Periodic Table as a Consequence of the Symmetric Existence Framework

Title:
Periodic Table as a Consequence of the Symmetric Existence Framework:
Derivation of Shell Structure, Orbital Geometry, and Period Pairing
from Two Structural Laws

Author:
Dr. Ashwini Kumar Sharma
Shant Sharma Aushadh Bhavan, Churu 331001, Rajasthan, India
Email: sef.sciences@gmail.com
ORCID: 0009-0002-4143-5660

Companion to:
Symmetric Existence Framework: Standard Edition-1
doi: 10.5281/zenodo.20528485

Description:
This paper shows that the structural content of the Periodic Table
of Elements follows from two laws of the Symmetric Existence Framework
(SEF), with no empirical inputs and no additional postulates.

The two laws are:
  PP Law 1 (Potency Conservation): Pc + Pd = P0 = constant
  PP Law 2 (Z2 Balance): Delta-Pc = -Delta-Pd

These two laws, together with three-dimensional space, are sufficient
to derive four formal theorems:

  Theorem PT-1 (Shell Capacity):
  The maximum number of quantum states in shell n is N(n) = 2n^2.
  This follows from the Z2 binary structure of PP Law 2 and the
  geometry of three-dimensional space. No empirical input is used.

  Theorem PT-2 (Pauli Exclusion Principle):
  No two fermions can occupy the same quantum state. This is derived
  as a consequence of PP Law 1: identical co-occupation would require
  the total potency at a node to equal 2*P0, violating conservation.
  The Pauli Exclusion Principle, a postulate in quantum mechanics,
  is here a theorem.

  Theorem PT-3 (Spin = 1/2):
  Matter particles are spinors because PP Law 2 defines a Z2 binary
  whose representation in the three-dimensional rotation group SO(3)
  is the spinor representation. Spin-1/2 is derived, not postulated.

  Theorem PT-4 (Period Pairing):
  The observed period lengths 2, 8, 8, 18, 18, 32, 32 arise because
  PP Law 1 requires every energy level to have both a Physical Universe
  (PU) realisation and a Conjugate Universe (CU) realisation. PU and
  CU are complementary branches governed by inverse potency rules,
  not mirror images of each other. Together they satisfy PP Law 1
  completely.

Additionally, orbital geometry (s, p, d, f types and their
(2l+1) orientations), the Aufbau filling principle, noble gas
chemical inertness, d-shell special stability, and lanthanide
contraction are all explained qualitatively from PP Law 2 alone.

What this paper does not claim:
Quantitative shell energies beyond helium require a PETN correction
factor f(n,l,Z) that has not yet been derived (Open Target PT-Q1).
Nuclear magic numbers (2, 8, 20, 28, 50, 82, 126) require a separate
derivation at the nuclear BRS scale (Open Target PT-Q3). These are
identified honestly as future work.

The Periodic Table is identified as the atomic-scale expression of
PP Law 2 acting on quantized levels in three-dimensional space.

Keywords:
periodic table, shell structure, Pauli exclusion principle, spin,
orbital geometry, Aufbau principle, period pairing, Conjugate Universe,
symmetric existence framework, PP Law 1, PP Law 2, Z2 symmetry,
potency conservation, shell capacity 2n^2, complementarity

License:
Creative Commons Attribution 4.0 International (CC BY 4.0)

DOI:
10.5281/zenodo.20570870

How to cite:
Sharma, A.K. (2026). Periodic Table as a Consequence of the Symmetric
Existence Framework: Derivation of Shell Structure, Orbital Geometry,
and Period Pairing from Two Structural Laws.
Zenodo. doi: 10.5281/zenodo.20570870