HINGED TROUCHET TILING FRACTALS

For the period B=001 and seed F=LLLR, the paper explicitly derives the self-similar decomposition X = 2^{-3} X⊕4 + 2^{-9/2} X⊕8 + 2^{-3/2} X⊕3 and the sim equation 4·(2^{-3})^s + 8·(2^{-9/2})^s + 3·(2^{-3/2})^s = 1, yielding the cubic 8λ^3 + 4λ^2 + 3λ − 1 = 0 with λ = 2^{-3s/2} and numerical value s ≈ 1.4128; see the text around Equation (1) and Figure 11, which also identifies the 3, 4, and 8 copies at scales 2^{-3/2}, 2^{-3}, 2^{-9/2} respectively . The candidate reproduces these counts and the same cubic. The paper verifies the Moran open set condition by choosing S = LLLRRRLRRLRR and taking U to be the interior of (001)^∞ S, with the 15 similarity images disjoint and contained in U as illustrated in Figures 12–13 ; the candidate selects the same U and argues disjointness in essentially the same way. The L-system rules used by the candidate (h→hLv/v→hRv for 0 and h→hRv/v→hLv for 1) match Table 1 of the paper . Finally, for the optional case B=0001, the paper states Y = (1/4)Y⊕5 + (1/16)Y⊕16 + (1/64)Y⊕16 with sim equation 16λ^3 + 16λ^2 + 5λ = 1 and s ≈ 1.4476, which exactly matches the candidate’s extension . Minor stylistic differences aside (the candidate writes explicit composed substitutions S_B(h), S_B(v) that the paper does not spell out), the substance and structure of the proofs coincide.