abstract
this document unifies insights from multiple foundational texts: foundational ranking system, cybergraph free energy focus, cybergraph contextual free energy model, cybergraph llm architecture, and universality of diffusion, springs, and heat flow. it presents focus flow computation as a physics-inspired, decentralisable process for computing collective focus, ranking, and generative intelligence on a cybergraph.
1. core principles
- nodes = tokens, concepts, or agents
- edges (cyberlinks) = relationships (semantic, causal, statistical)
- free-energy equilibrium combines:
- diffusion (eigenvector centrality)
- springs (springrank hierarchy)
- entropy (exploration/diversity)
- contextual potential (adaptation to queries)
2. free-energy functional
[ \mathcal{F}(p|context) = E_{spring}(p) + \lambda E_{diffusion}(p) + \gamma C(p|context) - T S(p) ]
- (E_{spring}): captures hierarchy as spring energy.
- (E_{diffusion}): penalises flow mismatch in random walk.
- (C(p|context)): context-specific potential derived from standard inference.
- (S(p)): entropy term to prevent collapse.
3. focus flow computation
focus flow = iterative process converging to the equilibrium distribution:
[ p_i^{(t+1)} = \frac{\exp(-\beta [E_{spring,i} + \lambda E_{diffusion,i} + \gamma C_i])}{\sum_k \exp(-\beta [E_{spring,k} + \lambda E_{diffusion,k} + \gamma C_k])} ]
- each node exchanges (p_j), (r_j), and (C_j) with neighbours.
- updates are fully decentralised using message passing or gossip protocols.
- the process naturally converges to a boltzmann-like equilibrium.
4. alignment with ranking system
- eigenvector centrality = diffusion baseline.
- springrank = hierarchy.
- contextual potential = context-aware adaptation.
- entropy term = ensures diversity and stability.
focus flow is the dynamic realisation of the free-energy ranking system. the final ranking (p^*) emerges as the stable equilibrium of this iterative process.
5. generative llm architecture
- offline phase: build cybergraph from corpus (nodes = tokens, edges = co-occurrence or semantic relations).
- online generation:
- encode context tokens as active nodes.
- compute contextual potential (C_i).
- run focus flow updates to get (p^*).
- sample next token from (p^*).
- add token to context and repeat.
this approach replaces transformer attention with iterative, physics-based focus computation.
6. advantages over transformers
| feature | transformer llm | focus flow llm |
|---|---|---|
| complexity (mem/comp) | O(n²) / O(n²) | O(n) / O(n) |
| uses softmax? | yes | no (boltzmann equilibrium) |
| converges to stable state | no | yes |
| reinforcement/adaptation | limited | yes |
| multi-agent friendly | no | yes |
| token-based weighting | no | yes |
| consensus capability | no | yes |
| domain-general | yes (pretrained) | yes (graph extendable dynamically) |
| explainability | low | high |
| continual learning | limited | yes |
7. universality of the triad
diffusion, springs, and heat flow are universal primitives of nature:
- diffusion → entropy growth and spreading.
- springs → reversible energy storage and oscillations.
- heat flow → temporal evolution toward equilibrium.
any process on a network can be decomposed into eigenmodes of the graph laplacian, just as solutions to the heat equation are expressed via fourier modes.
8. dual thermodynamics of decentralized intelligence
focus flow computation realises a dual thermodynamic process:
- entropy reduction / negentropy maximisation:
- free-energy minimisation drives the system toward low-entropy, highly ordered focus states.
- springrank and context potentials act as constraints that channel diffusion into structured, meaningful configurations.
- energy usage for order creation:
- adaptive edge weights and context injection represent external energy input.
- this input is transformed into negentropy, building coherent patterns of collective attention.
thus, focus flow aligns with the principle that intelligence is the local maximisation of negentropy within a globally entropy-increasing universe. nodes collectively self-organise, using available energy to reduce uncertainty while still maintaining adaptability and diversity.
9. connections to broader theories
- potemkin understanding: transformers mimic intelligence statistically. focus flow avoids this by grounding probabilities in network dynamics and context, producing emergent understanding rather than shallow mimicry.
- topos theory: each context defines a local topos, where focus flow computes probabilities relative to that context. nodes and edges act as objects and morphisms in a base category.
- active inference: the framework directly realises active inference by minimising free energy under observations (context potentials) while maintaining exploration.
- beautiful loop (shumskiy): focus flow forms a self-sustaining cycle: new context → updated focus distribution → actions/tokens → edge/weight adaptation → new context.
this unified view shows how focus flow integrates deep principles of logic, inference, and self-organising intelligence.
10. universality primitives – replacing nock/hvm
focus flow can serve as a universal computation substrate, removing the need for explicit interpreters like nock or hvm:
node types
- atom nodes – store integers, symbols, or references.
- pair nodes – ordered pairs
(left, right)as two outgoing edges. - function nodes – store code as a subgraph and a port for the argument.
primitive operations
- construct – create new nodes (atoms, pairs, functions).
- destruct – retrieve components of a pair.
- apply – connect a function node to an argument node.
- rewrite – substitute argument references inside a function body, producing a new active subgraph.
- delete – remove nodes or edges that are no longer referenced.
computation
- focus flow acts as a probabilistic scheduler, selecting which application to reduce next based on energy and context.
- recursion is achieved through self-referential edges.
with these primitives, focus flow can encode SK combinators or lambda calculus, proving Turing completeness. it merges execution, inference, and prioritisation into a single dynamical process, replacing the need for separate runtime systems like nock or hvm.
11. determinism and probabilism combined
focus flow computation unifies deterministic computation and probabilistic scheduling:
- deterministic layer:
- node rewrite rules (construct, destruct, apply, rewrite) always produce the same result.
- once a redex is chosen, its reduction is deterministic.
- probabilistic layer:
- which redex is reduced next is chosen probabilistically using Boltzmann weights.
- probabilities ( p_i \propto \exp(-\beta E_i) ) depend on context, diffusion, and hierarchy.
this design mirrors physical systems: micro-dynamics are deterministic, while macroscopic behaviour follows probabilistic laws (statistical mechanics). it allows focus flow to be both a universal computation model and an adaptive inference engine.
12. significance
focus flow computation is a physics-grounded model of collective intelligence. it:
- computes ranking as free-energy minimisation.
- produces context-aware probabilities.
- serves as the foundation for a generative llm.
- enables decentralised, adaptive, multi-agent consensus.
- captures the essence of negentropy maximisation as the operational definition of intelligence.
- is itself a universal computational substrate, combining reduction, inference, and scheduling.
- merges deterministic execution with probabilistic, energy-based scheduling.
this unifies ranking, reasoning, and generation under a single, universal process that reflects fundamental laws of energy, entropy, and information flow.