Symbolic operations on ∞, zero-class collapse, and finite-class stability.
SSM-Infinity replaces ambiguous infinite arithmetic with lawful, deterministic, reproducible rules.
Every operation is governed by three principles:
- Directional infinity:
(+∞, a)and(-∞, a)carry posture. - Class-safe outcomes: results fall only into infinite-class, zero-class, or finite-class.
- Alignment preservation: merging uses the hyperbolic lane (
atanh→ weighted sum →tanh).
The full behaviour is implemented in the core engine:
core/ssm_infinity_core.py
1. Addition & Subtraction
Same-direction infinities
(+∞, a1) + (+∞, a2) → (+∞, a_out)
(-∞, a1) + (-∞, a2) → (-∞, a_out)
Always infinite-class. Lane merges deterministically.
Opposite infinities
(+∞, a1) + (-∞, a2) → zero-class
(+∞, a1) - (+∞, a2) → zero-class
Classical ∞−∞ ambiguity collapses into the zero-class.
This is one of the strongest contributions of SSM-Infinity — zero randomness, fully reproducible.
Infinity with finite values
(+∞, a) + m → (+∞, a)
(-∞, a) + m → (-∞, a)
2. Multiplication
Infinity × positive finite
(+∞, a) * m (m>0) → (+∞, a)
(-∞, a) * m (m>0) → (-∞, a)
Infinity × negative finite
(+∞, a) * m (m<0) → (-∞, a)
(-∞, a) * m (m<0) → (+∞, a)
Infinity × zero
(+∞, a) * 0 → zero-class
(-∞, a) * 0 → zero-class
A second classical ambiguity (∞ * 0) resolved by the symbolic zero-class.
3. Division
Infinity ÷ finite
(+∞, a) / m (m>0) → (+∞, a)
(+∞, a) / m (m<0) → (-∞, a)
Finite ÷ infinity
m / (+∞, a) → zero-class
m / (-∞, a) → zero-class
Infinity ÷ infinity
(+∞, a1) / (+∞, a2) → zero-class
(-∞, a1) / (-∞, a2) → zero-class
(+∞, a1) / (-∞, a2) → zero-class
Another classical indeterminate (∞/∞) converted into a deterministic zero-class.
4. Exponentiation
Positive finite exponent
(+∞, a) ** k → (+∞, a)
Negative exponent
(+∞, a) ** (-k) → zero-class
This resolves the classical ambiguity of ∞**(-n).
Zero raised to negative infinity & related forms
Always collapses safely to zero-class.
5. Unary Operations
-(+∞, a) → (-∞, a)
-(-∞, a) → (+∞, a)
6. Full Implementation (Engine Code)
Below is the operator section of the engine for reference:
# ssm_infinity_core (excerpt)
def _add_same_direction(self, other):
if self.sign == other.sign:
return SSMInfinity(sign=self.sign,
align=tanh(atanh(self.align) + atanh(other.align)))
return ZERO_CLASS
def __add__(self, other):
if isinstance(other, SSMInfinity):
return self._add_same_direction(other)
return self if isinstance(self, SSMInfinity) else ZERO_CLASS
def __sub__(self, other):
if isinstance(other, SSMInfinity):
if self.sign != other.sign:
return SSMInfinity(sign=self.sign,
align=tanh(atanh(self.align) + atanh(other.align)))
return ZERO_CLASS
return self
(Code truncated for page readability; full version is in the uploaded file.)
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SSM-Infinity – Table of Contents
Disclaimer
Shunyaya Symbolic Mathematical Infinity (SSM-Infinity) is a research framework, not a numerical simulator or predictive tool.
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