Axiom 3: Oscillating Truths (Ontiparadox, Oscilloparadox, Divergiparadox)
Guideline: Paradoxes oscillate between states. A paradox may toggle between multiple possible truths without settling on one.
Mathematical Implication: In equations where multiple states exist (like complex number solutions), allow for oscillating values and use statistical or probabilistic models to handle multiple outcomes simultaneously.
Axiom 4: Absurd Solutions (Antiparadox, Absurdiparadox, Phantiparadox)
Guideline: Sometimes, the resolution of a paradox is absurd or unexpected, challenging initial expectations of logical conclusions.
Mathematical Implication: When encountering a seemingly unsolvable problem, introduce absurd variables or assumptions to test whether the paradox can be recontextualized — essentially allowing for variables that defy traditional logic to represent the paradox.
Axiom 5: Undefined Boundaries (Ambiviparadox, Reflectiparadox, Schismiparadox)
Guideline: Boundaries of paradoxes are inherently ambiguous. Any solution might be both correct and incorrect depending on how you reflect on it.
Mathematical Implication: Equations with ambiguous boundary conditions must be evaluated using fuzzy logic or tolerance ranges, allowing solutions to exist within a spectrum rather than being strictly defined.
Axiom 6: Recursive Inevitability (Futiliparadox, Nulliparadox, Velociparadox)
Guideline: Attempting to solve some paradoxes inherently leads to futility — they either repeat or their solution leads to the appearance of another paradox.
Mathematical Implication: Acknowledge inherent limitations within any mathematical approach to resolving paradoxes. When formulating algorithms, include termination points that acknowledge potential failure states or built-in timeouts to prevent infinite attempts at solutions.
Axiom 7: The Hidden Nature (Obfusiparadox, Intangiparadox, Elusiparadox)
Guideline: Some paradoxes only exist when observed, while others disappear once you start investigating them. They have an ephemeral, elusive nature.
Mathematical Implication: Mathematical laws involving probability or quantum states should reflect this concept. The act of measuring a value or observing a system can change the outcome (similar to the Heisenberg Uncertainty Principle).