Exact ordinary regularity at c=2

Exact statement

For c=2 the transported product is Baire class 1 and Riemann integrable on compact rectangles, its translations are cadlag, and it is discontinuous exactly when one coordinate is a positive rational and the other is positive; nevertheless no metric-group product can be separately continuous in the ordinary topology.

StatusProved and internally audited

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The intrinsic group geometry is clean while the ordinary order topology sees a dense arithmetic seam. Exact regularity locates that mismatch rather than describing the product simply as discontinuous.

Proof or evidence

The discontinuity set is (Q_>0 x (0,infinity)) union ((0,infinity) x Q_>0); both axes remain continuous. Finite truncations provide Baire-one approximation and the section analysis gives cadlag translations.

Ordinary regularity →

Verification notes

The axes, rational polarity, compact-rectangle integrability, translation jumps, and universal separate-continuity obstruction were independently reconciled. The decoder was corrected to Sigma^0_2/first Lebesgue class rather than standard Baire class 1.

Limitations

The exact arithmetic description is for c=2.
Baire class 1 of the binary product does not make the stream decoder Baire class 1.

Open work

Obtain descriptive-set-theory review of the exact discontinuity set and decoder complexity distinction.

Notes

The real-valued Baire-class theorem uses uniform finite-digit approximants. The decoder into N^N is instead Sigma^0_2-measurable or first Lebesgue class and is not standard Baire class 1; its exact discontinuity set is Q_{>0}. Every axis point is a continuity point of the binary product. These two corrections were found in the independent publication-level claim audit before first release.