Skip to content
Formal Philosophy
Menu
Tutorials Glossary About GitHub
Tutorial 32

Temporal label functions and staged bases

A narrow tutorial on temporal label functions, monitor-cell labels, and why a richer label basis can be useful only after the right first-stage carving.

View source Built 2026-04-01

Why this tutorial exists

Many neuro-symbolic loops talk about labels as if there were one natural basis.

Typical examples are:

  • pass or fail
  • accepted or rejected
  • flat trace classes

The temporal result shows that this picture is too simple.

Sometimes a richer temporal basis is not globally useful. But after the right first-stage carving, it becomes exact.

That makes temporal labels a staged basis-change tool.

Vocabulary Note

  • Quotient means the grouping of cases that remain indistinguishable under the current label basis.
  • Residue means the smaller family left after the first carve.
  • Carve means the first coarse split of the task family before switching to a finer second-stage basis.
  • Controller means the compact symbolic rule used after that carve, not a physical control device.

Part I: a label function is a choice of coordinates

A label function says how the loop sees the task family.

If the labels are too coarse, different hidden structures collapse together. If the labels are too fine, the loop may pay too much complexity too early.

The useful question is when a richer temporal basis becomes the right basis, not simply whether it is richer.

The easiest analogy is coordinate choice in geometry.

A polar coordinate system can be the right tool for one subproblem and the wrong tool for another. Temporal monitor labels can behave the same way.

Part II: the bounded result

The stable bounded temporal result in this repo is small but sharp:

  • raw monitor-cell labels strictly refine flat two-step trace labels on the full temporal controller family
  • after first-step carving, the two label functions become partition-equivalent

So the richer temporal basis really does carry more information.

But the same result also shows that the extra information is not automatically the right starting basis for the whole family.

Quick Logic Refresher

  • Partition-equivalent means two label functions cut the family into the same groups, even if they use different names.
  • Quotient means the grouped version of the family after cases that look the same under the current labels are merged.
  • Residue means the smaller family that remains after the first carve.

Temporal basis shift

Temporal label basis shift On the full temporal family, monitor-cell labels are strictly finer than flat trace labels. After the first-stage carve-out, the two label bases induce the same partition. Temporal label basis shift Richer labels are not always the right global basis. Sometimes they become exact only after the first carve. Full temporal family four controllers before any carve A1 A2 B1 B2 Flat trace basis coarse classes: {A1,A2} and {B1,B2} Monitor-cell basis strictly finer on the full family [A1] [A2] [B1] [B2] more information than the flat trace basis but also a more expensive global coordinate system v161 lesson: more information is not always the right first basis After first-stage carving the family has already been split once A1 A2 Partition-equivalent now flat labels and monitor labels induce the same remaining partition Basis shift should happen after the carve, not before it.
The richer temporal basis is strictly finer on the full family, but after the first carve it no longer buys a finer partition.

Interactive lab

Part III: what this changes in loop design

This result is small, but it changes how basis choice should be treated in two concrete ways.

First, loop comparison should ask three questions rather than one:

  • what label basis is being used?
  • what quotient does that basis induce?
  • at what stage does that basis become exact?

Asking whether one label set is “richer” than another is not enough. The bounded result shows that a strictly finer basis can become redundant after the right first-stage carve.

Second, it establishes that loop-space geometry includes label basis as a first-class axis alongside witness language and separator language. That warns against a common mistake: assuming the finest available basis is automatically the best global basis.

The temporal result suggests a staged design pattern:

  1. carve the task family with a coarse basis
  2. switch to the richer temporal basis on the residue
  3. compile or control the smaller residue