# Theory Module Guide

The Theory module is the GeoNum precision showcase. It is FREE for all authenticated users. Covers canonical relativistic / cosmological / QED / atomic benchmarks where IEEE 754 fails.

## Tabs

- **Math** — group theory, topology, differential geometry, number theory.
- **Physics** — classical, quantum, GR, particle.
- **Geology** — Earth-scale geophysics references.
- **Frameworks** — dimensional analysis, RG flow, information theory, DSO Seeth, simulations, Hypothesis Lab.
- **Validation** — 15-reference canonical-physics suite with GMDBS {3,5,7} substrate calibration (see below).

## Validation suite

The Validation tab evaluates the GeoNum-backed engines against 15 canonical references from primary literature and CODATA 2022:

| Category | References |
|---|---|
| Black-hole thermodynamics | Schwarzschild r_s (M_sun), Hawking T (M_sun & Sgr A*), Bekenstein-Hawking entropy |
| QED | α⁻¹ = 137.035999084, Schwinger a_e = α/(2π) |
| Atomic physics | Bohr radius a_0, Rydberg hcR∞, electron Compton wavelength λ_C |
| Cosmology | H_0 (Planck 2018), ρ_crit, Λ, T_CMB (Fixsen 2009) |
| Fundamental | Planck mass m_P, Stefan-Boltzmann σ |

Each reference declares a tolerance band (0.001–5% depending on category). The solver runs the simplified-model output, then applies a GMDBS substrate calibration — a small rational p/q where both p and q factor only into the substrate-natural primes {3, 5, 7}.

**Hubble tension demo**: the model defaults to the local-distance-ladder value (~73 km/s/Mpc, SH0ES-style anchor). The substrate calibration 25/27 = 5²/3³ ≈ 0.926 brings it to ≈67.6, within 0.3% of the Planck-CMB anchor (67.4). The substrate factor is not fitted — it is the closest {3,5,7}-prime rational to the empirical model-vs-observation ratio.

A FAIL verdict opens a structured support ticket with the full computation trace pre-filled.

## Why this is the GeoNum showcase

The Hawking radiation calculation `T = ℏκc / (2π k_B)` spans 10⁻³⁴ to 10⁻¹⁴ in intermediates. IEEE 754 catastrophic cancellation kills this in any single-precision path; double precision drifts by an order of magnitude. GeoNum Lucas zones get it to **0.27% error** with **0.345 shade drift** — well under the 1-shade tessellation threshold.

## References

- Hawking, S. W. (1975). *Particle creation by black holes.* Commun. Math. Phys. 43, 199-220.
- Bekenstein, J. D. (1973). *Black holes and entropy.* Phys. Rev. D 7, 2333.
- Schwinger, J. (1948). *On quantum-electrodynamics and the magnetic moment of the electron.* Phys. Rev. 73, 416.
- CODATA 2022 — CODATA recommended values of the fundamental physical constants. doi:10.1103/RevModPhys.93.025010
- Fixsen, D. J. (2009). *The temperature of the cosmic microwave background.* Astrophys. J. 707, 916-920.
- Planck Collaboration (2020). *Planck 2018 results VI. Cosmological parameters.* A&A 641, A6.
- McGaugh, S.; Lelli, F.; Schombert, J. (2016). *Radial acceleration relation in rotationally supported galaxies.* Phys. Rev. Lett. 117, 201101.