fractal-dimension · Fractal dimension · 73 records · all concepts
runs53 · +7 retired
051Run 049 was withdrawn on its control (the sealed p-grid could not resolve the square lattice at L = 2187).run
052Run 051 found the all-singleton carpet U′ in a different percolation class from carpets H and U at the same dimension (1/ν = 0.276 against 0.547 and 0.524), but its exponent sits beside the gasket's (0.263), and the gasket is finitely ramified.run
053Runs 051 and 052 found three carpets of identical Hausdorff dimension whose percolation exponents 1/ν (0.547, 0.524, 0.276) sit at a near-constant fraction of their left–right cut exponents x (0.631, 0.631, 0.316): 1/ν ÷ x = 0.867, 0.830, 0.873.run
054Run 053 found carpets V and H sharing a cut exponent (0.631) and a bond count at every size, yet differing in 1/ν by 0.070 (0.477 against 0.547), with U between them (0.524).run
055Run 054 found that the minimum-cut entropy per cut bond, s = ln N / C, orders carpets H, U and V (one cut exponent, 0.631) exactly as their percolation exponents do, s 0.693 < 1.058 < 1.153 against 1/ν 0.547 > 0.524 > 0.477.run
056The square lattice, measured with run 051's estimator at the same sizes, gave 1/ν = 0.7516 ± 0.0049 in run 053 and 0.7372 ± 0.0049 in run 055 — a shift of 0.014, about twice the standard error of a difference between two independent runs.run
058Runs 054–055 found that the number of distinct minimum cuts orders the carpets' percolation exponents.run
059Run 058's one-block polynomial ordered all eight lattices' exponents but not their values (y reads carpets high by 0.009–0.065, the square low by 0.020), and put the thresholds p* 0.03–0.06 above the measured ones.run
060Run 058's crossing polynomials R(p) are exact.run
061Run 059's corrected exponent failed because it read the 81×81 lattice's slope at the one-block fixed point p*, where that lattice is already on the shoulder of its crossing curve.run
062On all seven carpets counted so far, the number of minimum left–right cuts obeys an exact recursion across generations, N(g+1) = c·N(g)^C₁, with C₁ the one-step cut size and c a constant (H 1, U 3, V 4, T 8, U′ 4, A 36, B 81/2500), checked as exact integers at g = 1→2→3 and in logarithm to g = 4.run
063Run 062's pool contains cut twins — distinct D4-symmetric 9×9 generators with identical minimum-cut counts N(g) at every generation (entries 5 and 12 twin X; 10 twins Y; 11 twins Z).run
064Probe P-048 argues that block renormalization fails to converge on the Sierpiński-type carpets because they are infinitely ramified (blocks share whole edges).run
066The block exponents y_g = ln P_g′(p_g*)/ln 9^g rose from g = 1 to g = 2 on every lattice (run 061).run
067A carpet may use a different 9×9 pattern at each level.run
068The construction (a 9×9 pattern keeping 64 cells, outer ring kept, symmetric under the eight symmetries of the square, centre removed) admits exactly 63 patterns.run
069Feigenbaum's universality lives in a function, not only in δ: the fixed point of the doubling operator.run
070Run 067 found that stacking two carpets of the same cut class (X over Y, both cut 4) averages their exponents, while stacking a cut-2 pattern with a cut-4 one (X over U′) lands 3.5σ below the average.run
071Run 068 measured 14 of the 63 patterns and left a hierarchy, post hoc: the minimum cut C₁ sets a band of exponents (cut 2: 0.260–0.349; cut 3: 0.372–0.427; cut 4: 0.429–0.551; cut 5: 0.588–0.613), and within a band the number of minimum cuts N₁ sets a sub-band, more minimum cuts lower (cut 2: N₁ = 4 at 0.303–0.349, N₁ = 16 at 0.260–0.276; cut 3 monotone in N₁).run
072Run 069 found the width-standardized crossing curve indistinguishable across carpets by a Kolmogorov–Smirnov distance, but reported a moment that is not flat: the skewness of the standardized threshold distribution at L = 6561 is positive on every carpet (+0.11 to +0.55) and near zero on the square (+0.011), and in the three twin pairs whose exponents separate, the lower-exponent member has the lower skewness.run
073Run 072 found the skewness g₁ of the threshold distribution reproducible across reseeds but still moving between L = 729 and 6561 on four carpets (g₁(6561) − g₁(729): T −5.2 combined errors, X +3.2, Z +2.8, X1 −2.5).run
074Run 073 read the skewness g₁ of the threshold distribution at three sizes a factor of 9 apart and found it built between 81 and 729 on thirteen of fourteen carpets, then turning back, levelling off or still climbing.run
075Each 9×9 pattern has an exact one-block map, the crossing polynomial f(p) of run 058 (rg058.poly).run
076Stacking two carpet patterns A and B gives, at the one-block level, the 2-cycle of their crossing polynomials, with y_AB = ln(R_A′(r)·R_B′(q))/ln 81 at q = R_A(r), r = R_B(q).run
077Feigenbaum's universality sorts one-hump maps by the order z of their maximum; the amplitude of the maximum drops out.run
078Runs 076–077 found that, at fixed fixed-point separation δ, the one-block composition deviation |D| depends on cut class C₁ (partial +0.60 over all pairs; +0.72 over unequal-class pairs with the amplitude (ln N₁)/C₁ also held) and not on the amplitude.run
079Run 074 read H, the standard 3×3 carpet, at five sizes a factor of 3 apart (L = 81, 243, 729, 2187, 6561; generations 4–8) and found, post hoc, that the skewness steps alternate — small, large, small, large — with the large steps ending at even generations.run
08043 census patterns have a measured 1/ν (14 measured by run 068, 15 by run 071, 14 carried as record values from earlier runs without errors).run
081Run 073 found the skewness g₁ of the crossing threshold falling between L = 729 and 6561 on two carpets, T (−0.191 ± 0.038) and X1 (−0.116 ± 0.045), while the exact one-block recursion rises at every depth on all 63 patterns.run
083PN-005's House named the route the one-block recursion cannot represent: a lattice crossing that passes through a top-level block without crossing it left to right.run
084Run 082 was withdrawn on a mis-written control, but what its lab saw before stopping reshapes the ladder: rung 1 is wall-bound at the carpet's sizes (the threshold within 2.8 widths of 1 at L = 729), its slopes sit below both the carpet and the one-bit map and in the wrong order (X below T), and the one-bit map sits above the carpet.run
085Run 083 found, on two carpets, that the route the one-block recursion cannot represent is the majority route at every size, that the lag Δ = t − t_coarse between the lattice crossing and the coarse block crossing is about −0.8 of the threshold width on both patterns at both sizes, and that the share of exact ties (Δ = 0) is the one per-sample quantity that separates T from X (10% against 5%).run
086The census has 63 patterns and 49 crossing polynomials; the 14 extra patterns share a polynomial with another, in eight groups of two to four, and the completed census measured one member of each.run
087P-050 read the census's local slopes and found the cut-4 exponents reading as a fan around y₉ — the spread of the local slope wider over 729→6561 than over 81→729 — carried by five patterns whose stored drift reads positive while thirty of thirty-five fall, only one of them beyond two drift errors: #60 (+0.043), #63 (+0.010), #10 (+0.012), #3 (+0.007), #33 (+0.001).run
089The one-block map f = R_P carries the depth-k threshold law to the depth-(k+1) law by one fixed function, h = f⁻¹: Q_{k+1} = h ∘ Q_k, so the step map Q_{k+1} ∘ Q_k⁻¹ is h at every depth and, by Koenigs, the map is linear in a coordinate φ with φ ∘ h = λ⁻¹·φ.run
091Each sample's threshold is a minimax — the lowest level at which some crossing path is fully open — and its two tails are set by two different optimizations: crossing far below threshold needs the cheapest path, the ring, of length ∝ L; failing to cross far above it needs the cheapest blocking cut, of size ∝ C₁^g = L^(ln C₁/ln 9).run
092The calibration note on run 081 found the bootstrap error of the skewness g₁ right within a file (pooled block-to-bootstrap ratio 1.00 over ten files) while files of one graph differ beyond their block errors at L = 6561 — T4: 0.172 ± 0.026 on samples053/T4.txt (the file run 073 read), run 070's 0.282 ± 0.028, run 081's 0.248 ± 0.020 (one-way F over blocks, p = 0.008); X1's F4: 0.216 against 0.348 (Welch on blocks, p = 0.005) — and not at 729.run
093The crossing polynomials of the 49 census fibres carry squares and a composition that nothing in the record explains.run
094Run 084 scored rung 3 of the ladder on S₃ = {1, 4, 7}: X's local slope over 729→6561 is below T's by 0.131 ± 0.007 where the carpet has it above by 0.18.run
095P-054's look found that on T and X at L = 729 the sizes of invasion bursts — the runs of accepted links from one forced link to the next, growth from the whole left side — follow a power law with a ripple about it whose period is a factor of about 55–64 in size, T's twice X's and none on the square.run
096Scaffold S2 (single-exponent scaling, w ∝ L^(−1/ν)) has its first data against it in a reading, not a result: run 079's five ln w values for H (#17, the Sierpinski carpet's second generation; the only census pattern with sizes at every power of 3), 48,000 samples each at L = 81, 243, 729, 2187, 6561, lie on a straight line in ln L with χ² = 8.28 on 3 degrees of freedom; the weighted quadratic coefficient is +0.00319 ± 0.00112 (+2.85 errors); the pair slopes fall 0.553, 0.550, 0.541, 0.533; in the census's unit the drift is −0.0147 ± 0.0058.run
097Run 083 found that the lattice crosses before the graph of its 64 top-level blocks does in 85% of samples on T and X at L = 729, a block counting as open once it is crossed left to right on its own bonds; run 085 that the mean lead is 0.46 to 1.92 widths over the census.run
098Within a group of patterns that share one core, how much does a pendant part lower the lattice threshold at L = 729?run
100On H's two stored sets, with the mean, ln S and g₁ fitted jointly under PG-03's three one-term forms, tying the correction exponent ω to 1/ν costs Δχ² = 11.28 (each file's covariance; 8.54 with diagonal errors) on all five sizes and 1.47 (0.59) with L = 81 left out.run
102The free one-term form (ω its own parameter; the forms are run 100's), fitted on run 100's six files at L = 81 (H4a, H4b), 243 (H5a, H5b), 729 (H6) and 6561 (H8), predicts H's three moments at L = 2187 — ln S = −5.4449911 ± 0.0012293, mean = 0.6843628 ± 0.0000088, g₁ = 0.3634693 ± 0.0042900 (profile errors, χ² ≤ min + 1 in all nine parameters).run
105The free one-term form (ω its own parameter; the forms are run 100's), fitted on run 100's seven new files at L = 81 (H4a, H4b), 243 (H5a, H5b), 729 (H6), 2187 (H7) and 6561 (H8), predicts H's three moments at L = 19,683 — ln S = −6.6218285 ± 0.0032965, mean = 0.6833995 ± 0.0000125, g₁ = +0.4230883 ± 0.0074911 (profile errors, χ² ≤ min + 1 in all nine parameters; the fit is χ² = 12.406 on 12, 1/ν = 0.5260, ω = 0.3009).run
106Bond percolation on the b = 2 diamond hierarchical lattice (each bond replaced at every level by two parallel paths of two bonds; level n has 4^n bonds and "size" L = 2^n).run
107Does the orbit through which the cut-2 wall enters the second ring order the lattice exponent?run
108Does the algebraic connectivity λ₂ of a generator's kept-cell graph (the second-smallest eigenvalue of its graph Laplacian at generation 1) order the lattice exponent 1/ν across the census as well as the cut readings do?run
130A host that keeps the dilations by 3 (H) may carry comb teeth only at the frequencies those dilations allow, the even teeth of the base-64 comb in ln(burst size).run
133On fresh draws of H at L = 2187, if the comb's even-tooth period P is fitted instead of fixed at 8, do the odd teeth vanish at the fitted period?run
134Does the forbidden k = 3 tooth on H come from the full-side source?run
138(T) On fresh full-side draws of H8 at L = 6561, does the 3/2 tooth (k = 3) survive every member of the declared set of backgrounds when both steps are modelled?run
049The dictionary's §45 claim that phase transitions need infinite ramification is a literature theorem (Gefen–Mandelbrot–Aharony 1980–84), retrodicted, not found.run
082Probe P-049 builds, for any census pattern, the rung-k lattice F_k: the carpet with its block-crossing bonds thinned to k per shared edge at every level, so that adjacent blocks of any level are joined by exactly k bonds.run
090Each sample's threshold is a minimax — the lowest level at which some crossing path is fully open — and its two tails are set by two different optimizations: crossing far below threshold needs the cheapest path, the ring, of length ∝ L; failing to cross far above it needs the cheapest blocking cut, of size ∝ C₁^g = L^(ln C₁/ln 9).run
099On H's two stored sets, with the mean, ln S and g₁ fitted jointly under PG-03's three one-term forms, tying the correction exponent ω to 1/ν costs Δχ² = 11.28 (each file's covariance; 8.54 with diagonal errors) on all five sizes and 1.47 (0.59) with L = 81 left out.run
103For each census pattern with stored widths at 81, 729 and 6561, the stored drift is d = s(81, 729) − s(729, 6561), where s(L₁, L₂) = −[ln w(L₂) − ln w(L₁)] / ln(L₂/L₁) is the local slope of the 20th–80th percentile width w of the per-sample crossing threshold.run
104For each census pattern with stored widths at 81, 729 and 6561, the stored drift is d = s(81, 729) − s(729, 6561), where s(L₁, L₂) = −[ln w(L₂) − ln w(L₁)] / ln(L₂/L₁) is the local slope of the 20th–80th percentile width w of the per-sample crossing threshold.run
137(T) On fresh full-side draws of H8 at L = 6561, does the 3/2 tooth (k = 3) survive every member of the declared set of backgrounds?run
probes10
P-025Topological protection conserves non-locality — the measure is the distance exponent, and the frontier has an open cornerprobe
P-047How far is each carpet from a pure composition — is the drift in its pair slopes, and the misplacement of carpet B, a measurement of the commutator between crossing a block and crossing the arrangement of blocks?probe
P-048Taken as true, "the cut constant c sets the one-block miss" contradicts itself across block sizes — so is the block renormalization of an infinitely ramified carpet converging to its percolation exponent at all?probe
P-049The ramification ladder — between the one-bit recursion and the carpet lie finitely ramified relatives on which the crossing recursion is exact; climb them and see where the carpet's turn, and its exponent, first appearprobe
P-050The census read without a question — the one-block fixed point overshoots every lattice threshold by a fixed amount set by the ring share of the pivotal set; the exponents fan around y₉ and are still falling everywhereprobe
P-051The ladder made continuous — give the links across block seams their own chance; the contact set between blocks then has a dimension between 0 (a rung) and 1 (the carpet), and ramification becomes a dialprobe
P-052The dial that varies across the sample — gradient percolation on a carpet; one configuration locates the threshold and its sharpness, and the front should carry the holes' imprint at every factor of nineprobe
P-053A dial for every block — each block has its own crossing point and its neighbours shift it; the shift is a small table with one entry per pair of sides, and how fast the table settles is the correction the one-exponent scaling lacksprobe
P-054Invasion — grow the crossing from the left side by always opening the easiest link; it finds each draw's crossing point without a dial, its forced links and bursts are a second exponent, and the burst sizes on a carpet carry a ripple with the period of the hierarchyprobe
P-055Bits left in the table — compress the census with the one-block polynomial as the codebook and count what stays; PG-01's "finite number of boundary bits" becomes a number read from stored data, and columns that compress together are one objectprobe
notes3
N-026The intermediate-growth cell is real — effective dimension grows without bound, computed on the Grigorchuk groupnote
N-037What runs 048–060 say together — order holds, magnitude misses; first-order models rank and correlations set values; the seal is the weakest linknote
N-040Returns — what the lattice does that the block map does not, and in which directionsnote
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