A bounded, replayable trust dial: clamp → atanh → fuse → tanh.
11.0 Overview
align is the mathematical heart of the trust/stability lane in (-1,+1). It must be bounded, comparable across time/shards, replayable from first principles, and never alter the original value lane.
11.1 Requirements for align
- Bounded. Output strictly inside (-1,+1) to prevent infinities.
- Comparable. Same result whether streamed, batched, or sharded/merged.
- Replayable. Given the same raw history and declared knobs, two parties compute the same align.
- Policy-aware, not physics-breaking. We observe stability/risk without rewriting the original measurement.
11.2 Core transform pipeline (reference)
# Stage 1: Clamp (numerical safety)
a_c := clamp(a, -1+eps_a, +1-eps_a)
# Stage 2: Map to rapidity space (additive)
u := atanh(a_c)
# Stage 3: Weighted accumulation (order-invariant)
U += w * u
W += w
# Stage 4: Collapse back to bounded dial
align := tanh( U / max(W, eps_w) )
Declared knobs: eps_a, eps_w, and the weight rule w (e.g., w := 1, w := |m|^gamma, or w := criticality_score) must be published in the manifest.
11.3 Intuition behind each step
- Clamp protects everyone from edge explosions at ±1.
- atanh spreads extremes so strong signals can be fused fairly.
- (U, W) carry an order-invariant “fused opinion” that sharded systems can add.
- tanh returns to a human-readable, bounded dial.
11.4 Order-invariance and fairness over time
The (U,W) structure guarantees the same align across stream vs. batch vs. shard-merge, making the result audit-grade and independently verifiable.
11.5 Why this protects safety, audit, and humans
- align answers “Did we know it was getting bad?”
- band answers “What did policy require then?”
- manifest_id proves which rulebook applied.
- stamp proves when and in what order it was known.
Because the math and knobs are published, nobody can move goalposts after the fact.
One-line takeaway
align = tanh( U / max(W, eps_w) ), with U += w*atanh(clamp(a,-1+eps_a,+1-eps_a)) and W += w — a bounded, replayable trust dial any party can recompute identically.
Navigation
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Next — SSMDE — The Manifest: Freezing Policy in Time (12.0–12.6)
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SSMDE – Table of Contents
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