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

=== ## ===
* Platform name: nandi-platform (all platform references use this name).
* Core platform macro-apps: kogi-home, kogi-work, kogi-community, kogi-hub, kogi-os etc., as previously defined.
* y means confirm/continue when you need to continue tasks.
* All constructions must remain ASCII-friendly and compatible with the Obatala Monad, SDELang semantics, and category-theoretic foundations already produced.

==== An agent A is the tuple: ====

<syntaxhighlight>A = (S, M, P, C, F, Act, Ω, θ)

</syntaxhighlight>

Where:
* S: state space
* M: memory
* P: perception function (may be affinity-weighted)
* C: cognition (internal category)
* F: forward-simulation functor (Ran_{P}(Dynamics))
* Act: action/choice function
* Ω: goals/values/drives
* θ: social / embedding attributes (vectors used for affinity)

Agents live in category Agt. Affinities live in a value quantale V (practical: V = [0,1]).

Network adjacency (affinity matrix) W : Agt × Agt → V with entries W_{ij} = Aff(A_i,A_j).

==== ### ====
* Agt is a (small) V-enriched category: Hom(A,B) ∈ V (captures affinity or more general graded relation).
* Observations are modeled by a Grothendieck fibration: ``<code> π : Obs → World </code>`` each agent A has perception section P_A : World → Obs with π ○ P_A = Id.

===== - O : Agt → Agt with η : Id ⇒ O, μ : O O ⇒ O. =====
* O supports: - reflect ≡ η - μ flattening of nested agency - join_W : O A × O A → O A (affinity-parameterized merge) - sim : O A → Traj(A) (natural in A)

Monad laws hold for all A:

<syntaxhighlight>μ ∘ O η = id,    μ ∘ η_O = id,    μ ∘ O μ = μ ∘ μ_O

</syntaxhighlight>

===== - Aff : Agt^op × Agt → V (a V-profunctor). =====
* Composition defined by coend:

<syntaxhighlight>Aff^{(2)}(A,C) := (Aff ∘ Aff)(A,C) = ∫^{B∈Agt} Aff(A,B) ⊗ Aff(B,C).

</syntaxhighlight>
* Iteration:

<syntaxhighlight>Aff^{(n)}(A,Z) ≅ ∫^{B1...B_{n-1}} ∏_{i=0}^{n-1} Aff(B_i,B_{i+1})
where B_0=A, B_n=Z.

</syntaxhighlight>

This yields multiplicative / path-based transitive affinity when V is a quantale with ⊗ product and coends computed as joins (sup).

==== ### ====

<syntaxhighlight>S_{t+1} = C(S_t, P(E_t; W_t), M_t, Ω)
A_t = Act(S_t, M_t, Ω)
E_{t+1} = world_update(E_t, A_t)
M_{t+1} = memory_update(M_t, S_t, A_t, feedback; W_t)

</syntaxhighlight>

Perception uses affinity-weighting:

<syntaxhighlight>P_A(E) = normalize( ∑_j W_{A,j} · obs_j(E) )

</syntaxhighlight>

(where normalization uses chosen V-sum / join semantics).

===== For each ordered pair (i,j) use: =====

<syntaxhighlight>ΔW_{ij} = η · ( AffScore(A_i,A_j) - W_{ij} ) + ζ · Success_{ij} - δ · W_{ij}
W_{ij} ← clamp(W_{ij} + ΔW_{ij}, 0, 1)

</syntaxhighlight>

AffScore is a parameterized function (homophily vs heterophily mix):

<syntaxhighlight>AffScore(A,B) = σ( α·Sim(A,B) + β·Comp(A,B) + γ·Trust(A,B) + ... )

</syntaxhighlight>

===== Trust update after interaction: =====

<syntaxhighlight>Trust_{i→j, t+1} = (1-ρ) · Trust_{i→j,t} + ρ · Reward_{ij,t}

</syntaxhighlight>

Reward is performance outcome; influences AffScore via β·Trust.

==== All proofs below use standard enriched-category/coend calculus; assumptions: V is a commutative unital quantale (complete, ⊗ distributes over joins). ====

===== Aff: Agt^op ⊗ Agt → V obvious by construction (pairwise weight natural in both args). =====

===== Composition: (Aff ∘ Aff)(A,C) := ∫^{B} Aff(A,B) ⊗ Aff(B,C). =====
Associativity: by Fubini for coends and monoidality of ⊗:

<syntaxhighlight>(Aff ∘ (Aff ∘ Aff))(A,D) ≅ ∫^{B,C} Aff(A,B)⊗Aff(B,C)⊗Aff(C,D)
 ≅ ((Aff ∘ Aff) ∘ Aff)(A,D)

</syntaxhighlight>

Units: identity profunctor Id(A,B) := Agt(A,B) (or δ for discrete identity) satisfies:

<syntaxhighlight>Id ∘ Aff ≅ Aff ≅ Aff ∘ Id

</syntaxhighlight>

via enriched Yoneda / coend collapse.

(These proofs were given in full structured form earlier; this statement is the canonical summary.)

===== Transitive affinity reduces to: =====

<syntaxhighlight>Aff^{(n)}(a,z) = sup_{paths a=...=z} ∏ edge_weights

</syntaxhighlight>

This yields the familiar "best-path product" semantics.

==== ### ====

join_W : O A × O A → O A is parameterized by W and has algebraic behavior:

<syntaxhighlight>α_W(join_W(m,n)) = α_join( α_W(m), α_W(n); W_sub )

</syntaxhighlight>

where α_W : O A → A is an EM-algebra that interprets Trees into concrete organizational action using local affinities (W_sub = affinities among children).

===== Monad laws hold for O for any fixed admissible W (i.e., μ_W commutes with η and is associative); when W changes over time, semantics are parametric: we treat O as a family of monads O_W or as O with a parameter W_t passed to α at interpretation time (preferred for runtime stability). =====

===== α_W : O A → A (organization handler) must satisfy: =====

<syntaxhighlight>α_W ∘ η = id,    α_W ∘ μ = α_W ∘ O α_W

</syntaxhighlight>

and uses W to weight aggregation in the interpretation of Node(MergeOp, children).

==== ### ====

<syntaxhighlight>t ::= x | λx.t | t t | reflect t | bind x <- t in u | merge t t | sim t

</syntaxhighlight>

===== - reflect e → Leaf(lower(e)) =====
* merge a b → Node(MergeOp, [lower a; lower b])
* sim t → Node(SimOp, [lower t])
* bind x <- t in u → join( fmap (fun leaf -> lower(u)[x := leaf]) (lower t) )

===== <syntaxhighlight>bind x <- reflect v in u  →  u[x := v] =====
bind x <- (bind y <- m in n) in u  →  bind y <- m in (bind x <- n in u)
sim (reflect v)  →  sim_leaf(v)
sim (merge m n)  →  combine_sim(sim m, sim n; W)

</syntaxhighlight>

Combine_sim uses affinity W to compose trajectories/precedence.

==== ### ====
* Affinity Service: compute/store Aff(A,B), expose APIs.
* Matchmaker: composes O trees into candidate team manifests using homophily/heterophily policy parameters.
* Plan Compiler: SDELang → Tree IR → EM interpretation.
* EM Handler Registry: stores α_W implementations (policy + runtime code), authored in kogi-hub.
* Simulator: executes SimOp nodes; returns Traj.
* Coordinator: runs MergeOp using W and α_W.
* Network Dynamics Engine: steps affinity updates W_{t+1} using dynamics above.

Design principles:
* Persist Tree manifests immutably for audit/replay in kogi-state.
* Expose policy knobs: α (homophily strength), β (heterophily boost), decay δ, learning ρ.
* Enforce fairness thresholds in EM-algebras to guard against echo chambers.

==== I confirm this synthesis does not contradict any prior chat artifacts and is consistent with them: ====
* Obatala Monad: monad (O, η, μ) and Kleisli/EM accounts preserved; Tree instance and Coq sketch remain valid. (Matches your earlier Coq sketch.)
* Category theory: Agt, fibration π : Obs → World, enriched profunctor Aff and composition via coends — matches prior deep-category elaboration.
* SDELang: reflect/bind/merge/sim semantics, lowering to Tree IR, operational rules — consistent with earlier SDELang embedding and operational fragments.
'' Platform naming & mapping: uses nandi-platform, kogi-'' apps and mapping exactly as requested.
* Homophily/heterophily: defined as AffScore and integrated into perception, memory, update laws, Obatala join semantics, and EM-algebras (all prior chat guidance honored).
* Proofs & assumptions: quantale V assumption and Fubini/coend use are the same as in our category-theory notes; associativity/unit proofs are the standard enriched coend arguments previously delivered.

If you’d like, I will now:
* produce a single master ASCII file containing this entire synthesis plus the earlier Coq .v file and SDELang IR in one bundle, or
* generate a formal Coq/Lean formalization of the enriched profunctor proofs + Tree Obatala monad + simple EM algebra (machine-checkable), or
* implement a reference runtime prototype (language: pick OCaml / TypeScript / Python) that implements Aff composition, O as Tree, and the matchmaker + simulator pipeline.

Pick one of: master-file, coq-formalize, runtime-proto (python|ocaml|ts) — and I’ll emit it immediately.