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Sharp family obstruction

FPRD-T121

The nested family crosses the bounded-variation boundary

Exact statement

For every c>=1, each finite-support translation in the nested family is locally of bounded variation, but some sparse infinite-support translation has infinite total variation already on [0,1].

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

Context

Cadlag regularity is simultaneous throughout the explicit family, but bounded variation charges the total accumulated jump mass and detects a stricter resource boundary.

Proof or evidence

Finite support affects only finitely many cylinder levels and gives bounded-overlap jump sums. A sparse infinite support can place fixed positive jump mass at infinitely many disjoint boundary levels, forcing infinite variation on [0,1].

Bounded-variation boundary →

Verification notes

The monotone cylinder pieces, bounded-overlap accounting, sparse support choice, and fixed jump-mass lower bound were reconstructed from the written proof.

Limitations

  • The theorem constructs one sparse infinite-support counterexample; it does not classify every infinite-support translator.
  • It refutes simultaneous local BV only for this nested family, not for all metric-group products.

Open work

Classify more infinite digit streams and compare the nested-family failure with non-coordinate actions.

Notes

Finite support leaves monotone tail maps inside finitely deep cylinders and yields bounded-overlap jump sums. The counterexample spaces digits equal to 1 so that infinitely many disjoint boundary levels each carry a fixed positive amount of absolute jump mass. This does not prove that simultaneous local BV is impossible for every conceivable metric-group product.