ProofMesh
Network Propagation · Visualization Layer Math
ProofMesh is the living graph of proof interconnections — how signal propagates between nodes through interactions. It defines the network dynamics that make SignalNetwork more than a collection of isolated scores.
1. Resonance (Node Alignment)
$$\rho(u, v, t) = \max\!\left(0, \frac{\vec{s}_u(t) \cdot \vec{s}_v(t)}{\|\vec{s}_u(t)\| \; \|\vec{s}_v(t)\|}\right)$$
Clamped cosine similarity. Negative alignment (opposing signals) produces zero resonance — no destructive transfer.
2. Context-Fit Factor
Prevents off-topic amplification. The edge context embedding $\vec{c}_e$ must align with the sender's signal.
$$\kappa(e, t) = \max\!\left(0, \frac{\vec{s}_u(t) \cdot \vec{c}_e}{\|\vec{s}_u(t)\| \; \|\vec{c}_e\|}\right)$$
3. Edge-Type Weights
$$\omega(e) \in \{\omega_\text{cite}, \omega_\text{collab}, \omega_\text{witness}, \omega_\text{endorse}, \omega_\text{relay}\}$$
Default values: $\omega_\text{cite} = 0.3$ · $\omega_\text{collab} = 0.8$ · $\omega_\text{witness} = 1.0$ · $\omega_\text{endorse} = 0.6$ · $\omega_\text{relay} = 0.2$
4. One-Hop Transfer Law (Canonical) Core
For a single interaction event $e = (u \to v)$ at time $t$:
$$\Delta\vec{s}_v(t) = \eta \cdot \rho(u,v,t) \cdot \tau(u,t) \cdot \pi(e,t) \cdot \kappa(e,t) \cdot \omega(e) \cdot \vec{s}_u(t)$$
$\eta \in (0, 1)$ — global transfer coefficient
$\rho$ — resonance · $\tau$ — sender trust · $\pi$ — edge proof-weight
$\kappa$ — context-fit · $\omega$ — edge-type weight · $\vec{s}_u$ — sender signal vector
5. Full Mesh Dynamics (Inflow + Decay + Self-Production)
$$\vec{s}_v(t + \Delta t) = (1 - \lambda\Delta t)\vec{s}_v(t) + \sum_{e \in E_{\to v}(t)} \Delta\vec{s}_v^{(e)}(t) + \vec{g}_v(t)$$
Self-Generated Signal
$$\vec{g}_v(t) = \sum_{a \in A_v(t)} q(a) \cdot \vec{p}(a)$$
$\vec{p}(a)$ — proof vector contribution (maps artifact type to dimension impacts)
Dormancy Decay
$$\vec{s}_v(t) = \vec{s}_v(0) \cdot e^{-\lambda t}$$
6. Mesh Centrality (Network Importance)
Weighted Adjacency Matrix
$$A_{uv}(t) = \sum_{e: u \to v} \rho(u,v,t) \cdot \tau(u,t) \cdot \pi(e,t) \cdot \kappa(e,t) \cdot \omega(e)$$
Eigenvector Centrality
$$M_u(t) \propto \sum_v A_{vu}(t) \cdot M_v(t)$$
Computed via power iteration (20 iterations default). Network hubs emerge naturally.
7. Echo / Amplification (Relay Protocol)
$$\Delta S_v = \eta \cdot \rho(u,v) \cdot \tau(u) \cdot \pi(e) \cdot S_u$$
Echo Delay (Anti-Spam)
$$\eta(t) = \eta_0 \cdot \mathbb{1}[t - t_\text{last}(u \to v) \geq \Delta_\text{min}]$$
$\Delta_\text{min}$ — minimum time between relay events on the same edge (prevents spam amplification)
8. Anti-Mimic Detection Core
Mimicry = high similarity without proof. Detects nodes copying style without generating original work.
Style Similarity
$$\text{Sim}(u, v) = \frac{\vec{z}_u \cdot \vec{z}_v}{\|\vec{z}_u\| \|\vec{z}_v\|}$$
$\vec{z}_u$ — style/content embedding for node $u$
Mimic Risk Score
$$\mathcal{M}_u = \max_{v \neq u} \frac{\text{Sim}(u, v)}{P_u + \epsilon}$$
If $\mathcal{M}_u > \theta_M$ → quarantine / review.
New-Node Dampening Core
New nodes with low proof scores get artificially inflated mimic risk. Dampening prevents false positives.
$$\mathcal{M}'_u = \mathcal{M}_u \cdot \sigma(P_u - P_\text{threshold})$$
$P_\text{threshold}$ — minimum proof level before mimic detection activates meaningfully
Sigmoid dampens mimic risk for low-proof nodes, ramps up as proof accumulates
9. Edge Pruning / Decay V2
Edges representing outdated or disputed connections should decay, preventing the adjacency matrix from growing unbounded.
$$A_{uv}(t) = A_{uv}(t_0) \cdot e^{-\lambda_e(t - t_0)} \cdot (1 - D_{uv}(t))$$
$\lambda_e$ — edge decay rate
$D_{uv}(t) \in \{0, 1\}$ — dispute flag (1 if edge is disputed, 0 otherwise)
Edges below threshold $A_{uv} < \epsilon_e$ are pruned from active computation.
10. Sybil Resistance V2
Sybil attacks create many low-trust nodes to amplify signal. Resistance is enforced by proof-gating influence.
Influence Cap
$$\text{Influence}_u(t) = \min\!\left(\sum_{v: u \to v} \Delta\vec{s}_v, \; \Gamma \cdot P_u(t)\right)$$
$\Gamma$ — influence-to-proof ratio cap
A node cannot propagate more signal than its proof output justifies. No proof → no influence.
Connectivity Anomaly Detection
$$\text{Sybil}_u = \frac{|\{v : A_{uv} > 0\}|}{P_u + \epsilon}$$
High connectivity relative to proof output flags potential Sybil behavior.
If $\text{Sybil}_u > \theta_S$ → review.
11. Global vs Local Signal V2
Signal propagation scope can be tagged to distinguish network-wide events from community-local interactions.
$$\Delta\vec{s}_v^{(\text{scoped})} = \Delta\vec{s}_v \cdot \begin{cases} 1 & \text{if } e \text{ tagged global} \\ \rho_\text{community}(u, v) & \text{if } e \text{ tagged local} \end{cases}$$
$\rho_\text{community}$ — community-level resonance (subgraph alignment)
12. Practical Defaults
$\eta = 0.02$ · $\lambda = 0.001$/hr · $\theta_M = 0.85$ (mimic threshold) · $\Delta_\text{min} = 1$ hour · $\Gamma = 5.0$ (influence cap) · $\theta_S = 20$ (Sybil threshold) · $\lambda_e = 0.0005$/hr (edge decay) · $\epsilon_e = 0.01$ (edge prune threshold)