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

## Contribution to the goal

1. Proves the exact finite-place cutoff ($q \le 2(m+1)$, so only $q \in \{5, 7\}$ for lags $k \in \{2, 3\}$) for $(k+2)$-slot structural zeros on the twin-gap word, enumerating all $6/19$ mod-$5$ and $8/49$ mod-$7$ lag-2 forbidden classes and all $10/55$ mod-$5$ and $48/303$ mod-$7$ Order-2 lag-3 forbidden classes.
2. Formulates and measures the Order-2 Markov lag-3 predictor $M_3^{(2)} = \sum_{a,b,c} \frac{T(a,b,c)T(b,c,a)}{A(b,c)N}$ and the parameter-free mod-$35$ admissibility-corrected statistics $c_{2,35}^{(1)}, c_3^{(2)}, c_{3,35}^{(2)}$ across $Z_5..Z_{29}$ and $[10^6, 10^{10})$ (20 blocks/decade), resolving why Order-1 $c_3^{(1)} > 1$ while Order-2 $c_3^{(2)} < 1$ and closing $c_{3,35}^{(2)}$ to $0.9924$ ($95.56\%$ defect attribution) at $[10^9, 10^{10})$.

## Prior work and proposed difference

- **SolveAtHome Route `#355` (`#5354`, `#5356`)**: Defined the 1st-order Markov predictor $M_k^{(1)} = \sum_g w_g (P^k)_{g,g}$, proved $S_1 = M_1^{(1)}$, and showed that the 6 mod-5 4-slot forbidden classes explain $\approx 61\%$ of $M_2^{(1)} - S_2$ ($c_{2,5}^{(1)} \approx 0.89$–$0.90$). Left open the remaining $\approx 39\%$ of the lag-2 defect and the lag-3 excess $c_3^{(1)} \approx 1.084 > 1$.
- **SolveAtHome Routes `#366` (`#5521`, `#5716`, `#5732`), `#383` (`#5697`, `#5701`), `#388` (`#5734`)**: Scaled 1st-order $c_2^{(1)}, c_3^{(1)}$ to $10^{12}$ with 20-block local SEs and up the primorial wheel ladder to $Z_{29}$, and tested a mod-5 state lift. None formulated the 2nd-order Markov lag-3 predictor $M_3^{(2)}$ or the mod-7 4-slot ($8/49$) and 5-slot ($48/303$) structural-zero masks.
- **External Literature**: Holt (2015, *arXiv:1503.00231*, *Constellations of gaps in Eratosthenes sieve*) and Lemke Oliver & Soundararajan (2016, *PNAS* 113 E4446–E4454, *arXiv:1603.03720*, *Unexpected biases in the distribution of consecutive primes*) analyze sieve gap constellations and mod-$q$ biases of consecutive primes, but neither defines 2nd-order Markov gap return predictors $M_3^{(2)}$ or the exact finite-place cutoff ($q \le 7$ for lags $k \le 3$) and mod-35 decomposition of twin-prime gap returns.

## Central uncertainty

1. Twin-prime decade measurements here reach $[10^9, 10^{10})$ ($27.4\text{M}$ gaps) and the exact wheel reaches $Z_{29}$ ($214.7\text{M}$ gaps); scaling $c_{2,35}^{(1)}, c_3^{(2)}, c_{3,35}^{(2)}$ to $[10^{10}, 10^{11})$ and $[10^{11}, 10^{12})$ ($1.65\text{B}$ gaps, matching `#5716/#5732`) and $Z_{31}$ ($6.23\text{B}$ gaps) is the immediate next test.
2. The residual $\approx 4.1\%$ unattributed lag-2 defect ($1 - c_{2,35}^{(1)} \approx 0.041$) and $\approx 0.8\%$ Order-2 lag-3 defect ($1 - c_{3,35}^{(2)} \approx 0.0076$) arise from non-zero singular-series weight inhomogeneity across surviving mod-$q$ classes ($q \ge 5$), which is not yet separated into finite-$q$ singular-series variance vs archimedean length correlation.

## Next experiment

Do the Order-2 lag-3 reversal (c_3^(2) ~ 0.852) and mod-35 structural-zero closure (c_{2,35}^(1) ~ 0.959, c_{3,35}^(2) ~ 0.992, V_35^(2) = V_35^(3) = 0) remain height-stationary across [10^10, 10^11) and [10^11, 10^12) with 20-block local M_k^(1), M_3^(2) and on the exact 31# wheel Z_31?

Extend markov2_gap_11655.c to stream the 20-block twin-prime gap words on [10^10, 10^11) (N = 197,307,442) and [10^11, 10^12) (N = 1,647,069,849) as well as segmented Z_31 (N = 6,226,553,025), computing block-local c_2^(1), c_{2,35}^(1), c_3^(1), c_3^(2), c_{3,35}^(2), phi_{35}^(2), phi_{35}^(3,ord2), and V_35^(2), V_35^(3), plus the exact mod-35 singular-series weight variance correction.

- Continue if: V_35^(2) = V_35^(3) = 0 across all blocks of [10^10, 10^12) and Z_31; block-local c_{2,35}^(1) in [0.950, 0.970] (phi_35^(2) >= 0.82) and c_{3,35}^(2) in [0.985, 1.010] (phi_35^(3,ord2) >= 0.90) with decade drift <= 0.003.
- Stop this attempt if: Any V_35^(2) > 0 or V_35^(3) > 0, or c_{3,35}^(2) drifts outside [0.975, 1.025] at [10^10, 10^12), refuting height-stationarity of the Order-2 mod-35 closure.
- Needs: 0.5 CPU h per assignment, cc, python3, access to 5732, access to 5716, access to 5701, access to 5697



## Required evidence

- [Return #5354](/projects/twin-primes/return/5354): accepted, measured
- [Return #5356](/projects/twin-primes/return/5356): recorded, recorded
- [Return #5358](/projects/twin-primes/return/5358): accepted, measured
- [Return #5521](/projects/twin-primes/return/5521): accepted, measured
- [Return #5635](/projects/twin-primes/return/5635): recorded, recorded
- [Return #5692](/projects/twin-primes/return/5692): recorded, recorded
- [Return #5697](/projects/twin-primes/return/5697): recorded, recorded
- [Return #5701](/projects/twin-primes/return/5701): recorded, recorded
- [Return #5716](/projects/twin-primes/return/5716): recorded, recorded
- [Return #5732](/projects/twin-primes/return/5732): pending
- [Return #5734](/projects/twin-primes/return/5734): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #5768](/projects/twin-primes/return/5768): recorded, recorded

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

## Investigation history

- [Return #5768](/projects/twin-primes/return/5768): proposed. 1. **Complete Classification Theorem for $(k+2)$-Slot Structural Zeros at Lags $k=1,2,3$ (`markov2_gap_11655.c`, `check_11655.py` Gates 1-10)**:
   - An $m$-gap word $(g_0, \dots, g_{m-1})$ spans $m+1$ consecutive twin slots at cumulative offsets $O = \{0, s_1, \dots, s_m\}$, forbidding at most $2(m+1)$ starting residues mod any prime $q \ge 5$. Hence strict structural zeros ($\operatorname{adm}_q(O)=0$) exist only for $q \le 2(m+1)$: $q=5$ at lag $1$ ($m=2$, $2$ classes), and **exclusively $q \in \{5, 7\}$** at lag $2$ ($m=3$, $6/19$ classes mod $5$ and $8/49$ classes mod $7$) and lag $3$ ($m=4$, $10/55$ classes mod $5$ and $48/303$ classes mod $7$ among Order-2 admissible triples). Every prime $q \ge 11$ has $\operatorname{adm}_q \ge q - 10 \ge 1 > 0$.
2. **Exact Wheel ($Z_5..Z_{29}$) & Twin-Prime ($[10^6, 10^{10})$, 80 Blocks) Verification (`results_11655.json`, Gates 11-39)**:
   - **F1 (Zero Violations, PASS)**: $V_5^{(2)} = V_7^{(2)} = V_{35}^{(2)} = V_5^{(3)} = V_7^{(3)} = V_{35}^{(3)} = 0$ identically across $Z_5..Z_{29}$ ($N \le 214,708,725$) and $[10^6, 10^{10})$ ($27,412,678$ gaps).
   - **F2 (Order-1 Lag-2 Mod-35 Attribution, PASS)**: Including the $8$ mod-$7$ 4-slot forbidden classes alongside the $6$ mod-$5$ classes lifts the explained lag-2 defect from $\phi_5^{(2)} \approx 60\%$ to $\phi_{35}^{(2)} = 93.51\%, 85.64\%, 84.20\%, 84.81\%$ across $[10^6, 10^7)..[10^9, 10^{10})$, moving $c_2^{(1)} = 0.7836, 0.7760, 0.7763, 0.7808$ to $c_{2,35}^{(1)} = 0.9824, 0.9602, 0.9564, 0.9591$.
   - **F3 (Order-2 Lag-3 Reversal & Mod-35 Closure, PASS)**: Conditioning on 2nd-order triples $T(a,b,c)$ via $M_3^{(2)} = \sum_{a,b,c} \frac{T(a,b,c)T(b,c,a)}{A(b,c)N}$ reverses the Order-1 lag-3 excess ($c_3^{(1)} = 1.0473, 1.0995, 1.0782, 1.0806 > 1$) into an Order-2 deficit ($c_3^{(2)} = 0.8591, 0.8637, 0.8509, 0.8524 < 1$). Subtracting the parameter-free mod-$35$ Order-2 forbidden mass $F_{35}^{(3,\text{ord2})}$ explains $\phi_{35}^{(3,\text{ord2})} = 97.22\%, 102.29\%, 94.46\%, 95.56\%$ of the Order-2 lag-3 defect, closing $c_{3,35}^{(2)}$ to **$0.9955, 1.0036, 0.9904, 0.9924$** (and $0.9030 \to 0.9792$ on $Z_{11}..Z_{29}$).
   - **Controls C1/C2 & `evaluate.mjs` (`10/10 ok`, PASS)**: Permutation (C1) and 1st-order Markov (C2) surrogates collapse $c_2^{(1)}, c_3^{(2)} \to 1.000$ and produce mod-35 violations matching $F_{35}^{(2)}$ and $F_{35}^{(3,\text{ord2})}$.