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//! §Fase 87.f — the `TopologyBackend` port + the OSS reference TDA engine.
//!
//! The savant maps the *shape* of the ingested corpus so it can steer its
//! research toward epistemic gaps rather than toward semantic look-alikes (the
//! failure mode of vector-similarity RAG — paper §4). It approximates the corpus
//! as a Vietoris–Rips simplicial complex and reads its homology.
//!
//! The OSS reference computes the low-dimensional Betti numbers exactly over the
//! 1-skeleton (graph):
//! - `β₀` = connected components (union-find) — the number of disjoint
//! knowledge islands.
//! - `β₁` = the cyclomatic number `E − V + β₀` — independent 1-cycles (loops
//! of association with a "hole" in the middle).
//! - a **cycle-participation centrality**: per vertex, the number of incident
//! edges that lie on a cycle (i.e. survive 2-core reduction). This is the
//! reference proxy for Persistent-Homology Centrality (PHC, paper §4.2).
//!
//! Honest bound: the full persistence pairing across a filtration and the `β₂`
//! (void) analysis the paper leans on require the boundary-matrix reduction of
//! the enterprise engine (§87.i); the OSS reference is exact for `β₀`/`β₁` on the
//! graph and is the differential-test oracle. No advantage is claimed (§69) —
//! these are exact combinatorial invariants.
/// The low-dimensional Betti numbers of a graph (1-skeleton).
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
pub struct BettiNumbers {
/// `β₀` — connected components.
pub b0: usize,
/// `β₁` — independent 1-cycles (`E − V + β₀`).
pub b1: usize,
}
/// Euclidean distance between two equal-length points.
pub fn euclidean(a: &[f64], b: &[f64]) -> f64 {
a.iter()
.zip(b.iter())
.map(|(x, y)| (x - y) * (x - y))
.sum::<f64>()
.sqrt()
}
/// The Vietoris–Rips 1-skeleton at scale `threshold`: an edge for every pair of
/// points within `threshold`. Returns `(i, j)` with `i < j`.
pub fn vietoris_rips_edges(points: &[Vec<f64>], threshold: f64) -> Vec<(usize, usize)> {
let mut edges = Vec::new();
for i in 0..points.len() {
for j in (i + 1)..points.len() {
if euclidean(&points[i], &points[j]) <= threshold {
edges.push((i, j));
}
}
}
edges
}
// ── Union-Find (for β₀) ──────────────────────────────────────────────────────
struct UnionFind {
parent: Vec<usize>,
}
impl UnionFind {
fn new(n: usize) -> Self {
UnionFind {
parent: (0..n).collect(),
}
}
fn find(&mut self, x: usize) -> usize {
let mut root = x;
while self.parent[root] != root {
root = self.parent[root];
}
// Path compression.
let mut cur = x;
while self.parent[cur] != root {
let next = self.parent[cur];
self.parent[cur] = root;
cur = next;
}
root
}
fn union(&mut self, a: usize, b: usize) {
let ra = self.find(a);
let rb = self.find(b);
if ra != rb {
self.parent[ra] = rb;
}
}
fn components(&mut self, n: usize) -> usize {
let mut roots = std::collections::HashSet::new();
for i in 0..n {
let r = self.find(i);
roots.insert(r);
}
roots.len()
}
}
/// The TDA port (charter split R1). Enterprise mounts the full persistent-
/// homology engine (β₂ voids, persistence pairing) behind this trait (§87.i).
pub trait TopologyBackend {
/// `β₀`/`β₁` of the graph on `n_vertices` with the given edges.
fn betti(&self, n_vertices: usize, edges: &[(usize, usize)]) -> BettiNumbers;
/// Per-vertex cycle-participation centrality (PHC proxy): how many incident
/// edges lie on a cycle.
fn cycle_centrality(&self, n_vertices: usize, edges: &[(usize, usize)]) -> Vec<usize>;
}
/// The OSS reference: exact `β₀`/`β₁` over the 1-skeleton + a 2-core cycle-
/// participation centrality.
pub struct ReferenceTopology;
impl TopologyBackend for ReferenceTopology {
fn betti(&self, n_vertices: usize, edges: &[(usize, usize)]) -> BettiNumbers {
let mut uf = UnionFind::new(n_vertices.max(1));
for &(a, b) in edges {
uf.union(a, b);
}
let b0 = if n_vertices == 0 {
0
} else {
uf.components(n_vertices)
};
// Cyclomatic number: E − V + β₀ (independent cycles of the graph).
let e = edges.len() as isize;
let v = n_vertices as isize;
let b1 = (e - v + b0 as isize).max(0) as usize;
BettiNumbers { b0, b1 }
}
fn cycle_centrality(&self, n_vertices: usize, edges: &[(usize, usize)]) -> Vec<usize> {
// 2-core: iteratively strip vertices of degree < 2. The surviving edges
// are exactly the cycle edges; a vertex's centrality is its degree in
// that core.
let mut degree = vec![0usize; n_vertices];
let mut alive_edge = vec![true; edges.len()];
let mut removed = vec![false; n_vertices];
for &(a, b) in edges {
degree[a] += 1;
degree[b] += 1;
}
// Queue of vertices to peel.
let mut queue: Vec<usize> = (0..n_vertices).filter(|&v| degree[v] < 2).collect();
while let Some(v) = queue.pop() {
if removed[v] {
continue;
}
removed[v] = true;
for (ei, &(a, b)) in edges.iter().enumerate() {
if !alive_edge[ei] {
continue;
}
if a == v || b == v {
alive_edge[ei] = false;
let other = if a == v { b } else { a };
if !removed[other] && degree[other] > 0 {
degree[other] -= 1;
if degree[other] < 2 {
queue.push(other);
}
}
}
}
}
// Centrality = surviving (cycle) edges incident to each vertex.
let mut centrality = vec![0usize; n_vertices];
for (ei, &(a, b)) in edges.iter().enumerate() {
if alive_edge[ei] {
centrality[a] += 1;
centrality[b] += 1;
}
}
centrality
}
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn triangle_has_one_component_one_cycle() {
let t = ReferenceTopology;
let edges = vec![(0, 1), (1, 2), (0, 2)];
assert_eq!(t.betti(3, &edges), BettiNumbers { b0: 1, b1: 1 });
// Every vertex sits on the single cycle → 2 cycle-edges each.
assert_eq!(t.cycle_centrality(3, &edges), vec![2, 2, 2]);
}
#[test]
fn two_disjoint_edges() {
let t = ReferenceTopology;
let edges = vec![(0, 1), (2, 3)];
assert_eq!(t.betti(4, &edges), BettiNumbers { b0: 2, b1: 0 });
assert_eq!(t.cycle_centrality(4, &edges), vec![0, 0, 0, 0]);
}
#[test]
fn square_is_one_cycle() {
let t = ReferenceTopology;
let edges = vec![(0, 1), (1, 2), (2, 3), (3, 0)];
assert_eq!(t.betti(4, &edges), BettiNumbers { b0: 1, b1: 1 });
assert_eq!(t.cycle_centrality(4, &edges), vec![2, 2, 2, 2]);
}
#[test]
fn path_has_no_cycle() {
let t = ReferenceTopology;
let edges = vec![(0, 1), (1, 2), (2, 3)];
assert_eq!(t.betti(4, &edges), BettiNumbers { b0: 1, b1: 0 });
assert_eq!(t.cycle_centrality(4, &edges), vec![0, 0, 0, 0]);
}
#[test]
fn tadpole_isolates_the_cycle() {
// A triangle (0,1,2) with a tail 2-3-4. Only the triangle vertices carry
// cycle centrality; the tail is peeled by the 2-core.
let t = ReferenceTopology;
let edges = vec![(0, 1), (1, 2), (0, 2), (2, 3), (3, 4)];
assert_eq!(t.betti(5, &edges), BettiNumbers { b0: 1, b1: 1 });
let c = t.cycle_centrality(5, &edges);
assert_eq!(c[0], 2);
assert_eq!(c[1], 2);
assert_eq!(c[2], 2);
assert_eq!(c[3], 0);
assert_eq!(c[4], 0);
}
#[test]
fn isolated_points_are_their_own_components() {
let t = ReferenceTopology;
assert_eq!(t.betti(5, &[]), BettiNumbers { b0: 5, b1: 0 });
}
#[test]
fn vietoris_rips_gates_on_threshold() {
let pts = vec![vec![0.0, 0.0], vec![0.5, 0.0], vec![10.0, 10.0]];
// Small threshold: only the close pair connects → 2 components.
let near = vietoris_rips_edges(&pts, 1.0);
assert_eq!(near, vec![(0, 1)]);
assert_eq!(ReferenceTopology.betti(3, &near).b0, 2);
// Large threshold: everything connects.
let far = vietoris_rips_edges(&pts, 100.0);
assert_eq!(far.len(), 3);
assert_eq!(ReferenceTopology.betti(3, &far).b0, 1);
}
}