16 records · compact density · grouped by kind, state glyph on every row · all tags
probes9
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
notes1
N-040Returns — what the lattice does that the block map does not, and in which directionsnote
cases6
CASE-25A one-block polynomial ranks eight lattices' percolation exponents exactly and misses every value — what sets the size of the miss?case
CASE-26A slowly converging sequence of block estimates ranks exponents exactly at every block size — why does the order survive when the values do not?case
CASE-27Stacking two carpets averages their exponents when their cut classes match and falls below the average when they do not — why?case
CASE-28The 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?case
CASE-29Carpets that share a block polynomial share its rate but not always their threshold — what in the pattern, invisible to the polynomial, sets the threshold?case
CASE-31The 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?case
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