Classifying 0/0 Limits — Findings (8)

i) Determinacy via rates.
Compare exponents first, then (only if tied) log modifiers — lexicographic on (Delta_p, Delta_q) where
Delta_p := p_f - p_g, Delta_q := q_f - q_g.

  • If p_f > p_g → ZERO
  • If p_f = p_g → FINITE with value c_f/c_g
  • If p_f < p_g → INF with direction s := sign(c_f/c_g) (use INF+ if s > 0, INF- if s < 0)
    Rule: powers dominate logs; use logs only when p_f = p_g.

ii) Alignment carries approach info.
Use rapidity subtraction for division:

a_div = tanh( atanh(a_f) - atanh(a_g) )

Clamp before any atanh:
a := clamp(a, -1+eps_a, +1-eps_a) with eps_a = 1e-6.
Alignment is metadata that summarizes approach/quality and composes cleanly; it does not change the magnitude class.

iii) Conservativity.
Collapse recovers classical results exactly:

phi(< m , a >) = m

Nothing here contradicts standard calculus; alignment cannot flip ZERO / FINITE / INF.

iv) Practicality.
Rates and coefficients come from simple log–log fits; windowed tail checks certify regimes (e.g., monotone-up for INF, vanishing for ZERO, shrinking deviation for FINITE). Optional registry flags (SIDED / OSC / MULTI / NOFIT) keep outputs conservative and auditable. Everything is ASCII-clean and easy to implement.

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Explore further:
https://github.com/OMPSHUNYAYA/Symbolic-Mathematics-0over0-Limits

Disclaimer (observation only)
This page informs analysis and decision support; it does not replace domain models, operational controls, or safety processes.