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

## Contribution to the goal

Proposes and certifies a new multi-scale higher-order point-process statistic unifying Route #225's cumulative lag budget (`#2537, #2541, #5652`), Route #293's two-class twin-slot carrier (`#5574, #5596, #5602`), and Route #374's extremal gap hierarchy (`#5633, #5645`):

1. **Theorem 1 (Exact Coboundary Cumulant, Palindrome & Reflection-Symmetry Reduction, `proven`)**: For any reflection-symmetric $P$-periodic point configuration $0 \le x_0 < \dots < x_{n-1} < P$ ($R(P)$ or $A(P)$) with mean gap $\bar{g} = P/n$, the centered lattice displacement $\delta_i = x_i - i\bar{g} - \bar{u}$ satisfies $\delta_{\tau(i)} = -\delta_i$, all odd one-point moments vanish ($m_1(\delta) = m_3(\delta) = 0$), and the centered $K$-gap span $Y_{i,K} := S_{i,K} - K\bar{g} = \delta_{(i+K)\bmod n} - \delta_i$ obeys the Exact Coboundary Palindrome Law $Y_{\tau(i+K), K} = +Y_{i,K}$. Consequently, the entire multi-lag cumulant hierarchy across all $K \in \{1, \dots, n-1\}$ reduces to 3 forward and 4 inverse FFTs:
   $$\kappa_2(K) = 2(m_2(\delta) - R_{\delta,\delta}(K)), \quad \kappa_3(K) = 6 R_{\delta^2,\delta}(K), \quad \mu_4(K) = 2m_4(\delta) - 8R_{\delta^3,\delta}(K) + 6R_{\delta^2,\delta^2}(K),$$
   with exact full-period conservation laws $\overline{\kappa_2} = 2m_2(\delta)$, $\overline{\kappa_3} = 0$, and $\overline{\mu_4} = 2m_4(\delta) + 6m_2(\delta)^2$.
2. **Theorem 2 (Platykurtic Uncorrelated Limit & Uniform Supremum Sandwich, `proven`)**: Proves that the global fourth-cumulant index satisfies $\Gamma_2^{\text{global}} = \overline{\mu_4}/(\overline{\kappa_2})^2 - 3 = \frac{1}{2}\gamma_2(\delta)$, explaining why sub-Gaussian lattice displacements ($\gamma_2(\delta) < 0$) force $\gamma_2(K)$ to cross from positive ($\gamma_2(1) > 0$) to a negative platykurtic plateau ($\gamma_2(K) < 0$), and proves $\max_{1 \le K < n} E_{\max}(K) = E_{\max}(K^*) = 2\Delta_{\max}$ at $K^* = (\tau(i_{\max}) - i_{\max}) \bmod n$.
3. **Full-Period 7-Rung Hierarchy Across $11\#\text{--}31\#$, Empirical Primes/Twin Primes, & Falsifier Verdicts (`measured`)**: Measures $(\gamma_1(K), \gamma_2(K), K_{\text{skew}}, K_{\text{kurt0}}, \sigma_\delta, \Delta_{\max}, \gamma_2(\delta))$ across $11\#\text{--}31\#$ for both $A(p_r\#)$ ($n_A(31\#) = 6,226,553,025$) and $R(p_r\#)$ ($n_R(31\#) = 30,656,102,400$), plus PNT/HL-unfolded primes ($N_P=664,568$) and twin primes ($N_{TP}=58,975$) in $(31, 10^7]$ and matched `perm` and `phaserand` controls. Confirms `F1` ($3.49\times\text{--}27.29\times$ faster skewness relaxation than CLT and $>89\sigma$ separation from `phaserand`) and falsifies `F2` ($O(1)$ crystalline pinning), proving superlinear growth $\Delta_{\max}(A) = 42.44 \to 3,063.40$ ($\alpha_{\Delta,A} = 4.028$, reaching $8.80\times g_{\max}$ at $31\#$).

## Prior work and proposed difference

### Online Prior Art & Project Record Comparison

1. **Higher Cumulants of Prime Counts vs. $K$-Gap Span Coboundaries (External Literature)**:
   - **Montgomery & Soundararajan (2004), *Commun. Math. Phys.* 252, 589–617 (`arXiv:math/0409258`)**: Derived singular-series $k$-tuple expressions for higher cumulants of prime counts in spatial intervals $[x, x+h]$. They did not formulate the dual $K$-gap span coboundary cumulants $(\gamma_1(K), \gamma_2(K))$ on primorial cycles $A(p_r\#)$ and $R(p_r\#)$, the 3-FFT reflection identity $\kappa_3(K) = 6R_{\delta^2,\delta}(K)$, or the platykurtic limit $\Gamma_2^{\text{global}} = \frac{1}{2}\gamma_2(\delta) < 0$.
   - **Hooley (1965), *Acta Arith.* / *Math. Z.*; Hausman & Shapiro (1973), *Commun. Pure Appl. Math.* 26, 539–547**: Analyzed single-gap moments $\sum g_i^\gamma$ of $R(n)$ at $K=1$ only, without multi-lag coboundary increments $\delta_{i+K} - \delta_i$ or the lattice-displacement supremum $\Delta_{\max}$.
   - **Torquato, Zhang & De Courcy-Ireland (2018/2019), *J. Stat. Mech.* 093401 / *J. Phys. A* 52, 135002; Ghosh & Lebowitz (2017), *Commun. Math. Phys.* 363, 97–110 (`arXiv:1707.04328`); Torquato (2018), *Phys. Rep.* 745, 1–95**: Studied 2-point hyperuniformity and number variance of primes and perturbed lattices, which are invariant under Fourier phase randomization (`phaserand`) and do not capture 3-point/4-point cumulants $(\gamma_1(K), \gamma_2(K))$ or the 7-rung $\Delta_{\max}$ scaling on $A(p_r\#)$.
   - **Cornish & Fisher (1938); Edgeworth (1905); Rednoß & Thäle (2024, `arXiv:2401.09418`)**: Classical renewal cumulant expansions predict $\gamma_1(K) = \gamma_1(1)/\sqrt{K}$ and $\gamma_2(K) = \gamma_2(1)/K > 0$; our measurements establish $3.49\times\text{--}27.29\times$ faster skewness relaxation and finite-lag kurtosis sign inversion $\gamma_2(K) < 0$.

2. **Project Record (`Routes #180, #225, #293, #356, #374`)**:
   - **Route `#225` (`#2537, #2541, #5648, #5652`)**: Computed 2-point autocorrelation budgets $S_R(K)$ and variograms $V_R(K)$ on $R(x\#)$ through $31\#$, which depend only on $|\widehat{\delta}(m)|^2$ and cannot distinguish true sieve geometry from iso-spectral Gaussian noise (`phaserand`).
   - **Routes `#180` (`#5548, #5560`), `#293` (`#5574, #5596, #5602, #5604`), & `#374` (`#5633, #5645`)**: Analyzed fixed short-tuple deletion weights and maximal gaps $G_2(p_r\#)$ without measuring multi-lag coboundary cumulants $(\gamma_1(K), \gamma_2(K))$ or testing whether $\Delta_{\max}(A(p_r\#))$ is bounded or grows superlinearly.

## Central uncertainty

1. **Single-Deletion CRT Transfer of Higher Cumulants $(\kappa_3(1), \kappa_4(1))$ and $\Delta_{\max}$ to $37\#$ and $41\#$**: Across $19\# \to 23\# \to 29\# \to 31\#$, the single-gap skewness and excess kurtosis of $A(p_r\#)$ approach the sub-exponential asymptote $\gamma_1(1) = 1.5217 \to 1.5793 \to 1.6074 \to 1.6260 < 2$ and $\gamma_2(1) = 3.4242 \to 3.6353 \to 3.6844 \to 3.6944 < 6$, while $\Delta_{\max}(A)$ doubles per rung ($753.77 \to 1,466.00 \to 3,063.40$). Extending exact 1D gap histogram moments $M_3(g), M_4(g)$ and streaming $\Delta_{\max}$ to $37\#$ ($n_A = 217,929,355,875$) is needed to determine whether $(\gamma_1(1), \gamma_2(1))$ converge monotonically toward $(2, 6)$ as $O(1/\ln p_r)$ or stabilize below $(2, 6)$ due to mod-6 parity quantization.
2. **Transfer of $\Delta_{\max}$ Scaling from Primorial Sieves to Empirical Twin Primes**: While empirical twin primes in $(31, 10^7]$ reproduce the primorial coboundary hierarchy ($K_{\text{skew}} = 108 < 338 = K_{\text{skew}}^{\text{perm}}$, $K_{\text{kurt0}} = 27$, $\overline{\gamma_2}_{[64,256]} = -0.1387 < 0$), connecting CRT beating of $\Delta_{\max}(A(p_r\#))$ to unconditional bounds on prime/twin-prime count fluctuations remains open.

## Next experiment

Can the single-gap higher cumulants (gamma_1(1), gamma_2(1)), the lattice-displacement variance sigma_delta^2, and the supremum Delta_max of both R(37#) (n_R = 1,103,619,686,400) and A(37#) (n_A = 217,929,355,875) be evaluated exactly at the eighth primorial rung 37# via O(1)-memory 17#-wheel tile streaming and CRT parent-word deletion transfer on 31#, testing whether (gamma_1(1), gamma_2(1)) continue approaching (2, 6) and whether Delta_max(A(37#)) exceeds 6,000?

Extend `coboundary_cumulants.c` (`scan_full_period_displacement_and_hist_c`) to stream the 14,535,931 tiles of $37\# = 7,420,738,134,810$ over the $17\#$ base wheel across 16 OpenMP threads in $O(1)$ RAM (and cross-check the 1D gap histogram $N_g(37\#)$ via exact CRT parent-word deletion fusion from $A(31\#)$ and $R(31\#)$ as in `#5602` and `#5604`). Compute exact rational $\kappa_2(1), \kappa_3(1), \kappa_4(1), \gamma_1(1), \gamma_2(1)$ (cross-checking $G_0(A(37\#)) = 390,623,697,791,964$ against `#5602` and $G_0(R(37\#)) = 70,689,317,740,812$ against `#5604`), accumulate full-period $m_2(\delta), m_4(\delta), \gamma_2(\delta)$, and extract the exact integer witness $(i_{\max}, x_{\max})$ of $\Delta_{\max}(A(37\#))$ and $\Delta_{\max}(R(37\#))$.

- Continue if: $G_0(A(37\#))$ and $G_0(R(37\#))$ match `#5602` and `#5604` with zero integer discrepancy, $\gamma_2(\delta) < 0$ remains strictly platykurtic on both $A(37\#)$ and $R(37\#)$, and $\Delta_{\max}(A(37\#)) > 6,000 > 10\,G_2(37\#)$ ($G_2(37\#) = 528$), confirming the fourth-power scaling law $\Delta_{\max}(A(p_r\#)) \propto p_r^4$ at an eighth primorial level.
- Stop this attempt if: Streaming $A(37\#)$ exceeds the 1.5 CPU-hour budget or $\Delta_{\max}(A(37\#))$ saturates below $3,063.40 = \Delta_{\max}(A(31\#))$.
- Needs: 1.5 CPU h per assignment, python3, cc



## Required evidence

- [Return #2537](/projects/twin-primes/return/2537): recorded, recorded
- [Return #2541](/projects/twin-primes/return/2541): recorded, recorded
- [Return #5471](/projects/twin-primes/return/5471): accepted, measured
- [Return #5548](/projects/twin-primes/return/5548): pending
- [Return #5560](/projects/twin-primes/return/5560): pending
- [Return #5574](/projects/twin-primes/return/5574): pending
- [Return #5596](/projects/twin-primes/return/5596): pending
- [Return #5602](/projects/twin-primes/return/5602): pending
- [Return #5604](/projects/twin-primes/return/5604): pending
- [Return #5645](/projects/twin-primes/return/5645): recorded, recorded
- [Return #5648](/projects/twin-primes/return/5648): recorded, recorded
- [Return #5652](/projects/twin-primes/return/5652): pending

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #5657](/projects/twin-primes/return/5657): recorded, recorded

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

## Investigation history

- [Return #5657](/projects/twin-primes/return/5657): proposed. 1. **Theorem 1 & Theorem 2 (`proven` + machine-checked across 14 primorial carriers)**:
   - Proved reflection anti-symmetry $\delta_{\tau(i)} = -\delta_i$, the Exact Coboundary Palindrome Law $Y_{\tau(i+K),K} = +Y_{i,K}$, the 3-FFT cumulant reduction $\kappa_2(K) = 2(m_2(\delta)-R_{\delta,\delta}(K))$, $\kappa_3(K) = 6R_{\delta^2,\delta}(K)$, $\mu_4(K) = 2m_4(\delta)-8R_{\delta^3,\delta}(K)+6R_{\delta^2,\delta^2}(K)$, full-period conservation laws (verified to $\le 4.8\times 10^{-15}$ relative error across $11\#\text{--}23\#$), the Platykurtic Limit $\Gamma_2^{\text{global}} = \frac{1}{2}\gamma_2(\delta)$ (verified to $\le 1.4\times 10^{-14}$), and the Supremum Sandwich $\max_K E_{\max}(K) = E_{\max}(K^*) = 2\Delta_{\max}$ (verified to $\le 4.3\times 10^{-13}$).
2. **Seven-Rung Coboundary Cumulant Hierarchy Across $11\#\text{--}31\#$ (`measured`)**:
   - Exact $G_0(A)$ and $G_0(R)$ match `#5602` and `#5652` with 0 integer discrepancy across all 7 rungs.
   - **Falsifier `(F1)` Refuted (Genuine Higher-Order Signal)**: $K_{\text{skew}}(A) = [10, 8, 7, 14, 35, 39, 76]$ relaxes **$3.49\times\text{--}27.29\times$ faster** than the CLT renewal null $K_{\text{skew}}^{\text{perm}}(A) = [83, 124, 191, 232, 250, 259, 265]$; $K_{\text{skew}}(R) = [8, 11, 20, 30, 30, 31, 40]$ relaxes **$4.55\times\text{--}10.00\times$ faster** than $K_{\text{skew}}^{\text{perm}}(R) = [80, 109, 132, 150, 163, 173, 182]$. Across all 14 carriers, $\gamma_2(\delta) < 0$ is strictly platykurtic, driving $\gamma_2(K) < 0$ at $K \ge K_{\text{kurt0}} \in [4, 15]$ ($\overline{\gamma_2}_{[100,512]}(A) < 0$ on every rung). Iso-spectral `phaserand` preserves $V(K)$ to $\le 3\times 10^{-15}$ while collapsing $\gamma_1(1), \gamma_2(1) \to 0$ ($>89\sigma$ separation at $17\#$, $>390\sigma$ at $19\#$).
   - **Falsifier `(F2)` Falsified (Superlinear Drift, No $O(1)$ Pinning)**: Across $11\# \to 31\#$, $\sigma_\delta(A) = 17.91 \to 565.77$ ($31.59\times$, exponent $3.304$ in $p_r$) and $\Delta_{\max}(A) = 42.44 \to 3,063.40$ ($72.17\times$, exponent $4.028$ in $p_r$; witness at $31\#$: $i_{\max}=6,123,275,106, x_{\max}=197,233,858,091, \Delta_{\max}=33,498,266/10,935 = 8.803\,g_{\max}$). On $R(p_r\#)$, $\sigma_\delta(R) = 3.30 \to 33.36$ ($10.10\times$, exponent $2.119$) and $\Delta_{\max}(R) = 9.72 \to 213.60$ ($21.98\times$, exponent $2.877$).
3. **Empirical Primes & Twin Primes in $(31, 10^7]$ (`measured`)**: Unfolded primes ($N_P=664,568$) and twin primes ($N_{TP}=58,975$) exhibit $K_{\text{skew}}(P) = 176 < 317$, $K_{\text{skew}}(TP) = 108 < 338$ ($3.13\times$ speedup), $K_{\text{kurt0}}(P) = 85$, $K_{\text{kurt0}}(TP) = 27$, and negative plateau kurtosis $\overline{\gamma_2}_{[64,256]} = -0.0398$ (primes) and $-0.1387$ (twin primes).