@uor-foundation/geometry
Version:
Layer 5: Geometric manifolds - the shape of mathematical space
374 lines • 12.2 kB
JavaScript
"use strict";
Object.defineProperty(exports, "__esModule", { value: true });
exports.TopologicalInvariants = void 0;
const field_space_1 = require("./field-space");
class TopologicalInvariants {
constructor(topology, fieldSubstrate) {
this.topology = topology;
this.fieldSubstrate = fieldSubstrate;
this.fieldSpaceGeometry = new field_space_1.FieldSpaceGeometry(fieldSubstrate);
}
computeFundamentalGroup(basePoint, radius) {
const generators = this.findLoopGenerators(basePoint, radius);
const relations = this.findRelations(generators);
return {
basePoint,
generators,
relations,
};
}
findLoopGenerators(basePoint, _radius) {
const loops = [];
const fieldLoop = this.createFieldCycleLoop(basePoint);
if (fieldLoop)
loops.push(fieldLoop);
// For now, we'll create synthetic Lagrange points based on page structure
const pageNumber = this.getPageNumber(basePoint);
const lagrangePoints = this.findLagrangePointsInPage(pageNumber);
for (const lagrangePoint of lagrangePoints) {
const lagrangeLoop = this.createLagrangeLoop(basePoint, lagrangePoint);
if (lagrangeLoop)
loops.push(lagrangeLoop);
}
const pageBoundaryLoop = this.createPageBoundaryLoop(basePoint);
if (pageBoundaryLoop)
loops.push(pageBoundaryLoop);
return loops;
}
createFieldCycleLoop(basePoint) {
const path = [basePoint];
let current = basePoint;
for (let i = 0; i < 256; i++) {
current = current + 1n;
path.push(current);
const basePattern = this.fieldSubstrate.getFieldPattern(basePoint);
const currentPattern = this.fieldSubstrate.getFieldPattern(current);
if (this.patternsEqual(basePattern, currentPattern)) {
return {
basePoint,
path,
isClosed: true,
windingNumber: 1,
};
}
}
return null;
}
createLagrangeLoop(basePoint, lagrangePoint) {
if (basePoint === lagrangePoint)
return null;
const path = [basePoint];
const forward = this.createPath(basePoint, lagrangePoint);
path.push(...forward.slice(1));
const backward = this.createPath(lagrangePoint, basePoint);
path.push(...backward.slice(1));
return {
basePoint,
path,
isClosed: true,
windingNumber: this.computeWindingNumber(path),
};
}
createPageBoundaryLoop(basePoint) {
const pageNumber = this.getPageNumber(basePoint);
const pageStart = pageNumber * 48n;
const pageEnd = pageStart + 47n;
const path = [];
const toEnd = this.createPath(basePoint, pageEnd);
path.push(...toEnd);
path.push(pageEnd + 1n);
const toStart = this.createPath(pageEnd + 1n, pageStart);
path.push(...toStart.slice(1));
const backToBase = this.createPath(pageStart, basePoint);
path.push(...backToBase.slice(1));
return {
basePoint,
path,
isClosed: true,
windingNumber: 0,
};
}
computeHomologyGroups(maxDimension) {
const groups = [];
const h0 = this.computeH0();
groups.push(h0);
if (maxDimension >= 1) {
const h1 = this.computeH1();
groups.push(h1);
}
if (maxDimension >= 2) {
const h2 = this.computeH2();
groups.push(h2);
}
return groups;
}
computeH0() {
return {
dimension: 0,
rank: 1,
torsion: [],
generators: [[]],
};
}
computeH1() {
const fieldCycleRank = 8;
const pageCycleRank = 1;
const totalRank = fieldCycleRank + pageCycleRank;
const generators = [];
for (let i = 0; i < 8; i++) {
const loop = this.createFieldDimensionLoop(0n, i);
if (loop)
generators.push([loop]);
}
const pageLoop = this.createPageCycleLoop(0n);
if (pageLoop)
generators.push([pageLoop]);
return {
dimension: 1,
rank: totalRank,
torsion: [256],
generators,
};
}
computeH2() {
return {
dimension: 2,
rank: 0,
torsion: [],
generators: [],
};
}
computeEulerCharacteristic(region) {
const vertices = region.length;
let edges = 0;
for (let i = 0; i < region.length; i++) {
for (let j = i + 1; j < region.length; j++) {
if (this.areAdjacent(region[i], region[j])) {
edges++;
}
}
}
const faces = this.countFaces(region);
const chi = vertices - edges + faces;
return {
vertices,
edges,
faces,
chi,
};
}
computeBettiNumbers(maxDimension) {
const homologyGroups = this.computeHomologyGroups(maxDimension);
return homologyGroups.map((group) => group.rank);
}
isContractible(loop) {
if (!loop.isClosed)
return false;
if (loop.path.length <= 2)
return true;
const simplified = this.simplifyLoop(loop);
return simplified.path.length <= 1;
}
simplifyLoop(loop) {
const path = [...loop.path];
let changed = true;
while (changed) {
changed = false;
for (let i = 0; i < path.length - 2; i++) {
if (path[i] === path[i + 2]) {
path.splice(i, 3, path[i]);
changed = true;
break;
}
}
}
return {
...loop,
path,
};
}
computeGenus() {
const chi = this.computeEulerCharacteristic(this.getSampleRegion()).chi;
return (2 - chi) / 2;
}
detectHoles(region) {
const holes = [];
const boundary = this.findBoundary(region);
const components = this.findConnectedComponents(boundary);
if (components.length > 1) {
for (let i = 1; i < components.length; i++) {
holes.push(components[i]);
}
}
return holes;
}
findRelations(generators) {
const relations = [];
for (let i = 0; i < generators.length; i++) {
for (let j = i + 1; j < generators.length; j++) {
const commutator = this.computeCommutator(generators[i], generators[j]);
if (this.isContractible(commutator)) {
relations.push(`[g${i}, g${j}] = 1`);
}
}
}
for (let i = 0; i < generators.length; i++) {
const gen = generators[i];
if (gen.windingNumber > 0) {
relations.push(`g${i}^${256 / gen.windingNumber} = 1`);
}
}
return relations;
}
computeCommutator(loop1, loop2) {
const path = [loop1.basePoint];
path.push(...loop1.path.slice(1));
path.push(...loop2.path.slice(1));
const inverse1 = [...loop1.path].reverse();
path.push(...inverse1.slice(1));
const inverse2 = [...loop2.path].reverse();
path.push(...inverse2.slice(1));
return {
basePoint: loop1.basePoint,
path,
isClosed: true,
windingNumber: 0,
};
}
getPageNumber(n) {
return n / 48n;
}
findLagrangePointsInPage(pageNumber) {
const pageStart = pageNumber * 48n;
// Return the first two numbers of each page as synthetic Lagrange points
return [pageStart, pageStart + 1n];
}
patternsEqual(p1, p2) {
if (p1.length !== p2.length)
return false;
for (let i = 0; i < p1.length; i++) {
if (p1[i] !== p2[i])
return false;
}
return true;
}
createPath(from, to) {
const path = [];
const step = from < to ? 1n : -1n;
for (let current = from; current !== to; current += step) {
path.push(current);
}
path.push(to);
return path;
}
computeWindingNumber(path) {
if (path.length < 2)
return 0;
let totalAngle = 0;
for (let i = 0; i < path.length - 1; i++) {
const p1 = this.fieldSubstrate.getFieldPattern(path[i]);
const p2 = this.fieldSubstrate.getFieldPattern(path[i + 1]);
const angle = this.fieldSpaceGeometry.hammingDistance(p1, p2);
totalAngle += angle;
}
return Math.round(totalAngle / (2 * Math.PI));
}
createFieldDimensionLoop(basePoint, dimension) {
const path = [basePoint];
let current = basePoint;
for (let i = 0; i < 256; i++) {
current = current + 1n;
path.push(current);
const pattern = this.fieldSubstrate.getFieldPattern(current);
const basePattern = this.fieldSubstrate.getFieldPattern(basePoint);
if (pattern[dimension] === basePattern[dimension]) {
if (i === 127) {
return {
basePoint,
path,
isClosed: true,
windingNumber: 1,
};
}
}
}
return null;
}
createPageCycleLoop(basePoint) {
const path = [basePoint];
let current = basePoint;
// Create a cycle that goes through pages
for (let i = 0; i < 256; i++) {
current = current + 48n;
path.push(current);
}
// Close the loop
path.push(basePoint);
return {
basePoint,
path,
isClosed: true,
windingNumber: 256,
};
}
areAdjacent(a, b) {
const diff = a > b ? a - b : b - a;
return diff === 1n;
}
countFaces(_region) {
return 0;
}
getSampleRegion() {
const region = [];
for (let i = 0n; i < 100n; i++) {
region.push(i);
}
return region;
}
findBoundary(region) {
const regionSet = new Set(region.map((n) => n.toString()));
const boundary = [];
for (const point of region) {
const neighbors = [point - 1n, point + 1n];
for (const neighbor of neighbors) {
if (!regionSet.has(neighbor.toString())) {
boundary.push(point);
break;
}
}
}
return boundary;
}
findConnectedComponents(points) {
const components = [];
const visited = new Set();
for (const point of points) {
if (!visited.has(point.toString())) {
const component = this.exploreComponent(point, points, visited);
components.push(component);
}
}
return components;
}
exploreComponent(start, points, visited) {
const component = [];
const queue = [start];
const pointSet = new Set(points.map((p) => p.toString()));
while (queue.length > 0) {
const current = queue.shift();
if (visited.has(current.toString()))
continue;
visited.add(current.toString());
component.push(current);
const neighbors = [current - 1n, current + 1n];
for (const neighbor of neighbors) {
if (pointSet.has(neighbor.toString()) && !visited.has(neighbor.toString())) {
queue.push(neighbor);
}
}
}
return component;
}
}
exports.TopologicalInvariants = TopologicalInvariants;
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