Abstract
In SSM, every number is a pair (m, a). Here m is the classical magnitude and a ∈ [-1, +1] is a bounded alignment factor indicating how centred (stable) or edge-leaning (drifting) a value is. This page introduces the core terms—Pearo, Zearo, Nearo—and the practical rules for reading and declaring a.
🔄 What changes
In classical math, numbers are single scalars. In SSM, each value is a pair:
m— magnitude (the classical scalar)a— alignment in[-1, +1]that encodes closeness to centre (stability) vs drift (instability)
🧩 Key terms in use today
- Pearo (
a > 0): centre-leaning, stability-aligned. - Nearo (
a < 0): edge-leaning, drift-aligned. - Zearo (
a ≈ 0): neutral or undecided alignment.
Together, these three states let every number carry both its size and its context.
🔒 Why a is bounded (-1 to +1)
- Stability: keeps all operations in a safe, consistent range.
- Interpretability: a universal, comparable centre ↔ edge scale.
- Auditability: every result carries explanatory metadata, not just a raw value.
🧭 The five Z-states (larger framework)
For practice, we use Zearo, Pearo, and Nearo (computable, measurable). Quearo and Mearo are conceptual extensions for quantum/higher-order logic.
Name Symbol Description Domain examples
Zearo Z0 Neutral zero; undecided alignment Arithmetic, geometry
Pearo Z+ Positive-leaning zero; stability Calculus, signals
Nearo Z- Negative-leaning zero; drift Limits, chaos, transitions
Quearo Zq Quantum zero; zero-point uncertainty QFT, information theory
Mearo Zm Meta-zero; container of zero states Set theory, AI logic
(ASCII symbols are used intentionally: Z0, Z+, Z-, Zq, Zm.)
🗂️ Five-state lens (optional reporting)
These bands are for dashboards; core math remains continuous in a.
- Strong Nearo:
[-1.0, -0.6]→ high drift risk - Mild Nearo:
(-0.6, -0.2]→ emerging drift - Zearo:
(-0.2, +0.2)→ neutral / indecisive - Mild Pearo:
[+0.2, +0.6)→ emerging stability - Strong Pearo:
[+0.6, +1.0]→ high stability
🧾 Declaring alignment (a) in practice
Two lawful, commonly used mappings (declare one in the page Manifest):
Centered-from-earned-alignment: a = 2*SyZ_t - 1
Rapidity-from-drift-contrast: a = tanh( c * (A_t - Z_t) ) where c > 0
Include parameters and clamp policy with each dataset or example.
🧰 Edge-state handling (rapidity)
Compute in rapidity space and map back to keep |a| < 1:
u = atanh(a)
a' = tanh(u)
if |a| >= 1 - eps: a = sign(a) * (1 - eps) # default eps = 1e-6
📌 Why this matters (illustrative mini-examples)
- Same magnitude, different futures
Classical:64 = 64
Symbolic:(64, +0.8)vs(64, -0.3)→ one system strengthening, the other weakening. - Centre becomes explicit
Classical:10 - 4 = 6
Symbolic:(10, +0.2) ominus (4, -0.3) = (6, -0.1)→ result leans toward 9, exposing centre shift.
✅ Takeaway
Numbers retain their size (m) and now also declare their alignment (a). This minimal extension turns arithmetic from static calculation into predictive, auditable computation.
Transition note
Sections that follow build the algebraic, analytic, and geometric rules for symbolic numerals. Each rule is designed so that:
- Classical math is recovered when
a = +1(collapse guarantee). - SSM adds a second stability axis without breaking classical results.
Navigation
Previous → Introduction (1)
Next → Terminology & Notation (1.2)
Disclaimer
Observation only. Reproducible math; domain claims require independent peer review. Defaults: gamma=1, mult_mode=M2, clamp_eps=1e-6, |a|<1.
Explore further
https://github.com/OMPSHUNYAYA/Symbolic-Mathematics
Disclaimer
Observation-only; not for safety-critical decisions.
SHUNYAYA 