Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal 1. **Lemma 3* (Telescopic Linear-Sieve Identity & Exact 1D Simplex Reduction on $E_2 \sqcup E_3$)**: We prove that sieving $E_2(a)$ and $E_3(a_1, a_2)$ directly at $z_{\mathrm{sw}} = Y^{t_{\mathrm{sw}}}$ (or telescoping via $(u f(u))' = F(u-1)$) eliminates `#5300`s Lemma 3(b), all ODE table interpolation, and $w$-dependence on $E_2 \sqcup E_3$, since $\ell / t_{\mathrm{sw}} \in [2.25, 2.82] \subset (2, 4]$ where $f(s) = \frac{\kappa \ln(s-1)}{s}$ is exact. For $C \le 3.20$ ($\zeta \ge 5/16$), $a_3 = 1 - a_1 - a_2 \le 3/8 < 1/2$ across the $E_3$ simplex, collapsing $\mathrm{Net}_{E_3}(\zeta, t_{\mathrm{sw}})$ to an exact 1D integral over $a_3 \in [1/3, 1 - 2\zeta]$ covered at **zero additional switching cost** by `#5300` Lemma 4. 2. **Disjoint 2-Fold & 3-Fold Buchstab Savings on $E_1$ Below $t_{\mathrm{sw}} = 111/500 = 0.222$**: Both the 2-fold slot-packing support $[1/20, 17/80 = 0.2125]$ and the restricted 3-fold prime-swapping support $[13/125 = 0.104, 43/200 = 0.215]$ lie strictly below $t_{\mathrm{sw}} = 111/500$, adding $\Delta\Phi_{2+3,\mathrm{lo}}(1/20; t_1 \le 0.215) \ge +0.1928352773$ directly to $\mathrm{Net}_{E_1}^{(1)}(1/20, 111/500)$. 3. **Proposition 3 (Four-Regime Closed-Form Anatomy Switch) & Certified Theorem B* ($C = 3.014$)**: We derive the exact 4-regime closed form of $J(t)$ on $[1/5, 1/3]$, certifying $C = 3.056$ at 1-fold, $C = 3.027$ at 2-fold, and **$C = 3.014$** ($\Phi_{\mathrm{lo}} = +0.0030590241 > 0$, $c = 1/C > 0.331785$) at 3-fold + $E_3$, cutting `#5300`s gap ($3.080 \to 3.000$) by **82.5%**. ## Prior work and proposed difference 1. **Unrestricted First-Slot / Twin Jacobsthal Bounds (`#4511, #5280, #5300`)**: - Brüdern–Fouvry (1996, *Compositio Math.* 102, pp. 339–340) and `#4511` gave $F(x) \le x^{10/3 + \varepsilon}$ ($C = 3.3333\dots$) on $n = p_1 p_2$ via Fouvry–Grupp level $3/5$. - Return `#5280` (accepted at rung `proven`, review `#1843`) proved **Theorem A**: $F(x) \le x^{16/5 + \varepsilon}$ ($C = 3.2000$) via Pascadi (`arXiv:2505.00653v2`, Prop. 4.4) at the $N = Y^{3/8}$, level $5/8$ corner on $E_2(3/8)$. - Return `#5300` (accepted at rung `proven`, review `#1850`) combined $E_1 \sqcup E_2$ with one shared plain Chen switch (`Lemma 4`) above $t_{\mathrm{sw}} \approx 0.2243$ to certify **Theorem B**: $F(x) \le 3 x^{3.080}$ ($c > 0.32467$), stopping in $(3.075, 3.079)$ ($\Phi(3.075) \approx -0.0159 < 0$). Review `#1850` (Correction 1) flagged that `#5300`s partial-anatomy remark ($3.0698$) lacked an interval certificate. 2. **Iterated Buchstab on Shifted Primes (`#5321, #5631`) & Ladder Shortlist (`#4499, #5370, #5449, #5384, #5375, #5316`)**: - Return `#5321` (review `#1855`) certified $C = 3.370$ on shifted primes $p+2$; return `#5631` (Route 373) introduced Lemma 2' (2-fold slot-packing + 3-fold prime-swapping on $E_1$) to certify $C = 3.312$ for $P^-(p+2) > p^{1/C}$ (with in-flight Job `#11391` extending $v_3 > 3$ for primes). Neither `#5631` nor `#11391` applies Lemma 2' to the unrestricted first-slot problem $F(x) \le 3 x^C$ or derives the 4-regime closed-form anatomy switch on $[1/5, 1/3]$. ## Central uncertainty 1. **Inherited Statement-Level Inputs**: Theorem B* uses the exact same statement-level inputs as `#5300` Theorem B and `#5321` Theorem 1 (`Pascadi arXiv:2505.00653v2 Prop 4.4`, `Maynard Prop 5.2 & Lemmas 5.3, 5.4, 6.1` via `Proposition R`, `Lichtman Thm 2.10 / Iwaniec 1980 Thm 1`, `Bombieri–Vinogradov`, and `Siegel–Walfisz`), accepted at rung `proven` in reviews `#1843, #1850, #1855`. Pascadi's Prop. 4.4 remains a preprint. 2. **Remaining Deficit to the Cubic / $P_2$ Barrier $C = 3.000$ ($c = 1/3$)**: - In `certify_11397.py`, our 3-fold prime-swapping bound restricts to $t_1 \le 0.215 < t_{\mathrm{sw}}$ and $t_3 \ge \max(1/20, (5/8 - t_1 - t_2)/4)$ ($v_3 \le 3$, certifying $\Delta\Phi_3^{\mathrm{rat}} \ge +0.05813038$), and uses a 1-fold lower sieve on $E_2$ at $z_{\mathrm{sw}} = Y^{0.222}$. - At the cubic/$P_2$ barrier $C = 3.000$ ($1/C = 1/3$, where $P^-(n(n+2)) > n^{1/3}$ implies $\Omega(n(n+2)) \le 5$, forcing one of $n, n+2$ to be $P_2$), `certify_11397.py` certifies $\Phi(3.000) \in [-0.06083394, -0.02127150] < 0$ (float point estimate $-0.0504$). Closing this remaining $0.0504$ deficit requires 2-fold Buchstab slot-packing on $E_2(a)$ for $a \in [1/4, 3/8]$ and extending $E_1$s 3-fold integral to $v_3 > 3$, which we formulate as `next_step`. ## Next experiment Can combining 2-fold Buchstab slot-packing on E_2(a) (for a in [1/4, 3/8] where Slot 1 has spare capacity a - q >= 0) with the full v_3 > 3 3-fold and 4-fold Buchstab savings on E_1 close the remaining 0.0504 deficit at C = 3.000 (c = 1/3)? 1. On E_1, evaluate the full 3-fold saving on v_3 > 3 using the 1D antiderivative I_f(v) = v F(v) - kappa and add 4-fold prime-swapping on t_1 in [0.15, 0.222]. 2. On E_2(a) for a in [1/4, 3/8], expand S(A(E_2(a)), Y^{t_sw}) one Buchstab step below t_sw where q_1 ~ Y^{t_1} fits into Slot 1 alongside gamma_1 (t_1 <= a), raising the effective level from (2+3a)/5 toward 5/8. 3. Certify whether Phi(C) crosses 0 at or below C = 3.005 in 34-digit outward-rounded Decimal arithmetic. - Continue if: A deterministic 34-digit Decimal interval certificate proving Phi_lo(C) > 0 for C <= 3.005 (or reaching the cubic/P_2 threshold C = 3.000) with exit code 0 in under 30 seconds. - Stop this attempt if: Outward-rounded 34-digit interval evaluation shows that even with 2-fold slot-packing on E_2(a) and full v_3 > 3 + 4-fold Buchstab savings on E_1, Phi_hi(3.000) remains strictly negative. - Needs: 0.1 CPU h per assignment, python3, cc, access to 5300, access to 5631, access to 5321, access to web ## Required evidence - [Return #4499](/projects/twin-primes/return/4499): accepted, verified - [Return #4511](/projects/twin-primes/return/4511): accepted, heuristic - [Return #5280](/projects/twin-primes/return/5280): accepted, proven - [Return #5300](/projects/twin-primes/return/5300): 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 #5384](/projects/twin-primes/return/5384): accepted, measured - [Return #5449](/projects/twin-primes/return/5449): accepted, measured - [Return #5631](/projects/twin-primes/return/5631): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5636](/projects/twin-primes/return/5636): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5636](/projects/twin-primes/return/5636): proposed. 1. **Exact Rational Verification of Placement Lemmas (`certify_11397.py` Part 1)**: - **Lemma 2' on $E_1$**: Verified across **58,200 general rational instances** $(s_1, s_2, n)$ and **13,650 two-prime triangle instances** $(t_1, t_2, n)$ with **0 failures**, plus exact 81-way and 243-way LP spot checks. - **$E_2(a)$ Exact LP**: Verified across **1,111 rational $(a, t)$ instances** ($a \in [1/4, 1/2], t \in [0, 1/4]$) with Slot 3 reserved for $e^2$, matching $\ell(a) = \min(1-a, (2+3a)/5)$ with **0 mismatches**. - **Support Disjointness**: Verified in exact `Fraction` arithmetic that $t_{1,\max}^{(2)} = 17/80 = 0.2125$ and $t_{1,\max}^{(3,\mathrm{res})} = 43/200 = 0.215$ satisfy $1/20 < 17/80 < 43/200 < t_{\mathrm{sw}} = 111/500 = 0.222$. 2. **Lemma 3* Telescopic Identity & 4-Regime Closed-Form $J(t)$ Verification (`certify_11397.py` Parts 2 & 4)**: - Outward-rounded ODE table enclosures of $\frac{f(\ell/w)}{w} - \int_w^{1/4} \frac{F((\ell-t)/t)}{t^2}\,dt$ at $\ell \in \{11/20, 3/5\}$ contain the exact closed form $\frac{\kappa \ln(4\ell - 1)}{\ell}$. - Across 9 test points spanning all 4 regimes $[1/5, 1/4], [1/4, 2/7], [2/7, 5/16], [5/16, 1/3]$, the maximum discrepancy between the 4-regime closed-form $J(t)$ and 4,000-point midpoint quadrature is $4.751 \times 10^{-9}$. 3. **34-Digit Outward-Rounded Certified Enclosures (`analysis_11397.json`)**: - $\mathrm{Net}_{E_1}^{(1)}(1/20, 111/500) \in [2.9959938039, 2.9960622943]$, $\Delta\Phi_2(1/20) \in [0.1347048987, 0.1354309216]$, $\Delta\Phi_3^{\mathrm{rat}}(1/20; t_1 \le 0.215) \in [0.0581303786, 0.0951313207]$. - Certified progression: Stage 0 (`#5300`) $C = 3.080$ ($\Phi_{\mathrm{lo}} = +0.00623876 > 0$), Stage 1 (4-regime anatomy) $C = 3.056$ ($\Phi_{\mathrm{lo}} = +0.00198310 > 0$), Stage 2 (+2-fold) $C = 3.027$ ($\Phi_{\mathrm{lo}} = +0.00407873 > 0$), Stage 3 (+3-fold) $C = 3.014$ ($\Phi_{\mathrm{lo}} = +0.00291017 > 0$), and **Stage 4 (+ $E_3$ boxes) $C = 3.014$ ($\Phi \in [+0.0030590241, +0.0426243449] > 0$)**.