theoretical foundations

• 0. introduction and scope
• 1. primitives and notation
• 2. graph and stochastic model
• 3. theorem (collective focus theorem)
• 4. sensitivity and robustness
• 5. ranking primitives (minimal basis)
• 6. free-energy and focus-flow
• 7. incentive compatibility and governance
• 8. emergence thresholds and phase transitions
• 9. theory stack (connected theories)
• 10. system architecture
• 11. learning dynamics
• 12. scalability regimes
• 13. limitations and open questions
• 14. glossary
• appendix a. core equations
• appendix b. protocol sketch
• appendix c. implementation breadcrumbs

0. introduction and scope this document specifies the mathematical and systems foundations of the collective focus theorem (cft). the theorem states that a token-weighted random walk on an authenticated, strongly connected, aperiodic directed graph converges to a unique stationary distribution π, interpreted as society‑wide long‑run focus over particles (knowledge items).

1. primitives and notation

• particle: a content‑addressed node (e.g., an ipfs hash) representing a file or concept.
• neuron: an agent (public key) that signs edges between particles.
• cyberlink: a signed, timestamped, weighted directed edge i→j with weight w_ij ≥ 0.
• token and stake: non‑negative weight t_j associated with the destination j (or its controlling neuron).
• attention vs focus: attention is fast, local reweighting; focus is the slow, emergent equilibrium π over particles.

2. graph and stochastic model

• transition rule (token‑weighted walk): p_ij = (w_ij · t_j) / Σ_k (w_ik · t_k)
• conditions for ergodicity: the graph is strongly connected and aperiodic (or made so via standard teleportation).
• existence and uniqueness: there exists a unique stationary distribution π with πp = π and Σ_j π_j = 1. for any initial μ^(0), μ^(t) = μ^(0)p^t → π as t→∞.

3. theorem (collective focus theorem)

• statement: under the above conditions, π is the unique collective focus of the system; it is the limiting observation frequency of the token‑weighted random walk.
• interpretation: π_j measures the share of collective long‑run attention allocated to particle j. shifts in topology or stake alter π continuously within a stability margin.
• proof sketch: apply standard ergodic markov‑chain results; show irreducibility and aperiodicity; invoke perron–frobenius for the positive left eigenvector; normalise to obtain π.

4. sensitivity and robustness

• mixing time is controlled by the spectral gap of p. larger gap ⇒ faster convergence and greater robustness to noise.
• small perturbations Δw, Δt induce bounded changes Δπ (continuous dependence). practical reading: equilibrium focus is stable under graph churn.

5. ranking primitives (minimal basis)

• eigenvector centrality (diffusion): baseline global popularity at equilibrium.
• springrank (springs): convex energy model that yields an ordinal hierarchy from pairwise relations.
• heat‑kernel pagerank (heat flow): locality dial indexed by time t, interpolating local↔global views.
• composition: combine these signals into a context‑aware focus vector; use weights learned from downstream performance.

6. free‑energy and focus‑flow

• free‑energy functional: f(p | context) = e_spring + λ e_diffusion + γ c(context) − τ s(p) where s(p) is entropy and τ is temperature.
• focus‑flow iteration: a decentralised message‑passing update that descends the free‑energy surface toward p*; suitable for online adaptation when the graph changes.

7. incentive compatibility and governance

• influence ∝ stake and connectivity, providing skin‑in‑the‑game for quality linking.
• misbehaviour can be penalised economically (slashing, opportunity cost) and socially (reputation).
• anti‑capture measures include stake dispersion, rate limits, decay, and context‑specific caps.

8. emergence thresholds and phase transitions

• connectivity thresholds (average out‑degree, conductance) gate the onset of coherent global focus.
• token mixing and participation rates also act as control parameters; crossing critical values can yield sharp improvements in collective cognition.

9. theory stack (connected theories)

• markov chains and ergodic theory — guarantee existence/uniqueness and mixing; operational hook: keep the chain irreducible and aperiodic, monitor spectral gap.
• spectral graph theory — relate conductance/cheeger constants to mixing; hook: maximise conductance under cost.
• random walks and eigenvector centrality/pagerank — compute importance from local edges; hook: gpu power iterations for π.
• heat kernels and diffusion geometry — provide locality control; hook: heat‑kernel pagerank for zoomable focus.
• spring/electrical network models — extract hierarchies; hook: convex optimisation on graph laplacians.
• information theory and maximum entropy — justify free‑energy objectives; hook: tune temperature τ for exploration.
• variational inference and the free‑energy principle — cast focus as a variational posterior; hook: minimise kl divergences.
• stochastic approximation and reinforcement learning — adapt edge weights with regret guarantees; hook: hebbian‑style local updates plus exploration schedules.
• game theory and mechanism design — align incentives with epistemic accuracy; hook: stake‑weighted signals, proper scoring, peer prediction.
• market microstructure and prediction markets — treat focus as a price of attention; hook: cost‑function market makers for forecasting subgraphs.
• social choice and voting theory — explain limits of ballot aggregation and motivate probabilistic attention; hook: continuous, cardinal signals over ordinal ballots.
• cybernetics and control — feedback and homeostasis (ashby’s law); hook: error signals, integral control in focus‑flow.
• distributed consensus and state machine replication — ensure authenticated state under byzantine faults; hook: cometbft/tendermint and verifiable focus commits.
• cryptography (signatures, vrf, zkp, mpc) — integrity, randomness, privacy; hook: signature‑secured links, vrf sampling, optional zkp proofs of computation.
• identity and reputation — mitigate sybils; hook: blended stake, web‑of‑trust, proof‑of‑personhood with decay.
• percolation and phase transitions — predict critical connectivity; hook: push degree and clustering past thresholds.
• evolutionary dynamics — model selection among ideas/agents; hook: replicate/retire edges proportional to payoff and focus.
• robust statistics and adversarial learning — bound influence of heavy‑tailed noise; hook: median‑of‑means, trimming, influence functions.
• safety, alignment, governance — embed safeguards without centralisation; hook: rate limits, circuit breakers, transparency logs, citizen juries.
• complexity and scaling — guide multiscale decomposition; hook: hierarchical partitioning with inter‑level diffusion.
• semantics and information retrieval — map text/code/media into particle space; hook: hybrid lexical+embedding edges with learned mixtures.
• knowledge graphs and logic — add typed relations and constraints; hook: datalog/owl‑driven edge weighting.
• economics of attention and rational inattention — model cognitive budgets; hook: bound per‑agent outlinks and price scarce attention.
• queueing and load management — stabilise compute/memory; hook: admission control and fair schedulers for recomputation.
• causal inference — separate genuine signal from confounding; hook: off‑policy evaluation and link‑level intervention tests.

11. learning dynamics

• macro‑state evolution: s^(t+1) = f(s^(t), w^(t), t^(t)).
• local rules: hebbian reinforcement for successful links, exploration policies for novelty, decay for staleness.
• global re‑equilibration: π is recomputed (or tracked incrementally) after deltas to w and t.

13. limitations and open questions

• formal mixing‑time bounds for token‑weighted chains with dynamic weights.
• perturbation lemmas giving ||Δπ|| bounds under bounded ||Δw|| and ||Δt||.
• incentive proofs that long‑run stake tracks epistemic accuracy.
• transparency, interpretability, and earth‑aligned values at planetary scale.

14. glossary

• dkg: decentralised knowledge graph (abstract model).
• cybergraph: a merkle‑ised, token‑weighted dkg (on‑chain).
• particle, neuron, cyberlink: node, agent, signed edge primitives.
• attention vs focus: fast local weights vs slow global equilibrium π.
• truth vm: gpu virtual machine for convergent focus computation.

appendix a. core equations (at a glance)

• transition: p_ij = (w_ij · t_j) / Σ_k (w_ik · t_k).
• stationary: π = πp, Σ_j π_j = 1.
• convergence: μ^(t) = μ^(0) p^t → π.
• qualitative stability: small Δw, Δt ⇒ small Δπ.

appendix b. protocol sketch (ranking and composition)

• compute eigenvector centrality as baseline focus.
• compute springrank for hierarchy.
• compute heat‑kernel pagerank at several t for locality.
• combine signals into a context‑aware focus vector; use free‑energy or focus‑flow to reconcile under load.

appendix c. implementation breadcrumbs

• training stack: cometbft + truth vm + cosmwasm around a merkle cybergraph.
• inference stack: ipfs + cyb.ai + cozo/datalog/rune.
• current priority: raise connectivity and participation to cross predicted emergence thresholds; use link‑economy incentives and UX to grow edges per particle.