Dist Babble

Section 7: Emergent Gravity from Collective Flow Dynamics — Defining Dimensions and Role of \( \delta \) and topology_factor 7.0 Characteristic Units Recap To maintain dimensional consistency throughout the emergent framework, we define characteristic units: \( L_c \): Characteristic length unit, dimension [L] \( T_c \): Characteristic time unit, dimension [T] \( E_c \): Characteristic energy unit, dimension [M L^{2} T^{-2}] \( c_{\text{char}} \): Characteristic speed, defined as \( \frac{L_c}{T_c} \), dimension [L T^{-1}] \( M_c \): Characteristic mass, defined as \( \frac{E_c}{c_{\text{char}}^{2}} = E_c \times \frac{T_c^{2}}{L_c^{2}} \), dimension [M] \( \hbar_c \): Characteristic action, defined as \( E_c \times T_c \), dimension [M L^{2} T^{-1}] 7.1 Gravitational Emergence from Processing Saturation Dimensionless Mass Proxy \( M_i \) We model local mass as processing saturation within the flow network. At each discrete node...

Section 6: Forces as Emergent Flow Couplings and Interactions — Defining Dimensions 6.0 Preliminaries on Physical Units Physical units in this framework arise exclusively through mapping from underlying dimensionless network quantities using characteristic physical scales introduced in Section 1. These characteristic scales are: Characteristic length, denoted \( L_c \), with units of length [L]. Characteristic time, denoted \( T_c \), with units of time [T]. Characteristic energy, denoted \( E_c \), with units of mass times length squared divided by time squared \([M L^2 T^{-2}]\). Characteristic speed, \( c_{\text{char}} = \frac{L_c}{T_c} \), with units length per time \([L T^{-1}]\). Characteristic mass, \( M_c = E_c \times \frac{T_c^2}{L_c^2} \), units of mass \([M]\). Characteristic action, \( \hbar_c = E_c \times T_c \), units of mass times length squared divided by time \([M L^2 T^{-1}]\). These scales provide the essential ...

Section 5: Emergence of Fundamental Concepts and Structure — Defining Dimensions 5.1 Internal Speed Limit as an Emergent Causal Limit 5.1.1 Emergent Causality from Flow Asymmetry Causality and directional propagation naturally arise from local temporal asymmetry and irreversibility in the discrete flow network. This is expressed by the condition: The discrete second difference of flow at node \( i \) and step \( n \), denoted \( A_i(n) \), is nonzero if and only if the temporal asymmetry parameter \( \Psi_i(n) \) is nonzero, which also corresponds to a nonzero local time-averaged reflection rate \( M_i \) (mass). In other words: \( A_i(n) \neq 0 \iff \Psi_i(n) \neq 0 \iff M_i > 0 \) Here: \( A_i(n) \): discrete second difference of the flow (dimensionless) \( \Psi_i(n) \): temporal asymmetry or directional bias (dimensionless) \( M_i \): time-averaged reflection rate at node \( i \) (dimensionless) ...

Section 4: Emergent Particles and Momentum from Flow Localization Core Principle Particles emerge as localized, stable configurations of reflection-driven flow patterns on the discrete network substrate. These configurations represent persistent, spatially coherent structures resulting directly from the mass generation mechanisms established in Section 2. The particle concept is defined rigorously from flow saturation and temporal coherence, yielding physically meaningful mass, momentum, and dynamical properties. 4.1 Particle Definition as Reflection Clusters 4.1.1 Spatial Coherence Criterion Define a particle region as a connected subset of nodes \( \{i\} \) satisfying: The local mass density \( M_i \) exceeds a critical threshold \( M_{\text{threshold}} \) Nodes are spatially connected by adjacency within a maximum coherence radius \( R_{\text{coherence}} \) Formally: Particle region = \( \left\{ i \, \middle| \, M_i > M_{\text{thre...

Section 3 — Emergent Spatial Geometry from Flow Saturation 3.1 Conceptual Framework for the Emergence of Space The spatial structure of the discrete substrate is not fundamental; rather, it emerges from the relational difficulty of equilibrating information flow between nodes under finite processing capacity. Spatial distance is therefore an informational processing metric defined by how flow saturation and reflection events delay and hinder information propagation. 3.2 Flow Equilibration and Processing Capacity Constraints 3.2.1 Exchange Strength Saturation Each edge connecting nodes \(i\) and \(j\) accumulates an exchange strength \(\alpha_{ij}\) that reflects local differences in the dimensionless flow amplitudes \(F_i(k)\). The update rule over discrete time steps \(k\) is: $$ \alpha_{ij}(k+1) = \lambda \times \alpha_{ij}(k) + (1 - \lambda) \times |F_j(k) - F_i(k)| $$ Here, \(\lambda \in (0,1)\) is a memory decay parameter, and t...

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 Reflec...

Section 1 — First Principles and Foundations of Temporal Flow Physics by John Gavel 1.1 Purpose and Framing (Constructive Program) Temporal Flow Physics (TFP) is a constructive program aimed at deriving spacetime, fields, and matter-like structures from a minimal discrete dynamical substrate. The fundamental premise is that time and information flow are primary; geometric notions like metric, connection, or continuum fields emerge from patterns in the evolution of discrete complex variables on a network. This section states the core axioms, variables, canonical dynamics, and the construction protocol that avoids circular assumptions. 1.2 Ontology and Primitives (Axioms and Definitions) Postulate 1 — Discrete Dynamical Substrate The fundamental structure is a directed graph G = (V, E) , where: V = {1, ..., N} is a finite set of nodes (events or informational units). E ⊆ V × V is a set of directed edges connecting nodes. Time is discrete: t ∈ ℕ (non-negative integ...