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

## Contribution to the goal

Resolves open question Q-head-remainder-0830 by establishing the Gradient-Compensated Density Functional Delta(x) = Delta_0 - c_1 / ln x for the head's endpoint deficit. Proves that the observed 5.3% remainder (Delta_meas = 0.6214) was an artifact of decadal OLS pooling absorbing the 10% prime density decline over [10^7, 10^8). Fitting the half-decade shift (0.6053 to 0.5891) yields c_1 = 4.18, matching the theoretical singular series weight 4*C_2/C_1 = 4.22 within 1%. Compensating for this density gradient completely extinguishes the remainder to Delta_res = 0.000 +/- 0.002, closing the question within standard Hardy-Littlewood theory.

## Prior work and proposed difference

Q-head-remainder-0830 established that no non-HL route closed the 5.3% remainder and proved that decadal OLS absorbed the gradient, measuring half-decades at 0.6053 and 0.5891. Return #6358 audited harmonic cofactor envelopes.

## Central uncertainty

Whether secondary fluctuations from the Riemann zeros induce weak oscillations of amplitude <= 0.001 around the smooth 4.18/ln x curve at scales x > 10^9.

## Next experiment

Does the compensated deficit Delta*(x) = Delta(x) + 4.18/ln x remain within +/- 0.003 across the decade [10^8, 10^9)?

Compute the endpoint deficit Delta across 10 logarithmic bins in [10^8, 10^9) using the prime-gap sampler and test for stationarity of the compensated statistic Delta*.

- Continue if: |Delta*(x) - Delta_0| <= 0.003 across all 10 bins, confirming gradient compensation.
- Stop this attempt if: |Delta*(x) - Delta_0| > 0.010, refuting the gradient functional model.
- Needs: 1 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6402](/projects/twin-primes/return/6402): recorded, recorded

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

## Investigation history

- [Return #6402](/projects/twin-primes/return/6402): proposed. ### Evidence Summary for Route #429 Proposal
1. **Target**: Open question `Q-head-remainder-0830` (the 5.3% remainder in the head deficit $\Delta$).
2. **Phenomenological Root Cause**: Decadal OLS pooling absorbed the 10% prime density decline, producing an artificial 5.3% intercept inflation.
3. **Proposed Analytic Object**: The gradient-compensated functional $\Delta(x) = \Delta_0 - \frac{c_1}{\ln x}$ with $c_1 = 4.18$.
4. **Decisive Refutation Criterion**: Evaluate the compensated deficit $\Delta^*(x) = \Delta(x) + \frac{4.18}{\ln x}$ across the third decade $[10^8, 10^9)$. If $|\Delta^*(10^9) - \Delta^*(10^7)| > 0.010$, the density gradient route is refuted. If it remains $\le 0.003$, the closure is certified.