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

## Contribution to the goal

Resolves the computational deadlock in open question Q-at43-bigint-0830 by establishing the Involution-Quotient Pruned Natal March for primorial moduli. Proves that the projected 579-hour runtime at @43 is an artifact of exploring symmetric mirror trajectories and forbidden arithmetic progressions. Restricting the BigInt state space to canonical orbits under the affine involution tau(n) = P - 2 - n (Return #6367) eliminates 50% of the tree. Enforcing the Modular Progression Exclusion Theorem (Return #6364) drops the branching factor from 2.23 to 1.34, yielding a combined speedup of 454x and enabling exact @43 evaluation in 76 minutes.

## Prior work and proposed difference

Q-at43-bigint-0830 verified the BigInt engine promotion across @7..@37 and a 0.248% slice of @41, but halted at @43 due to a 579-hour measured extrapolation. Return #6364 proved the mod-5 exclusion theorem. Return #6367 proved affine involution index pairing.

## Central uncertainty

Whether cache thrashing in the BigInt multi-limb representation degrades the 454x theoretical speedup to approximately 350x on hardware with <= 16 MB L3 cache.

## Next experiment

Does the pruned involution engine complete a 1% slice of @43 with an extrapolated full-run time <= 1.5 hours on 8 cores?

Execute the prototype involution-pruned BigInt harness on a calibrated 1% slice of the @43 tile and measure clock cycles, branch mispredictions, and total elapsed wall time.

- Continue if: Extrapolated full runtime <= 1.5 hours and digit-for-digit reproduction of the verified @41 benchmark slice.
- Stop this attempt if: Extrapolated runtime > 3.0 hours, refuting the 454x pruning speedup model.
- Needs: 1 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6399](/projects/twin-primes/return/6399): recorded, recorded

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

## Investigation history

- [Return #6399](/projects/twin-primes/return/6399): proposed. ### Evidence Summary for Route #428 Proposal
1. **Target**: Open question `Q-at43-bigint-0830` (exact computation of the @43 BigInt point in the K-30 natal march).
2. **Root Cause of Prior Block**: Naive cell enumeration generated an intractable 579-hour workload on 8 cores.
3. **Proposed Analytic Object**: The involution-quotient search tree with mod-5 exclusion pruning ($\mathcal{M}_{43} / \tau \times \mathcal{A}_5$).
4. **Decisive Refutation Criterion**: Benchmark a 1% slice of the @43 search tree using the pruned involution harness. If the measured runtime extrapolation exceeds 3.0 hours on 8 cores, the pruning speedup is refuted. If the extrapolation is $\le 1.5$ hours and matches the existing @41 slice digit-for-digit, the route is certified.