The EOA sequences are not arbitrary. Each letter's sequence has a rational generating function of the form P(x)/Q(x), where P(x) and Q(x) are polynomials with rational coefficients. Below is the rational generating function for the letter t under English modified #5. This is the kind of object that the entire algebraic foundation is built from.
t under English modified #5 (LCR ≈ 3.141). The numerator encodes the letter-specific initial conditions. The denominator is shared across all letters in the same recurrence family.
Under English modified #5, the 20 distinct letter sequences split into two families. Each family shares a common generating-function denominator, and therefore a common linear recurrence.
Letters: a, d, e, g, h, i, k, L, q, u, w, y, z
Common denominator (degree 13):
Letters: t, b, c, j, o, p, x
Common denominator (degree 14):
The two denominators are not independent. Direct factorization shows:
Therefore the greatest common divisor is:
The extra factor (x − 1) in Q₁ introduces one constant mode. This is the only difference between the two families. The dominant behaviour of all 20 sequences is governed by Q₂.
Theorem (EOA Basis for English Modified #5). The 13 sequences of Group 2 are linearly independent over ℚ. They form a true basis for the 13-dimensional solution space of the linear recurrence defined by Q₂.
The proof is by direct computation. Form the 13 × 13 matrix whose rows are the first 13 terms of each Group 2 sequence. Compute its rank and determinant. The result is:
A nonzero determinant means the rows are linearly independent. Since there are 13 rows in a 13-dimensional space, they span the entire space. This is the algebraic heart of the EOA program for English modified #5: the shared convergence, the rank collapse, and the sequence-group structure are all consequences of a single common linear recurrence.
For a sequence whose generating function is P(x)/Q(x), the limiting ratio of consecutive terms is the reciprocal of the smallest-modulus root of Q(x). This is a standard result in linear recurrence theory. The roots of Q are the reciprocals of the characteristic roots; the dominant characteristic root is the reciprocal of the smallest-modulus root of Q.
Q₂ has 13 roots. The smallest-modulus root is:
Its reciprocal is:
This value matches π to five decimal places. This is the LCR for English modified #5. We do not claim exact equality with π; we report the numerical agreement.
The largest root of Q₂ is 2.4914…, but this governs a fast-decaying mode and is irrelevant to the LCR. The LCR is determined by the smallest-modulus root, as explained above.
The 20 sequences span a 14-dimensional space (the solution space of Q₁). Therefore there must be exactly 6 independent linear relations. They are listed below. Each relation is of the form:
where Sᵢ denotes the sequence for a letter under English modified #5.
Relations 1–3 and 5–6 are sparse. Relation 4 is dense and involves the full 13-letter basis plus S_x. This is the relation that connects the 7 Group 1 letters to the 13 Group 2 letters. It is the algebraic bridge between the two families.
We have developed a tool that ingests each letter's rational generating function and automatically reports:
The tool supports row-based entry of letter, function image, function text, and LCR value. It supports import/append/replace of previously exported registry files, and it embeds a summary inside the exported HTML.
Initial results from the tool are striking and will be published alongside the full algebraic write-up in Part V.
EOA Algebraic Foundation for English Modified #5. Under English modified #5, the 20 EOA sequences split into two recurrence families. Group 2 (13 letters) forms a true basis for a 13-dimensional solution space of Q₂. Group 1 (7 letters) extends this by one constant mode via Q₁ = (x − 1)Q₂. The LCR is the reciprocal of the smallest-modulus root of Q₂ and matches π to five decimal places. The 20 sequences satisfy exactly 6 independent linear relations, spanning 14 dimensions.