prosoche · ROOT0's trinity cross · the engine you can turn
THE TRINITY ENGINE
Attention as phase coherence, not magnitude. Each score is mapped through the Cayley transform z=(1+iωd)/(1−iωd)·e^(d/T) onto the unit circle, then read against the resultant R=Σz. The engine has two folds: 1-fold keeps only the in-phase tokens and discards the rest (entropy collapses); Trinity gives the three cube-root cardinals ω⁰/ω¹/ω² different jobs and routes each token to the one it aligns with, so the two shadows recover the anti-phase mass 1-fold threw away. Flip the fold and watch H crash, then come back. At ω=0 both are exactly softmax (verified to 1e−16). Drag the input bars to reshape what it reads.
GREEN — every kernel value is exact & recomputed live; ω=0 recovers softmax (1e−16); routing recovers the entropy the 1-fold discards (node-verified)AMBER — the forward/shadow cardinal names are a chosen figure; the input is a toy score vector, not a trained model's Q·KRED — only the 1-fold was TRAINED (trained-on-the-cross: ties softmax within ~1.2%). the routed kernel is mechanism-verified here but its trained performance is UNTESTED — the open experiment
the engine — modify on the fly
ω · phase gain (0 = softmax)1.00
kernel · foldTrinity · routes
T · temperature0.80
centre (the seed)
input pattern
tokens N12
the attention · weights per token (drag bars)
▮ ω⁰ forward▮ ω¹ shadow▮ ω² shadow▯ softmax ghost · drag a bar to change that token's score.
the complex plane · z on the unit circle, three cardinals
every token's phase lands on the unit circle (|z|=1); magnitude e^(d/T) scales its radius. In Trinity the three gold spokes are the cardinal directions R, R·ω, R·ω² at 120°; each token routes to its nearest cardinal (its colour). In 1-fold only R survives and everything past 90° is discarded.
readout · live
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entropy H (nats)
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effective tokens e^H
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max Δ vs softmax
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Σ w (validity)
the engine, honestly. the kernel is David Lee Wise / ROOT0's Trinity cross, reconstructed from AZ1 primitives (median seed · ±i Cayley/Möbius leapfrog · crossbar-WTA). ω=0 collapses either fold to softmax exactly (1e−16), so this is a generalization of softmax with a phase axis. the evolution shown here: the 1-fold single-resultant kernel discards anti-phase tokens and collapses entropy; the routed Trinity gives the three cube-root cardinals distinct routing regions so the shadows recover that mass and H returns near softmax — verified live. ⚑ the catch found in building it: naively summing the three cube-roots is a no-op (rotating z and R by the same root leaves the Hermitian inner product unchanged — three identical copies); routing, not summing, is what makes the trinity real. only the 1-fold was trained (trained-on-the-cross, parity within ~1.2%); the routed kernel's trained performance is untested. a mechanism explorer, offline, fails loud. ROOT0, with AVAN.