Shunyaya Symbolic Mathematics — Vectors & matrices (symbolic lifts) (2.18)

Abstract
Extends scalar SSM to higher dimensions. Vectors and matrices carry per-entry alignment a ∈ [−1, +1]. Sums use the associative U,W rapidity accumulator; products use M2 (rapidity-additive) alignment. Collapse recovers classical linear algebra on magnitudes; keeping a exposes how stability concentrates and propagates.

🧭 Conventions

  • Sum (⊕ / oplus): n-ary via U = Σ w_i * atanh(a_i), W = Σ w_i, a' = tanh(U/W) with w_i = |m_i|^gamma (default gamma = 1).
  • Product (⊗ / otimes): M2 per factor: (m1,a1) ⊗ (m2,a2) = (m1*m2, tanh(atanh(a1) + atanh(a2))).
  • Zero & one: (0,+1) is additive identity; multiplicative identity is (1,0).
  • Clamp: before any atanh, a <- clamp(a, −1+eps, +1−eps), eps = 1e−6.

📦 Symbolic vectors

A symbolic vector is a finite tuple of symbolic numerals:

v = ( (m1, a1), (m2, a2), …, (mn, an) )

Componentwise addition & negation

(v ⊕ w)_i = (mi, ai) ⊕ (ni, bi)
(−v)_i = −(mi, ai) = (−mi, ai)

Scalar multiplication (real r)

(r ⊙ v)_i = ( r*mi , ai )      # signs live on m; a is not scaled

Embedding for analytics (Φ_beta)

  • S_beta(m,a) = m * (1 − beta*(1 − a)), beta ∈ [0,1]
  • Φ_beta(m,a) = ( m , S_beta(m,a) )

Vector size functionals

  • Default 2D-embedding L2 size (matches scalar size):
    ||v||_beta = sqrt( Σ_i [ mi^2 + S_beta(mi,ai)^2 ] )
  • Strength-only (dashboard):
    ||v||_beta^S = sqrt( Σ_i S_beta(mi,ai)^2 )
  • Collapse: if all ai = +1, then S_beta = mi. For exact classical L2, set beta = 0.

Numeric example (default size)
v = ( (3,+1), (4,+0.5) ), beta = 1
S_1(3,+1)=3, S_1(4,+0.5)=2
||v||_1 = sqrt( 3^2+3^2 + 4^2+2^2 ) = sqrt(38) ≈ 6.164
(Strength-only: sqrt(3^2+2^2) = sqrt(13) ≈ 3.606.)

🧮 Symbolic matrices

A symbolic matrix is an array of pairs:

M = [ (m_ij, a_ij) ]   # i=1..p, j=1..n

Matrix addition (entrywise)
(M ⊕ N)_ij = M_ij ⊕ N_ij

Matrix–vector multiplication (⊗ with ⊕ accumulation)
For M ∈ R^{p×n} (symbolic) and v ∈ R^n (symbolic):

(M ⊗ v)_i =  ⊕_{k=1..n} ( M_{ik} ⊗ v_k )
  • Each term uses M2.
  • Accumulate the inner sum via U,W over k to preserve associativity.
  • Magnitudes follow classical matvec; alignment is the rapidity-mean of term alignments.

Matrix multiplication (⊗ with ⊕ accumulation)
For M ∈ R^{p×r}, N ∈ R^{r×n}:

(M ⊗ N)_{ij} =  ⊕_{k=1..r} ( M_{ik} ⊗ N_{kj} )
  • M2 per product; inner ⊕ via U,W.
  • Associativity of matmul inherits from M2 and the n-ary ⊕ accumulator.

Identity & zero matrices

  • Identity: diagonal (1, 0); off-diagonal (0, +1).
  • Zero: all entries (0, +1) (canonical zero-class).

🧪 Worked example (2×2, one entry in detail)

M = [ (1, 0)   (2, +0.9)
      (0, +1)  (1, −0.5) ]

N = [ (1, 0)   (0, +1)
      (3, +0.7) (1, 0) ]

Compute (M ⊗ N)_{11} = (1,0)⊗(1,0) ⊕ (2,+0.9)⊗(3,+0.7).

  • Term A: (1,0)⊗(1,0) = (1, tanh(0+0)) = (1, 0)
  • Term B: (2,+0.9)⊗(3,+0.7)
    u = atanh(0.9) ≈ 1.472, v = atanh(0.7) ≈ 0.867
    a' = tanh(u+v) = tanh(2.339) ≈ 0.981 → (6, +0.981)

U,W accumulation (γ=1):
U = 1*atanh(0) + 6*atanh(0.981) ≈ 0 + 6*2.323 ≈ 13.94
W = 1 + 6 = 7
a_sum = tanh(U/W) = tanh(1.991) ≈ +0.963
Magnitude: 1 + 6 = 7
So (M ⊗ N)_{11} ≈ (7, +0.963).

(Other entries proceed analogously.)

🧱 Induced matrix sizes (Frobenius-like)

  • Symbolic Frobenius (default 2D-embedding):
    ||M||_beta = sqrt( Σ_{i,j} [ m_ij^2 + S_beta(m_ij, a_ij)^2 ] )
  • Strength-only (dashboard):
    ||M||_beta^S = sqrt( Σ_{i,j} S_beta(m_ij, a_ij)^2 )
  • Collapse: if all a_ij = +1, then S_beta = m_ij. For exact classical Frobenius, set beta = 0.

Numeric example (strength-only, β=1)
M = [ (2,+1) (1,−0.5); (0,+1) (3,+0.8) ]
S_1 entries: 2, −0.5, 0, 2.4 → ||M||_1^S = sqrt(4 + 0.25 + 0 + 5.76) = sqrt(10.01) ≈ 3.17.

📐 Operator bounds (β-embedded)

Form the real matrix A_beta with entries A_beta[ij] = S_beta(m_ij, a_ij).

  • The classical operator norm ||A_beta||_2 upper-bounds strength propagation for one ⊗–⊕ layer.
  • For multi-layer symbolic nets (repeated matvec), track both magnitudes and ||A_beta||; declare the beta used.

🔁 Invertibility & determinants (notes)

  • Collapse: det_magnitude(M) = det( [m_ij] ) is classical.
  • Symbolic determinant alignment (M2): combine permutation-term alignments via ⊕ in rapidity; zero magnitudes or Zearo-heavy patterns yield fragile outcomes.
  • In practice, solves often use the classical solve on m while tracking alignment for diagnostics; exact symbolic inversion requires absence of zero-class obstructions and acceptable condition numbers (seen in A_beta).

💡 Interpretation & takeaway

  • A symbolic vector is a direction plus a stability profile over its components.
  • A symbolic matrix propagates both sizes and alignments; Nearo-heavy rows/columns can dominate behavior even when magnitudes look strong.
  • The U,W accumulator is essential: it preserves associativity of sums inside matvec/matmul.
    Bottom line: With M2 products and associative rapidity means for sums, symbolic linear algebra becomes centre-aware. Collapse gives standard results; keeping a reveals where stability concentrates and how fragility spreads.

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.