This page collects the non-negotiables that keep everything bounded, collapse-safe, and numerically sane, then shows what never changes under sensible rescalings. Use it as your quick audit anchor: do the outputs stay inside (−1, +1)? do classical balances remain intact? do benign unit/slope/weight changes preserve direction?
Invariants (what must always hold)
- Boundedness: the stability channel is strictly inside (−1, +1); the preference index stays bounded too.
- Collapse: classical magnitudes and balances are unchanged by symbolic machinery.
- Closure: covalent combine, ionic combine, pooling, priors, and the calm gate all map bounded inputs to bounded outputs under the standard clamps.
- Study-level clamp policy: apply the same alignment clamp everywhere before any inverse map or combine; publish the clamp margin in the manifest.
Scale/weight robustness (what never changes)
- Sign invariance: the sign of preference is unchanged by positive rescalings of energy units, the global slope, or the weight exponent (with non-negative weights).
- Order invariance: for fixed stoichiometry/weights, multiplying the contrast by any positive factor preserves the ranking (tanh is odd and strictly increasing).
- Stoichiometric scaling invariance: multiplying all coefficients by any k > 0 leaves the index unchanged for any non-negative weight exponent.
- Monotonicity reminder: for fixed stoichiometry/weights, the index is strictly increasing in the contrast.
Robustness grid (how to audit quickly)
Run a small grid over scale and weighting knobs and accept only if there are zero sign flips across the grid. Optionally, check that rankings match contrast rankings for a fixed weight exponent. Publish grid values, tolerance bands, and outcomes.
Plain ASCII formulas (copy-ready)
Boundedness, collapse, closure
# bounded inputs/outputs
|a| < 1 => |RSI| < 1
# collapse
phi(m, a) = m
# closure of operations
# M2 (covalent): rapidity-additive
a' = tanh( atanh(a1) + atanh(a2) ) # => a' in (-1, 1)
# M1 (ionic): multiplicative (then clamp)
a' = a1 * a2 # |a'| <= |a1||a2| < 1 -> clamp
# pooling (rapidity sums)
# inputs clamped: |a| <= 1 - eps_a => atanh(a) finite
# U_r, V_p, W_r finite; outer tanh keeps RSI in (-1, 1)
# priors (rapidity nudges)
a' = tanh( atanh(a) + delta_u ) # |delta_u| <= alpha_max => a' in (-1, 1)
# gate scaling
RSI_env = g_t * RSI # 0 <= g_t <= 1 => |RSI_env| <= |RSI| < 1
Clamp policy (study-level)
# apply before any atanh or combine
if abs(a) >= 1 - eps_a:
a := sign(a) * (1 - eps_a) # eps_a > 0 (e.g., 1e-12 .. 1e-6)
# publish eps_a and apply uniformly across the study
Scale/weight robustness (invariants)
# Sign invariance
# sign(RSI) unchanged under positive rescaling of:
# - E_unit (contrast scale)
# - c (global slope)
# - gamma (non-negative, so w = |m|^gamma >= 0 and W_r > 0)
# Order invariance (fixed stoichiometry/weights)
# e -> alpha * e with alpha > 0 preserves RSI ordering (RSI = tanh(k * e), k > 0)
# Monotonicity (reminder)
# For fixed stoichiometry/weights: RSI strictly increases with e
# Stoichiometric scaling invariance
# Multiply all coefficients by k > 0 -> RSI unchanged for any gamma >= 0
Recommended robustness grid (audit)
E_unit in {50, 100, 200}
c in {0.25, 0.5, 1.0}
gamma in {0, 1, 2}
Acceptance: zero sign flips across the full grid for each worked example.
Optional ranking check: for fixed gamma, RSI ranking == e ranking (Spearman rho = 1.0).
Minimal audit pseudocode (ASCII)
input:
cases = {case_k} # each has fixed stoichiometry/weights
grid_E = [50, 100, 200]
grid_c = [0.25, 0.5, 1.0]
grid_g = [0, 1, 2]
eps_a > 0, eps_w > 0
tau_zero = 1e-12 # tie tolerance for near-zero RSI
for each case in cases:
signs = []
for E_unit in grid_E:
for c in grid_c:
for gamma in grid_g:
# compute e with the chosen lens and canonical assignment (§3.2)
# compute RSI using guarded pools (§3.1): atanh_safe, W_r_safe = max(W_r, eps_w)
rsi = RSI(case, E_unit, c, gamma, eps_a, eps_w)
# robust sign with zero-tolerance band
s = 0 if abs(rsi) <= tau_zero else (1 if rsi > 0 else -1)
signs.append(s)
# acceptance criterion:
baseline = signs[0] # first grid point
assert all(s == baseline for s in signs)
# optional: ranking audit for fixed gamma values
for gamma in grid_g:
arr_e = [] # per-candidate contrasts (fixed lens)
arr_rsi = [] # RSI at (E_unit=100, c=0.5, this gamma)
for each candidate r in case.candidates:
e_r = contrast_of(r) # from the chosen lens
rsi_r = RSI(r, 100, 0.5, gamma, eps_a, eps_w)
arr_e.append(e_r)
arr_rsi.append(rsi_r)
assert rank(arr_rsi) == rank(arr_e) # Spearman rho = 1.0
Publish note (manifest)
Record the exact grid values used, tau_zero, and pass/fail per example.
If any failure occurs, publish the offending (E_unit, c, gamma) tuple and the computed RSI with its sign, plus corrective action (typically a data/guard issue).
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