Source
Sclocchi, Favero, Levi and Wyart (2024; ICLR 2025; JSTAT 2025), LIT-886, §3, App. A.2 (Eqs. 20–54), App. B, Figs. 2, 8–11, as read in NOTE-691. The model, the BP denoiser and the annealed map are those of LIT-884 (THEORY-199).
What was actually shown
The setting. The Random Hierarchy Model of LIT-877 with known rules, branching s, v symbols per level, m rules per symbol, in the regime where the class threshold of THEORY-199 exists: s m (v − 1)/(v^s − 1) < 1 and L → ∞. Leaves are noised (the ε-process: belief 1 − ε + ε/v on the true symbol), a new datum is sampled from the exact posterior by belief propagation, and each token is marked changed or not. The object is the connected correlation C(r, ε) of two tokens’ change marks over trajectories, averaged over data, and its sum, the dynamical susceptibility χ.
The calculation. In the annealed approximation, the joint law of two tokens’ changes is a product of 2 × 2 matrices through their lowest common ancestor (Eq. 51). Near the threshold ε* the upward belief p_ℓ stays near the repulsive fixed point p* of the map F for ℓ̃ ≈ −log|ε − ε*| / log F′(p*) levels before flowing to recovery or to chance. Equating the correlation length with the block a latent at that depth spans, ξ ≈ s^ℓ̃, gives ξ ∼ |ε − ε*|^(−ν), ν = log s / log F′(p*) (Eq. 54): the form of a renormalisation-group correlation-length exponent, with s as block factor.
What could have come out otherwise. Exact BP on rule draws at v = 32, m = 8, s = 2, L = 9 could have departed from the mean field or shown no peak; it matches the mean-field C(r, ε) and χ(ε), both peak at ε* ≈ 0.74 with C(r, ε*) ∼ r^(−1), and rescaling distances by Eq. 54’s ξ collapses the curves for ε between 0.60 and 0.90 (Figs. 2a, 9). With masking noise instead of the ε-process, the peak sits where the class reconstruction drops (Figs. 2b, 10), and it does not depend on whether sampling is by BP or by backward diffusion with the exact score (Fig. 8). For a Gaussian random field with spectrum ‖k‖^(−a) and exact score, the changes are the resampled modes above κ*(t) = (1/α_t − 1)^(−1/a), so the length grows monotonically and χ is largest at the end (App. B, Fig. 11): there, a peak at an intermediate noise does not occur.
What this does not say
- Not proved. The exponent comes from linearising an annealed map and identifying an escape depth with a correlation length; the critical decay exponent (a = 1) is fitted, not derived. The agreement with exact BP is one parameter set at s = 2. Recomputing from Eqs. 22 and 54 at that set gives ν ≈ 1.89 where the paper quotes ν ≃ 1.78 (NOTE-691, Limitations), an unexplained gap.
- Not about trained diffusion models. The denoiser knows the grammar, as in THEORY-199.
- Not that a χ peak implies a hierarchy. The only non-hierarchical contrast is the Gaussian field. A flat model with a class transition (a mixture of many classes) is not analysed, and could plausibly give χ a peak at its transition without a power law across scales. The discriminating signature in the model is the power-law C(r) at the peak, not the peak alone.
- Not that text or images are generated by such a grammar. The susceptibility peaks in a masked text diffusion model and an ImageNet diffusion model (read through CLIP patch embeddings) are consistent with it, over about one decade of distance and without a control.
- Not about attribute hierarchies. The hierarchy is part–whole; it leaves QUESTION-025 where it was.
- Not an instruction. It describes what an ideal denoiser’s changes look like, not how to build or use one.
Connections
It is a consequence of THEORY-199: the threshold that account places on the class becomes, here, a critical point with a diverging length. It is set beside THEORY-201, whose correlations are static, between tokens of the data; these are dynamical, between changes, and the paper’s Gaussian-field contrast is its case that static power-law correlations do not produce them. The exponent formula makes the level-to-level belief map a coarse-graining flow with one relevant direction, the kind of object THEORY-194 is about (which coarse-graining keeps the relevant variables), though here the flow is fixed by the grammar rather than chosen. The source of the observable is the physics of glasses, where a peaking dynamical susceptibility marks cooperative rearrangement.