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Universal density theorem

FPRD-T122

Every metric-group product has densely irregular sections at small scales

Exact statement

For every metric-group product and ordinary-open interval U, the parameters whose translation discontinuity set meets U form an intrinsic-open dense set; under intrinsic Baireness, whole-ray-dense sections are generic, and unconditionally every requested scale contains a smaller initial interval on which one translation has a dense discontinuity set.

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

Context

The closed-window theorem forces a discontinuity for each scale. The parameter topology and exact profile identity let those witnesses be inserted densely across parameter space.

Hypotheses and scope

  • Interval hitting and initial-interval density require only the metric-group axioms.
  • The generic whole-ray conclusion additionally assumes the intrinsic metric space is Baire.

Proof or evidence

Uniform profile distance makes interval hitting intrinsic-open; local-homeomorphism insertion makes it dense. A fixed-scale negation would turn the intrinsic topology into an admissible Baire extension, whose generic whole-ray sections contradict that negation.

Universal dense-section theorem →

Verification notes

The oscillation estimate, ball identity, local pullback, Baire transfer, direct nested-compact proof, and quantifier negation were independently reconstructed and hostilely reviewed.

Limitations

  • Unconditionally, both the parameter a and interval length epsilon may depend on the requested scale eta.
  • Whole-ray density without intrinsic Baireness remains open.
  • No conclusion about measure, cardinality, or nonmeagerness of universal exceptional unions is claimed.

Open work

Resolve whether every product has one whole-ray-dense section without assuming intrinsic Baireness.

Notes

The interval-hitting proof uses the exact translation-profile distance, the sharp closed-window theorem, and local-homeomorphism insertion; no Baire hypothesis is used there. The initial-interval theorem allows both a and epsilon to depend on eta. Whole-ray density remains conditional on intrinsic Baireness, and the claim does not assert that every product has a whole-ray-dense section or that universal exceptional unions are nonmeager, uncountable, or positive-measure. Chatyrko supplies the general admissible-extension transfer, not the metric-group pi-base or final theorem. No novelty claim is made.