Notation Quick Reference — Volume 3 (Matter Spectrum)

Volume 3 — The Matter Spectrum

Notation Quick Reference — Volume 3 (Matter Spectrum)

This reference complements the Volume 1 and Volume 2 Notation files. Symbols below are those most heavily used in V3 (Forces, Matter, Astrophysics, Experiments).

Standard Model Constants and Sectors

$\alpha$ , $\alpha^{-1}$ : Fine-structure constant. GCT bare value $\alpha^{-1}_{bare} = 360\phi^{-2} \approx 137.508$ (Tier 2 impedance mechanism + Tier 3 specific 360 multiplier, V1 Ch13); CODATA observed $\approx 137.036$ .
$\sin^2\theta_W$ : Weinberg angle. GCT bare volume-coupling input $= \phi^{-3} \approx 0.236$ ; GUT-scale Gram scalar $\rho_G=\phi^{-2}\approx0.382$ has normalized Cartan share $0.27639$ and is not itself the physical angle; Z-pole observed $0.23122$ (V3 Ch04).
$\theta_C$ : Cabibbo angle. GCT geometric value $\arcsin(\phi^{-3}(1-5\alpha))$ (V3 Ch10).
$\delta_{CP}$ : CP-violating phase. GCT geometric value $360°\phi^{-1}$ (V3 Ch10).

Mass Spectrum

$m_e$ : Electron mass. Formula $m_e = M_P \phi^{-107}(1-5\alpha)$ (Tier 2 mechanism + Tier 3 Coxeter-exponent handle + Tier 4 structural-link conjecture pending O.14; ~1006 ppm vs CODATA per engine output).
$m_\mu$ : Muon mass. Formula $m_\mu = m_e \phi^{11}(1+5\alpha+\phi^8\alpha^2)$ (Tier 2 mechanism + Tier 3 specific exponent + Tier 3 $+5\alpha$ coefficient normalization / pole-mass loop discipline -> Tier 3 headline precision; 21 ppm analytic residual).
$m_\tau$ : Tau mass. Formula $m_\tau = m_e \phi^{17}(1-3.6\alpha)$ (Tier 2 mechanism + Tier 3 specific exponent + Tier 3 combination rule, pending O.26b for tau coefficient stacking → Tier 3 headline coefficient; ~51 ppm against PDG 2024).
$m_p$ : Proton mass. Formula $m_p = m_e \phi^{15+\phi^{-1}}$ , prediction $\approx 938.417$ MeV against CODATA $938.272$ MeV (155 ppm residual; V3 Ch18 and App R §R.3).
$m_{p}/m_{e}$ ratio: $\phi^{15+\phi^{-1}} \approx 1836.437$ vs CODATA observed $1836.153$ .
$\Sigma m_\nu$ : Sum of neutrino masses. GCT normal-hierarchy scale $\Sigma m_\nu \approx 0.0853$ eV from $m_1=m_e\phi^{-36}$ plus oscillation gaps; Planck + DESI 2024 gives a $\Lambda$ CDM-context bound near $0.072$ eV at 95% CL, which is tension rather than the registered $<0.075$ eV / $>0.15$ eV exclusion band.

Lepton g-2 and Anomalies

$\Delta a_\mu$ : Muon anomalous magnetic moment correction. GCT vertex value $\approx 250.65 \times 10^{-11}$ (Tier 2 mechanism + Tier 3 $1/5$ coefficient + A3 + Tier 4 calibration-survival conjecture; Tension under WP2025, falsification conditional on long-term HVP-synthesis arbitration — see App V §P.5 for the HVP-Survival Condition).

Quasicrystal and Polaron Parameters

$\xi$ (Polaron Healing Length): $\xi = \hbar c/(\alpha^2 m_e c^2) = a_0/\alpha \approx 7.25$ nm (CODATA $\alpha$ ; the corrected GCT $\alpha^{-1}$ gives the same biological scale within the registered precision). Tier 1 textbook $a_0/\alpha$ scale + Tier 3 microtubule-lumen biological-scale match.
$\eta_{Zeno}$ (Zeno Coupling Efficiency): $\eta_{Zeno} = g_{single}^{eff}\sqrt{N_{bio}}/\kappa$ — the Tavis-Cummings cooperative-pumping ratio in the Identity Polaron's microtubule waveguide (V3 §13.1.2). It is the downstream robustness margin of a DMC-passing Polaron; the sharp Apperception boundary is the Dual Material Constraint gate.
$F_{CISS}$ (CISS enhancement factor): $\approx 2.0$ empirical import from spin-chemistry literature (Naaman et al. 2019, Nat. Rev. Chem. 3:250–260).
$N_{bio}$ : Effective number of cooperatively-pumped Trp dipoles in the microtubule lumen (V3 Ch13).

Experimental Anchors

3.55 keV line: Vacuum Quantum $E_{vac} = m_e/144 \approx 3.5486$ keV (Tier 2; V3 Ch11).
$\ell=6$ icosahedral PTA signature: $\Delta\Gamma_{max} \approx 1.73 \times 10^{-4}$ ( $\phi^{-18}$ ) (Tier 2; V3 Ch21).
GRB 090510 LIV bound: Abdo et al. 2009, Nature 462:331–334 — used as the empirical anchor for V3 Ch20 Protocol H.

Cross-Reference Conventions

See V2 Notation Reference for cross-reference conventions; identical here.
Volume-specific abbreviations: V3 §X.Y without further qualification refers to V3 Ch<X> §<X.Y>; cross-volume references are always written V<n> Ch<m> or App <letter>.

Volume 3 — The Matter Spectrum

Notation Quick Reference — Volume 3 (Matter Spectrum)

This reference complements the Volume 1 and Volume 2 Notation files. Symbols below are those most heavily used in V3 (Forces, Matter, Astrophysics, Experiments).

Standard Model Constants and Sectors

$\alpha$ , $\alpha^{-1}$ : Fine-structure constant. GCT bare value $\alpha^{-1}_{bare} = 360\phi^{-2} \approx 137.508$ (Tier 2 impedance mechanism + Tier 3 specific 360 multiplier, V1 Ch13); CODATA observed $\approx 137.036$ .
$\sin^2\theta_W$ : Weinberg angle. GCT bare volume-coupling input $= \phi^{-3} \approx 0.236$ ; GUT-scale Gram scalar $\rho_G=\phi^{-2}\approx0.382$ has normalized Cartan share $0.27639$ and is not itself the physical angle; Z-pole observed $0.23122$ (V3 Ch04).
$\theta_C$ : Cabibbo angle. GCT geometric value $\arcsin(\phi^{-3}(1-5\alpha))$ (V3 Ch10).
$\delta_{CP}$ : CP-violating phase. GCT geometric value $360°\phi^{-1}$ (V3 Ch10).

Mass Spectrum

$m_e$ : Electron mass. Formula $m_e = M_P \phi^{-107}(1-5\alpha)$ (Tier 2 mechanism + Tier 3 Coxeter-exponent handle + Tier 4 structural-link conjecture pending O.14; ~1006 ppm vs CODATA per engine output).
$m_\mu$ : Muon mass. Formula $m_\mu = m_e \phi^{11}(1+5\alpha+\phi^8\alpha^2)$ (Tier 2 mechanism + Tier 3 specific exponent + Tier 3 $+5\alpha$ coefficient normalization / pole-mass loop discipline -> Tier 3 headline precision; 21 ppm analytic residual).
$m_\tau$ : Tau mass. Formula $m_\tau = m_e \phi^{17}(1-3.6\alpha)$ (Tier 2 mechanism + Tier 3 specific exponent + Tier 3 combination rule, pending O.26b for tau coefficient stacking → Tier 3 headline coefficient; ~51 ppm against PDG 2024).
$m_p$ : Proton mass. Formula $m_p = m_e \phi^{15+\phi^{-1}}$ , prediction $\approx 938.417$ MeV against CODATA $938.272$ MeV (155 ppm residual; V3 Ch18 and App R §R.3).
$m_{p}/m_{e}$ ratio: $\phi^{15+\phi^{-1}} \approx 1836.437$ vs CODATA observed $1836.153$ .
$\Sigma m_\nu$ : Sum of neutrino masses. GCT normal-hierarchy scale $\Sigma m_\nu \approx 0.0853$ eV from $m_1=m_e\phi^{-36}$ plus oscillation gaps; Planck + DESI 2024 gives a $\Lambda$ CDM-context bound near $0.072$ eV at 95% CL, which is tension rather than the registered $<0.075$ eV / $>0.15$ eV exclusion band.

Lepton g-2 and Anomalies

$\Delta a_\mu$ : Muon anomalous magnetic moment correction. GCT vertex value $\approx 250.65 \times 10^{-11}$ (Tier 2 mechanism + Tier 3 $1/5$ coefficient + A3 + Tier 4 calibration-survival conjecture; Tension under WP2025, falsification conditional on long-term HVP-synthesis arbitration — see App V §P.5 for the HVP-Survival Condition).

Quasicrystal and Polaron Parameters

$\xi$ (Polaron Healing Length): $\xi = \hbar c/(\alpha^2 m_e c^2) = a_0/\alpha \approx 7.25$ nm (CODATA $\alpha$ ; the corrected GCT $\alpha^{-1}$ gives the same biological scale within the registered precision). Tier 1 textbook $a_0/\alpha$ scale + Tier 3 microtubule-lumen biological-scale match.
$\eta_{Zeno}$ (Zeno Coupling Efficiency): $\eta_{Zeno} = g_{single}^{eff}\sqrt{N_{bio}}/\kappa$ — the Tavis-Cummings cooperative-pumping ratio in the Identity Polaron's microtubule waveguide (V3 §13.1.2). It is the downstream robustness margin of a DMC-passing Polaron; the sharp Apperception boundary is the Dual Material Constraint gate.
$F_{CISS}$ (CISS enhancement factor): $\approx 2.0$ empirical import from spin-chemistry literature (Naaman et al. 2019, Nat. Rev. Chem. 3:250–260).
$N_{bio}$ : Effective number of cooperatively-pumped Trp dipoles in the microtubule lumen (V3 Ch13).

Experimental Anchors

3.55 keV line: Vacuum Quantum $E_{vac} = m_e/144 \approx 3.5486$ keV (Tier 2; V3 Ch11).
$\ell=6$ icosahedral PTA signature: $\Delta\Gamma_{max} \approx 1.73 \times 10^{-4}$ ( $\phi^{-18}$ ) (Tier 2; V3 Ch21).
GRB 090510 LIV bound: Abdo et al. 2009, Nature 462:331–334 — used as the empirical anchor for V3 Ch20 Protocol H.

Cross-Reference Conventions

See V2 Notation Reference for cross-reference conventions; identical here.
Volume-specific abbreviations: V3 §X.Y without further qualification refers to V3 Ch<X> §<X.Y>; cross-volume references are always written V<n> Ch<m> or App <letter>.