Title:Periodic Table as a Consequence of the Symmetric Existence Framework:Derivation of Shell Structure, Orbital Geometry, and Period Pairingfrom Two Structural Laws Author:Dr. Ashwini Kumar SharmaShant Sharma Aushadh Bhavan, Churu 331001, Rajasthan, IndiaEmail: sef.sciences@gmail.comORCID: 0009-0002-4143-5660 Companion to:Symmetric Existence Framework: Standard Edition-1doi: 10.5281/zenodo.20528485 Description:This paper shows that the structural content of the Periodic Tableof 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 sufficientto 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 gaschemical inertness, d-shell special stability, and lanthanidecontraction are all explained qualitatively from PP Law 2 alone. What this paper does not claim:Quantitative shell energies beyond helium require a PETN correctionfactor 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 separatederivation at the nuclear BRS scale (Open Target PT-Q3). These areidentified honestly as future work. The Periodic Table is identified as the atomic-scale expression ofPP 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 SymmetricExistence Framework: Derivation of Shell Structure, Orbital Geometry,and Period Pairing from Two Structural Laws.Zenodo. doi: 10.5281/zenodo.20570870