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Universal sharpness theorem

FPRD-T118

Universal discontinuity windows and the exact profile identity

Exact statement

For every metric-group product and a>0, each closed interval [r,r+a] contains a discontinuity of the translation T_a(x)=a star x, and the bound is sharp; moreover sup_b |T_b(x)-T_b(y)|=x star y.

StatusProved and internally audited
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Some irregularity is forced by the same-operation requirement, independently of the explicit nested family. Translation profiles recover the intrinsic metric exactly.

Proof or evidence

Distinct points in an ordinary continuity interval of T_a must be less than a apart, which forces a discontinuity in every length-a closed window. Translation invariance gives the upper profile bound and a suitable group element attains equality.

Universal translation bounds →

Verification notes

The strict inequality, closed-window endpoints, sharpness example, supremum equality, and continuity/equicontinuity equivalence were checked against the translation audit.

Limitations

  • The theorem forces dense-enough discontinuities but does not determine their exact set for an arbitrary product.
  • The profile fact is classical in origin as specialized here; no novelty is claimed.

Open work

Clarify the classical profile identity's provenance and seek review of the sharp interval construction.

Notes

The interval proof uses only involutivity, fixed-point freedom, the displacement bound |T_a(x)-x| <= a, and sign constancy on a connected continuity interval. Translation by 1 in the nested family swaps adjacent unit cells, and scaling gives sharpness for every a. The sup identity is the classical distance-profile isometry specialized to this numerical metric; no novelty is claimed for that general fact.