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A natural model of a source of randomness consists of a long stream of symbols X = X_1∘…∘X_t, with some guarantee on the entropy of X_i conditioned on the outcome of the prefix x_1,… ,x_{i-1}. We study unpredictable sources, a generalization of the almost Chor-Goldreich (CG) sources considered in [Doron et al., 2023]. In an unpredictable source X, for a typical draw of x ∼ X, for most i-s, the element x_i has a low probability of occurring given x_1,… ,x_{i-1}. Such a model relaxes the often unrealistic assumption of a CG source that for every i, and every x_1,… ,x_{i-1}, the next symbol X_i has sufficiently large entropy. Unpredictable sources subsume all previously considered notions of almost CG sources, including notions that [Doron et al., 2023] failed to analyze, and including those that are equivalent to general sources with high min entropy. For a lossless expander G = (V,E) with m = log |V|, we consider a random walk V_0,V_1,…,V_t on G using unpredictable instructions that have sufficient entropy with respect to m. Our main theorem is that for almost all the steps t/2 ≤ i ≤ t in the walk, the vertex V_i is close to a distribution with min-entropy at least m-O(1). As a result, we obtain seeded online condensers with constant entropy gap, and seedless (deterministic) condensers outputting a constant fraction of the entropy. In particular, our condensers run in space comparable to the output entropy, as opposed to the size of the stream, and even when the length t of the stream is not known ahead of time. As another corollary, we obtain a new extractor based on expander random walks handling lower entropy than the classic expander based construction relying on spectral techniques [Gillman, 1998]. As our main technical tool, we provide a novel analysis covering a key case of adversarial random walks on lossless expanders that [Doron et al., 2023] fails to address. As part of the analysis, we provide a "chain rule for vertex probabilities". The standard chain rule states that for every x ∼ X and i, Pr(x_1,… ,x_i) = Pr[X_i = x_i|X_[1,i-1] = x_1,… ,x_{i-1}] ⋅ Pr(x_1,… ,x_{i-1}). If W(x₁,… ,x_i) is the vertex reached using x₁,… ,x_i, then the chain rule for vertex probabilities essentially states that the same phenomena occurs for a typical x: Pr [V_i = W(x_1,… ,x_i)] ≲ Pr[X_i = x_i|X_[1,i-1] = x_1,… ,x_{i-1}] ⋅ Pr[V_{i-1} = W(x_1,… ,x_{i-1})], where V_i is the vertex distribution of the random walk at step i using X.

We present a randomness efficient version of the linear noise operator T_ρ from boolean function analysis by constructing a sparse linear operator on the space of boolean functions {0,1}ⁿ → {0,1} with similar eigenvalue profile to T_ρ. The linear operator we construct is a direct consequence of a generalization of ε-biased sets to the product distribution 𝒟_p on {0,1}ⁿ where the marginal of each coordinate is p = 1/2-1/2ρ. Such a generalization is a small support distribution that fools linear tests when the input of the test comes from 𝒟_p instead of the uniform distribution. We give an explicit construction of such a distribution that requires log n + O_{p}(log log n + log1/(ε)) bits of uniform randomness to sample from, where the p subscript hides O(log² 1/p) factors. When p and ε are constant, this yields a support size nearly linear in n, whereas previous best known constructions only guarantee a size of poly(n). Furthermore, our construction implies an explicitly constructible "sparse" noisy hypercube graph that is a small set expander.