Shunyaya Symbolic Mathematical Symbols – Pooling Weights (1.4)

What this page covers
How to form a stability-safe mean in rapidity space using nonnegative pooling weights. This is the backbone for s_sum (and other pools), keeps alignment bounded, and preserves collapse parity.

Definition (choose once per study)

w_i = |m_i|^gamma      # gamma >= 0, so w_i >= 0

Weighted rapidity mean (used by s_sum and other pools)

U      = sum_i [ w_i * atanh( clamp_a(a_i, eps_a) ) ]
W      = sum_i w_i
mean_u = U / max( W, eps_w )
a_pool = tanh( mean_u )

Collapse and bounds

  • Collapse is unaffected (pooling touches alignment only).
  • a_pool always satisfies |a_pool| < 1 (by clamp_a and tanh).

Domain guards

  • Use max(W, eps_w) to avoid division by zero.
  • All w_i >= 0 by construction (no sign flips via weights).

Notes

  • gamma = 0 gives equal weights; gamma > 0 emphasizes larger |m|.
  • Publish gamma (and eps_w) in the manifest.
  • Keep U and W as streaming accumulators to preserve associativity across batches/windows.

Navigation
Previous: Identities, Inverses, and Zero-Class Display (1.3)
Next: Collapse-Safety Contract (1.5)