Classical calculus treats many expressions of the form lim_{x->x0} f(x)/g(x) with f(x0)=g(x0)=0 as indeterminate. Shunyaya Symbolic Mathematics (SSM) makes this deterministic by (i) making rates of vanishing explicit and (ii) carrying direction/quality as a bounded alignment a in (-1,+1) composed in rapidity space. SSMS then prints compact, auditable one-liners.
Result in one line (ASCII).
Given local models near x0:
f ~ c_f * |x - x0|^(p_f)
g ~ c_g * |x - x0|^(p_g)
Alignment for division (rapidity subtraction; clamp before any atanh):
a_div = tanh( atanh(a_f) - atanh(a_g) )
Define for convenience:
s := sign(c_f / c_g) # assumes c_f != 0, c_g != 0
Magnitude class decided by powers first:
If p_f > p_g: f/g -> < 0 , a_div > # ZERO
If p_f = p_g: f/g -> < c_f / c_g , a_div > # FINITE
If p_f < p_g: f/g -> < s * inf , a_div > # INFINITY; +inf if s>0, -inf if s<0
Conservativity (collapse).
phi(< m , a >) = m
This reproduces the classical magnitude (0, finite ratio, or +/-inf). Alignment a never changes the headline class; it preserves approach/quality information that classical limits discard.
Optional extension (logs; lexicographic rule).
If local behavior includes logarithmic modifiers, define L(x) := log( 1 / |x - x0| ) and write
f ~ c_f * |x - x0|^(p_f) * L(x)^(q_f)
g ~ c_g * |x - x0|^(p_g) * L(x)^(q_g)
Let Delta_p := p_f - p_g, Delta_q := q_f - q_g, s := sign(c_f / c_g).
Decide lexicographically: compare Delta_p first; if Delta_p = 0 compare Delta_q; if both tie, return < c_f/c_g , a_div >.
If Delta_p > 0 -> < 0 , a_div >
If Delta_p < 0 -> < s * inf , a_div >
If Delta_p = 0 and Delta_q > 0 -> < s * inf , a_div >
If Delta_p = 0 and Delta_q < 0 -> < 0 , a_div >
(Powers dominate logs; deeper modifiers such as (log L)^r can be added by extending the vector.)
Print cue (SSMS one-liner, division).
< m_out , a_div > -> SSMS: {CLASS}@{A-tag} DIV[a_div]@{A-tag} {DIR?}
DIR is required for INF (DIR+ / DIR-), optional for FINITE, and omitted for exact ZERO.
Navigation
Prev: P3 — SSM — Why This Classification Matters & What It Unlocks (0C, 0D)
Next: Classical Context & Gap Addressed (2)
SHUNYAYA 