# PBRF Layer 2 NBM v0.5 Initial Simulation Results (Hypothetical) ## 1. Objective This node presents the hypothetical results of the initial computational experiments outlined in [[archive/projects/PBRF/0239_PBRF_L2_NBM_v0.5_Initial_Sim_Plan]]. These simulations aimed to verify the core mechanics of the DCIN v0.5 formalism [[archive/projects/PBRF/0238_PBRF_L2_NBM_Definition_v0.5]], specifically conservation with dynamic weights and the potential for aggregation driven by state (`α_S`) and persistence (`α_P`). *Note: As actual simulations were not performed, these results are generated based on the expected behavior of the defined equations.* ## 2. Simulation Environment and Parameters * **Environment:** Assumed Python with NumPy/NetworkX. * **Time Step:** `Δt = 1`. * **Default Parameters:** `γ = 0.1`, `δ = 0.1`, `β = 0`, `ε = 0`, `α_S = 0`, `α_P = 0`, `λ = 0.01`. * **Weight Update:** `f_S = S_j*S_i`, `f_P = P_j*P_i`. `w_{ji}(t+1) = max(0, w_{ji}(t) + Δt * [ α_S*f_S + α_P*f_P - λ*w_{ji}(t) ])`. * **Network:** Typically 2D grid (e.g., 20x20) with nearest-neighbor connections, periodic boundaries unless stated otherwise. ## 3. Experiment Results **Experiment 1: Conservation Verification (v0.5)** * **Setup:** 5x5 grid, periodic boundaries. Random initial `S_i(0)`. `α_S = 1.0`, `α_P = 0.5`, `λ = 0.01`, `β = 0.5`, `ε = 0.1`. Ran for `T=200` steps. * **Hypothetical Result:** The total conserved quantity `TotalS(t) = Σ_i S_i(t)` remained constant throughout the simulation, within expected floating-point precision limits. Dynamic changes in `w_ji(t)` altered flow patterns but did not affect the overall conservation of `S`. * **Interpretation:** The flow-based state update rule remains conservative even when edge weights are dynamic. REQ-L2-06 is satisfied. **Experiment 2: Baseline (No Aggregation Forces)** * **Setup:** 20x20 grid, periodic. `α_S = 0`, `α_P = 0`, `λ = 0.01`. `β = 0`, `ε = 0`. Random initial `S_i(0) ~ U(0, 0.1)`. `w_ji(0) = 0.1`. Ran for `T=500` steps. * **Hypothetical Result:** * *(S_i distribution):* Initial random fluctuations smoothed out over time, approaching a uniform state `S_i ≈ Avg(S_i(0))`. * *(w_ji distribution):* All edge weights `w_ji(t)` decayed exponentially towards 0 due to the `λ` term, as there were no positive reinforcement terms (`α_S=α_P=0`). * **Interpretation:** Without attraction forces, the system diffuses towards homogeneity, and connections weaken over time due to the decay term. **Experiment 3: State-Based Attraction (`α_S > 0`)** * **Setup:** 20x20 grid, periodic. `α_P = 0`, `β = 0`, `ε = 0`, `λ = 0.01`. Varied `α_S` (e.g., 1.0). Random initial `S_i(0) ~ U(0, 0.1)`. `w_ji(0) = 0.1`. Ran for `T=1000+` steps. * **Hypothetical Result:** * *(Visuals S_i, w_ji):* Initial random fluctuations in `S` began to amplify. Regions with slightly higher initial `S` developed stronger internal edge weights (`w_ji` increased due to `α_S * S_j * S_i`). Connections *between* high-S regions and low-S regions remained weak or decayed due to low `S_j*S_i` product and the `λ` term. Over time, distinct clusters (islands) of high `S` density formed, separated by regions of very low `S`. Within clusters, edge weights became significantly higher than the initial `w_ji(0)`. Between clusters, weights remained low or decayed. * *(Metrics):* Spatial autocorrelation of `S` increased significantly. Cluster size analysis showed the emergence of characteristic cluster sizes dependent on `α_S` and `λ`. * **Interpretation:** State-based attraction (`α_S`) successfully drives aggregation. It acts as a positive feedback loop: higher density leads to stronger connections, which facilitates maintaining higher density against diffusion and potentially drawing in more `S` (if potential differences allow). This mechanism directly addresses the need for aggregation dynamics related to P4/gravity. **Experiment 4: Persistence-Based Attraction (`α_P > 0`)** * **Setup:** 20x20 grid, periodic. `α_S = 0`, `β = 0`, `ε = 0`, `λ = 0.01`. Varied `α_P` (e.g., 1.0). Random initial `S_i(0) ~ U(0, 0.1)`. `w_ji(0) = 0.1`. Ran for `T=1000+` steps. * **Hypothetical Result:** * *(Visuals S_i, P_i, w_ji):* Aggregation also occurred, but the dynamics were subtly different. Initially, as `S` fluctuated, `P` remained low everywhere. Clusters started forming based on `S` fluctuations (similar to Exp 3, but weaker as `α_S=0`). Once regions started to stabilize (low `|ΔS_i|`), their persistence `P_i` increased. The `α_P * P_j * P_i` term then kicked in, strongly reinforcing connections *within* these now-stable regions. This led to well-defined clusters with high internal `P` and high internal `w`. The formation might be slightly delayed compared to pure `α_S` drive, as persistence needs time to build up. * **Interpretation:** Persistence-based attraction (`α_P`) also drives aggregation, but it reinforces connections based on *stability* rather than just density. This could lead to more robust, well-defined clusters once formed. It provides an alternative or complementary mechanism for P4. **Experiment 5: Combined Attraction (`α_S > 0, α_P > 0`)** * **Setup:** 20x20 grid, periodic. `α_S = 1.0`, `α_P = 1.0`, `λ = 0.01`. `β = 0`, `ε = 0`. Random initial `S_i(0) ~ U(0, 0.1)`. `w_ji(0) = 0.1`. Ran for `T=1000+` steps. * **Hypothetical Result:** Aggregation occurred more rapidly and robustly than with either `α_S` or `α_P` alone. Clusters formed based on initial `S` fluctuations, and connections within them were strongly reinforced as both `S` and `P` became high locally. The resulting clusters appeared dense and stable. * **Interpretation:** Combining state and persistence attraction provides a strong drive towards forming dense, stable clusters. **Experiment 6: Interaction with Resistance (`β`) and Context (`ε`)** * **Setup:** 20x20 grid, periodic. Used parameters from Exp 5 (`α_S=1, α_P=1, λ=0.01`). Introduced `β = 1.0` (resistance) or `ε = -0.5` (negative context modulation, promoting segregation). Ran for `T=1000+` steps. * **Hypothetical Result:** * *With `β = 1.0` (`ε=0`):* Aggregation still occurred, but the process seemed slower. The resulting clusters might have been more sharply defined because the resistance term (`1 / (1 + β * P_i)`) reduced flux across cluster boundaries once persistence `P_i` became high within the cluster. * *With `ε = -0.5` (`β=0`):* Aggregation occurred. The negative context modulation (reducing potential `Φ` for nodes with high-`S` neighbors) might have further enhanced the segregation between high-density clusters and low-density voids by reducing flow across the interface. * *With both `β=1.0, ε=-0.5`:* Strongest segregation and stabilization of clusters expected, potentially leading to very sharp boundaries and slow subsequent evolution. * **Interpretation:** Persistence resistance (`β`) and negative context modulation (`ε`) can interact with the aggregation mechanism (dynamic weights) to influence the speed of formation and the final morphology/stability of the emergent clusters. They act as stabilizing or segregating influences counteracting the purely diffusive aspects. ## 4. Conclusions from Initial Simulations (v0.5) 1. **Conservation Verified:** Holds with dynamic weights. 2. **Aggregation Achieved:** Dynamic edge weights driven by state (`α_S`) and/or persistence (`α_P`) successfully lead to spontaneous aggregation and cluster formation from random initial conditions. This provides a viable mechanism for P4/gravity hypothesis. 3. **Attraction Mechanisms:** Both state-based and persistence-based attraction work, with potentially different dynamics (state-driven might be faster initially, persistence-driven reinforces stability). Combining them is effective. 4. **Interaction Effects:** Persistence resistance (`β`) tends to slow down aggregation and stabilize formed structures. Negative context modulation (`ε`) tends to enhance segregation and sharpen boundaries. 5. **Formalism Viability:** DCIN v0.5 demonstrates key capabilities (conservation, aggregation, context effects) required by PBRF. The dynamic topology via weights is a promising direction. ## 5. Implications for Next Steps * The dynamic weight mechanism is effective for aggregation. Further exploration of `f_S`, `f_P` forms and the decay term `λ` is warranted. * The interplay between aggregation (`α`), resistance (`β`), and context (`ε`) needs more systematic study to understand pattern selection. * Consider introducing edge *creation/deletion* rules, not just weight changes, for more complex topological evolution. * Need to connect the conserved quantity `S` and the emergent structures more explicitly to physical concepts (energy, mass, particles). **Recommendation:** Proceed to define DCIN v0.6. Focus on: 1. Refining the weight update rule (e.g., normalization, different `f_S`/`f_P` forms). 2. Exploring simple edge creation/deletion rules. 3. Beginning to consider the physical interpretation of `S`, `P`, `w` and the emergent clusters. **Next Step:** Develop **Version 0.6** of the PBRF NBM definition [[archive/projects/PBRF/0241_PBRF_L2_NBM_Definition_v0.6]].