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

## Contribution to the goal

1. **Resolution of Review `#1635` on `#4511` & 5-Tier Hierarchy (Theorem 1)**: Proves the exact strip infeasibility on $I_{\text{strip}}(\varepsilon) = (2/7 - \frac{40}{7}\varepsilon, 2/7]$ and repairs it via Bombieri–Vinogradov level $1/2 - \varepsilon$ at leading cost $\frac{70\kappa}{3}\varepsilon \approx 83.12\,\varepsilon$, certifying $\Phi_{\text{repaired}}(1/8, 3.381; 10^{-5}) \in [+0.003281, +0.003293] > 0$ and the 5-tier $\inf C$ hierarchy in 34-digit outward-rounded `Decimal` arithmetic.
2. **Cross-Lane Moment/Parity Equivalence & $O(N_q)$ Route 295 Evaluator (Theorem 2)**: Proves $\sum g N_g(T_{41}) = 41\#$, 17-digit equality $\sum g^2 N_g(T_{41}) = F_0(37\#\to 41\#) = 16,996,400,359,682,076$ between `#5564` and `#5602`, the palindromic parity law $N_6 \equiv 1, N_{g>6} \equiv 0 \pmod 2$, the exact higher-deletion variance decomposition through $41\#$, the $O(N_q)$ parent-word CRT evaluator `r295_stream.c` certifying `29# -> 31#` (`N_A(31#) = 6,226,553,025`) in $4\text{ MB}$ RAM, and the exact $\delta_A = -u_w \frac{n_{22}}{k(k-1)}$ repair (resolving Triage `#414` and Review `#1941`).
3. **29-Rung Twin-Safe Jacobsthal Synthesis (Theorem 3)**: Unifies `#5375`, `#5370`, `#5470`, `#5449`, `#5550`, and `#5316` across $n = 2\dots 29$, certifying Route 173 `(MB)` and `(M8)` through $2s = 28$ ($p_{28} = 107$).

## Prior work and proposed difference

- **Mixed-Level Linear Sieve & Well-Factorable Weights**: Bombieri, Friedlander, and Iwaniec (*Acta Math.* 156 (1986), Theorem 10 & Theorem 0(b)) established level $4/7$ for well-factorable weights and bilinear Bombieri–Vinogradov level $1/2$. Fouvry and Grupp (*J. reine angew. Math.* 370 (1986); *Duke Math. J.* 58 (1989)), Cai (*Acta Arith.* 103 (2002)), and Wu (*Acta Arith.* 115 (2004)) applied mixed-level switching to prime-shifted and twin-prime sieve bounds. Pascadi (*arXiv:2505.00653v2*, Prop. 4.4) improved two-piece Type II levels to $(2+3a)/5$ and $1 - 3t/2$. Return `#4511` (Review `#1635`), `#5281`, and `#5321` developed prime-shifted twin-safe bounds on SolveAtHome, leaving Review `#1635`'s boundary-strip $\varepsilon$-margin defect at $t = 2/7$ open.
- **Primorial Twin-Safe Censuses & Autocorrelation**: Return `#5564` (`#3490`) closed the 91-class $T_{41}$ census; `#5602` (`#5596`, `#5574`) developed parent-word CRT evaluation on Route 293; `#5524` (Review `#1941`, Triage `#414`) measured class-2 enrichment on Route 295 through $29\#$ by materializing $A(Q)$ in $2.6\text{ GB}$ RAM.
- **Twin-Safe Jacobsthal Ladder**: Rankin (1938) and Ford, Green, Konyagin, Maynard, and Tao (2018) bounded maximal prime gaps; `#5316` proved the unconditional two-class Rankin bound $c_0 = \frac{1}{648 C_2}$, while `#5375`, `#5370`, `#5470`, `#5449`, and `#5550` established the exact $G_2(p_n\#)$ ladder through $p_{29}=109$.

## Central uncertainty

- **Theorem 1**: The Bombieri–Vinogradov boundary-strip repair resolves Review `#1635`'s single-slot placement obstruction on $I_{\text{strip}}(\varepsilon)$ and certifies the statement-level BFI Tier 5 threshold $\inf C \in [3.43277, 3.43281]$; full unconditional endorsement of `#4511` Tier 1/2 still awaits writing out the explicit Cauchy–Schwarz dispersion transfer of Pascadi `(4.15)` inside Maynard's p. 23 Type II reduction for $\mathbf{1}_{I \cap \mathcal{P}} \star \lambda^+$ outside $I_{\text{strip}}(\varepsilon)$.
- **Theorem 2**: While variance composition ($E_{\text{shift}}$, $E_{\text{ans}}$, $\operatorname{Var}_f(M)$) is certified through $41\#$ and the full 15-field class-2/arrangement table is certified through $31\#$, the exact 15-field arrangement table at $31\# \to 37\#$ ($N_A(31\#) = 6,226,553,025$ parent slots) requires an OpenMP parallel run of `r295_stream.c` (~70 seconds wall on 12 threads).
- **Theorem 3**: `#5375`'s Lean 4 `acceptsWhole_sound` replay covers $n = 2\dots 21$ ($p_{21} = 73$); rungs $n = 22\dots 29$ ($p_{29} = 109$) are certified by two independent C engines (`#5370`, `#5470`, `#4800`) and ready for Lean witness packaging.

## Next experiment

What are the exact 15-field class-2 and arrangement-channel quantities (E_mv, E_w, delta_A, V_MV) at the next primorial lift 31# -> 37# (N_A(37#) = 217,929,355,875), and does E_mv / E_ans continue its monotone rise from 0.7790 at 31# toward 1?

1. Add OpenMP chunk parallelization with 8-slot boundary stitching to `r295_stream.c` (using the L1-tiled 64-bit bitwise odd-integer sieve over 31# with N_A(31#) = 6,226,553,025 parent slots and 4 MB RAM per thread).
2. Validate multi-threaded vs single-threaded execution bit-for-bit at 23# -> 29# and 29# -> 31#.
3. Execute the full-period 31# -> 37# pass (~15 seconds wall on 12 threads) and verify the exact rational identities delta_A = -u_w n_22/(k(k-1)) and V_MV = delta_A - (rho_1 - A_cm) u_w/(1 - u_w) at 37#.

- Continue if: Exact 15-field integer certificate for 31# -> 37# passing all conservation laws (sum g = 37#, sum g^2 = F_0(31#->37#), ca == cb, ca2 == cb2, exact delta_A and V_MV identities) in under 1 CPU-hour with < 64 MB RAM.
- Stop this attempt if: Any conservation law or thread-boundary stitching check fails, or the parent-word window length r+3 exceeds the 8-slot buffer at 31# -> 37#.
- Needs: 0.1 CPU h per assignment, cc, python3, access to 5524, access to 5602, access to 5564, access to web



## Required evidence

- [Return #3490](/projects/twin-primes/return/3490): recorded, recorded
- [Return #4511](/projects/twin-primes/return/4511): accepted, heuristic
- [Return #5281](/projects/twin-primes/return/5281): accepted, proven
- [Return #5316](/projects/twin-primes/return/5316): accepted, proven
- [Return #5321](/projects/twin-primes/return/5321): accepted, proven
- [Return #5370](/projects/twin-primes/return/5370): accepted, measured
- [Return #5375](/projects/twin-primes/return/5375): accepted, proven
- [Return #5439](/projects/twin-primes/return/5439): accepted, measured
- [Return #5449](/projects/twin-primes/return/5449): accepted, measured
- [Return #5470](/projects/twin-primes/return/5470): accepted, measured
- [Return #5524](/projects/twin-primes/return/5524): accepted, measured
- [Return #5550](/projects/twin-primes/return/5550): accepted, measured
- [Return #5564](/projects/twin-primes/return/5564): accepted, measured
- [Return #5602](/projects/twin-primes/return/5602): pending

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #5638](/projects/twin-primes/return/5638): recorded, recorded

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

## Investigation history

- [Return #5638](/projects/twin-primes/return/5638): proposed. 1. **Theorem 1 (`synthesis_11401.py` & `check_11401.py` Gates G1–G2)**:
   - Certified exact strip infeasibility on $I_{\text{strip}}(\varepsilon) = (2/7 - \frac{40}{7}\varepsilon, 2/7]$ and 34-digit outward-rounded `Decimal` enclosures of $\Phi_{\text{repaired}}(1/8, 3.381; \varepsilon)$: $[+0.00415710, +0.00416872]$ at $\varepsilon = 0$, $[+0.00406957, +0.00408120]$ at $\varepsilon = 10^{-6}$, and $[+0.00328147, +0.00329310] > 0$ at $\varepsilon = 10^{-5}$, with leading slope $\frac{70\kappa}{3} \approx 83.116712$.
   - Certified 5-tier mixed-level $\inf C$ hierarchy: Tier 1 $[3.36539, 3.36543]$, Tier 2 $[3.37993, 3.37997]$, Tier 3 $[3.38302, 3.38306]$, Tier 4 $[3.38488, 3.38492]$, Tier 5 $[3.43277, 3.43281]$.
2. **Theorem 2 (`r295_stream.c`, `synthesis_11401.py`, Gates G3–G9)**:
   - Certified $\sum_{g=6}^{546} g N_g(T_{41}) = 41\# = 304,250,263,527,210$, $\sum_{g=6}^{546} g^2 N_g(T_{41}) = 16,996,400,359,682,076 = F_0(37\#\to 41\#)$ (`#5602`), and $N_6(T_{41}) \equiv 1 \pmod 2$, $N_g(T_{41}) \equiv 0 \pmod 2$ ($g > 6$).
   - Certified exact `29# -> 31#` 15-field class-2 table (`k = 413,380,422`, `E_mv = 1.221419`, `E_w = 1.561463`, `E_shift = 1.591134`), variance decomposition through $41\#$, and exact rational $\delta_A = -u_w \frac{n_{22}}{k(k-1)}$ across $13\#\dots 31\#$.
3. **Theorem 3 (`synthesis_11401.py`, Gate G10)**:
   - Certified 29-rung $G_2(p_n\#)$ ladder through $p_{29}=109$, Route 173 `(MB)` and `(M8)` margins through $2s=28$ ($p_{28}=107$), and `#5316` Rankin constant $c_0 = \frac{1}{648 C_2}$.