I study deterministic symbolic sequences generated by fixed recursive operators on phonetic encodings of letter names. My main object is a sequence over a 16-letter alphabet whose term lengths satisfy a linear recurrence of order 14, and whose dominant growth factor is an algebraic number of degree 13 that numerically approximates π to five decimal places. The sequence is a candidate test case for morphicity: it looks morphic, but I have not yet shown that it is.
Research summary
The Engineering of Alphabets (EOA) program treats alphabets, together with their spoken names, as deterministic symbolic systems. A single fixed operator is applied repeatedly to an initial letter name. Each iteration replaces the current string by a longer one, so the earlier terms remain embedded in the later ones. The program currently covers 13 main sequence families, plus several related cases.
Main object
For the letter a, the term lengths Ln satisfy a linear recurrence of order 14. The generating function is rational; its denominator factors as (x − 1) · G2(x), with G2 irreducible of degree 13. The dominant root of G2 is
which is algebraic of degree 13 and therefore cannot equal π, although it agrees with π to five decimal places.
Empirical observations on the prefix
| Quantity | Value (first 6260 symbols) |
|---|---|
| Alphabet size | 16 |
| Factor complexity p(2) | 55 |
| Factor complexity p(3) | 106 |
| Factor complexity p(4) | 150 |
| Factor complexity p(5) | 194 |
| Growth ratio Ln / Ln−1 | → 3.141593223578… |
| 25th term | 1,777,496,641,819 trillion letters |
Family structure
Of the 26 English letter names, 13 share the same degree-13 minimal polynomial:
The remaining 13 have a degree-14 polynomial equal to this one multiplied by (x - 1). All 26 therefore share a common structure; the first group is the case where the extra linear factor cancels.
Open problems
- Is the symbolic sequence morphic? If so, what is the substitution?
- Can the Perron eigenvalue be expressed in closed form or identified with a known algebraic number?
- Does the observed slow growth of factor complexity persist indefinitely, or does it change at longer block lengths?
- Does the 13/13 split of the letter names reflect a structural symmetry of the operator, or is it an artefact of the encoding?
Methods and tools
Sequences are generated by a fixed deterministic operator on phonetic letter names. The term-length recurrence and its characteristic polynomial are derived by fitting the first 14 terms and factoring the resulting polynomial with standard computer algebra. Factor complexity is computed by direct enumeration of subwords over a 6 260-symbol prefix. Code for factor-complexity counting and growth-factor extraction is public; the generating operator itself is withheld under staged disclosure.
Selected links
- ORCID profile
- Zenodo preprints and data
- GitHub repository
- Archive.org collection
- Press & Media (plain-language articles)
Keywords
symbolic dynamics combinatorics on words factor complexity morphic sequences automatic sequences substitution systems Perron–Frobenius eigenvalue algebraic generating functions deterministic sequences
Contact
Research and media enquiries: bdarghamneurolabs@gmail.com