Struct ActionFunctional 

pub struct ActionFunctional { /* private fields */ }

The action functional S[φ] on the 12,288-cell complex.

This is a simplified educational version. The actual action functional used to discover the Atlas is more complex and involves the full geometric structure.

Mathematical Definition

For a configuration φ: Cells → ℂ, the action is:

$$ S[\phi] = \sum_{c \in \text{Cells}} \phi(\partial c) $$

where ∂c denotes the boundary of cell c.

Discrete vs Continuous

Unlike continuous functionals (integrals), this is a discrete functional—a finite sum over finitely many cells. This makes it:

• Computable: Can be evaluated exactly
• Optimizable: Minima can be found algorithmically
• Verifiable: Results are reproducible

Implementations

impl ActionFunctional

pub const fn new( cells: Vec<CellId>, boundary: HashMap<CellId, Vec<(CellId, Ratio<i64>)>>, ) -> Self

Create a new action functional with the given cell structure.

Arguments

• cells - The list of cells in the complex
• boundary - For each cell, its boundary as a formal sum of lower-dimensional cells

pub fn evaluate(&self, config: &Configuration) -> Ratio<i64>

Evaluate the action functional S[φ] for a given configuration.

Mathematical Formula

$$ S[\phi] = \sum_{c \in \text{Cells}} \phi(\partial c) $$

where φ(∂c) is computed as the sum of φ evaluated on boundary cells.

Examples

use atlas_embeddings::foundations::action::{ActionFunctional, Configuration};
use std::collections::HashMap;
use num_rational::Ratio;

// Simple example: 2 cells, cell 1 has cell 0 as boundary
let cells = vec![0, 1];
let mut boundary = HashMap::new();
boundary.insert(1, vec![(0, Ratio::from_integer(1))]);

let functional = ActionFunctional::new(cells, boundary);

let mut config = Configuration::new();
config.insert(0, Ratio::from_integer(2));
config.insert(1, Ratio::from_integer(3));

let action = functional.evaluate(&config);
// action depends on boundary structure

pub fn cell_count(&self) -> usize

Get the number of cells in the complex.

Trait Implementations

impl Clone for ActionFunctional

fn clone(&self) -> ActionFunctional

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for ActionFunctional

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.