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

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arXiv:2608.12791v1 Announce Type: cross 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 $\Phi_{\mathrm{fit}}$, the record-correlation stock $J_{D}=I(M;D)$, an update-side search ledger $\sigma_{M}$, 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 $\mathrm{flat}^{*}$ regime, data-free updates never increase $V$. (II) Capitalization ledger: an exact $\mathrm{flat}^{*}$ extraction identity and a universal ledger identity give, for (F5$'$)-stable $M$-local updates under a no-discarded-record-correlation condition (f), the bound $\eta_{\mathrm{cap}}\le 1$ for the capitalization efficiency $\eta_{\mathrm{cap}}=\Delta V/(k T\,\sigma_{M})$, together with necessary and sufficient conditions for equality. (III) Value retention: for the retention gap $L_{\mathrm{gen}}$ and retention ratio $\rho_{\mathrm{gen}}$ (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';D\mid …

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