semilattice-discovery Skill

semilattice-discovery

Find, compose, and verify skills using the free commutative idempotent semigroup.

Trit: 0 (Ergodic - coordinator)
Bundle: core

The Algebra

Every skill has a SemilatticeFiber: {color, trit, hash}.

hash: SplitMix64(name) — deterministic, O(1)
trit: hash mod 3 - 1 ∈ {-1, 0, +1} — GF(3) conservation
color: hash → RGB → OKLAB — perceptual similarity metric

Two skills compose via fiber join:

fiberJoin(a, b) = {
  color: OKLAB midpoint (idempotent, commutative)
  trit:  findBalancer(a.trit, b.trit)  — conserves GF(3)
  hash:  mix64(a.hash + b.hash)
}

Commands

# Compute fiber for a skill
bb skill-semilattice.bb fiber bisimulation-game

# Join two skills — shows conservation
bb skill-semilattice.bb join bisimulation-game sheaf-cohomology

# Find the best third skill to complete a conserved triad
bb skill-semilattice.bb triad bisimulation-game sheaf-cohomology

# Discover skills by fiber proximity to a query
bb skill-semilattice.bb discover "color theory oklab"

# Verify global GF(3) conservation across all 1425 skills
bb skill-semilattice.bb conserved?

# Show skill clusters by trit and color proximity
bb skill-semilattice.bb cluster 5

Semilattice Laws (verified)

| Law | Substrate | Braid | Color | |-----|-----------|-------|-------| | a ⊔ a = a | trit balancer idempotent | merge(v,v) = v | blend(c,c,0.5) = c | | a ⊔ b = b ⊔ a | + is commutative | merge commutes | blend commutes | | (a⊔b)⊔c = a⊔(b⊔c) | + is associative | merge associative | blend ≈ associative |

Origin

Three independent discoveries of the same algebra:

Flavors (Symbolics, 1979): mixin merge
CRDTs (Kleppmann, 2011): state merge
OKLAB (Ottosson, 2020): perceptual blend

Jaffer's SLIB contains the three pieces (colorspace.scm, modular.scm, random.scm) never composed. Gay.jl and nanoclj-zig/src/semilattice.zig are the first known compositions.

Implementation

Zig: nanoclj-zig/src/semilattice.zig (211 lines, 9 tests)
Babashka: asi/skill-semilattice.bb (walks all 1425 skills)

Results (2026-04-05)

Skills: 1425
Trit distribution: +1=478  0=481  -1=466
Trit sum: 12
GF(3) balanced: YES ✓ (sum mod 3 = 0)

Related Skills

join-semilattice — the abstract algebra
skill-dispatch — GF(3) triadic task routing
gf3-conservation-oracle — conservation verification
bisimulation-game — behavioral equivalence
implicit-coordination — stigmergic merge (distance 0.0000 from "semilattice crdt merge")

Agent Skills: semilattice-discovery

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