Turn signed contrasts into safe, bounded signals.
3.2 Symmetric Maps → Alignment
Purpose. Convert signed contrasts e_in (penalties) and e_out (support) into bounded alignments a_in, a_out in (-1,+1) with a single gain knob c > 0. This keeps evidence interpretable, comparable, and edge-safe.
Definition (two-channel map).
a_in := tanh( -c * e_in )
a_out := tanh( +c * e_out ) # c declared per lens
• Monotone & odd: larger support ⇒ larger a_out; larger penalties ⇒ more negative a_in.
• Implicit clamp: outputs are strictly inside (-1,+1).
Why symmetric?
• Sign clarity: support and penalty push in opposite directions by construction.
• Scale robustness: one c governs both channels; tuning stays simple and transparent.
• Bounded behavior: no matter how large |e| gets, |a| < 1.
Choosing c (practical).
• Aim for typical |c*e| in [0.3, 1.2] so tanh is responsive but not saturated.
• Start with c = 1.0 if e is normalized (Unit = 1).
• If |a| saturates > 0.9 often, reduce c or increase Unit; if a hovers near 0, increase c or reduce Unit.
Single-channel variant (simple lenses).
a := tanh( c * e ) # feed to pooling directly
# or set a_out := a, a_in := 0
Worked minis (calculator-fast).
• Mild support: e_out = 0.4, c = 1 → a_out ≈ tanh(0.4) ≈ 0.379949.
• Strong penalty: e_in = 1.0, c = 0.8 → a_in ≈ tanh(-0.8) ≈ -0.664037.
• Balanced: e_out = 0.7, e_in = 0.5, c = 1 → a_out ≈ 0.604368, a_in ≈ -0.462117.
Pseudocode (drop-in).
def map_to_alignment(e_in=0.0, e_out=0.0, c=1.0):
a_in = tanh(-c * e_in)
a_out = tanh(+c * e_out)
return a_in, a_out
def align_single(e, c=1.0):
return tanh(c * e) # in (-1,+1)
QA checklist (pass/fail).
• Monotonicity: increase e_out ⇒ a_out increases; increase e_in ⇒ a_in decreases.
• Bounds: |a_in| < 1 and |a_out| < 1 for all finite e.
• Gain sanity: adjusting c scales responsiveness without breaking bounds.
One-line takeaway. Use tanh with a single gain c to convert contrasts into clean, bounded alignments—ideal inputs for order-invariant pooling and the universal RSI chooser.
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