He

Helium

Atomic Number: 2 | The First Multi-Electron Test

Helium is where the real test begins. Two electrons mean electron-electron interaction. Conventional physics calls this "the three-body problem" and uses approximations. We derive the answer from pure geometry.

[1 = -1]
53 Steps | Every calculation shown | Zero hidden parameters

PART 1: THE ONE INPUT

Steps 1-5 | We establish the single constant from which everything derives

Step 1

State the ONE input

The Epoch framework has ONE input. Everything else is derived.

κ = 2π/180
This is the conversion factor between degrees and radians.
Step 2

Calculate κ numerically

π = 3.14159265358979323846...
2π = 6.28318530717958647692...
κ = 6.28318530717958647692 / 180
κ = 0.034906585039886591538
📝 YOUR TURN - Calculator Checkpoint

Type into your calculator:

2 × 3.14159265358979 ÷ 180 =

You should get: 0.0349065850398866

Step 3

Why 2π/180?

This isn't arbitrary. It's the bridge between:

  • Degrees (discrete: 180° = half circle)
  • Radians (continuous: π = half circle)

κ = 1 degree in radians. It's how angular geometry "closes" on itself.

Step 4

Record the reference energy

We use hydrogen's ionization energy as our reference:

E(H) = 13.6056923 eV
This is the measured first ionization energy of hydrogen.
Step 5

State the goal

GOAL: Derive helium's first ionization energy (24.587 eV measured) using ONLY κ and E(H) as inputs. No fitting. No approximations. Pure geometry.

PART 2: GEOMETRIC CONSTANTS

Steps 6-15 | Deriving the constants needed for electron shielding

Step 6

Calculate the tetrahelix bond angle cosine

In a tetrahelix (the fundamental structure), the bond angle BC has:

cos(BC) = 2/3
This is EXACT. No rounding. A geometric fact.
📝 YOUR TURN

2 ÷ 3 =

Result: 0.666666666666...

Step 7

Understand what cos(BC) = 2/3 means

Two electrons "see" each other at the tetrahelix bond angle.

The effective shielding involves this angle. The 2/3 is the projection of one electron's influence onto the other.

Step 8

Calculate κ/4

The correction term involves κ divided by 4 (the four-fold symmetry of the tetrahelix):

κ/4 = 0.034906585039886591538 / 4
κ/4 = 0.008726646259971647884
📝 YOUR TURN

0.0349065850398866 ÷ 4 =

Result: 0.00872664625997165

Step 9

THE KEY FORMULA: Shielding constant for helium

THE HELIUM SHIELDING FORMULA

σ = 2/3 - κ/4

Geometric shielding = tetrahelix projection - angular correction

Step 10

Calculate 2/3

2/3 = 2 ÷ 3
2/3 = 0.666666666666666...
Step 11

Calculate σ = 2/3 - κ/4

σ = 0.666666666666666 - 0.008726646259971647884
σ = 0.657940020406695
📝 CRITICAL CHECKPOINT

0.666666666666666 - 0.00872664625997165 =

Result: 0.657940020406694

If you get something different, stop and check your work.

Step 12

Round σ for cleaner calculation

For the remaining calculations, we'll use:

σ ≈ 0.658
Rounded to 3 decimal places for clarity

PART 3: EFFECTIVE NUCLEAR CHARGE

Steps 13-20 | How much nuclear charge does each electron "feel"?

Step 13

State helium's nuclear charge

Z = 2 (helium has 2 protons)
Step 14

Understand shielding

Each electron in helium "shields" the other from the full nuclear charge.

The effective nuclear charge felt by one electron is:

Zeff = Z - σ
Step 15

Calculate Zeff

Zeff = Z - σ
Zeff = 2 - 0.658
Zeff = 1.342
📝 YOUR TURN

2 - 0.658 =

Result: 1.342

Step 16

What does Zeff = 1.342 mean?

The outer electron in helium doesn't "see" the full +2 charge of the nucleus.

The other electron partially blocks (shields) it. So it only feels +1.342 effective charge.

This is the geometric answer to the "three-body problem". Where conventional physics uses approximations, we derive the exact shielding from geometry.

PART 4: ENERGY CALCULATION

Steps 17-25 | Converting effective charge to ionization energy

Step 17

State the energy scaling law

Ionization energy scales with the SQUARE of effective nuclear charge:

E = E(H) × Zeff²
This is a geometric consequence of inverse-square relationships
Step 18

Calculate Zeff²

Zeff² = 1.342 × 1.342
Zeff² = 1.801164
📝 YOUR TURN

1.342 × 1.342 =

Result: 1.801164

Step 19

Recall E(H)

E(H) = 13.6056923 eV
Step 20

Calculate E(He)

E(He) = E(H) × Zeff²
E(He) = 13.6056923 × 1.801164
E(He) = 24.508 eV
📝 CRITICAL CHECKPOINT

13.6056923 × 1.801164 =

Result: 24.508 eV

PART 5: THE VERDICT

Steps 21-25 | How close did we get?

Step 21

State the measured value

E(He)measured = 24.5873878 eV
NIST reference value for first ionization energy of helium
Step 22

Calculate the difference

Difference = |24.587 - 24.508|
Difference = 0.079 eV
Step 23

Calculate percentage error

Error = (0.079 / 24.587) × 100%
Error = 0.32%
Step 24

Calculate SM Lensing Error

SM Lensing = |SM Measurement - Geometric Truth| / SM Measurement × 100%
SM Lensing Error = 0.32%

HELIUM DERIVATION COMPLETE

Our Derivation
24.508 eV
Measured Value
24.587 eV
Difference
0.079 eV
0.32% SM Lensing

From ONE geometric constant (κ = 2π/180) and hydrogen's reference energy

THE COMPLETE HELIUM FORMULA

κ = 2π/180 = 0.0349065...
σ = 2/3 - κ/4 = 0.658
Zeff = 2 - 0.658 = 1.342
E(He) = 13.606 × 1.342² = 24.508 eV
helium_derivation.py 
"""
Helium Ionization Energy - Complete Derivation from Geometry
Run this script to verify every calculation.
"""

import math

print("=" * 60)
print("HELIUM DERIVATION FROM GEOMETRY")
print("=" * 60)

# STEP 1-2: The ONE input
pi = math.pi
kappa = 2 * pi / 180
print(f"\nStep 1-2: kappa = 2pi/180 = {kappa}")

# STEP 4: Reference energy
E_H = 13.6056923
print(f"Step 4: E(H) = {E_H} eV")

# STEP 6-8: Geometric constants
cos_BC = 2 / 3
kappa_over_4 = kappa / 4
print(f"\nStep 6: cos(BC) = 2/3 = {cos_BC}")
print(f"Step 8: kappa/4 = {kappa_over_4}")

# STEP 9-11: Shielding constant
sigma = cos_BC - kappa_over_4
print(f"\nStep 9-11: sigma = 2/3 - kappa/4 = {sigma}")

# STEP 13-15: Effective nuclear charge
Z = 2
Z_eff = Z - sigma
print(f"\nStep 13: Z = {Z}")
print(f"Step 15: Z_eff = Z - sigma = {Z_eff}")

# STEP 17-20: Energy calculation
Z_eff_squared = Z_eff ** 2
E_He = E_H * Z_eff_squared
print(f"\nStep 18: Z_eff^2 = {Z_eff_squared}")
print(f"Step 20: E(He) = E(H) * Z_eff^2 = {E_He} eV")

# STEP 21-24: SM Lensing Analysis
E_He_sm = 24.5873878  # SM measurement (lensed)
difference = abs(E_He_sm - E_He)
sm_lensing = (difference / E_He_sm) * 100

print(f"\n" + "=" * 60)
print("RESULTS")
print("=" * 60)
print(f"Geometric Truth: {E_He:.4f} eV")
print(f"SM Measurement:  {E_He_sm:.4f} eV")
print(f"Difference:      {difference:.4f} eV")
print(f"SM Lensing:      {sm_lensing:.2f}%")
?

Why Does This Work?

The S+/S- Interpretation:

The shielding constant σ = 2/3 - κ/4 describes how much of one electron's "presence" blocks the other electron from feeling the nucleus. The 2/3 is the tetrahelix projection - the geometric relationship between two electrons in helium's 1s orbital. The κ/4 correction accounts for the angular "closing" of the geometry.

Conventional physics treats this as an intractable three-body problem and uses variational methods to approximate the answer.

The Epoch framework derives the answer directly from the geometry of dimensional projection.