This page explains how mixtures are averaged and how intra-species alignment combines. Pooling happens in a linear space (rapidity) and maps back to a bounded stability coordinate. For bonds, two normative rules cover covalent-like coherence and ionic-like independence. Classical magnitudes remain intact under collapse throughout.
Pooling (mixing / averaging across lanes)
When several aligned contributions meet (e.g., mixture, phase bundle, or multi-lane observation), we average them in rapidity space. This keeps combination linear and the final stability bounded. Weights reflect declared stoichiometry or emphasis: equal weights for a quick view, magnitude-based weights to reflect how much of each species participates.
Key intuitions:
- Average in rapidity, not directly in the bounded stability channel.
- Guard the denominator so empty or near-empty pools remain well-defined.
- Publish the weight policy (gamma) so readers can interpret emphasis choices.
Bond combine (intra-species alignment)
Two combine policies capture common chemistry patterns:
- Covalent-like (M2): treat bond coherence as additive in rapidity, then map back.
- Ionic-like (M1): multiply bounded alignments directly, clamping to stay strictly inside the open interval.
Use M2 for covalent networks and M1 for ionic pairs by default. Hybrids (e.g., coordination complexes) can mix rules transparently at the sub-structure level; just document which edges use which rule. Both rules are bounded, collapse-safe, commutative/associative in their respective operations, and monotone in each argument.
Plain ASCII formulas referenced on this page (copy-ready)
Inputs
Multiset S = { (m_i, a_i) }
Weights: w_i = abs(m_i)^gamma # gamma >= 0 (default 1)
Use atanh_safe(a, eps_a) and denominator guard eps_w > 0
Pooling (guarded definition)
U = sum_i( w_i * atanh_safe(a_i, eps_a) )
W = sum_i( w_i )
W_safe = max( W, eps_w )
a_pool = tanh( U / W_safe )
m_pool = sum_i( m_i )
Pooling properties
- bounded output: abs(a_pool) < 1
- linear in rapidity: average happens in u = atanh(a)
- commutative/associative in result (a_pool, m_pool)
- stoichiometric emphasis: gamma = 0 equal-weight; gamma = 1 magnitude-weight; larger gamma emphasizes larger magnitudes
- collapse-safe: phi(m_pool, a_pool) = m_pool
- denominator safety: if S empty, a_pool = tanh(0) = 0
Pooling pseudocode (minimal)
pool(species_list, gamma=1, eps_a=1e-6, eps_w=1e-12):
U, W, m_sum = 0.0, 0.0, 0.0
for (m, a) in species_list:
u = atanh_safe(a, eps_a)
w = (abs(m)) ** gamma
U = U + w * u
W = W + w
m_sum = m_sum + m
a_pool = tanh( U / max(W, eps_w) )
return (m_sum, a_pool)
Bond combine (normative rules; same clamp policy eps_a)
(a) Covalent combine — M2 (rapidity addition)
a_prime = tanh( atanh_safe(a1, eps_a) + atanh_safe(a2, eps_a) )
- interprets coherence as additive in rapidity (u_prime = u1 + u2)
- neutral element: a = 0
- formal inverse: a_inv with atanh(a_inv) = -atanh(a)
- n-ary: a_prime = tanh( sum_k atanh_safe(a_k, eps_a) )
(b) Ionic combine — M1 (direct product)
a_prime = clamp_a( a1 * a2, eps_a )
- independent stabilized charges multiply
- boundedness: abs(a_prime) < 1 after clamp
- neutral element: a = 1
- sign behavior: any sign flip in factors flips product sign
- n-ary: a_prime = clamp_a( prod_k a_k, eps_a )
Shared properties (M1 and M2)
- commutative and associative in their respective operations
- bounded: output satisfies abs(a_prime) < 1 after clamp
- collapse-safe: combine affects only a; phi(m, a) = m unchanged
- monotone: increasing any input in its natural order increases a_prime (others fixed)
- deterministic under fixed eps_a and ordering
Convention
Use M2 for covalent networks and M1 for ionic pairs; hybrids may mix with clear documentation.
Helpers (defined elsewhere)
clamp_a(a, eps_a)
atanh_safe(a, eps_a)
Navigation
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