Notation Quick Reference — Volume 2 (Cosmic Architecture)

Volume 2 — Cosmic Architecture

Notation Quick Reference — Volume 2 (Cosmic Architecture)

This reference complements the Volume 1 Notation (01_Volume_1.../99_Backmatter/02_Notation_Ref.md) and the global Parameter Ledger (00_Global_Frontmatter/03_Parameter_Ledger.md). Symbols below are those most heavily used in V2 (Crystallography, Light, Gravity, Dark Sector).

Lattice and Manifolds

$\mathbb{Z}^6$ : The 6D parent integer lattice (Crystallography Ch01–Ch03).
$H_3$ , $H_4$ : Coxeter root systems associated with the icosahedral cut.
$E_8$ : The maximal exceptional Lie root system, parent of the icosahedral 6D slice.
$E_\parallel$ , $E_\perp$ : Parallel (physical 3D) and perpendicular (internal 3D) projections of the 6D lattice.
$\Sigma_\infty$ : The Adelic Solenoid (V1 Ch07; reused in V2 Ch05 as the topological identity carrier under acoustic-metric refraction).
$W$ (Acceptance Window): Rhombic Triacontahedron (RT) — the cut-and-project filter in $E_\perp$ .

Phason and Phonon Fields

$\Phi_\perp$ (Phason field): The internal-manifold soft mode; the propagator of graviton-like excitations in the GCT supersolid.
$u_\parallel$ (Phonon field): The physical-manifold acoustic mode; sets the rigid-rod scale.
$K_\parallel$ , $K_\perp$ : Phonon and phason stiffness, with the GCT-postulated ratio $K_\perp/K_\parallel = \phi^{-18}$ (Tier 2 postulate + Tier 3 specific exponent, App K §K.4; uniqueness of $D=18$ pending Open Problem O.15). $K_\perp$ has SI units J/m³; in natural units it is the GUT-scale energy $\sim 10^{15}$ GeV (V2 Ch04 §4.2.3).
$c$ (Acoustic light speed): The maximal causal-organization speed in the supersolid, identified with the second-sound (phason) branch $c = (K_\perp/\rho_{eff})^{1/2}$ (V2 Ch05 §5.2.3, Ch06 §6.2.2). The first-sound (phonon) speed $(K_\parallel/\rho)^{1/2}$ is the bare lattice speed $v_{\text{bare}} \gg c$ but transmits no topological information (Ch06 §6.2.4).
$\hat{c} = \phi^{-9}$ (dimensionless lattice speed): The φ-scaling carried by the speed of light under the Galois transform.

Acoustic Metric and Gravity

$g_{\mu\nu}$ : The effective Lorentzian metric on the lattice (V2 Ch08).
$\Phi$ (Newtonian potential): Identified with the localized phason strain density $\delta\rho/\rho_0$ in the Newtonian limit (V2 Ch08 §8.1.4).
$G$ (Newton's constant): Jacobson/acoustic thermodynamic mechanism with Tier 4 Planck-link inheritance from O.14 and Tier 3 $m_e$ dimensional anchoring; precision against CODATA is reported in App R.
$a_6$ : The 6D lattice unit-cell spacing (taken $\equiv 1$ in lattice units).
$\ell_P, M_P$ : Planck length and mass (CODATA).

Dark Sector (Vols 2 + 3)

$\Lambda$ (Cosmological constant): Re-interpreted as the locking potential of the phason vacuum.
$\rho_\Lambda \approx 5.4 \times 10^{-10}$ J/m³ (Observed): Anchored to current cosmological data.
$\chi = \pi(c/(H_0\ell_P))^2 \approx 2.27 \times 10^{122}$ : Susceptibility from the holographic horizon-area ratio at Planck 2018 $H_0 = 67.4$ km/s/Mpc (V2 Ch14 §14.1.5); the numerical evaluation also uses CODATA constants.
$w(z)$ : The dark-energy equation-of-state parameter; the sustained Phantom Phase ( $w(z) < -1$ , asymptoting to $-1$ from below per the IMP-01 pipeline) is a Tier 3 phenomenological signature in sign-opposite tension with DESI CPL fits. The operative V7' Class-2 envelope is $|w+1|\in[2,5]\times10^{-5}$ near the $z\approx0.28$ low-redshift sensitivity marker; this marker is not a transition redshift or mathematical zero of $w+1$ (see V2 Ch14 §14.6.3).

Topological Order Parameters

$\eta_g$ (Glass order parameter, V2 Ch11): Distinct from $\eta_{Zeno}$ (V3 Ch13) and $\eta_{stiff} = \phi^{-18}$ (App K).
$\eta_{visc}$ (Viscosity, V2 Ch13): Hydrodynamic dissipation coefficient.

Cross-Reference Conventions

§X.Y or §X.Y.Z without volume prefix refers to a section within the current chapter/volume.
V1 Ch<n> / V2 Ch<n> / V3 Ch<n> specifies cross-volume references explicitly.
App <letter> refers to a Global Appendix.
For any symbol with cross-volume usage, the canonical definition is the entry above and any per-chapter restatement either matches it or explicitly notes a temporary re-binding (e.g., the bare $\kappa$ vs $\kappa_{ij}$ disambiguation in V1 §17.1.3).

Volume 2 — Cosmic Architecture

Notation Quick Reference — Volume 2 (Cosmic Architecture)

Lattice and Manifolds

$\mathbb{Z}^6$ : The 6D parent integer lattice (Crystallography Ch01–Ch03).
$H_3$ , $H_4$ : Coxeter root systems associated with the icosahedral cut.
$E_8$ : The maximal exceptional Lie root system, parent of the icosahedral 6D slice.
$E_\parallel$ , $E_\perp$ : Parallel (physical 3D) and perpendicular (internal 3D) projections of the 6D lattice.
$\Sigma_\infty$ : The Adelic Solenoid (V1 Ch07; reused in V2 Ch05 as the topological identity carrier under acoustic-metric refraction).
$W$ (Acceptance Window): Rhombic Triacontahedron (RT) — the cut-and-project filter in $E_\perp$ .

Phason and Phonon Fields

$\Phi_\perp$ (Phason field): The internal-manifold soft mode; the propagator of graviton-like excitations in the GCT supersolid.
$u_\parallel$ (Phonon field): The physical-manifold acoustic mode; sets the rigid-rod scale.
$K_\parallel$ , $K_\perp$ : Phonon and phason stiffness, with the GCT-postulated ratio $K_\perp/K_\parallel = \phi^{-18}$ (Tier 2 postulate + Tier 3 specific exponent, App K §K.4; uniqueness of $D=18$ pending Open Problem O.15). $K_\perp$ has SI units J/m³; in natural units it is the GUT-scale energy $\sim 10^{15}$ GeV (V2 Ch04 §4.2.3).
$c$ (Acoustic light speed): The maximal causal-organization speed in the supersolid, identified with the second-sound (phason) branch $c = (K_\perp/\rho_{eff})^{1/2}$ (V2 Ch05 §5.2.3, Ch06 §6.2.2). The first-sound (phonon) speed $(K_\parallel/\rho)^{1/2}$ is the bare lattice speed $v_{\text{bare}} \gg c$ but transmits no topological information (Ch06 §6.2.4).
$\hat{c} = \phi^{-9}$ (dimensionless lattice speed): The φ-scaling carried by the speed of light under the Galois transform.

Acoustic Metric and Gravity

$g_{\mu\nu}$ : The effective Lorentzian metric on the lattice (V2 Ch08).
$\Phi$ (Newtonian potential): Identified with the localized phason strain density $\delta\rho/\rho_0$ in the Newtonian limit (V2 Ch08 §8.1.4).
$G$ (Newton's constant): Jacobson/acoustic thermodynamic mechanism with Tier 4 Planck-link inheritance from O.14 and Tier 3 $m_e$ dimensional anchoring; precision against CODATA is reported in App R.
$a_6$ : The 6D lattice unit-cell spacing (taken $\equiv 1$ in lattice units).
$\ell_P, M_P$ : Planck length and mass (CODATA).

Dark Sector (Vols 2 + 3)

$\Lambda$ (Cosmological constant): Re-interpreted as the locking potential of the phason vacuum.
$\rho_\Lambda \approx 5.4 \times 10^{-10}$ J/m³ (Observed): Anchored to current cosmological data.
$\chi = \pi(c/(H_0\ell_P))^2 \approx 2.27 \times 10^{122}$ : Susceptibility from the holographic horizon-area ratio at Planck 2018 $H_0 = 67.4$ km/s/Mpc (V2 Ch14 §14.1.5); the numerical evaluation also uses CODATA constants.
$w(z)$ : The dark-energy equation-of-state parameter; the sustained Phantom Phase ( $w(z) < -1$ , asymptoting to $-1$ from below per the IMP-01 pipeline) is a Tier 3 phenomenological signature in sign-opposite tension with DESI CPL fits. The operative V7' Class-2 envelope is $|w+1|\in[2,5]\times10^{-5}$ near the $z\approx0.28$ low-redshift sensitivity marker; this marker is not a transition redshift or mathematical zero of $w+1$ (see V2 Ch14 §14.6.3).

Topological Order Parameters

$\eta_g$ (Glass order parameter, V2 Ch11): Distinct from $\eta_{Zeno}$ (V3 Ch13) and $\eta_{stiff} = \phi^{-18}$ (App K).
$\eta_{visc}$ (Viscosity, V2 Ch13): Hydrodynamic dissipation coefficient.

Cross-Reference Conventions

§X.Y or §X.Y.Z without volume prefix refers to a section within the current chapter/volume.
V1 Ch<n> / V2 Ch<n> / V3 Ch<n> specifies cross-volume references explicitly.
App <letter> refers to a Global Appendix.
For any symbol with cross-volume usage, the canonical definition is the entry above and any per-chapter restatement either matches it or explicitly notes a temporary re-binding (e.g., the bare $\kappa$ vs $\kappa_{ij}$ disambiguation in V1 §17.1.3).