Infinity with structure. Deterministic, class-safe, alignment-preserving.
1. What Makes Directional Infinity Unique
In classical mathematics, infinity is a monolith — a single symbol with no internal structure.
SSM-Infinity replaces this with a fully defined object:
<sign, a>
sign → +1 (positive infinity) or -1 (negative infinity)
a → alignment lane in (-1, +1)
This gives infinite quantities direction, posture, and behavioural consistency.
2. Why Direction Matters
A runaway physical or mathematical process does not simply “go infinite.”
It diverges with orientation:
- +∞ = outward expansion
- −∞ = inward collapse
- Alignment
a= the shape or posture of the divergence
SSM-Infinity captures this in a clean symbolic form that preserves classical outcomes but allows finer analysis.
3. Three Symbolic Classes
Every operation in SSM-Infinity returns one of three lawful result categories:
🟦 1. infinite-class
A new directional infinity is the output.
Form:
("infinite-class", SymbolicInfinity(sign, a))
Example:(+∞) + (+∞) → infinite-class (positive infinity strengthened)
🟧 2. zero-class
Both infinities perfectly cancel.
Form:
("zero-class", a)
Example:(+∞) + (−∞) → zero-class (alignment extracted from merge)
🟩 3. finite-class
A structured finite value emerges.
Form:
("finite-class", a)
Example:(+∞) / (+∞) → finite-class (alignment determines category)
These three outcomes replace classical “undefined” results with lawful, deterministic structure.
4. Alignment-Preserving Merging (atanh → tanh)
Just like the rest of Shunyaya Symbolic Mathematics, SSM-Infinity uses a proven stable merge model:
# alignment merge
u1 = atanh(a1)
u2 = atanh(a2)
U = u1 + u2
a_out = tanh(U)
This ensures:
- order-invariance
- stability
- reversibility
- bounded output in (−1,+1)
No chaos. No randomness. No numerical simulation.
5. A Clean Example: ∞ − ∞
Classically
∞ − ∞ → undefined
SSM-Infinity
from ssm_infinity_core import SymbolicInfinity
x = SymbolicInfinity(+1, 0.5)
y = SymbolicInfinity(+1, 0.5)
result = x - y
print(result)
Output (deterministic):
("zero-class", 0.0)
Because the two infinities have matching direction and matching posture, they cancel cleanly.
A different alignment would yield finite-class or infinite-class.
No undefined outcomes.
6. Example: ∞ / ∞
Classically
∞ / ∞ → indeterminate
SSM-Infinity
x = SymbolicInfinity(+1, -0.3)
print(x / x)
Output:
("finite-class", 0.0)
Division produces a structured finite remainder, not undefined chaos.
7. Example: +∞ + (−∞)
x = SymbolicInfinity(+1, 0.8)
y = SymbolicInfinity(-1, 0.4)
print(x + y)
Output is deterministic:
("zero-class", merged_alignment)
Or if posture mismatch is strong:
("infinite-class", SymbolicInfinity(sign, a))
No ambiguity, no undefined behaviour.
Navigation
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SSM-Infinity – Table of Contents
Disclaimer
This page explains an observation-only mathematical model derived from Shunyaya Symbolic Mathematical Infinity (SSM-Infinity) and is not intended for critical or operational decision-making.
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