The Monster Curvature Principle

by [Gabriel Brennan], July 16, 2025

Introduction

What began as a speculative conversation about photons and geometry has unfolded into a surprisingly coherent framework—one that connects quantum emergence, curvature, group theory, and even the elusive Monster group. This article presents a new conceptual tool: The Monster Curvature Principle—a proposal that the geometry of all possible worlds is encoded in the curved surface of a monstrous, high-dimensional symmetry object. We arrived here through a series of insights, beginning with the interaction of photon bivectors and culminating in a new interpretation of scattering amplitudes, supersymmetry, and emergent physical laws.

Photon Collisions and the Birth of Mass

The journey began by examining photon-photon collisions. Normally, we regard photons as massless, propagating entities that travel at the speed of light. But under the right conditions, particularly when two photon-like objects interact via geometric algebra, something astonishing can happen: a trapped toroidal configuration emerges.

By modeling this interaction with bivectors and a Baker-Campbell-Hausdorff (BCH) series, we observed that two space-like objects (photons) could produce a time-like evolution. The resulting object—a multivector structure—exhibits closed trigonometric cycles and hyperbolic evolution components. It’s a rotor with both spin and boost characteristics, forming what can be visualized as a torus. The electron field we wrote down looked like:

This multivector captures how energy, spin, and momentum might condense into a rest-mass particle such as an electron. The sinusoidal and hyperbolic parts reflect internal circular symmetries (spin) and external time evolution (momentum), respectively.

Rotors Are Not Fundamental

Upon deeper reflection, we realized that these rotors—the building blocks of transformations in geometric algebra—are themselves emergent phenomena. They arise statistically from correlation cycles among fundamental “events” or fluctuations. The sine and cosine functions that appear in wavefunctions or rotation operators are not primitive structures, but the result of patterns in large numbers.

This leads us to a remarkable conjecture: the trigonometric and hyperbolic functions emerge from the statistical behavior of large numbers of correlations, perhaps through permutations or interaction rules.

The classical world emerges as a sort of “consensus geometry,” where enough micro-interactions repeat with enough structure that persistent patterns arise. These patterns stabilize into things like spinors, rotors, and conserved quantities such as mass and momentum.

Minimal Action as a Statistical Attractor

This statistical view gives new life to the principle of least action. Why does nature follow the path of minimal action? Perhaps not because it’s chosen a perfect optimization, but because those paths are the most statistically stable. Trigonometric correlation patterns (like sine waves) might act as statistical attractors—the natural outcome of large sets of fluctuating, minimally constrained systems.

When we observe a system evolving along the path of least action, we may be seeing the path of highest statistical persistence—a form of emergent order arising from the sea of microscopic chaos.

The Hyperbolic and Trigonometric Dance

An important insight emerged: closed trigonometric cycles (like sine functions) do not persist on their own. For them to evolve or propagate, they must “ride” on hyperbolic evolution—open-ended structures like sinh⁡\sinhsinh and cosh⁡\coshcosh, which are characteristic of wave propagation in spacetime.

This insight reveals the deep link between wave functions and the d’Alembertian operator in physics. The d’Alembertian is a hyperbolic operator, and the solutions it governs are sine waves. Thus, in a very real sense, a trigonometric cycle requires a hyperbolic path to survive—a trigonometric wave function continues to exist by evolving through hyperbolic spacetime.

Emergent Geometry and the Law of Large Numbers

We conjectured that these emerging correlations might be shaped by kissing number configurations (from sphere packing) and even prime number factorizations. These discrete mathematical structures can serve as the scaffolding for how connections form and decay—how complexity gives rise to structure.

We are not claiming that the universe literally counts primes or arranges spheres in tight packs—but rather that these mathematical phenomena mirror the types of combinatorial rules that underlie emergent geometry. In the large number limit, the apparently chaotic behavior of fundamental fluctuations coalesces into smooth trigonometric fields—what we call quantum fields.

The Amplituhedron: A Portal to Geometry

At this stage, we turned our attention to scattering amplitudes. A revolutionary idea in theoretical physics is the Amplituhedron, a geometric object proposed to compute particle interactions without relying on spacetime or Lagrangians.

Definition: The Amplituhedron

The amplituhedron is a mathematical object in projective space that encodes scattering amplitudes of certain quantum field theories, especially in planar supersymmetric Yang-Mills theory. Its volume corresponds to the probability amplitude of a given process, bypassing traditional Feynman diagrams.

Each facet or “corner” of the amplituhedron is labeled by permutation data—combinatorial structures describing how particles interact. These corners are points of geometric coherence, where symmetry and conservation laws converge.

The Monster Group: The Hidden Symmetry

Our journey now arrives at one of the most mysterious and beautiful mathematical objects ever discovered: the Monster group.

Definition: The Monster Group

The Monster group is the largest of the sporadic simple groups in group theory. It contains over 8x10^53 elements and exhibits deep connections with number theory, modular functions, string theory, and even quantum gravity. The phenomenon known as Monstrous Moonshine links this group to modular forms and conformal field theories in a completely unexpected way.

What if the amplituhedron is not the whole story, but just one local patch of a larger, more intricate monster polytope whose corners and symmetries are governed by the Monster group?

The Monster Curvature Principle

Here, we arrive at the central thesis:

The curvature observed in particle interactions, the anomalies in symmetry, and the emergence of spacetime dynamics are all reflections of our local universe being a curved patch on the surface of a Monster-like symmetry polytope.

We propose the Monster Curvature Principle, which asserts:

1. Each amplituhedron is a facet of a much larger hyperdimensional object.

2. SUSY-breaking and anomaly phenomena arise when the local patch fails to close—signaling curvature in this broader polytope.

3. The “surface” of this polytope, if it can be defined, encodes the configuration space of all possible worlds.

4. The entirety of this polytope is governed by the Monster group and its permutations.

This reinterpretation transforms the amplituhedron into a window—a local geometric patch on a far vaster, highly symmetric structure. It suggests that what we call physical curvature may be the geometric residue of global permutation symmetry breaking in this monster space.

What is SUSY?

Supersymmetry (SUSY) is a theoretical principle proposing a deep symmetry in nature that connects two fundamentally different classes of particles: bosons (force carriers) and fermions (matter particles). SUSY allows transformations between these particles, meaning a boson can be transformed into its fermionic superpartner and vice versa. This symmetry extends the Standard Model by linking matter and forces, offering solutions to outstanding problems in physics such as hierarchy issues and dark matter candidates. Although not yet observed experimentally, SUSY provides a powerful framework for understanding particle interactions and the fundamental fabric of reality.

Implications and the Future

The implications of this principle are extraordinary:

• Quantum fields, amplitudes, and interactions are local expressions of permutation geometry.

• Curvature, in this context, is no longer just a property of spacetime, but of symmetry surfaces in a meta-geometric configuration space.

• Multiverse interpretations gain a mathematical foundation: each universe is a corner of the monster polytope.

• Our current understanding of physics—based on Lie algebras, differential geometry, and Hilbert spaces—could be emergent shadows of a larger combinatorial symmetry.

Conclusion: A Kaleidoscope of Reality

What emerges is a vision of the universe as a hyperdimensional kaleidoscope: infinite, symmetric, and ever shifting. We perceive only a sliver—our local amplituhedron—and its slight asymmetries reveal the larger curvature of the Monster.

Our proposal is both poetic and precise: the laws of nature emerge as statistical attractors from a space of all possible correlations, and their irregularities reflect the deeper, rich tapestry of Monster symmetry. The amplituhedron is the lens. The Monster is the source. And we, observers within this kaleidoscope, are fragments of a living geometry.

Article by [Gabriel Brennan]July 16, 2025Contact: [gabriel.b.bcrv@gmail.com]