Quick, self-contained examples that mirror the document rules. All formulas are pure ASCII; alignment is metadata (print once via DIV[...]).
Conventions (reminder).
a_div := tanh( atanh(a_f) - atanh(a_g) ) # clamp a_f, a_g before any atanh
phi(<m,a>) = m # collapse recovers classical magnitude
Bands: A++, A+, A0, A-, A-- # from the value of a_div
Equal rates (finite ratio)
Setup. f = 2*|x|^3, g = 5*|x|^3 (near x0 = 0)
Rates. p_f = p_g = 3, so Delta_p = 0; value ratio c_f/c_g = 2/5.
Verdict. < 0.4 , a_div >
SSMS. VAL[0.400]@A0 DIV[a_div]@A0 DIR+
(Neutral priors example: a_f = a_g = 0 -> a_div = 0 -> A0.)
Numerator slower (goes to zero)
Setup. f = |x|^4, g = 2*|x|^2
Rates. p_f = 4 > p_g = 2
Verdict. < 0 , a_div >
SSMS. Z@A0 DIV[a_div]@A0
Numerator faster (blows up)
Setup. f = 3*|x|^2, g = |x|^5
Rates. p_f = 2 < p_g = 5, sign(c_f/c_g) = +
Verdict. < +inf , a_div >
SSMS. INF+@A0 DIV[a_div]@A0 DIR+
Side dependence (true SIDED finite)
Setup. f = |x|, g = 2*x
Observation. f/g = |x|/(2x) = (1/2)*sign(x)
One-sided prints.
x -> 0+ : < +0.5 , a_div > -> SSMS: VAL[0.5]@A0 DIV[a_div]@A0 DIR+
x -> 0- : < -0.5 , a_div > -> SSMS: VAL[-0.5]@A0 DIV[a_div]@A0 DIR-
Headline policy. REG=SIDED (print both one-sided lines).
Log tie-break (p ties, logs decide)
Setup. f = |x|^2, g = |x|^2 * log(1/|x|) as x -> 0+
Rates. p_f = p_g = 2 (p-tie), but Delta_q = 0 - 1 = -1 < 0
Verdict. < 0 , a_div >
SSMS. Z@A0 DIV[a_div]@A0
Essential-type decay beats powers (rare but illustrative)
Setup. f = exp( -|x|^(-0.5) ), g = |x|^2
Heuristic. Essential decay dominates any power as x -> 0.
Verdict. < 0 , a_div >
SSMS. Z@A0 DIV[a_div]@A0
(If both sides fit essential with same r, compare alphas per the model-chooser rule.)
Alignment channel (applies to all examples)
a_div = tanh( atanh(a_f) - atanh(a_g) )
a_f := clamp(a_f, -1+eps_a, +1-eps_a)
a_g := clamp(a_g, -1+eps_a, +1-eps_a)
eps_a = 1e-6
Print once as DIV[a_div]@{A-tag}; alignment does not change the headline magnitude.
Navigation
Prev: Classifying 0over0 Limits — Acceptance Tests — Operational diagnostics and registry (ties, flags, print grammar) (6B)
Next: Classifying 0over0 Limits — Findings (8)
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.
SHUNYAYA 