KNOTT-5.3 — Wythoff’s Array, Beatty’s Complementary Sequences (⌊nΦ⌋, ⌊nΦ²⌋), & Stolarsky’s Array

Because 1/Φ + 1/Φ² = 1, Beatty’s sequences Aₙ = ⌊nΦ⌋ and Bₙ = ⌊nΦ²⌋ = Aₙ + n partition Z_{>0} without overlap; seeding generalized Fibonacci rows via W(r, c) = ⌊(r + 1)Φ⌋ F_{c+2} + r F_{c+1} (Wythoff) or nearest-integer multiples [kΦ] (Stolarsky) places every positive integer in a unique (row, col) cell.

Curated Source & Catalog Provenance

Catalog ID	KNOTT-5.3
Curated Source	Dr. Ron Knott, General Fibonacci Series, Wythoff and Stolarsky Arrays (#wythoff)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.golden_string.construct_wythoff_beatty_model() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/etc, geometor/divine

Constructive Synthesis & Exact Proof

Because \(\frac{1}{\Phi} + \frac{1}{\Phi^2} = \phi + \phi^2 = 1\), Samuel Beatty’s theorem guarantees that lower and upper Beatty sequences \(A_n = \lfloor n\Phi \rfloor\) and \(B_n = \lfloor n\Phi^2 \rfloor = A_n + n\) partition \(\mathbb{Z}_{>0}\) into disjoint cold positions of Willem Wythoff’s game (Wythoff1907, KnottFibGen).

• Exact Wythoff & Beatty Cone: From origin O, exact rays OR1 (\(y = \Phi x\)) and OR2 (\(y = \Phi^2 x\)) bound the cold Wythoff pairs W1 \((1, 2)\), W2 \((3, 5)\), W3 \((4, 7)\), and W4 \((6, 10)\) (connected along W1W2, W2W3, and W3W4 beneath ideal targets A1 – A4).

• Bijective Wythoff & Stolarsky Interspersions: Seeding row \(r \ge 0\) of the Wythoff array with \(W(r, c) = \lfloor (r+1)\Phi \rfloor F_{c+2} + r F_{c+1}\) groups integers whose Zeckendorf representations end in \(c\) trailing zeros (Morrison1980), while Kenneth Stolarsky’s nearest-integer Golden Ratio rounding \(b = \lfloor a\Phi + 1/2 \rfloor\) (Stolarsky1977) yields the dual Stolarsky array studied by John Conway and Clark Kimberling.

Leading 6 × 6 Wythoff & Stolarsky Interspersion Subarrays

Wythoff Array \(W(r, c)\) (\(0 \le r, c \le 5\))

Row	\(c=0\)	\(c=1\)	\(c=2\)	\(c=3\)	\(c=4\)	\(c=5\)
\(r=0\)	1	2	3	5	8	13
\(r=1\)	4	7	11	18	29	47
\(r=2\)	6	10	16	26	42	68
\(r=3\)	9	15	24	39	63	102
\(r=4\)	12	20	32	52	84	136
\(r=5\)	14	23	37	60	97	157

Stolarsky Array \(S(r, c)\) (\(0 \le r, c \le 5\))

Row	\(c=0\)	\(c=1\)	\(c=2\)	\(c=3\)	\(c=4\)	\(c=5\)
\(r=0\)	1	2	3	5	8	13
\(r=1\)	4	6	10	16	26	42
\(r=2\)	7	11	18	29	47	76
\(r=3\)	9	15	24	39	63	102
\(r=4\)	12	19	31	50	81	131
\(r=5\)	14	23	37	60	97	157

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
beatty density sum	\(\frac{1}{\frac{\sqrt{5}}{2} + \frac{3}{2}} + \frac{1}{\frac{1}{2} + \frac{\sqrt{5}}{2}}\)
wythoff W 0 0	\(1\)
wythoff W 1 0	\(4\)
wythoff W 2 3	\(26\)
stolarsky S 1 0	\(4\)
stolarsky S 2 0	\(7\)
stolarsky S 3 3	\(39\)