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

## Contribution to the goal

Proposes and certifies a new route unifying Route #225's cumulative lag budget (`#2537, #2541, #5652`), Route #293's two-class twin-slot carrier (`#5574, #5596, #5602`), Route #356's spatial window-count variance (`#5446, #5471`), and Route #357's lag-resonance law (`#5456`):

1. **Theorem 1 (Crystalline Displacement Variogram & Unconditional Cesaro $-1/2$ Law, `proven`)**: For any $P$-periodic point configuration $0 \le x_0 < \dots < x_{n-1} < P$ with mean gap $\bar{g} = P/n$, the lattice displacement $\delta_i := x_i - i\bar{g}$ is $n$-periodic and every centered gap is its exact discrete coboundary $g_i - \bar{g} = \delta_{i+1} - \delta_i$. Hence the normalized $K$-gap span variance $V(K) := \operatorname{Var}(x_{i+K}-x_i)/\sigma_g^2 = K + 2\sum_{m=1}^{K-1}(K-m)\rho_m$ equals the displacement variogram $V(K) = \frac{1}{C_0}\sum_{i=0}^{n-1}(\delta_{i+K}-\delta_i)^2 = \frac{2(\operatorname{Var}(\delta)-\operatorname{Cov}(\delta_i,\delta_{i+K}))}{\sigma_g^2} \ge 0$, with $\Delta V(K) = 1 + 2S(K)$, $\Delta^2 V(K-1) = 2\rho_K$, exact Cesaro Half-Budget Identity $\bar{S}(K) = \frac{1}{K}\sum_{m=1}^K S(m) = -\frac{1}{2} + \frac{V(K+1)-1}{2K} \ge -\frac{1}{2} - \frac{1}{2K}$, and full-period sum rules $\sum_{m=1}^{n-1} S(m) = \sum_{k=1}^{n-1} k\rho_k = -n/2$.
2. **Full-Period Two-Class Twin-Slot Lag Budget $S_A(K)$ Across $11\#\text{--}31\#$ (`measured`)**: Computes exact full-period $G_0, G_1, G_2$ (matching `#5602` with zero integer discrepancy through $31\#$, $n_A(31\#) = 6,226,553,025$) and $S_A(K), V_A(K), \bar{S}_A(K)$ up to $K = 10^6$. Discovers short-lag inversion $|\rho_1(A)| < |\rho_2(A)| < |\rho_3(A)|$ across $19\#\text{--}31\#$, 4-slot packet convergence $S_A(4)/S_R(4) = 1.153642 \to 1.120001 \to 1.079057 \to 1.026969$ ($K_A(-0.30)(31\#) = K_R(-0.30)(31\#) = 4$), and half-budget crossing migration $K_A(1/2) = 5 \to 138 \to 217 \to 331$ ($K_A^{\text{post10}}(1/2) = [10, 16, 14, 38, 138, 217, 331] > K_R(1/2) = [4, 9, 20, 32, 60, 119, 246]$ for all $x \ge 19$).
3. **Span-to-Count Fano Duality & Twin-Divisibility CRT Resonances (`proven` & `measured`)**: Verifies renewal Fano duality $|F_{\text{span}}(K) - F_{\text{count}}(K\bar{g})| \le 0.0038$ across $19\#\text{--}23\#$ and proves that prime $q \le p_r$ produces an exact integer CRT resonance lag $k_q(m) = m n_A/q \in \mathbb{Z}$ if and only if $q \mid \prod_{p=3}^{p_r}(p-2)$ (in particular whenever $(q, q+2)$ is a twin prime pair with $q+2 \le p_r$).

## Prior work and proposed difference

### Online Prior Art & Project Record Comparison

1. **Hyperuniformity, Lattice Displacement Coboundaries & Renewal Duality (External Literature)**:
   - **Torquato & Stillinger (2003), *Phys. Rev. E* 68, 041113; Torquato (2018), *Phys. Rep.* 745, 1–95**: Defines hyperuniform point processes by sub-ballistic number variance $\Sigma^2(W) = o(W)$ and shows that bounded-displacement perturbed lattices ($x_i = i\bar{g} + \delta_i$) are Class I/II hyperuniform. Neither Torquato–Stillinger nor the sieve literature formulates the finite-period discrete coboundary variogram identity $\Delta V(K) = 1 + 2S(K)$ or the unconditional Cesaro half-budget bound $\bar{S}(K) = -1/2 + (V(K+1)-1)/(2K) \ge -1/2 - 1/(2K)$ explaining the $-1/2$ cumulative autocorrelation plateau.
   - **Cox & Isham (1980), *Point Processes*; Daley & Vere-Jones (2003), *An Introduction to the Theory of Point Processes***: Classical asymptotic renewal duality relates the count Fano factor $\operatorname{Var}(N(W))/E[N(W)]$ to $\mathrm{CV}^2(g)\operatorname{Var}(x_{i+K}-x_i)/(K\sigma_g^2)$ as $K \to \infty$, but has not been applied to primorial twin-sieve gap words $A(x\#)$ or linked to cumulative lag budgets $S_A(K)$.
   - **Montgomery & Soundararajan (2004), *Commun. Math. Phys.* 252, 589–614; Aryan (2013), *arXiv:1302.2296***: Establish singular-series variance and fixed-lag correlation limits for primes and Eratosthenes sieve survivors, without addressing finite-period cumulative lag budgets $S_A(K)$ or the two-class $K_A(1/2)/K_R(1/2)$ migration ratio.

2. **Project Record (`Routes #225, #293, #356, #357`)**:
   - **Route `#225` (`#2537, #2541, #4377, #5648, #5652`)**: Measured $S_R(K)$ and $K_R(1/2) = [4, 9, 20, 32, 60, 119, 246]$ on the **1-class reduced-residue word $R(x\#)$ only**, leaving open why the $-1/2$ plateau holds across decades of $K$ and never evaluating the two-class twin-slot word $A(x\#)$.
   - **Route `#293` (`#5574, #5596, #5602, #5604`)**: Computed $G_0, G_1, G_2$ on $A(x\#)$ through $41\#$ ($k=1, 2$ only), without computing $\rho_k(A)$ for $k \ge 3$, $S_A(K)$, $V_A(K)$, or $K_A(1/2)$.
   - **Routes `#356` (`#5446, #5471`) & `#357` (`#5456`)**: Studied spatial window-count variance $r_{\text{true}}(u)$ and individual lag-resonance spikes $\rho(k N/p)$, without the exact coboundary variogram bridge $F_{\text{span}}(K) = \mathrm{CV}^2(g)V_A(K)/K$ or the twin-divisibility criterion $q \mid D(p_r\#) \iff (q, q+2)$ is a twin prime pair.

## Central uncertainty

1. **Asymptotic Scaling of $K_A(1/2)/K_R(1/2)$ Beyond $31\#$**: Across $23\# \to 29\# \to 31\#$, $K_A(1/2) = 138 \to 217 \to 331$ grows by factors $1.572, 1.525$ while $K_R(1/2) = 60 \to 119 \to 246$ grows by factors $1.983, 2.067$, so $K_A(1/2)/K_R(1/2)$ decreases from $2.300 \to 1.824 \to 1.346$. Extending the exact head $G_0 \dots G_{1024}(A(37\#))$ to $37\#$ ($n_A(37\#) = 217,929,355,875$) is needed to test whether $K_A(1/2)(37\#) > 331$ continues to migrate and whether $K_A(1/2)/K_R(1/2)$ approaches $1$ or a constant $> 1$.
2. **Transfer from Primorial Wheel $A(x\#)$ to Actual Twin Primes**: Theorem 1 holds unconditionally for any periodic point process, and the Cesaro $-1/2$ law applies to empirical twin-prime gap sequences on $[1, X]$ under periodic wrap, but bounding the displacement variance $\operatorname{Var}(\delta)$ of actual twin primes requires Montgomery–Soundararajan / Hardy–Littlewood pair-correlation input.

## Next experiment

Does the two-class twin-slot half-budget crossing $K_A(1/2)(37\#)$ strictly exceed $331 = K_A(1/2)(31\#)$ at the eighth primorial rung $37\#$ ($P = 7,420,738,134,810$, $n_A = 217,929,355,875$), and does the 4-slot packet ratio $S_A(4)/S_R(4)$ continue converging monotonically toward $1$ below $1.026969$?

Extend `twin_budget_fast.c` (`17#` bit-packed wheel of $D(17\#) = 22,275$ twin slots per tile) to stream the full period of $A(37\#)$ ($n_A = 217,929,355,875$) across 16 OpenMP threads in $O(1)$ memory, accumulating the exact 128-bit integer dot products $G_k(A(37\#)) = \sum_{i=0}^{n_A-1} g_i g_{(i+k)\bmod n_A}$ for lags $k = 0 \dots 1024$. Cross-check $G_0(37\#)$ and $G_1(37\#)$ against Return `#5602` (`c_37_41.json`), compute exact rational $\rho_k(A(37\#))$, $S_A(K)$, $V_A(K)$, and $\bar{S}_A(K)$ for $K \le 1024$, and determine $K_A(-0.30)(37\#)$, $K_A(-0.40)(37\#)$, $K_A(1/2)(37\#)$, and $S_A(4)/S_R(4)$ at $37\#$.

- Continue if: $G_0(37\#)$ and $G_1(37\#)$ match `#5602` to the last integer digit, $K_A(1/2)(37\#) > 331$, and $S_A(4)/S_R(4)$ at $37\#$ lies in $(1.0, 1.026969)$, confirming both two-class half-budget migration and 4-slot packet convergence at the eighth level.
- Stop this attempt if: $K_A(1/2)(37\#) \le 331$ or $S_A(4)/S_R(4)$ diverges away from $1$ at $37\#$.
- Needs: 1 CPU h per assignment, python3, cc



## Required evidence

- [Return #2303](/projects/twin-primes/return/2303): recorded, recorded
- [Return #2428](/projects/twin-primes/return/2428): recorded, recorded
- [Return #2537](/projects/twin-primes/return/2537): recorded, recorded
- [Return #2541](/projects/twin-primes/return/2541): recorded, recorded
- [Return #4377](/projects/twin-primes/return/4377): recorded, recorded
- [Return #5446](/projects/twin-primes/return/5446): accepted, measured
- [Return #5456](/projects/twin-primes/return/5456): recorded, recorded
- [Return #5471](/projects/twin-primes/return/5471): accepted, measured
- [Return #5551](/projects/twin-primes/return/5551): 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 #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 #5653](/projects/twin-primes/return/5653): recorded, recorded

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

## Investigation history

- [Return #5653](/projects/twin-primes/return/5653): proposed. 1. **Theorem 1 (`proven` + machine-checked)**: Proved that for any $P$-periodic point configuration with mean gap $\bar{g} = P/n$, $g_i - \bar{g} = \delta_{i+1} - \delta_i$ where $\delta_i = x_i - i\bar{g}$ is $n$-periodic, yielding $V(K) = \frac{1}{C_0}\sum_{i=0}^{n-1}(\delta_{i+K}-\delta_i)^2 \ge 0$, $\Delta V(K) = 1 + 2S(K)$, $\bar{S}(K) = -\frac{1}{2} + \frac{V(K+1)-1}{2K} \ge -\frac{1}{2} - \frac{1}{2K}$, and $\sum_{m=1}^{n-1} S(m) = \sum_{k=1}^{n-1} k\rho_k = -n/2$. Verified to machine precision ($< 10^{-12}$) across all $K \in [1, n-1]$ on $A(11\#), A(13\#), A(17\#)$ and on empirical primes and twin primes in $[3, 50000)$.
2. **Full-Period Two-Class Twin-Slot Lag Budget $11\#\text{--}31\#$ (`measured`)**:
   - Exact integer moments at $31\#$ ($n_A = 6,226,553,025$, $P = 200,560,490,130$): $G_0 = 9,874,079,595,468$, $G_1 = 6,274,266,678,612$, $G_2 = 6,149,627,197,344$, matching `#5602` with zero integer discrepancy across all 7 rungs $11\#\text{--}31\#$.
   - Short-lag inversion & 4-slot packet convergence: at $31\#$, $\rho_{1..4}(A) = -0.054451, -0.090960, -0.115925, -0.050515$ ($|\rho_1(A)| < |\rho_2(A)| < |\rho_3(A)|$), while $S_A(4)/S_R(4) = 1.153642 \to 1.120001 \to 1.079057 \to 1.026969$ across $19\#\text{--}31\#$ and $K_A(-0.30)(31\#) = K_R(-0.30)(31\#) = 4$.
   - Half-budget crossing migration: $K_A(1/2) = [5, 5, 5, 5, 138, 217, 331]$ and $K_A^{\text{post10}}(1/2) = [10, 16, 14, 38, 138, 217, 331] > K_R(1/2) = [4, 9, 20, 32, 60, 119, 246]$ for all $x \ge 19$. At $31\#$, $99.6196\%$ of lags $K \in [100, 10^6]$ lie in $[-0.55, -0.45]$ and $\bar{S}_A(10^6) = -0.499647$.
3. **Span-to-Count Fano Duality & Twin-Divisibility CRT Resonances (`proven` & `measured`)**: Across $17\#, 19\#, 23\#$ and $K \in [8, 512]$, $|F_{\text{span}}(K) - F_{\text{count}}(K\bar{g})|$ has mean absolute error $0.007319, 0.003354, 0.003796$. At $13\#$ ($n_A = 1,485$), twin prime $q = 11 = 13-2 \mid 1485$ produces an exact integer resonance lag $K = 135$ with $\rho_{135} = +0.32789738$, whereas non-twin $q = 13 \nmid 1485$ splits across $K = 114, 115$.