Symbolic numeral.
x := < m , a >
m : classical magnitude (extended reals allowed: 0, finite, +inf, -inf)
a in (-1, +1) : alignment (metadata; composed via rapidities)
Alignment rapidity (with clamp).
a := clamp(a, -1 + eps_a, +1 - eps_a)
u := atanh(a)
recommended: eps_a = 1e-6
Alignment composition (many causes).
a := tanh( sum_i atanh(a_i) )
Division alignment (quotient).
a_div := tanh( atanh(a_f) - atanh(a_g) )
Interpretation of alignment a (metadata only).
a ~ +1-> highly centered / stable / cleana ~ 0-> neutral / unknown qualitya ~ -1-> strongly off-center / unstable
(Alignment never alters the classical magnitude after collapse: phi(<m,a>) = m.)
Why tanh/atanh?
- keeps
abounded in(-1, +1) - turns composition into addition in rapidity space (stable, symmetric)
- subtraction in rapidity naturally models division:
a_div := tanh( atanh(a_f) - atanh(a_g) )
Default mappings (practical).
From R2 in [0,1]:
linear map (default): a := 2*R2 - 1
contrast map (when R2 clusters near 0 or 1): a := tanh( c*(R2 - 0.5) )
with c in [0.5, 2.0], default c = 1.0 # publish c
micro-calibration (3-point): choose low/med/high R2 exemplars;
target bands ~ A- (~-0.6), A0 (~0.0), A+ (~+0.6); record in manifest
Reference channel default (e.g., time): a_g := +0.80 # avoid saturation at +1.0; then clamp
Tiny manifest to publish with results (example).
{ eps_a: 1e-6,
map: linear | contrast(c=1.0),
calibration: { targets: {A-: -0.6, A0: 0.0, A+: +0.6} },
a_g_default: 0.80,
bands: { A++: a>=0.90, A+: 0.60<=a<0.90, A0: -0.20<=a<0.20,
A-: -0.60<=a<-0.20, A--: a<=-0.60 } }
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