation stock $J</em>{D}=I(M;D)$, an update-side search ledger $σ<em>{M}$, and an operational capital value $V(M;T,b)$. This value is the work gap between an informed protocol class and a blind class obtained by deleting the memory-read port and re-optimizing from scratch. (I) Separation: for every $n$, there is a device family on which record correlation and world correlation grow by $n\ln 2$ while the capital gain is exactly zero. In the $\mathrm{flat}<sup>{*}$ regime, data-free updates never increase $V$. (II) Capitalization ledger: an exact $\mathrm{flat}<sup>{*}$ extraction identity and a universal ledger identity give, for (F5$&#39;$)-stable $M$-local updates under a no-discarded-record-correlation condition (f), the bound $η</em>{\mathrm{cap}}\le 1$ for the capitalization efficiency $η<em>{\mathrm{cap}}=ΔV/(k T\,σ</em>{M})$, together with necessary and sufficient conditions for equality. (III) Value retention: for the retention gap $L_{\mathrm{gen}}$ and retention ratio $ρ_{\mathrm{gen}}$ (the former carries no sign constraint; the latter is defined for positive training-side value and is not confined to $[0,1]$) we give a two-layer alignment domain: an exact exchange rate between value and the side-information-adjusted record fit $I(M&#39;;D\mid Y)$ without any record-side-information independence assumption, and a raw record-stock exchange rate under a joint side-information neutrality condition $(M,D)\perp Y$, whose boundary is marked by an explicit one-time-pad witness. These are statements about finite-device value retention under task-distribution shift, not a theory of statistical generalization.