David J. Schwab and Pankaj Mehta (2016), arXiv preprint — ARXIV-1609.03541
Key takeaways
- Conceded: a preserved free energy is not a preserved distribution. Lin and Tegmark’s v1 appendix builds a joint Hamiltonian with the right partition function and the wrong marginal. The comment accepts this in full, says the authors “never claimed otherwise”, and quotes Kadanoff, Houghton and Yalabik (1976) that variational bounds pertain to the free energy and give “no guarantee that the derivatives will be accurate”.
- Conceded: Eq. 8 of LIT-882 is one-directional. The trace condition Tr_h e^{T(v,h)} = 1 for every v implies ΔF = 0; the converse fails, and the ”⟺” is called “the typo in eq. (8)”. It was more than a slip of notation: LIT-882 writes “Thus, for any exact RG transformation, we know that” the trace condition holds, inferring it from ΔF = 0 through the withdrawn direction. With exactness defined as the trace condition instead, nothing downstream of Eq. 9 changes.
- Argued, and right: the counterexample violates the trace condition. Under T = −E + H the counterexample has −T(y, y′) = H(y′) + K(y) + ln Z̃, so Tr_{y′} e^T = Z e^{−K(y)}/Z̃, which is constant only if K is. It therefore says nothing about LIT-882’s Eq. 22, which assumes the trace condition.
- Asserted: the trace condition is how variational RG keeps more than the free energy. It “implies that the microscopic probability distribution is unperturbed by the introduction of the auxiliary variables”. True, but it constrains only the visible marginal, and the comment does not say what makes the hidden variables a coarse-graining. It declines to address Lin and Tegmark’s other claims about RG and neural networks.
Standing in the record
Filed on 2026-10-09 at the owner’s request, as one of the works cited by the batch on hierarchy and hyperbolic geometry that neither record held (nucleation#115, via LIT-887 and NOTE-683). The reading of Lin, Tegmark and Rolnick (LIT-887, NOTE-688) read it and named it as held in neither, and the reading of Mehta and Schwab’s mapping (NOTE-683) and THEORY-197 cite its concession by arXiv id. Read on its own merits (NOTE-694), with the record’s description of the concession checked against the text.
It is the middle move of a three-step exchange: Lin and Tegmark’s v1 counterexample (29 August 2016), this reply (12 September 2016), and Lin and Tegmark’s v2 (28 September 2016), which accepts the reply and adds that the trace condition is met by systems that do not interact at all. The reply wins the step it answers and does not reach the next one; THEORY-202 holds that next step. The record’s descriptions of the concession in NOTE-683 and THEORY-197 are accurate.
It does not bear on QUESTION-025: it concerns the exactness condition of a spatial coarse-graining, not attributes that imply one another.