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Simultaneous regularity classification

FPRD-T119

Complete translation regularity for the nested family

Exact statement

For every c>=1 in the nested family, all translations are cadlag and share the countable dense discontinuity set B_c^+=E_c(Fin)\{0}; infinite-support parameters jump at all of B_c^+, while a finite nonzero parameter has an exact closed scattered jump set of Cantor-Bendixson height s+1 and largest gap h_c^s(2^r).

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

Context

The universal window theorem says irregularity must occur. The digit presentation identifies exactly where, how densely, and with what scattered depth each translation jumps.

Hypotheses and scope

  • The parameter c satisfies c>=1.
  • For finite support, s is the final support coordinate, A the final digit, and r=v_2(A).

Proof or evidence

Zero has no jumps; infinite support jumps at every positive finite-code point; finite nonzero support produces an exact scattered subset with height s+1 and largest gap h_c^s(2^r). Their union is B_c^+, leaving a co-countable common continuity set.

Translation classification →

Verification notes

The zero, finite-support, and infinite-support cases; density; arithmetic identification; gap formula; and shared-family regularity claims were separately reconciled and replayed with finite regression probes.

Limitations

  • The equality B_c=Q_>=0 holds for positive integer c, not arbitrary real c.
  • The equi-Baire-one statement is a shared-gauge family property and does not promote the decoder to standard Baire class 1.

Open work

Seek specialist review of the finite-support Cantor-Bendixson calculation and simultaneous-family regularity terminology.

Notes

If the final support is s and the final digit A has r=v_2(A), the discontinuity set has height s+1 and exact maximum complementary diameter h_c iterated s times at 2^r. For every positive integer c, B_c is the nonnegative rationals; this arithmetic identification is not asserted for arbitrary real c. Equi-Baire 1 refers only to the cited shared-LTZ-gauge family notion and does not make the decoder standard pointwise-limit Baire class 1 into N^N.