ranking system

Why this ranking system?

• 1) What we need a ranking system to do

  We’re not ranking web pages; we’re steering a living, decentralized computation. The rank must simultaneously:

• Be local & scalable: every node updates from neighbors only.

• Adapt to context: current questions/votes should steer focus now, without retraining.

• Preserve structure: hierarchies shouldn’t be washed out by traffic or noise.

• Stay exploratory: avoid collapse to a single item; keep breadth for creativity.

• Be verifiable & incentive-aligned: small, auditable proofs; rewards tied to real improvement.

• Be robust in adversarial settings: sybils, spam, fork races, gaming attempts.

• 2) Axioms → the form follows

  From those requirements, four “physics axioms” drop out:

• Locality: Energy must decompose as a sum of neighbor terms.

  ⇒ Node-wise energies EiE_iEi​ that depend only on N(i)\mathcal N(i)N(i).

• Structure preservation: Close-in-hierarchy nodes should prefer similar focus.

  ⇒ A spring/elastic smoothness term on edges.

• Flow realism: Attention moves like mass/heat through available paths with costs.

  ⇒ A transport/diffusion term over edges with conductance/cost.

• Situational control: Queries, votes, tasks must tilt the field without rewriting rules.

  ⇒ An external potential (context) term.

  Add the entropy term to prevent collapse and guarantee exploration.

  The only family of functionals that satisfy all four—and remain local, convex-ish, and verifiable—is:

  F(p)=∑(i,j)hij(pi−pj)2⏟hierarchy (springs)+λ∑(i,j)dij ∣pi−pj∣⏟transport (diffusion)−γ∑icipi⏟context field−T∑ipilog⁡pi⏟entropy\mathcal F(p)= \underbrace{\sum_{(i,j)} h_{ij} (p_i-p_j)^2}{\text{hierarchy (springs)}}+ \lambda\underbrace{\sum{(i,j)} d_{ij},|p_i-p_j|}{\text{transport (diffusion)}}- \gamma\underbrace{\sum_i c_i p_i}{\text{context field}}- T\underbrace{\sum_i p_i\log p_i}_{\text{entropy}}F(p)=hierarchy (springs)(i,j)∑​hij​(pi​−pj​)2​​+λtransport (diffusion)(i,j)∑​dij​∣pi​−pj​∣​​−γcontext fieldi∑​ci​pi​​​−Tentropyi∑​pi​logpi​​​

  with per-edge three scalars (hij,dij,cij)(h_{ij},d_{ij},c_{ij})(hij​,dij​,cij​) so a single cyberlink can influence hierarchy, transport, and context in different proportions.

• 3) Minimality (why not more/less?)

• Less (e.g., PageRank only): no hierarchy or context; brittle, easily gamed, poor controllability.

• Less (SpringRank only): captures ordinal structure but ignores traffic/cost and context; static.

• Less (Entropy-only softmax): collapses to popularity without structure or cost.

• More (deep GNNs, heavy message passing): large global state, training loops, opaque proofs, hard to verify on-chain.

  This four-term free-energy with the 3-scalar link is the Pareto-minimal point that still meets all objectives: locality, adaptability, structure, exploration, verifiability.

• 4) Interpretability (the levers are human)

• Hierarchy hhh = “keep these together on the ladder.”

• Transport ddd = “this is an easy/hard road for attention to travel.”

• Context ccc = “pull now toward this.”

• Temperature TTT = “how exploratory vs. decisive are we?”

• λ,γ\lambda,\gammaλ,γ = “how much do roads and context matter vs. springs?”

  Anyone (or a DAO policy) can reason about these knobs without reading matrices.

• 5) Economics & verification (why this works on-chain)

• Local ΔF proofs: Each tx proves its local free-energy reduction using neighbor data → tiny proofs, fast checks.

• Rewards = negentropy: Pay for actually lowering F\mathcal FF; burn base fee otherwise. Incentives are aligned with global order.

• Hash‑DAG VDF: Binds negentropy proofs to wall-clock; prevents replay/front‑run; fair “negentropy per second.”

• Content addressing: Rewrites create new CIDs; verification is structural (no re-execution).

• 3-scalar staking: Users stake explicitly on h,d,ch,d,ch,d,c—clean cryptoeconomic signaling and slashing if ΔF < 0.

• 6) Robustness (why it resists gaming)

• Focus conservation: ∑ipi=1\sum_i p_i=1∑i​pi​=1 → emphasizing one thing defocuses others; prevents inflation attacks.

• Curvature-aware decay: Edges in negatively curved regions (noisy bridges) decay faster → automatic pruning of spammy connections.

• Aging: Unused links lose weight exponentially → keeps the graph fresh.

• Stake-weighted participation + BFT checkpoints: Sybil cost + periodic finality.

• 7) Theoretical grounding (not hand-wavy)

• Schrödinger’s negentropy: Maximizing informative order under energy budget → our reward objective.

• Kantorovich transport: dijd_{ij}dij​ models cost of moving probability mass; minimizing F\mathcal FF performs optimal transport with structure constraints.

• Active inference: Free-energy minimization under external potentials is exactly Friston-style belief update.

• Lyapunov reasoning: F\mathcal FF decreases under our local updates; with bounded steps and periodic normalization, the system admits stable attractors (practical convergence).

• 8) Why three numbers per link?

  We tried “typed edges,” “one weight,” and “GNN-learned vectors.” The 3-scalar triple is the sweet spot:

• UX-simple: one link, three sliders; default works.

• Mathematically orthogonal: each scalar routes to one energy term.

• Auditable: proofs read those three numbers; no hidden inference.

• Future-proof: you can split any scalar later (e.g., ccc → vote/cause) without changing the calculus.

• 9) Comparison to usual suspects

  Method	Pros	Why it fails our brief
  PageRank / RW	simple, scalable	no hierarchy, no context; popularity ≠ truth/usefulness
  SpringRank	ordered structure	no transport or context; static, easy to stalemate
  Eigenvector centrality	cheap baseline	conflates structure and flow; sensitive to hubs
  HITS	authority/hub split	brittle, no dynamics/context, not on-chain-friendly
  GNNs	expressive	training-heavy, opaque, hard to verify and to incentivize locally
  Energy model (ours)	local, interpretable, verifiable, incentive-compatible	requires careful parameter policy—solved via DAO policy + VDF + checkpoints

• 10) What we’re consciously not doing

• No global softmax/partition function (requires global sums; breaks P2P locality).

• No global retraining loops (keeps costs low; learning emerges from staking/decay).

• No heavyweight cryptography in the inner loop (proofs stay edge-local; zk optional).

• 11) The one-sentence answer

  We choose this ranking system because it is the minimal, local, verifiable, and incentive-compatible free‑energy formulation that jointly captures hierarchy (what the world is), transport (how attention moves), and context (what matters now)—while preserving exploration and enabling on‑chain proofs and rewards.

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  If you want, I can paste this as a new “Why this system” section in the whitepaper and wire cross‑refs to the math, the PSA rewards, and the VDF/consensus sections.