Shunyaya Symbolic Mathematics — Algebraic structures (2.13)

Abstract
Examines SSM under ⊕ and ⊗. Under ⊕ the pairs (m,a) form a commutative group. Under ⊗ (M2, default) the nonzero-magnitude slice is an abelian group, while the full space has zero-divisors. Exact distributivity over ⊕ does not hold on the alignment channel (M2), so SSM is not a ring/semiring. M1 is recorded as a non-normative alternative and also does not restore global distributivity.

➕ Additive structure (⊕ / oplus)

  • Closure: (m1, a1) ⊕ (m2, a2) ∈ R × [−1, +1].
  • Associativity: exact for finite multisets via the weighted rapidity mean (streaming U,W).
  • Commutativity: follows from symmetry of weights.
  • Identity: zero class 0_S = { (0, a) : a ∈ [−1, +1] }, displayed canonically as (0, +1).
  • Inverse: for every (m, a), −(m, a) = (−m, a).

Mini-example
(3, +0.5) ⊕ (−3, +0.5) = (0, +1) # zero-class display

Conclusion: under ⊕, symbolic numerals form a commutative group.

✖️ Multiplicative structure (⊗ / otimes)

Default M2 (rapidity-additive alignment)

Let u_i = atanh(a_i).

(m1, a1) ⊗ (m2, a2) = ( m1*m2 , tanh(u1 + u2) )
  • Closure: alignment remains in (−1, +1) for finite u; magnitudes multiply classically.
  • Associativity & commutativity: inherited from addition in u and real multiplication.
  • Identity: (1, 0).
  • Inverses (for m ≠ 0): (m, a)^{-1} = (1/m , −a).

Mini-examples
(4, +0.5) ⊗ (2, −0.5) = (8, tanh(atanh(0.5)+atanh(−0.5))) = (8, 0)
(4, +0.5)^{-1} = (0.25, −0.5)

Conclusion: on { (m, a) : m ≠ 0 }, multiplication is abelian and every element has a multiplicative inverse; with m = 0 present, zero-divisors exist (as in classical arithmetic).

Alternative M1 (direct product, non-normative)

(m1, a1) ⊗_M1 (m2, a2) = ( m1*m2 , a1*a2 )
  • Closure holds but alignment can exit [-1, +1] without clamping.
  • Inverse exists only when m ≠ 0 and a ≠ 0; near a = 0 it is unstable.
  • Distributivity over ⊕ still fails in general (see next section).

Mini-example
(4, +0.5)^{-1}_M1 = (0.25, 2.0) # out of bounds

➗ Distributivity (mixed ⊗ over ⊕)

For all x, y, z, compare x ⊗ (y ⊕ z) vs (x ⊗ y) ⊕ (x ⊗ z).

  • Magnitude distributivity holds exactly: π_m respects classical distributivity.
  • Alignment distributivity (M2) is not exact because ⊕ is a weighted tanh-mean while ⊗ adds rapidities; these interact nonlinearly.
  • Alignment distributivity (M1) is also not exact against the same ⊕ (weighted rapidity mean).
  • Exact distributivity returns under collapse (a ≡ +1), or in certain special cases (equal alignments, balanced weights, or small-variation regimes).

Mini-example (schematic)
Let x = (2, +0.8), y = (3, +0.5), z = (4, −0.5).
π_m side: 2*(3+4) = 14 = 2*3 + 2*4.
Alignment side: the two expressions yield slightly different a′, but both stay in (−1, 1).

⭕ Zero, units, and zero-divisors

  • Additive identity: (0, +1).
  • Multiplicative identity: (1, 0).
  • Zero-divisors: (0, a) ⊗ (m, b) has magnitude 0 for any m; alignment follows the chosen multiplication (display typically canonicalized to (0, +1)).

🧩 Field-like properties (clarified)

  • With M2: multiplicative channel on m ≠ 0 is group-like (inverse exists), but the whole space has zero-divisors (m = 0) and non-distributive alignment → not a field, not a ring/semiring.
  • With M1: despite a simpler a-product, pairing it with the same ⊕ still fails global distributivity; inverses fail at a = 0 → likewise not a ring/semiring.

✅ Takeaway

SSM forms a commutative group under ⊕ and, under M2, a commutative, bounded, invertible-on-m≠0 multiplicative structure. Because the alignment channel does not distribute exactly over ⊕, the full system is not a ring/semiring (nor a field). This mirrors reality: neutral or collapsing states resist universal algebraic inversion even when magnitudes look well-behaved.

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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.