Section 2: 1D Chain Networks and Emergent Mass via Reflection (Enhanced Version)

Section 2 — Emergent Structures and Dynamics in the 1D Chain Model

2.1 Definition of the 1D Chain Substrate and Flow Variables

Substrate: The system is an infinite 1D chain of nodes indexed by integers i ∈ ℤ.

Each node i connects to exactly two neighbors:

$$ N(i) = \{i - 1, i + 1\} $$

This topology encodes a discrete spatial dimension.

Each node i ∈ ℤ of the 1D chain carries a complex flow variable F_i(t) ∈ ℂ at discrete integer time steps t = 0, 1, 2, \ldots

Flow Variables: At each discrete time step k, each node i carries a complex flow variable

$$ F_i^{(k)} \in \mathbb{C} $$

which can be expressed in amplitude-phase form:

$$ F_i^{(k)} = A_i^{(k)} \times \exp\left(i \times \varphi_i^{(k)}\right) $$

where amplitude A_i^{(k)} \geq 0 and phase \varphi_i^{(k)} \in [0, 2\pi).

Temporal Discreteness: Time is discrete with integer steps k=0,1,2,\ldots. The fundamental time step is set to 1 unit.

2.2 Discrete Local Update Dynamics with Temporal Asymmetry and Reflection

The update rule for flow at node i from step k to k+1 is

$$ F_i^{(k+1)} = F_i^{(k)} + G_i\big(\{F_{i-1}^{(k)}, F_{i+1}^{(k)}\}, \gamma_i^{(k)}\big) - \beta \times \frac{\partial V_{int}}{\partial F_i^{(k)}} $$

where:

• G_i models flow exchange with neighbors weighted by edge exchange strengths \alpha_{ij}^{(k)} and a sign \sigma_i^{(k)} encoding reflection modes.
• \beta > 0 scales a nonlinear internal restoring potential V_{int}.
• \gamma_i^{(k)} is the local temporal asymmetry parameter defined below.

Exchange Strength \alpha_{ij} Dynamics:

$$ \alpha_{ij}^{(k+1)} = \lambda \times \alpha_{ij}^{(k)} + (1 - \lambda) \times |F_j^{(k)} - F_i^{(k)}| $$

with memory decay 0 < \lambda < 1, e.g. \lambda=0.9.

Temporal Asymmetry Parameter \gamma_i^{(k)}:

$$ \gamma_i^{(k)} = \frac{1}{2} \times \left[1 + \mathrm{sign}\left((F_{i+1}^{(k)} - F_i^{(k)}) \times (F_i^{(k)} - F_{i-1}^{(k)})\right)\right] $$

This equals 1 if local flow differences are monotonic (forward temporal flow), 0 if node i is a local extremum (potential reflection point).

Processing Mode Sign \sigma_i^{(k)}:

Calculate total exchange demand

$$ D_i^{(k)} = \alpha_{i,i-1}^{(k)} + \alpha_{i,i+1}^{(k)} $$

Then

$$ \sigma_i^{(k)} = \begin{cases} +1 & \text{if } D_i^{(k)} \leq 1 \quad \text{(normal processing)} \\ -\gamma_i^{(k)} & \text{if } D_i^{(k)} > 1 \quad \text{(reflection mode)} \end{cases} $$

Local Flow Update Functional G_i:

$$ G_i = \sigma_i^{(k)} \times \big[ \alpha_{i,i-1}^{(k)} (F_{i-1}^{(k)} - F_i^{(k)}) + \alpha_{i,i+1}^{(k)} (F_{i+1}^{(k)} - F_i^{(k)}) \big] $$

Internal Potential V_{int}(F_i):

$$ V_{int}(F_i) = (F_i - F_0)^2 \times (F_i - F_1)^2 $$

with stable vacuum points F_0 = -1 and F_1 = +1.

Gradient:

$$ \frac{\partial V_{int}}{\partial F_i} = 2 \times (F_i - F_0) \times (F_i - F_1) \times \big[ 2F_i - (F_0 + F_1) \big] $$

Interpretation: Reflection mode reverses the sign of updates weighted by \gamma_i^{(k)}, enabling localized feedback loops in flow that underpin mass emergence.

2.3 Correlation Functions, Normalized Transport, and Operational Distance

Correlation Function C_{ij}(\tau):

For lag \tau,

$$ C_{ij}(\tau) = \frac{1}{T} \sum_{k=0}^{T-1} \left[ F_i^{(k)} \times \overline{F_j^{(k+\tau)}} \right] $$

Measures amplitude-phase correlations over time window T.

Normalized Transport T_{j \to i}(\tau):

$$ T_{j \to i}(\tau) = \frac{C_{ij}(\tau)}{\sqrt{C_{ii}(0) \times C_{jj}(0)}} $$

Quantifies directed influence from j to i at lag \tau.

Symmetrized Transport T_{i \leftrightarrow j}:

$$ T_{i \leftrightarrow j} = \sqrt{\left| T_{i \to j}(\tau^*) \times T_{j \to i}(\tau^*) \right|} $$

where \tau^* maximizes |T_{j \to i}(\tau)|.

Operational Distance d_{ij}:

$$ d_{ij} = -\log \big( T_{i \leftrightarrow j} \big) $$

Defines an effective distance between nodes based on dynamical coherence.

Interpretation: This distance incorporates both amplitude and phase information and serves as the basis for emergent geometry.

2.4 Emergence of Metric Tensor from Local Embeddings

Using distances d_{ij} from correlations over a local neighborhood N(i), embed nodes into a pseudo-Riemannian manifold patch with coordinates x^\mu.

Define perturbation tensor

$$ \epsilon_{\mu \nu}(i) = \alpha \times \sum_{j \in N(i)} \frac{F_j^\mu \times F_j^\nu}{\|F_j\|^2} $$

where \alpha is a dimensionless coupling parameter, and F_j^\mu are components of the flow vectors embedded as coordinate tangent vectors.

The emergent metric is

$$ g_{\mu \nu}(i) = \eta_{\mu \nu} + \epsilon_{\mu \nu}(i) $$

where \eta_{\mu \nu} = \mathrm{diag}(-1,1,1,1) is the Minkowski background.

Coherence Length L_c:

Defined as the scale over which \epsilon_{\mu \nu} varies smoothly and local flow patterns maintain stability.

Interpretation: The metric tensor arises naturally from the alignment and coherence of local flow vectors, linking discrete network dynamics to continuous geometry.

2.5 Curvature from Dynamical Loops and Holonomy

The 1D chain substrate lacks topological loops, but the dynamics induce recursive coherence motifs or dynamical loops through flow feedback and reflection modes. A reflection event at a node, where flow reverses direction (for example, A \to B \leftarrow C), is a prime example of such a motif.

This localized flow oscillation creates a phase mismatch, measurable as a non-zero holonomy.

Holonomy from Dynamical Loops

The operational edge phases, defined as

$$ a_{j \to i} = \arg\big(T_{j \to i}(\tau^*)\big) $$

represent the discrete phase connection between nodes. The closed-loop holonomy H_C for a recursive motif measures the net phase accumulated.

In the 1D chain, this is computed around the oscillation motif:

$$ H_C = \arg \big( T_{i-1 \to i} \times T_{i \to i+1} \times T_{i+1 \to i-1} \big) $$

A reflection event introduces a non-zero phase twist, so that

$$ H_C \neq 0 \quad \bmod 2\pi $$

Connection to Curvature

In differential geometry, a non-zero holonomy means that parallel transporting a vector around a closed loop fails to return it to its original state — the hallmark of intrinsic geometric curvature.

Thus, in our model, discrete reflection-induced phase twisting acts as a source of curvature in the emergent geometry, even though no topological loops exist in the underlying 1D substrate. From this emergent metric g_{\mu \nu} and connection \Gamma^{\lambda}_{\mu \nu}, we can compute the Ricci tensor R_{\mu \nu} and scalar curvature R using standard formulas, demonstrating how microscopic dynamics generate macroscopic geometric properties.

2.6 Informational Friction and Topology Factor

Informational Friction \delta:

Linearize update map G around steady state flow F^*.

Calculate Jacobian J with entries

$$ J_{ij} = \frac{\partial G_i}{\partial F_j} \bigg|_{F^*} $$

Spectral radius \rho(J) measures perturbation growth.

$$ \delta = 1 - \rho(J) $$

Interpretation: \delta > 0 implies stable decay of perturbations; \delta measures dissipative information loss ("friction").

Topology Factor TF:

Defined as weighted sum over recursive coherence motifs (loops and osculations):

$$ TF = \frac{1}{|V|} \times \sum_{\text{loops } P} \prod_{(i \to j) \in P} \alpha(i,j) $$

where \alpha(i,j) are edge coupling weights.

Relationship: Higher TF correlates with greater recursive constraint density, stabilizing coherence length L_c and enhancing flow alignment.

2.7 Emergent Effective Physical Constants and Einstein Equation

Effective Gravitational Coupling:

$$ G_{eff} = \kappa \times (\delta \times TF) \times L_c^2 $$

where \kappa is a dimensional constant chosen for unit consistency.

Einstein Field Equation:

$$ R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R = \frac{8 \pi G_{eff}}{c^4} \times T_{\mu \nu} $$

Stress-energy tensor T_{\mu \nu} derived from flow densities and fluxes.

Interpretation: Gravitational coupling emerges as a coarse-grained effective parameter controlled by microscopic network stability and topology.

2.8 Emergence of Maximal Speed and Causality

Locality Axiom (Axiom A):

Information propagates only via local neighbor updates per time step \Delta t.

Emergent Maximal Speed:

$$ c_{internal} = \frac{\Delta x}{\Delta t} $$

with \Delta x average physical distance between nodes.

Interpretation: This speed sets the emergent causal cone and is identified with the speed of light c in the continuum limit.

2.9 Locality of Temporal Direction and Arrow of Time

Mechanism:

Temporal asymmetry \gamma_{asym} and informational friction \delta produce local irreversible flow of information and entropy gradients.

Entropy Functional S_{loc}:

At node i,

$$ S_{loc}(i) = - \sum_k P(F_k^{(i)}) \times \log P(F_k^{(i)}) $$

where P(F_k^{(i)}) is the probability distribution over flow states.

Interpretation: Local entropy increases in coarse-grained frames produce a local arrow of time, even if the global network evolution is non-monotonic or quasi-recurrent.