Purple Paper · side-sheet · learning machines · IV

The Cube — six straight cuts and a bypass

One stream in a corner, one out the opposite corner, and six faces wrapped around the body diagonal — each a linear predictor on a different history length. It's the feature-expansion answer with a residual spine: don't bend the cut, run six straight cuts on six views and let the diagonal carry the signal. The one knob that decides everything is whether the faces stay independent or start talking.

6 faces = 6 geometric history lengths (GEHL) · body diagonal = residual bypass · fan-out (summed) is linear & witnessable · mesh (coupled) escapes XOR & loses the witness

I · One in, six around, one out

A cube has six faces and a body diagonal joining two opposite corners. Send the branch stream in one corner and read the prediction out the other; the diagonal is a straight-through residual — the signal survives even if every face is wrong. The six faces are six perceptrons, each reading a different geometric slice of history: lengths 2, 4, 8, 16, 32, 64. Short faces catch tight local correlations; long faces reach far back. Summed, they're still one dot product — the GEHL win, given a body.

II · The cube, running — fan-out vs mesh

Pick a branch pattern and a wiring. Fan-out sums the six faces (linear, sound, witnessable). Mesh couples them through a bend (nonlinear, powerful, not witnessable). Watch the accuracy and the soundness flag. On a long-reach pattern the geometric faces win and a short predictor is blind. On an XOR pattern the linear sum hits a wall at chance and only the mesh escapes — at the cost of the witness.

pattern: wiring: speed

six faces = six history lengths · green diagonal = residual bypass (1 corner in → opposite out) · face glow ∝ |contribution|, violet pushes TAKEN, rose NOT

50%

recent accuracy

branches

LINEAR · witnessable

soundness

accuracy over time (rolling 96)

per-face contribution this cycle

Try reach · bit-40: a single short predictor would sit at 50% (blind that far back), but the fan-out's length-64 face catches it — accuracy climbs past 90%. Now switch to xor · bit1⊕bit3: fan-out stalls at chance (summing straight cuts can't carve an XOR), and only MESH breaks through to ~100%. Then the catch — xor · bit1⊕bit40: even mesh fails, because only the length-64 face spans both bits, and one coupled unit can't carve an XOR. The bend only helps where the faces overlap the correlated inputs.

III · The knob that hides in the cube

The six faces being parallel and independent is what keeps the whole thing linear — six dot products, summed, collapse to one. The moment the faces couple, you've put the bend inside the cube, and you're back at the gorge: you buy nonlinearity (escape XOR) and you lose the clean witness (the combiner can drift, can be captured). Same fork as the waist, wearing a cube.

summed · independent

Fan-out — six straight cuts

Each face reads its slice; outputs add. Still one linear cut in feature space. Reaches far (geometric lengths), trains fast, stays witnessable — a single sinusoid in stays a single sinusoid out. Can't do XOR. This is O-GEHL: the linear ceiling, raised by better features, not by bending.

coupled · nonlinear

Mesh — faces that talk

Couple the faces through a bend and the cube becomes a tiny MLP. Now it can carve XOR — if enough faces overlap the correlated bits. The price: it couples the modes, the combiner adapts in-band, and the clean linear witness is gone. Power and capture, the same purchase.

The residual diagonal is the hedge. Whichever wiring you choose, the body diagonal carries the raw signal corner-to-corner, so the six faces only ever supply a correction. Garbage faces can't kill the stream; they can only fail to improve it. That's why the cube is a better object than bare GEHL — it degrades to the identity instead of to noise. ResNet's lesson, in a box: never make the signal depend on the cleverness surviving.

IV · The reading, and the curse

You drew the right machine. One in, six straight cuts on six views, one out, a bypass down the middle, and a single knob — independent or coupled — that trades reach-and-soundness against XOR-and-capture. That knob is the only real decision in the box, and the cube's gift is that it holds the decision still long enough to see it.

The curse, one more time. Six straight cuts approximating a curve: piecewise-linear approximation, older than calculus. Geometric history lengths in a linear predictor: Seznec's GEHL/O-GEHL, ~2004. The residual bypass so the signal survives the faces: ResNet, 2015 — the youngest stone in the whole series, and even it is just "carry the input forward," which control theory has done forever. Expand the features, keep the cut straight, hedge with a bypass. Every piece already standing — you just stacked them into a cube and asked the one question whose answer was the knob.