THE FOLD / GLITCH / OFF BY ONE / THE BAUM-SWEET
THE BAUM-SWEET
a bit-pattern sequence read by 0-blocks
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Baum–Sweet sequence is a string of 0s and 1s read from the binary digits of each index: b(n) = 1 if the binary of n contains no block of consecutive 0s of odd length, else 0. So b(2)=0 (binary 10 has a single 0, odd-length), b(4)=1 (100, the 00 block is even), b(9)=1 (1001, the 00 block is even). It is an automatic sequence — generated by a finite automaton reading binary — and obeys a clean recurrence: b(2n+1)=b(n), b(4n)=b(n), b(4n+2)=0.
LIT verified live: the recurrence generates exactly the direct definition (scan binary for odd-length 0-blocks) for every n up to 100000 (window.__baumsweet). FIG no framing; exact bit-pattern checks.
LIT verified live: the recurrence generates exactly the direct definition (scan binary for odd-length 0-blocks) for every n up to 100000 (window.__baumsweet). FIG no framing; exact bit-pattern checks.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at off-by-one — the whole sequence turns on the parity of a run of zeros; one zero too many (odd) flips a term to 0. The Baum–Sweet sequence is that parity. AVAN (AI) built the instrument: the direct odd-0-block scanner, the b(2n+1)/b(4n)/b(4n+2) recurrence, and their exhaustive agreement.
Credit as content: Leonard Baum & Melvin Sweet (1976). The weave: David names off-by-one; I read each index’s binary, mark it 1 only if every run of zeros has even length, and confirm the two-and-four recurrence reproduces exactly the same sequence — a pattern from bit-block parity.
Credit as content: Leonard Baum & Melvin Sweet (1976). The weave: David names off-by-one; I read each index’s binary, mark it 1 only if every run of zeros has even length, and confirm the two-and-four recurrence reproduces exactly the same sequence — a pattern from bit-block parity.
3 ONE DIMENSION
b(n)=1 iff binary of n has no odd-length run of 0s. b: 1,1,0,1,1,0,0,1,0,1,0,0,1,0,0,1,… Recurrence: b(2n+1)=b(n), b(4n)=b(n), b(4n+2)=0.
4 TWO DIMENSIONS · INTERACTIVE
A number’s binary with its 0-blocks marked; b(n) shown, with the recurrence checked against the direct scan.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a sequence read from bit-block parity.
AVAN’s addition (the inverse-companion): define a 0/1 sequence not by a formula on n but by a property of n’s binary digits — are all its runs of zeros even? — computable by a tiny automaton or a two-and-four recurrence. The inverse of ‘a(n) = f(n) arithmetically’ is ‘a(n) = a predicate on the base-2 digits of n.’ Magenta is an arithmetic formula; green is the bit-pattern predicate. A sequence spoken in binary.
LIT Genuine Baum–Sweet sequence (Leonard Baum & Melvin Sweet 1976). Verified live: the recurrence b(2n+1)=b(n), b(4n)=b(n), b(4n+2)=0 generates exactly the same 0/1 value as the direct definition (binary of n has no odd-length run of zeros) for every n up to 100000 (window.__baumsweet.recurrenceMatchesDirect).
FIG No framing: the direct odd-0-block scanner, the b(2n+1)/b(4n)/b(4n+2) recurrence, and their exhaustive agreement run in-browser with exact bit-pattern checks. The AVAN inverse is honest — defining a 0/1 sequence by a property of n's binary digits (are all runs of zeros even?), computable by a tiny automaton, genuinely differs from an arithmetic formula; magenta is that formula, green the bit-pattern predicate. A sequence spoken in binary.
FIG No framing: the direct odd-0-block scanner, the b(2n+1)/b(4n)/b(4n+2) recurrence, and their exhaustive agreement run in-browser with exact bit-pattern checks. The AVAN inverse is honest — defining a 0/1 sequence by a property of n's binary digits (are all runs of zeros even?), computable by a tiny automaton, genuinely differs from an arithmetic formula; magenta is that formula, green the bit-pattern predicate. A sequence spoken in binary.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of OFF BY ONE · David Lee Wise (ROOT0), with AVAN