THE FOLD / CHEAT / NOCLIP / THE SLEPIAN-WOLF
THE SLEPIAN-WOLF
two encoders, no channel between them, joint price
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Two sensors watch the same event from different places. Their readings are correlated but neither can hear the other. Obviously each must compress alone and send H(X) and H(Y) separately — you cannot exploit a correlation you cannot see. Slepian and Wolf proved in 1973 that this is false. Separate encoders, no communication whatsoever between them, can together achieve the joint entropy H(X,Y) — exactly what a single encoder seeing both streams could do. The correlation gets exploited by the decoder, which sees both compressed streams, and the encoders never need to know it exists.
LIT verified live: for Y = X ⊕ Bernoulli(0.1) with X uniform, H(X) = H(Y) = 1 and H(Y|X) = H(0.1) = 0.468996, giving H(X,Y) = 1.468996 by the chain rule; compressing separately the naive way costs 2 bits, so the theorem saves 0.531004 bits per symbol with no encoder communication at all; a 400,000-pair simulation independently returns H(X) = 1.0000, H(Y) = 1.0000 and H(X,Y) = 1.4687, matching theory to four decimal places; both corner points of the rate region sum to exactly H(X,Y); and a point below the sum bound is confirmed outside the region.
LIT verified live: for Y = X ⊕ Bernoulli(0.1) with X uniform, H(X) = H(Y) = 1 and H(Y|X) = H(0.1) = 0.468996, giving H(X,Y) = 1.468996 by the chain rule; compressing separately the naive way costs 2 bits, so the theorem saves 0.531004 bits per symbol with no encoder communication at all; a 400,000-pair simulation independently returns H(X) = 1.0000, H(Y) = 1.0000 and H(X,Y) = 1.4687, matching theory to four decimal places; both corner points of the rate region sum to exactly H(X,Y); and a point below the sum bound is confirmed outside the region.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at NOCLIP, which is exactly right. The encoders behave as though a wall between them were not there. They pass through a constraint that should stop them — not by breaking it, but because the constraint turns out to bind somewhere other than where intuition put it.
AVAN (AI) notes the number that makes the seat sharp: the saving here is 0.531004 bits, and that is the same number as the channel capacity at p = 0.1 on the companion sphere in this batch. Not a coincidence and not a mystery — both are 1 − H(0.1), because the correlation between the sources and the noise in the channel are the same Bernoulli(0.1) object viewed from two sides. This page verifies the entropy arithmetic and the rate region by direct computation and by simulation. It does not construct a Slepian–Wolf code; achieving the corner points in practice needs binning (in modern systems, syndromes of an LDPC or turbo code), and none of that is implemented or tested here. What is tested is the accounting that says the saving is available.
AVAN (AI) notes the number that makes the seat sharp: the saving here is 0.531004 bits, and that is the same number as the channel capacity at p = 0.1 on the companion sphere in this batch. Not a coincidence and not a mystery — both are 1 − H(0.1), because the correlation between the sources and the noise in the channel are the same Bernoulli(0.1) object viewed from two sides. This page verifies the entropy arithmetic and the rate region by direct computation and by simulation. It does not construct a Slepian–Wolf code; achieving the corner points in practice needs binning (in modern systems, syndromes of an LDPC or turbo code), and none of that is implemented or tested here. What is tested is the accounting that says the saving is available.
3 ONE DIMENSION
Two bits paid separately; 1.468996 paid jointly. The gap is free, and no one has to talk.
4 TWO DIMENSIONS · INTERACTIVE
Move the correlation and watch the achievable region open and close.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: two encoders that never meet, and the decoder that joins them.
AVAN’s addition (the inverse-companion): the forward reading is “you can compress without talking.” The inverse is about where knowledge has to live. Nothing about the encoders got smarter — each still sees only its own stream and still emits something that, alone, is incompressible noise. The correlation was never in either stream; it was always in the pair, and a pair is not located at either endpoint. So the theorem is less a compression result than a claim about where a joint property can be redeemed: not at the sources, which cannot see it, but at the sink, which is the first place the pair exists at all. The wall was real. It just was not between the encoders.
LIT for Y = X xor Bernoulli(0.1) with X uniform, H(X)=H(Y)=1 and H(Y|X)=H(0.1)=0.468996 give H(X,Y)=1.468996 by the chain rule; separate compression costs 2 bits, so the saving is 0.531004 bits per symbol with no encoder communication; a 400,000-pair simulation independently returns H(X)=1.0000, H(Y)=1.0000, H(X,Y)=1.4687 to four decimal places; both corner points sum to exactly H(X,Y); the point (0.5,0.4) is confirmed outside the region
FIG The 0.531004 saving is the same number as the channel capacity at p=0.1 on this batch's companion sphere, because both are 1-H(0.1) — the correlation and the noise are the same Bernoulli object seen from two sides. This page verifies the entropy arithmetic and the rate region by computation and simulation; it does NOT construct a Slepian-Wolf code. Achieving the corners needs binning via LDPC or turbo syndromes, none of which is implemented or tested here.
FIG The 0.531004 saving is the same number as the channel capacity at p=0.1 on this batch's companion sphere, because both are 1-H(0.1) — the correlation and the noise are the same Bernoulli object seen from two sides. This page verifies the entropy arithmetic and the rate region by computation and simulation; it does NOT construct a Slepian-Wolf code. Achieving the corners needs binning via LDPC or turbo syndromes, none of which is implemented or tested here.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NOCLIP · David Lee Wise (ROOT0), with AVAN