The Field Knows: Cross-Dimensional Geometry from Navigation to Black Holes

Summary

The paper introduces a continuous metric field framework trained by a single causal contrastive loss. The framework encodes a scene into coefficients of a fixed symmetric matrix basis, assembles them into a Lie algebra element, and exponentiates the result to a Riemannian or Lorentzian metric. Across dimensions, this field discovers the full spectrum of geometric structures: from obstacle-avoiding geodesics in robot navigation across planar and manipulator configuration spaces, to event horizons of black holes in Lorentzian spacetime. Extensive zero-shot generalization studies demonstrate that the field captures transferable geometric structure rather than memorizing specific configurations. In the black hole setting, the causal loss spontaneously evolves genuine black-hole-like structures with the correct Lorentzian signature. The same loss, the same architecture, and the same training protocol produce the full range of geometric phenomena across dimensions. The field knows geometry, and geometry knows physics.

The paper presents four dimensional regimes: 2D planar (a continuous metric field for point robot navigation, with zero-shot generalization across systematically varied test suites), 6D manipulator (a 6-DOF arm in a cluttered workspace, where the configuration-space metric generalizes from single-scene training to fully random environments), 3D Lorentzian (BTZ-like black hole emergence, where the metric spontaneously develops an event horizon under the causal constraint), and 4D Lorentzian (Schwarzschild-like black hole emergence with suppressed cross-terms and near-spherical symmetry).

The contributions are: (1) a continuous metric field framework that deepens spatial intelligence from discrete generators to a unified geometric object spanning every point in space, and extends it across dimensions from robot navigation to black hole horizons, all under a single causal contrastive loss; (2) systematic zero-shot generalization studies in both planar and 6-DOF manipulator spaces, demonstrating that the continuous field captures transferable geometric structure rather than memorizing specific configurations; (3) demonstration that the causal loss spontaneously evolves genuine black-hole-like structures with the correct Lorentzian signature in 3D and 4D spacetime, including the identification of a mechanism that prevents trivial collapse and forces genuine horizon formation.

The architecture has three stages. Stage 1: Scene Encoding — the scene is discretized into a grid and encoded by a lightweight backbone network (a 2D CNN for planar scenes, a 3D CNN for the 3D obstacles projected to the 6D joint configuration space, and a 3D and 4D CNN for 3D and 4D spacetime respectively). At any spatial query point x, the feature map is linearly interpolated to produce a feature vector, augmented with a positional encoding of x, and passed to an MLP. Stage 2: Basis Matrix Assembly — the MLP outputs m = dim Sym(n) = n(n+1)/2 scalar coefficients per spatial point, which are the coordinates of H(x) in a fixed basis Xk ⊂ Sym(n), forming a Lie algebra element H(x) ∈ gl(n). The basis follows a unified construction for any dimension n: X0 = I (identity matrix, controlling the isotropic trace component), n−1 traceless diagonal matrices controlling anisotropic deformations along independent axes, and n(n−1)/2 symmetric off-diagonal matrices controlling shear between dimensions. This yields m = n(n+1)/2 basis matrices for any n (e.g., m = 3 for 2D, m = 6 for 3D, m = 10 for 4D, and m = 21 for 6D). Stage 3: Metric Assembly — before exponentiation, the Lie algebra element H(x) passes through a Cartan spectral clamp to prevent numerical overflow: H̃(x) = c(x)·H(x), where c(x) = tanh(∥H(x)∥/L) / (∥H(x)∥/L + ε) > 0, with L the clamp scale, ∥H(x)∥ the spectral norm, and ε = 10−12. Since c(x) > 0, H̃(x) is a positive scalar multiple of H(x) whose spectral radius does not exceed L. From H̃(x), the group element E(x) = exp(H̃(x)) ∈ GL(n) is obtained via the matrix exponential. The metric field is then assembled in a unified form across all dimensions: G(x) = E(x)T η E(x), where η is the metric signature matrix. For Riemannian geometry (2D and 6D obstacle avoidance), η = I and G = E2 = exp(2H̃). For Lorentzian geometry (3D and 4D black hole), η = diag(−1, +1,..., +1). By Sylvester's law of inertia, sig(G) = sig(η) as long as E(x) is non-singular. For Riemannian metrics, a final minimum eigenvalue shift of max(0, 0.05 − λmin) is applied to prevent numerical instability; this shift is omitted for Lorentzian metrics since the negative eigenvalue of the time component is physically meaningful.

The entire field is trained by a single causal contrastive loss: L = Lpos/(Lneg + 0.1) + 0.001·Lpos, where path costs are computed as discrete Riemannian (or Lorentzian) lengths: Lpath = Σ δiT G(mi) δi, with δi = pi+1 − pi as displacement vectors and mi = (pi + pi+1)/2 as segment midpoints. The first term is the ratio Lpos/(Lneg + 0.1), which drives the field to separate positive and negative path costs. The second term is the regularization term 0.001·Lpos, which serves as an anchor providing gentle downward pressure on the absolute magnitude of Lpos. In the 2D and 6D obstacle avoidance settings, Lpos is the mean cost of collision-free paths and Lneg is the mean cost of obstacle-penetrating paths. In the 3D/4D black hole setting, Lpos is the mean cost of ingoing (infalling) paths and Lneg is the mean cost of outgoing (escaping) paths. A path is ingoing if it starts at a radius r > rmax/2 and ends at r < rmax/4; outgoing if the reverse. The constraint is purely causal: it is easier to fall in than to climb out. No mass, no curvature, no Einstein equations are provided; only the relative cost of ingoing versus outgoing geodesics.

The ratio loss admits a degenerate solution: the optimizer can shrink all metric eigenvalues toward zero, making Lpos and Lneg both arbitrarily small while maintaining a small ratio, without learning any geometric structure. This is prevented by enforcing t0 > 0 on the coefficient of the identity basis matrix X0 = I: t0 = softplus(t raw 0) + 10−6, which guarantees tr(H) = n·t0 > 0 pointwise (all other basis matrices Xk, k ≥ 1, are traceless). The mechanism follows from the identity det(exp(M)) = exp(tr(M)). Under the Cartan clamp, H̃ = c·H with c > 0, so tr(H̃) = c·tr(H) > 0. This yields det(G) = exp(2 tr(H̃)) > 1. For Riemannian geometry, det(G) = exp(2 tr(H̃)) directly. For Lorentzian geometry, det(G) = det(E)2 det(η) = −exp(2 tr(H̃)), yielding the same absolute value. Since det(G) > 1, the eigenvalues of G cannot all shrink toward zero simultaneously, so the metric cannot be uniformly collapsed; the ratio loss is forced to work through genuine geometric structure rather than through metric shrinkage.

The Cartan spectral clamp serves two purposes. Mathematically, it bounds the spectral radius of H(x), preventing the matrix exponential from producing numerically singular metrics. Physically, it controls the maximum curvature the field can express: a larger L permits stronger local variations; a smaller L produces smoother fields. In the 3D and 4D experiments, spatial variation in L is introduced. Let r be the radial distance from the center. The clamp parameter becomes: L(r) = Lf + (Lc − Lf)·σ((rtr − r)/w), where Lc (center clamp) allows strong curvature near the origin, Lf (far clamp) tightly constrains the far region, σ is the sigmoid function, rtr is the transition radius, and w controls the sharpness of the decay. This radial profile enables the Cartan seesaw mechanism. For 2D and 6D, uniform Cartan clamping (L = constant) is sufficient.

In the 2D experiments, a point robot learns to avoid obstacles in a 3×3 world, rasterized into a 64×64 density grid. The metric is assembled via G = E2 = exp(2H̃) with η = I. Hyperparameters: Cartan clamp L = 12, learning rate 3×10−3, 400 epochs, Adam optimizer, gradient clipping at 10.0. The t0 > 0 constraint is applied. Training uses a single scene (T1) with two circular obstacles of radius 0.3 at (0, 0) and (0.6, 0.6), with start and goal positions randomly sampled for each training path. The training pool consists of 600 collision-free and 1,800 obstacle-penetrating paths generated via A* planning and straight-line interpolation. A random subset of 80 positive and 160 negative paths is sampled per epoch. After training, the model is evaluated on a comprehensive zero-shot test suite spanning 5 difficulty levels: L1 (start/goal, 10 scenes with same obstacles but new start and goal positions), L2 (position, 10 scenes with obstacles shifted to new locations), L3 (count, 10 scenes with 1–7 obstacles), L4 (shape, 10 scenes with non-circular obstacles: rectangles, L-shapes, corridors), and L5 (random, 50 fully randomized scenes). All test scenes are filtered to guarantee that the straight-line path intersects at least one obstacle. Paths are extracted via A* search on the learned metric field, using only the metric cost δTGδ with no hard collision checking or explicit obstacle representation. The T1-trained model achieves 100% pass rate on all 90 test scenes across all 5 levels. The learned metric field, visualized as local ellipses representing the inverse metric G−1(x), expands in obstacle regions and contracts in free space, guiding the A* geodesic to naturally avoid obstacles without any explicit collision constraint.

In the 6D experiments, the framework is extended to a 6-DOF serial manipulator operating in a 3D workspace with spherical obstacles. The manipulator is defined by standard Denavit-Hartenberg parameters (Table I: joint 1 with d=0.5, a=0.0, α=π/2; joint 2 with d=0.0, a=0.8, α=0; joint 3 with d=0.0, a=0.7, α=0; joint 4 with d=0.3, a=0.0, α=π/2; joint 5 with d=0.0, a=0.0, α=−π/2; joint 6 with d=0.15, a=0.0, α=0). The 3D workspace (1.5×1.5×1.5) is voxelized into a 32×32×32 grid. Obstacles are spherical (r ∈ [0.08, 0.18]), placed randomly within the manipulator's reachable workspace. Collision checking is performed via forward kinematics at 20 interpolation points per path, with a link radius of 0.08. Paths are extracted via A* search on a discrete grid followed by geodesic optimization. The metric field is parameterized by Sym(6) with m = 21 basis matrices (6 diagonal, 15 off-diagonal). The Cartan clamp uses L = 3.0 uniformly. Training uses 400 epochs, Adam with learning rate 3×10−3, feature dimension 16. Training uses a single scene (T1) containing two spherical obstacles: one at (0.07, 0.28, 0.29) with radius 0.16, and the other at (0.10, −0.09, −0.16) with radius 0.09. The training pool consists of 19,200 paths (4,800 collision-free, 14,400 colliding), an 8× scale-up from the 2D T1 data volume, necessary to suppress overfitting in the 21-dimensional parameter space of Sym(6). Each training epoch randomly samples 80 positive and 160 negative paths. After training, evaluation is on a comprehensive zero-shot test suite spanning 6 difficulty levels across 91 total scenes, with 1,000 paths per scene: L0 (baseline, identical to training scene), L1 (perturbation, 10 scenes with obstacle positions perturbed by ±0.15), L2 (position, 10 scenes with obstacles regenerated at new random locations), L3 (count, 10 scenes with 1–6 randomly placed obstacles), L4 (size, 10 scenes with obstacle radii r ∈ [0.05, 0.20]), and L5 (random, 50 fully randomized scenes). All test scenes are filtered to ensure the straight-line joint-space path intersects at least one obstacle. Paths are extracted via A* search on the learned 6D metric field, using only the metric cost δTGδ with no hard collision checking. The T1-trained model achieves 100% pass rate on all 91 test scenes across all 6 levels, with a global median separation ratio of 3.07× and 89.0% of scenes exceeding 2.0× (Table II). The L0 baseline achieves 6.16×, confirming that the learned metric generalizes well beyond the specific training paths. The separation ratio degrades gracefully under distribution shift, remaining above 2.7× median across all perturbation levels.

In the 3D experiments, the architecture is extended to 3D Lorentzian spacetime. The 3D experiments use a 12×12×12 spatial grid. The metric field is parameterized by Sym(3) with m = 6 basis matrices. The Lorentzian basis is η = diag(−1, +1, +1). Ingoing paths start at r > 0.5 and end at r < 0.25 (pos); outgoing paths go the reverse direction (neg). Training uses 400 epochs, Adam with learning rate 3×10−3, feature dimension fd = 20, and 200 ingoing / 400 outgoing paths per batch. The Cartan clamp uses the radial profile with Lc = 6.0, Lf = 0.5, rtr = 0.35, and w = 0.01. The field discovers a black-hole-like metric in the optimal configuration. The defining signature of a black hole is the sign flip of the time-time component Gtt: Gtt 0 in the interior. The radius where Gtt crosses zero is the event horizon rh, a surface beyond which causal structure is trapped, where ingoing paths have significantly lower cost than outgoing paths. The learned field exhibits exactly this behavior: at the center Gtt(0) = +0.189 (positive, signature flipped), while in the far region Gtt ≈ −0.419 (negative, Lorentzian preserved), with the sign flip occurring at rh ≈ 0.184. The separation ratio between outgoing and ingoing path costs reaches 569×, demonstrating a strong causal asymmetry. The spatial components Gxx and Gyy remain positive definite. Crucially, this is a sign flip, a metric with one event horizon, analogous to the BTZ black hole.

The sharpness of the Cartan decay, controlled by the width w, is the decisive parameter for horizon emergence. A scan of 12 configurations crossing four widths (w ∈ 0.08, 0.04, 0.02, 0.01) and three transition radii (rtr ∈ 0.35, 0.20, 0.10), with Lc = 6.0, Lf = 0.5 fixed, reveals three qualitative patterns (Table III). First, sharper decay promotes horizon formation: at w = 0.01 and w = 0.02, black-hole-like solutions (with a single sign flip) appear at both rtr = 0.35 and rtr = 0.20, while coarser decay (w = 0.04, 0.08) yields at most one success. Second, rtr = 0.10 universally fails regardless of w, producing deeply negative Gtt at the center; the transition is too close to the origin, leaving insufficient room for the metric to develop curvature. Third, among the successful configurations, the two transition radii produce different regimes: rtr = 0.35 consistently yields large separation (460–569×) with a moderate Gtt(0) ∈ [+0.19, +0.29], while rtr = 0.20 yields more variable outcomes — a weak horizon at coarse decay (Gtt(0) = +0.013) but an extreme one at sharp decay (Gtt(0) = +1.402), the latter approaching the shell-flip regime. The sharp decay effectively decouples the center and far regions: the center enjoys full expressive freedom while the far region is tightly anchored, leaving the optimizer no choice but to produce a genuine sign flip.

Under suboptimal Cartan configurations (insufficient Lc, or uniform clamping), a distinct failure mode appears: the shell flip, where Gtt changes sign twice along the radial direction: negative in the far region, positive in an intermediate shell, and negative again at the center. This corresponds to two event horizons, with separation ratios of 9× (uniform L = 5.0) and 152× (uniform L = 4.0). The shell flip can appear under uniform clamping and is suppressed by the Cartan seesaw.

In the 4D experiments, the architecture is extended to Sym(4) with m = 10 basis matrices. The grid is 12×12×12×12. The Lorentzian basis is η = diag(−1, +1, +1, +1). Training uses feature dimension fd = 20, 400 epochs, Adam with learning rate 3×10−3, and 200 ingoing / 400 outgoing paths per batch. The optimal configuration uses the radial profile with Lc = 8.0, Lf = 0.5, rtr = 0.20, and w = 0.01. The 4D field discovers a Schwarzschild-like black hole with separation ratio 628×, characterized by two signatures. First, an event horizon: Gtt is positive at the center (+96.5) and negative in the far region (−0.536), crossing zero at rh = 0.211; inside rh the time direction is trapped, outside it is free, consistent with the sign pattern of the Schwarzschild metric. Second, emergent spherical symmetry: the three off-diagonal time-space components (Gtx, Gty, Gtz) are all suppressed below 4×10−2 without any symmetry constraint, indicating that the m = 10 basis provides sufficient independent control to suppress cross-terms while maintaining the Gtt sign flip. The spatial diagonal components remain close to unity (≈ 1.04) at the far region, close to the Minkowski value.

An Lc scan (Table IV) shows the effect of increasing Lc while keeping Lf = 0.5. For Lc ≤ 6.0, Gtt(0) is deeply negative, and the center remains causally free, with no horizon forming. At Lc = 7.0, Gtt(0) barely crosses zero (+0.017), producing a weak horizon. At Lc = 8.0, the center becomes strongly positive (+72.3), producing a robust horizon with 623× separation. The seesaw requires a sufficiently large Lc to overcome the anchoring effect of Lf = 0.5.

The paper concludes that a continuous metric field, constructed via a fixed basis of symmetric matrices and trained by a single causal contrastive loss, discovers the full cross-dimensional spectrum of geometric structures, from zero-shot obstacle avoidance in 2D and 6D, to BTZ-like horizons in 3D, to Schwarzschild-like black holes in 4D, using only causal signals. The same loss function and assembly principle produce this range of phenomena across dimensions. The broader implication is that causal principles alone suffice to drive the emergence of diverse geometric structures: a geometry satisfying good paths are cheaper than bad paths will, under sufficient parametric freedom, develop the geometric structures appropriate to its dimension.

Based on the paper, I can identify several concrete improvements to AI systems, particularly in motion planning, geometric representation learning, and physics-informed neural networks.

Improvement 1: Continuous Metric Field Encoder for Motion Planning

Replace discrete obstacle-based cost maps with a learned continuous metric field G(x) in SPD(n) generated by a three-stage architecture: (a) a CNN backbone encoding the scene into a feature grid, (b) an MLP mapping interpolated features plus positional encoding to m = n(n+1)/2 coefficients of a fixed symmetric matrix basis, (c) assembly via G(x) = (H(x)) eta (H(x)) where H(x) = sum k t k(x) X k. This eliminates per-obstacle generators and routers, producing a unified field defined at every point.

What the improved system can do:

• Achieve 100% zero-shot pass rate on unseen obstacle configurations (shifted positions, varying counts, non-circular shapes, fully random scenes) after training on a single scene, in both 2D (90 test scenes) and 6D manipulator configuration spaces (91 test scenes).

• Extract geodesics via A* search using only delta G delta as cost, with no hard collision checking, achieving median separation ratios of 3.07× (6D) and perfect generalization across perturbation levels.

Improvement 2: Causal Contrastive Loss with Structural Anti-Collapse Constraint

Train the metric field with the loss L = L pos/(L neg + 0.1) + 0.001 times L pos, where L pos is the mean length of good paths (collision-free, or ingoing) and L neg is the mean length of bad paths (colliding, or outgoing). Enforce t 0 = softplus(t 0 raw) + 10-6 on the identity basis coefficient, guaranteeing tr(H) > 0 and hence (G) > 1 pointwise, preventing trivial metric collapse.

Improvement 3: Cartan Spectral Clamp with Radial Profile for Curvature Control

Replace uniform spectral clamping with a radial profile L(r) = L f + (L c - L f) times sigma((r tr - r)/w), where L c permits strong curvature near the origin and L f tightly anchors the far region. Apply (x) = c(x) H(x) with c(x) = (H(x)/L(r)) / (H(x)/L(r) + epsilon), bounding the spectral radius of H before exponentiation.

Improvement 4: Cross-Dimensional Transferable Architecture

Use a single unified construction for the symmetric matrix basis X k for any dimension n: X 0 = I, n-1 traceless diagonal matrices diag(1,, 1, -k, 0,, 0), and n(n-1)/2 symmetric off-diagonal matrices with unit entries. Apply the same loss, same training protocol (400 epochs, Adam, lr 3 times 10-3), and same assembly rule G = E eta E across 2D, 3D, 4D, and 6D.

Improvement 5: Failure-Mode Detection via Double Sign-Flip Characterization

Monitor the radial profile of G tt(r) during training to detect the shell flip failure mode (two sign changes: negative far, positive intermediate, negative center), which indicates suboptimal Cartan configuration (insufficient L c or uniform clamping). Use this as a diagnostic to automatically adjust L c or w.