Editing Openai/693343d7-a38c-8012-a67c-11cbed4c0fd9 (section)

=== We define an algebraic theory Th_O that axiomatizes Obatala operations. Consider signature Σ containing: ===
* Nullary: none
* Unary: η : A → OA (reflect)
* Binary: join : O A × O A → O A (merge/organization)
* Unary: sim : O A → Traj(A) (simulation projection)
* Unary: flatten : O(O A) → O A (μ)
* N-ary operations as needed for aggregation/consensus

Equations (selected core):
# flatten( reflect(x) ) = x  (unit law)
# flatten( map(reflect)(m) ) = m  (left identity) — written more categorically as μ ∘ O η = id.
# flatten( flatten(m) ) = flatten( map(flatten)(m) )  (associativity) — μ ∘ μ_O = μ ∘ O μ.
# For join: join(join(x,y),z) = join(x,join(y,z)) (associativity)
# join(x, reflect(y)) = join(reflect(y), x) = ... — laws consistent with commutativity if desired.
# sim( map(f)(m) ) = lift_traj(f)( sim(m) )  (functoriality of simulation)

map(f) denotes O f (lifting of f by functoriality). lift_traj(f) is the induced map on trajectories.

These equations form the equational theory for reasoning about Obatala computations.