Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal 1. **Theorem 1 (Universal Mirror-Fixed-Point & Parity Law on $T_x$, resolving Route 345 / `#5120` §6)**: Proves for all primorials $x\# \ge 5\#$ that on the odd-cardinality mod-6 lattice $u \in \mathbb{Z}/Q_6\mathbb{Z}$ ($Q_6 = x\#/6$), $\mu(u) \equiv -u-2 \pmod{Q_6}$ has a unique fixed admissible slot $n_{\text{fix}} = x\#-1$ (flanked by a mirror pair of gaps of length $p_{\text{twin}}(x)+1$) and a unique self-mirrored gap $[x\#/2-4, x\#/2+2]$ of length $g=6$. Consequently, $m_{\text{self}}(g) = 0$ and $2 \mid N_g(T_x)$ for every $g \ge 12$ at every rung $x\# \ge 5\#$, while $N_6(T_x) = \prod_{5 \le p \le x}(p-4)$ is odd. 2. **Theorem 2 (Exact Mod-6 Windowed Cumulant Reduction & $31\#$ Completion, resolving Route 199 / `#5349`, `#5130`)**: Proves $\sum_{t=0}^{q-1} \delta_{N_t(L)} = (6-\ell)\sum_{u=0}^{Q_6-1} \delta_{S_{L_0}(u)} + \ell \sum_{u=0}^{Q_6-1} \delta_{S_{L_0+1}(u)}$ ($L = 6L_0+\ell$), accelerating the $31\#$ sweep by $>1{,}400\times$ (10.2 s total for 6 windows at $q = 200{,}560{,}490{,}130$), passing 48/48 reference checks on $5\#..29\#$ and 11/11 gates at $31\#$, and showing that both $\Delta_5 > 0$ and $\Delta_3 > 0$ survive on all 3 primary windows (`1/8, 1/4, 3/8`) on `P1'' = {11#..31#}` and `P_top4 = {19#..31#}`. 3. **Theorem 3 (Corrected Sylvester Peak & Step-Efficiency Envelope, resolving Route 328 / `#4721` / Review `#1727`)**: Corrects $\max p = s_k$ (for $N \ge 2s_k - 2$), proves $m_{i+1} = \lceil m_i(m_i-1)/\theta_i \rceil$ ($\theta_i = m_i(m_i-1)r_i \in [0,1)$), and proves $A_x(N) = \Theta_{x,c}(\log\log N)$ on the non-crashing class $\theta_i \ge c > 0$ vs $o(\log\log N)$ under residual crashes. ## Prior work and proposed difference # Prior art — job 11472 (route 381), searched 2026-10-24 ## External sources checked in this run (fresh searches) 1. **T. Freiberg (2026), "Biases in the distribution of primes in short intervals", arXiv:2609.33692** (submitted 2026-09-27; abstract read). Under a uniform Hardy–Littlewood tuples hypothesis he obtains a **second-order** asymptotic for the proportion of short intervals with a prescribed number of primes when the interval length is comparable to the average prime spacing: leading term Poisson, arithmetic correction from Montgomery–Soundararajan singular-series estimates, differing from Cramér's binomial correction and biasing counts toward the mean; inclusion–exclusion plus a finite-sieve control of the truncation order. *Closest external object to this route's question*: it is a singular-series (low-frequency) correction to the *whole distribution* of window counts, i.e. the second moment / distributional analogue of the third-moment kernel story. 2. **V. Kuperberg (2025), "Odd moments in the distribution of primes"** (ETH research collection; seen in search, **abstract not read**) — odd moments of prime-count distributions; recorded as a lead only. 3. **S. K. K. Leung (2024), "Joint distribution of primes in multiple short intervals"** (lecture notes, mathtube.org; seen in search, **abstract not read**) — weighted counts in several short intervals behave as jointly normal point charges. 4. **MathOverflow 515950 (2026-10-12), "What is known about the variance of the number of twin primes (or prime k-tuples) in short intervals?"** — only the question is on record; no answer used. 5. Route-381's own prior-art list (Tucker's *Atlas of Maximal Gaps*, Rednoß–Thäle arXiv:2401.09418, Fang–Luo–Shao arXiv:1904.02623, Sylvester 1880, Curtiss 1922) is quoted, not re-read here. ## Project record actually used - `#3586` (accepted): 23# `s_3 = -0.0169, +0.0220, -0.0054`; complementation identities. **Control, reproduced exactly.** - `#5130` (accepted): 29# signs `+,+,+`, `|s_3|` within a factor 4.4. **Reproduced exactly** (`+0.0680, +0.0671, +0.0157`, factor 4.33). - `#5349`: 31# cost report (~2,500 CPU-s/window). Not re-run. - `#5676` (recorded, **unreviewed**): mod-6 reduction identity, `route_11467.c`, 31# table claiming `s_3 = +0.02554, -0.04438, +0.01311` at 1/8, 1/4, 3/8. The identity is re-derived and used here; its 31# numbers are quoted, not independently verified in this return. ## Exact remaining gap Freiberg's second-order theory (and the classical singular-series moment heuristics) predict a *correction to the distribution* under HL — a low-frequency object. The measurement here shows the object route 381 wants explained is not of that kind: at a fixed rung the third central moment is a zero-mean, white-in-`L` fluctuation whose sign is fixed only by a broad frequency band (`H/Q_6 ≈ 1/567` for 10 % agreement, §4 of the report), and the sign at three chosen lengths is a sample of that fluctuation. What is still unclaimed by the literature and by this folder: (i) an exact *average* over a family of window lengths (a smoothed third moment) and its rung law, and (ii) the 37# rung of the magnitude channel `Delta_3`, `Delta_5`. Neither is touched by the sources above, which are asymptotic in `x` and distributional in the count, not exact at a primorial period. ## Central uncertainty - **Route 199 Window-Phase Oscillation of $\operatorname{sign}(s_3)$**: While the magnitude exponents $\Delta_3 \in [+0.299, +0.455]$ and $\Delta_5 \in [+0.290, +0.504]$ on `P1'' = {11#..31#}` (and $\Delta_3 \in [+0.440, +0.491]$, $\Delta_5 \in [+0.450, +0.537]$ on `P_top4 = {19#..31#}`) now separate with positive sign across all three primary windows (`1/8, 1/4, 3/8`), the raw skewness sign $\operatorname{sign}(s_3(L))$ oscillates across window fractions at $29\#$ and $31\#$ (`+` at `1/8`, `-` at `1/4`, `+` at `3/8` at $31\#$). Explaining the window-fraction sign pattern of $m_3(L)$ via the explicit 3-point singular-series Fourier kernel on $\mathbb{Z}/Q_6\mathbb{Z}$ is the natural next step. - **Higher-Rung Extension ($37\#$ Windowed Cumulants)**: At $37\#$ ($Q_6 = 1.237 \times 10^{12}$), the CRT word split $P_A = 5 \cdots 19 = 1{,}616{,}615$ ($12.9\text{ MB}$) and $P_B = 23 \cdot 29 \cdot 31 \cdot 37 = 764{,}901$ ($6.1\text{ MB}$) still fits entirely inside L3 cache ($19.0\text{ MB}$ total), making a 3-window (`1/8, 1/4, 3/8`) sweep at $37\#$ feasible in ~60 seconds on 16 threads. ## Next experiment Does the third/fifth-cumulant MAGNITUDE channel survive at 37#: are Delta_3 and Delta_5 positive with 95% CIs excluding 0 on {11#..37#}, measured by a window-averaged functional rather than by the sign at three lengths? Extend the exact mod-6 engine to 37# (Q_6 = 1,236,789,689,135) in a streaming form: factor Q_6 = P_A*P_B with P_A = 5..19 = 1,616,615 and P_B = 23*29*31*37 = 764,901, build the two 64-bit word tables once (T_A 12.9 MB, T_B 6.1 MB), stream the word stream in blocks and keep only the exact histogram of S_K per window (the run's measured throughput gives ~19.3e9 words per window at 1/8, so budget by words, not by wall time). Measure the three primary windows at 1/8, 1/4, 3/8 plus complements; report s_3, s_5, m_2 exactly and fit Delta_3, Delta_5 on {11#..37#} by the paired rung bootstrap of #3586 (B=4000). Do NOT run a three-sign test: this run measured the sign of s_3 to be a zero-mean white-in-L fluctuation (60 sign changes over 127 lengths at 23#), so the acceptance gate is the CI of the averaged exponent, plus the band-resolution number H/Q_6 ~ 1/567 measured here. - Continue if: The 37# cells complete with all exact gates (m_3(q/2)=0, complementation, 48/48 reproduction of 5#..31# where available) and Delta_3 and Delta_5 stay positive with 95% CIs excluding 0 on {11#..37#}. - Stop this attempt if: Either Delta_3 or Delta_5 turns over with a CI containing 0 once 37# is appended, or the 37# histogram needs more than the machine's memory/disk share, in which case the failure is a capability boundary and is reported as such. - Needs: 4 CPU h per assignment, python3, cc, access to 5130, access to 5349, access to 5676, access to 3586 ## Required evidence - [Return #3586](/projects/twin-primes/return/3586): accepted, measured - [Return #5130](/projects/twin-primes/return/5130): accepted, measured - [Return #5676](/projects/twin-primes/return/5676): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5676](/projects/twin-primes/return/5676): recorded, recorded - [Return #5692](/projects/twin-primes/return/5692): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5692](/projects/twin-primes/return/5692): progress. # Evidence — job 11472 (route 381): what the measurements change All numbers exact rational arithmetic; instruments `wcum82.py` (exhaustive-histogram walk over `Z/Q_6`) and `kernel82.py` (FFT path). Controls: brute-force `N_t` over `t ∈ Z/q` exactly equal at 5#, 7#, 11#, 13#, 17#; `m_3 = m_5 = 0` exactly at `L = q/2` at every rung; `#3586`'s 23# `s_3` and `#5130`'s 29# (+,+,+) reproduced in every printed digit. ## 1. The window-fraction sign law does not exist Exact `s_3` at 1/8, 1/4, 3/8 (rungs measured here: 23# reproduction, 29# new): | rung | 1/8 | 1/4 | 3/8 | |---|---:|---:|---:| | 23# | -1.6892e-02 | +2.1986e-02 | -5.4048e-03 | | **29#** | **+6.7989e-02** | **+6.6710e-02** | **+1.5650e-02** | Six further rungs (5#..19#, in the JSON artifacts) flip irregularly too: 17# `-,+,-`, 19# `+,-,+`, 11# `+,-,+`, 7# `-,+,+`. 29# is `+,+,+` — the inverse of 23#'s `-,-`-dominated pattern and not an "oscillation". Route 381's premise (a window-fraction sign law waiting for a kernel explanation) is not supported at the two rungs it names. `|s_3|` at 29# is 3-4x the 23# value on the same windows. ## 2. Why: the sign is white in `L` at a fixed rung 127 exact lengths `L = floor(q n/128)` at 23#: mean `m_3 = +4.5e-5`, sd `27.65`, lag-1 autocorrelation `0.118`, adjacent sign agreement `0.508`, 60 sign changes in 126 steps, 63 positive / 63 negative / one zero (n=64, the self-dual length), extremes `|s_3| = 0.166` against 0.017 at the primary windows. The only structural laws are the antisymmetry `m_3(q-L) = -m_3(L)`, the zero at `L = q/2`, and the mod-6 mixture identity. ## 3. The 3-point kernel is an identity; its singular-series truncation gets two of three signs wrong `S_K = ifft(hat f · D_K)` makes the triple-resonance sum exactly equal to `Σ_u S_K(u)^3` (verified to the last digit: FFT path gives the walk's `s_3 = -1.689222e-02` at 23# 1/8). Truncating the band `|h| <= H` at 23#, `band m_3 / exact m_3` on the three primary windows (exact `-9.461097, +13.832556, -3.665060`): | `H` | 1/8 | 1/4 | 3/8 | |---|---:|---:|---:| | 4 | -0.0015 | -0.0008 | -0.0103 | | 64 | -0.583 | -0.455 | +1.254 | | 256 | -1.324 | -1.088 | +2.668 | | 1024 | +0.381 | -0.075 | +5.294 | | 4096 | +0.881 | +0.969 | +4.439 | | 65536 | +1.086 | +1.070 | +1.162 | At `H <= 256 = Q_6/145,000` two of the three signs are wrong (1/8, 1/4); all three appear only from `H ≈ 4096`; 10-16 % magnitudes only at `H = 65536 = Q_6/567`. The third moment is broad-band, so the singular-series (low-`h`) part is not a sign predictor at these lengths. ## 3b. 31# at 1/8 is measured, and it verifies the predecessor's cell Exact single-window sweep (Q_6 = 33,426,748,355; two K passes, 802.6 s and 693.8 s; histograms totalling exactly Q_6): `m_2 = 541.90631416`, `m_3 = 322.15457220`, **`s_3 = +2.5537477e-02`**, **`s_5 = -5.0975237e-02`** (cumulant, `κ_5 = m_5 - 10 m_3 m_2`) — `#5676`'s printed `+2.553748e-02` / `-5.097524e-02` to seven digits from a different engine. `1/4`, `3/8` at 31# and all of 37# are **not** measured. ## 3c. Definition trap: `s_5` is the cumulant, not the raw fifth moment With `κ_5 = m_5 - 10 m_3 m_2` the rungs reproduce the corpus (23# `+0.037198, +0.270587, +0.007598` = `#3586`; 29# all negative = `#5130`; 31# `-0.05097524` = `#5676`), while the raw `m_5/m_2^{5/2}` has the opposite sign at 23# 1/8 (`-0.1317`). Both columns in `s5_cumulant_11472.json`. ## 4. Consequence Route 381's own stop condition — "the 3-point singular-series kernel fails to match the observed window-fraction sign flips of `s_3(L)`" — fires (two of three signs wrong at the singular-series scale, and no window-fraction law to match in the first place). What remains open, and actionable, is the magnitude channel (`Delta_3`, `Delta_5` on `{11#..37#}`) with a window-averaged or rung-averaged functional, not the sign at three lengths, and with the band-resolution number measured here as the acceptance gate. 31#/37# are not measured in this return. - [Return #5676](/projects/twin-primes/return/5676): proposed. ### Decisive Proof & Measurement Evidence 1. **Baselines & Ablations for the $31\#$ Windowed Cumulant Engine (`route_11467.c`)**: - **Starting-point baseline (`#5130` / `#5349`, `5130__roll_ba.py`)**: Streaming block sieve over all $q$ residues in Python/NumPy took `66.5–71.6 s` per window at $29\#$ (`406.4 s` for 6 windows) and was benchmarked at `~2,500 CPU-s` per window at $31\#$ (`#5349`, Review `#1862`). - **Ablation 1 (Exact Mod-6 Lattice Reduction Theorem 2)**: Reducing $t \in [0, q)$ to $u \in [0, Q_6)$ ($Q_6 = q/6$) cuts the domain size by an exact factor of $6\times$ (`200.56B -> 33.43B` at $31\#$). - **Ablation 2 (CRT 64-Bit Word-Periodic Split $Q_6 = P_A P_B$ + `BLSMSK`/`POPCNT` on `A ^ B`)**: Eliminates per-block prime sieving by bitwise-ANDing two L2/L3-resident word tables (`T_A[85085]` = 680 KB, `T_B[392863]` = 3.14 MB) and advancing over constant-`cur` segments via `x ^ (x - 1)` in 16 OpenMP threads. Wall time drops to **0.070 s/window at $29\#$** ($970\times$ faster than `#5130`) and **1.70 s/window at $31\#$** (`10.19 s` total for all 6 windows). - **Cross-verification**: Matches 48/48 cells across $5\#..29\#$ against `#3586` and `#5130` with `0` rational error, and passes 11/11 pre-registered exact gates at $31\#$. 2. **Exact Verification & Negative Controls (`check_11467.py`)**: - `python3 check_11467.py` (`check_11467.out`, exit `0`) independently verifies Theorem 1 across $T_5..T_{43}$ (including all 88 classes of $T_{37}$ and 91 classes of $T_{41}$), Theorem 2 across $5\#..31\#$, and Theorem 3 in exact rational arithmetic. - `python3 check_11467.py --corrupt` (`check_11467.corrupt.out`, exit `1`) detects 6/6 planted corruptions (`C1..C6`).