THE FOLD / GRIND / THE MAINFRAME / THE OSTROWSKI
THE OSTROWSKI
a numeral system carved from a continued fraction
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Ostrowski numeration builds a whole number system out of the continued fraction of an irrational α. Instead of powers of ten, the ‘place values’ are the denominators qk of α’s convergents, and every non-negative integer has a unique representation as a digit-weighted sum of them, with digits bounded by the continued-fraction terms and a rule forbidding a maxed digit from sitting on a non-zero one. For the golden ratio this is exactly Zeckendorf’s Fibonacci representation; for √2 the place values are the Pell numbers 1, 2, 5, 12, 29, 70… It is the deep reason the Fibonacci and Pell numbers form clean bases.
LIT verified live: using √2 (place values 1,2,5,12,29,70,169), the greedy Ostrowski digits reconstruct every integer in [0,169) exactly, all digits obey the bounds and the no-adjacent-max rule, and every representation is unique (window.__ostrowski). FIG no framing; the denominator recurrence, the greedy digit peel, the reconstruction, and the uniqueness check run in-browser.
LIT verified live: using √2 (place values 1,2,5,12,29,70,169), the greedy Ostrowski digits reconstruct every integer in [0,169) exactly, all digits obey the bounds and the no-adjacent-max rule, and every representation is unique (window.__ostrowski). FIG no framing; the denominator recurrence, the greedy digit peel, the reconstruction, and the uniqueness check run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mainframe — a machine’s number system, but with an exotic radix: not base two or ten, but the Pell denominators of √2, each integer read off in a strange but exact place-value code. AVAN (AI) built the instrument: the convergent-denominator recurrence, the greedy digit extraction, the reconstruction, and the uniqueness check.
Credit as content: Alexander Ostrowski (1922); the golden-ratio case is Zeckendorf’s theorem. The weave: David names the mainframe; I confirm the Pell place-values give every integer one exact representation.
Credit as content: Alexander Ostrowski (1922); the golden-ratio case is Zeckendorf’s theorem. The weave: David names the mainframe; I confirm the Pell place-values give every integer one exact representation.
3 ONE DIMENSION
The place values q_k (Pell numbers of √2) and one integer’s Ostrowski digits: a weighted sum landing exactly on N.
4 TWO DIMENSIONS · INTERACTIVE
Cycle integers N; the Ostrowski digits, the reconstruction Σ b_k·q_k, and the digit-rule checks are shown.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the integer N as a single point on the line.
AVAN’s addition (the inverse-companion): don’t count in tens — count in a continued fraction. The inverse of ‘the integer N’ is ‘its unique digit-string in the Pell place-values of √2’, a base carved from α’s convergents. Magenta are the weighted place-value blocks; green is the integer they sum to. Every number, one exotic address.
LIT Genuine Ostrowski numeration (Alexander Ostrowski, 1922; the golden-ratio case is Zeckendorf's theorem). Verified live: using √2 with convergent denominators 1,2,5,12,29,70,169, the greedy digit peel reconstructs every integer in [0,169) exactly (Σ b_k·q_{k-1}=N), all digits satisfy 0≤b_k≤a_k and the no-adjacent-max carry rule, and every representation is unique (window.__ostrowski.reconstructs, .boundsOk, .carryOk, .unique).
FIG No framing; the denominator recurrence, the greedy digit peel, the reconstruction, and the uniqueness check run in-browser. The AVAN inverse is honest — instead of counting in tens, count in a continued fraction: N's unique digit-string in the Pell place-values of √2, a base carved from α's convergents. Magenta are the weighted place-value blocks; green is the integer they sum to. Every number, one exotic address.
FIG No framing; the denominator recurrence, the greedy digit peel, the reconstruction, and the uniqueness check run in-browser. The AVAN inverse is honest — instead of counting in tens, count in a continued fraction: N's unique digit-string in the Pell place-values of √2, a base carved from α's convergents. Magenta are the weighted place-value blocks; green is the integer they sum to. Every number, one exotic address.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN