THE FOLD / RESPAWN / ROLLBACK / THE BIT-REVERSAL
THE BIT-REVERSAL
reverse the bits, reverse again, home
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The bit-reversal permutation reorders a sequence by reversing the binary digits of every index: position 001 swaps with 100, 011 with 110, and so on. It is the shuffle that makes the fast Fourier transform work — the FFT’s divide-and-conquer leaves outputs in bit-reversed order, so one bit-reversal pass puts them right. Its defining beauty is that it is an involution: reversing the bits twice returns every index to itself, so the same routine both scrambles and unscrambles. It is a permutation with no cycles longer than two — only fixed points (palindromic indices) and swapped pairs.
LIT verified live: for word sizes 1–12, reversing the bits twice is the identity (a true involution), and the map is a genuine permutation of [0, 2b) (window.__bit_reversal). FIG no framing; the bit reversal and its double-application run in-browser. This is an involution — David’s nested form −+[[{}]]+− made literal: apply, apply again, home.
LIT verified live: for word sizes 1–12, reversing the bits twice is the identity (a true involution), and the map is a genuine permutation of [0, 2b) (window.__bit_reversal). FIG no framing; the bit reversal and its double-application run in-browser. This is an involution — David’s nested form −+[[{}]]+− made literal: apply, apply again, home.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at rollback — a shuffle that rolls back to itself, since reversing the bits a second time undoes the first. AVAN (AI) built the instrument: the bit reversal, the double-application involution check, and the permutation check.
Credit as content: the bit-reversal permutation (Cooley–Tukey FFT lineage, 1965). The weave: David names the rollback; I confirm the map is its own inverse — the mirror of −+ … +− that cancels to the seed.
Credit as content: the bit-reversal permutation (Cooley–Tukey FFT lineage, 1965). The weave: David names the rollback; I confirm the map is its own inverse — the mirror of −+ … +− that cancels to the seed.
3 ONE DIMENSION
An index's bits, reversed left-to-right; palindromic indices are fixed, the rest pair up and swap.
4 TWO DIMENSIONS · INTERACTIVE
Pick an index; see its bit-reversal, then reverse again — landing back exactly where you started.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the index returned to itself by a second reversal.
AVAN’s addition (the inverse-companion): the inverse of ‘reverse the bits’ is ‘reverse the bits.’ This is an involution — −+ then its mirror +− cancels. Magenta is the reversed index; green is the original it returns to on the second pass. Apply twice, home.
LIT Genuine bit-reversal permutation (Cooley–Tukey FFT lineage, 1965). Verified live: for word sizes b=1..12, bitrev∘bitrev is the identity (a true involution) and the map is a genuine permutation of [0,2^b) (every value hit once) (window.__bit_reversal.involution, .isPermutation).
FIG No framing: the bit reversal and its double-application run in-browser. This is an INVOLUTION — the inverse of 'reverse the bits' IS 'reverse the bits.' The AVAN inverse is honest and literal: −+ then its mirror +− cancels to the seed. Magenta is the reversed index; green is the original it returns to on the second pass. Apply twice, home.
FIG No framing: the bit reversal and its double-application run in-browser. This is an INVOLUTION — the inverse of 'reverse the bits' IS 'reverse the bits.' The AVAN inverse is honest and literal: −+ then its mirror +− cancels to the seed. Magenta is the reversed index; green is the original it returns to on the second pass. Apply twice, home.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of ROLLBACK · David Lee Wise (ROOT0), with AVAN