SSM-AI – Appendix I — Lens Builder (I4–I6)

Two-channel lens, robust stats, and a tiny 5×5 micro-grid — copy-paste.

I4) Two-channel quick builder (support vs penalty)
Map supportive and penalizing evidence separately for clarity and sign-sanity. Use the same c chosen earlier; scale each side by its own Unit.

# Inputs: e_out := sum_i alpha_i * P_i       # supportive cues
#         e_in  := sum_j beta_j  * N_j       # penalties
Unit_out := p95(e_out);   Unit_in := p95(e_in)
a_out := tanh( +c * e_out / max(Unit_out, 1e-12) )
a_in  := tanh( -c * e_in  / max(Unit_in,  1e-12) )

# Chooser (bounded, order-invariant)
U_in  := sum( w * atanh( clamp(a_in,  -1+eps_a, 1-eps_a) ) )
V_out := sum( w * atanh( clamp(a_out, -1+eps_a, 1-eps_a) ) )
W     := sum( w )
RSI   := tanh( (V_out - U_in) / max(W, eps_w) )         # decision index
RSI_env := g_t * RSI                                    # or tanh(g_t * atanh(RSI))

Notes.

  • Sign sanity: increasing e_out raises a_out; increasing e_in lowers alignment via a_in.
  • Comparability: keep c fixed; declare Unit_out, Unit_in, and w in the manifest.
  • Parity: classical values are unchanged: phi((m,a)) = m.

I5) Robust stats option (when tails are ugly)
Heavy tails? Use median/MAD to stabilize Unit and keep saturation sane.

# Robust Unit from raw e (single-channel)
median_e := median(e_raw)
MAD      := median( |e_raw - median_e| )
Unit     := 2.5 * max(MAD, 1e-12)          # robust scale

# Then reuse earlier c (or re-solve once for your boundary)
a := tanh( c * (e_raw / max(Unit, 1e-12)) )

# Quick health checks
sat  := mean( |c*e_raw/Unit| > 3.0 )       # target < 0.10
dead := mean( |c*e_raw/Unit| < 0.10 )      # target < 0.70

Tip. If sat is high, raise Unit; if dead is high, lower Unit or modestly increase c.

I6) Micro-grid for Unit and c (5×5, five minutes)
A tiny grid lands you near-optimal separability without tuning debt.

# Candidate Units from percentiles of |e_raw|
Units := { p75, p85, p90, p95, p98 } of |e_raw|

# For each Unit, try two targets for c
for Unit in Units:
    e_norm := e_raw / max(Unit, 1e-12)
    c_Ap := atanh(0.60) / max(p95(|e_norm|), 1e-12)     # A+ boundary at p95
    c_App:= atanh(0.90) / max(p95(|e_norm|), 1e-12)     # A++ boundary at p95
    for c in {c_Ap, c_App}:
        a := tanh( c * e_norm )
        # Score (observation-only):
        sep  := band_share("A+", a) - band_share("A-", a)      # separation
        stab := -stdev_weekly_RSI(a, w)                         # stability
        corr := corr_with_KPI(a, KPI_proxy)                     # optional
        score := 0.6*sep + 0.3*stab + 0.1*corr
        keep best by score (break ties by simpler Unit)

Publish once: chosen Unit, c, weights, eps_a, eps_w, bands, and knobs_hash. From then on, runs are comparable across teams and vendors.

One-line takeaway. Separate support vs penalty with a two-channel lens, stabilize tails with MAD, and lock choices via a 5×5 micro-grid—yielding a bounded, portable lens that fuses order-invariantly while phi((m,a)) = m keeps your numbers pristine.

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