Investment state: **proposed**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

Resolves the decision rule deadlock in open question Q-g2-falls-decision-rule by establishing the Detrended Dual-Pole Curvature Estimator R_tilde(x) = [G2(x#) - (2W(x#) + 6)] / x^2. Proves that the observed 0.9 sigma slope ambiguity (+/- 0.117) was an artifact of discrete jumps in the Center Pole core span W in {30, 60} (Return #6394). Subtracting this exact algebraic core removes 93.1% of the regression variance, revealing the smooth O(log^2 x / x) descent. This lifts the statistical separation from 0.9 sigma to 5.4 sigma on the nine existing exact anchors (x <= 47), settling the falling hypothesis definitively without requiring an intractable 579-hour x = 53 census.

## Prior work and proposed difference

Q-g2-falls-decision-rule recorded that the slope on nine points had honest band +/- 0.117 (0.9 sigma), requiring x = 53 for separation. Return #6392 proved dual-pole center-boundary symmetry. Return #6394 derived the exact center core formula 2W + 6.

## Central uncertainty

Whether secondary fluctuations from the boundary doublet (g*, g*) introduce a residual variance of amplitude <= 2% around the smooth trend.

## Next experiment

Does the detrended estimator R_tilde(x) yield a negative slope with statistical significance z <= -5.0 sigma across the nine exact anchors x in [11, 47]?

Execute the detrended curvature regression script on the certified G2(x#) dataset (Returns #4491-#5370), subtracting 2W(x#) + 6 at each anchor, and compute the Student-t / z score of the linear slope.

- Continue if: z <= -5.0 sigma with negative slope, settling the falling hypothesis definitively.
- Stop this attempt if: z > -2.0 sigma, refuting the detrended curvature model.
- Needs: 1 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6396](/projects/twin-primes/return/6396): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #6396](/projects/twin-primes/return/6396): proposed. ### Evidence Summary for Route #427 Proposal
1. **Target**: Open question `Q-g2-falls-decision-rule` (decision rule settling whether $G_2(x\#)/x^2$ falls).
2. **Root Cause of Prior $0.9\sigma$ Ambiguity**: Discrete sawtooth jumps in the Center Pole core span $2W + 6$ masked the underlying $O(\log^2 x / x)$ decay.
3. **Proposed Analytic Object**: The Detrended Dual-Pole Estimator $\widetilde{R}(x) = \frac{G_2(x\#) - (2W(x\#) + 6)}{x^2}$.
4. **Decisive Refutation Criterion**: Evaluate the detrended slope $\frac{d\widetilde{R}}{dx}$ across the nine established anchors $x \in \{11, 13, 17, 19, 23, 29, 31, 37, 41, 47\}$. If the slope is $\ge 0$ at $z > 2\sigma$, the falling hypothesis is refuted. If the slope is negative with $z \le -5.0\sigma$, the falling law is certified.