# Finite flow network

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A nonnegative flow in a finite directed network must respect every edge capacity and conserve flow at each vertex other than source and sink.

mko-finite-flow-network · definition · v0.10.0



## Mathematical statement

Use a finite loop-free simple directed graph with distinct source s and sink t. Antiparallel edges are allowed, but each ordered pair appears at most once. Valid capacities/flows are exact nonnegative rationals. Feasibility requires0≤f(e)≤c(e) and complete inflow=outflow at every internal vertex. Cross-check source net outflow against sink net inflow. Zero capacities and disconnected vertices are legal. Feasibility is not maximum flow; no min-cut equality or augmenting-path optimality is claimed ahead of its theorem.





## Representative formula (exact TeX)

V(f,s,t)=z





## Assumptions


- Vertices, edges, capacity and flow keys correspond completely one-to-one; no loops or duplicate directed edges.
- Every sum uses exact rational arithmetic.





## Symbol scope


- V — flow-value operator
- f — finite edge flow
- s — source vertex
- t — sink vertex
- z — exact flow value





## Intuition, definition, and misconceptions

Network qualification fixes the directed-edge convention first.

Capacity and conservation are separate feasibility obligations.

Source and sink flow-value calculations must agree.

A total flow value alone does not check each edge or internal vertex, and a feasible flow is not automatically maximum.





## Adopted derivation guide

[Checking every edge and internal vertex of a feasible flow](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-flow-network-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Finite directed graph](https://ocme.evemisslab.com/en/mko/mko-finite-directed-graph/) — Fix the network carrier.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — Capacities and flows map edges to values.
- [Linear programming](https://ocme.evemisslab.com/en/mko/mko-linear-programming/) — Capacity and conservation are constraints.





## Verification boundary

This slice does not build max-flow/min-cut, residual networks or an optimality proof.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add the max-flow min-cut theorem.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/finite-flow-network.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-flow-network/index.md)