Primitive Temporal Flow

Primitive Temporal Flow

By John Gavel

Discrete Temporal Progression: $T = \{ t \in \mathbb{N} \mid t \ge 0 \}$. For all $t \in T$, there exists a unique $t+1 \in T$. No $\tau \notin T$ satisfies $t < \tau < t+1$.

Primitive Asymmetry (Binary Difference): Existence manifests as binary-signed asymmetry $F \in \{ +, - \}$. This asymmetry is not between pre-existing entities — it is a self-referential primitive fact. Binary character is irreducible; it is the minimal structure capable of supporting self-differentiating propagation.

Primitive Substrate

Let $S = \{ s_i \}$ be a discrete substrate. Each element $s_i$ carries a state variable $X_i$. Each element has a neighborhood $N(i)$ defined relationally.

Minimal Comparative Structure

Define a minimal comparative triple $(s_0, s_1, s_2)$ where $s_0$ is the reference ("self") and $s_1, s_2 \in N(s_0)$ are relational neighbors.

Informational Weight Distribution: $$ w_{\text{self}} = \frac{1}{|N(s_0)| + 1}, \quad w_{\text{rel}} = \frac{|N(s_0)|}{|N(s_0)| + 1} $$ For a 1D lattice, $|N(s_0)| = 2$, giving $w_{\text{self}} = 1/3$ and $w_{\text{rel}} = 2/3$.

Generalization to $d$ dimensions with coordination number $z = 2d$: $$ w_{\text{self}}(d) = \frac{1}{2d + 1}, \quad w_{\text{rel}}(d) = \frac{2d}{2d + 1} $$

Primitive Flow Update Rules

Forward and backward flows at site $i$ evolve via: $$ F_i^+(n+1) = F_i^+(n) + \alpha A_i - \delta, \quad F_i^-(n+1) = F_i^-(n) - \alpha A_i - \delta $$ where $A_i = F_i^+ - F_i^-$ is asymmetry and $\delta$ enforces global anchoring.

Neighbor coupling adds: $$ A_i(n+1) = (1 + 2\alpha - \beta k_i) A_i + \beta \sum_j R(i,j) A_j $$ with $k_i = \sum_j R(i,j)$ the relational degree.

Emergent Proto-Time

The ordered update operator $U$ defines proto-time: $n \prec n+1$. The asymmetry dynamics create a direction of evolution before any metric time exists. Early dimensionality emerges as relational rank: $$ D_i = k_i $$

Emergent Spatial Structure

Correlation-based operational distance: $$ C_{ij}(\Delta\tau) = \frac{1}{T} \sum_{t=0}^{T-1} F_i(t) F_j(t+\Delta\tau) \Big/ \sqrt{\langle F_i^2 \rangle \langle F_j^2 \rangle} $$ $$ T_{i \leftrightarrow j} = \max_{\Delta\tau} |C_{ij}| \cdot \max_{\Delta\tau} |C_{ji}|, \quad d_\text{eff}(i,j) = -\log T_{i \leftrightarrow j} $$

Adaptive neighbor set: $$ N_\text{eff}(i) = \{ j \mid d_\text{eff}(i,j) < \epsilon_\text{coh} \} $$ Weights updated via: $$ \alpha_{ij}(k+1) = \lambda \alpha_{ij}(k) + (1-\lambda) |C_{ij}(0;k,T)| $$

Discrete → Continuum Mapping

Spatial Taylor expansion: $$ F_{i\pm1}(t) = F(x \pm a, t) = F \pm a F_x + \frac{a^2}{2} F_{xx} + \frac{a^3}{6} F_{xxx} + O(a^4) $$ Temporal Taylor expansion: $$ F(t+\tau) - 2F(t) + F(t-\tau) = \tau^2 F_{tt} + \frac{\tau^4}{12} F_{tttt} + O(\tau^6) $$ Resulting PDE: $$ F_{tt} = c_\text{eff}^2 F_{xx} + \gamma_x F_{xxxx} - \mu_\text{eff}^2 F - \gamma_t F_{tttt} + O(a^6, \tau^6) $$ with $c_\text{eff}^2 = \alpha a^2 / \tau^2$, $\gamma_x = \alpha a^4 / (12 \tau^2)$, $\gamma_t = \tau^2 / 12$.

Motif Spin and Curvature Proxies

Local spin proxy: $$ s_i = F_i^+ \frac{\partial F_i^-}{\partial x} - F_i^- \frac{\partial F_i^+}{\partial x}, \quad \frac{\partial F}{\partial x} \approx \frac{F_{i+1}-F_{i-1}}{2} $$ Discrete holonomy for loops: $$ H_p = \sum_{(u \to v) \in p} \arg S_{uv}(\omega^*) \mod 2\pi $$ Curvature proxy: $$ R_\text{proxy}(i) = \frac{1}{\#\text{loops through i}} \sum_{p \ni i} \frac{H_p}{A_p}, \quad R_\text{phys}(i) = R_\text{proxy}(i)/L_c^2 $$

Experimental Signatures

- Bell-type correlations probe the minimal comparative structure. - Local motif relaxation times $\tau_M$ define effective decoherence. - Running couplings and cluster coherence affect emergent masses $M_\text{cluster}(l)$.

Cluster mass law: $$ M_\text{cluster}(l) = E_\text{cluster}(l) \exp[\chi(l)], \quad E_\text{cluster}(l) = \sum_{i \in \text{cluster}} (|F_i^+|^2 + |F_i^-|^2) $$

Summary

• Primitive asymmetry and flows → proto-time and relational rank
• Correlation-based distances → emergent space
• Discrete → continuum mapping → PDEs
• Motif spin, holonomy → curvature proxies
• Experimental predictions: Bell tests, decoherence, mass spectra