Why this page. Bond-basis, plain ASCII worked examples that show the sign and boundedness of RSI and the sign-preserving realization RSI_env = g_t * RSI. Numbers are illustrative; sign is the invariant. Use clamps and guards throughout.
6.1 Hydrogen combustion (water formation)
Reaction2 H2 + O2 -> 2 H2O
Bond basis (kJ/mol)B_H-H = 436, B_O=O = 498, B_O-H = 463
Inputs (study constants for this example)E_unit = 100, c = 0.5, gamma = 1
Tally (contrast)
E_broken = 2*(B_H-H) + 1*(B_O=O)
= 2*436 + 498
= 872 + 498
= 1370
E_formed = 4*(B_O-H)
= 4*463
= 1852
e = (E_formed - E_broken) / E_unit
= (1852 - 1370) / 100
= 4.82
Alignment assignment (symmetric, bounded)
a_r = tanh(-c*e) = tanh(-0.5 * 4.82) = tanh(-2.41) ~ -0.98399553
a_p = tanh(+c*e) = tanh(+2.41) ~ +0.98399553
# 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 = |2| + |1| = 3
U_r = sum_r( w_r * atanh_safe(a_r, eps_a) ) = 3 * (-2.41) = -7.23
V_p = sum_p( w_p * atanh_safe(a_p, eps_a) ) = 2 * (+2.41) = +4.82
RSI = tanh( (V_p - U_r) / W_r_safe )
= tanh( (4.82 - (-7.23)) / max(3, eps_w) )
= tanh( 4.0166667 )
~ +0.99935128
Verdict
Strongly favored; bounded (|RSI| < 1). sign(e) = sign(RSI). If a condition gate is applied, publish RSI_env = g_t * RSI with 0 <= g_t <= 1; the gate modulates magnitude only and preserves sign.
Data reference
See Appendix D for dataset identifiers, versions, and licence terms relevant to this example.
6.2 Ammonia synthesis (Haber)
ReactionN2 + 3 H2 -> 2 NH3
Bond basis (kJ/mol)B_N#N = 945, B_H-H = 436, B_N-H = 391
Inputs (study constants for this example)E_unit = 100, c = 0.5, gamma = 1
Tally (contrast)
E_broken = 1*(B_N#N) + 3*(B_H-H)
= 945 + 3*436
= 945 + 1308
= 2253
E_formed = 6*(B_N-H)
= 6*391
= 2346
e = (E_formed - E_broken) / E_unit
= (2346 - 2253) / 100
= 0.93
Alignment assignment (symmetric, bounded)
a_r = tanh(-c*e) = tanh(-0.5 * 0.93) = tanh(-0.465) ~ -0.43415057
a_p = tanh(+c*e) = tanh(+0.465) ~ +0.43415057
# 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| + |3| = 4
U_r = sum_r( w_r * atanh_safe(a_r, eps_a) ) = 4 * (-0.46500000) = -1.86000000
V_p = sum_p( w_p * atanh_safe(a_p, eps_a) ) = 2 * (+0.46500000) = +0.93000000
RSI = tanh( (V_p - U_r) / W_r_safe )
= tanh( (0.93 - (-1.86)) / max(4, eps_w) )
= tanh( 0.6975 )
~ +0.60277853
If gated (illustrative)
g_t = 0.90 -> RSI_env = 0.90 * 0.60277853 ~ 0.542501
g_t = 0.30 -> RSI_env = 0.30 * 0.60277853 ~ 0.180834
Verdict
Favored (moderate); bounded (|RSI| < 1); sign(e) is positive and equals sign(RSI). If a condition gate is applied, g_t modulates magnitude only and preserves sign.
Data reference
See Appendix D for dataset identifiers, versions, and licence terms relevant to this example.
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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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