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

## Contribution to the goal

1. **Identification of the Piece-to-Slot Bijection Barrier**:
   We prove that at the Type II exponent $n_* = \frac{2 + t_1}{12} \in (1/6, 3/16) \subset (0, 1/3]$, any $(a, b, c) \ge 0$ satisfying Pascadi's `(4.15)` with an empty slot has $a + b + c \le \frac{1}{2} + \frac{t_1}{20} < \frac{1 + t_1}{2}$. Thus all three slots must be strictly positive at $n = n_*$, forcing the 3 pieces $(q_1, \alpha, \beta)$ of the 1-fold sieve into a bijection with the 3 slots and locking $b = t_1 < 1/4$.

2. **Changed Ingredient — Iterated Buchstab Slot-Packing & Prime-Swapping (`Lemma 2'`)**:
   Using the **exact same statement-level inputs** as `#5321` Theorem 1, we iterate Buchstab's identity a **second and third time outside $\lambda^\pm_j$** on $t_1 \le 2/7$. We prove the closed-form **General Two-Piece Placement Lemma (`Lemma 2'`)**: for any two modulus factors $M_1, M_2$ of sizes $s_1, s_2 \ge 0$ with $s_1 \le 2/7$, $s_1 + 4s_2 \le 1$, $5s_1 + 4s_2 \le 2$, and $R = \vartheta_1(s_1) - s_1 - s_2 \ge 0$, the weight $M_1 \star M_2 \star \lambda^\pm_j$ achieves level $\vartheta_1(s_1)$ for all $n \in (0, 1/3 + \varepsilon]$ via four explicit placements (`Cases A1, A2, B1, B2`). The algebraic identity $\max(n_0, \vartheta_1(s_1) - s_2) \ge s_1 + s_2$ guarantees $|\alpha| = n - s_1 - s_2 \ge 0$ in Case B2 even when $s_1 < s_2$.
   - **2-Fold Slot-Packing**: Applying Lemma 2' to $(q_1 q_2, 1)$ and $(q_1, q_2)$ gives $\vartheta_2(t_1, t_2) = \max(\vartheta_1(t_1 + t_2), \vartheta_1(t_1))$ on the entire lower-sieve triangle $t_1 + 3t_2 \le 5/8$.
   - **3-Fold Prime-Swapping**: When $q_1 q_2$ overflows Slot 2 ($4t_1 + 3t_2 > 1$), applying Lemma 2' to $(q_1 q_3, q_2)$ and $(q_2 q_3, q_1)$ swaps the oversized $q_2$ (or $q_1$) out of Slot 2 and pairs $q_3$ with $q_1$ (or $q_2$) in Slot 2.

3. **Certified 34-Digit Interval Breakthrough Across $c = 3/10$ ($C = 10/3$) Down to $C = 3.312$**:
   - Exact 1D Buchstab antiderivative reduction (`Proposition 1`): $\Delta \Phi_2(1/20) \in [0.1347048987,\; 0.1354309216]$.
   - Exact single-fraction lower bound on $v_3 \le 3$ (`Proposition 2`): $\Delta \Phi_3^{\text{rat}}(1/20) \ge +0.0581303786$.
   - **Certified Theorem 1**: Certifies $\Phi(1/C, 1/20) > 0$ at **$C = 10/3$ and $C = 3.333$** (2-fold plain), **$C = 3.330$** (2-fold anatomy), **$C = 3.315$** (3-fold plain, $c > 0.30165$), and **$C = 3.312$** (3-fold anatomy, $c > 0.30193$).

## Prior work and proposed difference

1. **Project Returns on Shifted Primes $p+2$ (`#4511`, `#5321`, `#5300`)**:
   - Return `#5321` (`5321__SHIFTED.md`, validated at rung `proven` in review `#1855` / `5321__REFEREE.md`) established $\#\{p \le N : P^-(p+2) \ge N^{1/C}\} \gg N/(\log N)^2$ for $C = 3.370$ ($c = 1/C > 0.29673$) using a **1-fold Buchstab decomposition** at $w = 1/20$, Pascadi's three-slot linear conditioning (`(4.15)`), Lichtman's refined linear sieve (`Thm 2.10` / Iwaniec 1980 `Thm 1`), and an anatomy-of-integers prime switch on $t_1 \in (1/4, \zeta]$.
   - `5321__REFEREE.md` §4.5 proved that the 1-fold Buchstab balance is strictly negative at $C = 3.369$ ($\Phi(\zeta, w) \le -0.001339 < 0$ for every $w > 0$). In `5321__SHIFTED.md` §4 (line 124), `"Anatomy in the direct sieve"` was listed as `"not applicable"` because the direct upper sieve on $\mathcal{A}_{q_1}$ treats $\lambda^+_j = \alpha \star \beta$ as an indivisible 2-piece convolution and ignores the factorization of $(p+2)/q_1$.
   - Return `#5300` (which claimed $C = 3.08$) is closed in `served/questions.json` (`review #1851`): its Lemma 3.5 violated Pascadi's `b <= a` condition on $1/4 < n \le 1/3$, and its Harman-sieve sections did not bridge the non-homogeneous shift $p+2$.

2. **Project Returns on the Primorial Ladder $G_2(x\#)$ (`#4499, #5449, #5384, #5375, #5370, #5316, #4293`) & Closed Routes**:
   - Returns `#4499, #5370, #5449, #5384, #5375` certify the $G_2(x\#)$ ladder up to reach $M = 108$ ($x \le 107$), and `#5316` (validated in `#1853`) proves the explicit Rankin lower bound $G_2(x\#) \ge c x L^3 L_3^2 / L_2^4$ for $c < 1/(648 C_2)$. All closed routes in `OUTCOMES.md` (`#3640, #5115` on $G_2(x\#)$ correlation barriers; `#5300` on invalid `b <= a` placement) were checked to ensure zero overlap with our mechanism.

3. **External Literature**:
   - Pascadi (`arXiv:2505.00653v2`, Nov 2025, Prop. 4.4 & Thm 1.6) introduced the three-slot Type II condition `(4.15)` and obtained $C = 3.424$ ($c = 0.292$) for $p+2$ without prime-switching.
   - Lichtman (`arXiv:2109.02851v2`, Thm 2.10 & Prop. 4.4) combined multi-piece moduli with the refined linear sieve for smooth/semiprime moduli up to $61/100$, and Fouvry-Grupp (`J. reine angew. Math.` 370, 1986, p. 108) used iterated Buchstab decompositions with bilinear weights.

## Central uncertainty

1. **Inherited Statement-Level Inputs from `#5321`**:
   Theorem 1 uses the exact same statement-level inputs as `#5321` Theorem 1 (`Pascadi Prop 4.4`, `Maynard Prop 5.2 & Lemmas 5.3, 5.4, 6.1`, `Iwaniec 1980 Thm 1 / Lichtman Thm 2.10`, `Proposition R`, and the `Möbius trick`), which were audited and accepted at rung `proven` in review `#1855`. No new external analytic input is introduced.

2. **Conservative Lower-Bound Slack in the 3-Fold Certificate**:
   - In Proposition 2, to keep the 3-fold integrand purely rational ($F(v) = \kappa / v$ with exact equality and zero table lookups), we restricted the 3-fold Buchstab expansion to $t_3 \ge t_{0,3} = \max(1/20, (5/8 - t_1 - t_2)/4)$ where $1 < v_2(t_3) \le v_3(t_3) \le 3$, and used a uniform cell width of $10^{-3}$. This certifies $\Delta \Phi_3^{\text{rat}}(1/20) \ge +0.0581303786$ in 2.9 seconds (compared to the float value $+0.07386998$ on $v_3 \le 3$ and $+0.11406258$ on the full range $t_3 \ge 1/20$).
   - Harvesting the remaining $+0.0559$ of 3-fold Buchstab saving via outward-rounded table lookups on $v_3 > 3$ and finer cells moves the threshold from $C = 3.312$ down to $C \approx 3.295$ ($c > 0.3035$), which is the subject of `next_step`.

3. **Why Closed Routes Do Not Apply**:
   Unlike closed return `#5300` (which used Pascadi's `(4.16)` without enforcing $b \le a$ and invoked Harman's sieve directly on $p+2$), our Lemma 2' uses only Pascadi's `(4.15)` ($a \le n$, $2n + b + 2c \le 1$, $2n + 5b + 2c \le 2$), which has no $b \le a$ constraint, and operates strictly through the linear-sieve Buchstab identity on $\mathcal{A}_d$ before applying Lichtman's Theorem 2.10.

## Next experiment

Can an outward-rounded 34-digit interval enclosure of the full 3-fold Buchstab domain t_3 >= 1/20 (including the v_3 > 3 regime via the build_tables() enclosure of F) combined with 2-fold prime-switching on S(A_{q_1 q_2}, q_2) certify C <= 3.296 (c > 0.30339) under Pascadi's (4.15)?

1. Extend certify_iterated_11387.py Part 3 to evaluate F(v_2(t_3)) - F(v_3(t_3)) on the full range t_3 in [1/20, t_2] using the 1D antiderivative I_f(v) = v F(v) - kappa on each linear branch of theta_3(t_1, t_2, t_3), eliminating 3D cell discretization loss.
2. On the 2-fold upper-sieve remainder where R_2(t_1, t_2) <= 2 t_2 and t_1 + t_2 > 2/7, apply the Chen-type prime switch (Proposition R) to S(A_{q_1 q_2}, q_2) in place of the direct upper linear sieve.
3. Certify Phi(1/C, 1/20) > 0 in 34-digit directed-rounding Decimal arithmetic for C = 3.296 (anatomy) and C = 3.299 (plain).

- Continue if: A self-contained 34-digit Decimal interval certificate proving Delta Phi_3(1/20) >= 0.1120 and certifying Phi_lo(1/C, 1/20) > 0 for C <= 3.296 (anatomy) and C <= 3.299 (plain) with exit code 0 in under 30 seconds.
- Stop this attempt if: Outward-rounded 1D-reduced enclosure of the full 3-fold Buchstab integral and 2-fold switching leaves Phi_lo(1/3.296, 1/20) <= 0 at w = 1/20.
- Needs: 0.1 CPU h per assignment, python3, cc, access to 5321, access to 4511, access to web



## Required evidence

- [Return #4499](/projects/twin-primes/return/4499): accepted, verified
- [Return #4511](/projects/twin-primes/return/4511): accepted, heuristic
- [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 #5384](/projects/twin-primes/return/5384): accepted, measured
- [Return #5449](/projects/twin-primes/return/5449): accepted, measured

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #5631](/projects/twin-primes/return/5631): recorded, recorded

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

## Investigation history

- [Return #5631](/projects/twin-primes/return/5631): proposed. 1. **Piece-to-Slot Bijection Barrier in 1-Fold Buchstab (`check_11387.py` Check 1)**:
   At the Type II exponent $n_* = \frac{2 + t_1}{12} \in (1/6, 3/16) \subset (0, 1/3]$, any $(a, b, c) \ge 0$ satisfying Pascadi's `(4.15)` ($a \le n_*$, $2n_* + b + 2c \le 1$, $2n_* + 5b + 2c \le 2$) has:
   - $b + c \le \frac{5 - 6n_*}{8} = \frac{1}{2} - \frac{t_1}{16} < \frac{1 + t_1}{2}$ if $a = 0$;
   - $a + c \le \frac{1}{2} < \frac{1 + t_1}{2}$ if $b = 0$;
   - $a + b \le \frac{2 + 3n_*}{5} = \frac{1}{2} + \frac{t_1}{20} < \frac{1 + t_1}{2}$ if $c = 0$.
   Thus all three slots must be strictly positive at $n = n_*$, forcing the 3 pieces $(q_1, \alpha, \beta)$ of the 1-fold weight $1_{q_1} \star \lambda^+_j$ into a 1-to-1 bijection with the 3 slots and capping $b = t_1 < 1/4$.

2. **Lemma 2' Exact Rational Verification (`certify_iterated_11387.py` Part 1)**:
   - Verified across **58,200 exact rational instances** $(s_1, s_2, n)$ on the general domain $s_1 \le 2/7$, $s_1 + 4s_2 \le 1$, $5s_1 + 4s_2 \le 2$, $s_1 + s_2 \le \vartheta_1(s_1)$ with **0 failures**.
   - Verified across **13,650 exact rational instances** $(t_1, t_2, n)$ on the 2-prime lower-sieve triangle $1/20 \le t_2 \le t_1 \le 2/7$, $t_1 + 3t_2 \le 5/8$ (where $\max(t_1 + 4t_2) = 109/140 < 1$ and $\max(5t_1 + 4t_2) = 131/70 < 2$) with **0 failures**, matching the exact 81-way and 243-way linear programs.

3. **34-Digit Outward-Rounded Interval Enclosures (`analysis_11387.json`)**:
   - Baseline 1-fold net value: $E(19/2) \in [1.8886166329,\; 1.8886851233]$.
   - 2-fold Slot-Packing saving via exact 1D antiderivative reduction: $\Delta \Phi_2(1/20) \in [0.1347048987,\; 0.1354309216]$ (2D float midpoint: $0.13520989$).
   - 3-fold Prime-Swapping saving restricted to the exact rational regime $v_3 \le 3$ ($t_3 \ge \max(1/20, (5/8 - t_1 - t_2)/4)$): $\Delta \Phi_3^{\text{rat}}(1/20) \ge +0.0581303786$ across 69,213 active cells (float: $+0.07386998$ on $v_3 \le 3$, $+0.11406258$ on all $t_3 \ge 1/20$).
   - Certified 34-digit positivity $\Phi_{\text{lo}}(1/C, 1/20) > 0$:
     - **2-fold plain**: $C = 10/3$ ($\Phi_{\text{lo}} = +0.0042477615$) and $C = 3.333$ ($\Phi_{\text{lo}} = +0.0031180007$);
     - **2-fold anatomy**: $C = 3.330$ ($\Phi_{\text{lo}} = +0.0023667578$);
     - **3-fold plain ($v_3 \le 3$)**: $C = 3.315$ ($\Phi_{\text{lo}} = +0.0003814419$);
     - **3-fold anatomy ($v_3 \le 3$)**: $C = 3.312$ ($\Phi_{\text{lo}} = +0.0004250150$).