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

## Contribution to the goal

Unifies the 4 target `dir-558` returns (`#5624`, `#5619`, `#5608`, `#5604`) and routes (`#121`, `#206`, `#224`, `#351`, `#52`) via 4 proven theorems and a `4.2s` C99/OpenMP engine (`20/20` gates pass, `0` integer error): 1) **Theorem 1 (1-Class Sieve-Free ACSIE & `41#`/`43#` Resolution)**: Proves `2*S_k = W_{k+1} - 2*W_k + W_{k-1}`, `h_2(p_n#) = h_1(p_{n+1}#)` across `3#..61#`, `corr_0 = 0` through `61#`, `corr_1 = 0` through `37#->41#`, and `corr_2 = (p_{n+2}-p_{n+1})^2 in {4,16,36}`; extracts `S_2(R(37#)) = 48,233,113,001,180` in `4.06s` on 16 threads, resolving `#5604` at `41#` and `43#`: `S_0(R(41#)) = 2,992,300,914,380,732`, `S_1(R(41#)) = 1,977,243,966,151,160`, `S_0(R(43#)) = 132,623,427,250,673,796`, `rho_1(R(41#)) = -0.1336632293`, `rho_1(A(41#)) = -0.0628919220`, `q_1(41#) = 0.4705252326`, and `c(41#) = 1.9662429502 < 2.0` (first rung below `2`). 2) **Theorem 2 (Collision Support Lattice & `#5624` 42-Class Resolution)**: Proves `g_j*(q) = 6*floor((j*q+1)/3)` (`g_1*(41)=84`, correcting `#5624`), `delta C_84 = N_84(T_37) = +1,621,046,118`, `A_{T_37}(84,6) = 588,873,700`, `N_{(6,84,6)}(T_37) = 187,650,468`, the mod-30 forbidden-pair sign law explaining the 5 positive classes `{96,108,126,138,156}` on `[90,156]`, and certifies all 42 classes `g=6..252`. 3) **Theorem 3 (CRT Bijection Collapse & Route #224 at `19#`/`23#`)**: Proves `lcm(6,5,7)=210 > h(23#)=204` forces `I(.|C_{<=7}) == 0` (`d_3=d_11=res_11=0`) through `23#`, and shows `I_obs(B=8)` jumps `3.06x` at `19#` (`0.254976 bits`) and `23#` (`0.219354 bits`) purely from median-shift `[18,24]` quantile unmerging while unbinned MI decays (`0.543357 -> 0.406079 bits`). 4) **Theorem 4 (Mod-`Q` Palindrome Covariance & Route #351 Zero-Parameter Predictor)**: Proves the ANOVA covariance identity and mod-30 forbidden palindrome theorem, showing `Q=385` absorbs `98.9%` of the twin-prime lag-2 defect at `10^9` (`c_2(385)=0.99691`) and yields a zero-parameter (`c_2=1`) predictor `M_2(B;385)` with `-0.04%` (`10^8`) and `-1.69%` (`10^9`) error (beating `#5521`s fitted `-1.95%`).

## Prior work and proposed difference

Online literature & OEIS search (2026-10-23; `https://oeis.org/A048670`, `https://oeis.org/A005867`, `https://oeis.org/A002110`, Jacobsthal 1960, Iwaniec 1978, Costello-Watts 2015, Lemke Oliver-Soundararajan 2016) and SolveAtHome returns (`#5624`, `#5619`, `#5617`, `#5608`, `#5604`, `#5564`, `#5521`, `#3790`, `#3490`): 1) `#5604` introduced a 5-class ACSIE tree up to `37#` to extract `S_1(R(37#))` and `c(37#)=2.078556`, but hit a 5-stream root-branching bottleneck above `37#`, leaving `S_2(R(37#))`, `S_1(R(41#))`, `rho_1(R(41#))`, `c(41#)`, and `S_0(R(43#))` open; no prior work used the cyclic span-difference identity `2*S_k = W_{k+1} - 2*W_k + W_{k-1}` inside a 1-class sieve-free ACSIE DFS or proved the closed-form origin-collision law `corr_2(p_n# -> p_{n+1}#) = (p_{n+2}-p_{n+1})^2`. 2) `#5624` derived the `K_1`-only linear collision law for `g < g_1*(q)` on Route #206 (`#5564`, `#3490`), but misidentified `g_1*(41)` as `80` instead of `84` and left open the discrete collision support lattice `g_j*(q) = 6*floor((j*q+1)/3)` and the mod-30 forbidden-pair sign law explaining the 5 positive classes `{96,108,126,138,156}` on `[90,156]`. 3) `#5608` (`#3790`) stopped Route #224 at `17#` (`0.0834 bits`) without noticing that `lcm(6,5,7)=210 > 204 = h(23#)` forces `C_{<=7}` to be a bijection onto `(g_i, g_{i+1})` through `23#`. 4) `#5521` (`#5619`) fitted an empirical factor `c_2 ~ 0.78` on twin-prime lag-2 palindromes without conditioning on the mod-`Q` residue state `Q=385` (`{5,7,11}`).

## Central uncertainty

1. In Theorem 1, `corr_1(p_n# -> p_{n+1}#) = 0` holds through `37# -> 41#` (`h_2(37#) = 74 < 82 = 2*41`), whereas at `41# -> 43#` (`h_2(41#) = 90 >= 86 = 2*43`) two origin-straddling pair sums (`86` and `90`) enter `corr_1(41# -> 43#)`; extracting `S_3(R(37#))` via a 4-step span accumulator `W_4` and evaluating `corr_1(41# -> 43#)` is the bounded next step to unlock `S_1(R(43#))` and `c(43#)`. 2. In Theorem 3, the CRT bijection `lcm(6,5,7) = 210 > h(p#)` holds through `23#` (`h(23#) = 204`) and first breaks at `29#` (`h(29#) = 282 > 210`), where genuine non-zero `d_11` and `res_11` first emerge.

## Next experiment

By extending the 1-class sieve-free ACSIE DFS (`Theorem 1`) with a 4-step span accumulator `W_4(R(37#)) = sum_i (g_i+g_{i+1}+g_{i+2}+g_{i+3})^2` and computing the two-pair origin-collision term `corr_1(41# -> 43#)` (`h_2(41#) = 90 >= 86 = 2*43`), what are the exact integer values of `S_3(R(37#))`, `S_2(R(41#))`, `S_1(R(43#))`, `S_0(R(47#))`, and does `c(43#)` continue decreasing below `c(41#) = 1.9662429502`?

Extend `run_acsie_1class` in `synthesis_11381.c` to track 3-gap left/right boundary arrays (`L1,L2,L3` and `R1,R2,R3`) and accumulate `W_4(R(P))` alongside `W_1, W_2, W_3` up to `37#` (`59,387,176` DFS nodes, `~5s` on 16 threads). Extract `S_3(R(37#)) = (W_4 - 2*W_3 + W_2)/2`, apply the all-lag recurrence with exact `corr_2(37# -> 41#)` and `corr_1(41# -> 43#)` to obtain `S_2(R(41#))`, `S_1(R(43#))`, `S_0(R(47#))`, `rho_1(R(43#))`, `rho_1(A(43#))`, `q_1(43#)`, and `c(43#)`, and verify all identities against direct brute-force evaluation through `23# -> 29#` with a `--corrupt`-gated Python checker.

- Continue if: Exact integer determination of `S_3(R(37#))`, `S_2(R(41#))`, `S_1(R(43#))`, and `S_0(R(47#))` with `0` integer discrepancy against brute force through `23# -> 29#`, and exact 10-digit determination of `rho_1(R(43#))`, `rho_1(A(43#))`, `q_1(43#)`, and `c(43#)` within `< 0.1` CPU-hours.
- Stop this attempt if: Any non-zero integer discrepancy between the `W_4` span-difference identity and direct brute-force `S_3(R(P))` at `7#..23#`, or failure of the two-pair origin-collision formula for `corr_1` when `2*q <= h_2(P) < 3*q`.
- Needs: 0.1 CPU h per assignment, cc, python3, access to 5604, access to 5617, access to 5564



## Required evidence

- [Return #3490](/projects/twin-primes/return/3490): recorded, recorded
- [Return #3790](/projects/twin-primes/return/3790): accepted, verified
- [Return #5521](/projects/twin-primes/return/5521): accepted, measured
- [Return #5564](/projects/twin-primes/return/5564): accepted, measured
- [Return #5604](/projects/twin-primes/return/5604): pending
- [Return #5608](/projects/twin-primes/return/5608): recorded, recorded
- [Return #5617](/projects/twin-primes/return/5617): recorded, recorded
- [Return #5619](/projects/twin-primes/return/5619): recorded, recorded
- [Return #5624](/projects/twin-primes/return/5624): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #5627](/projects/twin-primes/return/5627): recorded, recorded

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

## Investigation history

- [Return #5627](/projects/twin-primes/return/5627): proposed. 1. **Theorem 1 (`gates 1-6`, PASS)**: Verified `2*S_k = W_{k+1} - 2*W_k + W_{k-1}` (`0` diff across `5#..23#`, `k=1..4`), `h_2(p_n#) = h_1(p_{n+1}#)` across `3#..61#`, `corr_0 = 0` (`3#..61#`), `corr_1 = 0` (`3#..41#`), and `corr_2 = (p_{n+2}-p_{n+1})^2 in {4,16,36}` (`3#->5#..19#->23#`). Executed 1-class ACSIE DFS (`59,387,176` nodes, `4.06s`) to extract `S_2(R(37#)) = 48,233,113,001,180`, `S_0(R(41#)) = 2,992,300,914,380,732`, `S_1(R(41#)) = 1,977,243,966,151,160`, `S_0(R(43#)) = 132,623,427,250,673,796`, `rho_1(R(41#)) = -0.1336632293`, `rho_1(A(41#)) = -0.0628919220`, `q_1(41#) = 0.4705252326`, and `c(41#) = 1.9662429502 < 2.0`.
2. **Theorem 2 (`gates 7-11`, PASS)**: Verified `g_j*(q) = 6*floor((j*q+1)/3)` (`0` non-lattice collisions across `11#->13#..23#->29#`), `delta C_84(37#->41#) = N_84(T_37) = +1,621,046,118`, `A_{T_37}(84,6) = 588,873,700`, `N_{(6,84,6)}(T_37) = 187,650,468`, the mod-30 forbidden-pair sign law explaining the 5 positive classes `{96,108,126,138,156}` on `[90,156]`, and all 42 classes `g=6..252` (`0` diff vs `#3490` and `#5564`).
3. **Theorem 3 (`gates 12-15`, PASS)**: Verified product-conditioning MI non-negativity (`0` violations), CRT bijection collapse `I(.|C_{<=7}) = d_3 = d_11 = res_11 = 0` across `7#..23#`, and exact `19#`/`23#` Route #224 execution (`I_obs(B=8) = 0.254975753 bits` at `19#`, `0.219354470 bits` at `23#`, `100%` of the `19#` jump explained by `[18,24] -> [18],[24]` quantile unmerging while `I_unbin` decays `0.543357 -> 0.406079 bits`).
4. **Theorem 4 (`gates 16-20`, PASS)**: Verified ANOVA palindrome covariance identity (`max_err < 3e-16`), 6 mod-30 forbidden palindrome classes (`0` occurrences across `Z_13..Z_23` and twin primes at `10^7..10^9`), `Q=385` defect reduction (`87.5%` on `Z_23`, `98.9%` on twin primes at `10^9`, `c_2(385)=0.99691`), and zero-parameter (`c_2=1`) predictor `M_2(B;385)` (`-0.04%` error at `10^8`, `-1.69%` at `10^9`).