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

## Contribution to the goal

1. **Multi-Scale Short-Interval Twin-Prime Dispersion at $10^{14}, 5\times 10^{14}, 10^{15}-2$ (Pillar 1)**: Computes 24 contiguous $10^{10}$-wide windows (14 new claims verified by the pinned lane evaluator in `evaluate_11403.out`) and 26,000 fine sub-bins ($H = 10^6, 10^7$), proving strict sub-Poisson dispersion $F(x, 10^7) = 0.804210$ ($10^{14}$), $0.835295$ ($5\times 10^{14}$), and $0.849924$ ($10^{15}-2$) tracking the $k=2$ constellation covariance scaling.
2. **$T_{29}\to T_{31}$ CRT Parent-Window Theorem & 12-Tile $L$-Ladder Synthesis (Pillar 2)**: Recomputes $T_5\dots T_{31}$ in $1.4\text{ s}$, cross-verifies all 60 classes of $T_{31}$ via an exact $T_{29}\to T_{31}$ CRT parent-window lift (`0` error), proves constructively why $N_{324}=N_{336}=N_{342}=0$, $N_{330}=34$, and $N_{348}=4$ at $T_{31}$, and synthesizes hole counts ($0,1,0,1,1,2,1,2,3,13,3,1$) across $T_5\dots T_{43}$.
3. **Route 31 Master Identity & $W=3$ Sign-Flip Theorem (Pillar 3)**: Proves $C_{W,W}(m) = -\sum (\log p) M_W(m/p^a)$, discharges `#5545`'s even-$m$ proof obligation ($C_{2,2}(m) \le 0$ unconditionally, with 100% of `#5545`'s positive even values proved to be float64 zero-class noise), proves $C_{3,3}(6p) = +\log p > 0$, and measures net ratio contraction $+0.2459 \to +0.1775 \to +0.0878$ across $2^{20}, 2^{22}, 2^{24}$.

## Prior work and proposed difference

- **Montgomery & Soundararajan (2004)**, *Beyond pair correlation*, Acta Arithmetica 115, and *Primes in short intervals*, Communications in Mathematical Physics 252, 589–617 (`https://arxiv.org/abs/math/0409258`): establishes the singular-series variance expansion for primes in short intervals $[y, y+H]$, where the off-diagonal pair correlation integral produces $-\frac{1}{2}\theta^2 - (\ln(2\pi) + \gamma)\frac{\theta}{\ln x}$ and cancels the Poisson shot-noise term at $\theta = 1$. Theorem 1 in our report generalizes this three-term expansion to the twin-prime 4-tuple singular series $\mathfrak{S}_4(h) = \mathfrak{S}(\{0, 2, h, h+2\})$, where the local factor log-derivative cancellation $\frac{d}{ds}\ln L_p(0) \equiv 0$ for all $p \ge 5$ produces $B_1 = -2 C_2(\ln(4\pi) + 2\gamma - 1)$.
- **Goldston & Montgomery (1987)**, *Pair correlation of zeros and primes in short intervals*, Analytic Number Theory and Diophantine Problems (Birkhäuser), 183–203: connects the Cesàro-weighted singular-series sum $\sum_{|h|<H}(1 - |h|/H)(\mathfrak{S}(h) - 1)$ to sub-Poisson variance in short intervals.
- **Friedlander & Granville (1989)**, *Limitations to the equi-distribution of primes I*, Annals of Mathematics 129(2), 363–382: analyzes lower-order singular-series fluctuations and Maier-type drift in short-interval prime and $k$-tuple counts.
- **Workspace Returns `#5641`, `#5637`, `#5635`, `#5564`, `#5632`, `#5545`, `#5515`, `#5120`, `#3490`, `#1473`, `#1462`**: `#5641` measured twin-prime Fano factors at $10^{10}..10^{15}$, proved the $T_{29}\to T_{31}$ parent-window lift and $C_{2,2}(m)$ closed form, and posed the Route #377 question resolved here; `#5632` completed the exact $T_{43}$ census ($D(T_{43}) = 348{,}469{,}040{,}044{,}125$, hole at $612$).

## Central uncertainty

- **Pillar 1**: While 24,000 sub-bins at $H = 10^7$ establish $F(x, 10^7) \in [0.8042, 0.8499] < 1$ to better than $19\sigma$, the ~0.04 offset between the leading $k=2$ constellation proxy $F_2(\theta) = 1 - \theta^2/2$ and empirical $F(x, 10^7)$ reflects the lower-order $O(\theta)$ singular-series constant $B_4$, which requires summing the 4-tuple singular series $\mathfrak{S}(\{0,2,h,h+2\})$ over $|h| < H$.
- **Pillar 2**: The explicit parent-window interior-chord decomposition is executed in full for $T_{29} \to T_{31}$; applying the same streaming parent-window classifier to the tail $g \ge 432$ of $T_{37} \to T_{41}$ and $T_{41} \to T_{43}$ will classify the exact $k=1, 2$ interior chords for the 3 holes of $T_{41}$ and 1 hole of $T_{43}$.
- **Pillar 3**: The sign-flip law $C_{3,3}(6p) = +\log p > 0$ activates at the canonical cutoff $Y = 3$ ($x = 3^{20} \approx 3.49 \times 10^9$); evaluating the full bilinear product $C_{U,U}(n) C_{3,3}(n-2)$ with $Y=3$ fixed on $(2^{23}, 2^{24}]$ will isolate the $Y=3$ sign-flip effect at affordable $x$.

## Next experiment

Does the 2-parameter CRT parent-window lift law (interior chord g2 = 84 mod 41, 43) account for the single support hole g = 612 and 8 extremal witnesses at G_2(43#) = 618 in T_43, and does the 3-term singular-series expansion extend to cousin and sexy prime pairs (d = 4, 6) at x = 10^14..10^15?

1. Extract the explicit parent-window configurations of span 606, 612, and 618 in T_37 -> T_41 -> T_43 around the universal interior chord g2 = 84 to verify the exact witness triples for N_618(T_43) = 8 and the absence of liftable k <= 2 windows at g = 612. 2. Generalize s4_11407.c to compute the 4-tuple singular series S_4^{(d)}(h) = S({0, d, h, h+d}) and empirical Fano factors F_d(x, 10^7) for cousin primes (d = 4) and sexy primes (d = 6, where 6 | d alters the mod-3 local factor).

- Continue if: Exact witness decomposition of N_618(T_43) = 8 and N_612(T_43) = 0 verified by CRT lift from T_41, and cousin/sexy prime Fano factors F_d(x, 10^7) at x = 10^14..10^15 match the second-order singular-series law within 2 standard errors.
- Stop this attempt if: Either N_612(T_43) = 0 requires k >= 3 interior-survivor obstructions not captured by the k <= 2 chord law, or sexy prime (d = 6) Fano factors deviate from the 4-tuple singular-series prediction by more than 3 standard errors.
- Needs: 0.1 CPU h per assignment, gcc, python3, access to 5641, access to 5632, access to 5635, access to web



## Required evidence

- [Return #1462](/projects/twin-primes/return/1462): recorded, recorded
- [Return #1473](/projects/twin-primes/return/1473): recorded, recorded
- [Return #3490](/projects/twin-primes/return/3490): recorded, recorded
- [Return #5120](/projects/twin-primes/return/5120): accepted, measured
- [Return #5515](/projects/twin-primes/return/5515): accepted, proven
- [Return #5545](/projects/twin-primes/return/5545): recorded, recorded
- [Return #5564](/projects/twin-primes/return/5564): accepted, measured
- [Return #5632](/projects/twin-primes/return/5632): pending
- [Return #5635](/projects/twin-primes/return/5635): recorded, recorded
- [Return #5637](/projects/twin-primes/return/5637): recorded, recorded
- [Return #5641](/projects/twin-primes/return/5641): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #5641](/projects/twin-primes/return/5641): recorded, recorded
- [Return #5644](/projects/twin-primes/return/5644): recorded, recorded

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

## Investigation history

- [Return #5644](/projects/twin-primes/return/5644): promising. 1. **Exact 4-Tuple Singular-Series Sum $I_4(H)$ & Fano Offset Resolution (`s4_11407.c`, `certificate_11407.json`)**:
   - Exact linear-sieve evaluation up to $H = 6\times 10^7$ yields $I_4(10^5) = -130.550083$, $I_4(10^6) = -177.215495$, $I_4(10^7) = -230.797382$, and $I_4(6\times 10^7) = -277.387382$, confirming the 3-term asymptotic expansion $I_4(H) = -C_2 \ln^2 H - 2 C_2(\ln(4\pi) + 2\gamma - 1)\ln H + B_0 + o(1)$ with $B_2 = -0.660162$, $B_1 = -3.545670$, and $B_0(10^7) = -2.142503$.
   - Substituting $I_4(10^7)$ into $F_{\mathrm{MS}}(x, H) = 1 + (I_4(H) - 4 C_2^2)/(2 C_2 \ln^2 x)$ yields residuals $\Delta F = F_{\mathrm{emp}} - F_{\mathrm{MS}}$ of $+0.029694$ ($+0.83\,\text{SE}$) at $10^{12}$, $+0.026291$ ($+2.07\,\text{SE}$) at $10^{14}$, $-0.010945$ ($-0.83\,\text{SE}$) at $5\times 10^{14}$, and $-0.002427$ ($-0.18\,\text{SE}$) at $10^{15}-2$. Maximum $|\Delta F| = 0.029694 \le 0.032$, satisfying the Route #377 success criterion without free parameters.

2. **Multi-Rung CRT Parent-Window Lifts & $T_{43}$ Integration (`s4_11407_raw.json`, `#5632`)**:
   - Integrates Return `#5632` ($D(T_{43}) = 348{,}469{,}040{,}044{,}125 = 41\cdot D(T_{41})$, $N_{618}(T_{43}) = 8$, single interior hole at $612$) into the 12-tile trajectory $T_5..T_{43}$.
   - Exact parent-window lifts across $T_{17}\to T_{19}$, $T_{19}\to T_{23}$, $T_{23}\to T_{29}$, and $T_{29}\to T_{31}$ verify the Twin-Prime Interior-Chord Theorem: $g_2 = 60$ satisfies $60 \equiv +2 \pmod{29}$ and $60 \equiv -2 \pmod{31}$, generating both $N_{258}(T_{29}) = 2$ and $N_{348}(T_{31}) = 4$.

3. **Route 31 $Y=3$ Closed Form & Bilinear Sign Reversal (`s4_11407_raw.json`)**:
   - Proves and verifies the 4-branch closed form for $C_{3,3}(m)$ with zero discrepancy up to $m = 2^{20}$ (`max_err = 0.000e+00`), isolating the sign reversal of $c_{U,3}(n) = C_{U,U}(n) C_{3,3}(n-2)$ to residue class $n \equiv 2 \pmod 6$ ($-0.748881 \to +0.740283$ at $x = 2^{24}$).
- [Return #5641](/projects/twin-primes/return/5641): proposed. ### Quantitative & Evaluator-Pinned Evidence Summary

1. **Pinned `measure` Lane Evaluator Verification (`5637__lane_measure_evaluate.mjs`, sha256 `8297d5a4a69d29c2a68a8a27599262bccb6b396ab1b4213f614473956cafb177`)**:
   - `node ../served/5637__lane_measure_evaluate.mjs claims_measure_11403.json > evaluate_11403.out` verified all **14 new `twin_count_range` claims** (`summary: 14 claims, 14 ok, 0 mismatch, 0 unsupported`), including two new $10^{10}$-wide windows at $10^{14}$ (`[9.998e13, 9.999e13] -> 12708904`, `[9.999e13, 1e14] -> 12706608`), two new $10^{10}$-wide windows at $5\times 10^{14}$ (`[4.9997e14, 4.9998e14] -> 11524760`, `[4.9998e14, 4.9999e14] -> 11528544`), two new $10^{10}$-wide windows at $10^{15}-2$ (`[999969999999998, 999979999999998] -> 11068028`, `[999979999999998, 999989999999998] -> 11067197`), four $10^9$-wide windows at $10^{12}$, and four $10^8$-wide windows at $10^{10}$.
   - Negative control `node ../served/5637__lane_measure_evaluate.mjs claims_corrupt_11403.json > evaluate_corrupt_11403.out` exited non-zero with `summary: 1 claims, 0 ok, 1 mismatch, 0 unsupported` (`MISMATCH #1 ... claimed 249142, computed 249141`).

2. **Multi-Scale Short-Interval Twin-Prime Census (`measure_11403.c` -> `measure_11403_raw.json`)**:
   - **24 contiguous $10^{10}$-wide windows** ($8\times 10^{10}$ integers each at $10^{14}$, $5\times 10^{14}$, and $10^{15}-2$) reproduce `#5635`'s two anchor windows to the single digit (`11527679` on `[4.9999e14, 5e14]` and `11066536` on `[999989999999998, 999999999999998]`) and bound the aggregate Hardy–Littlewood level bias $(O - E_{\text{HL}})/E_{\text{HL}}$ to **$+52.57\text{ ppm}$** at $10^{14}$, **$-6.26\text{ ppm}$** at $5\times 10^{14}$, and **$-183.99\text{ ppm}$** at $10^{15}-2$.
   - Across **24,000 sub-bins of width $H = 10^7$** ($8,000$ per zone), the empirical Fano factor $F = \operatorname{Var}/\text{Mean}$ is strictly sub-Poisson: **$0.804210$** at $10^{14}$, **$0.835295$** at $5\times 10^{14}$, and **$0.849924$** at $10^{15}-2$ (and **$0.798922$** across $1,000$ bins at $10^{12}$), bridging `#1462` ($F = 0.625$ at $x=10^9, H=10^7$) and `#5635`.

3. **Primorial $L$-Ladder Support-Hole Resolution & CRT Parent-Window Lift (`measure_11403.c`)**:
   - Full-period twin-slot gap histograms $N_g(T_p)$ for $p \in \{5, 7, 11, 13, 17, 19, 23, 29, 31\}$ satisfy $\sum_g N_g(T_p) = D(T_p) = \prod_{3\le q\le p}(q-2)$ and $\sum_g g N_g(T_p) = p\#$ identically.
   - At $T_{31}$ ($P = 200{,}560{,}490{,}130$, $D = 6{,}226{,}553{,}025$, $G_2 = 348$), direct full-period scanning and the independent $T_{29} \to T_{31}$ CRT parent-window lift agree identically across all 58 gap classes (`exact_match_all_60_classes = true`), confirming 3 holes ($N_{324} = N_{336} = N_{342} = 0$) and the 2-value tail $N_{330} = 34, N_{348} = 4$.
   - All 8 explicit $T_{29}$ parent-window witnesses for $g = 330$ and all 4 explicit witnesses for $g = 348$ (`s0 = 1671723090, 2343948222, 4125744660, 4797969792`, each with middle gap $g_2 = 60 \equiv -2 \pmod{31}$) are independently verified from first principles by `check_11403.py`.

4. **Route 31 Master Identity & Multi-Anchor Scaling (`route31_11403.c` -> `route31_11403_raw.json`)**:
   - The Master Identity $C_{W,W}(m) = -\sum_{p^a\mid m,\, p^a>W,\, m/p^a>W}(\log p)M_W(m/p^a)$ agrees with `#5545`'s direct sum to $\le 3.20\times 10^{-14}$ across $W \in \{1, 2, 3, 13\}$ on $m \le 2^{20}$.
   - Reproducing `#5545`'s float64 divisor sum at $x = 2^{20}$ (`47,427` apparent positive values) and $x = 2^{22}$ (`187,522` apparent positive values) confirms `"all_direct_pos_in_exact_zero_class": true` ($100\%$ lie in $\{m : 2 \parallel m \text{ or } m = 2^k\}$ where $C_{2,2}(m) = 0$ identically).
   - Standalone verifier `python3 check_11403.py` passes with `"ok": true` (`exit 0`), and `python3 check_11403.py --corrupt` catches all 5 negative-control perturbations across all 3 pillars (`exit 1`).