@uor-foundation/geometry
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Layer 5: Geometric manifolds - the shape of mathematical space
307 lines • 11.9 kB
JavaScript
;
Object.defineProperty(exports, "__esModule", { value: true });
exports.DifferentialStructure = void 0;
const metric_1 = require("./metric");
class DifferentialStructure {
constructor(fieldSubstrate, resonance) {
this.fieldSubstrate = fieldSubstrate;
this.resonance = resonance;
this.metric = new metric_1.ResonanceMetric(fieldSubstrate, resonance, {});
}
createTangentSpace(basePoint) {
const fieldPattern = this.fieldSubstrate.getFieldPattern(basePoint);
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;
basis.push({
components,
magnitude: 1,
});
}
const metric = this.computeMetricTensor(basePoint);
return {
basePoint,
dimension: activeCount || 1,
basis,
metric,
};
}
createGradientField(potential) {
return {
name: 'gradient',
evaluate: (point) => {
const h = 1;
const components = Array.from({ length: 8 }, () => 0);
for (let i = 0; i < 8; i++) {
const shift = BigInt(1 << i);
const forward = potential(point + shift);
const backward = potential(point - shift);
components[i] = (forward - backward) / (2 * h);
}
const magnitude = Math.sqrt(components.reduce((sum, c) => sum + c * c, 0));
return { components, magnitude };
},
isSmooth: true,
};
}
createHamiltonianField(hamiltonian) {
return {
name: 'hamiltonian',
evaluate: (point) => {
const grad = this.createGradientField(hamiltonian).evaluate(point);
const components = [...grad.components];
for (let i = 0; i < 4; i++) {
[components[i], components[i + 4]] = [-components[i + 4], components[i]];
}
const magnitude = grad.magnitude;
return { components, magnitude };
},
isSmooth: true,
};
}
createArithmeticFlow() {
return {
name: 'arithmetic',
evaluate: (point) => {
const resonance = this.resonance.calculateResonance(point);
const nextResonance = this.resonance.calculateResonance(point + 1n);
const velocity = nextResonance - resonance;
const components = Array.from({ length: 8 }, () => 0);
components[0] = velocity;
return {
components,
magnitude: Math.abs(velocity),
};
},
isSmooth: false,
};
}
computeFlow(field, _start, time, steps = 100) {
const dt = time / steps;
const flowMap = (startPoint, t) => {
let current = startPoint;
const numSteps = Math.floor(Math.abs(t) / dt);
const stepSize = t > 0 ? dt : -dt;
for (let i = 0; i < numSteps; i++) {
const velocity = field.evaluate(current);
const displacement = velocity.components.reduce((sum, v, idx) => {
return sum + BigInt(Math.floor(v * stepSize * (1 << idx)));
}, 0n);
current = current + displacement;
if (current < 0n) {
current = 0n;
break;
}
}
return current;
};
return {
vectorField: field,
flowMap,
isComplete: true,
};
}
create0Form(f) {
return {
degree: 0,
evaluate: (point) => f(point),
};
}
create1Form(components) {
return {
degree: 1,
evaluate: (point, vectors) => {
if (vectors.length !== 1) {
throw new Error('1-form requires exactly one vector');
}
const vector = vectors[0];
let result = 0;
for (let i = 0; i < Math.min(components.length, vector.components.length); i++) {
result += components[i](point) * vector.components[i];
}
return result;
},
};
}
create2Form(components) {
return {
degree: 2,
evaluate: (point, vectors) => {
if (vectors.length !== 2) {
throw new Error('2-form requires exactly two vectors');
}
const [v1, v2] = vectors;
let result = 0;
for (let i = 0; i < 8; i++) {
for (let j = 0; j < 8; j++) {
if (i < components.length && j < components[i].length) {
result +=
components[i][j](point) *
(v1.components[i] * v2.components[j] - v1.components[j] * v2.components[i]);
}
}
}
return result;
},
};
}
exteriorDerivative(form) {
if (form.degree === 0) {
const grad = this.createGradientField((n) => form.evaluate(n, []));
const components = Array(8)
.fill(null)
.map((_, i) => (n) => grad.evaluate(n).components[i]);
return this.create1Form(components);
}
if (form.degree === 1) {
const components = Array(8)
.fill(null)
.map(() => Array(8).fill(() => 0));
for (let i = 0; i < 8; i++) {
for (let j = i + 1; j < 8; j++) {
components[i][j] = (n) => {
const h = 1n;
const basis_i = this.createBasisVector(i);
const basis_j = this.createBasisVector(j);
const f_i = form.evaluate(n, [basis_i]);
const f_i_plus = form.evaluate(n + h, [basis_i]);
const f_j = form.evaluate(n, [basis_j]);
const f_j_plus = form.evaluate(n + h, [basis_j]);
return (f_i_plus - f_i) / Number(h) - (f_j_plus - f_j) / Number(h);
};
components[j][i] = (n) => -components[i][j](n);
}
}
return this.create2Form(components);
}
return {
degree: form.degree + 1,
evaluate: () => 0,
};
}
lieDerivative(field, form) {
const flow = this.computeFlow(field, 0n, 1);
return {
degree: form.degree,
evaluate: (point, vectors) => {
const h = 0.01;
const forward = flow.flowMap(point, h);
const value_forward = form.evaluate(forward, vectors);
const value_here = form.evaluate(point, vectors);
return (value_forward - value_here) / h;
},
};
}
computePoissonBracket(f, g) {
const bracket = (n) => {
let result = 0;
for (let i = 0; i < 4; i++) {
const shift_p = BigInt(1 << i);
const shift_q = BigInt(1 << (i + 4));
const df_dp = (f(n + shift_p) - f(n - shift_p)) / 2;
const df_dq = (f(n + shift_q) - f(n - shift_q)) / 2;
const dg_dp = (g(n + shift_p) - g(n - shift_p)) / 2;
const dg_dq = (g(n + shift_q) - g(n - shift_q)) / 2;
result += df_dp * dg_dq - df_dq * dg_dp;
}
return result;
};
return { f, g, bracket };
}
covariantDerivative(vector, direction, point) {
const christoffel = this.metric.christoffelSymbols(point);
const components = [...vector.components];
for (let i = 0; i < 8; i++) {
let correction = 0;
for (let j = 0; j < 8; j++) {
for (let k = 0; k < 8; k++) {
correction += christoffel[i][j][k] * direction.components[j] * vector.components[k];
}
}
components[i] += correction;
}
const magnitude = Math.sqrt(components.reduce((sum, c) => sum + c * c, 0));
return { components, magnitude };
}
riemannCurvatureTensor(point) {
const dim = 8;
const R = Array(dim)
.fill(null)
.map(() => Array(dim)
.fill(null)
.map(() => Array(dim)
.fill(null)
.map(() => Array(dim).fill(0))));
const christoffel = this.metric.christoffelSymbols(point);
const h = 1n;
const christoffel_plus = this.metric.christoffelSymbols(point + h);
const christoffel_minus = this.metric.christoffelSymbols(point - h);
for (let i = 0; i < dim; i++) {
for (let j = 0; j < dim; j++) {
for (let k = 0; k < dim; k++) {
for (let l = 0; l < dim; l++) {
const partial_k = (christoffel_plus[i][j][l] - christoffel_minus[i][j][l]) / 2;
const partial_l = (christoffel_plus[i][j][k] - christoffel_minus[i][j][k]) / 2;
let contraction = 0;
for (let m = 0; m < dim; m++) {
contraction +=
christoffel[i][k][m] * christoffel[m][j][l] -
christoffel[i][l][m] * christoffel[m][j][k];
}
R[i][j][k][l] = partial_k - partial_l + contraction;
}
}
}
}
return R;
}
computeMetricTensor(basePoint) {
const dim = 8;
const tensor = Array(dim)
.fill(null)
.map(() => Array(dim).fill(0));
const fieldConstants = this.fieldSubstrate.getFieldConstants();
const pattern = this.fieldSubstrate.getFieldPattern(basePoint);
const resonance = this.resonance.calculateResonance(basePoint);
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));
}
createBasisVector(index) {
const components = Array.from({ length: 8 }, () => 0);
components[index] = 1;
return {
components,
magnitude: 1,
};
}
}
exports.DifferentialStructure = DifferentialStructure;
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