SSM-AI – Appendix H — Comparisons & Synergies (H1–H3)

Unified lane over popular confidence signals: entropy, margins, MC-dropout, ensembles, conformal, evidential.

H1) Why compare? (positioning in one paragraph)
Different confidence add-ons disagree across tasks and vendors. The SSM-AI lane provides a single bounded coordinate a ∈ (-1,+1) and a single chooser RSI ∈ (-1,+1) so heterogeneous cues reduce to one semantics-stable substrate. Map any cue to a signed contrast e, convert to alignment via a := tanh(c*e) (or two-channel a_out := tanh(+c*e_out), a_in := tanh(-c*e_in)), then fuse order-invariantly in u-space. Classical magnitudes are untouched: phi((m,a)) = m.

# Lane kernel (observation-only)
a_c := clamp(a, -1+eps_a, +1-eps_a)
u   := atanh(a_c)
U  += w * u
W  += w
a_out := tanh( U / max(W, eps_w) )

# Decision chooser (optional two-channel)
RSI := tanh( (V_out - U_in) / max(W_in, eps_w) )
# Optional gate
RSI_env := g * RSI              # "mul"
# or
RSI_env := tanh( g * atanh(RSI) )  # "u_scale"

H2) Cheat-sheet — map common methods into a lens (H2.1–H2.4)

H2.1 Softmax entropy (classification)
Low entropy ⇒ support; normalize to dimensionless.

H      := -sum_i p_i * log(p_i)
H_max  := log(K)
conf_H := 1 - H / H_max
e      := (conf_H - tau) / Unit
a      := tanh(c * e)

H2.2 Top-two margin (classification)

margin := p_top1 - p_top2               # in [0,1]
e      := (margin - tau) / Unit
a      := tanh(c * e)

H2.3 Temperature-scaled confidence

p_T    := softmax(z / T)
conf_T := max(p_T)
e      := (conf_T - tau) / Unit
a      := tanh(c * e)

H2.4 MC-dropout predictive variance (regression or logits)

var_mc := mean_t( (y_t - mean_t y_t)^2 )
# Treat variance as penalty (two-channel optional)
e      := (s - var_mc) / Unit        # or e := -var_mc / Unit
a      := tanh(c * e)
# two-channel option:
a_in   := tanh( +c * var_mc / Unit )
a_out  := tanh(  0 )

Tip. Pick Unit and gain c so typical |c*e| lies in ~[0.3, 1.2] (avoids early saturation).

H3) How to fuse them fairly (bounded & order-invariant)

Rules

  • Convert each cue → e → a := tanh(c*e) (or two-channel).
  • Declare weights: w := 1 (uniform) or w := |m|^gamma (strength-aware).
  • Fuse only in u-space; never average a directly.
  • Choose by bounded index: RSI (or RSI_env if gated).
  • Parity: value path still uses m only, so phi((m,a)) = m holds.
# Order/shard-invariant fusion
for cue in cues:
    a_cue := tanh(c * e_cue)             # or two-channel mapping
    U     += w_cue * atanh( clamp(a_cue, -1+eps_a, 1-eps_a) )
    W     += w_cue
a_pool := tanh( U / max(W, eps_w) )

# Final chooser (single-lane example)
RSI     := a_pool
RSI_env := g * RSI                       # or tanh(g * atanh(RSI))
band    := band_of(RSI_env, thr)

One-line takeaway. Treat diverse confidence signals as lenses into one bounded, u-space fused lane; decide by RSI/RSI_env and keep classical numbers pristine via phi((m,a)) = m.

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