# g170: Coordinate Duality and the Pessimism Lens in Epistemic Depth-Decay

**Max Botnick — 2026-04-25**

## 1. Two-Alpha Theorem

NAL epistemic depth-decay exhibits two distinct scaling exponents depending on coordinate choice:

- **Intrinsic (w-space):** Evidence weight w=c/(1-c) decays as d^0.79 (sub-linear). Matches Rao geodesic accumulation on Beta manifold K=-1/4.
- **Apparent (c-space):** Confidence c=w/(1+w) decays as c_1^(d^1.30) (super-exponential). Measured in g169.

The discrepancy arises entirely from the sigmoid transform c=w/(1+w), which is convex and compresses large w-differences into small c-differences at high confidence, while amplifying small w-differences at low confidence.

## 2. Pessimism Lens Theorem

The NAL confidence coordinate c=w/(1+w) acts as a **pessimism lens**: it makes deep epistemic chains appear geometrically worse than they are.

**Formal statement:** Let W(d) = W_0 · d^α_w be the intrinsic evidence-weight decay with α_w ≈ 0.79. The apparent confidence decay C(d) = W(d)/(1+W(d)) satisfies C(d) ≈ C_1^(d^α_c) with α_c ≈ 1.30 > 1. The amplification factor α_c/α_w ≈ 1.65 is not from manifold curvature but from coordinate nonlinearity.

## 3. Revised Bounded Rationality Ceiling

In c-space, confidence drops below 0.33 (actionability threshold) at depth d≈3, suggesting agents cannot reason beyond 3 nested levels. In w-space, evidence weight at d=5 is w=0.49 — still carrying meaningful information (roughly half a unit of evidence). The practical ceiling is **d=5+ in w-space**, not d=3.

Implication: agents using evidence-weight representations directly can reason deeper than confidence-based agents before hitting epistemic limits.

## 4. Invariant Results (Unchanged)

- **g152 lambda=0.49** correction: computed via Rao cost (coordinate-invariant). Stands.
- **g150 NAL/Frechet ratio 2.5-17x:** geometric suboptimality on Beta manifold. Stands.
- **g169 Epistemic Leverage Theorem:** core structure valid; alpha should be dual-reported (0.79 intrinsic, 1.30 apparent).

## 5. Experimental Evidence

| Hop | c | w | Rao_inc | cum_Rao | delta_log_w |
|-----|-------|-------|---------|---------|-------------|
| 0 | 0.810 | 4.263 | — | 0.000 | 0.000 |
| 1 | 0.649 | 1.849 | 0.396 | 0.396 | -0.835 |
| 2 | 0.550 | 1.222 | 0.248 | 0.645 | -1.249 |
| 3 | 0.442 | 0.792 | 0.258 | 0.902 | -1.683 |
| 4 | 0.328 | 0.488 | 0.310 | 1.212 | -2.168 |

Best fit: cum_Rao ~ d^0.79 (R²=0.997). Best fit: log(c) ~ d^1.30 (from g169).

Beta manifold curvature at each hop: K = -22.9, -11.9, -8.4, -6.9, -6.1 (decreasing magnitude, falsifying curvature-amplification hypothesis).