Shunyaya Symbolic Mathematics — Norms & distances (2.12)

Abstract
Centre-aware measures for comparing symbolic numerals. We define a size functional built from the ordering score S_beta, then embed pairs into a 2D space to obtain an Euclidean distance that captures both magnitude and alignment. ASCII-only.

📏 Size functional (“norm”)

Classical size

|| (m, a) ||_m = |m|

Centre-aware size (pick beta ∈ [0, 1])

S_beta(m, a) = m * (1 - beta * (1 - a))
|| (m, a) ||_beta = sqrt( m^2 + S_beta(m, a)^2 )
  • If beta = 0 (ignore alignment): || (m, a) ||_0 = sqrt( m^2 + m^2 ) = sqrt(2) * |m|
  • If beta = 1 (full weighting): || (m, a) ||_1 = sqrt( m^2 + (m*a)^2 )

Example (equal size, opposite alignment; beta = 1)
|| (10, +1) ||_1 = || (10, -1) ||_1 = sqrt(100 + 100) = sqrt(200) ≈ 14.14
→ Equal size; the distance below will separate them.

Example (beta switch)
x = (8, -0.5)
||x||_0 ≈ sqrt(2)*8 ≈ 11.314
||x||_1 = sqrt(64 + 16) = sqrt(80) ≈ 8.944
→ As beta increases, alignment penalty can reduce size when |a| < 1.

Terminology note: ||·||_beta is a size functional on pairs. Because (m, a) is not a linear vector space in the second coordinate, we do not claim ||·||_beta is a norm in the strict linear-algebra sense.

📐 Distance between two numerals

Embedding

Phi_beta(m, a) = ( m , S_beta(m, a) )

Euclidean distance in the embedding

d_beta( (m1, a1), (m2, a2) )
  = sqrt( (m1 - m2)^2 + ( S_beta(m1, a1) - S_beta(m2, a2) )^2 )
  • Captures both magnitude gap and centre-aware strength gap.
  • If magnitudes match but alignments differ, d_beta > 0.
  • If alignments match and only magnitudes differ, behaves like classical distance.

Example (same magnitude, opposite alignment; beta = 1)
x = (10, +0.9), y = (10, -0.9)
S_1(x) = 9.0, S_1(y) = -9.0
d_1(x,y) = sqrt(0^2 + (9 - (-9))^2) = 18

Example (beta switch on the same pair)
x = (8, -0.5), y = (7, +0.9)
S_0: 8, 7 → d_0 = sqrt(1^2 + 1^2) = sqrt(2) ≈ 1.414
S_1: -4, 6.3 → d_1 = sqrt( 1^2 + ( -4 - 6.3 )^2 ) = sqrt(1 + 10.3^2) ≈ 10.35
→ With beta = 0 the pair looks similar; with beta = 1 the drift difference dominates.

🎯 Centre distance delta_c

Given a finite set X = { (m_i, a_i) }, define the centre functional

C_hat(X) = ( Σ m_i * a_i ) / ( Σ |a_i| ),   with C_hat(X) = 0 if Σ|a_i| = 0

For x = (m, a), define the centre distance

delta_c(x; X) = m - C_hat(X)

This is the displacement along the m-axis from x to the system’s estimated centre. It can be viewed as d_beta between x and a reporting proxy at ( C_hat(X), a_centre ) for some chosen a_centre.

Example (centre-aware deviation)
X = { (10, +0.9), (9, +0.8), (12, -0.6) } gives C_hat ≈ 3.91 (see §2.9).
For x = (10, +0.9): delta_c ≈ 6.09
For y = (12, -0.6): delta_c ≈ 8.09
→ The Nearo-leaning larger value sits further from the centre, signalling risk.

📌 Properties (at a glance)

  1. d_beta ≥ 0, and d_beta = 0 iff both m and S_beta match.
  2. For fixed beta, d_beta is monotone in |m1 − m2| and in |S_beta(m1,a1) − S_beta(m2,a2)|.
  3. As beta → 0, d_beta approaches a rescaled classical metric on m (since S_beta → m).
  4. For beta = 1, distance fully reflects alignment via S_1 = m*a.
  5. Triangle inequality: d_beta is Euclidean in Phi_beta, so it satisfies the triangle inequality.
  6. Topology: metrics induced by L1 or L2 on Phi_beta are topologically equivalent; either can be used for proofs/analytics.

✅ Takeaway

Symbolic size and distance are centre-aware by construction: they reuse S_beta, the same functional that governs ordering. Two values identical in magnitude can be far apart symbolically if their alignments differ—a crucial predictor of divergent futures that classical metrics ignore.

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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.