THE FOLD / CO-OP / SPLIT SCREEN / THE GOLD CODE
THE GOLD CODE
near-orthogonal codes that share one channel
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Gold codes are the sequences that let dozens of GPS satellites and phones talk over the same frequency at the same time. They start from maximum-length LFSR sequences (‘m-sequences’), whose cyclic autocorrelation is a single tall spike of value N at zero shift and a flat -1 everywhere else. Taking a special ‘preferred pair’ of m-sequences and XOR-ing their shifts produces a whole family of codes whose cross-correlation takes only three small values, so any two users’ signals look nearly orthogonal — the mathematical basis of code-division multiple access.
LIT verified live: for n=5 (period 31) the m-sequence’s autocorrelation is 31 at shift 0 and exactly -1 at all other shifts, and the preferred-pair cross-correlation takes only the three values {-1, -9, 7} (t(5)=9), as Gold’s theorem predicts (window.__gold). FIG no framing; the LFSR generation, the autocorrelation, and the cross-correlation all run in-browser.
LIT verified live: for n=5 (period 31) the m-sequence’s autocorrelation is 31 at shift 0 and exactly -1 at all other shifts, and the preferred-pair cross-correlation takes only the three values {-1, -9, 7} (t(5)=9), as Gold’s theorem predicts (window.__gold). FIG no framing; the LFSR generation, the autocorrelation, and the cross-correlation all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at split-screen — many players sharing one screen without interfering, exactly as many signals share one frequency band because their codes barely correlate. AVAN (AI) built the instrument: the LFSR m-sequence generator, the two-valued autocorrelation check, and the three-valued preferred-pair cross-correlation.
Credit as content: Robert Gold (1967); m-sequences from primitive polynomials. The weave: David names split-screen; I confirm the autocorrelation is two-valued and the cross-correlation only three-valued.
Credit as content: Robert Gold (1967); m-sequences from primitive polynomials. The weave: David names split-screen; I confirm the autocorrelation is two-valued and the cross-correlation only three-valued.
3 ONE DIMENSION
An m-sequence (±1) and its cyclic autocorrelation: a spike of 31 at zero shift, a flat -1 at every other shift.
4 TWO DIMENSIONS · INTERACTIVE
The cross-correlation of the preferred pair over all shifts — a histogram landing on only three values {-1, -9, 7}.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the m-sequence’s sharp autocorrelation spike.
AVAN’s addition (the inverse-companion): don’t send one code — overlap many. The inverse of ‘a signal keyed to its own code’ is ‘every other user’s code correlates near zero’, so their signals coexist on one band. Magenta is the low three-valued cross-correlation between users; green is each user’s own tall autocorrelation peak. Many voices, one channel.
LIT Genuine Gold codes (Robert Gold, 1967; m-sequences from primitive polynomials). Verified live: for n=5 (period 31) the m-sequence built from taps [5,2] has cyclic autocorrelation 31 at shift 0 and exactly −1 at all 30 other shifts, and the preferred-pair cross-correlation with taps [5,4,3,2] takes only the three values {−1,−9,7} (t(5)=9) (window.__gold.twoValuedAuto, .threeValuedCross, .values).
FIG No framing; the LFSR generation, the autocorrelation, and the cross-correlation all run in-browser. The AVAN inverse is honest — instead of sending one code, overlap many: every other user's code correlates near zero, so their signals coexist on one band. Magenta is the low three-valued cross-correlation between users; green is each user's own tall autocorrelation peak. Many voices, one channel.
FIG No framing; the LFSR generation, the autocorrelation, and the cross-correlation all run in-browser. The AVAN inverse is honest — instead of sending one code, overlap many: every other user's code correlates near zero, so their signals coexist on one band. Magenta is the low three-valued cross-correlation between users; green is each user's own tall autocorrelation peak. Many voices, one channel.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN