Appendix D: Glossary — The ν‑Framework

D.1 Latin Symbols

A Mass number (nucleon count) of a nuclide; integer 1 ≤ A ≤ 295 in ν‑matrix.

B₀ Reference magnetic field, defined as 1 T exactly, used to fix π_spin and f_RMN scales.

C Charge-conjugation operator; acts on π‑space as M_C = diag(1, −1, 1, −1, −1).

c Speed of light in vacuum; exact value 299 792 458 m·s⁻¹ (CODATA 2022).

E Measurable energy (rest, binding, excitation, or decay width) expressed in joules.

e Elementary charge; exact value 1.602 176 634 × 10⁻¹⁹ C (CODATA 2022).

f Frequency in hertz (Hz) obtained by f = E/h; universal currency of ν/π‑vectors.

G_F Fermi coupling constant; 1.166 378 7 × 10⁻⁵ GeV⁻² (PDG 2024).

g g‑factor (dimensionless) relating magnetic moment to spin.

h Planck constant; exact value 6.626 070 15 × 10⁻³⁴ J·Hz⁻¹ (CODATA 2022).

ħ Reduced Planck constant; ħ = h / 2π.

I Nuclear spin quantum number (integer or half-integer).

J Total angular momentum quantum number (used in T‑transformation phase).

m Mass in kilograms; converted to frequency via π_grav = m c² / h.

n Principal quantum number (referenced in saw‑tooth discussion).

P Parity operator; acts on π‑space as M_P = diag(1, 1, 1, 1, −1).

Q Electric quadrupole moment in barns (1 b = 10⁻²⁸ m²).

Q² Squared electric charge in units of e² (Q = +2/3 → Q² = 4/9).

T Time-reversal operator; anti-unitary with spin-phase factor e^(iπJ).

Z Proton number (atomic number); integer 1 ≤ Z ≤ 118 in ν‑matrix.

D.2 Greek Symbols

α Fine-structure constant; α = e² / (4π ε₀ ħ c) ≈ 7.297 × 10⁻³ (CODATA 2022).

α′ Kinetic-mixing parameter for dark-photon models; 0 ≤ α′ ≪ α.

δ Residual or relative uncertainty; δx / x denotes fractional 1σ error.

Δs Uniform logarithmic spacing of cosmic curve; Δs = 0.102 ± 0.008 dex per proton.

ε₀ Vacuum permittivity; exact value via c and μ₀.

Λ_QCD QCD confinement scale; 200 ± 10 MeV (PDG 2024).

μ Magnetic moment in J·T⁻¹; related to π_spin via μ = g (e ħ / 2m).

μ_B Bohr magneton; μ_B = e ħ / (2 m_e) ≈ 9.274 × 10⁻²⁴ J·T⁻¹.

μ_N Nuclear magneton; μ_N = e ħ / (2 m_p) ≈ 5.051 × 10⁻²⁷ J·T⁻¹.

ν Composite state vector; 7‑tuple (f_grav, f_forte, f_EM, f_RMN, f_quad, f_mag, f_beta).

π Elementary state vector; 5‑tuple (π_grav, π_EM, π_strong, π_weak, π_spin).

φ_EM EM phase angle; 0 for Q > 0, π for Q < 0, undefined for Q = 0.

γ Gyromagnetic ratio in rad·s⁻¹·T⁻¹; f_RMN = γ B₀ / (2π).

Γ Decay width in joules; f_beta = Γ / h.

τ½ Partial half-life in seconds; related to Γ via Γ = ħ ln 2 / τ½.

D.3 Subscript & Functional Notation

f_grav Gravitational/mass‑energy frequency of nucleus; f_grav = m_nucleus c² / h.

f_forte Strong‑force excitation frequency; f_forte = E(2₁⁺) / h ≤ 2 MeV.

f_EM Electromagnetic/optical frequency; f_EM = c / λ_air (strongest visible line).

f_RMN Nuclear magnetic resonance frequency at B₀ = 1 T; f_RMN = γ B₀ / (2π).

f_quad Electric quadrupole interaction frequency; f_quad = e · Q · ⟨V_zz⟩ / [h · 4I(2I − 1)].

f_mag Magnetic moment interaction frequency at B₀ = 1 T; f_mag = μ · B₀ / (I · ħ).

f_beta Weak‑decay frequency; f_beta = Γ_β / h = ħ / (τ½ · h · ln 2).

π_grav Elementary mass‑energy frequency; π_grav = m_particle · c² / h.

π_EM Elementary EM coupling frequency; π_EM = α · Q² · e^(iφ_EM).

π_strong Elementary strong‑charge frequency; binary flag Λ_QCD / h or 0.

π_weak Elementary weak‑charge frequency; π_weak = G_F · E_weak² / (ħ · h).

π_spin Elementary magnetic frequency; π_spin = g · (e ħ / 2m) · B₀ / h (leptons only).

ν(Z) Cosmic‑curve function; parametric 1‑D trajectory through ν‑space.

φ Mapping φ : SM → π‑space; deterministic, invertible, parameter‑free.

F Unknown frequency‑algebraic function; postulated ν = F(π₁, π₂, ..., π_A; κ).

κ Binding kernel encoding many‑body correlations in F (non‑perturbative).

D.4 Operational Axioms & Criteria

Axiom 1 Every measurable energy E of a stable state (τ½ > 10⁻⁸ s) shall be expressed as f = E / h.

Axiom 2 Uniqueness of signature: ‖ν(s₁) − ν(s₂)‖_Σ⁻¹ > 3 for any two distinct states.

Axiom 3 Frequency‑space components are closed under C, P, T transformations (Table 2.1).

Criterion 1 Injectivity: no two SM particles share indistinguishable π‑vectors within 3σ.

Criterion 2 Categorical separation: SI > 3σ between quarks/leptons, charged/neutral, bosons/fermions.

Criterion 3 Symmetry preservation: residuals δ < 10⁻⁴% under C, P, T operations.

Criterion 4 Completeness: 37 unique π‑vectors cover entire SM census (30 types + 7 gluon colours).

Criterion 5 Residue‑free antiparticle duality: C(π) = π(antiparticle) within 10⁻⁴%.

D.5 Specialized & Model‑Independent Terms

BSM Beyond Standard Model; any particle not in PDG 2024 census.

CMB Cosmic microwave background; used to constrain ν_s and A′ masses.

dex Decimal exponent; 1 dex = factor 10; used for logarithmic spacing Δs.

FAP False‑alarm probability in Lomb‑Scargle test; p < 10⁻⁶ implies > 5σ significance.

HDF5 Hierarchical Data Format v5; archival format for ν‑ and π‑matrices.

IUPAC International Union of Pure and Applied Chemistry; defines terrestrial isotopic abundances.

LISA Laser Interferometer Space Antenna; future gravitational‑wave probe of Planck‑scale ν₀.

NMR‑silent Element with no stable isotope having I ≠ 0; f_RMN = null.

null vs. zero null = structurally excluded channel; zero = permitted but inactive (f_i = 0).

PDG Particle Data Group; authoritative 2024 compilation used for π‑vector derivation.

SI Separation Index = ‖μ_A − μ_B‖ / (σ_A + σ_B); metric for categorical disjointness.

saw‑tooth 8.07‑unit periodic modulation in log₁₀(f_EM) versus Z; encodes aufbau principle.

sterile neutrino Hypothetical ν_s with π_weak = 0, π_EM = 0, π_grav < 10¹⁴ Hz.

dark photon Hypothetical A′ with π_EM ≠ 0, π_weak = 0, π_strong = 0; forbidden in SM π‑space.

cosmic curve 1‑D manifold capturing 89.3% of variance in {f_grav, f_EM, f_mag}; slope Δs = 0.102 dex/proton.

harmonic anchors Z = 12 (Mg) and 20 (Ca) where saw‑tooth residuals vanish within 0.5%.

binding kernel κ Unknown non‑perturbative operator in F that produces 100‑fold mass cancellation.

frequency algebra F Postulated map ν = F(π₁, π₂, ..., π_A; κ); constrained but not yet solved.

ν₀‑vector Speculative vacuum boundary vector at f_Hubble ≈ 2.3 × 10⁻¹⁸ Hz.

Planck frequency f_Planck = E_Planck / h = 1.856 × 10⁴³ Hz; upper bound for physical frequencies.

D.6 Acronyms & Database Identifiers

AME2020 Atomic Mass Evaluation 2020 via NNDC (f_grav source).

ASD v5.8 NIST Atomic Spectra Database v5.8 (f_EM source).

ENSDF Evaluated Nuclear Structure Data File (f_forte, f_beta, Q values).

CODATA Committee on Data of the International Science Council (fundamental constants).

ETCSL Electronic Text Corpus of Sumerian Literature (historical cross‑check).

LHC Large Hadron Collider; probes dark‑photon resonances.

DUNE Deep Underground Neutrino Experiment; tests π_weak = 0 for ν_s.

KATRIN Karlsruhe Tritium Neutrino Experiment; kinematic ν‑mass bound.

CMB‑S4 Stage‑4 CMB survey; acoustic‑peak constraint on A′ mass.