600 Terms, a Degree-14 Rational Generating Function, and a Limit Near π

I have been working on a deterministic operator called (EOA). It maps the spoken names of letters into integer sequences.

For one English encoding, I generated over 600 terms of one sequence. The final term has 249 digits. I then used gfun to guess a rational generating function. It returned a rational function of degree about 14.

That means the sequence satisfies a linear recurrence with constant coefficients. If the recurrence is exact, the sequence goes on forever and its limiting ratio is algebraic.

The limiting ratio is numerically close to π to five decimal places. It is not exactly π. Because the generating function is rational with integer coefficients, the limit is algebraic, while π is transcendental. The interesting result is the algebraic structure, not the digits.

The generation code is not public. It is available to collaborators. The data and the generating function are public so anyone can check the recurrence.

What is EOA?
EOA stands for Engineering of Alphabets. It is a research program that studies deterministic recursive operators on phonetic letter names.

The process is simple:

1. Take the spoken name of a letter.
2. Apply a fixed deterministic operator.
3. Get an integer sequence.

There is no training data, no randomness, and no statistical optimization.

Under one encoding, English Modified #5, the system produces 20 distinct sequences. These sequences split into two families. The larger family contains 13 sequences. They share a common denominator and satisfy the same linear recurrence.

I focus here on one sequence from that family.

The data
I published over 600 terms of this integer sequence. The final term has 249 digits.

The data is archived on Zenodo:

https://doi.org/10.5281/zenodo.22738062

Access is available upon request. You can request access to the dataset and run your own checks.

The tool: gfun
gfun is a Maple package for guessing generating functions from finite sequences. It takes the first N terms and tries to find a rational function that matches them.

I ran gfun on the 600+ terms. It returned a rational generating function of degree about 14.

Here is the denominator of that rational function:

This is the common denominator for the larger family of 13 sequences. The numerator for this specific sequence is a polynomial of degree 13.

A rational generating function means the sequence satisfies a linear recurrence with constant coefficients. The order of the recurrence is the degree of the denominator, which is 13.

What this means
If the recurrence is exact, then:

• The sequence is infinite. You can generate as many terms as you want from the recurrence and the initial conditions.
• The limiting ratio of consecutive terms is algebraic. It is the reciprocal of the smallest-modulus root of the denominator polynomial.
• The limiting ratio cannot be exactly π. But π is transcendental, so it cannot be the root of a polynomial with integer coefficients.

The numerical agreement with π is therefore a coincidence. It is an interesting one, but it is not a proof of exact equality.

The real result is the algebraic structure: a degree-13 recurrence that generates the sequence.

Verification and collaboration
The generation code is not public. I am seeking collaborators to independently verify and extend the work.

If you are a mathematician or developer, you can:

1. Request access to the data on Zenodo.
2. Check whether the recurrence holds for all 600+ terms.
3. Try to extend the sequence to more terms
4. Report any errors or improvements.

If you want access to the generation pipeline for verification, contact me. The code can be shared under a collaboration agreement.

The recurrence itself is public. The denominator is above. The data is public. That is enough to check whether the sequence satisfies the recurrence.

Possible applications and domains for further investigation
If the recurrence is exact, it opens several potential directions. None of these are demonstrated yet. They are domains where I believe the structure could be useful, and where I am looking for collaborators.

• Symbolic dynamics. The operator produces sequences that satisfy a linear recurrence. This connects to finite-state systems and shift spaces. A natural question is whether the operator defines a sofic shift or a subshift of finite type.
• Linear recurrences and generating functions. The degree-13 recurrence is a concrete algebraic object. It could be studied for its own sake: factorisation of the denominator, distribution of roots, and the algebraic nature of the limit.
• Combinatorics on words. The mapping from letter names to sequences is a morphism or substitution. The combinatorial properties of this mapping, such as fixed points, recognisability, and factor complexity, are open.
• Coding theory and compression. If a long sequence is generated by a short recurrence, it can be compressed. The recurrence coefficients and initial conditions are enough to reconstruct the entire sequence. This is a form of lossless compression for this specific data.
• Cryptography and pseudorandom generation. Linear recurrences over finite fields are the basis of LFSRs. If the EOA operator can be adapted to finite fields, it could serve as a deterministic pseudorandom generator. This is speculative and needs careful study.
• Machine learning and representation learning. The EOA sequences are deterministic, training-free embeddings. They could be used as structural priors or as fixed features in models that process language or symbols. The rank collapse observed in word vectors built from these sequences suggests a strong inductive bias.
• Mathematical linguistics. The operator acts on phonetic encodings. It raises questions about the relationship between the sound structure of language and algebraic structure. Is the recurrence stable across languages and encodings? What does the limit say about the encoding?

If you work in any of these areas, I want to hear from you. The goal is to test the structure, extend it, and find out whether it is useful.

What is next
I am looking for independent checks and collaborators.

If the recurrence is correct, it strengthens the result. If there is an error, it clarifies what the system is actually doing.

Contact me if you want to collaborate on verification, extension, or application.

GitHub repository | Zenodo archive | Research site | Internet Archive