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Equivalence theorem and restricted no-go results

FPRD-T124

The exact frontier for simultaneous local bounded variation

Exact statement

A metric-group product with all translations locally BV exists exactly when the half-line admits a regular elementary abelian 2-action by locally BV permutations satisfying |h_a(x)-x|<=a; locally finite interval-exchange actions cannot do this, and every dyadic-only repair of binary labels gives infinite variation to infinitely alternating translators.

StatusProved frontier; general existence remains open
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The theorem converts the remaining analytic question into an exact action-realization problem, then removes two natural but insufficient representation doctrines.

Hypotheses and scope

  • The action equivalence selects a regular elementary abelian 2-subgroup; it does not assume all locally BV permutations are closed under composition.
  • The binary obstruction preserves ordinary binary labels away from eventually constant names.

Proof or evidence

Evaluation at zero transports a regular action to the group law, and the displacement bound gives metric domination. T122 rules out locally finite interval exchanges. In the binary architecture, each adjacent bit change contributes more than 1/2 to finite-prefix variation, so infinitely alternating translators have infinite variation.

Simultaneous local-BV frontier →

Verification notes

The action transport, displacement equivalence, locally finite-IET contradiction, finite-prefix XOR variation formula, and endpoint repair boundary were independently re-derived; the finite formula has exact regression support.

Limitations

  • Infinite support alone is not the binary obstruction; infinitely many adjacent bit changes are required.
  • Dense discontinuities can still have summable jumps and bounded variation.
  • The general regular locally-BV action is neither constructed nor refuted.

Open work

Construct or obstruct a regular locally-BV Boolean action beyond coordinate readouts and locally finite interval exchanges.

Notes

The binary obstruction applies only to bijections that preserve ordinary binary labels away from eventually constant names and uses infinitely many adjacent changes, not infinite support alone. Dense discontinuities are compatible with bounded variation, so the universal initial-density theorem does not refute the general regular action. The all-one endpoint calibration supplies no metric law or simultaneous BV family; a different ordering or non-coordinate action is outside the binary theorem. The two remaining stronger questions are unconditional whole-ray density and existence of the general locally BV regular action. Głąb--Kaufmann-- Pellegrini provide dense-jump BV calibration only. No novelty claim is made.