The Atanasov--Ranganathan construction on row 14 #
Row 14 is K₃,₃ with the edge 0--1 deleted and replaced by the path
0 -- 6 == 7 -- 1, whose middle step is a banana. The bipartition of the
K₃,₃ part is {0,3,5} against {2,1,4}.
Put one chip on each of 0, 3, 5, 7. This is exactly the divisor
Atanasov--Ranganathan display for this family (Figure 8 scope 14; the
straightforward-cases figure marks a = 0, d = 3, f = 5, h = 7 as the
chip vertices), and exactly the divisor of the generated fixed cover. The
chip-free vertices are 1, 2, 4, 6, and the paper's own edge patterns split
them as:
- dashed -- vertex
2, whose three slots end on the chips0, 3, 5; - dotted -- vertex
4, whose three slots end on the chips5, 0, 3; - plain -- vertices
1and6, forming AR's sixth local picture:3 -- 1 -- 5, then1 -- 7, then the banana7 == 6, then6 -- 0.
Vertex 1 is read here as a third configuration-2 tripod rather than as the
arm vertex of that sixth picture -- its three slots also end on chips
(7, 3, 5) -- which leaves only the far end 6 of the banana to be done by
hand. So three of the four centres come straight from
LowGenus/ConfigurationTwo.lean, and the fourth from
LowGenus/ConfigurationBananaTail.lean, which carries
AR's sixth picture generically in the core.
The two families are combined exactly as in GenusFiveRow12: each names its
own centres, and between them they cover every chip-free vertex.
The divisor #
Indicator weight placing one chip at each guarding-set vertex.
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The row-14 graph divisor obtained by pushing the four guarding-set chips to their core equivalence classes.
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The three configuration-2 tripods #
The chip-free vertices all of whose slots end on a chip.
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First tripod arm at centres 1, 2, and 4, respectively slots 1, 4, and 8; other inputs use zero.
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Second tripod arm at centres 1, 2, and 4, respectively slots 6, 5, and 9; other inputs use zero.
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Third tripod arm at centres 1, 2, and 4, respectively slots 7, 10, and 11; other inputs use zero.
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Chip at the end of the first tripod arm for centres 1, 2, and 4, respectively vertices 7, 0, and 5.
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Chip at the end of the second tripod arm for centres 1, 2, and 4, respectively vertices 3, 3, and 0.
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Chip at the end of the third tripod arm for centres 1, 2, and 4, respectively vertices 5, 5, and 3.
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The one chip each tripod centre does not touch.
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Row 14 read as three AR configuration-2 pictures.
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The class-sum form of the displayed divisor agrees with the four-chip form the configuration-2 family uses.
The nested-min heights of AR's sixth picture at vertex 6 #
la = |3--1| and lb = |1--5| are the two arms, w = |7--1| the middle
slot, p, q = |7--6| the banana, and u = |0--6| the tail slot.
The shorter external arm of the banana-tail configuration, the minimum of lengths 6 and 7.
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- AtanasovRanganathan.GenusFiveRow14.armMin d = min (d.length 6) (d.length 7)
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The shorter parallel banana slot, the minimum of lengths 2 and 3.
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- AtanasovRanganathan.GenusFiveRow14.parMin d = min (d.length 2) (d.length 3)
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The height at the centre 6.
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The height at the chip 7 at the near end of the banana.
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The height at the chip-free arm vertex 1.
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Unquotiented banana-tail height: arm height at vertex 1, middle height at vertex 7, end height at vertex 6, and zero elsewhere.
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Integer-valued firing potential given by the negative of the unquotiented banana-tail height.
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Reading the raw profile at the canonical representative makes class invariance definitional.
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Redistributing chips inside contracted classes #
The conditional transfer that pays for the two banana chips when the middle slot has collapsed.
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The four-chip weight redistributed across contracted arms 6 and 7 and tail 0, with a further transfer from 1 to 7 across contracted slot 1 when the height comparison requires it.
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Which vertex of the centre's class carries the delivered chip #
Target representative for centre 6: choose vertex 7 when a parallel slot contracts and the tail is longer than the shorter arm plus middle slot 1; choose vertex 6 otherwise.
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Endpoint accounting on the contracted face #
The per-vertex coefficient of the local residual #
Expanded coefficients of the allocated divisor after the banana-tail height script, with each incident slot read in its prescribed orientation.
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The local residual at the centre 6 #
Combining the two families #
The tripod table and the banana centre between them name every chip-free vertex.
Row 14 as a guarding set. Chips on 0, 3, 5, 7; the tripod table
covers 1, 2, 4 and the banana tail covers 6. The closing step is the
generic Guarding.GuardingSet.closedConstruction, exactly as on row 06, whose
decomposition is this one with the multiplicities exchanged.
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AR configurations 2 and 6 on row 14, valid simultaneously on the open cell and every nonloopy forest face.