Shunyaya Symbolic Mathematics — Symbolic Probability and Measure (2.23)

Abstract
We lift probability theory to pairs (m, a) so randomness captures both size and stability. Expectations, variances, LLN/CLT, and characteristic functions follow by working on the magnitude channel and an alignment-aware strength channel, and collapse exactly to classical probability when a = +1.

Symbolic probability space

A symbolic probability space is a triple (Omega, F, P_s) where:

  • Omega ⊆ R × [-1, +1] contains symbolic numerals (m, a).
  • F is a sigma-algebra on Omega (e.g., Borel on R × [-1, +1]).
  • P_s : F -> [0, 1] is a probability measure (classical axioms).

Collapse consistency. Under phi(m, a) = m, if a = +1 almost surely, then P_s ∘ phi^{-1} is an ordinary probability measure on R.

Symbolic random variables and projections

A symbolic r.v. is a measurable map X : Omega -> R × [-1, +1], X(omega) = ( m(omega), a(omega) ).

  • Classical part: pi_m(X) = m.
  • Alignment part: pi_a(X) = a.

In prose we write “a” for alignment; inside pairs (m, a) we just write a.

Expectations (two defaults)

  1. Pair expectation (componentwise)
E_s[X] = ( E[m] , E[a] )

Collapse: if a = +1 a.s., then E_s[X] = ( E[m] , +1 ).

  1. Strength expectation (scalar, choose beta in [0, 1])
S_beta(m, a) = m * (1 - beta*(1 - a))     # S_0 = m, S_1 = m*a
E_beta[X]    = E[ S_beta(m, a) ]

Worked example. P{X=(2,+1)}=0.5, P{X=(2,0)}=0.5.
E[m]=2, E[a]=0.5 ⇒ E_s[X]=(2, 0.5).
E_1[X]=E[m*a]=0.5*(2*1)+0.5*(2*0)=1.

Variance, covariance, and risk views

  • Symbolic variance (pair): Var_s[X] = ( Var[m] , Var[a] ).
  • Cross-covariance: Cov(m, a) = E[(m - E[m]) (a - E[a])].
  • Strength variance: Var_beta[X] = Var( S_beta(m, a) ).

Hidden-instability illustration.
If X alternates between (10, +0.9) and (10, -0.9) with equal probability:
Var[m]=0, Var[a]=0.81, but Var_1[X]=Var(10*a)=100*0.81=81 — large risk visible only via alignment.

Distributions (model in rapidity)

Let u = atanh(a) with standard clamp in applications.

  1. Symbolic normal (u–Gaussian).
    X ~ N_s( mu=(m0, a0), sigmas=(sigma_m, sigma_u) ) with density in (m, u) proportional to
exp( -0.5 * [ (m - m0)^2 / sigma_m^2 + (u - u0)^2 / sigma_u^2 ] ),  u0 = atanh(a0)

(If parameterized in a, include the Jacobian factor 1/(1 - a^2).)

  1. Symbolic Bernoulli (alignment-coded).
    P{ X = (1, +1) } = p, P{ X = (1, -1) } = 1 - p.
  2. Zearo law.
    Support on I = { (m, 0) }: neutral-but-fragile outcomes concentrated at a = 0.

Independence and conditioning

  • Independence is classical on Omega: two pairs are independent if their joint law factors.
  • Conditional expectation (componentwise):
E_s[ X | G ] = ( E[m|G] , E[a|G] )
E_beta[ X | G ] = E[ S_beta(m, a) | G ]

Tower and linearity laws hold in each channel.

Law of Large Numbers (LLN) — symbolic sample mean

For i.i.d. X_i = (m_i, a_i), aggregate with the associative evaluation rule:

w_i   = |m_i|^gamma            # default gamma = 1
M_n   = sum_i m_i
U_n   = sum_i w_i * atanh( clamp(a_i, -1+eps, +1-eps) )
W_n   = sum_i w_i
A_n   = tanh( U_n / max(W_n, tiny) )
mean  = ( M_n / n , A_n )

If E|m| < ∞, E[w * |atanh(a)|] < ∞, and E[w] > 0, then a.s.

M_n / n -> E[m],          A_n -> tanh( E[w*atanh(a)] / E[w] )

Collapse: if a = +1 a.s., then A_n -> +1 and M_n / n -> E[m] (classical LLN).

Central Limit Theorem (CLT) — strength embedding

Let Y = ( m , S_beta(m, a) ).
If Y has finite covariance, the classical 2D CLT applies to

(1/sqrt(n)) * sum_i ( Y_i - E[Y] )

This yields Gaussian limits for both magnitude and beta-strength channels, enabling standard inference with alignment awareness.

Characteristic functions (beta-embedding)

phi_X(t1, t2) = E[ exp( i * ( t1*m + t2*S_beta(m, a) ) ) ]

Spectral methods and LLN/CLT proofs reduce to classical arguments on the embedded real vector.

Tail and concentration (practical note)

If |m| is sub-Gaussian and |atanh(a)| is sub-exponential (typical with clamping), then S_beta is sub-exponential. Bernstein/Hoeffding-type bounds apply to empirical strengths, and to A_n via the delta method in u-space.

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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.