Physics · Primitives

Gradient

The directional rate of change of a scalar field — substrate for hill-climbing, attention flow, and credit assignment

Definition

Mechanism

The directional rate of change of a scalar field across a differentiable space. For any sufficiently smooth field f over a differentiable space, the gradient ∇f at a point is the vector whose direction is the one of steepest ascent of f and whose magnitude is the rate of increase in that direction. Substrate property: gradients exist wherever fields exist, whether or not anyone measures them. Specific cases include Entropy gradients (the direction of maximum disorder increase), attention gradients, credibility gradients, information-density gradients. Higher patterns use gradients for hill-climbing search, attention allocation, credit assignment, and flow routing.

Invariants

  1. Existence: every sufficiently smooth field on a differentiable space has a gradient defined at every interior point.
  2. Direction: ∇f points in the direction of steepest ascent of f.
  3. Magnitude: |∇f| is the rate of change in the direction of steepest ascent.

Risks

Failure modes

None stated.

Relationships

Builds on

References

Used by

References

Infrastructure

Judge

Scalar evaluation of merit

Society

MemeticSeed

Viral propagation of semantic standards via economic subsidy

Mind

Optimize

Maximize utility against a metric

Reference

Identity

Ref
Gradient#ecfc
Sema ID
sema:Gradient#mh:SHA-256:ecfc2cd2d9eb95778ef04ede8a7e872c33822f174a9e3273fa7bba92b3a30e36
Ring
0
Tier
1
Updated
2026-11-11
Supersedes
  • sema:Gradient#mh:SHA-256:dcf04816f4c8639244be889bb5030889bcd786600046a64a67b3ad59a679bb96