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      lex: "Measured: 4 rules among (e)-(h) that are complex/random-like by the pre-registered criteria (n = 4). Interpretation: \'Fluctuations ... complex and in many respects random\' (p. 130-131) is read as: the increment-sign sequence of the detrended fluctuations is not compressible to a small fraction of its entropy bound by a general-purpose compressor (nested/digit-based fluctuations compress to < 0.01 bits/symbol), and positive and negative fluctuations are balanced (Wolfram\'s own stated \'respect\' for case (f)). The claim is existential (\'can produce\'), so one qualifying rule suffices; the book\'s nested rules (c) and (d) serve as internal controls that must be detected as regular. The OEIS b-files provide independent public values of two of the sequences to verify the implementation. Trend n/2 for (c)-(g) follows the book\'s plots; the power-law trend for (h) follows the book\'s caption (0.42 n^0.916)."
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      lex: "Pre-registered. Rules (all with f[1]=f[2]=1, from NKS pp. 129-130): (c) f[n]=f[f[n-1]]+f[n-f[n-1]] (OEIS A004001); (d) f[n]=f[n-f[n-1]]+f[n-f[n-2]-1]; (e) f[n]=f[n-f[n-1]]+f[n-f[n-2]] (OEIS A005185); (f) f[n]=f[n-f[n-1]-1]+f[n-f[n-2]-1]; (g) f[n]=f[f[n-1]]+f[n-f[n-2]-1]; (h) f[n]=f[f[n-1]]+f[n-2f[n-1]+1]. Compute f[n] for n <= N = 2^20 (stop a rule if an index < 1 occurs). Residual r[n] = f[n] - n/2 for (c)-(g); for (h) r[n] = f[n] - a n^b with (a,b) a least-squares power-law fit in log-log. Statistics: (i) bits per symbol = 8 * lzma(preset 9|EXTREME) compressed size of the increment-sign sequence sign(r[n]-r[n-1]) in {-1,0,1} stored as one byte each, divided by N-1; (ii) sign balance |P(increment>0) - P(increment<0)|; (iii) dyadic self-similarity corr(r[n], r[2n]) for n in [N/4, N/2) (reported). A rule is \'complex/random-like\' if it is well defined up to N, (i) >= 0.5 bits/symbol and (ii) <= 0.05. The claim PASSES if at least one of rules (e)-(h) is complex/random-like AND the nested controls (c) and (d) both have (i) < 0.15 bits/symbol (validity of the measure). Calibration from controls outside the family: binary digit sum s2(n) gives 0.002 bits/symbol, its partial sums 0.005, a symmetric random walk 1.10."
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