Abstract
Classical extensions of R—complex numbers, quaternions, and Clifford algebras—lift arithmetic into richer spaces. In SSM we lift them with alignment, keeping collapse exact (phi(m,a)=m) while carrying stability through each axis via M2 (rapidity-additive products) and the associative U,W scheme for sums.
Symbolic complex numbers
Representation
Carry alignment in each channel:
z = ( (m1, a1), (m2, a2) ) # real and imag components
Addition (⊕ / oplus, componentwise)
z1 ⊕ z2 = ( (m1, a1) ⊕ (n1, b1) , (m2, a2) ⊕ (n2, b2) )
Multiplication (⊗ / otimes, M2 default; bilinear with symbolic ops)
Re(z1 ⊗ z2) = (m1, a1) ⊗ (n1, b1) ⊖ (m2, a2) ⊗ (n2, b2)
Im(z1 ⊗ z2) = (m1, a1) ⊗ (n2, b2) ⊕ (m2, a2) ⊗ (n1, b1)
Products use M2 (add rapidities in u = atanh(a)); sums use the U,W accumulator (gamma-weighted rapidity mean).
Conjugation and identities
conjugate: z_bar = ( (m1, a1), ( -m2, a2 ) )
additive zero: ( (0, +1), (0, +1) )
multiplicative one: ( (1, 0), (0, +1) ) # 1 + 0i with M2 identity (1,0)
Modulus and argument (declare one centre-aware choice)
|z|_m = sqrt( m1^2 + m2^2 ) # collapse-true
S_beta(x) = x_m * (1 - beta*(1 - x_a))
|z|_beta = sqrt( S_beta(m1,a1)^2 + S_beta(m2,a2)^2 )
theta_beta = atan2( S_beta(m2,a2), S_beta(m1,a1) )
Under collapse (a1=a2=+1), |z|_beta -> |z|_m and theta_beta -> classical arg.
Worked outlinez1 = ( (2, +0.9), (1, -0.5) ), z2 = ( (1, +1), (3, +0.8) ).
Compute four M2 products, then combine with ⊖/⊕ via U,W.
Collapse phi recovers (2 + i)(1 + 3i) = -1 + 7i; alignment channels expose stability flow.
Implementation tip (2×2 matrix form)
Represent z as a 2×2 symbolic matrix and reuse Section 2.18 matmul:
z ↦ [ (m1,a1) (-m2,a2) ]
[ (m2,a2) (m1,a1) ]
Symbolic quaternions
Let
q = ( (m0,a0), (m1,a1), (m2,a2), (m3,a3) )
with Hamilton rules i^2 = j^2 = k^2 = ijk = -1. Extend bilinearly by replacing scalar + and * with ⊕ (U,W) and ⊗ (M2). Basis non-commutativity is preserved; each coefficient carries its own alignment.
Conjugate, size, inverse (practical forms)
q_bar = ( (m0,a0), (-m1,a1), (-m2,a2), (-m3,a3) )
|q|_m^2 = m0^2 + m1^2 + m2^2 + m3^2 # collapse-true
|q|_beta^2 = sum_k S_beta(mk, ak)^2 # centre-aware
q^{-1} ≈ (1/|q|_m^2) ⊗ q_bar # when |q|_m != 0; use M2 division entrywise
Zero-class obstructions behave as in scalars: if any critical magnitude collapses or Nearo dominates, inversion becomes fragile.
Interpretation
Quaternions with alignment attach stability per rotational axis. Nearo components flag fragile modes even when the classical norm is large.
Clifford-like symbolic algebras (sketch)
Given basis {e_i} with e_i e_j + e_j e_i = 2*delta_ij, promote scalars to symbolic numerals and extend bilinearly:
- Multiply coefficients with
⊗(M2). - Sum multivector parts with
⊕via theU,Wrule. - Collapse (
a == +1) returns the classical Clifford algebra; symbolic channels make stability part of the geometry.
Collapse consistency
If all component alignments are +1, every construction above (complex, quaternionic, Clifford) reduces exactly to its classical counterpart. Alignment augments analysis; it never alters collapsed results.
Navigation
Previous → Entropy and Energy Functionals
Next → Functional Analysis in Symbolic Space
Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: gamma = 1, mult_mode = M2, clamp_eps = 1e-6, |a| < 1.
SHUNYAYA 