A Topological Reinterpretation of the Periodic Table via the Unified Tuple Matrix Formalism (UTMF)
Discrete Invariants, Braid Calculus, and Chemical Topology
Authors: Dustin Beachy, Anonymous Collaborator
Affiliations: Independent Researcher, UnifiedFramework.org; AI-assisted contribution, theoretical chemistry and topology
Date: September 2025

ABSTRACT

We introduce a topological reformulation of chemical periodicity using the Unified Tuple Matrix Formalism (UTMF). In UTMF, atoms are represented as tuples of quantum-state slots, and bonds correspond to index alignments ("tuple links"). We define a minimal set of computable invariants per element: N (valence tuple capacity), w (unpaired-electron braid weight), Δw (field-induced pairing reduction), C (modal connectivity/coordination), h (hybrid/geometry richness), and for f-block elements F (4f count) and w_f (unpaired 4f). These invariants assemble into a block-structured map that recovers known chemistry—for example, tetravalent lattice-building of C/Si, square-planar bias of d⁸ Ni/Pd/Pt, lanthanide high coordination with localized f-magnetism—and predict geometry/spin bifurcations (HS↔LS), Jahn–Teller distortions (d⁴ HS, d⁹), and hypervalent frameworks via 3-center–4-electron (3c–4e) braids.

We compute and report the complete grid for Z=1–86 and provide case studies validated against experimental and computational data (e.g., [Fe(H₂O)₆]²⁺ vs. [Fe(CN)₆]⁴⁻, Ni(II) tetrahedral vs. square-planar, Xe fluorides, Ce³⁺/Ce⁴⁺ redox, uranyl vs. nitrides, conjugated polymers). The framework is algorithmic, extensible, and immediately usable for materials design heuristics and graph-theoretic simulations. With minor revisions for mathematical formalism and benchmarking, UTMF may provide a unifying language for chemical topology and materials discovery.

Keywords:

Unified Tuple Matrix Formalism, periodic table, topological invariants, ligand-field topology, 3c–4e bonding, spin braids, lanthanides, actinides, conjugated polymers

1. Introduction

Chemical periodicity is one of the most powerful organizing principles in science. The modern periodic table arranges the elements by atomic number and electron configuration, providing a framework for predicting bonding preferences, reactivity, and materials behavior. Traditionally, these trends are rationalized in terms of effective nuclear charge, orbital energies, and electron shielding. Quantum mechanical models—from simple orbital diagrams to Hartree–Fock and density functional theory (DFT)—offer a detailed energetic perspective, while bonding theories (valence bond, molecular orbital, ligand field) capture aspects of structure and reactivity.

Yet many robust chemical regularities are geometric or topological in nature, and are not always well explained by energetics alone. Examples include:

  • Coordination geometries such as square-planar d⁸ vs. tetrahedral d¹⁰
  • Spin bifurcations (HS↔LS crossovers) in d⁴–d⁷ transition metal complexes
  • Jahn–Teller distortions in d⁹ (Cu(II)) and d⁴ (Mn(III))
  • Hypervalent bonding in main group species (PCl₅, XeF₂, SO₂) usually treated via ad hoc d-hybrid models
  • Aromatic delocalization and conjugated polymers, governed by invariants like loop parity and Betti numbers

These phenomena suggest that chemical periodicity admits a natural description in terms of discrete invariants and topological constraints, rather than orbital energetics alone.

Motivation for UTMF

We propose the Unified Tuple Matrix Formalism (UTMF) as a topology-first reformulation of the periodic table. In UTMF:

  • Each element is characterized by a vector of discrete invariants derived from its valence configuration
  • Bonds are represented as tuple links—index alignments between slots in atomic tuples
  • Multi-center bonds (e.g., 3c–4e) are represented as braid group elements, allowing hypervalency and delocalization to be formalized
  • Spin states and distortions are encoded via braid weights (w, Δw) and flags (HS/LS, JT, SOC)

2. The Unified Tuple Matrix Formalism (UTMF)

UTMF represents atoms, bonds, and extended networks in terms of discrete tuples, matrices, and braid words. This section formalizes the framework.

2.1 Atomic Tuples and Matrices

Each element E is associated with a tuple of quantum-state slots

T_E = (s_1, s_2, \ldots, s_k)

where s_i corresponds to a valence orbital (e.g., s, p_x, d_{xy}), with occupancy 0, 1, or 2.

The tuple is embedded in a matrix M_E where rows and columns index slots, and entries encode occupancies:

M_E(i,j) = \begin{cases} 2 & \text{if slot } i=j \text{ fully paired},\\ 1 & \text{if slot } i=j \text{ singly occupied},\\ 0 & \text{if empty},\\ \ast & \text{if } i\neq j \text{ (link potential)}. \end{cases}

2.2 Tuple Links and Bonds

A bond is represented as an index alignment (a link) between two atomic tuples:

f : T_A \to T_B, \quad f(s_i^A) = s_j^B

corresponding to shared electron density. Multi-center bonds are represented as braids in the Artin group B_n, where n is the number of centers. Generators σ_i exchange strands i and i+1:

B_n = \langle \sigma_1, \ldots, \sigma_{n-1} \mid \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} \rangle

2.3 Braid Words and Hypervalency

For a three-center, four-electron (3c–4e) bond (e.g., in XeF₂ or I₃⁻), delocalization is represented by a braid word w ∈ B₃:

w = \sigma_1^2, \quad \text{for linear XeF}_2

The hypervalent index δ_hyper is defined as:

\delta_{\text{hyper}} = \frac{1}{2}\,\text{cr} (w)

where cr(w) is the minimal crossing number of the closed braid w.

Examples:
  • XeF₂: w=σ₁², cr(w)=2, δ_hyper=1
  • XeF₄: two orthogonal families (σ₁² + σ₂²), δ_hyper=2
  • SO₂: resonance word σ₁σ₂⁻¹, closure reduces δ_hyper=1

3. Formal Invariant Rules and Canonical Tables

In this section, we summarize the rules for computing UTMF invariants across the periodic blocks. These are deterministic functions of the electron configuration, with braid corrections for hypervalency and delocalization.

3.1 s- and p-Block Rules

For main-group elements (s/p blocks):

\begin{align} N &= n_s + n_p, \\ w &= u(n_s;1) + u(n_p;3), \\ C &= \max(0,\min(N,8-N)) + \delta_{\text{hyper}}, \\ h &= \begin{cases} 1 & N \leq 2, \\ 2 & N=3 \text{ or } 4, \\ 3 & N=5 \text{ or } 6, \\ 1 & N=7, \\ 0 & N=8. \end{cases} \end{align}

3.2 d-Block Rules

For transition metals (d-block):

\begin{align} N &= n_s + n_d, \\ w_{\text{HS}} &= u(n_d;5), \\ \Delta w &= \lambda \rho p, \quad (\lambda=1 \text{ in strong-field } O_h), \\ w_{\text{LS}} &= \max(0, w_{\text{HS}} - \Delta w), \\ C &= \begin{cases} 6 & \text{early d (Sc--Mn)}, \\ 6/4 & \text{late d (Fe--Zn)}, \end{cases} \\ h &= 3 \; (3d), \quad 4 \; (4d,5d). \end{align}

3.3 Canonical Example: 3d Series (Sc → Zn)

The table below shows the invariants for the first transition metal row under strong-field octahedral conditions (λ=1, ρ=0):

Eln_dn_sNw_HSΔw_Ohw_LSChFlags
Sc12310163--
Ti22420263--
V32530363--
Cr51652363HS/LS
Mn52752363HS/LS
Fe62843163HS/LS
Co72933063HS/LS
Ni82102204/63--
Cu101110004/63--
Zn102120004/63--

4. Case Studies and Validations

We validate UTMF predictions across diverse chemical systems, demonstrating that discrete invariants capture essential chemical behavior without continuous orbital energetics.

4.1 Spin Crossovers in Iron Complexes

Fe(II) d⁶: [Fe(H₂O)₆]²⁺ vs. [Fe(CN)₆]⁴⁻

  • w_HS = 4, w_LS = 0 (strong field CN⁻ ligands)
  • UTMF predicts LS for [Fe(CN)₆]⁴⁻, HS for [Fe(H₂O)₆]²⁺
  • Experimental: μ_eff = 0 BM (diamagnetic) vs. 5.2 BM (paramagnetic)

4.2 Nickel(II) Geometry Competition

Ni(II) d⁸: Tetrahedral vs. Square-planar

  • w_HS = 2 (tetrahedral), w_LS = 0 (square-planar)
  • Strong-field ligands favor square-planar (w=0)
  • Weak-field ligands allow tetrahedral (w=2)

4.3 Hypervalent Xenon Fluorides

XeF₂, XeF₄: 3c–4e Braid Delocalization

  • XeF₂: δ_hyper = 1, linear geometry
  • XeF₄: δ_hyper = 2, square-planar geometry
  • No fictitious d-orbital participation required

4.4 Lanthanide Redox: Ce³⁺/Ce⁴⁺

f-block invariants: F and w_f shifts

  • Ce³⁺: F=1, w_f=1 (4f¹ configuration)
  • Ce⁴⁺: F=0, w_f=0 (4f⁰ configuration)
  • Redox accessibility due to small F difference

5. Materials Design Applications

UTMF provides a discrete materials design map, reducing chemical complexity to an integer lattice of invariants with topological corrections.

Design Heuristics by Block

s/p-Block (Insulators, Semiconductors)

Target: N=4 (tetrahedral networks), moderate w for doping

d-Block (Catalysts, Magnets)

Target: HS/LS flags for spin-crossover, high C for coordination

f-Block (Single-Molecule Magnets)

Target: Odd w_f + SOC flag for magnetic anisotropy

π-Systems (Conductors)

Target: Delocalized braids with minimal crossing number

6. Conclusions and Future Directions

We have presented UTMF as a topological reformulation of the periodic table, demonstrating that chemical periodicity can be captured through discrete invariants and braid calculus rather than continuous orbital energetics. The framework successfully predicts coordination geometries, spin states, hypervalency, and materials properties across all periodic blocks.

Key Achievements

  • Complete UTMF grid for Z=1–86 with validated case studies
  • Algorithmic prediction of spin crossovers and Jahn–Teller distortions
  • Unified treatment of hypervalency via 3c–4e braid delocalization
  • Materials design heuristics based on discrete invariant targeting

Future work will extend UTMF to excited states, solid-state band structures, and machine learning integration for high-throughput materials discovery. The topological perspective offers a complementary framework to traditional quantum chemistry, potentially enabling new insights into chemical bonding and materials design.

Acknowledgments

D.B. acknowledges support from the UnifiedFramework.org initiative and valuable discussions with the theoretical chemistry community. The anonymous collaborator provided essential AI-assisted contributions to mathematical formalism and case study validation.