Hydrogen

Atomic Number: 1 | The Simplest Atom

Hydrogen is the foundation. One proton, one electron. The simplest possible atom. We establish the reference energy here that all other derivations build upon.

[1 = -1]

Why Hydrogen is the Foundation

Hydrogen is unique among atoms. With only one proton and one electron, there is NO electron-electron interaction to complicate things. This makes hydrogen the perfect reference point.

The measured first ionization energy of hydrogen is:

E(H) = 13.6056923 eV

This is the energy required to remove the one electron from hydrogen

This value is also called the Rydberg energy (R_H). It is one of the most precisely measured values in all of physics.

The Geometric Foundation

In the Epoch framework, the hydrogen ionization energy is not a mysterious "fundamental constant" - it emerges from geometry. Here's how:

Start with κ:

κ = 2π/180

κ = 2 × 3.14159265358979 / 180

κ = 6.28318530717958 / 180

κ = 0.0349065850398866

📝 YOUR TURN - Checkpoint 1

Get your calculator and type:

2 × 3.14159265358979 / 180 =

Expected result: 0.0349065850398866

If you don't get this number, something is wrong. Stop and check your calculator.

The Hydrogen-Kappa Relationship

The hydrogen ionization energy relates to κ through the fine structure constant α. In the Epoch framework:

α ≈ (κπ/15) × (1 - κ²)

α ≈ 0.00730190...

See the Fine Structure Constant white paper for full derivation

The hydrogen energy emerges from the relationship:

E(H) = (1/2) × m_e × c² × α²

Where m_ec² = 511 keV (electron rest mass energy)

Let's verify this:

m_ec² = 511,000 eV = 511 keV

α² = (0.0072973525693)² = 0.0000532502...

E(H) = (1/2) × 511,000 × 0.0000532502

E(H) = 255,500 × 0.0000532502

E(H) = 13.604 eV

📝 YOUR TURN - Checkpoint 2

Verify the calculation:

0.5 × 511000 × 0.0000532502 =

Expected result: ~13.6 eV

The Simple Truth About Hydrogen

Here's the honest statement about hydrogen in the Epoch framework:

The hydrogen ionization energy (13.606 eV) is used as the REFERENCE POINT.

We do not claim to derive 13.606 eV from scratch using only κ. What we claim is:

The fine structure constant α DOES emerge from κ (see companion white paper)
The hydrogen energy relates to α² geometrically
Once we have E(H), we can derive all other atomic energies from geometry

The power of the Epoch framework is not in deriving hydrogen - it's in deriving EVERYTHING ELSE from hydrogen using geometric relationships.

Hydrogen as the S+/S- Boundary

In the S+/S- framework, the hydrogen atom represents the simplest case of an electron existing at the boundary between manifested (S+) and potential (S-) reality.

The bound electron: Exists in S+ - it has a definite energy state, a probability distribution, measurable properties.

The ionization energy: Is the energy required to move the electron from S+ (bound) to S- (free/potential).

The 13.606 eV represents the "height" of the S+/S- boundary for a single electron bound to a single proton. This is why it serves as the reference:

For Z = 1 (hydrogen): E = R_H × Z² = 13.606 × 1 = 13.606 eV

For Z = 2 (helium-like): E = R_H × Z² = 13.606 × 4 = 54.424 eV

But helium has TWO electrons, so shielding reduces Z_eff

Preparing for Multi-Electron Atoms

The real power of the Epoch framework appears when we move beyond hydrogen. We need to calculate the geometric constants that will determine electron shielding:

The Shadow Constant (κ_shadow)

κ_shadow = 1/κ

κ_shadow = 1 / 0.0349065850398866

κ_shadow = 28.6478897565412

📝 YOUR TURN - Checkpoint 3

1 / 0.0349065850398866 =

Expected: 28.6478897565412

The Projection Factor (P)

P = √3 / (2π)

P = 1.73205080756888 / 6.28318530717958

P = 0.275664407370958

The Tetrahelix Ratio (cos BC = 2/3)

cos(BC) = 2/3

cos(BC) = 0.666666...

This is EXACT - the tetrahelix bond angle cosine

The Helix Overlap (σ = 5/16)

σ = 5/16

σ = 0.3125

This is EXACT - no rounding

HYDROGEN SUMMARY

Reference Energy

13.606 eV

Status

BASELINE

Hydrogen establishes the reference. The real test comes with helium and lithium.

KEY FORMULAS FOR MULTI-ELECTRON ATOMS

E(atom) = E(H) × Z_eff²

Where Z_eff = Z - σ (nuclear charge minus geometric shielding)

For Helium: σ = 2/3 - κ/4

For Lithium: Uses shell structure + P factor

                hydrogen_reference.py
                Python Verification Script
            

"""
Hydrogen Reference Values - Verification Script
The foundation for all atomic derivations.
"""

import math

print("=" * 60)
print("HYDROGEN REFERENCE VALUES")
print("=" * 60)

# The ONE input
pi = math.pi
kappa = 2 * pi / 180
print(f"\nκ = 2π/180 = {kappa}")

# Derived constants
kappa_shadow = 1 / kappa
P = math.sqrt(3) / (2 * pi)
cos_BC = 2 / 3
sigma = 5 / 16

print(f"\nκ_shadow = 1/κ = {kappa_shadow}")
print(f"P = √3/(2π) = {P}")
print(f"cos(BC) = 2/3 = {cos_BC}")
print(f"σ = 5/16 = {sigma}")

# Hydrogen reference
E_H = 13.6056923  # eV - measured value
print(f"\nHydrogen ionization energy: {E_H} eV")
print("This is the REFERENCE for all other derivations.")

# Fine structure constant (derived from kappa)
alpha_derived = (kappa * pi / 15) * (1 - kappa**2)
alpha_measured = 0.0072973525693
print(f"\nα (derived) = {alpha_derived}")
print(f"α (measured) = {alpha_measured}")
print(f"Agreement: {100 - abs(alpha_derived - alpha_measured)/alpha_measured * 100:.4f}%")

print("\n" + "=" * 60)
print("READY FOR HELIUM AND LITHIUM DERIVATIONS")
print("=" * 60)
            