Radix Expansions and Superdeterminism

The decimal expansions of rational numbers obviously exhibit many patterns and regularities, and they necessarily become periodic, often with complementary parts. For example, we have

The period of a unit fraction like this is the order of the base (in this case, 10) modulo the denominator (in this case 7). The residues of 10^k (mod 7) are 1, 3, 2, 6, 4, 5, 1, …, so the order is six, and we note that 132+645=999. Similarly the first three digits 142 of the expansion are complementary to the next three digits 857 in the sense that 142+857 = 999. As a result, we have

which implies that 10³ – 1 = (7)(143). Naturally we also have the geometric series

and thus 10⁶ – 1 = (7)(142857). In general the digits in base b of a fraction N/D < 1 can easily be computed recursively by setting r₀ = N and then computing

for j = 1, 2, …  where d_j is the coefficient of b^–j. In addition to the periodicity and (sometimes) complementarity, some fractions also have expansions containing palindromes with uniformly ascending or descending digits. For example, with N=1 and D=(10¹+1)(10+1) in the base 10 we have

For another example, with N=1 and D=(10³+1)(10+1) in the base b=10 we have the expansion

Thus the expansion begins with the three digits 000, and then a 30-digit palindrome consisting of the sequential series of palindromic and self-complementary triples 090, 818, 272, 636, 454, 454, 636, 272, 818, and 090. Next comes the complement of all these digits, consisting of the triple 999 followed by the 30-digit complementary palindrome. This completes the overall period of 66 repeating digits. To highlight the braid of triples (and the similarity to the previous example) we can also write this expansion as

In general, for any even base b and odd exponent k the expansion of 1/[(b^k+1)(b+1)] takes this same form, with two complementary palindromic braids of palindromic self-complementary k-tuples, with an overall period of 6(b+1). For example, with b=6 and k=3 we have the expansion

which has a period of 42. It is straight-forward (albeit laborious) to verify the algebraic identity underlying this pattern. Taking the palindromic braid by columns, we have

where

To facilitate generalization, we can write A and B in the form

Hence we have

This generalizes to any even base b by replacing each 6 with b and each 5 with b–1. (The restriction to even bases is due to the fact that with odd bases the expression for the alternating geometric series has a plus sign in the numerator.) We can then evaluate the expressions

The generalized expression for 1/11011 (base 6) is

Inserting the previous expressions and simplifying, we confirm the algebraic identity

Again, this corresponds to the expansion in the base b just if b is even, because of the form of the truncated alternating geometric series. For odd bases the expansion of 1/11011 consists of just a single braid, separated by null triples. For example, with the odd base b=7 we have

This has period 3(b+1) = 24. The expansion can be written as

where

Similar to the previous derivation, we generalize this using the truncated geometric series to give

The signs in the numerators of the second two terms of these expressions are different than in the previous case because of the odd parity of b. From this we get (in the general case for odd base b)

Inserting this into the generalized expression for 1/11011, we confirm the algebraic identity

As mentioned above, similar structures occur in the expansions of 1/(b^k+1)(b+1)] for higher odd values of k. Instead of producing a braid of triples they produce a braid of k-tuples. For example, with the base b=9 and k=5 we have

Likewise for the base b=10 and k=7 we have

It’s interesting how both local and non-local correlations (complementarity and palindromicity) are enforced by this purely local recurrence. Each digit d_j is a function of just the previous digit d_j–1 and the previous residue r_j–1, the latter being a “hidden variable”. The resonances between the consecutive powers of the base b modulo D and the reduction to the set of just b digits (by the recurrence described previously) can produce seemingly non-trivial structures for very simply-defined fractions. This might be seen as a primitive example of a kind of superdeterminism, i.e., not only are the digits purely deterministic (obviously), following from a simple local law of evolution, but they exhibit persistent and seemingly intricate symmetries on multiple levels, which one might think could only result from some implausibly precise “fine-tuning” of the initial conditions. In a sense these example have indeed been fine-tuned, merely by the choice of the denominator of the form (b^k+1)(b+1) with base b, but the structure of the denominator greatly restricts the possible choices.

There are also example of “imperfections” in the evidently simple patterns of some expansions. For example, in the base 10 the expansion of 1/[(10³+1)(10²–1)] is

Each column appears to be repetitions of the natural sequence of ascending and descending digits, but since there are 11 rows and 10 digits, each column has either double 0 or double 1. The staggering of the columns, as well as the choice of whether to double the 0 or the 1 might appear to be random, even though it is actually given by the simple radix recurrence.

These radix expansion also give an interesting example of how some transitional structures might plausibly occur in nature. Recall that the production of all the higher elements in stars was possible only because of a certain resonance of the carbon atom. Similarly one could speculate that the production of a self-replicating molecule might require the existence of certain resonances in naturally occurring patterns, such as the braids produced by the simple radix expansion recurrence with resonant denominators and bases.

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