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ResourceFunction["MultiwayFunctionSystem"][nMost[Divisors[n]],{100},10,"StatesGraph"]
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ResourceFunction["MultiwayFunctionSystem"][nMost[Divisors[n]],{1000},10,"StatesGraph",AspectRatio1/2]
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TransitiveReductionGraph[%]
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TransitiveReductionGraph[ResourceFunction["MultiwayFunctionSystem"][nMost[Divisors[n]],{5040},10,"StatesGraph",AspectRatio1/2]]
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Graph3D[%]
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Only need to go 2 steps...
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TransitiveReductionGraph[ResourceFunction["MultiwayFunctionSystem"][nMost[Divisors[n]],{5040},2,"StatesGraph",AspectRatio1/2]]
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PrimePi[1000]
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168
Total number of nodes in the graph is number of divisors:
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DivisorSigma[0,1000]
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16
Total “weight” of graph is
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DivisorSigma[1,1000]
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2340

Edge count

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Table[EdgeCount[ResourceFunction["MultiwayFunctionSystem"][nMost[Divisors[n]],{n},2,"StatesGraph"]],{n,100}]
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{0,1,1,3,1,5,1,6,3,5,1,12,1,5,5,10,1,12,1,12,5,5,1,22,3,5,6,12,1,19,1,15,5,5,5,27,1,5,5,22,1,19,1,12,12,5,1,35,3,12,5,12,1,22,5,22,5,5,1,42,1,5,12,21,5,19,1,12,5,19,1,48,1,5,12,12,5,19,1,35,10,5,1,42,5,5,5,22,1,42,5,12,5,5,5,51,1,12,12,27}
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Table[PrimePi[n],{n,100}]
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{0,1,2,2,3,3,4,4,4,4,5,5,6,6,6,6,7,7,8,8,8,8,9,9,9,9,9,9,10,10,11,11,11,11,11,11,12,12,12,12,13,13,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,18,18,18,18,18,18,19,19,19,19,20,20,21,21,21,21,21,21,22,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,25,25,25,25}
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ListLinePlot[{%97,%98}]
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Table[DivisorSigma[0,n],{n,100}]
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{1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,6,4,4,4,9,2,4,4,8,2,8,2,6,6,4,2,10,3,6,4,6,2,8,4,8,4,4,2,12,2,4,6,7,4,8,2,6,4,8,2,12,2,4,6,6,4,8,2,10,5,4,2,12,4,4,4,8,2,12,4,6,4,4,4,12,2,6,6,9}
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ListLinePlot[%]
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Related to Riemann hypothesis
https://en.wikipedia.org/wiki/Divisor_function
https://mathworld.wolfram.com/RobinsTheorem.html

Relation between spectral properties of “Hamiltonian” and cycle-like structure in MW graph

What is the transitive reduction doing at each node?

Dividing by the prime factors.....
Depth is determined by largest exponent in prime factorization.....
Prime factorization is trying to find the predecessors of 1...
The presence of multiplication in the number is like the concatenation of rules to the multiway system
Number of distinct prime factors gives dimension....

Analogy with harmonic oscillator

Causal edges

[[ In the code, a causal connection is made between shared prime factors ]]
[ Creator and destroyer events of prime factors ... ]

Branchial structure

[[ Related to equivalence of binary quadratic forms ???? ]]

Direct Riemann zeta

Interpreting multiway systems as having real and imaginary values

The coordinatization of branchial space is then the assignment of a phase

​

Path weights are determined by the multiplicities of factors, so e.g. the final primes have weights that are their exponents in the factorization

Real vs complex integer functions

This is a pure real function iteration:
A “complete superposition” of a particular branchlike hypersurface can be thought of as a sum of the complex numbers obtained from state weights and phases
Across Pascal’s triangle, we are assigning each number a complex phase

Imagine that every node has some small vector of integers [or just a single integer]

What is the analog between choice of branch cuts and choice of foliations?

Each sheet in Riemann surface is like a hypersurface

[ Something different : ]

When it branches, it’s like e.g. solving for something, or taking a square root

(For the divisor multiway function, it’s like “solving for what divides”)

Claim: any multiway system graph can be generated from a complex function iteration

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