The Atanasov--Ranganathan construction on row 09 #
Row 09 is the last scope of AR's straightforward cases figure. Its core is two vertex-disjoint triangles joined by two cross edges and by a banana path,
e0 : 0 -> 2 \ e5 : 3 -> 4 \
e1 : 2 -> 1 | near triangle e6 : 4 -> 5 | far triangle
e2 : 1 -> 0 / e7 : 5 -> 3 /
e3 : 2 -> 5 cross e8 : 0 -> 6 near banana leg
e4 : 1 -> 4 cross e9 : 3 -> 7 far banana leg
e10, e11 : 6 == 7
The formalization uses the following length-independent divisor:
D = [1] + [2] + [3] + [7]
It is core supported, has degree four, and is valid on the whole closed nonloopy forest orthant: one chamber, no interior chip, no marks, no symmetry transport.
Its four chip-free vertices fall into two already-formalized local pictures.
{4, 5}-- the chipped triangle ofConfigurationChippedTriangle: the far triangle3-4-5with the chip sitting on3and one chip arm from each vertex. Read at the target4withal = |e4|,be = |e3|,ga = |e9|,p = |e5|,q = |e6|,r = |e7|, and at the target5withal ↔ be,p ↔ r.{0, 6}-- AR's Fifth picture,ConfigurationFive: outer centre0, inner centre6, cycle vertex7, boundary armsla = |e0|andlb = |e2|, middlec = |e8|, parallel slots|e10|, |e11|, far slotb = |e9|.
Both pictures see all four chips, and they share the slot e9 and the chip at
7; nothing forbids that, since only D_v ≤ D is ever asked. Every height is
a nested minimum of slot lengths, hence constant across a collapsed slot, so one
script covers the open cell and every nonloopy forest face at once.
Each triangle of core 09 is a directed 3-cycle, so the two boundary arms at
the outer centre 0 are read in opposite orientations (e0 forward, e2
backward). That is why ConfigurationFive.outerTarget_center_nonneg and
innerTarget_center_nonneg carry two independent SlotLedgers.
The endpoint plumbing is ConfigurationMarkedRow's, instantiated at the
identically-zero mark noMark, which recovers the ordinary one-ramp script
definitionally; no slot of row 09 carries a chip in its interior.
No marks #
Row 09's divisor is core supported, so every slot is an ordinary single ramp.
Passing the identically-zero mark through ConfigurationMarkedRow recovers
exactly that, and buys the whole residual-effectivity wrapper unchanged.
The zero mark: no slot of row 09 carries an interior chip.
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A height profile with no marks needs only class constancy.
The divisor #
The row-09 divisor obtained by placing one chip at each of vertices 1, 2, 3, and 7 and passing to core equivalence classes.
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The nested-min heights #
Two pictures, four readings, all four verbatim from auxiliary calculations §3.3.
a = min |e0| |e2|, the outer centre's arm minimum.
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- AtanasovRanganathan.GenusFiveRow09.armMin d = min (d.length 0) (d.length 2)
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m = min |e10| |e11|, the banana minimum.
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- AtanasovRanganathan.GenusFiveRow09.parMin d = min (d.length 10) (d.length 11)
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Outer-target reading: o = min a (b + m + c).
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Outer-target reading: i = min o (b + m).
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Outer-target reading: e = min i b.
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Inner-target reading: i = min (a + c) (b + m).
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Inner-target reading: o = min a i.
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Inner-target reading: e = min b i.
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The chipped triangle read at the target 4:
al = |e4|, be = |e3|, ga = |e9|, p = |e5|, q = |e6|, r = |e7|.
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Auxiliary side height for the chipped-triangle script targeting vertex 4, using arms 4 and 3 and slots 9 and 7.
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- AtanasovRanganathan.GenusFiveRow09.h0T4 d = AtanasovRanganathan.ConfigurationChippedTriangle.sideHeight (d.length 4) (d.length 3) (d.length 9) (d.length 7)
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Height at target vertex 4 in the chipped-triangle script, with connecting slots 5 and 6 and partner slot 7.
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- AtanasovRanganathan.GenusFiveRow09.htT4 d = AtanasovRanganathan.ConfigurationChippedTriangle.targetHeight (d.length 4) (d.length 3) (d.length 9) (d.length 5) (d.length 6) (d.length 7)
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Height at partner vertex 5 in the chipped-triangle script targeting vertex 4.
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- AtanasovRanganathan.GenusFiveRow09.hvT4 d = AtanasovRanganathan.ConfigurationChippedTriangle.partnerHeight (d.length 4) (d.length 3) (d.length 9) (d.length 5) (d.length 6) (d.length 7)
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The same picture read at the target 5: al ↔ be, p ↔ r.
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Auxiliary side height for the chipped-triangle script targeting vertex 5, with the two arms exchanged.
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- AtanasovRanganathan.GenusFiveRow09.h0T5 d = AtanasovRanganathan.ConfigurationChippedTriangle.sideHeight (d.length 3) (d.length 4) (d.length 9) (d.length 5)
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Height at target vertex 5 in the chipped-triangle script, exchanging the arms and the roles of slots 5 and 7.
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- AtanasovRanganathan.GenusFiveRow09.htT5 d = AtanasovRanganathan.ConfigurationChippedTriangle.targetHeight (d.length 3) (d.length 4) (d.length 9) (d.length 7) (d.length 6) (d.length 5)
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Height at partner vertex 4 in the chipped-triangle script targeting vertex 5.
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- AtanasovRanganathan.GenusFiveRow09.hvT5 d = AtanasovRanganathan.ConfigurationChippedTriangle.partnerHeight (d.length 3) (d.length 4) (d.length 9) (d.length 7) (d.length 6) (d.length 5)
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The four height vectors #
Configuration 5 read at the outer centre 0.
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Configuration 5 read at the inner centre 6.
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The chipped triangle read at the target 4.
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The chipped triangle read at the target 5.
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The endpoint ledger, vertex by vertex #
Each of the eight core vertices is trivalent, so contribForm has three terms
per vertex; they are the twenty-four slot ends of row 09 sorted by vertex.
The sum of incident tail and head contributions at each row-09 core vertex for a prescribed height profile.
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Chip allocations #
A collapsed arm puts its chip in the centre's contracted class; the chipped
triangle additionally lends the chip on 3 to its partner when the slot between
them collapses. Every transfer moves weight inside a class, so no class sum
changes.
The allocation both configuration-5 readings use.
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The allocation of the chipped triangle read at 4.
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The allocation of the chipped triangle read at 5.
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The outer centre 0 #
Representative of the outer target at vertex 0: move it to vertex 6 only when slot 8 is contracted and the shorter arm exceeds the tail plus the shorter parallel slot.
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Coefficients after firing the outer-target script and subtracting one chip at its chosen representative from the allocated divisor.
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The inner centre 6 #
Representative of the inner target at vertex 6: across contracted slot 8 choose vertex 0 when the shorter arm is at most the tail plus the shorter parallel slot.
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Coefficients after firing the inner-target script and subtracting one chip at its chosen representative from the allocated divisor.
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The chipped triangle at the target 4 #
Expanded coefficients after applying the chipped-triangle height script targeting vertex 4 to its allocated divisor.
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Representative receiving the vertex-4 chip: use vertex 4 when the script delivers there, otherwise vertex 3 across contracted slot 5 without lending, and vertex 5 in the remaining case.
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The chipped triangle at the target 5 #
Expanded coefficients after applying the chipped-triangle height script targeting vertex 5 to its allocated divisor.
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Representative receiving the vertex-5 chip: use vertex 5 when the script delivers there, otherwise vertex 3 across contracted slot 7 without lending, and vertex 4 in the remaining case.
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Every contracted core class is reached #
AR row 09. The length-independent divisor [1]+[2]+[3]+[7], valid
simultaneously on the open cell and every nonloopy forest face -- one chamber,
the whole closed orthant.
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