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Research draft

invariant

vr.tr.invariant · XCT.QLT

Let an agent explain invariants by field, transformation, examples and use in proofs, classification and program correctness.

Thing Registry Cross-cutting context

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

Let an agent explain invariants by field, transformation, examples and use in proofs, classification and program correctness.

A property, quantity or object that remains unchanged when a specified transformation or operation is applied, used throughout mathematics, physics and computer science; examples include the determinant under similarity, the Euler characteristic, knot invariants, conserved quantities and loop invariants.

What it is for: Classifying objects, proving results and checking correctness.

It can be identify invariants under a transformation; use invariants to show two objects differ; explain loop invariants in algorithms; relate invariants to conservation laws.

Distinguishing features

Defined relative to transformations

Used to distinguish objects

Complete invariants classify fully

Links to symmetry

What it looks like

Not physical; formulas, numbers or structures in mathematics and code.

How it is recognised

Unchanged under a stated transformation

Euler characteristic and knot polynomials

A constant is not necessarily an invariant of anything

Related models

is a kind of - category

mathematical property

is defined by - context

transformation group

includes - example

Euler characteristic

is related to - concept

symmetry

In practice

Families and kinds

algebraic invariants

topological invariants

knot and link invariants

physical invariants and conserved quantities

loop and class invariants in programming

Standards and regulation

No specific regulation; formal verification standards use invariants

Failure modes and hazards

Not stating the transformation

Treating an incomplete invariant as a classifier

Also called

adiabatic invariantfocal parameterp-quantilehomologylink invariantde Rham cohomologytree propertyDolbeault cohomologyinvariant of geometric transformationirregularity of a surfaceGromov–Witten invariantcohomological invariantInvariant of a binary formj-multiplicityweak dimensionGopakumar-Vafa invariantknot invariantGriffiths groupsystole6-j symbolgeneral covarianceétale fundamental groupdegree of a continuous mappingexponent of a groupBerezinianCarminati–McLenaghan invariantscardinal functionBass numberbirational invariantChow ringcohomological dimensioncohomotopy setComplex dimensioncovering numberDeligne cohomologyDeviation of a local ringdifferential invariantDonaldson–Thomas theoryFitting idealHilbert–Samuel function

+24

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Concept What it means.

Transformations matter.

Transformation

Under what.

Transformation

Transformation.

  1. Under which transformations is this quantity invariant? definition
  2. Is it invariant under all of them? boundary

Completeness

Classification.

Completeness

Completeness.

  1. Is the invariant complete, so that equal values imply equivalence? boundary
  2. Which counterexamples exist? provenance
Examples Fields.

Invariants appear widely.

Topology

Shapes.

Topology

Topological invariants.

  1. How does the Euler characteristic distinguish surfaces? definition
  2. How is it computed? measurement

Knots

Knot theory.

Knots

Knot invariants.

  1. Which knot invariants can show two knots are different? definition
  2. How are they calculated? measurement
Physics Conservation.

Symmetry gives invariants.

Conservation

Conserved quantities.

Conservation

Conserved quantities.

  1. How does Noether theorem link symmetries to conserved quantities? definition
  2. Which quantity corresponds to which symmetry? definition

Adiabatic

Adiabatic invariants.

Adiabatic

Adiabatic invariants.

  1. What is an adiabatic invariant, and where is it used? definition
  2. How approximate is it? boundary
Computing Programs.

Invariants prove correctness.

Loop

Loop invariants.

Loop

Loop invariants.

  1. What loop invariant proves this algorithm correct? definition
  2. Does it hold at initialisation, maintenance and termination? boundary

Verification

Formal methods.

Verification

Formal verification.

  1. Which tools check invariants in code? provenance
  2. How are they specified? definition

What the second pass must settle

  • Should each invariant be a separate entry?
  • How should fields be linked?
  • How should programming invariants be split off?
Draft as JSON