Skip to content

math.md

Latest commit

 

History

History
83 lines (52 loc) · 2 KB

math.md

File metadata and controls

83 lines (52 loc) · 2 KB
Raw

Mathematical Notes

Directional Relational Manifold

A Directional Relational Manifold is described by:

  • a space M;
  • a set of active directions D(p) at each point p;
  • an effective local dimension dimD(p) = |D(p)| or a soft relaxation by gates;
  • a relational metric g_p over admissible directions;
  • admissible curves whose velocities lie in span(D(p)).

In this implementation, p is represented by a latent state vector z.

Active Directions

The model learns directions:

V_i(z) in T_z M
a_i(z) in [0, 1]
dimD(z) = sum_i a_i(z)

The directions may be non-orthogonal. Gates provide a soft active set instead of a fixed fundamental dimension.

Relational Metric

The learned metric is:

G(z) = diag(softplus(d(z)) + eps) + U(z)U(z)^T

For a velocity v, metric energy is:

g_z(v, v) = v^T G(z) v

The low-rank term gives the metric a learned coupling structure between state coordinates and directions.

Admissible Curves

A rollout is admissible when:

dz_t in span(D(z_t))

The implementation enforces this by constructing:

dz_t = sum_i a_i(z_t) c_i(z_t, e_t) V_i(z_t)

Action

The discrete action proxy is:

A(z_0:T) = sum_t dt * g_z_t(dz_t, dz_t)

Training can penalize mean action. This is a variational bias, not an exact geodesic solver.

State Update

The MVP uses an Euler update:

z_{t+1} = z_t + dt * dz_t

When bounded_state is enabled, norm clipping and tanh projection keep the coordinate state in a compact region.

Future Metrics

A future version could define relational metrics through pullbacks from decoder distributions or Fisher information. That would connect the geometry more tightly to emitted language distributions.

Toroidal Convergence

Toroidal convergence is not assumed. It is only a possible hypothesis under strong conditions such as boundedness, recurrence, and structural stability. The optional toroidal representation is a coordinate utility, not evidence of toroidal topology.