Abstract
We evaluate the integer sequences generated across 36 human writing systems under the Engineering of Alphabets (EOA) operator. Auditing all 1,056 resulting sequences against the On-Line Encyclopedia of Integer Sequences (OEIS), three facts survive every check we could devise. First, the growth rates of these sequences cluster on a small set of exact algebraic numbers 2.618…, 3.414…, 2.414…, 2, and 3, with 2.618… (φ²) the single largest cluster. Second, several of the generated sequences are, term for term, classics of mathematics: the powers of two, the Fibonacci bisections, the Lucas bisection, the Pell family, the Mersenne numbers, and the bare counting numbers, each verified against the full published OEIS record (up to 90 terms) from our 500-term exact sequences. Third, 388 distinct generated sequences appear in no catalogue, although every one satisfies an explicit integer linear recurrence. To test whether these observations arise by chance, we evaluate two random-baseline controls: random alphabets of matched size never form the tiny cores that natural scripts form (0 of 250 simulations), and the English E5 spectrum splits into two equal halves, a pattern random alphabets never produce (0 of 100 simulations). We release the full census, the matrices, and the verification code, and we pose two challenges to the sequence community.
1. THE RULE, IN ONE PARAGRAPH
Every letter has a name. Apply a fixed recursive expansion rule to each letter name, and repeat round after round. This produces an endless string of letters. Count how long the string becomes at each round: that growing list of numbers is the letter’s sequence. The counts obey a fixed recipe fixed by the script’s letter names alone; no tuning, no search. The mathematical engine behind every sequence is the script’s incidence matrix, and the growth rate of the counts is the Perron eigenvalue of that matrix. The matrices for all 36 scripts are open data; everything below was recomputed from them in exact integer arithmetic.
2. WHAT WE MEASURED
This census covers 36 different writing systems, totaling 1,056 individual letters. We chose these scripts based on their documented historical names and to include a wide mix of different writing styles from unconnected parts of the world. We did not cherry-pick this list to force a result. We deliberately kept all the 'failures' in our dataset: 8 scripts that matched nothing in the OEIS at all (seven formed massive, tangled graphs, while Georgian collapsed), along with several scripts that produced completely trivial, flat sequences. If we had filtered the data to keep only the 12 scripts that generated the non-trivial mathematical families, two-thirds of the dataset would have been thrown out. Because we kept them in, you can trust that the patterns we found emerged naturally from the raw data.
The 1,056 letter-sequences collapse into 402 distinct sequences, because letters that share a naming pattern share their sequence (Appendix, Tables A1–A2). For each distinct sequence we computed its exact recurrence order and its growth rate. Two summaries:
• Recurrence orders run from 1 to more than 14. The most common orders are 13 (47 sequences), 5 (41), 4 (29), and 11 (23); 176 sequences need order above 14.
• Growth rates cluster tightly: 78 sequences grow at φ² = 2.618…, 44 at 2+√2 = 3.414…, 27 at 1+√2 = 2.414…, 20 near 3.1416 (English), 19 near 4.21 (Ogham), 19 at exactly 2 (the Mersenne regime), 8 at exactly 3 (Korean), and 30 distinct sequences are flat (rate 1), matched by 176 letters total.
This census is not a sample; it is the whole population of distinct sequences the 36 scripts can produce. Growth at φ² is the most common outcome in the entire dataset.
3. WHAT MATCHED THE OEIS, AND SURVIVED VERIFICATION
While all our sequences were generated to 500 terms, published OEIS records cut off much earlier; we kept only identities that hold term-by-term across the entire available OEIS entry (up to 90 terms) under an exact recurrence check. These are exact structural identities, not coincidental prefix agreements or numerical approximations. Six families pass:
• The powers of two, A000079: generated by four Hebrew letters (bet, tet, mem, nun), and by the same naming pattern in Phoenician, Aramaic, Syriac, Arabic, Japanese-pronounced Latin, Sanskrit and Amharic. (1, 2, 4, 8, 16 … to 2⁸⁹, exact.)
• The Fibonacci bisection, A001906: generated in Ge’ez and Thai (1, 3, 8, 21, 55 …). Its recurrence t(n+2) = 3•t(n+1) − t(n) is the polynomial x²−3x+1 , the same equation behind the φ² growth above. The odd Fibonacci numbers, A001519, appear the same way in both scripts.
• The odd-index Lucas numbers, A002878: two Thai letters (1, 4, 11, 29, 76 …), same polynomial, different seeds.
• The Pell family, A094723 and A048739: generated in Arabic. Their growth rate is the silver ratio 1+√2 , which is exactly why the Arabic core grows at 1+√2.
• The Mersenne numbers 2ⁿ−1, A000225: generated in Japanese-pronounced Latin, Aramaic and Sanskrit.
• The bare counting numbers: one Syriac letter group generates 1, 2, 3, 4, 5 … with no offset and no error, through all 90 published OEIS terms and continuing through all 500 generated terms.
Table A1 lists all fourteen catalogued sequences, including the further matches beyond these six (the shifted counting variant, the odd numbers, the cake numbers A000125, A000295, and the Burmese silver-square recurrence A007070).
4. WHAT MATCHED NOTHING, THE UNCATALOGUED HALF
388 distinct sequences have no OEIS entry (Table A2 lists the per-script breakdown; every sequence is released with its recurrence in census.csv and uncatalogued_sequences.csv). Each one satisfies an explicit integer linear recurrence of small order, derived from its matrix, and each is released with that recurrence, its matrix, its script and its letters. They concentrate in the large-core scripts: every letter of English E5, Greek, Gothic, Runic, Ogham and Glagolitic is uncatalogued. We make no claim that they are important; we claim only that they are generated, that their recurrences are known, and that no catalogue records them. Whether that gap is worth filling is precisely Challenge I below.
5. TWO CONTROLS THAT MAKE THE ABOVE MEAN SOMETHING
(a) The small-core anomaly. The six verified families come from tiny closed loops of letters: Hebrew and Arabic have dominant cores of 3 letters, Thai of 2, Japanese of 4. Is that usual? We built 250 random 26-letter alphabets with the same name lengths (2–5 letters per name) and computed their cores. Every random alphabet entangled into one giant loop: core sizes ran from 21 to 26, and not one fell below 21 (0 of 250). Natural scripts, in other words, do something random name systems never do: they isolate their growth into loops of two to four letters. This is the quantitative backbone of our name-structure conjecture, and it is reproducible with one script run.
(b) The English split. The English E5 matrix factors as thirteen linear factors and one irreducible polynomial of degree exactly 13, a spectrum split into equal halves. We generated 100 random 26-letter alphabets of matched shape: none produced such a split (0 of 100). Generic random matrices have one large irreducible factor of degree 20 to 26 and almost no linear factors. The English split is a genuine structural anomaly, not a presentation choice.
6. THE CHALLENGES
I. To the uncatalogued 388. Each is released with its recurrence, matrix and attestation. We ask the community for an adjacency audit: is a sequence’s recurrence family already represented in the OEIS? Our own flagship example was, the Hebrew pair dalet and lamed: they generate 1, 4, 12, 34, 94, 256, 690 …, which is 2•F(2n) − 2^(n−1): an order-3 cousin of two catalogued families, whose sibling F(2n+2) − 2ⁿ is A105693. We had called it missing; the record corrected us, and we print the correction. Where a family is genuinely unrepresented and of independent interest, we support inclusion. Where it is catalogued, we ask only that the census record the correspondence. A reasoned rejection will be published with the same care as a match.
II. To the matched six families. Is the hit rate itself remarkable, or do small integer matrices match the OEIS at high rates regardless? We designed a random-matrix control, density-matched, row-shuffled, and degree-preserving, at match windows from 10 to 40 terms, and released the protocol and code. We have not run it to completion. The interpretation rules are fixed in advance: if real scripts match above all controls, naming structure selects catalogued sequences; if they match at control levels, the census rests on the verified identities and the controls alone; if real beats the density control but not the degree-preserving one, the driver is name length and degree structure. Whichever outcome occurs, we pre-commit to publishing it.
7. WHAT WE CLAIM, AND WHAT WE DO NOT
We present an objective measurement: the first complete census of the integer sequences generated by 36 human writing systems, with two verified structural anomalies (small cores, and the English split) and six exact term-for-term matches to named sequences. We do not claim that ancient scribes intentionally encoded mathematics into letter names, nor that these mathematical matches are surprising in number, or that the uncatalogued 388 are profound. The honest open question is the one the census poses and cannot answer: why does the same handful of polynomials, x²−3x+1 above all, organize naming systems that never met?
VERIFICATION AND MATERIALS
Every sequence was computed to 500 terms in exact BigInt arithmetic from the open matrices, and identities in Section 3 were verified against the full published OEIS entries up to 90 terms. The pre-computed census tables (census.csv, uncatalogued_sequences.csv, census_summary.json) and verification scripts ship directly with the release: audit.py (generates the 402-sequence census), verify_wows.py (reproduces the term-exact OEIS identity checks), baseline_oeis_lookup.py (hit-rate control protocol), and e5_split_test_sage.sage (E5 spectral split null).
Matrices: https://doi.org/10.5281/zenodo.23440579.
Main technical paper: https://doi.org/10.5281/zenodo.23449728.
Appendices: https://doi.org/10.5281/zenodo.23695751.