Editing Openai/693b7cce-38f4-800d-92f7-5e56a467bfa7 (section)

=== Single knowledge item (fact) indexed by kkk. We'll omit kkk where obvious. ===
* NNN — total number of agents (NPCs).
* i,j∈{1..N}i,j\in\{1..N\}i,j∈{1..N} — agent indices.
* bi(t)∈[0,1]b_i(t)\in[0,1]bi(t)∈[0,1] — agent iii’s belief strength that fact is true at time ttt. (1 = fully believes true; 0 = fully believes false.)
* ci→j(t)∈[0,1]c_{i\to j}(t)\in[0,1]ci→j(t)∈[0,1] — agent iii’s belief about agent jjj’s belief (meta-belief). We use these to compute recursion / common knowledge.
* Aij∈{0,1}A_{ij}\in\{0,1\}Aij∈{0,1} — adjacency (edge) presence in social graph (1 if a social tie exists). Optionally weighted by frequency of contact.
* dij∈{1,…,6}∪{∞}d_{ij}\in\{1,\dots,6\}\cup\{\infty\}dij∈{1,…,6}∪{∞} — shortest social distance (degree) from iii to jjj. If no path within 6, set ∞\infty∞.
* Tij∈[0,1]T_{ij}\in[0,1]Tij∈[0,1] — trust weight agent iii places on messages from jjj.
* V∈[0,1]V\in[0,1]V∈[0,1] — base visibility of the knowledge item (publicity). (public ≈ 1, secret ≈ 0)
* α>0\alpha>0α>0 — degree attenuation exponent (higher → faster attenuation by degree).
* M∈[0,1]M\in[0,1]M∈[0,1] — mutation / telephone mutation rate (higher → more distortion per hop).
* λi≥0\lambda_i\ge 0λi≥0 — memory decay rate for agent iii.
* ηi(t)\eta_i(t)ηi(t) — external source input (e.g., official broadcast) influencing bib_ibi.
* Personality & emotion scalar traits for agent iii: - EiE_iEi extraversion ∈ [0,1] - AiA_iAi agreeableness ∈ [0,1] - CiC_iCi conscientiousness ∈ [0,1] - NiN_iNi neuroticism ∈ [0,1] - OiO_iOi openness ∈ [0,1] - ai(t)a_i(t)ai(t) arousal/intensity ∈ [0,1] (emotion state variable)
* Pij(t)∈[0,1]P_{ij}(t)\in[0,1]Pij(t)∈[0,1] — probability agent iii will share/transmit to jjj at time ttt. (function of personality, emotion, trust)
* μij(t)\mu_{ij}(t)μij(t) — additive mutation noise in message from jjj to iii (modeled as stochastic process). Typical distribution: μ∼N(0,σij2)\mu\sim \mathcal{N}(0,\sigma_{ij}^2)μ∼N(0,σij2).

Define attenuation function by degree:

Sij={Vdijαif dij≤60if dij>6S_{ij} = 
\begin{cases}
\displaystyle \frac{V}{d_{ij}^{\alpha}} & \text{if } d_{ij}\le 6\\[6pt]
0 & \text{if } d_{ij} > 6
\end{cases}Sij=⎩⎨⎧dijαV0if dij≤6if dij>6
(For direct neighbors dij=1d_{ij}=1dij=1, Sij=VS_{ij}=VSij=V.)