Why this page. Two plain ASCII, sign-oriented worked examples using equilibrium and bond lenses. They demonstrate bounded RSI and the sign-preserving realization RSI_env = g_t * RSI. Numbers are illustrative; sign is the invariant. Use clamps and guards throughout.
6.7 Ksp-driven precipitation check (AgCl) — Ag+(aq) + Cl-(aq) -> AgCl(s)
(sign-only, equilibrium lens, 298.15 K)
Equilibrium-constant lens
K_sp(AgCl) ~ 1.8e-10
DeltaG0 = -R*T*ln(K_sp) = +R*T*ln(1/K_sp)
R*T ~ 2.47896 kJ/mol, ln(1/K_sp) ~ 22.438
DeltaG0 ~ +55.623 kJ/mol
Contrast (Gibbs lens, normalized)
# dissolution: AgCl(s) -> Ag+ + Cl-
e_diss = (-DeltaG0) / G_unit = -(+55.623) / 100 = -0.55622997
# precipitation (reverse direction)
e_precip = +0.55622997
Alignment assignment (symmetric, bounded; c = 0.5)
ce = 0.5 * 0.55622997 = 0.27811499
a_r = tanh(-ce) = tanh(-0.27811499) = -0.27115956
a_p = tanh(+ce) = tanh(+0.27811499) = +0.27115956
# clamp before any atanh:
|a| <= 1 - eps_a
RSI computation (stoichiometric weights, gamma = 1)
Reactants: Ag+ (1), Cl- (1) -> W_r = |1| + |1| = 2
U_r = 2 * atanh_safe(a_r, eps_a) = 2 * (-0.27811499) = -0.55622999
V_p = 1 * atanh_safe(a_p, eps_a) = +0.27811499
s = (V_p - U_r) / max(W_r, eps_w) = (0.27811499 - (-0.55622999)) / 2 = 0.41717248
RSI = tanh(s) = tanh(0.41717248) = +0.39454573
Verdict
Precipitation is favored at standard state: e > 0 -> RSI > 0. Boundedness holds (|RSI| < 1); sign matches the inequality.
Optional gate (illustrative)
If g_t = 0.75 -> RSI_env = 0.75 * 0.39454573 = 0.29590930
Data reference
See Appendix D for dataset identifiers, versions, and licence terms relevant to this example.
6.8 Esterification (organic synthesis, sign-only with data-light bond lens)
ReactionCH3COOH + C2H5OH -> CH3COOC2H5 + H2O
Bond basis (kJ/mol, illustrative)B_O-H = 463, B_C-O = 358
Inputs (study constants for this example)E_unit = 100, c = 0.5, gamma = 1
Tally (contrast)
E_broken = 2 * B_O-H
= 2 * 463
= 926
E_formed = 1 * B_O-H + 1 * B_C-O
= 463 + 358
= 821
e = (E_formed - E_broken) / E_unit
= (821 - 926) / 100
= -1.05
Alignment assignment (symmetric, bounded)
a_r = tanh(-c * e) = tanh(-0.5 * -1.05) = tanh(+0.525) ~ +0.48154980
a_p = tanh(+c * e) = tanh(+0.5 * -1.05) = tanh(-0.525) ~ -0.48154980
# clamp before any atanh: enforce |a| <= 1 - eps_a
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( |m_r|^gamma * atanh_safe(a_r, eps_a) ) = 2 * 0.52500000 = +1.05000000
V_p = sum_p( |m_p|^gamma * atanh_safe(a_p, eps_a) ) = 2 * (-0.52500000) = -1.05000000
RSI = tanh( (V_p - U_r) / max(W_r, eps_w) )
= tanh( (-1.05 - 1.05) / 2 )
= tanh( -1.05 )
~ -0.78180636
Optional prior (transparent nudge, e.g., water-removal or phase-separation prior on the ester product)
delta_u = alpha * Index = 0.25 * 0.80 = +0.20, applied to the ester term only.
V_p' = (-0.52500000) [water] + (-0.52500000 + 0.20) [ester] = -0.85000000
s' = (V_p' - U_r) / W_r = (-0.85 - 1.05) / 2 = -0.95
RSI' = tanh(-0.95) ~ -0.73978305
Verdict
Disfavored at these table-light settings (sign only): e < 0 -> RSI < 0. A small, declared prior nudges magnitude but does not flip sign (as intended).
Gate note (illustrative)
If g_t = 0.70 -> RSI_env ~ 0.70 * (-0.78180636) = -0.54726445
If g_t = 0.30 -> RSI_env ~ 0.30 * (-0.78180636) = -0.23454191
Data posture
This is a data-light, sign-oriented check using generic bonds; real equilibria depend on solvent, water removal, and catalysis. Publish any priors and lens choices in the manifest.
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.
SHUNYAYA 