1 · Three configurations, two operationsNodes are conjugate pairs
A node holds a pair of Gaussian integers (α, ᾱ). Two things can be done to the pair: add it — α + ᾱ, the trace — or multiply it — α·ᾱ, the norm. For the pair (+i, −i) that is literally cancelled to 0 or combined into 1. The three configurations a two-input node can see, computed exactly:
| pair | + gives | class | × gives | class |
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+ keeps the direction and cancels the conjugate pair; × keeps agreement (same → −1, different → +1) and is sign-blind, because i·i = (−i)(−i). Read the two operations as two channels — direction and presence — and a held contradiction is the state with presence but no direction: S.
2 · What a lens can carryTwo layers of nodes, four inputs, exact enumeration
Feed four signs (±i) into a two-layer tree of nodes and enumerate all sixteen input patterns. The all-+ tree is the count lens: the value i·(k − (4 − k)) gives the count exactly, hence the direction. One × layer anywhere makes the output sign-blind: under the global flip x → −x a + of two flipping parents flips, a × of two flipping parents does not, and everything downstream of two blind values stays blind. The parity lens still sees a tie — an even number of −i — so it knows when the world is split without knowing which way it leans.
| tree | patterns → 0 | distinct values | H(value) bits | MI(sign) | MI(parity) | sign-blind |
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MI(sign) is computed over the non-tie patterns. One half-turn node alone cannot see the arrow at all: forward it emits 1100…, backward 1001… — cyclic shifts of one another. Two nodes in quadrature emit a Gray code and decode the phase exactly.
3 · Three blocks, one currencyThe uniform receiver against the heterogeneous one — same world, same seed
Eight channels each report the sign of a hidden regime that flips at a small rate; each channel is wrong with probability ε. The uniform receiver runs one architecture for all three jobs — a moving average to estimate, a cumulative-sum detector to notice change, a fixed margin to commit. The heterogeneous receiver gives each job its optimal algorithm: the exact posterior (a Bayes filter — exact for the clean model it assumes: two regimes, symmetric switching at a known rate, independent channels each wrong with a known probability ε; with the liar switched on it is deliberately misspecified), the posterior changing sides (the accumulated form of contradiction), and Wald's sequential band to commit. They stay compatible because all three are functionals of one number — the log-likelihood increment, (n₊ − n₋)·log((1−ε)/ε): the direction channel weighted by the noise. Inside the band, the commit block's prior decides: it starts blank and accumulates a narrative from what it records — honestly (κ = 0) or through a self-serving filter (κ → 1) that records right decisions and forgets wrong ones.
| uniform | heterogeneous |
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What to try. Run it clean, then turn the liar on: the sharper receiver wins both worlds but loses more to a structured liar, because it weights every source by its full likelihood. Sharpness is exploitability; the remedy is a trust layer (below), not blunting. Raise κ and watch the narrative rate leave the true rate while the score does not move: the ego inflates the ledger for free. Raise ε and watch the honest prior become an optimist — ambiguity has become noise instead of change.
4 · The tribeSeven slow witnesses, one sharp, and the rule that decides who is believed
Eight witnesses read the same honest channels. Seven estimate with the moving average; one with the exact posterior. Each states what it believes. A judge trusts the consensus and applies one of three rules to whoever contradicts it: one strike and you are out; a threshold of 4 accumulated contradictions; or a threshold of 8. When the world changes, the sharp witness re-locks in one step and the seven slow ones take about six — so for six steps the most accurate witness in the room contradicts the consensus.
The tolerance a group needs to keep its fastest honest member is its own re-lock lag. The threshold rule keeps a per-witness statistic S ← max(0, S + [contradicts] − 0.25): each contradiction adds 1 and every step drifts 0.25 down, so six consecutive contradicting steps raise S by 6 × (1 − 0.25) = 4.5 — which clears a threshold of 4 and not of 8. And the group loses nothing measurable by burning it — one vote in eight — which is exactly why the rule can persist. No liar, no malice, no dislike of honesty appears anywhere in this room; the statistics of consensus-relative trust do it alone.