# IO v4.0 Formalism: Specific GA Proposals for Dynamics and Transition ## 1. Objective Building on the operationalization of IO principles in GA [[0152_IO_GA_Principles_Op2]] and the candidate evolution equation structure [[releases/archive/Information Ontology 3/0156_IO_GA_Derivation]], this node proposes **specific mathematical forms** using Geometric Algebra ($\mathcal{G}(1,3)$) for the key dynamic terms and the κ → ε transition rule. These proposals aim to provide a concrete starting point for implementation and testing [[releases/archive/Information Ontology 3/0157_IO_GA_Feasibility]]. *(Disclaimer: These are initial proposals, likely requiring significant refinement based on theoretical analysis and simulation results.)* ## 2. State Representation * `Ψ(i, t)`: Multivector field in $\mathcal{G}(1,3)$. For simplicity initially, assume real components. `Ψ = s + v + B + P` where `s`=scalar, `v`=vector, `B`=bivector, `P`=pseudoscalar. * `Θ_val(i, t)`: Scalar stability value. * `w(j → i, t)`: Scalar causal weights. ## 3. Proposed GA Forms for Dynamic Terms in `∂_t Ψ(i)` Recall the structure: `∂_t Ψ(i) = F_intrinsic + F_Η + F_Θ + F_Interaction` **3.1. Intrinsic Dynamics (`F_intrinsic`)** * **Goal:** Represent mass, self-interaction. Needs to be Lorentz covariant if spacetime is Minkowskian. * **Proposal:** Inspired by Dirac-Hestenes equation (spin-1/2) and non-linear terms. * `F_intrinsic = -Ψ(i) B_int i c / ħ_I - (m_0 c^2 / ħ_I) Ψ(i) γ_0 - (λ / ħ_I) Ψ(i) ⟨Ψ(i) \tilde{Ψ}(i)⟩₀` * `B_int`: An internal constant bivector (related to Zitterbewegung/spin?). * `i`: Pseudoscalar $I = \gamma_0\gamma_1\gamma_2\gamma_3$. * `m_0`: Scalar mass parameter. * `λ`: Non-linear self-interaction strength. * `ħ_I`: IO fundamental constant (action units). * *(Justification: Combines linear mass term and non-linear term common in soliton-forming equations, expressed covariantly using GA products).* **3.2. Entropy (`F_Η`)** * **Goal:** Stochastic drive, suppressed by stability. * **Proposal:** * `F_Η = σ(Θ_val(i)) * η(i, t)` * `σ(Θ_val) = σ₀ / (1 + α_H * Θ_val(i, t))` (Stability suppresses noise amplitude). * `η(i, t)`: Multivector white noise (each of the 16 components gets independent Gaussian noise `N(0,1)` scaled by `dt^(-1/2)` if using discrete steps). **3.3. Theta (`F_Θ`)** * **Goal:** Resist change based on `Θ_val`. * **Proposal:** Implement Θ primarily by modulating Η (as above) and via the `dΘ_val/dt` equation. Avoid adding strong damping directly to `∂_t Ψ` to prevent trivial freezing. * `F_Θ ≈ 0` (Direct damping term removed for now, effect is via `σ(Θ_val)`). * `dΘ_val/dt = a - b * ||∂_t Ψ(i, t)|| - c * Θ_val` (Using a suitable GA norm `||...||`, e.g., scalar part `|⟨(∂_t Ψ) \widetilde{(∂_t Ψ)}⟩₀|^(1/2)`). **3.4. Interaction (`F_Interaction`)** * **Structure:** `Σ_{j | (j→i)∈E} w(j→i) * f_K(K(i,j)) * [ F_{int}(Ψ(i), Ψ(j)) + F_{M}(Ψ(i), Ψ(j)) ]` * **Contrast K:** [[0154_IO_GA_Contrast_Definition]] * **Proposal:** Use grade-specific differences. Start simple: * `K(i, j) = α_0 |⟨Ψ(i)⟩₀ - ⟨Ψ(j)⟩₀| + α_2 ||⟨Ψ(i)⟩₂ - ⟨Ψ(j)⟩₂||` (Scalar and Bivector contrast). Requires norm `||...||` for bivectors. * **Gating `f_K`:** Sigmoid function: `f_K(K) = 1 / (1 + exp(-k_slope * (K - K_min)))`. * **Core Interaction `F_int`:** Represents basic influence/exchange. * **Proposal:** Diffusion-like term based on difference: `g * (Ψ(j) - Ψ(i))`. (Strength `g`). * **Mimicry `F_M`:** Alignment term. * **Proposal:** Pull towards *difference* from neighbor, weighted by neighbor's state (more complex than simple average): `p_M * (Ψ(j) - Ψ(i)) Ψ(j)`? Or perhaps projection-based: `p_M * (Project_{Ψ(j)}(Ψ(i)) - Ψ(i))`? Needs more thought. Let's start with the simple diffusion `F_int` only and set `F_M = 0` initially, adding M later if needed. **3.5. Causal Weight Update (`dw/dt`)** * **Goal:** Reinforce links based on correlation and stability [[0118_IO_Formalism_Refinement]]. * **Proposal:** * `dw(j→i)/dt = Δw_base * Corr(Ψ(j,t), Ψ(i,t+dt)) * f_Θ_corr(Θ_val(i,t+dt)) - decay_rate * w` * **Correlation `Corr`:** How to define correlation between multivectors? * *Option:* Use scalar part of relative state: `Corr = ⟨Ψ̂(j, t) Ψ̂(i, t+dt)⟩₀` (where `Ψ̂` are normalized). Positive if aligned, negative if anti-aligned. * **Stability Weighting `f_Θ_corr`:** `f_Θ_corr = (1 + Θ_val(i, t+dt) / theta_max)`. ## 4. Proposed κ → ε Transition Rule * **Trigger:** Interaction `j → i` occurs if `K(i, j) > K_min` (or probabilistically via `f_K`). An Η fluctuation provides the attempt probability `P_attempt`, modulated by `Θ_val(i)`. `P_actual_attempt = P_attempt * f_K * f_Θ`. * **Context & Projectors `{P_k}`:** The state `Ψ(j)` of the interacting neighbor (or probe) defines the basis/projectors `{P_k}` relevant to the interaction [[releases/archive/Information Ontology 3/0155_IO_GA_Actualization]]. *(Mechanism for deriving P_k from Ψ(j) and interaction type is still needed - major challenge)*. * **Probability `Prob(k)`:** * **Proposal:** `Prob(k) = ⟨(P_k Ψ(i)) \widetilde{(P_k Ψ(i))}⟩₀ / Σ_l ⟨(P_l Ψ(i)) \widetilde{(P_l Ψ(i))}⟩₀`. (Normalized magnitude squared of the projection). * **Update:** Sample outcome `k` based on `Prob(k)`. Update state: `Ψ(i) → Ψ_ε = P_k Ψ(i) / ||P_k Ψ(i)||`. Reset `Θ_val(i)`. Update `w` based on correlation with outcome `k`. ## 5. Synthesized Dynamics (Conceptual Summary) 1. `Ψ(i)` evolves continuously via the GA differential equation incorporating intrinsic dynamics, Η noise (modulated by Θ), and K-gated interactions (diffusion `F_int`, potentially Mimicry `F_M`) weighted by dynamic CA links `w`. 2. `Θ_val(i)` evolves based on the rate of change `||∂_t Ψ(i)||`. 3. `w(j → i)` evolves based on stability-weighted correlation between `Ψ(j)` and the subsequent `Ψ(i)`. 4. Probabilistic κ → ε transitions (projections `P_k`) can occur, triggered by interactions meeting K/Η/Θ criteria, interrupting the continuous evolution and actualizing the state `Ψ` into `Ψ_ε`. ## 6. Conclusion: A Concrete (but Complex) Starting Point This node provides specific, albeit preliminary and complex, proposals for the mathematical forms of the IO principles within the GA network formalism (v4.0). It defines candidate terms for the evolution equation of `Ψ`, the dynamics of `Θ_val` and `w`, and the probabilistic κ → ε transition rule. Key challenges remain, particularly in rigorously deriving the interaction terms, the projection operators `P_k` from context, and the Born