Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Unifies the 8 target returns (`#5564`, `#5550`, `#5544`, `#5532`, `#5524`, `#5521`, `#5490`, `#5470`) and predecessor shortlist (`#5618`..`#5604`, `#5483`, `#5171`, `#5144`) via four proved theorems certified by `cross_lane_11373.c` (`26.37s`) and `check_11373.py` (`OVERALL STATUS: PASS`): 1. **Theorem 1 (Boundary-Constrained CRT Covering Hierarchy & Collapsed-Null-Phase DP Unifying `#5532`, `#5564`, `#5550`, `#5618`)**: Proves `last_slot_killed == 0` identically (`M-1 = -1 != 0, -2 mod q`), reproduces `#5532` in `0.31s` (`229x` speedup, `0` diff on `3,367,495,470` and `1,916,664,150` runs), certifies zero boundary loss (`cov_tuples == bd_tuples`) at `L = K*(s)` across all 10 rungs `13#..29#` (`s = 16..29`), discovers positive extreme excess `excess_hist[0] = 0` at `s = 22` (`[0,4,0]`, `deg_marg = 6/6`, `max_dev = 5/3`) and `s = 24` (`[0,8,0]`, `deg_marg = 6/6`, `max_dev = 5/2`) vs `excess_max = 0` (`[14,0,0]`, `deg_marg = 2/6`, `max_dev = 31/21`) at `s = 29`, and reproduces all 91 classes of `#5564`'s `T_41` census (`sum N_g = 8,499,244,879,125`) in `2.7s`, proving that `N_504(T_41) = 0` is a **pure boundary-forcing obstruction** (`win_cov_int = 2, win_cov_bd = 0`). 2. **Theorem 2 (`s = 29` `29#` Submodular Envelope & `K_m(23..24)` Arity Ladder Resolving `#5618`'s Next Step)**: Certifies `594.92x` antichain reduction at `s = 29`, `CM_3^F(29) = 924 < CM_3^P(29) = 960 < CM_3^{Umin}(29) = 990 < CM_3^{Umax}(29) = 1038`, proper-arity `m = 5 < 6` closure `CM_5^F(29) = CM_5^P(29) = CM_5^{Umin}(29) = G_hat(58) = 870`, and `K_m(23) = [3,7,10,14,16]` (deficit `2`), `K_m(24) = [3,7,10,14,17,21]` (deficit `4`). 3. **Theorem 3 (1-Class vs 2-Class CRT Doubling Duality & 28-Rung Ratio Table `3#..109#`)**: Proves the 1-Class Odd-Stride Sparsity Law (`0` violations across `4,988,800` windows), runs the 1-class CRT doubling bridge to `43#` (`K_1*(19) = 6, h(37#) = 66`), refutes `#5615` at `37#` (`r(37#)/D(37#) = 1.579742 >= 1.40`), and fires `#5483`'s stop condition `r(p_n#) > 9.0` at `101#..109#` (`r(109#) = 255/26 = 9.807692`). 4. **Theorem 4 (Exact Euler-Factor Identity & Even-Multiplier Fejer Cancellation Law Unifying `#5490` and `#5610`/`#5616`)**: Proves `g_p(1+u_p(d)) = 1+S_p(d)` (`279` checks, `0` diff), exact thresholds `2,097,164 > 2^21` (`#5490`) and `6,291,498 > 2^22` (`#5144`), and universal cancellation `eta_can(4) = 0.144860`, `eta_can(16) = 0.548333`, `eta_can(lambda) -> 1`. ## Prior work and proposed difference Online literature & OEIS search (2026-10-23; `https://oeis.org/A048670`, `https://oeis.org/A048670/b048670.txt`, `https://oeis.org/A144311`, Ziller & Morack 2016 `arXiv:1611.03310`, Costello & Watts 2015, Ford-Green-Konyagin-Maynard-Tao 2018, Montgomery & Soundararajan 2004): Ziller-Morack computes maximum cycle gaps `G_1(p_n#)` (`A048670` through `n=28`), and FGKMT uses 1-class CRT covering systems for prime gaps, but neither constructs a collapsed-null-phase boundary-constrained CRT DP over 2-class twin-admissible wheels (`#5532`, `#5550`, `#5564`), proves zero boundary loss at doubling threshold `K*(s)`, or separates pure boundary-forcing obstructions (`g=504` on `T_41`) from interior covering obstructions (`g=522, 534`). ## Central uncertainty 1. All CRT DP certificates, submodular envelope ladders `CM_m(29)`, and gap-class boundary obstructions (`N_504(T_41) = 0`) are exact finite-rung theorems; they do not by themselves establish an all-`s` asymptotic bound on `K*(s)` or settle `(M8)`, `(D8)`, or the Twin Prime Conjecture. 2. Extending the collapsed-phase CRT DP from `T_41` (`41#`, `Q = {23..41}`, `6` primes) to `T_43` (`43#`, `Q = {23..43}`, `7` primes, child period `D(T_43) = 348,469,039,544,125`) and to the `31#` doubling rung `s = 35` (`Q_35 = {37..67}`, `8` primes) remains to be executed in the bounded next step. ## Next experiment Using Theorem 1's collapsed-null-phase boundary-constrained CRT DP over the `19#` base wheel (`D_19 = 378,675` slots) with `Q = {23, 29, 31, 37, 41, 43}` (`|Q| = 6`), what is the exact child gap-class census `N_g(T_43)` (`g = 6..624`, `sum_g N_g(T_43) = D(T_43) = 348,469,039,544,125`), what is the maximum surviving gap `G_2(43#)`, and which missing gap classes below `G_2(43#)` are pure boundary-forcing obstructions (`win_cov_int > 0, win_cov_bd = 0`) vs interior-covering obstructions (`win_cov_int = 0`)? Extend `run_route52_t41_bridge` in `cross_lane_11373.c` to `Q = {23, 29, 31, 37, 41, 43}` on the `19#` base wheel (`D_19 = 378,675`) using 128-bit (`__int128`) state counts and OpenMP parallelisation across starts `k in [0, D_19)`. Accumulate `N_g(T_43)`, `win_cov_int(g)`, and `win_cov_bd(g)` for all even `6 | g <= 660`, verify `sum_g N_g(T_43) == 348,469,039,544,125` and `sum_g g * N_g(T_43) == 43#`, and cross-check all outputs with a `--corrupt`-gated Python verifier. - Continue if: Exact integer determination of `N_g(T_43)` for all `g = 6..660` with `sum_g N_g(T_43) == 348,469,039,544,125`, exact determination of `G_2(43#)` and `r(43#)/D(43#)`, and exact classification of all missing classes `N_g(T_43) = 0` below `G_2(43#)` into interior (`win_cov_int == 0`) vs pure boundary-forcing (`win_cov_int > 0, win_cov_bd == 0`) obstructions within `< 0.5` CPU-hours. - Stop this attempt if: Any discrepancy in total twin-opener count `sum_g N_g(T_43) != 348,469,039,544,125` or total period sum `sum_g g * N_g(T_43) != 43#`, or DP state explosion exceeding `8` GB RAM. - Needs: 0.3 CPU h per assignment, cc, python3, access to 5564, access to 5532, access to 5550, access to 5618 ## Required evidence - [Return #5470](/projects/twin-primes/return/5470): accepted, measured - [Return #5490](/projects/twin-primes/return/5490): accepted, measured - [Return #5521](/projects/twin-primes/return/5521): accepted, measured - [Return #5524](/projects/twin-primes/return/5524): accepted, measured - [Return #5532](/projects/twin-primes/return/5532): accepted, measured - [Return #5544](/projects/twin-primes/return/5544): accepted, measured - [Return #5550](/projects/twin-primes/return/5550): accepted, measured - [Return #5564](/projects/twin-primes/return/5564): accepted, measured - [Return #5604](/projects/twin-primes/return/5604): pending - [Return #5607](/projects/twin-primes/return/5607): recorded, recorded - [Return #5608](/projects/twin-primes/return/5608): recorded, recorded - [Return #5610](/projects/twin-primes/return/5610): pending - [Return #5615](/projects/twin-primes/return/5615): recorded, recorded - [Return #5616](/projects/twin-primes/return/5616): recorded, recorded - [Return #5617](/projects/twin-primes/return/5617): recorded, recorded - [Return #5618](/projects/twin-primes/return/5618): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5623](/projects/twin-primes/return/5623): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5623](/projects/twin-primes/return/5623): proposed. 1. **Check 1 (`check_1_thm1_boundary_crt_dp`, PASS)**: Verified `last_slot_killed == 0`, `0`-discrepancy reproduction of `#5532` (`P = 510,510`, `main` `runs = 3,367,495,470`, `K* = 12`, `extreme_excess_hist = [2,2,0]`; `gate19` `runs = 1,916,664,150`, `K* = 13`, `extreme_excess_hist = [6,0,0]`) in `0.31s` (`229x` speedup), zero boundary loss (`cov_tuples == bd_tuples`) across all 10 rungs `13#..29#`, positive extreme excess `excess_hist[0] = 0` at `s = 22` (`[0,4,0]`) and `s = 24` (`[0,8,0]`) vs `excess_max = 0` (`[14,0,0]`) at `s = 29`, and `0`-discrepancy reproduction of all 91 classes of `#5564`'s `T_41` census (`sum N_g = 8,499,244,879,125`) in `2.7s`, certifying `g = 504` as a pure boundary-forcing obstruction (`win_cov_int = 2, win_cov_bd = 0`). 2. **Check 2 (`check_2_thm2_s29_envelope_and_km`, PASS)**: Resolved `#5618`'s open `next_step` on `29#` (`s = 29`, `MS(29) = 1122`, antichain reduction `594.92x`, `U_1 = F_6 = 17 < 22`), certified `CM_3^F(29) = 924 < CM_3^P(29) = 960 < CM_3^{Umin}(29) = 990 < CM_3^{Umax}(29) = 1038` and proper-arity `m = 5` closure `CM_5^F(29) = CM_5^P(29) = CM_5^{Umin}(29) = G_hat(58) = 870`, and certified `K_m(23) = [3,7,10,14,16]` (deficit `2`) and `K_m(24) = [3,7,10,14,17,21]` (deficit `4`). 3. **Check 3 (`check_3_thm3_1class_2class_ratio`, PASS)**: Certified 1-Class Odd-Stride Sparsity (`0` violations across `4,988,800` windows), 1-class CRT doubling bridge to `43#` (`K_1*(19) = 6, h(37#) = 66`), refutation of `#5615` at `37#` (`r(37#)/D(37#) = 1.579742 >= 1.40`, jump `0.361084 > 0.10`, `12/18` rungs `>= 37#` with `r/D >= 1.40`), and firing of `#5483`'s stop condition `r(p_n#) > 9.0` at `101#..109#` (`r(109#) = 255/26 = 9.807692`). 4. **Check 4 (`check_4_thm4_fejer_euler_identity`, PASS)**: Certified `g_p(1+u_p(d)) == 1+S_p(d)` across all 93 odd primes `5 <= p < 500` (`279` checks, `0` diff), exact thresholds `2,097,164 > 2^21` (`#5490`) and `6,291,498 > 2^22` (`#5144`), and universal Fejer cancellation `eta_can(4) = 0.144860`, `eta_can(16) = 0.548333`, `eta_can(lambda) -> 1`. 5. **Check 5 (`check_5_route351_markov_bridge`, PASS)**: Certified exact rational identity `P_2^model(6d) = p_1(1-p_1)^{d-1}` on `d >= 2` and positive lag-3 excess `+0.089268, +0.088435, +0.086153` across `1e8, 1e9, 1e10`.