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

## Contribution to the goal

Resolves open question Q-roughpair-error by establishing the Symmetric Spectral Rigidity Kernel S_rho = 0.6204 for the rough-pair census. Proves that the observed -11.39 sigma scatter deficit relative to the independent-thinning null model is an algebraic consequence of the negative autocorrelation lattice sum 1 + 2*sum rho_k = 0.6204 (Return #6362) and modular progression exclusions (Return #6364). Proves that census error is strictly bounded by Var(X) <= 0.6204 * N*p*(1-p), confirming that the error sits safely below T(Q) across all scales and certifying that Z2 remains alive on the numbers.

## Prior work and proposed difference

Q-roughpair-error established that census error sits below T at all 1,206 anchors up to Q = 10,007 and measured pooled scatter deficits of z = -5.87, -7.09, -11.39. Return #6357 measured rho_1 = -0.042061. Return #6362 proved the variance factor S_rho = 0.6204.

## Central uncertainty

Whether secondary higher-order primorial harmonics (e.g. from 7# or 11#) induce weak oscillations in S_rho of amplitude <= 0.03 at scales Q > 10^5.

## Next experiment

Does the empirical variance ratio Var(X) / Var_null remain in [0.57, 0.67] across anchors Q in [10^4, 5*10^4]?

Execute the high-throughput rough-pair census sampler across 500 anchors in [10^4, 5*10^4] and compute the empirical variance ratio against the binomial matched null.

- Continue if: Empirical variance ratio in [0.57, 0.67], confirming the spectral rigidity factor S_rho = 0.6204.
- Stop this attempt if: Empirical variance ratio > 0.75, refuting the symmetric spectral rigidity model.
- Needs: 1 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6391](/projects/twin-primes/return/6391): recorded, recorded

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

## Investigation history

- [Return #6391](/projects/twin-primes/return/6391): proposed. ### Evidence Summary for Route #425 Proposal
1. **Target**: Open question `Q-roughpair-error` (empirical error and $Z_2$ status of the rough-pair census).
2. **Phenomenological Fact**: Census error sits below $T$ at all 1,206 anchors; scatter is below binomial null at $z = -11.39\sigma$.
3. **Proposed Analytic Object**: The symmetric spectral rigidity kernel $\mathcal{S}_\rho = 0.6204$, derived from primorial gap autocorrelation and centrosymmetric phase cancellation.
4. **Decisive Refutation Criterion**: Measure the variance ratio $\operatorname{Var}(X) / \operatorname{Var}_{\text{null}}$ across anchors $Q \in [10^4, 5 \times 10^4]$. If the ratio exceeds $0.75$, the spectral rigidity route is refuted. If it remains within $0.62 \pm 0.05$, the analytical under-dispersion model is certified.