The periodic table in 4 steps

Demonstration

Four steps, zero parameter

1. 1

   Pauli ⇒ two states per orbit

   The {1 ↔ 2} involution forbids identical fermions from sharing an orbit.

   A direct consequence of T1 (mod-3 forbidden transitions) read on the binary channel p = 2. Each magnetic-quantum-number m_ℓ orbit hosts at most two occupants (spin ↑ and spin ↓). No parameter, no ansatz.

   → Universal factor 2 across all shells.

2. 2

   Shell capacity: 2(2ℓ + 1)

   The Z/(2p)Z channel carries 2(2ℓ + 1) distinct states.

   The orbit ℓ admits (2ℓ + 1) magnetic orientations (C1 reading of the Z/(2ℓ+1)Z polygon). Multiplied by the factor 2 (step 1) gives the capacity.

   → s: 2 · 1 = 2 · p: 2 · 3 = 6 · d: 2 · 5 = 10 · f: 2 · 7 = 14

3. 3

   Which channels are active?

   At μ* = 15, only primes {3, 5, 7} satisfy γ_p > 1/2.

   γ_3 = 0.808, γ_5 = 0.696, γ_7 = 0.595, but γ_11 = 0.426 < 1/2. So p = 11 is inactive. No g block (ℓ = 4) because that would need p = 11 to be active.

   → Four blocks total: s (ℓ=0), p (ℓ=1), d (ℓ=2), f (ℓ=3). No fifth.

4. 4

   Period L(k): forced combinatorics

   L(k) = 2⌈k/2⌉² yields 2, 8, 8, 18, 18, 32, 32.

   Count the new shells activated each period (Aufbau / Madelung) and sum the capacities. The parity of k yields the closed formula.

   → Reproduces exactly the observed periodic-table periods.

Not exceptions

Anomalies are consequences

Cr, Cu, Mo, Pd, Ag, Au, lanthanides… are not ad-hoc "exceptions". They are informational stabilities forced by the geometry of the channel polygons.

Reproducibility

Python code: 8 lines to derive everything

The ptc.periodic module implements the 4 steps in fewer than 100 lines. Canonical usage:

from ptc.periodic import (
    period_length, block_for_Z, aufbau_config, capacity_for_l
)

# Step 2: closed capacities
for l, name in [(0, 's'), (1, 'p'), (2, 'd'), (3, 'f')]:
    print(f"{name} (ℓ={l}): {capacity_for_l(l)} states")

# Step 4: periods 1 to 7
for k in range(1, 8):
    print(f"Period {k}: {period_length(k)} elements")

# Block and configuration from Z
for Z in [1, 6, 26, 47, 92, 118]:
    print(f"Z={Z:3d}: block {block_for_Z(Z)} | {aufbau_config(Z)}")

# Expected output (excerpt):
#   s (ℓ=0): 2 states
#   p (ℓ=1): 6 states
#   d (ℓ=2): 10 states
#   f (ℓ=3): 14 states
#   Period 1: 2 elements
#   Period 2: 8 elements
#   ...
#   Z=  6: block p | 1s² 2s² 2p²
#   Z= 26: block d | [Ar] 3d⁶ 4s²
#   Z= 47: block d | [Kr] 4d¹⁰ 5s¹     ← half-closure anomaly
#   Z= 92: block f | [Rn] 5f³ 6d¹ 7s²

Integrity tests

# tests/test_periodic.py
def test_capacities():
    assert capacity_for_l(0) == 2
    assert capacity_for_l(1) == 6
    assert capacity_for_l(2) == 10
    assert capacity_for_l(3) == 14

def test_periods():
    assert [period_length(k) for k in range(1, 8)] == [2, 8, 8, 18, 18, 32, 32]

def test_no_g_block():
    # p=11 inactive at fixed point μ*=15 ⇒ no g block
    from ptc.gamma import gamma_p
    assert gamma_p(11, mu=15) < 0.5

def test_anomalies():
    # Cu = [Ar] 3d¹⁰ 4s¹ (pentagon closure)
    assert aufbau_config(29) == "[Ar] 3d¹⁰ 4s¹"
    # Au = [Xe] 4f¹⁴ 5d¹⁰ 6s¹ (pentagon closure)
    assert aufbau_config(79).endswith("5d¹⁰ 6s¹")

Score

Comparison with experiment

Going further

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