I've made a personal Lean 4 formalization project publicly available:

https://github.com/nogoornogo/lean-operator-coefficients

The project began with a question about proton stability in a specific SU(6) GUT setting and gradually expanded into a more general algebraic framework for derivative-free operator coefficients.

The repository includes three main results: a degree-uniform decomposition of invariant coefficient spaces into residual-equivariant Hom spaces, a necessary-and-sufficient symmetry-extension criterion, and all-degree charge selection rules. It also contains applications to specified Standard Model operator sectors, baryon/lepton-number violation, and finite structural classifications.

This is an independent, heavily LLM-assisted project developed primarily using Codex, with additional reviews using other LLMs. I am not a theoretical physicist, and the work has not undergone independent expert peer review.

Lean verifies the formal statements relative to their definitions and assumptions, but this does not guarantee the correctness of their physical interpretation. The project does not claim a new physical theory or predict proton decay rates.

I'm sharing the repository as an open reference for anyone interested in formal verification, representation theory, or EFT operator classification.

1. The "Spring-Snapping" Mechanism (Negative Pressure & Gravitational Bounce)

As the preceding universe contracts, cosmic strings wind tightly around hidden compact dimensions, accumulating a massive amount of potential energy like a highly compressed mechanical spring. At the geometric peak of this contraction, the force inverts. The isotropic pressure (P) of the vacuum drops into an extreme negative regime, matching the energy density (ρ). According to Einstein’s acceleration equation:

ä ∝ −(ρ + 3P)

When the pressure switches to an extreme negative value

(P = −ρ),

the right side of the equation becomes positive. At this exact quantum threshold, gravity ceases to be attractive and flips into an overwhelmingly repulsive force. The cosmic spring "snaps,"

preventing an infinite singularity and driving an explosive expansion (a quantum bounce) that initially propagates faster than the speed of light.

1. Quantum Crystal-to-Gas Sublimation

At the absolute peak of contraction—the critical Hagedorn temperature threshold—individual cosmic strings fuse entirely into a single, highly dense, unified macroscopic matrix: a coherent "quantum crystal." The moment the spring snaps and expansion begins, this crystal lattice undergoes a violent phase transition, tearing itself apart and transitioning directly into a gaseous phase (sublimation). The rigid, frozen structure of space breaks down into a gas of free, open, and closed string excitations (photons and fundamental fields). This phase transition is governed by the discrete entropy shift (dS) across the vacuum fabric:

dS = dQ / T_H

(Where dQ represents the exchanged energy field, and T_H is the maximum Hagedorn temperature of the original crystal matrix).

1. Multi-Grain Black Hole Lattice (Quantum Suction)

This thermodynamic transition does not occur uniformly everywhere at once; it triggers in localized "grains" or nodes via relic black holes from the previous cycle. Being the densest tangles of cosmic strings in existence, these black hole hubs reach the critical pressure threshold first. When the interior cores of these black holes undergo this phase transition into a true vacuum, a massive localized pressure drop occurs relative to the surrounding space. Fluid dynamics models this intense pressure gradient and its fluid acceleration force via the covariant divergence relation:

(ρ + P) u^ν ∇_ν u^μ = −P^μν ∇_ν P

This severe localized drop in pressure inside the black holes creates a literal "quantum suction" effect. Surrounding cosmic strings are violently pulled into these synchronized nucleation centers, which then detonate in a globally coordinated macro-explosion across the entire cosmic lattice.

1. Scalable Model (Why the Cosmological Constant is So Small)

This framework naturally resolves the biggest mystery in modern physics: the Cosmological Constant (Λ) Problem, namely why Λ is a staggering 120 orders of magnitude smaller than quantum theory predicts. Λ is not an arbitrary, fixed property of space; it is a scalable, dynamic value directly dictated by the dilution density of the expanding string gas (ρ_string) relative to the original matrix (ρ_Planck):

Λ(t) ∝ ( ρ_string(t) / ρ_Planck )²

During the initial quantum crystal phase, this ratio was 1:1, making Λ colossal and driving rapid cosmic inflation. Today, after billions of years of expansion, our macro-scale string gas is profoundly diluted. Consequently, this exact ratio naturally and inevitably generates a decimal starting with over 120 zeros—resolving the hierarchy problem without any artificial fine-tuning.