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Definition and structural reduction

FPRD-D35

Metric-group product on the nonnegative half-line

Exact statement

A metric-group product is one binary operation star on the nonnegative reals that both makes the set a group and makes d(x,y)=x star y a metric.

StatusSelf-contained definition
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The same operation must carry the group law and directly return numerical distances. This is much more rigid than merely placing an invariant metric on an abstract group.

Definitions

  • The carrier is [0,∞)[0,\infty), the identity is 00, and d(x,y)=x⋆yd(x,y)=x\star y.

Proof or evidence

The group and metric axioms force a commutative Boolean group, and the triangle inequality becomes the single numerical constraint x star y <= x+y.

Definition and forced structure →

Verification notes

The definition, forced identity, self-inverse law, commutativity, and subadditivity reduction were reconciled against the theorem-claims and construction audits.

Limitations

  • This definition does not require ordinary continuity of the group operation.
  • It is not equivalent to choosing an invariant metric on a pre-existing continuum group.

Open work

Add broader examples and compare this same-operation requirement with ordinary invariant metrics on abstract groups.

Notes

This is the Conway/Klene problem shape, not an FPRD novelty claim. It is stronger than merely choosing an invariant metric on an abstract group whose underlying set has continuum cardinality.