roots 049 · N-026 · P-025 · 133 records · depth 35 · 22 unanticipated · 33 falsified · 36 confirmed · 11 withdrawn · index density (§5)
049 The dictionary's §45 claim that phase transitions need infinite ramification is a literature theorem (Gefen–Mandelbrot–Aharony 1980–84), retrodicted, not found.
N-026 The intermediate-growth cell is real — effective dimension grows without bound, computed on the Grigorchuk group
P-025 Topological protection conserves non-locality — the measure is the distance exponent, and the frontier has an open corner
030 On the intermediate-growth host Cay(G/St(4)) (n = 4096), is the domain-wall free-energy barrier controlled by the host's local, low-dimensional structure — or does the entropy of near-minimal cuts eat it?
051 Run 049 was withdrawn on its control (the sealed p-grid could not resolve the square lattice at L = 2187).
P-026 Primes upstairs, zeros downstairs — the growth trichotomy as a prime-number-theorem / Riemann-hypothesis dichotomy on the host graph
P-043 A growth axis on one tree — five self-similar groups on the binary tree, the nucleus as the coordinate, and the claim that nucleus size predicts which instruments move
031 Does the minimum bisection of Cay(G/St(n)) track the Schreier-fibre count |G_n|/2^n — a separator of order n/log n whose exponent climbs toward 1 — or does the exponent 0.76 measured between n = 128 and n = 4096 (run 030) hold?
032 Run 030 found two metastable families of near-minimal cuts on Cay(G/St(4)) (≤ 224 edges from METIS, 270 from disorder).
033 The Ihara primes of the host are its relators.
035 Expanders are globally synchronising (the Kuramoto flow has no stable equilibrium but synchrony — no twisted states, no U(1) loop-closing sectors).
036 Expander codes (Sipser–Spielman, Tanner) need unique-neighbour expansion of small sets; amenability kills expansion only asymptotically.
052 Run 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.
N-035 The sequential-circuit reading of self-similar groups — latch, register, carry, counter — and why "which types exist" is a 3×3 grid, not a line
034 Run 032 identified the glassy walls of run 030 with the minimum-bisection landscape of the ⟨b,c,d⟩-coset graph, a spin glass by Fu–Anderson.
038 At level 4 the sign of the 176-cut's projection onto the λ₂ irreducible (the sibling-odd representation, dimension 8) is a balanced cut of exactly 192 = (3/4)·256, and the bands up to dimension 72 reproduce 176, so 11/16 = 3/4 − 1/16.
041 Run 036 measured edge expansion of the ⟨b,c,d⟩-coset graph at 0.78 of a random 4-regular graph's at s = N/8, and P-026 read the intermediate cell's code ceiling from it.
053 Runs 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.
P-045 A quantum walk on the intermediate-growth cell — is the boundary a quasicrystal and the bulk a flat-band localizer, or is the analogy decorative?
039 At level 4 the 176 minimum bisection of the ⟨b,c,d⟩-coset graph is exactly the sign of its projection onto the lowest 73 Laplacian eigenvectors — the constant, the λ₂ = 4−4cos(π/8) irrep (multiplicity 8, share 0.697), two bands it is orthogonal to (4−4cos(π/6), multiplicity 8, and 2−√2, multiplicity 16; together +0.0002), and the 4−4cos(π/5) band (multiplicity 40, +0.189) — with agreement 1.000 and the cut 176 on the nose.
040 Run 034's instrument (Franz–Parisi at T = 1) failed its control and was withdrawn, but its second look stood unsealed — at T = 0, greedy pair-swap paths between minimum bisections of the level-4 ⟨b,c,d⟩-coset graph climb ~50 edges above the endpoints (28% of the 176 cut), against ~2 on a random 4-regular graph of the same size and ~0 on the level-3 coset graph.
043 Five self-similar groups on the binary tree — odometer (nucleus 3, polynomial growth), Grigorchuk (5, intermediate), IMG(z² + i) (4, intermediate), Basilica (7, exponential, amenable), Aleshin (not contracting, free, non-amenable) — measured by the instruments of runs 033–042 at matched vertex count.
044 The quantum reading of the intermediate cell (P-045) — the boundary as a quasicrystal, the bulk as a flat-band localizer — is either a transport signature or a decoration.
054 Run 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).
N-034 The coset graph is a union of squares, and squares kill unique-neighbour expansion — no expander code on the intermediate-growth host at any level
042 Two "two-flavor" splits have appeared on the intermediate-growth host.
045 Run 044 sealed the graph Laplacian as the walk Hamiltonian, and on the Schreier graph the Laplacian cancels every loop — which is exactly where the graph's aperiodicity lives (which of b, c, d fixes a leaf follows the substitution).
055 Run 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.
064 Probe P-048 argues that block renormalization fails to converge on the Sierpiński-type carpets because they are infinitely ramified (blocks share whole edges).
088 Run 044's bulk clause was confirmed — on the Grigorchuk level-4 coset graph the quantum walker stays near its origin at long times (PR/N = 0.096, return floor 76/N) against a random 4-regular graph's PR/N = 0.99 — and P-045 reads it as a transport signature of the intermediate-growth cell.
N-036 Two routes to zero unique-neighbour expansion on a Cayley graph, both through g² — the Klein-four squares of N-034 and the cosets of ⟨g²⟩ found on Basilica and Aleshin
046 Runs 044 and 045 walked the Grigorchuk Schreier graph under isotropic operators (the Laplacian, then the equal-weight Hecke operator) and found a ballistic path both times — every leaf carries exactly one loop, so equal weights make the loop potential constant and the operator periodic.
056 The 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.
065 Run 064 found that the ⟨b,c,d⟩-coset tower is not an exact renormalization under φ(μ) = μ(8 − μ)/2 (18 of 24 new level-4 eigenvalues orphans).
101 On the sixteen stored minimum bisections of the level-5 ⟨b,c,d⟩-coset graph (cut 90,112), read on 2026-10-04, exactly 40,960 free cosets are independently flippable on every one, and they lie over the map to level 4 the same way on every one.
CASE-30 Every lowest-cut bisection found on a self-similar graph carries exactly the same number of independently flippable vertices — what fixes that number, and is there a ground state without it?
P-036 A Tanner code on the squared coset graph — the empirical code ceiling of the intermediate-growth host, now that N-034 rules out an expander code on the graph itself
P-044 The seam orbit as a register — capacity and write cost of the host's memory, from the group action instead of quenches
047 Run 046 found the Grigorchuk boundary walks like a quasicrystal under the anisotropic Hecke operator (β = 0.800) and run 044 found a random expander's walk saturates by t ≈ 2 and a path's is ballistic.
048 Run 040 produced seventeen level-5 minimum bisections at exactly 90,112 and run 042 sorted them into ten classes (twins at |q| ≈ 0.953 are one class — a balanced random half of the free cosets flipped), with the overlaps between classes on a ladder 0, 0.33, 0.48, 0.64, 0.80.
057 Birch and Swinnerton-Dyer read the rank of an elliptic curve as a growth exponent — the slope of log ∏_{p≤X} N_p/p against log log X. That is the same instrument runs 051–056 use on carpets: a global exponent fitted from local counts across a few decades of scale, where pair slopes drift.
050 Run 048 found the level-5 minimum bisections are not one small orbit — 16 seeds, 16 orbits, ≥ 4,096 states each, no stabiliser within reach — and that the generator d carries the seam to its twin at zero barrier, so part of the free-spin layer is symmetry.
058 Runs 054–055 found that the number of distinct minimum cuts orders the carpets' percolation exponents.
059 Run 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.
060 Run 058's crossing polynomials R(p) are exact.
P-047 How 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?
061 Run 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.
CASE-25 A one-block polynomial ranks eight lattices' percolation exponents exactly and misses every value — what sets the size of the miss?
N-037 What runs 048–060 say together — order holds, magnitude misses; first-order models rank and correlations set values; the seal is the weakest link
062 On 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.
066 The block exponents y_g = ln P_g′(p_g*)/ln 9^g rose from g = 1 to g = 2 on every lattice (run 061).
063 Run 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).
P-048 Taken 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?
067 A carpet may use a different 9×9 pattern at each level.
069 Feigenbaum's universality lives in a function, not only in δ: the fixed point of the doubling operator.
CASE-26 A slowly converging sequence of block estimates ranks exponents exactly at every block size — why does the order survive when the values do not?
N-038 A reductio pass over the whole record outside the carpet thread — stale back-links, controls that do not control, agreements built in by construction, two-level laws, and misquoted theorems
P-049 The 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 appear
068 The 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.
070 Run 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.
072 Run 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.
CASE-27 Stacking two carpets averages their exponents when their cut classes match and falls below the average when they do not — why?
P-050 The 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 everywhere
P-051 The 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 dial
071 Run 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).
073 Run 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).
076 Stacking 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).
P-052 The 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 nine
P-053 A 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 lacks
074 Run 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.
075 Each 9×9 pattern has an exact one-block map, the crossing polynomial f(p) of run 058 (rg058.poly).
080 43 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).
P-054 Invasion — 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 hierarchy
P-055 Bits 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 object
077 Feigenbaum's universality sorts one-hump maps by the order z of their maximum; the amplitude of the maximum drops out.
079 Run 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.
086 The 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.
093 The crossing polynomials of the 49 census fibres carry squares and a composition that nothing in the record explains.
103 For 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.
104 For 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.
108 Does 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?
078 Runs 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.
081 Run 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.
082 Probe 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.
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).
CASE-28 The skewness of the crossing threshold turns back on some carpets between L = 729 and 6561, while the exact one-block recursion's keeps rising — where does the turn come from?
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.
084 Run 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.
092 The 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.
096 Scaffold 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.
098 Within a group of patterns that share one core, how much does a pendant part lower the lattice threshold at L = 729?
CASE-29 Carpets that share a block polynomial share its rate but not always their threshold — what in the pattern, invisible to the polynomial, sets the threshold?
085 Run 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%).
089 The 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.
094 Run 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.
099 On 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.
107 Does the orbit through which the cut-2 wall enters the second ring order the lattice exponent?
090 Each 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).
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.
097 Run 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.
100 On 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.
109 For each stored sample, the second crossing t₂ — the fraction of bonds open when left and right are first joined after the single bond that completed the first crossing is deleted and the same bond order continues — and the gap (t₂ − t)/w in widths of the first crossing's pile.
119 On the 19 cut-2 census patterns, does flipping one cell of the 15-cell wedge break boundary isotropy visibly — the left–right and top–bottom transfer operators acquiring different singular values — for every flip that removes the pattern's D4 symmetry, and does the size of the break order by the flipped cell's symmetry class?
091 Each 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).
102 The 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).
105 The 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).
122 On census generators run 119 did not build, does a single-cell flip change the block transfer operator T exactly when the cell lies on a simple path between two distinct cells of the left and right columns?
130 A 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).
N-040 Returns — what the lattice does that the block map does not, and in which directions
106 Bond 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).
126 On the 63 census generators, does the number of one-way pairs of the block transfer operator equal |U|·(511 − |U|), with U fixed by the right-column holes alone?
132 On five census blocks no operator run has built (20, 35, 36, 39, 44), does the graph-only core rule say exactly which single-cell toggles give the same block operator?
133 On 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?
CASE-31 The crossing threshold is the weakest link of the strongest wall — does the number of near-strongest walls set the lean of the threshold pile, and its turn with size?
110 On the diamond hierarchical lattices (b, s) with recursion F_{n+1} = 1 − (1 − F_n^s)^b, does the sign of the lean g₁ of the threshold distribution at level n = 24 follow the rule "positive iff b < s" on the three rungs the House named?
127 On the census generators other than generator 1, does the number of one-way pairs of the block transfer operator equal |U|·(511 − |U|), with U fixed by the right-column holes alone?
134 Does the forbidden k = 3 tooth on H come from the full-side source?
111 On the diamond hierarchical lattice with recursion R_{b,s}(p) = 1 − (1 − p^s)^b, is λ(b, s) = R′_{b,s}(p*) equal to λ(s, b) on the pairs (2,5)/(5,2), (3,5)/(5,3), (2,6)/(6,2)?
131 On sixteen fresh 9 × 9 blocks, none in the census and half with a kept column-7 cell cut off from the left, does the number of one-way pairs of the block transfer operator equal |U′|·(511 − |U′|), with U′ from the corrected flood-fill rule?
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?
114 For a hierarchical lattice whose rule is a Boolean function f on k independent children, the RG map is R(p) = Σ_j c_j p^j (1 − p)^{k−j} with c_j the number of accepted inputs of weight j, 0 ≤ c_j ≤ C(k, j).
135 On the nine blocks with no one-way pairs, is the block operator reversible in the detailed-balance sense in some diagonal metric?
136 On one row of nine blocks (a 9 × 81 strip) of each of H, U, U′, T and X, how far does chaining the block's left-to-right operator T fall short of the strip's exact crossing?
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?
117 On the exclusive hierarchical rules of run 114 (not-all-equal and the band rules [a, b], k = 2 … 12), does the distance to period three O(R) increase with k along NAE, and is the stretch |λ| at the interior fixed point above 1.8 on every rule that has a period-3 orbit?
120 With the distance to period three measured on the map's core in both orientations (O₂), is O₂'s sign exactly the period-3 test on run 114's 286 exclusive rules, and does O₂ increase along the not-all-equal rules?
121 Does O₂'s sign — the Li–Yorke slack on a hump map's core, in either orientation — separate period three exactly on hierarchical rules no run has computed: the NAE and band rules at k = 13 … 16, and random rules from the full box at k = 7 and 8?
123 Off the entropy plateaus, does the Li–Yorke slack O₂ rank the period-3 band rules at each k as their topological entropy does?
CASE-32 Does the sign of the Li–Yorke slack on the core separate period three exactly, and why?
124 Is the Li–Yorke slack O₂ a reading of the period-3 saddle-node's unfolding, ranking the period-3 band rules at each k as the split between their two 3-cycles' multipliers does?
125 On the continuous band family, is the stable-born 3-cycle's multiplier at or below −8 at every point where the map reaches the log 2 ceiling, the crisis R²(c) = q?
128 Past k = 30, does the band family's closest approach to −8 at the crisis keep approaching from below at the rate k⁻², while the other cycle approaches +8 at the slower rate k⁻¹?
129 At k = 72 to 128, do the closest approach's four scaled quantities extrapolate to the derived limits, d₋k² → 27.2532, d₊k → 8.28684, (b − a)/√k + k^{−1/2} → 0.931582 and (a + b)/(2k) → 0.683325?