Full Technical Derivation

The recurrence, the verse, the identity — step by step

The recurrence (Reeds 1998/2006)

Each of the Book of Soyga’s 36 tables is a 36×36 grid of letters from the 23-letter Latin alphabet:

Alphabet: a b c d e f g h i k l m n o p q r s t u x y z Ordinals: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 (No j, v, or w. Arithmetic is mod 23; position 23+1 wraps to position 1.)

Left margin. Each table is keyed to a 6-letter code word W₁W₂W₃W₄W₅W₆. The left column of the 36-row grid is the sequence: W₁W₂W₃W₄W₅W₆W₆W₅W₄W₃W₂W₁ (word then its reverse), repeated three times to fill 36 rows.

Top row (no cell above). For columns 2–36 of row 1:

X = W + f(W) (mod 23)

where W is the cell to the left. Only the left-neighbour matters; there is no cell above.

Interior cells (rows 2–36, columns 2–36):

X = N + f(W) (mod 23)

where N is the cell above and W is the cell to the left.

The crucial point: in both cases the function f is applied to the left (west) cell, and its output is added to the north cell (or, in the top row, to the left cell itself).

Reeds’s empirical f (Table II of his 1999 paper):

W	a	b	c	d	e	f	g	h	i	k	l	m	n	o	p	q	r	s	t	u	x	y	z
f(W)	2	2	3	5	14	2	6	5	14	15	20	22	14	8	13	20	11	8	8	15	15	15	2

Reeds noted: “f is known to us only by a table of values determined empirically." Its origin was open from 1998 until 2026.

The Section 18 verse and its number-words

Book of Soyga, Section 18 (Aldaraia sive Soyga vocor). Kupin 2014 edition, Latin text lines 8151–8172, English translation lines 8318–8337.

Heading: “Versibus ostendam quid monstret quaeque figura” — “In verses I will show what each figure denotes.”

The verse assigns each letter a value V(W) through Latin number-words. Parsing letter by letter:

Verse fragment (Latin)	Letter(s)	Number-word	V(W)
Tres numeros z, b; numeros tres continet a, f	z, b, a, f	tres = 3	3
H bis tres numeros	h	bis tres = 2×3 = 6	6
Septem G	g	septem = 7	7
O plenum cum T, comis est, S ter tria portat	o, t, s	ter tria = 3×3 = 9	9
Bis sex, R	r	bis sex = 2×6 = 12	12
Bis septem P	p	bis septem = 2×7 = 14	14
E cum n tenet I, numeros ter quinque	e, n, i	ter quinque = 3×5 = 15	15
V cum K ut x, y, dat bis octo	u, k, x, y	bis octo = 2×8 = 16	16
L ut Q, numeros ter vii	l, q	ter vii = 3×7 = 21	21
M xxti tria portat	m	xx + tria = 20+3 = 23	23
Terque novem per se C	c	ter × novem = 3×9 = 27	27
Vigintique novem D dat cum sumitur ampla	d	viginti + novem = 20+9 = 29	29

Note on o, t = 9: The line “O plenum cum T, comis est, S ter tria portat” is fully legible in the printed Kupin edition (second independent review, 2026-06-09, ~line 8165). The values o = t = s = 9 are attested in the printed text, not inferred. The same independent OCR pass of the Internet Archive source concurs.

Letter-by-letter verification of f(W) = V(W) − 1 (mod 23)

Letter	V(W) (verse)	V(W)−1	(V(W)−1) mod 23	Reeds f(W)	Match?
a	3	2	2	2	✓
b	3	2	2	2	✓
c	27	26	3	3	✓ (mod-23 wrap)
d	29	28	5	5	✓ (mod-23 wrap)
e	15	14	14	14	✓
f	3	2	2	2	✓
g	7	6	6	6	✓
h	6	5	5	5	✓
i	15	14	14	14	✓
k	16	15	15	15	✓
l	21	20	20	20	✓
m	23	22	22	22	✓
n	15	14	14	14	✓
o	9	8	8	8	✓
p	14	13	13	13	✓
q	21	20	20	20	✓
r	12	11	11	11	✓
s	9	8	8	8	✓
t	9	8	8	8	✓
u	16	15	15	15	✓
x	16	15	15	15	✓
y	16	15	15	15	✓
z	3	2	2	2	✓
Total matches				23 / 23

The mod-23 wrap. For c: V = 27, so V−1 = 26, and 26 mod 23 = 3 = f(c). For d: V = 29, so V−1 = 28, and 28 mod 23 = 5 = f(d). No other letter requires the wrap; c and d alone have V > 23.

Uniqueness. The offset k = 1 (i.e., f = V−1) is the only constant giving 23/23 matches. Every other integer k from 0 to 22 gives 0/23. This rules out coincidence.

The −1 as inclusive/exclusive counting

The verse says (in the author’s terms): the V-th letter, counting from and including the letter above, is the new value. Reeds’s recurrence says: advance f steps past the cell above. The gap between “the Nth item counting inclusively from position P” and “advance N steps from P” is exactly 1. Under the reading V is an inclusive ordinal and f is an exclusive step count, the tables regenerate exactly. This gloss is ours, not a quoted instruction from the manuscript; under it the arithmetic is perfect.

Regeneration of all 36 tables and 46,656 cells

Applying f(W) = V(W) − 1 (mod 23) to the recurrence reproduces:

All 36 tables (2×12 zodiac + 7 planets + 4 elements + 1 “Magistri”)
All 46,656 cells (36 × 36 × 36)
NISRAM (Tabula 1, Aries) — 216/216 cells against Reeds’s full typeset transcription (the 2 glyph ambiguities are both the italic b/h pair, not substantive errors)
ORRASE (Tabula 5, Leo) — 466/468 cells against Reeds’s transcription; the 2 differences are Reeds’s own catalogued original manuscript errors, which the algorithm correctly does not reproduce
UISEUA (Tabula 31, Moon) — margin + column 1, 24/24 cells

NISRAM ground-truth rows

Tabula 1, code word NISRAM, rows 1–3 (verified against Reeds’s full typeset transcription):

Row 1: n d i z b d i z b d i z b d i z b d i z b d i z b d i z b d i z b d i z
Row 2: i s r l y t r l y t r l y t r l y t r l y t r l y t r l y t r l y t r l
Row 3: s c u c b x i b a x i b a x i b a x i b a x i b a x i b a x i b a x i b

The top-row pattern (n→d→i→z→b→d→i→z→…) is an instance of the two-cycle attractor structure explained in the Verification section.

The 36-table inventory

Tables	Keys
T1–T12	Zodiac (Aries NISRAM through Pisces)
T13–T24	Zodiac reversed (T13 = MARSIN, the exact reverse of NISRAM; all 12/12)
T25–T31	Seven planets
T32–T35	Four elements
T36	Magistri (MOYSES — Moses)

T13–T24 code words are the exact reverses of T1–T12. However, a reverse code word does not produce a geometric mirror of its partner table: T1 (NISRAM) and T13 (MARSIN) differ in 1,233 of 1,296 cells (95.1%). The recurrence is non-linear; only the seed is reversed.

The Magistri table (MOYSES). Tabula 36 is the only table whose code word is a proper name. Its famous missing row is demonstrably not an algorithmic artifact: regenerating MOYSES yields zero rule violations across all interior cells. The blank row is a deliberate scribal reservation — and the two manuscripts (Bodley 908 and Sloane 8) blank different rows (Bodley row 13, Sloane row 36 after closing ranks), proving the scribes knew a row was reserved but disagreed which.

Error model

Reeds’s 1999 paper catalogued errors in the manuscript copies: 13 original transcription errors and 7 transposition errors. The verification implementation diverges from the manuscript precisely at those coordinates, and nowhere else. This is an additional confirmation that the generation algorithm and f are correct.