Abstract
This page defines the rapidity map u = atanh(a) for stable alignment calculus and the weight axiom that forces w(m) = |m|^gamma. Together they underpin n-ary addition, streaming evaluation, and consistent units across applications.
🧭 Rapidity map (alignment on a stable axis)
Use rapidity to linearize the alignment channel and keep computations numerically safe:
u = atanh(a)
a' = tanh(u)
Clamp before atanh (edge safety):
a_clamped = clamp(a, -1+eps, +1-eps) # default eps = 1e-6
Tiny numeric illustration (edge clamp). With eps = 1e-6, if a is supplied as +1:
a_clamped = 1 - 1e-6 = 0.999999
u = atanh(0.999999) ≈ 7.254328619
…a finite value instead of +∞.
Notes. Edge states a = ±1 correspond to |u| = ∞. All continuity statements are taken uniformly on any band where |a| ≤ 1 − eps for some small eps ∈ (0,1).
⚖️ Weight axiom (scale invariance)
Axiom W. For any c > 0 and any nonzero m, n,
w(c*m) / w(c*n) = w(m) / w(n)
Consequence. The only solutions are power laws:
w(m) = |m|^gamma with gamma ≥ 0
Default. gamma = 1.
Interpretation. Weights depend only on relative magnitude, not on chosen units.
Tiny numeric illustration (unit invariance). With gamma = 1:
w(3)/w(6) = 3/6 = 1/2
w(30)/w(60) = 30/60 = 1/2 # same ratio after a 10× unit change
⭕ Zero-class convention (canonical additive zero)
All pairs with zero magnitude form the zero class:
0_S = { (0, a) : a ∈ (-1, +1) }
Canonical representative: (0, +1). This ensures:
- Any
(0, a)acts as an additive identity becausew(0) = 0. - Results with
m = 0are canonicalized to(0, +1)for algebraic consistency. - Edge inputs
a = ±1are treated as limits and clamped beforeatanh(·).
Compatibility note. These algebraic choices do not alter your centre/zero semantics or five-band reporting; they provide a neutral backbone the rest of the book can reference.
📌 Reference for later use
The two primitives defined here are assumed throughout subsequent pages:
Rapidity: u = atanh(a) # compute/average in u-space; map back with tanh
Weights: w(m) = |m|^gamma # default gamma = 1; declare if different
They drive the U,W streaming scheme for exact associativity of addition and appear in proofs, ordering functionals, and vector/matrix lifts.
Navigation
Previous → Algebraic structure
Next → Primitive objects
Disclaimer
Observation only. Reproducible math; domain claims require independent peer review. Defaults: gamma=1, mult_mode=M2, clamp_eps=1e-6, |a|<1.
SHUNYAYA 