Nomenclature & Symbol Table

This table defines the standard notation used throughout the Quantum Braid Dynamics (QBD) monograph.

Symbol	Description	First Used
$A, B$	Generic propositions within a logical schema	§1.1.2
$\vdash$	Syntactic derivability (provability within a formal system)	§1.1.2
$\models$	Semantic entailment (truth within a model)	§1.1.2
$\Gamma$	A set of premises or axioms	§1.1.2
$\theta$	A derived theorem or conclusion	§1.1.2
$S_n$	The $n$-th statement in a formal proof sequence	§1.1.2
$F$	A consistent, effectively axiomatized formal system	§1.1.3
$\mathcal{G}$	The Gödel sentence ("This statement is unprovable in F")	§1.1.3
$Con(F)$	The statement asserting the consistency of system $F$	§1.1.3
$t_{phys}$	Physical Time (emergent, geometric, continuous, local)	§1.2.1
$t_L$	Global Logical Time (fundamental, discrete, integer-valued)	§1.2.1
$\mathbb{N}_0$	The set of non-negative integers $\{0, 1, 2, \dots\}$	§1.2.2
$U_{t_L}$	The global state of the universe at logical time step $t_L$	§1.2.2
$\mathcal{U}$	The Universal Evolution Operator	§1.2.2
$\hat{H}$	Total Hamiltonian operator	§1.2.2
$\Psi$	The wavefunction of the universe	§1.2.2
$n$	Generation step (Cellular Automaton context)	§1.2.2.1
$\mu$	Renormalization scale (or mean in statistical contexts)	§1.2.2.1
$\tau$	Fictitious or imaginary time parameter	§1.2.2.1
$\mathcal{T}$	Unimodular Time variable	§1.2.2.3
$\hat{P}$	Permutation operator (Cellular Automaton Interpretation)	§1.2.2.2
$\vert \psi(t)\rangle$	The Ontic State vector	§1.2.2.2
$\Lambda, \hat{\Lambda}$	Cosmological constant (and corresponding operator)	§1.2.2.3
$\hbar$	Reduced Planck constant	§1.2.2.3
$S(U_{t_L})$	Entropy of the state $U_{t_L}$	§1.2.3
$O(\cdot)$	Big O notation (asymptotic upper bound)	§1.2.3
$\ell_P$	Planck length ($\approx 1.6 \times 10^{-35}$ m)	§1.2.3
$N_P$	Number of Planck voxels	§1.2.3
$\vert \Omega_n \vert$	Cardinality of the state space at step $n$	§1.2.3
$k_B$	Boltzmann constant	§1.2.3
$E$	Energy	§1.2.3
$T$	Temperature	§1.2.3
$d_{t_L}$	Dimension of the Hilbert space at step $t_L$	§1.2.3
$\mathcal{R}$	Rule set for evolution	§1.2.3
$s_{t_L}$	Number of active rewrite sites at step $t_L$	§1.2.3
$b$	Branching factor (outcomes per site)	§1.2.3
$\delta, \gamma$	Scaling constants for site growth	§1.2.3
$\sigma^2$	Variance	§1.2.4.1
$\mathbb{E}[\cdot]$	Expected value operator	§1.2.4.1
$\mathbb{P}(\cdot)$	Probability measure	§1.2.4.1
$\mathbb{Z}_{<0}$	The set of negative integers	§1.2.7.1
$\prec$	Strict precedence relation (causal ordering)	§1.2.5.1
$c$	Speed of light in vacuum	§1.2.6.1
$G_{\mu\nu}$	Einstein tensor	§1.2.6.2
$T_{\mu\nu}$	Stress-energy tensor	§1.2.6.2
$G$	Gravitational constant	§1.2.6.2
$R_s$	Schwarzschild radius	§1.2.6.2
$R_n$	The $n$-th Grim Reaper in the paradox sequence	§1.2.7.2
$\Omega$	Universal State Space (set of all admissible graphs)	§1.3.1
$G$	A specific Causal Graph configuration	§1.3.1
$V$	The set of Vertices (Abstract Events)	§1.3.1
$E$	The set of Edges (Causal Relations)	§1.3.1
$H$	The History Function (Timestamp map)	§1.3.1
$v_i, u, w$	Individual vertices (events)	§1.3.1
$e$	An individual directed edge	§1.3.1
$\mathbb{N}$	Natural numbers (codomain of $H$)	§1.3.1
$\mathfrak{T}$	Elementary Task Space (set of kinematic transformations)	§1.4.1.1
$\mathfrak{T}_{add}$	Edge Creation Task	§1.4.1
$\mathfrak{T}_{del}$	Edge Deletion Task	§1.4.1
$\Delta F$	Change in Free Energy	§1.4.4
$V_A, V_B$	Disjoint vertex sets in a bipartite graph	§1.5.1
$\to$	Directionality indicator in a path (e.g., $v \to w$)	§1.5.2
$v_i \to v_j$	Directed causal link	§2.1.1
$\nrightarrow$	Negation of directed link	§2.1.1
$\mathcal{R}$	Rewrite Rule	§2.2.1
$S$	Entropy	§2.2.3
$\le$	Effective Influence Relation	§2.2.3
$\gamma$	Geometric Quantum (Directed 3-cycle)	§2.3.2
$\Pi(u,v)$	Set of directed paths from u to v	§2.3.3.1
$\Phi(G)$	Lexicographic Potential	§2.3.4
$L$	Length of longest simple cycle	§2.3.4
$N_L$	Number of cycles of length L	§2.3.4
$\langle k \rangle$	Mean cycle length	§2.7.3
$R$	Radius of local computational patch	§2.7.3
$\rho$	Graph density	§2.7.3
$\xi$	Correlation length	§2.7.4
$G_0$	Initial state ($t_L=0$)	§3.1.1
$V_0, E_0, H_0$	Initial vertex, edge, and history sets	§3.1.1
$d_{depth}(v)$	Vertex depth from root	§3.1.5
$\Aut(G)$	Automorphism group of graph G	§3.1.6
$V_{even}, V_{odd}$	Depth parity partition sets	§3.1.8
$k_{deg}$	Internal coordination number (Bethe fragment)	§3.2.1
$\mathcal{O}(G; \lambda)$	Structural Optimality Score	§3.2.9
$H_S(G)$	Shannon entropy of orbit size distribution	§3.2.9
$\mathcal{S}_{sites}(G)$	Set of compliant rewrite sites	§3.3.2
$\varphi$	Automorphism mapping	§3.3.2
$e_{tunnel}$	Symmetry-breaking tunneling edge	§3.4.2
$\mathbb{P}_{ign}$	Ignition probability	§3.4.5
$\chi(\vec{\sigma}_e)$	Syndrome-response function	§3.4.5
$\mathcal{H}$	Hilbert configuration space $(\mathbb{C}^2)^{\otimes K}$	§3.5.2
$\mathcal{C}$	Valid codespace	§3.5.1.1
$Z_{uv}, X_{uv}$	Pauli operators on edge qubit	§3.5.2
$\Pi_{cycle}$	2-Cycle Constraint Projector	§3.5.3
$\Pi_{local}$	Locality Constraint Projector	§3.5.3
$K_{uv}$	Triplet Check Operator	§3.5.4