This page provides a compact, dependency-free reference implementation of the core building blocks used throughout this series. Everything is bounded, collapse-safe, and plain ASCII. Use these helpers as the stable substrate for pooling, bond-combines, and downstream preference (RSI) and gating.
What’s included (and when to use it)
- sign, clip — tiny utilities to keep behavior deterministic and bounded.
- clamp_a, atanh_safe — enforce the strict open interval for alignment and provide a numerically safe rapidity inverse.
- phi — collapse map to recover classical magnitudes exactly.
- pool — mixes species by averaging in rapidity with stoichiometric weights and guards.
- combine_M2 / combine_M1 — covalent (rapidity-additive) and ionic (multiplicative) alignment combines with clamps.
Implementation notes
- Use double precision; apply
clamp_abefore everyatanh_safeand after each combine. - Keep
eps_a,eps_wconsistent with your study manifest. - If
species_listis empty,poolreturns(0.0, 0.0)before clamping (outer tanh guards the division). - Determinism: fix seeds and threads at the application level if you rely on solver backends.
Plain ASCII pseudocode (copy-ready)
# sign helper (no +0 / -0 ambiguity)
sign(x):
return 1 if x >= 0 else -1
# clip helper (bounded outputs)
clip(x, lo, hi):
return min(max(x, lo), hi)
# alignment clamp (keeps |a| <= 1 - eps_a)
clamp_a(a, eps_a):
if abs(a) >= 1 - eps_a:
return sign(a) * (1 - eps_a)
return a
# safe atanh in rapidity space (uses the same clamp policy)
atanh_safe(a, eps_a):
ac = clamp_a(a, eps_a)
return 0.5 * log((1 + ac) / (1 - ac))
# collapse map (recover classical magnitude)
phi(m, a):
return m
# pooling (mix/average in rapidity space), returns a single pooled species
# species_list = [ (m, a), ... ], w = |m|^gamma with gamma >= 0
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 += w * u
W += w
m_sum += m
a_pool = tanh( U / max(W, eps_w) )
return (m_sum, clamp_a(a_pool, eps_a))
# covalent combine (M2): additive in rapidity
combine_M2(a1, a2, eps_a=1e-6):
u1 = atanh_safe(a1, eps_a)
u2 = atanh_safe(a2, eps_a)
return clamp_a( tanh(u1 + u2), eps_a )
# ionic combine (M1): product in alignment, then clamp
combine_M1(a1, a2, eps_a=1e-6):
a1c = clamp_a(a1, eps_a)
a2c = clamp_a(a2, eps_a)
return clamp_a( a1c * a2c, eps_a )
Result
These primitives are bounded and collapse-safe. They provide the base for pooling, bond-combines (M1/M2), and downstream use in preference (RSI) and condition gating (g_t).
Navigation
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Next – Reaction Stability Index — Definition (3, 3.1)
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