Why this page. Two plain ASCII worked examples that illustrate negative RSI cases (endothermic or weakly dissociative lenses) and how a condition gate g_t modulates magnitude while preserving sign. Use clamps and guards throughout.
6.5 Thermal NO formation (disfavored)
ReactionN2 + O2 -> 2 NO
Bond basis (kJ/mol, ASCII)B_N#N ~ 945, B_O=O = 498, B_N=O ~ 607
Inputs (study constants for this example)E_unit = 100, c = 0.5, gamma = 1
Tally (contrast)
E_broken = 1*(B_N#N) + 1*(B_O=O)
= 945 + 498
= 1443
E_formed = 2*(B_N=O)
= 2*607
= 1214
e = (E_formed - E_broken) / E_unit
= (1214 - 1443) / 100
= -2.29
Alignment assignment (symmetric, bounded)
a_r = tanh(-c*e) = tanh(-0.5 * -2.29) = tanh(+1.145) ~ +0.81609090
a_p = tanh(+c*e) = tanh(+0.5 * -2.29) = tanh(-1.145) ~ -0.81609090
# guard before any atanh:
a_r = clamp_a(a_r, eps_a)
a_p = clamp_a(a_p, eps_a)
RSI computation (stoichiometric weights)
W_r = |1| + |1| = 2
U_r = sum_r( w_r * atanh_safe(a_r, eps_a) ) = 2 * (+1.14500000) = +2.29000000
V_p = sum_p( w_p * atanh_safe(a_p, eps_a) ) = 2 * (-1.14500000) = -2.29000000
RSI = tanh( (V_p - U_r) / W_r_safe )
= tanh( (-2.29 - 2.29) / max(2, eps_w) )
= tanh( -2.29 )
~ -0.97969840
Verdict
Disfavored (endothermic lens); bounded (|RSI| < 1); sign(e) < 0 equals sign(RSI) < 0. Under low calm (g_t small; spikes, shear, poor mixing), traces of NO can persist as edge-lift. The unified calm gate in §5 scales magnitude only: RSI_env = g_t * RSI with 0 <= g_t <= 1, sign preserved.
Data reference
See Appendix D for dataset identifiers, versions, and licence terms relevant to this example.
6.6 Weak-acid dissociation (acetic acid) — CH3COOH(aq) ⇌ H+(aq) + CH3COO−(aq) (sign-only, equilibrium lens)
Equilibrium constant lens (298.15 K)K_a(acetic) ~ 1.8e-5 -> DeltaG0 = -R*T*ln(K_a) ~ +27.083 kJ/mol
Contrast (Gibbs lens, normalized)
e = (-DeltaG0) / G_unit
= -(+27.083 kJ/mol) / 100 kJ/mol
= -0.27082950
Alignment assignment (symmetric, bounded)
a_r = tanh(-c*e) = tanh(-0.5 * -0.27082950) = tanh(+0.13541475) ~ +0.13459307
a_p = tanh(+c*e) = tanh(+0.5 * -0.27082950) = tanh(-0.13541475) ~ -0.13459307
# guard before any atanh:
a_r = clamp_a(a_r, eps_a)
a_p = clamp_a(a_p, eps_a)
RSI computation (stoichiometric weights)
W_r = |1| = 1
U_r = 1 * atanh_safe(a_r, eps_a) = +0.13541475
V_p = 1 * atanh_safe(a_p for H+) + 1 * atanh_safe(a_p for acetate)
= 2 * (-0.13541475)
= -0.27082950
RSI = tanh( (V_p - U_r) / W_r_safe )
= tanh( (-0.27082950 - 0.13541475) / max(1, eps_w) )
= tanh( -0.40624425 )
~ -0.38527929
Verdict
Disfavored dissociation at standard state (weak acid): e < 0 => RSI < 0. Sign matches classical acid strength (K_a << 1).
Optional gate
If g_t = 0.80 -> RSI_env = 0.80 * (-0.38527929) ~ -0.30822343
Data reference
See Appendix D for dataset identifiers, versions, and licence terms relevant to this example.
Navigation
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Disclaimer (observation-only). All formulas and results are observation-only—not predictive or operational—and require peer validation and governance before any deployment.
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