Shunyaya Symbolic Mathematics — Algebraic Structure of SSM (2.2)

Abstract
Corrected claims. With the M2 product and the standard rapidity–weighted sum, SSM has: (i) an abelian group under ⊕; (ii) a commutative monoid under ⊗ on all pairs and an abelian group under ⊗ on the subset m ≠ 0; and (iii) exact distributivity of ⊗ over ⊕. With the operational zero-canonicalization, SSM is a commutative ring with unity (1, 0); on m ≠ 0 it forms a commutative group under ⊗ (hence a field with these operations). The collapse phi(m,a) = m is a ring homomorphism to the classical reals.

Definition (symbolic numeral and operations)

An element is a pair x = (m, a) with m in R and a in (-1, +1). Let u = atanh(a) and choose gamma >= 0 (default gamma = 1).

Addition (⊕ / oplus, streaming rule). For two terms:

(m1, a1) ⊕ (m2, a2) = ( m1 + m2 , a_sum )
u_sum = ( |m1|^gamma * u1 + |m2|^gamma * u2 ) / ( |m1|^gamma + |m2|^gamma )
a_sum = tanh( u_sum )
Zero-canonicalization: if m1 + m2 = 0 (exact), return (0, +1).

The same (U,W) accumulator gives associativity for any finite multiset.

Multiplication (⊗ / otimes, M2 default).

(m1, a1) ⊗ (m2, a2) = ( m1 * m2 , a_prod )
u_prod = u1 + u2
a_prod = tanh( u_prod )

Collapse. phi(m, a) = m.

Additive channel (⊕): abelian group

  • Closure. R × (-1, +1) is closed under ⊕.
  • Associativity. Exact for any finite multiset via the (U,W) accumulator.
  • Commutativity. Symmetric in the arguments.
  • Identity. 0_S = (0, +1) (operational canonical representative of the zero-class).
  • Inverse. -(m, a) = (-m, a) and x ⊕ ( -x ) = (0, +1) by zero-canonicalization.

Conclusion: (SSM, ⊕) is a commutative (abelian) group.

Multiplicative channel (⊗, M2): monoid on all pairs; group on m ≠ 0

  • Closure. Magnitudes multiply classically; u adds, so a = tanh(u) remains in (-1, +1).
  • Commutativity/associativity. From real multiplication and rapidity addition.
  • Identity. (1, 0) since tanh(atanh(0) + u) = a.
  • Inverses (nonzero magnitudes). For m ≠ 0, (m, a)^{-1} = ( 1/m , -a ) (m, a) ⊗ ( 1/m , -a ) = ( 1 , 0 ).

Conclusion: on S* = { (m, a) : m ≠ 0 }, ⊗ is an abelian group; on all pairs (including m = 0) it is a commutative monoid with the absorbing element (0, +1).

(Contrast) M1 (direct alignment product) would be (m1*m2, a1*a2). It is non-normative here: it lacks stable inverses when a = 0 and can require explicit clamps to keep a in [-1, +1].

Distributivity (exact with M2 and standard weights)

Let w(m) = |m|^gamma. For any x, y, z:

x ⊗ ( y ⊕ z ) = ( x ⊗ y ) ⊕ ( x ⊗ z ).

Magnitude channel. Classical distributivity:
pi_m( x ⊗ (y ⊕ z) ) = m_x * ( m_y + m_z ) = m_x*m_y + m_x*m_z.

Alignment (rapidity) channel. Write u_x, u_y, u_z and w_y = w(m_y), w_z = w(m_z).
Left-hand side rapidity:

u_L = u_x + ( w_y*u_y + w_z*u_z ) / ( w_y + w_z ).

Right-hand side rapidity (weights scale by |m_x|^gamma, which cancels):

u_R = ( |m_x|^gamma * w_y * (u_x + u_y) + |m_x|^gamma * w_z * (u_x + u_z) )
      / ( |m_x|^gamma * (w_y + w_z) )
    = u_x + ( w_y*u_y + w_z*u_z ) / ( w_y + w_z )
    = u_L.

Hence distributivity is exact for the M2 product paired with the rapidity–weighted ⊕.

Ring and field statements

  • With the operational zero-canonicalization, (SSM, ⊕, ⊗) is a commutative ring with unity (1, 0).
  • On S* = { m ≠ 0 }, every element has a multiplicative inverse, so S* forms an abelian multiplicative group. Together with the additive group, (SSM, ⊕, ⊗) behaves as a field under these operations.

Collapse homomorphism

phi : SSM -> R given by phi(m,a) = m satisfies:

phi( x ⊕ y ) = phi(x) + phi(y)
phi( x ⊗ y ) = phi(x) * phi(y)
phi( (1, 0) ) = 1,     phi( (0, +1) ) = 0.

Thus phi is a ring homomorphism; restricting to S* gives a group homomorphism under ⊗. SSM is a conservative extension of R.

Interpretation

  • Think of ⊕ as an associative, commutative aggregator with scale-aware weighting w(m) = |m|^gamma.
  • The M2 product ⊗ keeps alignment bounded (u adds) and guarantees stable inverses for m ≠ 0.
  • Exact distributivity ensures the symbolic algebra aligns with classical algebra on magnitudes while transparently transporting stability.

Summary

Abelian group under ⊕; commutative monoid under ⊗ on all pairs and abelian group under ⊗ on m ≠ 0; exact distributivity of ⊗ over ⊕; ring homomorphism phi(m,a)=m. With these, SSM forms a commutative ring with unity (1, 0) and behaves as a field on the nonzero subset.

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Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: gamma = 1, mult_mode = M2, clamp_eps = 1e-6, |a| < 1. All formulas are presented in plain text.