General-dyadic extraction identity above a budget floor

Establish whether the exact flat-task extraction identity can be restored for general dyadic conditional distributions when the evaluation budget exceeds an appropriate budget floor.

Background

The exact extraction identity is proved only when each relevant conditional distribution is a point mass or uniform on a subset of the alphabet. The paper explains that deeper dyadic distributions can force negative-work branches, which violate the pathwise budget constraint at zero budget.

The authors leave open whether a sufficiently large budget floor restores exact attainment for general dyadic states. A noisy binary-symmetric-channel example is given as a concrete special case, with a conjectured threshold for attainment.

References

A general dyadic version above a budget floor $b{*}$ is a recorded future slot (Sec.~\ref{sec:discussion}).

— Thermodynamics of Learning: A Typed Four-Component Accounting of Memory, Fit, and Value  (2608.12791 - Sudo, 13 Aug 2026) in Section 3.3.1, Remark 3.10; Section 9 (Discussion and limitations); Appendix B, subsection 'Witness A'

while attainment for $b\ge b{*}(\epsilon)=k T\ln\bigl(1/(2\epsilon)\bigr)$ is a conjecture (the same open slot as the general-dyadic budget floor, Remark~\ref{rem:flat-scope}(v)).

— Thermodynamics of Learning: A Typed Four-Component Accounting of Memory, Fit, and Value  (2608.12791 - Sudo, 13 Aug 2026) in Appendix D, subsection 'Witness A: device AC$'$'

It is therefore unclear whether a sufficient flatness condition can be established at a fixed level $(m,d)$ to extract a source-transfer realization, or equivalently a mixed-model realization.

— Quantum tree networks: complete semidefinite hierarchies and the source-transfer model  (2609.19310 - Xu et al., 16 Sep 2026) in Remark 10.6, Section 10.3, “Inflation-NPA SDP hierarchy and its convergence”