Abstract
The core unit in SSM is the symbolic numeral (m, a): m is the classical magnitude, a ∈ [-1, +1] is a bounded alignment coordinate that encodes centre vs edge. This page fixes the literal forms, projections, and practical notes you will use everywhere.
🧱 Symbolic numeral (core unit)
A value is the ordered pair (m, a) where:
m ∈ R→ magnitude (the classical scalar)a ∈ [-1, +1]→ alignment (centre–edge coordinate;+1= Pearo,-1= Nearo,0 ≈Zearo)
🔎 Projections (readability / interop)
pi_m(m, a) = m # classical magnitude view
pi_a(m, a) = a # alignment view
🔢 Canonical literals (convention)
0 == (0, +1) # calm zero (additive identity)
1 == (1, 0) # true multiplicative identity (M2)
-1 == (-1, +1)
These canonical forms keep proofs and code deterministic.
🧩 Optional encoding trick (not default)
Some systems with a minimal unsigned alphabet may encode -1 as:
-1 → (1, -1)
If you adopt this non-canonical representation, declare it in the Manifest and apply it consistently.
🔒 Why alignment is bounded (a ∈ [-1, +1])
- Stability: algebra stays inside safe bounds (no runaway).
- Interpretability: universal, comparable centre ↔ edge scale.
- Auditability: cross-domain comparability of alignment.
🧾 Notes
mmay be raw or normalized (domain-specific). Robust/log transforms are allowed; the primitive remains(m, a).- How
ais obtained is declared in the reproducibility Manifest (method, params, clamps). - Units attach to
m;ais unitless and comparable across domains.
🧪 Worked example (illustrative literals)
(5, +1) # magnitude 5, perfectly stable (Pearo)
(5, -1) # same magnitude, full drift (Nearo)
Both collapse to m = 5, but symbolically they differ via pi_a:
pi_a(5, +1) = +1
pi_a(5, -1) = -1
Calm zero remains additive identity:
(5, -1) oplus (0, +1) = (5, -1)
✅ Takeaway
The pair (m, a) enriches every classical scalar with a stability coordinate. This single extension underpins all subsequent operations, ordering, calculus, and applications in SSM.
Navigation
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Disclaimer
Observation only. Reproducible math; domain claims require independent peer review. Defaults: gamma=1, mult_mode=M2, clamp_eps=1e-6, |a|<1.
SHUNYAYA 