Shunyaya Symbolic Mathematics — Centre estimators: making the hidden centre visible (2.9)

Abstract
Defines centre estimators that reveal the true balance point of a collection of symbolic numerals. The default functional weights magnitudes by alignment; variants include gamma-weighted and rapidity-weighted centres. Also defines centre distance and an optional dispersion for dashboards.

🎯 Definition — centre functional `C_hat`

For a finite set X = { (m_i, a_i) }, define

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

If most a_i ≈ +1 (Pearo), then C_hat ≈ the average magnitude.
If many a_i < 0 (Nearo), the centre shifts away from the nominal midpoint.
No eps is needed once the zero-denominator case is explicitly declared.

Streaming implementation (numerically stable)

N := Σ (m_i * a_i)
D := Σ |a_i|
return 0 if D == 0 else N / D

🎛️ Optional variants (declare exactly one in the Manifest if used)

Gamma-weighted centre (scale-invariant weights)

C_hat_gamma(X) = ( Σ |m_i|^gamma * m_i * a_i ) / ( Σ |m_i|^gamma * |a_i| ),    gamma ≥ 0   # default gamma = 1

Rapidity-weighted centre (robust near edges)

a_i_clamped = clamp(a_i, -1+eps, +1-eps)
u_i = atanh(a_i_clamped)
C_hat_u(X) = ( Σ m_i * u_i ) / ( Σ |u_i| )

Pick one policy per application and declare it (with parameters) in the Manifest.

📏 Definition — centre distance `delta_c`

For a numeral x = (m, a) relative to set X:

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

delta_c > 0 → value lies above the system’s centre.
delta_c < 0 → value lies below the centre.
|delta_c| measures how far a value leans from balance.

📐 Properties (sketch)

Linearity (Pearo slice). If all a_i = +1, then C_hat(X) equals the arithmetic mean of {m_i}.
Sensitivity to drift. A single strongly negative a_i can pull C_hat sharply, even with moderate m_i.
Collapse. On the collapsed subset a_i = +1, delta_c reduces to the classical deviation from the mean.
Symmetry. If alignments are symmetric (roughly Σ a_i ≈ 0), then C_hat → 0, flagging a fragile centre.
Units/scale. Scaling all magnitudes by k > 0 scales C_hat by k (unit-consistent).

🧪 Worked examples

Example 1 — Asymmetric mix

X = { (10, +0.9),  (9, +0.8),  (12, -0.6) }

Numerator N = 10*0.9 + 9*0.8 + 12*(-0.6) = 9 + 7.2 - 7.2 = 9.0
Denominator D = |0.9| + |0.8| + |−0.6| = 2.3

C_hat = N / D = 9.0 / 2.3 ≈ 3.91

Even though magnitudes are 9–12, the true centre ≈ 3.91 because one value drifts strongly Nearo.
Classical mean (10+9+12)/3 = 10.33 hides this instability.

Example 2 — Symmetric alignments

X = { (5, +0.7), (5, -0.7) }

N = 5*0.7 + 5*(-0.7) = 3.5 - 3.5 = 0
D = |0.7| + |−0.7| = 1.4

C_hat = 0 / 1.4 = 0

The centre collapses to 0 even though both magnitudes are 5—opposing alignments cancel, revealing fragility.

📊 Optional dispersion around the centre (dashboards)

Alignment-weighted absolute deviation:

Sigma_c(X) = ( Σ | m_i - C_hat(X) | * |a_i| ) / ( Σ |a_i| )

This quantifies spread while respecting alignment emphasis.

✅ Takeaway

Centre estimation is a functional, not a heuristic. It surfaces where balance actually sits—not merely the arithmetic mean—and enables early detection of systemic drift or fragility (storms, heartbeats, markets, signals). If you adopt a variant (gamma-weighted or rapidity-weighted), declare it once in the Manifest for reproducibility.

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.

🎯 Definition — centre functional C_hat