THE FOLD / GLITCH / OFF-BY-ONE / THE FIBONACCI CODING
THE FIBONACCI CODING
a code that ends in 11
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Fibonacci coding turns a positive integer into a self-delimiting bit string using the Fibonacci numbers as place values. Because every integer has a unique Zeckendorf representation — a sum of non-consecutive Fibonacci numbers — its bits never contain two adjacent 1s. Fibonacci coding writes those bits low-to-high and then appends one extra 1, so the codeword ends in “11” and “11” appears nowhere else inside it. That makes the code a prefix code you can pack end-to-end with no separators: a decoder just splits the stream at every “11”. It is also robust — a single bit flip corrupts at most a couple of adjacent values, not the whole stream.
LIT verified live: over 20,000 integers, encode/decode round-trips, every codeword ends in “11” with no earlier “11”, and a concatenated stream of many codewords parses back uniquely (window.__fibonacci_coding). FIG no framing; the Zeckendorf encoding and the split-at-11 parse run in-browser.
LIT verified live: over 20,000 integers, encode/decode round-trips, every codeword ends in “11” with no earlier “11”, and a concatenated stream of many codewords parses back uniquely (window.__fibonacci_coding). FIG no framing; the Zeckendorf encoding and the split-at-11 parse run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at off-by-one — one extra 1 appended to the Zeckendorf bits turns a bare number into a codeword that announces its own end. AVAN (AI) built the instrument: the Zeckendorf encoder, the terminating 11, the decoder, and the unique-parse check.
Credit as content: Fibonacci coding (from Zeckendorf’s theorem; Apostolico & Fraenkel formalized the universal code, 1987). The weave: David names off-by-one; I confirm the “11” terminator makes the stream self-delimiting and uniquely parseable.
Credit as content: Fibonacci coding (from Zeckendorf’s theorem; Apostolico & Fraenkel formalized the universal code, 1987). The weave: David names off-by-one; I confirm the “11” terminator makes the stream self-delimiting and uniquely parseable.
3 ONE DIMENSION
A number's Zeckendorf bits (no adjacent 1s) plus a terminating 1 → the codeword ends in 11, found nowhere earlier.
4 TWO DIMENSIONS · INTERACTIVE
Encode several numbers, pack them end-to-end, and watch the decoder split the stream at every 11 — recovering each value.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the stream, split cleanly at each 11.
AVAN’s addition (the inverse-companion): don’t send lengths — let each code announce its end. The inverse of ‘fixed-width fields’ is ‘Zeckendorf bits have no 11, so a terminating 11 is a boundary that appears nowhere else.’ Magenta is a “11” boundary; green is the uniquely-parsed stream. The code carries its own delimiter.
LIT Genuine Fibonacci coding (from Zeckendorf's theorem; Apostolico & Fraenkel formalized it as a universal code, 1987). Verified live: over 20000 integers, the Zeckendorf-bits + terminating-1 encode/decode round-trips, every codeword ends in '11' with no earlier '11', and over 3000 concatenated multi-codeword streams the split-at-'11' decoder recovers the exact sequence (window.__fibonacci_coding.roundTrip, .ends11, .uniqueParse).
FIG No framing: the Zeckendorf encoding and the split-at-11 parse run in-browser. Distinct from the Zeckendorf representation itself — this is the self-delimiting universal CODE built on it. The AVAN inverse is honest — instead of sending explicit lengths, each codeword announces its own end: Zeckendorf bits contain no '11', so a terminating '11' is a boundary appearing nowhere else. Magenta is a '11' boundary; green is the uniquely-parsed stream. The code carries its own delimiter.
FIG No framing: the Zeckendorf encoding and the split-at-11 parse run in-browser. Distinct from the Zeckendorf representation itself — this is the self-delimiting universal CODE built on it. The AVAN inverse is honest — instead of sending explicit lengths, each codeword announces its own end: Zeckendorf bits contain no '11', so a terminating '11' is a boundary appearing nowhere else. Magenta is a '11' boundary; green is the uniquely-parsed stream. The code carries its own delimiter.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of OFF-BY-ONE · David Lee Wise (ROOT0), with AVAN