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

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 Phifit, the record-correlation stock JD=I(M;D), an update-side search ledger σM, 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 nln 2 while the capital gain is exactly zero. In the flat* regime, data-free updates never increase V. (II) Capitalization ledger: an exact flat* extraction identity and a universal ledger identity give, for (F5')-stable M-local updates under a no-discarded-record-correlation condition (f), the bound ηcap≤ 1 for the capitalization efficiency ηcap=Delta V/(k T,σM), together with necessary and sufficient conditions for equality. (III) Value retention: for the retention gap Lgen and retention ratio ρgen (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';Dmid 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.