Source
Lin and Tegmark (2016), LIT-887: arXiv v1, Section III E and Appendix A; arXiv v2, Section III E and Appendix B, including the response to Schwab and Mehta; as read in NOTE-688. The appendix is absent from v3, v4 and the Journal of Statistical Physics text. Schwab and Mehta’s comment on v1 (arXiv:1609.03541), not held, was read for that note. Mehta and Schwab (2014), LIT-882, Eqs. 18–22, for the identification under which the trace condition is an exact fit of the data, as read in NOTE-683.
What was actually shown
A matched partition function is not a matched distribution (v1 Appendix A, kept in v2). For a system with Hamiltonian H(y) and any non-constant K(y), the joint Hamiltonian H(y, y′) = H(y) + H(y′) + K(y) + ln Z̃, with Z̃ = Σ_y e^{−H(y)−K(y)}, has Z_tot = Z but marginal e^{−H−K}/Z̃, not e^{−H}/Z. Adding a parameter dependence keeps every derivative of ln Z matched as well, so every macroscopic observable agrees while the distribution does not. Checked. Schwab and Mehta accepted this, called the biconditional of their Eq. 8 a typo, and noted the example violates the trace condition Tr_h e^{T(v,h)} = 1.
The trace condition does not select the coarse variables either (v2, argued in one sentence: “any two systems that do not interact with each other will trivially satisfy their trace condition”). Written out under LIT-882’s identification T = −E + H: take E(v, h) = H(v) + H′(h) + ln Z′ with Z′ = Tr_h e^{−H′(h)}. Then T = −H′(h) − ln Z′, Tr_h e^T = 1 for every v, the visible marginal equals the data exactly, and the coarse Hamiltonian is H′, chosen freely and unrelated to the system. LIT-882 states that its identity holds for any Boltzmann machine, so the construction lies inside the scope of its “exact” step. The working-out is the reader’s; the observation is the paper’s.
What could have come out otherwise: the trace condition could have implied a coupling between hidden and visible variables, or constrained the coarse Hamiltonian beyond normalization; it does neither.
What this does not say
- Not that Kadanoff’s schemes are unsound. Kadanoff’s kernels have a fixed form that couples each hidden spin to a block, which rules out the non-interacting construction. The claim is about the two conditions taken as criteria; whether they suffice inside his kernel families is QUESTION-205.
- Not that an exact restricted Boltzmann machine keeps irrelevant variables. An RBM proper has no visible–visible couplings, so it cannot fit interacting data with W = 0; the second construction needs a general Boltzmann machine. For an RBM, exactness forces coupling but still says nothing about which functions of the data the hidden units carry.
- Not that renormalization must be supervised. The paper concluded that; THEORY-194 and LIT-873 show a target can come from the data’s spatial structure (the environment beyond a buffer) without labels. This account says only that some target has to be stated.
- Not a correction of LIT-882’s Eqs. 18–22, which use the pointwise condition and stand; it extends THEORY-197’s “content only at the exact point” to “and not enough there either”.
- Nothing about attribute hierarchies or embeddings, so it leaves QUESTION-025 as it was; and no instruction for machine learning.
Connections
Together with THEORY-197 (the correspondence is an identity of parametrizations) and THEORY-194 (the variables a compression keeps are set by what it must stay informative about), it closes the triangle of the 2014–2018 dispute: the formalisms correspond, neither exactness nor partition-function matching chooses the coarse variables, and an objective with a relevance target does.
Note of 2026-10-20: both counterexamples, re-derived
Re-derived for the promotion audit (batch 049, ADR-047 v2).
A matched partition function. Let Z = Σ_y e^{−H(y)}, take any non-constant K, and put Z̃ = Σ_y e^{−H(y)−K(y)} and H(y, y′) = H(y) + H(y′) + K(y) + ln Z̃. Then Z_tot = (1/Z̃) Σ_y e^{−H(y)−K(y)} · Σ_{y′} e^{−H(y′)} = Z, while the marginal of y is e^{−H−K}/Z̃, not e^{−H}/Z. If H carries a parameter, the identity Z_tot = Z holds at every value, so every derivative of ln Z matches too.
The trace condition. Under Mehta and Schwab’s identification T(v, h) = −E(v, h) + H(v), take E(v, h) = H(v) + H′(h) + ln Z′ with Z′ = Σ_h e^{−H′(h)} and H′ arbitrary. Then T = −H′(h) − ln Z′, so Tr_h e^{T} = 1 for every v. The visible marginal is proportional to e^{−H(v)}, the data exactly, and the coarse Hamiltonian −ln Tr_v e^{T(v,h)−H(v)} = H′(h) + ln Z′ − ln Z is H′ up to a constant, chosen freely. The construction needs a visible–visible term H(v) in E, which a restricted Boltzmann machine lacks, and it couples no hidden spin to any block, which Kadanoff’s kernels require. The title is held to general Boltzmann machines for that reason.
A candidate target (2026-10-20). THEORY-742 (Raju, Machta and Sethna) proposes keeping the singular part of the Fisher metric of the bare couplings. This record’s non-interacting construction fails that target, since its coarse variables carry no Fisher information about the couplings. The identity map passes it, so it is necessary, not sufficient. It does not meet this account’s promote_when, which asks about variational RG and the trace condition.