This at-a-glance page lists the neutral elements and inverses used across pooling and combine rules, plus the canonical “absence” placeholder and collapse neutrality.
Key points (plain language)
- Pooling neutrality: a term with zero pooling weight leaves the pooled rapidity and totals unchanged. With magnitude-weighting, any zero-magnitude term contributes no weight; with equal-weight pooling, explicitly set its weight to zero if your implementation allows it.
- Covalent combine (M2): the stability zero acts as the identity; the inverse of a stability value flips its sign in rapidity, canceling to zero.
- Ionic combine (M1): the multiplicative identity is stability one; product sign flips when any factor’s sign flips. True multiplicative inverses are generally out-of-bounds and not used.
- Absence placeholder:
(0, +1)denotes “absent but well-aligned”; it contributes nothing to classical balances and (with magnitude-weighting) no pooling weight. - Collapse neutrality: all identities commute with collapse, so classical stoichiometric sums and products remain exactly as in textbook chemistry.
Plain ASCII formulas (copy-ready)
Pooling neutrality (weights)
- w = abs(m)^gamma
- If gamma > 0 and m = 0 ⇒ w = 0
- If gamma = 0 (equal weights) then m = 0 does not imply w = 0 (use explicit w = 0 only if implementation supports)
Covalent combine — M2 (rapidity identity and inverse)
tanh( atanh(a1) + 0 ) = a1
# inverse in rapidity:
atanh(a_inv) = -atanh(a) => a_inv = -a
tanh( atanh(a) + atanh(a_inv) ) = 0
Ionic combine — M1 (multiplicative identity and sign)
1 * a1 = a1 # identity is a = 1
# sign behavior:
# a_product = a1 * a2 ... ; flipping any factor’s sign flips overall sign
# true multiplicative inverse 1/a is generally out of bounds (|1/a| > 1) -> not used
Magnitude zero (absence)
(0, +1)
# for gamma > 0 it contributes no pooling weight
# under collapse: phi(0, +1) = 0
Collapse neutrality
phi(m, a) = m
# classical sums/products and stoichiometric balances are preserved under collapse
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