Problem framing. In classical analysis, 0/0 as a literal division is undefined, and as a limit it is indeterminate because outcomes depend on relative rates of vanishing and the approach path/quality — neither is explicitly represented.
Aim. Provide a conservative, deterministic classification by:
- (i) making rates explicit for the magnitude class, and
- (ii) attaching a bounded alignment signal
a in (-1,+1)that records approach quality.
Conservativity. Under collapse, phi(<m,a>) = m reproduces the classical outcome when it exists (0, finite, +inf, -inf).
When classical limits do not exist. Instead of forcing a single number, report the regime with optional qualifiers:
ZERO / FINITE / INF+ / INF-— magnitude class from rate comparison (sign for infinity reflectssign(c_f/c_g)).SIDED(L/R)— left- and right-sided analyses differ (in class or sign).OSC— oscillatory behavior persists in shrinking neighborhoods ofx0.MULTI— competing best-fit families/rate-vectors disagree (conservative handling).NOFIT— available models fail basic adequacy checks.
Machine-friendly metadata. The alignment channel a carries bounded, composable approach/quality information (path, sidedness pooling, stability) without altering the recovered classical magnitude after collapse. Alignment composes via rapidity add/subtract and is reported once per division as DIV[a_div] with a band tag; for infinity outcomes, include DIR+ / DIR- in the print.
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