@uor-foundation/geometry
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Layer 5: Geometric manifolds - the shape of mathematical space
317 lines • 12.3 kB
JavaScript
;
Object.defineProperty(exports, "__esModule", { value: true });
exports.MathematicalManifold = void 0;
class MathematicalManifold {
constructor(fieldSubstrate, resonance, topology, _operators, _algebra) {
this.fieldSubstrate = fieldSubstrate;
this.resonance = resonance;
this.topology = topology;
this._operators = _operators;
this._algebra = _algebra;
}
getMetric(a, b) {
const fieldDistSquared = this.fieldSpaceDistanceSquared(a, b);
const resonanceBarrierSquared = this.resonanceBarrierSquared(a, b);
const topologicalDistSquared = this.topologicalDistanceSquared(a, b);
return Math.sqrt(fieldDistSquared + resonanceBarrierSquared + topologicalDistSquared);
}
findGeodesic(start, end) {
const path = [];
let current = start;
while (current !== end) {
path.push(current);
const candidates = this.getNeighbors(current);
let bestNext = current;
let bestMetric = Infinity;
for (const candidate of candidates) {
const metricToEnd = this.getMetric(candidate, end);
if (metricToEnd < bestMetric) {
bestMetric = metricToEnd;
bestNext = candidate;
}
}
if (bestNext === current) {
current = current < end ? current + 1n : current - 1n;
}
else {
current = bestNext;
}
if (path.length > Number(end - start) * 2) {
break;
}
}
path.push(end);
return path;
}
getCurvature(n) {
const h = 1n;
const resonanceHere = this.resonance.calculateResonance(n);
const resonancePlus = this.resonance.calculateResonance(n + h);
const resonanceMinus = this.resonance.calculateResonance(n - h);
// These values were computed but not used in the curvature calculation
// const resonancePlusPlus = this.resonance.calculateResonance(n + 2n * h);
// const resonanceMinusMinus = this.resonance.calculateResonance(n - 2n * h);
const firstDerivative = (resonancePlus - resonanceMinus) / 2;
const secondDerivative = resonancePlus - 2 * resonanceHere + resonanceMinus;
const scalar = secondDerivative / Math.pow(1 + firstDerivative * firstDerivative, 1.5);
const fieldPattern = this.fieldSubstrate.getFieldPattern(n);
const ricciMatrix = this.computeRicciCurvature(n, fieldPattern);
const sectionalCurvatures = new Map();
for (let i = 0; i < 8; i++) {
for (let j = i + 1; j < 8; j++) {
const key = `${i}-${j}`;
sectionalCurvatures.set(key, this.computeSectionalCurvature(n, i, j));
}
}
return {
scalar,
ricci: ricciMatrix,
sectional: sectionalCurvatures,
};
}
getPageManifold(pageNumber) {
const basePoint = pageNumber * 48n;
// const pageInfo = this.topology.getPageInfo(Number(pageNumber));
const lagrangePoints = [];
let totalCurvature = 0;
for (let i = 0n; i < 48n; i++) {
const n = basePoint + i;
const curvature = this.getCurvature(n);
totalCurvature += curvature.scalar;
}
const averageCurvature = totalCurvature / 48;
const activeDimensions = new Set();
for (let i = 0n; i < 48n; i++) {
const pattern = this.fieldSubstrate.getFieldPattern(basePoint + i);
pattern.forEach((active, idx) => {
if (active)
activeDimensions.add(idx);
});
}
return {
basePoint,
dimension: activeDimensions.size,
curvature: averageCurvature,
lagrangeWells: lagrangePoints,
};
}
getFieldManifold() {
const allPatterns = [];
for (let i = 0; i < 256; i++) {
const pattern = Array.from({ length: 8 }, () => false);
for (let j = 0; j < 8; j++) {
pattern[j] = (i & (1 << j)) !== 0;
}
const activeFields = pattern
.map((active, idx) => (active ? idx : -1))
.filter((idx) => idx >= 0);
allPatterns.push({ pattern, activeFields });
}
const metric = (p1, p2) => {
let distance = 0;
for (let i = 0; i < 8; i++) {
if (p1.pattern[i] !== p2.pattern[i]) {
distance += 1;
}
}
return Math.sqrt(distance);
};
const curvature = this.computeFieldSpaceCurvature();
return {
dimension: 8,
points: allPatterns,
metric,
curvature,
topology: 'torus',
};
}
getTangentSpace(n) {
const fieldPattern = this.fieldSubstrate.getFieldPattern(n);
const activeCount = fieldPattern.filter((b) => b).length;
const basis = [];
for (let i = 0; i < 8; i++) {
const components = Array.from({ length: 8 }, () => 0);
components[i] = 1;
const magnitude = 1;
basis.push({ components, magnitude });
}
const metric = this.computeMetricTensor(n, fieldPattern);
return {
basePoint: n,
dimension: activeCount || 1,
basis,
metric,
};
}
getUniversalBundle() {
const projection = (n, _fields) => {
return n;
};
const parallelTransport = (v, path) => {
let current = { ...v };
for (let i = 0; i < path.length - 1; i++) {
const from = path[i];
const to = path[i + 1];
const connection = this.computeConnection(from, to);
const transported = this.applyConnection(current, connection);
current = transported;
}
return current;
};
const curvatureForm = this.computeCurvatureForm();
return {
base: 'NumberLine',
fiber: 'FieldSpace',
projection,
connection: {
parallelTransport,
curvatureForm,
},
};
}
fieldSpaceDistanceSquared(a, b) {
const patternA = this.fieldSubstrate.getFieldPattern(a);
const patternB = this.fieldSubstrate.getFieldPattern(b);
let distance = 0;
for (let i = 0; i < 8; i++) {
if (patternA[i] !== patternB[i]) {
distance += 1;
}
}
return distance;
}
resonanceBarrierSquared(a, b) {
const resA = this.resonance.calculateResonance(a);
const resB = this.resonance.calculateResonance(b);
return Math.pow(resA - resB, 2);
}
topologicalDistanceSquared(a, b) {
const pageNumberA = Number(a / 48n);
const pageNumberB = Number(b / 48n);
const pageInfoA = this.topology.getPageInfo(pageNumberA);
const pageInfoB = this.topology.getPageInfo(pageNumberB);
const pageA = pageInfoA.pageNumber;
const pageB = pageInfoB.pageNumber;
const pageDiff = Number(pageB > pageA ? pageB - pageA : pageA - pageB);
return pageDiff * pageDiff;
}
getNeighbors(n) {
const neighbors = [];
neighbors.push(n - 1n, n + 1n);
// Check if this is within first two positions of a page (synthetic Lagrange points)
const position = Number(n % 48n);
if (position === 0 || position === 1) {
// This is likely a Lagrange point, add nearby points as neighbors
const nearbyPoints = [n - 48n, n + 48n].filter((p) => p >= 0n);
neighbors.push(...nearbyPoints);
}
return neighbors.filter((neighbor) => neighbor >= 0n);
}
computeRicciCurvature(_n, pattern) {
const dim = 8;
const ricci = Array.from({ length: dim }, () => Array.from({ length: dim }, () => 0));
for (let i = 0; i < dim; i++) {
for (let j = 0; j < dim; j++) {
if (pattern[i] && pattern[j]) {
const fieldConstants = this.fieldSubstrate.getFieldConstants();
ricci[i][j] = (fieldConstants[i] * fieldConstants[j]) / 100;
}
}
}
return ricci;
}
computeSectionalCurvature(n, i, j) {
const pattern = this.fieldSubstrate.getFieldPattern(n);
if (!pattern[i] || !pattern[j])
return 0;
const fieldConstants = this.fieldSubstrate.getFieldConstants();
const resonance = this.resonance.calculateResonance(n);
return (fieldConstants[i] * fieldConstants[j]) / (resonance * resonance);
}
computeFieldSpaceCurvature() {
const scalar = Math.PI / 4;
const ricci = Array.from({ length: 8 }, () => Array.from({ length: 8 }, () => 0));
for (let i = 0; i < 8; i++) {
ricci[i][i] = 1 / 8;
}
const sectional = new Map();
for (let i = 0; i < 8; i++) {
for (let j = i + 1; j < 8; j++) {
sectional.set(`${i}-${j}`, 1 / 16);
}
}
return { scalar, ricci, sectional };
}
computeMetricTensor(n, pattern) {
const dim = 8;
const tensor = Array.from({ length: dim }, () => Array.from({ length: dim }, () => 0));
const fieldConstants = this.fieldSubstrate.getFieldConstants();
const resonance = this.resonance.calculateResonance(n);
for (let i = 0; i < dim; i++) {
for (let j = 0; j < dim; j++) {
if (i === j) {
tensor[i][j] = pattern[i] ? fieldConstants[i] / resonance : 1;
}
else if (pattern[i] && pattern[j]) {
tensor[i][j] = (fieldConstants[i] * fieldConstants[j]) / (resonance * resonance * 10);
}
}
}
const determinant = this.computeDeterminant(tensor);
return { tensor, determinant };
}
computeDeterminant(matrix) {
const n = matrix.length;
if (n === 1)
return matrix[0][0];
if (n === 2)
return matrix[0][0] * matrix[1][1] - matrix[0][1] * matrix[1][0];
let det = 0;
for (let i = 0; i < n; i++) {
const minor = this.getMinor(matrix, 0, i);
det += Math.pow(-1, i) * matrix[0][i] * this.computeDeterminant(minor);
}
return det;
}
getMinor(matrix, row, col) {
return matrix.filter((_, r) => r !== row).map((row) => row.filter((_, c) => c !== col));
}
computeConnection(from, to) {
const dim = 8;
const connection = Array.from({ length: dim }, () => Array.from({ length: dim }, () => 0));
const patternFrom = this.fieldSubstrate.getFieldPattern(from);
const patternTo = this.fieldSubstrate.getFieldPattern(to);
for (let i = 0; i < dim; i++) {
for (let j = 0; j < dim; j++) {
if (patternFrom[i] !== patternTo[i]) {
connection[i][j] = patternTo[i] ? 0.1 : -0.1;
}
}
}
return connection;
}
applyConnection(v, connection) {
const newComponents = [...v.components];
for (let i = 0; i < 8; i++) {
let delta = 0;
for (let j = 0; j < 8; j++) {
delta += connection[i][j] * v.components[j];
}
newComponents[i] += delta;
}
const magnitude = Math.sqrt(newComponents.reduce((sum, c) => sum + c * c, 0));
return { components: newComponents, magnitude };
}
computeCurvatureForm() {
const form = Array.from({ length: 8 }, () => Array.from({ length: 8 }, () => 0));
for (let i = 0; i < 8; i++) {
for (let j = i + 1; j < 8; j++) {
form[i][j] = Math.sin(((i + 1) * (j + 1) * Math.PI) / 16);
form[j][i] = -form[i][j];
}
}
return form;
}
}
exports.MathematicalManifold = MathematicalManifold;
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