Shunyaya Symbolic Mathematics — Terminology & notation (1.2)

Abstract
Quick reference for SSM. Defines the pair (m, a), collapse phi, alignment/rapidity, Z-states, operators, functions, weights, pipelines, zero-class, and includes mini worked examples. ASCII-only, copy-paste ready.

🧱 Core objects

  • Symbolic numeral: (m, a) — m = classical magnitude; a ∈ [-1, +1] = alignment (stability vs drift).
  • Collapse (conservative extension): phi(m, a) = m.

🧭 Alignment & rapidity

  • Alignment: a — dimensionless, bounded in [-1, +1].
  • Rapidity (stable compute space): u = atanh(a) a' = tanh(u)
  • Clamp (edge handling): a_clamped = clamp(a, -1+eps, +1-eps) # default eps = 1e-6

🧩 Z-states (naming)

  • Z0 (Zearo): neutral (a ≈ 0)
  • Z+ (Pearo): stability-leaning (a > 0)
  • Z- (Nearo): drift-leaning (a < 0)
  • Zq (Quearo): quantum zero (conceptual)
  • Zm (Mearo): meta-zero (conceptual)

➕ Operators (symbols & ASCII)

  • ⊕ (addition) — ASCII: oplus
  • ⊗ (multiplication) — ASCII: otimes
  • ⊘ (division) — ASCII: odiv
  • Default policy: ⊗ and ⊘ use M2 (rapidity-additive) unless stated otherwise.

📚 Functions (ASCII)

  • log, exp — natural logarithm, exponential
  • tanh, atanh — hyperbolic tangent and inverse
  • abs(x) — absolute value
  • clamp(x, lo, hi) — bounds x to [lo, hi]

⚖️ Weights & scale

  • Weight: w(m) = |m|^gamma
  • Default: gamma = 1 (declare others if used)

🧵 Pipelines (alignment sources)

  • SyZ_t — earned alignment from SYASYS-core
  • A_t, Z_t — drift/stability signals from ZEOZO-core
  • Lawful mappings (declare one in manifest): a = 2*SyZ_t - 1 a = tanh( c * (A_t - Z_t) ) # c > 0

⭕ Zero-class

  • Family: 0_S = { (0, a) : a ∈ [-1, +1] }
  • Canonical representative: (0, +1)

🧪 Mini worked examples

1) Addition (⊕, n-ary via rapidity averaging)
Example: (10, +0.6) ⊕ (5, −0.2)

m_total = 10 + 5 = 15
U = |10|*atanh(0.6) + |5|*atanh(-0.2)
W = |10| + |5| = 15
a' = tanh(U/W) ≈ +0.3753
Result: (15, +0.3753)

2) Multiplication (⊗, default M2)
Example: (4, +0.5) ⊗ (3, −0.4)

m = 4*3 = 12
a' = tanh( atanh(0.5) + atanh(-0.4) ) ≈ +0.125
Result: (12, +0.125)

3) Division (⊘, default M2)
Example: (20, +0.8) ⊘ (5, +0.2)

m = 20 / 5 = 4
a' = tanh( atanh(0.8) - atanh(0.2) ) ≈ +0.714
Result: (4, +0.714)

Note: All operations preserve collapse; phi of each result equals the classical result.

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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.